Fundamentals of Hydraulic Engineering Systems Solutions Manual

Table of contents :
PREFACE xi
ACKNOWLEDGMENTS xv
INTRODUCTION xix
1 FUNDAMENTAL PROPERTIES OF WATER 1
1.1 The Earth's Atmosphere and Atmospheric Pressure 2
1.2 The Three Phases of Water 2
1.3 Mass (Density) and Weight (Specific Weight) 3
1.4 Viscosity of Water 5
1.5 Surface Tension and Capillarity 7
1.6 Elasticity of Water 8
1.7 Forces in a Fluid Field 10
Problems 10
vi Contents
2 WATER PRESSURE AND PRESSURE FORCES 14
2.1 The Free Surface of Water 14
2.2 Absolute and Gauge Pressures 14
2.3 Surfaces of Equal Pressure 17
2.4 Manometers 19
2.5 Hydrostatic Forces on Flat Surfaces 23
2.6 Hydrostatic Forces on Curved Surfaces 28
2.7 Buoyancy 31
2.8 Flotation Stability 33
Problems 37
3 WATER FLOW IN PIPES 54
3.1 Description of Pipe Flow 54
3.2 The Reynolds Number 55
3.3 Continuity and Momentum Equations in Pipe Flow 57
3.4 Energy in Pipe Flow 60
3.5 Loss of Head from Pipe Friction 63
3.5.1 Friction Factor for Laminar Flow 64
3.5.2 Friction Factor for Turbulent Flow 65
3.6 Empirical Equations for Friction Head Loss 71
3.7 Friction Head Loss–Discharge Relationships 74
3.8 Loss of Head in Pipe Contractions 75
3.9 Loss of Head in Pipe Expansions 78
3.10 Loss of Head in Pipe Bends 79
3.11 Loss of Head in Pipe Valves 81
3.12 Method of Equivalent Pipes 84
3.12.1 Pipes in Series 84
3.12.2 Pipes in Parallel 85
Problems 87
4 PIPELINES AND PIPE NETWORKS 94
4.1 Pipelines Connecting Two Reservoirs 94
4.2 Negative Pressure Scenarios (Pipelines and Pumps) 98
4.3 Branching Pipe Systems 103
4.4 Pipe Networks 110
4.4.1 The Hardy–Cross Method 111
4.4.2 The Newton Method 122
Contents vii
4.5 Water Hammer Phenomenon in Pipelines 125
4.6 Surge Tanks 134
4.7 Pipe Network Modeling 136
4.7.1 The EPANET Model 137
Problems 141
5 WATER PUMPS 155
5.1 Centrifugal (Radial Flow) Pumps 155
5.2 Propeller (Axial Flow) Pumps 161
5.3 Jet (Mixed-Flow) Pumps 164
5.4 Centrifugal Pump Characteristic Curves 165
5.5 Single Pump and Pipeline Analysis 166
5.6 Pumps in Parallel or in Series 169
5.7 Pumps and Branching Pipes 173
5.8 Pumps and Pipe Networks 176
5.9 Cavitation in Water Pumps 177
5.10 Specific Speed and Pump Similarity 181
5.11 Selection of a Pump 183
Problems 187
6 WATER FLOW IN OPEN CHANNELS 197
6.1 Open-Channel Flow Classifications 199
6.2 Uniform Flow in Open Channels 201
6.3 Hydraulic Efficiency of Open-Channel Sections 207
6.4 Energy Principles in Open-Channel Flow 210
6.5 Hydraulic Jumps 216
6.6 Gradually Varied Flow 219
6.7 Classifications of Gradually Varied Flow 221
6.8 Computation of Water Surface Profiles 224
6.8.1 Standard Step Method 225
6.8.2 Direct Step Method 227
6.9 Hydraulic Design of Open Channels 234
6.9.1 Unlined Channels 236
6.9.2 Rigid Boundary Channels 238
6.10 Open Channel Flow Modeling 239
6.10.1 The HEC-RAS Model 240
Problems 245
viii Contents
7 GROUNDWATER HYDRAULICS 253
7.1 Movement of Groundwater 255
7.2 Steady Radial Flow to a Well 258
7.2.1 Steady Radial Flow in Confined Aquifers 259
7.2.2 Steady Radial Flow in Unconfined Aquifers 261
7.3 Unsteady Radial Flow to a Well 263
7.3.1 Unsteady Radial Flow in Confined Aquifers 263
7.3.2 Unsteady Radial Flow in Unconfined Aquifers 267
7.4 Field Determination of Aquifer Characteristics 270
7.4.1 Equilibrium Test in Confined Aquifers 271
7.4.2 Equilibrium Test in Unconfined Aquifers 273
7.4.3 Nonequilibrium Test 275
7.5 Aquifer Boundaries 279
7.6 Surface Investigations of Groundwater 285
7.6.1 The Electrical Resistivity Method 285
7.6.2 Seismic Wave Propagation Methods 285
7.7 Seawater Intrusion in Coastal Areas 286
7.8 Seepage Through Dam Foundations 291
7.9 Seepage Through Earth Dams 294
Problems 296
8 HYDRAULIC STRUCTURES 307
8.1 Functions of Hydraulic Structures 307
8.2 Dams: Functions and Classifications 308
8.3 Stability of Gravity and Arch Dams 311
8.3.1 Gravity Dams 311
8.3.2 Arch Dams 314
8.4 Small Earth Dams 316
8.5 Weirs 318
8.6 Overflow Spillways 323
8.7 Side-Channel Spillways 326
8.8 Siphon Spillways 328
8.9 Culverts 331
8.10 Stilling Basins 336
Problems 340
Contents ix
9 WATER PRESSURE, VELOCITY, AND DISCHARGE MEASUREMENTS 347
9.1 Pressure Measurements 347
9.2 Velocity Measurements 349
9.3 Discharge Measurements in Pipes 352
9.4 Discharge Measurements in Open Channels 357
9.4.1 Sharp-Crested Weirs 357
9.4.2 Broad-Crested Weirs 360
9.4.3 Venturi Flumes 361
Problems 366
10 HYDRAULIC SIMILITUDE AND MODEL STUDIES 371
10.1 Dimensional Homogeneity 372
10.2 Principles of Hydraulic Similitude 373
10.3 Phenomena Governed by Viscous
Forces: Reynolds Number Law 378
10.4 Phenomena Governed by Gravity
Forces: Froude Number Law 381
10.5 Phenomena Governed by Surface
Tension: Weber Number Law 383
10.6 Phenomena Governed by Both
Gravity and Viscous Forces 384
10.7 Models for Floating and Submerged Bodies 384
10.8 Open-Channel Models 386
10.9 The Pi Theorem 388
Problems 392
11 HYDROLOGY FOR HYDRAULIC DESIGN 397
11.1 Watershed Delineation 399
11.2 Design Storms 400
11.2.1 Storm Hyetograph 401
11.2.2 Intensity–Duration–Return
Period Relationships 402
11.2.3 Design-Storm Selection 402
11.2.4 Synthetic Block Design-Storm Hyetograph 404
11.2.5 Soil Conservation Service Hyetographs 405
11.3 Losses from Rainfall and Rainfall Excess 408
11.3.1 Green and Ampt Infiltration Model 410
11.3.2 Soil Conservation Service Method 413
x Contents
11.4 Design Runoff Hydrographs 415
11.4.1 Time of Concentration 416
11.4.2 Unit Hydrograph 419
11.4.3 Total Runoff Hydrograph 424
11.5 Storage Routing 426
11.6 Hydraulic Design: The Rational Method 435
11.6.1 Design of Stormwater-Collection Systems 437
11.6.2 Design of Stormwater Pipes 439
11.7 Hydrologic Modeling 443
11.7.1 The HEC-HMS Model 444
11.7.2 The EPA-SWMM Model 447
Problems 451
12 STATISTICAL METHODS IN HYDROLOGY 463
12.1 Concepts of Probability 464
12.2 Statistical Parameters 464
12.3 Probability Distributions 468
12.3.1 Normal Distribution 468
12.3.2 Log-Normal Distribution 469
12.3.3 Gumbel Distribution 469
12.3.4 Log-Pearson Type III Distribution 470
12.4 Return Period and Hydrologic Risk 472
12.5 Frequency Analysis 473
12.5.1 Frequency Factors 473
12.5.2 Testing Goodness of Fit 476
12.5.3 Confidence Limits 478
12.6 Frequency Analysis Using Probability Graphs 481
12.6.1 Probability Graphs 481
12.6.2 Plotting Positions 481
12.6.3 Data Plotting and Theoretical Distributions 483
12.6.4 Estimating Future Magnitudes 484
12.7 Rainfall Intensity-Duration-Frequency Relationships 485
12.8 Applicability of Statistical Methods 488
Problems 488
SYMBOLS 493
ANSWERS TO SELECTED PROBLEMS 496
INDEX 502

COMMON CONSTANTS AND CONVERSIONS Physical Properties of Water Units System* SI BG

SI BG

Specific Weight (γ)

Density (ρ)

Kinematic Viscosity (ν)

Viscosity (μ)

Surface Tension (σ)

Vapor Pressure

@ Normal conditions [20.2°C(68.4°F) and 760 mm Hg (14.7 lb/in2)] 9,790 N/m3 998 kg/m3 1.00 * 10 -3 7.13 * 10 -2 1.00 * 10 -6 2 2 N/m N # s/m m /s 3 3 -5 -5 62.3 lb/ft 1.94 slug/ft 4.89 * 10 -3 1.08 * 10 2.09 * 10 2 2 lb/ft lb # s/ft ft /s

2.37 * 103 N/m2 3.44 * 10 -1 lb/in2

@ Standard conditions [4°C(39.2°F) and 760 mm Hg (14.7 lb/in2)] 9,810 N/m3 1,000 kg/m3 1.57 * 10 -3 7.36 * 10 -2 1.57 * 10 -6 N/m N # s/m2 m2/s 3 3 -5 -5 62.4 lb/ft 1.94 slug/ft 5.04 * 10 -3 3.28 * 10 1.69 * 10 2 2 lb/ft lb # s/ft ft /s

8.21 * 102 N/m2 1.19 * 10 -1 lb/in2

* Le Système International d'Unitès (SI) or the British gravitational system of units (BG)

Bulk Modulus of Elasticity, Specific Heat, Heat of Fusion/Vaporization Bulk Modulus of Elasticity (water)* = 2.2 * 109 N/m2 (3.2 * 105 lb/in.2, or psi) Specific Heat of Water** = 1 cal/g # °C (1.00 BTU/lbm # °F) Specific Heat of Ice** = 0.465 cal/g # °C (0.465 BTU/lbm # °F) Specific Heat of Water Vapor = 0.432 cal/g # °C (at constant pressure) Specific Heat of Water Vapor = 0.322 cal/g # °C (at constant volume) Heat of Fusion (Latent Heat) = 79.7 cal/g (144 BTU/lbm) Heat of Vaporization = 597 cal/g (1.08 * 104 BTU/lbm) * For typical pressure and temperature ranges. ** Under standard atmospheric pressure.

Common Constants Design Constants Standard Atmospheric Pressure

Gravitational Constant

SI 1.014 * 105 N/m2 (Pascals) 760 mm Hg 10.3 m H2O 9.81 m/s2

BG 14.7 lb/in2 29.9 in. Hg 33.8 ft H2O 32.2 ft/s2

Useful Conversions 1 N (kg # m/s2) = 100,000 dynes (g # cm/s2) 1 acre = 43,560 ft2 1 ft3 = 7.48 gallons 1 ft3/s (cfs) = 449 gallons/min (gpm)

1 hectare = 10,000 m2 (100m by 100m) 1 mi2 = 640 acres 1 hp = 550 ft # lb/s

Conversion Factors: SI Units to BG Units Unit Measure Area

Density Force Length

Mass Power Energy Pressure Specific Weight Temperature Velocity Viscosity Viscosity (kinematic) Volume Volume flow rate (discharge)

To convert from

m2 cm2 hectares kg/m3 N m cm km kg W kW N # m (Joule)

ft2 in2 acres slugs/ft3 lb ft in. mi slug ft # lb/s hp ft # lb

N/m2 (Pascal) N/m2 (Pascal) N/m3 °C m/s N # s/m2 m2/s m3 liter

lb/ft2 (psf) lb/in2 (psi) lb/ft3 °F ft/s lb # s/ft2 ft2/s ft3 gal

m3/s m3/s

ft3/s (cfs) gal/min (gpm)

Multiply by 1.076 * 101 1.550 * 10 -1 2.471 1.940 * 10 -3 2.248 * 10 -1 3.281 3.937 * 10 -1 6.214 * 10 -1 6.852 * 10 -2 7.376 * 10 -1 1.341 7.376 * 10 -1 2.089 * 10 -2 1.450 * 10 -4 6.366 * 10 -3 Tf = 1.8 Tc + 32° 3.281 2.089 * 10 -2 1.076 * 101 3.531 * 101 2.642 * 10 -1 3.531 * 101 1.585 * 104

Fifth Edition

Fundamentals of Hydraulic Engineering Systems Robert J. Houghtalen Rose-Hulman Institute of Technology, Emeritus Engineering & Management Consultant, East Africa

A. Osman Akan Old Dominion University, Emeritus American University of Sharjah

Ned H. C. Hwang James L. Knight Professor University of Miami, Emeritus

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DEDICATION To my grandchildren, Kyah, Karis, and Peanut; the next generation of engineers. Robert J. Houghtalen To my colleagues and friends at the American University of Sharjah, United Arab Emirates. A. Osman Akan To my wife Maria, and my sons, Leon and Leroy and their families. Ned H. C. Hwang

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Contents

PREFACE

ACKNOWLEDGMENTS

INTRODUCTION

xix

1 FUNDAMENTAL PROPERTIES OF WATER 1.1 1.2 1.3 1.4

The Earth's Atmosphere and Atmospheric Pressure The Three Phases of Water 2 Mass (Density) and Weight (Specific Weight) 3 Viscosity of Water 5

1.5 1.6 1.7

Surface Tension and Capillarity Elasticity of Water 8 Forces in a Fluid Field 10 Problems 10

1 2

Contents

2 WATER PRESSURE AND PRESSURE FORCES 2.1

The Free Surface of Water

2.2 2.3

Absolute and Gauge Pressures 14 Surfaces of Equal Pressure 17

2.4

Manometers

2.5

Hydrostatic Forces on Flat Surfaces

2.6

Hydrostatic Forces on Curved Surfaces

2.7

Buoyancy

2.8

Flotation Stability Problems 37

19 23 28

31 33

3 WATER FLOW IN PIPES 3.1 3.2 3.3 3.4 3.5

3.6 3.7 3.8 3.9 3.10 3.11 3.12

Description of Pipe Flow 54 The Reynolds Number 55 Continuity and Momentum Equations in Pipe Flow Energy in Pipe Flow 60 Loss of Head from Pipe Friction 63 3.5.1 Friction Factor for Laminar Flow 64 3.5.2 Friction Factor for Turbulent Flow 65 Empirical Equations for Friction Head Loss 71 Friction Head Loss–Discharge Relationships 74 Loss of Head in Pipe Contractions 75 Loss of Head in Pipe Expansions 78 Loss of Head in Pipe Bends 79 Loss of Head in Pipe Valves 81 Method of Equivalent Pipes 84

3.12.1 Pipes in Series 84 3.12.2 Pipes in Parallel 85 Problems 87 4 PIPELINES AND PIPE NETWORKS 4.1 4.2 4.3 4.4

Pipelines Connecting Two Reservoirs 94 Negative Pressure Scenarios (Pipelines and Pumps) Branching Pipe Systems 103 Pipe Networks 110 4.4.1 The Hardy–Cross Method 111 4.4.2 The Newton Method 122

94 98

Contents

vii

4.5 4.6

Water Hammer Phenomenon in Pipelines Surge Tanks 134

4.7

Pipe Network Modeling 4.7.1 Problems

125

136

The EPANET Model

137

141

5 WATER PUMPS

155

5.1

Centrifugal (Radial Flow) Pumps

155

5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11

Propeller (Axial Flow) Pumps 161 Jet (Mixed-Flow) Pumps 164 Centrifugal Pump Characteristic Curves 165 Single Pump and Pipeline Analysis 166 Pumps in Parallel or in Series 169 Pumps and Branching Pipes 173 Pumps and Pipe Networks 176 Cavitation in Water Pumps 177 Specific Speed and Pump Similarity 181 Selection of a Pump 183 Problems 187

6 WATER FLOW IN OPEN CHANNELS

197

6.1 6.2 6.3

Open-Channel Flow Classifications 199 Uniform Flow in Open Channels 201 Hydraulic Efficiency of Open-Channel Sections

6.4 6.5

Energy Principles in Open-Channel Flow Hydraulic Jumps 216

6.6 6.7 6.8

Gradually Varied Flow 219 Classifications of Gradually Varied Flow 221 Computation of Water Surface Profiles 224 6.8.1 Standard Step Method 225 6.8.2 Direct Step Method 227 Hydraulic Design of Open Channels 234 6.9.1 Unlined Channels 236

6.9

6.10

6.9.2 Rigid Boundary Channels 238 Open Channel Flow Modeling 239 6.10.1 The HEC-RAS Model 240 Problems 245

210

207

viii

Contents

7 GROUNDWATER HYDRAULICS

253

7.1

Movement of Groundwater

7.2

Steady Radial Flow to a Well 258 7.2.1 Steady Radial Flow in Confined Aquifers 7.2.2

7.3

7.4

7.5 7.6

7.7 7.8 7.9

255

Steady Radial Flow in Unconfined Aquifers

Unsteady Radial Flow to a Well

261

263

7.3.1

Unsteady Radial Flow in Confined Aquifers

7.3.2

Unsteady Radial Flow in Unconfined Aquifers

263 267

Field Determination of Aquifer Characteristics 270 7.4.1 Equilibrium Test in Confined Aquifers 271 7.4.2 Equilibrium Test in Unconfined Aquifers 273 7.4.3 Nonequilibrium Test 275 Aquifer Boundaries 279 Surface Investigations of Groundwater 285 7.6.1 The Electrical Resistivity Method 285 7.6.2 Seismic Wave Propagation Methods 285 Seawater Intrusion in Coastal Areas 286 Seepage Through Dam Foundations 291 Seepage Through Earth Dams 294 Problems 296

8 HYDRAULIC STRUCTURES

307

8.1 8.2

Functions of Hydraulic Structures 307 Dams: Functions and Classifications 308

8.3

Stability of Gravity and Arch Dams 8.3.1 Gravity Dams 311 8.3.2 Arch Dams 314 Small Earth Dams 316 Weirs 318 Overflow Spillways 323 Side-Channel Spillways 326 Siphon Spillways 328 Culverts 331 Stilling Basins 336 Problems 340

8.4 8.5 8.6 8.7 8.8 8.9 8.10

259

311

Contents

9 WATER PRESSURE, VELOCITY, AND DISCHARGE MEASUREMENTS 9.1

Pressure Measurements

9.2 9.3

Velocity Measurements 349 Discharge Measurements in Pipes

9.4

Discharge Measurements in Open Channels

347 352

9.4.1

Sharp-Crested Weirs

357

9.4.2

Broad-Crested Weirs

360

9.4.3

Venturi Flumes

Problems

357

361

366

10 HYDRAULIC SIMILITUDE AND MODEL STUDIES 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

Dimensional Homogeneity 372 Principles of Hydraulic Similitude 373 Phenomena Governed by Viscous Forces: Reynolds Number Law 378 Phenomena Governed by Gravity Forces: Froude Number Law 381 Phenomena Governed by Surface Tension: Weber Number Law 383 Phenomena Governed by Both Gravity and Viscous Forces 384 Models for Floating and Submerged Bodies Open-Channel Models 386 The Pi Theorem 388 Problems 392

371

384

11 HYDROLOGY FOR HYDRAULIC DESIGN 11.1

Watershed Delineation

11.2

Design Storms 400 11.2.1 Storm Hyetograph 401 11.2.2 Intensity–Duration–Return Period Relationships 402 11.2.3 Design-Storm Selection 402 11.2.4 Synthetic Block Design-Storm Hyetograph 404 11.2.5 Soil Conservation Service Hyetographs 405 Losses from Rainfall and Rainfall Excess 408 11.3.1 Green and Ampt Infiltration Model 410 11.3.2 Soil Conservation Service Method 413

11.3

347

399

397

Contents

11.4

Design Runoff Hydrographs 415 11.4.1 Time of Concentration 416 11.4.2

Unit Hydrograph

419

11.4.3

Total Runoff Hydrograph

424

11.5

Storage Routing

11.6

Hydraulic Design: The Rational Method 11.6.1 11.6.2

11.7

426 435

Design of Stormwater-Collection Systems Design of Stormwater Pipes 439

437

Hydrologic Modeling 443 11.7.1 The HEC-HMS Model 444 11.7.2 The EPA-SWMM Model 447 Problems 451

12 STATISTICAL METHODS IN HYDROLOGY 12.1 12.2 12.3

12.4 12.5

12.6

12.7 12.8

Concepts of Probability 464 Statistical Parameters 464 Probability Distributions 468 12.3.1 Normal Distribution 468 12.3.2 Log-Normal Distribution 469 12.3.3 Gumbel Distribution 469 12.3.4 Log-Pearson Type III Distribution Return Period and Hydrologic Risk 472 Frequency Analysis 473 12.5.1 Frequency Factors 473 12.5.2 Testing Goodness of Fit 476 12.5.3 Confidence Limits 478 Frequency Analysis Using Probability Graphs 12.6.1 Probability Graphs 481

463

470

481

12.6.2 Plotting Positions 481 12.6.3 Data Plotting and Theoretical Distributions 12.6.4 Estimating Future Magnitudes 484 Rainfall Intensity-Duration-Frequency Relationships Applicability of Statistical Methods 488 Problems 488

483 485

SYMBOLS

493

ANSWERS TO SELECTED PROBLEMS

496

INDEX

502

Preface

This book provides a fundamental treatment of engineering hydraulics. It is primarily intended to serve as a textbook for undergraduate engineering students. However, it will serve as a very useful desk reference for practicing engineers who want to review basic principles and their applications in hydraulic engineering systems. Engineering hydraulics is an extension of fluid mechanics in which many empirical relationships are applied and simplifying assumptions made to achieve practical engineering solutions. Experience has shown that many engineering students, possibly well versed in basic fluid mechanics, have trouble solving practical problems in hydraulics. This book is intended to bridge the gap between fundamental principles and the techniques applied to the design and analysis of hydraulic engineering systems. As such, the reader is exposed to many problems commonly encountered in practice and various solution scenarios including effective design procedures, equations, graphs/tables, and computer software that can be used to advantage. This book contains twelve chapters. The first five chapters cover the fundamentals of fluid statics, fluid dynamics, and pipe flow. The first chapter discusses fundamental properties of water as a fluid. In this chapter, the basic differences between the SI system (Le Système International d'Unitès) and British units are discussed. The second chapter presents the concepts of water pressure and pressure forces on surfaces in contact with liquids. Chapter 3 introduces the basic principles of water flow in pipes. These principles are applied to the practical problems xi

xii

Preface

of pipelines and pipe networks in Chapter 4 with an emphasis on hydraulic systems. Chapter 5 discusses the theory, analysis, and design aspects of water pumps. The systems approach is again emphasized with detailed instruction on pump analysis within pipelines, branching pipe systems, and pipe networks, as well as pump selection and design considerations. The next three chapters cover open channel flow, ground water, and the design of various hydraulic structures. Water flow in open channels is presented in Chapter 6. Detailed discussions of uniform flow (normal depth), rapidly varied flow (hydraulic jumps), and gradually varied flow (classifications and water surface profiles) are included, along with open channel design. The hydraulics of wells and seepage problems are two key topics in Chapter 7 on ground water. Well hydraulics includes coverage of equilibrium and non-equilibrium conditions in confined and unconfined aquifers. Chapter 8 introduces some of the most common hydraulic structures such as dams, weirs, spillways, culverts, and stilling basins. Functionality, hydraulic principles, practical considerations, and design procedures are provided. The book ends with four ancillary topics: measurements, model studies, hydrology for hydraulic design, and statistical methods in hydrology. Chapter 9 provides information on the measurement of water pressure, velocity, and discharge in pipes and open channels. The proper use of scaled models is an essential part of hydraulic engineering. As such, the use of hydraulic models and the laws of engineering similitude are covered in Chapter 10. Flow rates are required for the design of all hydraulic structures; many are obtained using the principles of hydrology. The last two chapters introduce common techniques used for obtaining hydrologic design flows. Deterministic procedures are covered in Chapter 11 and statistical methods are covered in Chapter 12. In addition, the design of stormwater collection, transport, and storage systems (routing) is introduced in Chapter 11. New to this Edition Significant revisions have been made to this new edition based on feedback from practitioners, university professors, and book reviewers. These revisions include: š More than half of the end-of-chapter problems provided in the book are new or revised from the previous edition. A solutions manual and a test/examination manual are available to university professors who adopt the book for their class. š Topical PowerPoint slides are available to explain most of the major sections in the book and include active (classroom) learning exercises. The solutions manual, test manual, and topical PowerPoint slides can be obtained under the book's instructor resources on the publisher's website. š Chapter 11 is completely revised to emphasize urban hydrology and the most widely used analysis techniques for stormwater management. In addition, there is a section on hydrologic modeling that describes the features of the HEC-HMS model and EPASWMM model. The use of these models is encouraged in some of the end-of-chapter problems. š Chapter 9 includes weir equations for SI units to go along with BG counterparts. š Chapter 7 includes an example problem on unsteady flow in unconfined aquifers. š Chapter 6 includes a new section on open channel flow modeling, describes the features of the HEC-RAS model, and encourages the use of models in many of the end-of-chapter problems.

Preface

xiii

š Chapter 4 includes a new section on pipe network modeling, describes the features of the EPANET model, and encourages the use of models in many of the end-of-chapter problems. In addition, many clarifications to nomenclature and process have been made along with better solution explanations in Examples 4.4, 4.6, and 4.8 š Chapter 3 includes a more refined development of the momentum equation, a more elegant proof of the energy equation, and an updated diffusor loss equation. š Chapter 2 includes new example problems to clarify the concepts of equal pressure surfaces and floatation stability. š Chapter 1 now contains water property tables in both BG and SI units. š Removal of errors and clarification of concepts have been enacted throughout this new edition. Many new and clarified figures have been added as well. Chapters 11 and 12 (hydrology) in concert with earlier chapters on pipe flow and pumps will accommodate professors who teach hydrology in combination with hydraulics. For those professors who teach a separate hydraulics and hydrology class sequence, Chapters 1 through 6, 10, and 9 (first three sections) in some combination form a nice package for hydraulics. Chapters 11, 12, 7, 8, and 9 (last section) form a nice package for hydrology. Faculty Information and Resources This book is primarily used for a one-semester (16 weeks), three-hour course in the undergraduate engineering curriculum. This edition continues to provide example and chapter problems in both SI and British units. A prerequisite in fluid mechanics is not necessary, but is highly recommended. One of the authors has used the text for a course in hydraulic engineering (fluid mechanics as a prerequisite) and begins in Chapter 4 after a quick review of the first three chapters. It is not possible to cover the book completely in a semester. However, many of the later chapters (ground water, hydraulic structures, model studies, and hydrology) can be covered or deleted from the syllabus according to instructor preferences without a loss of continuity. Some adopters have used the book for a two semester sequence of hydraulics followed by hydrology. PowerPoint slides to accompany the book with active learning exercises for students are available for most of the chapters under the book's instructor resources on the publisher's website. There are 119 example problems and 596 homework problems, providing coverage for every major topic in the book. In general, the homework problems are sequenced according to Bloom's taxonomy; the earlier problems in a section measure comprehension and application and the later problems measure analysis and some synthesis. A solutions manual is available for university professors who adopt the book for their class. Three significant figures were used almost exclusively in all problem solutions. In addition, a test manual is available to those same professors to assist them in quickly creating tests for student assessment. The test manual includes short-answer questions and two or three problems from each major section of the book (193 problems in all). The test manual could be used to assign supplemental homework problems. The authors have included many topics which would benefit greatly from the use of offthe-shelf computer software and student-generated spreadsheet analysis. Some of these topics include: energy balance in pipelines (Sections 3.4–3.12); pipeline analysis, branching pipe systems, and pipe networks (Sections 4.1–4.4); pump/pipeline system analysis (Sections 5.4–5.8); pump selection (Section 5.11); normal and critical depth in open channels (Sections 6.2–6.4);

xiv

Preface

water surface profiles in open channels (Section 6.8); culvert analysis (Section 8.9); hydrologic design techniques (Chapter 11); and statistical methods in hydrology (Chapter 12). In engineering practice, an abundance of computer software is available in these topic areas to accelerate and simplify the design and analysis process. Some companies make use of in-house computer software or spreadsheet solutions written by their technical staff. Other companies use software that is readily available from private software vendors and governmental agencies. The authors have written spreadsheets in some topic areas (evident in the solutions manual and available to course instructors who adopt the textbook) and occasionally used Internet freeware or governmental software to check desktop methods used in the solutions manual. The authors encourage course instructors to have their students use off-the-shelf software and, in some cases, program their own solutions in spreadsheets or computer algebra packages (e.g., Mathcad, Maple, or Mathematica). Recommendations for task-specific, nonproprietary packages include "EPANET" (Environmental Protection Agency) for pipe networks, "HECRAS" (U.S. Army Corps of Engineers) for normal and critical depths and water surface profiles, "HEC-HMS" (U.S. Army Corps of Engineers) for storm hydrographs and reservoir routing, and "EPA-SWMM" (Environmental Protection Agency) for urban stormwater collection systems. Other software is freely available on the Internet for pressure pipe flow, open channel flow, and pump analysis/design. Internet searches can uncover an abundance of freeware that will prove very useful. In addition, there is an abundance of proprietary packages that solve specific hydraulic problems. Almost all of the solution techniques are amenable to spreadsheet and/or computer algebra package programming and would make great student projects. (Please contact the authors, [email protected] or [email protected] if you need assistance.) A number of homework problems in the book encourage the use of computer software. In addition, a few classroom computer exercises and some homework problems are included to introduce appropriate software and its capabilities. These exercises can be accomplished during class or lab time. They are meant to promote a cooperative learning environment where student teams are actively engaged in engineering analysis/design to promote some rich classroom discussion. The primary objective is the development of a deeper understanding of the subject matter, but they do require student laptops loaded with the appropriate software. The senior author uses teams of two or three students to stimulate dialogue and enhance the learning process. Alternatively, students can proceed through the exercises for homework and bring printouts of their results to class for discussion. Despite the gentle nudge to acquaint students with hydrologic and hydraulic software, the book does not become a slave to it. As previously mentioned, a number of topics encourage the use of software. The first few problems at the end of each of these sections (topics) require hand calculations. After the students are aware of the solution algorithms, more complex problems are introduced which can be solved with software. Consequently, the students will be in a position to anticipate data that the software needs to solve the problem and understand what the software is doing computationally. In addition, the instructor can ask many "what if " questions of the students in connection with these homework problems. This will greatly enhance their understanding of the subject matter without burdening them with tedious calculations.

Acknowledgments

I was honored to be asked by Dr. Ned Hwang to participate in writing the third edition of this textbook over two decades ago. Ned did a terrific job in writing the first edition in the early 1980s. Writing a textbook is a massive undertaking, and it is to his credit that that the book has been used by many universities for over three decades. At one time, the fourth edition was being used by almost 60 American universities with many international sales in Asia, Canada, the Middle East, and the United Kingdom. Our second author, Dr. A. Osman Akan, has published extensively in the hydrology and hydraulics literature. He is an excellent teacher and a scholar, having published numerous research papers and other textbooks in these two fields. I was fortunate to have coauthored another book with him in the area of urban stormwater management. I am so blessed to have had the opportunity to work with Osman and Ned over the years; they have been good friends and mentors to me as I develop as a person and a professional. It takes a great deal of effort to write and revise a textbook. I am thankful to my wife, Judy, and my three sons for supporting me in these efforts through the years. The fourth edition was written while I was in the Sudan on sabbatical working for International Aid Services (IAS), a Swedish humanitarian organization that works tirelessly in the areas of water and sanitation. Leif and Daniel Zetterlund from IAS were instrumental in helping me to see the altruistic side of engineering outreach. This edition was written in part while spending time teaching and doing xv

xvi

Acknowledgments

humanitarian work in Africa. The need for water and supporting projects continues to be monumental in this great continent. I continue to be grateful to North Carolina State University and Colorado State University for providing me with an engineering education and scholarship assistance. Dr. Rooney Malcom and Dr. Jim Loftus were instrumental in my early professional development. I am also grateful to the late Jerome Normann, my friend and mentor, who shared with me the fine art of professional practice in hydraulics and hydrology. I am grateful to Siavash Beik (Vice President, Christopher B. Burke Engineering, LLC) for his help in my professional growth. We taught ASCE continuing education classes together on the topics covered in this book. His firm supplied some of the chapter opening images. I would also like to thank my friends and colleagues at Rose-Hulman Institute of Technology for all their support. I spent 21 years teaching there, a truly great place to work, grow, and engage with students and scholars. Finally, I would like to thank the many reviewers of this book through the years and the many professors and practitioners who have pointed out errors and made suggestions to improve the content of the book. Robert J. Houghtalen Rose-Hulman Institute of Technology, Emeritus Engineering & Management Consultant, East Africa

It was a great honor for me to be asked by Robert and Ned to join them again for writing this new edition. Ned deserves the most credit for the existence of this book. He wrote the earlier editions with Robert's participation in the third edition. Many professors, including myself, and students used these editions as a textbook or a reference. Robert led the effort for the fourth and fifth editions and did an excellent job. I had the opportunity to co-author another book with him in the recent past. He is a true scholar, a dedicated educator, and a fine individual. I feel very fortunate for having him as a colleague and friend for decades. I made my contributions to this edition while serving as Head of the Department of Civil Engineering at the American University of Sharjah, United Arab Emirates. This is a great university to teach and conduct research. I am thankful for the support I received from my colleagues, the Dean's Office, and the university administration. I am indebted to all the professors I had at Middle East Technical University, Ankara, Turkey, as an undergraduate student and at the University of Illinois as a graduate student. However, the late Ben C. Yen, my thesis and dissertation supervisor, always has a special place in my heart. I learned so much from him. Dr. Yen, a scholar and gentleman, remained my teacher, mentor, and friend until he passed away in 2001. I am most fortunate to have a wonderful family and grateful to my wife, Güzin, and my son, Doruk, both engineers, for their love and support during this project and always. A. Osman Akan Old Dominion University, Emeritus American University of Sharjah

Acknowledgments

xvii

The first attempt to arrange my lecture notes into a textbook was made in 1974. My colleagues at the University of Houston, particularly Professors Fred W. Rankin Jr. and Jerry Rogers, had provided valuable suggestions and assistance, which formed the foundation of this book. Dr. Rogers also carefully reviewed the first edition text after its completion. I am also deeply indebted to my students, Drs. Travis T. Stripling, John T. Cox, James C. Chang, and Po-Ching Lu. All provided assistance during the various stages of preparation. Dr. Ahmed M. Sallam, who used a first draft of this book in a hydraulics course, provided many suggestions. Dr. Carlos E. Hita, who was co-author of the second edition, offered valuable suggestions and also provided many of the sample problems used in the text. My dear friend, Dr. David R. Gross, an ever-inquisitive physiologist (as was Jean Louise Poiseuille, 1799–1869) with a great interest in hydraulic matters, reviewed the first edition and offered many irrefutable criticisms. During the preparation of the first edition, I was ill for some time. The continuous encouragement, loyalty, and love of Maria, Leon, and Leroy kept me on track during the dark hours that have since passed. To them also, I dedicate this book. Ned H. C. Hwang James L. Knight Professor University of Miami, Emeritus

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Introduction

Hydraulic systems are designed to transport, store, or regulate water. All hydraulic systems require the application of fundamental principles of fluid mechanics. However, many also require an understanding of hydrology, soil mechanics, structural analysis, engineering economics, geographic information systems, and environmental engineering for proper planning, design, construction, and operation. Unlike some branches of engineering, each hydraulic project encounters a unique set of physical conditions to which it must conform. There are no standard solutions or simple handbook answers. Hydraulic engineering relies on fundamental knowledge that must be applied to meet the special conditions of each project. The shape and dimensions of hydraulic systems may vary from a small flow meter a few centimeters in size to a levee several hundred kilometers long. Generally, however, hydraulic structures are relatively massive when compared with the products of other engineering disciplines. For this reason, the design of large hydraulic systems is site specific. It is not always possible to select the most desirable location or material for a particular system. Commonly, a hydraulic system is designed to suit the local conditions, which include topography, geology, ecology, environmental protection, social concerns, and the availability of native materials. All of these considerations and many more make up what is more recently referred to as Integrated Water Resources Management (IWRM). xix

Introduction

Hydraulic engineering is as old as civilization itself, testifying to the importance of water to human life. There is much evidence that hydraulic systems of considerable magnitude existed several thousand years ago. For example, a large-scale drainage and irrigation system built in Egypt can be dated back to 3200 BC. Rather complex water supply systems, including several hundred kilometers of aqueducts, were constructed to bring water to ancient Rome. Dujonyen, a massive irrigation system in Siechuan, China, built nearly 2500 years ago, is still in effective use today. The abundant knowledge resulting from these and other, more recent practical applications of hydraulic engineering is indispensable. In addition to the analytical approach, some modern hydraulic system design and operation depends on empirical formulas that produce excellent results in water works. Unfortunately, most of these empirical formulas cannot be analyzed or proven theoretically. In general, they are not dimensionally homogeneous. For this reason, the conversion of units from the British system to SI units and vice-versa is more than just a matter of convenience. Sometimes the rigorous form (e.g., the Parshall equations for water flow measurements) must be maintained with its original units. In these cases, all quantities should be converted to the original units specified by the equation for computations. The book leans toward the use of SI units. It is the most widely used unit system in the world and is particularly dominant in commerce and science. As the world has "flattened," using the language of Thomas Friedman, the move to SI units has accelerated. The European Union has started to ban non-SI markings on imported goods (e.g., pipes, pumps, etc.). Roughly 40% of the problems in the book are written in British units to accommodate the mixture of units (BG and SI) used in some countries. However, the chapters on hydrology may use a higher percent of problems in BG units. The transition to SI units in the field of hydrology in the U.S. does not seem to be progressing very quickly. A detailed table of conversions is provided on the inside front cover to assist the student.

Fundamental Properties of Water

The word hydraulic comes from two Greek words: "hydor" (meaning water) and "aulos" (meaning pipe). Through the years, the definition of hydraulics has broadened beyond merely pipe flow. Hydraulic systems are designed to accommodate water at rest and in motion. The fundamentals of hydraulic engineering systems, therefore, involve the application of engineering principles and methods to the planning, control, transportation, conservation, and utilization of water. It is important that the reader understand the physical properties of water to solve properly the various problems in hydraulic engineering systems. For example, the density, the surface tension, and the viscosity of water all vary, in one way or another, with water temperature. Density is a fundamental property that directly relates to the operation of a large reservoir. For example, change of density with temperature causes water to stratify in the summer, with warmer water on top of colder water. During the late fall, the surface water temperature drops rapidly, and water begins to sink toward the reservoir bottom. The warmer water near the bottom rises to the surface, resulting in "fall overturn" in northern climates. In the winter, the surface water freezes while warmer water remains insulated below the ice. The winter stratification is followed by a "spring overturn." The ice melts, and the surface water warms to 4° Celsius (C) (highest water density) and sinks to the bottom as the warmer water below rises. Similarly, variation of surface tension directly affects the evaporation loss from a large water body in storage; the variation of water viscosity with temperature is important to all problems involving water in motion. 1

Fundamental Properties of Water

Chap. 1

This chapter discusses the fundamental physical properties of water that are important to problems in hydraulic engineering systems. 1.1 The Earth's Atmosphere and Atmospheric Pressure The Earth's atmosphere is a thick layer (approximately 1,500 km) of mixed gases. Nitrogen makes up approximately 78% of the atmosphere, oxygen makes up approximately 21%, and the remaining 1% consists mainly of water vapor, argon, and trace amounts of other gases. Each gas possesses a certain mass and consequently has a weight. The total weight of the atmospheric column exerts a pressure on every surface it contacts. At sea level and under normal conditions, the atmospheric pressure is approximately equal to 1.014 * 105 N/m2, or approximately 1 bar.* The pressure unit 1 N/m2 is also known as 1 pascal, named after French mathematician Blaise Pascal (1623 to 1662). Water surfaces that come in contact with the atmosphere are subjected to atmospheric pressure. In the atmosphere, each gas exerts a partial pressure independently of the other gases. The partial pressure exerted by the water vapor in the atmosphere is called the vapor pressure. 1.2 The Three Phases of Water The water molecule is a stable chemical bond of oxygen and hydrogen atoms. The amount of energy holding the molecules together depends on the temperature and pressure present. Depending on its energy content, water may appear in solid, liquid, or gaseous form. Snow and ice are the solid forms of water; liquid is its most commonly recognized form; and moisture, water vapor in air, is water in its gaseous form. The three different forms of water are called its three phases. To change water from one phase to another phase, energy must either be added or taken away from the water. The amount of energy required to change water from one phase to another is known as a latent energy. This amount of energy may be in the form of heat or pressure. One of the most common units of heat energy is the calorie (cal). One calorie is the energy required to increase the temperature of 1 gram (g) of water, in liquid phase, by 1°C. The amount of energy required to raise the temperature of a substance by 1°C is known as the specific heat of that substance. The latent heat and specific heat of all three phases of water are discussed next. Under standard atmospheric pressure, the specific heat of water and ice are, respectively, 1.0 and 0.465 cal/g # °C. For water vapor, the specific heat under constant pressure is 0.432 cal/g # °C, and at constant volume it is 0.322 cal/g # °C. These values may vary slightly depending on the purity of the water. To melt 1 g of ice, changing water from its solid to liquid phase, requires a latent heat (heat of fusion) of 79.7 cal. To freeze water, the same amount of heat energy must be taken out of each gram of water, thus the process is reversed. Evaporation, the changing of liquid-phase water into its gaseous phase, requires a latent heat (heat of vaporization) of 597 cal/g. Evaporation is a rather complex process. Under standard atmospheric pressure, water boils at 100°C. At higher elevations, where the atmospheric pressure is less, water boils at * 1 atmospheric pressure = 1.014 * 105 N/m2 = 1.014 * 105 pascals = 1.014 bars = 14.7 lb/in.2 = 760 mm Hg = 10.33 m H2O

Sec. 1.3

Mass (Density) and Weight (Specific Weight)

Table 1.1 Vapor Pressure of Water

Temperature (°C) -5 0 5 10 15 20 25 30 35 40 45 50

(°F) 23 32 41 50 59 68 77 86 95 104 113 122

Vapor Pressure Atm 0.004162 0.006027 0.008600 0.01210 0.01680 0.02304 0.03122 0.04183 0.05545 0.07275 0.09453 0.1217

* 103 N/m2 0.4210 0.6110 0.8730 1.227 1.707 2.335 3.169 4.238 5.621 7.377 9.584 12.33

Temperature

Vapor Pressure

lb/in.2

(°C)

(°F)

Atm

0.06118 0.08860 0.1264 0.1779 0.2470 0.3387 0.4590 0.6149 0.8151 1.069 1.390 1.789

55 60 65 70 75 80 85 90 95 100 105 110

131 140 149 158 167 176 185 194 203 212 221 230

0.1553 0.1966 0.2468 0.3075 0.3804 0.4674 0.5705 0.6919 0.8342 1.000 1.192 1.414

*103 N/m2

lb/in.2

15.75 19.92 25.02 31.17 38.56 47.37 57.82 70.13 84.55 101.4 120.8 143.3

2.283 2.889 3.628 4.521 5.592 6.871 8.386 10.17 12.26 14.70 17.53 20.78

temperatures lower than 100°C. This phenomenon may be explained best from a molecularexchange viewpoint. At the gas–liquid interface, there is a continual interchange of molecules leaving the liquid to the gas and molecules entering the liquid from the gas. Net evaporation occurs when more molecules are leaving than are entering the liquid; net condensation occurs when more molecules are entering than are leaving the liquid. Equilibrium exists when the molecular exchange at the gas–liquid interface are equal over a time interval. Vapor molecules in the air exert a partial pressure on any contact surface that is known as the vapor pressure. This partial pressure combined with the partial pressures created by other gases in the atmosphere makes up the total atmospheric pressure. If the temperature of a liquid is increased, the molecular energy is raised, causing a large number of molecules to leave the liquid. This, in turn, increases the vapor pressure. When the temperature reaches a point at which the vapor pressure is equal to the ambient atmospheric pressure, evaporation increases significantly, and boiling of the liquid takes place. The temperature at which a liquid boils is commonly known as the liquid's boiling point. For water at sea level, the boiling point is 100°C. The vapor pressure of water is shown in Table 1.1. In a closed system (e.g., pipelines or pumps), water vaporizes rapidly in regions where the pressure drops below the vapor pressure. This phenomenon is known as cavitation. The vapor bubbles formed in cavitation usually collapse in a violent manner when they move into higher pressure regions. This may cause considerable damage to a system. Cavitation in a closed hydraulic system can be avoided by maintaining the pressure above the vapor pressure everywhere in the system. 1.3 Mass (Density) and Weight (Specific Weight) In the International System of Units (SI),* the unit of measurement for mass is either gram or kilogram (kg). The density of a substance is defined as the mass per unit volume. It is a property inherent in the molecular structure of the substance. This means that density depends not only on the size and weight of the molecules but also on the mechanisms by which the molecules are * From the French Le Système International d'Unités.

Fundamental Properties of Water Table 1.2

Chap. 1

Density and Specific Weight of Water

Temperature

Density (ρ)

Specific Weight (γ)

(°C)

(°F)

kg/m3

slug/ft3

N/m3

0 (ice) 0 (water) 4 10 20 30 40 50 60 70 80 90 100

32 32 39.2 50 68 86 104 122 140 158 176 194 212

917 999 1,000 999 998 996 992 988 983 978 972 965 958

1.78 1.94 1.94 1.94 1.94 1.93 1.92 1.92 1.91 1.90 1.89 1.87 1.86

8,996 9,800 9,810 9,800 9,790 9,771 9,732 9,692 9,643 9,594 9,535 9,467 9,398

lb/ft3 57.3 62.4 62.4 62.4 62.3 62.2 62.0 61.7 61.4 61.1 60.7 60.3 59.8

bonded together. The latter usually varies as a function of temperature and pressure. Because of its peculiar molecular structure, water is one of the few substances that expands when it freezes. The expansion of freezing water when contained in a closed container causes stresses on the container walls. These stresses are responsible for the bursting of frozen water pipes, the creation of cracks and holes in road pavement, and the weathering of rocks in nature. Water reaches a maximum density at 4°C. It becomes less dense when further chilled or heated. The density of water is shown as a function of temperature in Table 1.2. Note that the density of ice is different from that of liquid water at the same temperature. We observe this phenomenon when we see ice float on water. Seawater (or ocean water) contains dissolved salt. The molecules that make up the salt have more mass than the molecules they displace. Therefore, the density of seawater is about 4% more than that of freshwater. Thus, when freshwater meets seawater without sufficient mixing, as in Chesapeake Bay, salinity will increase with depth. In the SI system, the weight of an object is defined by the product of its mass (m, in grams, kilograms, etc.) and the gravitational acceleration (g = 9.81 m/s2 on Earth). The relationship* may be written as W = mg

(1.1)

Weight in the SI system is usually expressed in the force units of newtons (N). One newton is defined as the force required to accelerate 1 kg of mass at a rate of 1 m/s2. The specific weight (weight per unit volume) of water (γ) can be determined by the product of the density (ρ) and the gravitational acceleration (g). The ratio of the specific weight of any liquid at a given temperature to that of water at 4°C is called the specific gravity of that liquid. Note that the specific weight of water is shown as a function of temperature in Table 1.2. *In the British (Imperial) system, the mass of an object is defined by its weight (ounce or pound) and the gravitational acceleration (g = 32.2 ft/s2 on Earth). The relationship is written as m = W/g

(1.1a)

Sec. 1.4

Viscosity of Water

The unit of mass in the British system is the slug. One slug is defined as the mass of an object that requires 1 lb of force to reach an acceleration of 1 ft/s2. Example 1.1 An aquarium holds 0.5 m3 of water. The weight of the aquarium is 5,090 N when full and 200 N when empty. Determine the temperature of the water. Solution The weight of the water in the aquarium is: W (water) = 5,090 N - 200 N = 4,890 N The specific weight of the water is: γ = 4,890N/(0.5 m3) = 9,780 N/m3 The temperature of the water can be obtained from Table 1.2: T ? 25°C

1.4 Viscosity of Water Water responds to shear stress by continuously yielding in angular deformation in the direction of the shear as shown in Figure 1.1. This leads to the concept of viscosity. The schematic diagram in Figure 1.1 represents the physical basis of viscosity. Consider that water fills the space between two parallel plates (lightweight plastic) that are a distance y apart. A horizontal force T is applied to the upper plate and moves it to the right at velocity v while the lower plate remains stationary. The shear force T is applied to overcome the water resistance R, and it must be equal to R because there is no acceleration involved in the process. The resistance per unit area of the upper plate (shear stress, τ = R/A = T/A) has been found to be proportional to the rate of angular deformation in the fluid, dθ>dt. The relationship may be expressed as dx/dy dθ dx/dt dv τ ∝ = = = dt dt dy dy where v = dx/dt is the velocity of the fluid element. Alternatively, τ = μa

Moving plate

dv b dy

(1.2)

T V v + dv

v u

dv dy

Stationary plate Figure 1.1

Shearing stresses in fluids

Fundamental Properties of Water

Chap. 1

Table 1.3 Viscosities of Water and Air

Water Temperature (°C)

0 5 10 15 20 25 30 40 50 60 70 80 90 100

(°F)

32 41 50 59 68 77 86 104 122 140 158 176 194 212

Viscosity (μ)

Air Kinematic Viscosity (ν)

Viscosity

K. Viscosity

* 10 -3 N # s/m2

* 10 -5 lb # s/ft2

* 10 -6 m2/s

* 10 -5 ft2/s

* 10 -5 N # s/m2

* 10 -5 m2/s

1.781 1.518 1.307 1.139 1.002 0.890 0.798 0.653 0.547 0.466 0.404 0.354 0.315 0.282

3.721 3.171 2.730 2.379 2.093 1.859 1.667 1.364 1.143 0.973 0.844 0.740 0.658 0.589

1.785 1.519 1.306 1.139 1.003 0.893 0.800 0.658 0.553 0.474 0.413 0.364 0.326 0.294

1.921 1.634 1.405 1.226 1.079 0.961 0.861 0.708 0.595 0.510 0.444 0.392 0.351 0.316

1.717 1.741 1.767 1.793 1.817 1.840 1.864 1.910 1.954 2.001 2.044 2.088 2.131 2.174

1.329 1.371 1.417 1.463 1.509 1.555 1.601 1.695 1.794 1.886 1.986 2.087 2.193 2.302

The proportionally constant, μ, is the absolute viscosity of the fluid. Equation 1.2 is commonly known as Newton's law of viscosity. Most liquids abide by this relationship and are called Newtonian fluids. Liquids that do not abide by this linear relationship are known as non-Newtonian fluids. These include most house paints and human blood. Absolute viscosity has the dimension of force per unit area (stress) multiplied by the time interval considered. It is usually measured in the unit of poise (named after French engineerphysiologist J. L. M. Poiseuille). The absolute viscosity of water at room temperature (20.2°C) is equal to 1 centipoise (cP), which is one one-hundredth (1/100) of a poise: 1 poise = 0.1 N # s/m2 = 100 cP or (1 N # s/m2 = 1,000 cP) The absolute viscosity of air is approximately 0.018 cP (roughly 2% of water). In engineering practice, it is often convenient to introduce the term kinematic viscosity, ν, which is obtained by dividing the absolute viscosity by the mass density of the fluid at the same temperature: ν = μ/ρ. The kinematic viscosity carries the unit of cm2/s (with the unit of stokes, named after British mathematician G. G. Stoke). The absolute viscosities and the kinematic viscosities of pure water and air are shown as functions of temperature in Table 1.3. Example 1.2 A flat plate of 50 cm2 is being pulled over a fixed flat surface at a constant velocity of 45 cm/s (Figure 1.1). An oil film of unknown viscosity separates the plate and the fixed surface by a distance of 0.1 cm. The force (T) required to pull the plate is measured to be 31.7 N, and the viscosity of the fluid is constant. Determine the viscosity (absolute). Solution The oil film is assumed to be Newtonian, therefore the equation τ = μ(dv/dy) applies: μ = τ/(dv/dy) = (T/A)/(∆v/∆y)

Sec. 1.5

Surface Tension and Capillarity

because τ = T/A and the velocity–distance relationship is assumed to be linear. Therefore, μ = [(31.7 N)/(50 cm2)]/[(45 cm/s)/0.1 cm)] μ = 1.41 * 10-3 N # s/cm2 [(100 cm)2/(1 m)2] = 14.1 N # s/m2

1.5 Surface Tension and Capillarity Even at a small distance below the surface of a liquid body, liquid molecules are attracted to each other by equal forces in all directions. The molecules on the surface, however, are not able to bond in all directions and therefore form stronger bonds with adjacent liquid molecules. This causes the liquid surface to seek a minimum possible area by exerting surface tension tangent to the surface over the entire surface area. A steel needle floating on water, the spherical shape of dewdrops, and the rise and fall of liquid in capillary tubes are the results of surface tension. Most liquids adhere to solid surfaces. The adhesive force varies, depending on the nature of the liquid and of the solid surface. If the adhesive force between the liquid and the solid surface is greater than the cohesion in the liquid molecules, then the liquid tends to spread over and wet the surface as shown in Figure 1.2 (a). If the cohesion is greater, then a small drop forms as in Figure 1.2 (b). Water wets the surface of glass, but mercury does not. If we place a small-bore vertical glass tube into the free surface of water, the water surface in the tube rises. The same experiment performed with mercury will show that the mercury falls. These two typical cases are schematically presented in Figures 1.3 (a) and 1.3 (b). The phenomenon is commonly known as capillary action. The magnitude of the capillary rise (or depression), h, is determined by the balance of adhesive force between the liquid and solid surface and the weight of the liquid column above (or below) the liquid-free surface. The angle θ at which the liquid film meets the glass depends on the nature of the liquid and the solid surface. The upward (or downward) motion in the tube will cease when the vertical component of the surface-tension force around the edge of the film equals the weight of the raised (or lowered) liquid column. When the very small volume of liquid above (or below) the base of the curved meniscus is neglected, the relationship may be expressed as (σπD) sin θ =

πD2 (γh) 4 Air

a < 90°

Air

Water

a > 90°

Wetting surface (a)

Nonwetting surface (b)

Figure 1.2 Wetting and nonwetting surfaces

Fundamental Properties of Water

Chap. 1

s u

h Water

Mercury

u (a)

(b) Figure 1.3 Capillary actions

Hence, h =

4σ sin θ γD

(1.3)

where σ and γ are the surface tension and the unit weight of the liquid, respectively, and D is the inside diameter of the vertical tube. The surface tension of a liquid is usually expressed in the units of force per unit length. Its value depends on the temperature and the electrolytic content of the liquid. Small amounts of salt dissolved in water tend to increase the electrolytic content and, hence, the surface tension. Organic matter (such as soap) decreases the surface tension in water and permits the formation of bubbles. The surface tension of pure water as a function of temperature is listed in Table 1.4. 1.6 Elasticity of Water Water is commonly assumed to be incompressible under ordinary conditions. In reality, it is about 100 times more compressible than steel. It is necessary to consider the compressibility of water when water hammer issues are possible (see Chapter 4). The compressibility of

Table 1.4 Surface Tension of Water

Temperature (°C/°F) Surface Tension, σ * 10-2 N/m dynes/cm * 10-3 lb/ft

104

122

140

158

176

194

7.416 7.279 7.132 6.975 6.818 6.786 6.611 6.436 6.260 6.071 74.16 72.79 71.32 69.75 68.18 67.86 66.11 64.36 62.60 60.71 5.081 4.987 4.887 4.779 4.671 4.649 4.530 4.410 4.289 4.160

Sec. 1.6

Elasticity of Water

water is inversely proportional to its volume modulus of elasticity, Eb, also known as the bulk modulus of elasticity. The pressure–volume relationship may be expressed as ∆P = -Eb a

∆Vol b Vol

(1.4)

where Vol is the initial volume, and ∆P and ∆Vol are the corresponding changes in pressure and volume, respectively. The negative sign means that a positive change in pressure (i.e., pressure increase) will cause the volume to decrease (i.e., negative change). The bulk modulus of elasticity of water varies with both temperature and pressure. In the range of practical applications in typical hydraulic systems, a value of 2.2 * 109 N/m2, or in BG* units, 3.2 * 105 lb/in2 (psi) can be used.

Example 1.3 The density of seawater is 1,026 kg/m3 at sea level. Determine the density of seawater on the ocean floor 2,000 m deep, where the pressure is approximately 2.02 * 107 N/m2. Solution The change of pressure at a depth of 2,000 m from the pressure at the water surface is ∆P = P - P atm = 2.01 * 107 N/m2 From Equation 1.4 we have ∆P = -Eb a

so that

Because

∆Vol - ∆P -2.01 * 107 b = a b = = -0.00914 Volo Eb 2.20 * 109 ρ = a

then

so that

∆Vol b Volo

∆Vol = a ρ = §

m b Vol

m m b - a b ρ ρo

6 Vol = a 6

m b ρ

ρo ∆Vol = a b - 1 ρ Volo

1,026 kg/m3 ρo ¥ = ¢ ≤ = 1,040 kg/m3 ∆Vol 1 - 0.00914 1 + Volo

*British gravitational system of units

Fundamental Properties of Water

Chap. 1

1.7 Forces in a Fluid Field Various types of forces may be exerted on a body of water at rest or in motion. In hydraulic practice, these forces usually include the effects of gravity, inertia, elasticity, friction, pressure, and surface tension. These forces may be classified into three basic categories according to their physical characteristics: 1. body forces, 2. surface forces, and 3. line forces (or forces over a solid–liquid contact distance). Body forces are forces that act on all particles in a body of water as a result of some external body or effect but not because of direct contact. An example of this is the gravitational force. It acts on all particles in any body of water as a result of the Earth's gravitational field, which may not be in direct contact with the particular water body of interest. Other body forces that are common in hydraulic practice include inertial forces and forces resulting from elastic effects. Body forces are usually expressed as force per unit mass (N/kg) or force per unit volume (N/m3). Surface forces act on the surface of the water body by direct contact. These forces may be either external or internal. Pressure forces and friction forces are examples of external surface forces. The viscous force inside a fluid body may be viewed as an internal surface force. Surface forces are expressed as force per unit area (N/m2). Line forces act at the liquid surface normal to a line drawn in the surface. They often act along a linear solid–liquid interface. An example of a line force is surface tension. Line forces are expressed as force per unit length (N/m). Problems (secTion 1.2) 1.2.1. Determine the amount of energy released (in calories) when steam at 110°C (constant pressure) condenses and cools to produce 500 liters of water at 50°C? 1.2.2. Water needs to be turned into steam in a high altitude lab where the atmospheric pressure is 84.6 kPa. Compute the heat energy (in calories) required to evaporate 900 g of water at 15°C under these conditions. 1.2.3. There are only 2.00 * 108 calories of energy available in a certain laboratory process. What pressure must be maintained in a closed vessel to turn 300 liters of water at 20°C entirely into water vapor using the available energy? 1.2.4. In a thermal container, 10 grams of ice at - 6°C is mixed with 0.165 liters of water at 20°C. Determine whether or not all the ice will be melted. If the ice is melted, determine the final temperature once equilibrium is established. 1.2.5. Determine the final temperature of a water–ice bath that is allowed to establish equilibrium in a thermal container. The water–ice bath is produced when 5 slugs of ice (specific heat = 0.46 BTU/lbm # °F) at 20°F is mixed with 10 slugs of water (specific heat = 1.0 BTU/lbm # °F) at 120°F. (Note: 1 slug = 32.2 lbm and the heat of fusion is 144 BTU/lbm.) 1.2.6. A large kitchen pot initially contains 7.5 kg of water at 20°C. A burner is turned on and heat is added to the water at a rate of 500 cal/s. Determine how many minutes it will take for one-third of the mass to evaporate at standard atmospheric pressure.

Problems

(secTion 1.3) 1.3.1. Derive the relationship between specific weight and density starting with Newton's 2nd Law (F = ma or alternatively, W = mg). 1.3.2. Mexican crude oil has a specific gravity of 0.976 at 15.5°C. Determine its specific weight (in N/m3) and density (in kg/m3) at that temperature. 1.3.3. The specific weight of a liquid is 55.5 lb/ft3. Determine the weight, density, and specific gravity of the liquid if it occupies a volume of 20 ft3. Provide results in both the British system and S.I. units. 1.3.4. A container has a 5m3 volume capacity and weighs 1500 N when empty and 47,000 N when filled with a liquid. What is the mass density and specific gravity of the liquid? 1.3.5. A 2.5-ft-diameter cylindrical water tank (Figure P1.3.5) is suspended vertically by its sides. The tank contains water (20°C) to a depth d. The force on the bottom of the tank is 920 pounds. Determine the depth of the water in the tank.

Figure P1.3.5

1.3.6. A rocket carrying a specific volume of water weighing 8.83 kN on Earth lands on the Moon where the gravitational acceleration is one-sixth of that of the Earth. Find the mass and the Moon-weight of the water. 1.3.7. Determine the change in volume of 100 m3 of water contained in a swimming pool as it warms from 10°C in the spring to 30°C in the summer. 1.3.8. The unit of energy in the SI system is a Newton-meter (or Joule). Determine the conversion factor for one unit of energy in this system to the B.G. system (foot-pound). Check your result by looking in the front jacket of the book. 1.3.9. The unit of pressure in the SI system is a Pascal (N/m2). Determine the conversion factor for one unit of pressure in this system to the British system (pounds per square inch, psi). Check your result by looking in the front jacket of the book.

(secTion 1.4) 1.4.1. Determine the conversion factors that are required when changing: (a) absolute viscosity units of lb # s/ft2 (BG) to poise (SI) and (b) kinematic viscosity units of ft2/s (BG) to stoke (SI). 1.4.2. Compare the ratios of the absolute and the kinematic viscosities of air and water at (a) 0°C and (b) 100°C. Explain how the ratios change with temperature? 1.4.3. Convert the absolute and kinematic viscosity of water (Table 1.3) at 68°F (20°C) to the British system equivalent. (Check your result by looking in the front endpapers of the book.)

Fundamental Properties of Water

Chap. 1

1.4.4. Velocity measurements are made along a cross section of a fluid flow field. The velocity at two points (2 cm apart) are 4.8 m/s and 2.4 m/s, respectively. What is the magnitude of the shear stress at this location if the velocity profile is linear and the fluid is water at 20°C? 1.4.5. A very thin plate measuring 6 in. by 18 in. is being pulled between two stationary plates (Figure P1.4.5) at a speed of 1.5 ft/s? The 0.5-in. space between the two plates is filled with lubricating oil with a dynamic viscosity of 0.0065 [email protected]/ft2. How much force (F) is required to pull the thin plate.

0.5 in.

Figure P1.4.5

1.4.6. A flat plate weighing 100 N slides down a 15° incline with a velocity of 2.5 cm/s. A thin film of oil with a viscosity of 1.52 N # s/m2 separates the plate from the ramp. If the plate is 80 cm by 90 cm, calculate the film thickness in mm. 1.4.7. The 25-cm diameter ram in a hydraulic lift slides in a 25.015-cm diameter cylinder. The viscosity of the oil filling the gap is 0.04 N # s/m2. If the speed of the ram is 15 cm/s, determine the frictional resistance force when a 3 m length of the ram is engaged in the cylinder. 1.4.8. A liquid flows with velocity distribution V = y 2 - 3y, where V is given in ft/s and y in inches. Calculate the shear stresses at y = 0, 1, 2, 3, 4 if the viscosity is 8.35 * 10-3 [email protected]/ft2. 1.4.9. A flat circular disk of radius 1.0 m is rotated at an angular velocity of 0.65 rad/s over a fixed flat surface. An oil film separates the disk and the surface. If the viscosity of the oil is 16 times that of water (20°C) and the space between the disk and the fixed surface is 0.5 mm, determine the torque required to rotate the disk. 1.4.10. Fluid viscosity can be measured by a rotating-cylinder viscometer, which consists of two concentric cylinders with a uniform gap between them. The liquid to be measured is poured into the gap between the two cylinders. For a particular liquid, the inner cylinder rotates at 2000 rpm and the outer cylinder remains stationary and measures a torque of 1.10 ft # lb. The inner cylinder diameter is 2.0 in., the gap width is 0.008 in., and the liquid fills up to a height of 1.6 in. in the cylindrical gap. Determine the absolute viscosity of the liquid in lb # s/ft2.

(secTion 1.5) 1.5.1. A line force contains the units of force per unit length. This differs from a surface force (like pressure with units of force per unit area) or a body force (like specific weight with units of force per unit volume). Surface tension is considered a line force. Based on the development of Equation 1.3, explain why it is logical to consider surface tension a line force. 1.5.2. Mercury (SG = 13.6) is used in a glass tube to measure pressure. If the surface tension is 0.57 N/m and the contact angle ranges from 40° to 50°, determine the minimum diameter of the tube so that the measurement error in the manometer does not exceed 1.0 mm. 1.5.3. A capillary rise experiment is proposed for a high school physics class. The students are told that for water in clean glass tubes, the contact angle between liquid and glass (θ) is 90°. The students are asked to measure capillary rise in a series of tube diameters (D = 0.02, 0.04, 0.06, 0.08, and 0.10 in.). They are then asked to graph the results and determine the approximate tube diameters that would produce capillary rises of 1.5, 1.0, and 0.5 in. Predict the results if the water used in the experiment is at 20°C.

Problems

1.5.4. A small amount of solvent is added to the ground water to change its electrolytic content. As a result, the contacting angle, θ, representing the adhesion between water and soil material, is increased from 30° to 50° while the surface tension decreases by 12% (Figure P1.5.4). If the soil has a uniform pore size of 0.8 mm, determine the percentage change of capillary rise in the soil.

30º– 50º

0.8 mm Figure P1.5.4

1.5.5. Vertical glass tubes (piezometers) can be used to measure the pressure in pipes. However, capillary action can create a measurement error. Determine the error (in cm) resulting from the use of 1.2-cm-diameter piezometers in a pipe conveying saltwater (SG = 1.03) that has a surface tension 20% greater than fresh water (T = 35°C) if the contact angle is 35°. 1.5.6. The pressure inside a droplet of water is greater than the pressure outside. Split a droplet in half and identify forces. The bursting force is the pressure difference times the area that is balanced by the surface tension acting on the circumference. Derive an expression for the pressure difference.

(secTion 1.6) 1.6.1. Determine the percentage change in the density of water (40°F) when the pressure is increased from 1 atm to 220 lb/in.2. 1.6.2. A steel tank holds 120 ft3 of water with a weight of 7,490 lb at atmospheric pressure. Determine its current density and the new density if the pressure is raised to 1470 lb/in.2. 1.6.3. Lake Baikal in Siberia, Russia is thought to be the deepest lake in the world at 1,645 m. Determine the percentage change in the specific weight of water at the surface of the lake as compared to the water at the bottom where the pressure is roughly 1.61 * 107 N/m2. Also determine the specific weight of the water at the bottom and compare it to the surface water. Assume a water temperature of 4°C. 1.6.4. The pressure in a 150-cm diameter pipe 2000 m long is 30 N/cm2. Determine the additional volume of water that is able to enter the pipe if the pressure increases to 30 bars. Assume that the pipe is rigid and does not increase its volume.

Robert Houghtalen

Water Pressure and Pressure Forces

2.1 The Free Surface of Water When water fills a containing vessel, it automatically seeks a horizontal surface on which the pressure is constant everywhere. In practice, a free water surface is one that is not in contact with an overlying vessel cover. A free water surface may be subjected to atmospheric pressure (open vessel) or any other pressure that is exerted within the vessel (closed vessel). 2.2 Absolute and Gauge Pressures A water surface in contact with the Earth's atmosphere is subjected to atmospheric pressure, which is approximately equal to a 10.33-m-high column of water at sea level. In still water, any object located below the water surface is subjected to a pressure greater than atmospheric pressure. This additional pressure is often referred to as hydrostatic pressure. More precisely, it is the force per unit area acting in a normal direction on the surface of a body immersed in the fluid (in this case water). To determine the variation of hydrostatic pressure between any two points in water (with a specific weight of γ), we may consider two arbitrary points A and B along an arbitrary x-axis, as shown in Figure 2.1. Consider that these points lie in the ends of a small prism of water having a cross-sectional area dA and a length L. PA and PB are the pressures at each end, where the 14

Sec. 2.2

Absolute and Gauge Pressures

x PA A L h = L sinu

y u B PB Figure 2.1

Hydrostatic pressure on a prism

cross-sectional areas are normal to the x-axis. Because the prism is at rest, all forces acting on it must be in equilibrium in all directions. For the force components in the x-direction, we may write Σ Fx = PA dA - PB dA + γLd A sin θ = 0 Note that L # sin θ = h is the vertical elevation difference between the two points. The above equation reduces to PB - PA = γh

(2.1)

Therefore, the difference in pressure between any two points in still water is always equal to the product of the specific weight of water and the difference in elevation between the two points. If the two points are on the same elevation, h = 0 and PA = PB. In other words, for water at rest, the pressure at all points in a horizontal plane is the same. If the water body has a free surface that is exposed to atmospheric pressure, Patm, we may position point A on the free surface and write (PB)abs = γh + PA = γh + Patm

(2.2)

This pressure, (PB)abs, is commonly referred to as the absolute pressure. Pressure gauges are usually designed to measure pressures above or below the atmospheric pressure. Pressure so measured, using atmospheric pressure as a base, is called gauge pressure, P. Absolute pressure is always equal to gauge pressure plus atmospheric pressure: P = Pabs - Patm

(2.3)

Water Pressure and Pressure Forces

Absolute pressure (water column, m)

solute pressure Ab 20 30 10 10 20 0 ge pres s u ure a G 30 0 −10.33

Meters H2O 40 70 50 60 80 60 70

Chap. 2

14.00 +P1 Atmospheric pressure

10.33

−P2 10

8.00

solute pressure Ab 20 30 10 20 0 press

u ge Ga

0 −10.33

+(P1)abs

ure

Meters H2O 40 70 50 60 80 60 70

+(P2)abs

30 50

Absolute vacuum 0 Figure 2.2

Absolute and gauge pressure

Figure 2.2 diagrammatically shows the relationship between the absolute and gauge pressure and two typical pressure-gauge dials. Comparing Equations 2.2 and 2.3, we have or

P = γh

(2.4)

P γ

(2.5)

h =

Here the pressure is expressed in terms of the height of a water column h. In hydraulics it is known as the pressure head. Equation 2.1 may thus be rewritten in a more general form as PB PA = ∆h γ γ

(2.6)

Sec. 2.3

Surfaces of Equal Pressure

meaning that the difference in pressure heads at two points in water at rest is always equal to the difference in elevation between the two points. From this relationship we can also see that any change in pressure at point B would cause an equal change at point A, because the difference in pressure head between the two points must remain the same value ∆h. In other words, a pressure applied at any point in a liquid at rest is transmitted equally and undiminished in all directions to every other point in the liquid. This principle, also know as Pascal's law, has been made use of in the hydraulic jacks that lift heavy weights by applying relatively small forces. Example 2.1 The diameters of cylindrical pistons A and B are 3 cm and 20 cm, respectively. The faces of the pistons are at the same elevation, and the intervening passages are filled with an incompressible hydraulic oil. A force P of 100 N is applied at the end of the lever, as shown in Figure 2.3. What weight W can the hydraulic jack support?

P 80 cm

20 cm W B

A 3 cm

20 cm

F Measurements in cm Figure 2.3 Hydraulic jack

Solution Balancing the moments produced by P and F around the pin connection yields (100 N)(100 cm) = F(20 cm) Thus, F = 500 N From Pascal's law, the pressure PA applied at A is the same as that of PB applied at B. Therefore, PA =

PB =

[(π # 32)/4] cm2 500 N 2

7.07 cm 6 W = 500 N a

[(π # 202)/4] cm2

W 314 cm2

314 cm2 7.07 cm2

b = 2.22 * 104 N

2.3 Surfaces of Equal Pressure The hydrostatic pressure in a body of water varies with the vertical distance measured from the free water surface. In general, all points on a horizontal surface in a static body of water are subjected to the same hydrostatic pressure, according to Equation 2.4. For example, in

Water Pressure and Pressure Forces

Chap. 2

ES = equal pressure surface NES = nonequal pressure surface

Oil

NES 10 8

9 7

NES ES

H2O Valve closed (a)

(b) Figure 2.4

(c)

Hydraulic pressure in vessels

Figure 2.4 (a), points 1, 2, 3, and 4 have equal pressure, and the horizontal surface that contains these four points is a surface of equal pressure. However, in Figure 2.4 (b), points 5 and 6 are on the same horizontal plane but the pressures are not equal. This is because the water in the two tanks is not connected and the overlying depths to the free surfaces are different. Applying Equation 2.4 would produce different pressures. Figure 2.4 (c) displays tanks filled with two immiscible liquids of different densities. (Note: Immiscible liquids do not readily mix under normal conditions.) The horizontal surface (7, 8) that passes through the interface of the two liquids is an equal pressure surface. Applying Equation 2.4 at both points leads to the same pressure; we have the same fluid (water) at both locations (just below the interface at point 8), and both points are the same distance beneath the free water surface. However, points 9 and 10 are not on an equal pressure surface because they reside in different liquids. Verification would come from the application of Equation 2.4 using the different depths from the free surface to points 9 and 10 and the different specific weights of the fluids. In summary, a surface of equal pressure requires that (1) the points on the surface be in the same liquid, (2) the points be at the same elevation (i.e., reside on a horizontal surface), and (3) the liquid containing the points be connected. The concept of equal pressure surface is a useful method in analyzing the strength or intensity of the hydrostatic pressure at various points in a container, as demonstrated in the following section. Example 2.2 Suppose the oil surface is 0.02 m above the water surface in Figure 2.4 (c). The specific weight of water is γw = 9,810 N/m3 and that of oil is γo = 8,660 N/m3. (a) Determine the distance between section 7 and the water surface. (b) Section 9 is 0.10 m below the water surface. Determine the pressure at sections 9 and 10. Solution (a) Sections 7 and 8 are on an equal pressure surface. Therefore, by using Equation 2.4, we can write

Sec. 2.4

Manometers

γwhw = γoho or a9,810

N 3

bh7 = a8,660

N m3

b(h7 + 0.02)

where h7 denotes the distance between section 7 and water surface. Solving this equation, we obtain h7 = 0.15 m. (b) Sections 9 and 10 are not on an equal pressure surface. Therefore, applying Equation 2.4 to these sections separately we obtain P9 = (9,810 N/m3)(0.10 m) = 981 N/m2 and P10 = (8,660 N/m3) (0.10 m + 0.02 m) = 1,040 N/m2.

2.4 Manometers A manometer is a pressure-measurement device. It usually is a tube bent in the form of a "U" that contains a fluid of known specific gravity. The difference in elevations of the liquid surfaces under pressure indicates the difference in pressure at the two ends. Basically, there are two types of manometers: 1. An open manometer has one end open to atmospheric pressure and is capable of measuring the gauge pressure in a vessel. 2. A differential manometer has each end connected to a different pressure tap and is capable of measuring the pressure difference between the two taps. The liquid used in a manometer is usually heavier than the fluids to be measured. It must form a distinct interface—that is, it must not mix with the adjacent liquids (i.e., immiscible liquids). The most frequently used manometer liquids are mercury (sp. gr. = 13.6), water (sp. gr. = 1.00), alcohol (sp. gr. = 0.9), and other commercial manometer oils of various specific gravities (e.g., from Meriam* Red Oil, sp. gr. = 0.827 to Meriam No. 3 Fluid, sp. gr. = 2.95). Figure 2.5 (a) shows a schematic of a typical open manometer; Figure 2.5 (b) shows a schematic of a typical differential manometer. It is obvious that the higher the pressure in vessel A, the

Water A

Atmosphere Water A

Water B

y y

D h

h 2

C 2

Liquid M

Liquid M (a)

(b)

Figure 2.5 Types of manometers: (a) open manometer and (b) differential manometer

*Meriam Process Technologies, Cleveland, Ohio 44102

Water Pressure and Pressure Forces

Chap. 2

larger the difference, h, in the surface elevations in the two legs of the manometer. A mathematical calculation of pressure in A, however, involves the densities of the fluids and the geometry involved in the entire measuring system. A simple step-by-step procedure is suggested for pressure computation. Step 1. Make a sketch of the manometer system, similar to that in Figure 2.5, and approximately to scale. Step 2. Draw a horizontal line through the lower surface of the manometer liquid (point 1). The pressure at points 1 and 2 must be the same since the system is in static equilibrium. Step 3. (a) For open manometers, the pressure on 2 is exerted by the weight of the liquid M column above 2; and the pressure on 1 is exerted by the weight of the column of water above 1 plus the pressure in vessel A. The pressures must be equal in value. This relation may be written as follows: γMh = γy + PA

PA = γMh - (γy)

(b) For differential manometers, the pressure on 2 is exerted by the weight of the

liquid M column above 2, the weight of the water column above D, and the pressure in vessel B, whereas the pressure on 1 is exerted by the weight of the water column above 1 plus the pressure in vessel A. This relationship may be expressed as: γMh + γ(y - h) + PB = γy + PA or ∆P = PA - PB = h(γM - γ) Either one of these equations can be used to solve for PA. Of course, in the case of the differential manometer, PB must be known. The same procedure can be applied to any complex geometry, as demonstrated in the following example. Example 2.3 A mercury manometer (sp. gr. = 13.6) is used to measure the pressure difference in vessels A and B, as shown in Figure 2.6. Determine the pressure difference in pascals (N/m2). Solution The sketch of the manometer system (step 1) is shown in Figure 2.6. Points 3 and 4 (P3, P4) are on a surface of equal pressure (step 2) and so are the vessel A and points 1 and 2 (P1, P2): P3 = P4 PA = P1 = P2 The pressures at points 3 and 4 are, respectively (step 3), P3 = P2 + γ (27 cm) = PA + γ (27 cm) P4 = PB + γ (135 cm) + γM (15 cm) Now P3 = PA + γ (27 cm) = P4 = PB + γ (135 cm) + γM (15 cm)

Sec. 2.4

Manometers

y = 123 cm A

6 5

12 cm h = 15 cm

Figure 2.6

and noting that γM = γ (sp. gr.) ∆P = PA - PB = γ (135 cm - 27 cm) + γM (15 cm) ∆P = γ [108 + (13.6)(15)]cm = (9790 N/m3)(3.12 m) ∆P = 30,500 N/m2 (pascals) or 30.5 [email protected]

The open manometer, or U-tube, requires readings of liquid levels at two points. In other words, any change in pressure in the vessel causes a drop of liquid surface at one end and a rise in the other. A single-reading manometer can be made by introducing a reservoir with a larger cross-sectional area than that of the tube into one leg of the manometer. A typical single-reading manometer is shown in Figure 2.7. Because of the large area ratio between the reservoir and the tube, a small drop of surface elevation in the reservoir will cause an appreciable rise in the liquid column of the other leg. If there is an increase in pressure, ∆PA will cause the liquid surface in the reservoir to drop by a small amount ∆y. Then A∆y = ah

(2.7)

where A and a are cross-sectional areas of the reservoir and the tube, respectively. Applying step 2 to points 1 and 2, we may generally write γA(y + ∆y) + PA = γB(h + ∆y)

(2.8)

Simultaneous solution of Equations 2.7 and 2.8 give the value of PA, the pressure in the vessel, in terms of h. All other quantities in Equations 2.7 and 2.8—A, a, y, γA, and γB—are quantities predetermined in the manometer design. A single reading of h will thus determine the pressure.

Water Pressure and Pressure Forces

Chap. 2

h A

y 1

∆y Figure 2.7

Single-reading manometer

Because ∆y can be made negligible by introducing a very large A/a ratio, the above relationship may be further simplified to γAy + PA = γBh

(2.9)

Thus, the height reading h is a measure of the pressure in the vessel. The solution of practical hydraulic problems frequently requires the difference in pressure between two points in a pipe or a pipe system. For this purpose, differential manometers are frequently used. A typical differential manometer is shown in Figure 2.8. The same computation steps (steps 1, 2, and 3) suggested previously can be readily applied here, too. When the system is in static equilibrium, the pressure at the same elevation points, 1 and 2, must be equal. We may thus write γA(y + h) + Pc = γBh + γAy + Pd

Hence, the pressure difference, ∆P, is expressed as ∆P = Pc - Pd = (γB - γA)h

c Liquid A a

h 2

Liquid B Figure 2.8

A differential manometer installed in a flow-measurement system

(2.10)

Sec. 2.5

Hydrostatic Forces on Flat Surfaces

2.5 Hydrostatic Forces on Flat Surfaces Determining the total (or resultant) hydrostatic force on structures produced by hydrostatic pressure is often critical in engineering design and analysis. To determine the magnitude of this force, let's examine an arbitrary area AB (Figure 2.9) on the back face of a dam that inclines at an angle θ. Next, place the x-axis on the line where the surface of the water intersects with the dam surface (i.e., into the page) with the y-axis running downward along the surface or face of the dam. Figure 2.9 (a) shows a plan (front) view of the area and Figure 2.9 (b) shows the projection of AB on the dam surface. We may assume that the plane surface AB is made up of an infinite number of horizontal strips, each having a width of dy and an area of dA. The hydrostatic pressure on each strip may be considered constant because the width of each strip is very small. For a strip at depth h below the free surface, the pressure is P = γh = γy sin θ The total pressure force on the strip is the pressure times the area dF = γy sin θ dA The total pressure force (resultant force) over the entire AB plane surface is the sum of pressure on all the strips F =

dF =

γy sin θ dA = γ sin θ

ydA

= γ sin θ Ay

(2.11)

u y y p

C.G.

C.P.

B (a)

(b) Figure 2.9

Hydrostatic pressure on a plane surface

Water Pressure and Pressure Forces

Chap. 2

ydA/A is the distance measured from the x-axis to the centroid (or the center of LA gravity, C.G.) of the AB plane (Figure 2.9). where y =

have

Substituting h, the vertical distance of the centroid below the water surface, for y sin θ, we F = γhA

(2.12)

This equation states that the total hydrostatic pressure force on any submerged plane surface is equal to the product of the surface area and the pressure acting at the centroid of the plane surface. Pressure forces acting on a plane surface are distributed over every part of the surface. They are parallel and act in a direction normal to the surface. These parallel forces can be analytically replaced by a single resultant force F of the magnitude shown in Equation 2.12. The resultant force also acts normal to the surface. The point on the plane surface at which this resultant force acts is known as the center of pressure (C.P., Figure 2.9). Considering the plane surface as a free body, we see that the distributed forces can be replaced by the single resultant force at the pressure center without altering any reactions or moments in the system. Designating yp as the distance measured from the x-axis to the center of pressure, we may thus write Fyp =

ydF

Hence, ydF LA yp = F

(2.13)

Substituting the relationships dF = γy sin θ dA and F = γ sin θ Ay, we may write Equation 2.13 as LA Ay

y 2dA

yp =

(2.14)

LA moment of the plane surface AB with respect to the x-axis. Therefore,

in which

y 2dA = Ix and A y = Mx are, respectively, the moment of inertia and the static

yp =

Ix Mx

(2.15)

With respect to the centroid of the plane, this may be written as yp =

I0 + Ay 2 I0 = + y Ay Ay

(2.16)

Sec. 2.5

Hydrostatic Forces on Flat Surfaces

where I0 is the moment of inertia of the plane with respect to its own centroid, A is the plane surface area, and y is the distance between the centroid and the x-axis. The center of pressure of any submerged plane surface is always below the centroid of the surface area (i.e., yp 7 y). This must be true because all three variables in the first term on the right-hand side of Equation 2.16 are positive, making the term positive. That term is added to the centroidal distance (y). The centroid, area, and moment of inertia with respect to the centroid of certain common geometrical plane surfaces are given in Table 2.1. Table 2.1

Shape

Surface Area, Centroid, and Moment of Inertia of Certain Simple Geometrical Plates

Area

Centroid

Rectangle

1 b 2 1 y = h 2 x =

y x h

C.G.

Moment of Inertia About the Neutral x-Axis I0 =

1 3 bh 12

I0 =

1 3 bh 36

I0 =

1 πd 4 64

Triangle

1 bh 2

y c x

y x B

C.G.

b + c 3 h y = 3

x =

Circle

1 2 πd 4

1 d 2 1 y = d 2 x =

d = 2r r

C.G. B

h(a + b) 2

Trapezoid y

y =

h(2a + b) 3(a + b)

I0 =

h3(a2 + 4ab + b2) 36(a + b)

x h C.G.

π bh

Ellipse y h C.G. b

x = b y = h

I0 =

π 3 bh 4

h b

(continued)

Water Pressure and Pressure Forces

Shape

Area

Centroid

Semiellipse

π bh 2

x = b 4h y = 3π

h B

2 bh 3

y y = h( 1– x

x2 ) b2

Semicircle

1 2 πr 2

y C.G. B

(9π2 - 64) 3 bh 72π

2 h 5

x =

3 b 8

y =

4r 3π

I0 =

8 bh3 175

y =

C.G.

I0 =

Parabolic section

Moment of Inertia About the Neutral x-Axis

C.G.

Chap. 2

I0 =

(9π2 - 64)r 4 72π

Example 2.4 A vertical trapezoidal gate with its upper edge located 5 m below the free surface of water is shown in Figure 2.10. Determine the total pressure force and the center of pressure on the gate.

1m Figure 2.10

Sec. 2.5

Hydrostatic Forces on Flat Surfaces

Solution The total pressure force is determined using Equation 2.12 and Table 2.1. F = γhA = 9,790c 5 +

2[(2)(1) + 3] 2(3 + 1) dc d 3(1 + 3) 2

= 2.28 * 105 N = 228 kN The location of the center of pressure is yp =

I0 + y Ay

where (from Table 2.1) 23[12 + 4(1)(3) + 32] I0 = = 1.22 m4 36(1 + 3) y = 5.83 m A = 4.00 m2 Thus, yp =

1.22 + 5.83 = 5.88 m 4(5.83)

below the water surface.

Example 2.5 An inverted semicircular gate (Figure 2.11) is installed at 45° with respect to the free water surface. The top of the gate is 5 ft below the water surface in the vertical direction. Determine the hydrostatic force and the center of pressure on the gate.

45° _ h

5 ft

5 ft hp

_ y

4 ft C.G.

F yp

C.P.

Figure 2.11

Water Pressure and Pressure Forces

Chap. 2

Solution The total pressure force is F = γy sin θA where A =

1 [π(4)2] = 25.1 ft2 2

and y = 5 s 45° +

4(4) = 8.77 ft 3π

Therefore, F = 62.3 (sin 45°)(8.77)(25.1) = 9,700 lb This is the total hydrostatic force acting on the gate. The location of the center of pressure is yp =

I0 + y Ay

where (from Table 2.1) I0 =

(9π2 - 64) 4 r = 28.1 ft4 72π

Therefore, yp =

28.1 + 8.77 = 8.90 ft 25.1(8.77)

This is the inclined distance measured from the water surface to the center of pressure.

2.6 Hydrostatic Forces on Curved Surfaces The hydrostatic force on a curved surface can be analyzed best by resolving the total pressure force on the surface into its horizontal and vertical components. (Remember that hydrostatic pressure acts normal to a submerged surface.) Figure 2.12 shows the curved wall of a container gate that has a unit width normal to the plane of the page. Because the water body in the container is stationary, every part of the water body must be in equilibrium or each of the force components must satisfy the equilibrium conditions—that is, ΣFx = 0 and ΣFy = 0. In the free body diagram of the water contained in ABA′, equilibrium requires the horizontal pressure exerted on plane surface A′B (the vertical projection of AB) to be equal and opposite the horizontal pressure component FH (the force that the gate wall exerts on the fluid). Likewise,

Sec. 2.6

Hydrostatic Forces on Curved Surfaces

Free body WAA′

WAA′

A′

A WABA′ FA′B

WABA′ FA′B

B Figure 2.12

Hydrostatic pressure on a curved surface

the vertical component, FV, must equal the total weight of the water body above gate AB. Hence, the horizontal and vertical pressure force on the gate may be expressed as ΣFx = FA′B - FH = 0 6 FH = FA′B ΣFy = FV - (WAA′ + WABA′) = 0 6 FV = WAA′ + WABA′ Therefore, we may make the following statements. 1. The horizontal component of the total hydrostatic pressure force on any surface is always equal to the total pressure on the vertical projection of the surface. The resultant force of the horizontal component can be located through the center of pressure of this projection. 2. The vertical component of the total hydrostatic pressure force on any surface is always equal to the weight of the entire water column above the surface extending vertically to the free surface. The resultant force of the vertical component can be located through the centroid of this column. Example 2.6 Determine the total hydrostatic pressure and the center of pressure on the 5-m-long, 2-m-high quadrant gate in Figure 2.13. Solution The horizontal component is equal to the hydrostatic pressure force on the projection plane A′B. 1 FH = γ h A = (9,790 N/m3) a (2 m) b[(2 m)(5 m)] = 97,900 N 2

Water Pressure and Pressure Forces 0

Chap. 2

1.33 m FV

0.85

A′ Figure 2.13

The location of the horizontal component is yp = I0/Ay + y, where A = 10 m2 (projected area) and I0 = [(5 m)(2 m)3]>12 = 3.33 m4, yp = (3.33 m4)>[(10 m2)(1 m)] + 1 m = 1.33 m below the free surface. The vertical component is equal to the weight of the water in the volume AOB. The direction of this pressure component is downward. 1 FV = γ(Vol) = (9,790 N/m3) a π(2 m)2 b(5 m) = 154,000 N 4

The pressure center is located at 4(2)>3π = 0.85 m (Table 2.1), and the resultant force is F = 2(97,900)2 + (154,000)2 = 182,000 N θ = tan-1 a

FV 154,000 b = tan-1 = 57.6° FH 97,900

Example 2.7 Determine the total hydrostatic pressure and the center of pressure on the semicylindrical gate shown in Figure 2.14. Solution The horizontal component of the hydrostatic pressure force on the projection plane A′B′ per unit width can be expressed as FH = γ h A = γa

H 1 b(H) = γH 2 2 2

The pressure center of this component is located at a distance of H/3 from the bottom. The vertical component can be determined as follows. The volume AA′C over the upper half of the gate, AC, produces a downward vertical pressure force component: FV1 = -γa

H2 πH 2 b 4 16

Sec. 2.7

Buoyancy

31 A gh h FV1

A′ B

FH gH

B′

B FV2

Figure 2.14

The vertical pressure force component exerted by the water on the lower half of the gate, CB, is upward and equivalent to the weight of water replaced by the volume AA′CB: FV2 = γa

H2 πH 2 + b 4 16

By combining these two components, one can see that the direction of the resultant vertical force is upward and equal to the weight of the water replaced by the volume ACB. FV = FV1 + FV2 = γJ - a

The resultant force is then

H2 πH 2 H2 πH 2 π b + a + b R = γ H2 4 16 4 16 8

F = γH 2

1 π2 + 64 A4

θ = tan-1

FV π = tan-1 a b = 38.1° FH 4

Because all pressure forces are concurrent at the center of the gate, point O, the resultant force must also act through point O.

2.7 Buoyancy Archimedes discovered (∼250 b.c.) that the weight of a submerged body is reduced by an amount equal to the weight of the liquid displaced by the body. Archimedes' principle, as we now call it, can be easily proven by using Equation 2.12.

Water Pressure and Pressure Forces S1

Chap. 2

FVa

B FH

F′H

A b N FVb Figure 2.15

Buoyancy of a submerged body

Assume that a solid body of arbitrary shape, AB, is submerged in water as shown in Figure 2.15. A vertical plane MN may then be drawn through the body in the direction normal to the page. One observes that the horizontal pressure force components in the direction of the paper, FH and F′H, must be equal because they both are calculated using the same vertical projection area MN. The horizontal pressure force components in the direction normal to the page must also be equal for the same reason; they share the same projection in the plane of the page. The vertical pressure-force component can be analyzed by taking a small vertical prism ab with a cross-sectional area dA. The vertical pressure force on top of the prism (γh1dA) acts downward. The vertical force on the bottom of the prism (γh2dA) acts upward. The difference gives the resultant vertical force component on the prism (buoyancy force) FV = γh2dA - γh1dA = γ(h2 - h1)dA c which is exactly equal to the weight of the water column ab replaced by the prism. In other words, the weight of the submerged prism is reduced by an amount equal to the weight of the liquid replaced by the prism. A summation of the vertical forces on all the prisms that make up the entire submerged body AB gives the proof of Archimedes' principle. Archimedes' principle may also be viewed as the difference of vertical pressure forces on the two surfaces ANB and AMB. The vertical pressure force on surface ANB is equal to the weight of the hypothetical water column (volume of S1ANBS2) acting upward; and the vertical pressure force on surface AMB is equal to the weight of the water column S1AMBS2 acting downward. Because the volume S1ANBS2 is larger than the volume S1AMBS2 by an amount exactly equal to the volume of the submerged body AMBN, the net difference is a force equal to the weight of the water that would be contained in the volume AMBN acting upward. This is the buoyancy force acting on the body. A floating body is a body partially submerged resulting from a balance of the body weight and buoyancy force.

Sec. 2.8

Flotation Stability

2.8 Flotation Stability The stability of a floating body is determined by the relative positions of the center of gravity of body G and the center of buoyancy B, which is the center of gravity of the liquid volume replaced by the body, as shown in Figure 2.16. The body is in equilibrium if its center of gravity and its center of buoyancy lie on the same vertical line, as in Figure 2.16 (a). This equilibrium may be disturbed by a variety of causes (e.g., wind or wave action), and the floating body is made to heel or list through an angle θ as shown in Figure 2.16 (b). When the floating body is in the heeled position, the center of gravity of the body remains unchanged, but the center of buoyancy, which is now the center of gravity of area a′cb′, has been changed from B to B′. The buoyant force γ # Vol, acting upward through B′, and the weight of the body W, acting downward through G, constitute a couple, W # X, which resists further overturning and tends to restore the body to its original equilibrium position. By extending the line of action of the buoyant force through the center of buoyancy B′, we see that the vertical line intersects the original axis of symmetry c-t at a point M. The point M is known as the metacenter of the floating body, and the distance between the center of gravity and the metacenter is known as the metacentric height. The metacentric height is a measure of the flotation stability of the body. When the angle of inclination is small, the position of M does not change materially with the tilting position. The metacentric height and the righting moment can be determined in the following way. Because tilting a floating body does not change the total body weight, the total displacement volume is not changed. The roll through an angle θ only changes the shape of the displaced volume by adding the immersion wedge bob′ and subtracting the emersion wedge aoa′. In this new position, the total buoyancy force (γ # Vol) is shifted through a horizontal distance S to B′. This shift creates a couple F1 and F2 because of the new immersion and emersion wedges. The moment of the resultant force (γ # Vol)B′ about point B must equal the sum of the moments of the component forces: (γVol)B′(S) = (γVol)B(zero) + moment of the force couple = 0 + γVolwedgeL L

t a W

W O

a′

O a

X S

b′

B g ⋅Vol

B′

g ⋅Vol

c (a)

(b) Figure 2.16

Center of buoyancy and metacenter of a floating body

Water Pressure and Pressure Forces

Chap. 2

or (γVol)B′S = γVolwedgeL S =

Volwedge Vol

(a)

where Vol is the total volume submerged, Volwedge is the volume of wedge bob′ (or aoa′), and L is the horizontal distance between the centers of gravity of the two wedges. But, according to the geometric relation, we have S = MB sin θ

MB =

S sin θ

(b)

Combining Equations (a) and (b), we get MB =

VolwedgeL Vol sin θ

For a small angle, sin θ ≈ θ, the previous relationship may be simplified to MB =

VolwedgeL Volθ

The buoyancy force produced by wedge bob′, as depicted in Figure 2.17, can be estimated by considering a small prism of the wedge. Assume that the prism has a horizontal area, dA, and is located at a distance x from axis of rotation O. The height of the prism is x(tan θ). For a small angle θ, it may be approximated by xθ. Thus, the buoyancy force produced by this small prism is γxθdA. The moment of this force about the axis of rotation O is γx2θdA. The sum of the moments produced by each of the prisms in the wedge gives the moment of the immersed wedge. The moment produced by the force couple is, therefore, γVolwedgeL = FL =

γx2 θd A = γθ

x O

x2 dA

dA b′

Figure 2.17

Sec. 2.8

Flotation Stability

LA body about the axis of rotation O.

But

x2 dA is the moment of inertia of the waterline cross-sectional area of the floating

I0 =

x2 dA

Hence, we have VolwedgeL = I0θ For small angles of tilt, the moment of inertia for upright cross section aob, and the inclined cross section a′ob′ may be approximated by a constant value. Therefore, MB =

I0 Vol

(2.17)

The metacentric height, defined as the distance between the metacenter M and the center of gravity G, can be estimated: GM = MB { GB =

I0 { GB Vol

(2.18)

The distance between the center of gravity and the center of buoyancy GB in the upright position, shown in Figure 2.16, can be determined by the sectional geometry or the design data of the vessel. The { sign indicates the relative position of the center of gravity with respect to the center of buoyancy. For greater flotation stability, it is advantageous to make the center of gravity as low as possible. If G is lower than B, then GB would be added to the distance to MB and produce a larger value of GM. The righting moment, when tilted as depicted in Figure 2.16 (b), is M = WGM sin θ

(2.19)

The stability of buoyant bodies under various conditions may be summarized as follows. 1. A floating body is stable if the center of gravity is below the metacenter. Otherwise, it is unstable. 2. A submerged body is stable if the center of gravity is below the center of buoyancy.

Example 2.8 A 3 m * 4 m rectangular box caisson is 2 m deep (Figure 2.18). It has a draft of 1.2 m when it floats in an upright position. Compute (a) the metacentric height and (b) the righting moment in seawater (sp. gr. = 1.03) when the angle of heel (list) is 8°.

Water Pressure and Pressure Forces

8°

G B

Chap. 2

O B′

3m Figure 2.18

Solution From Equation 2.18 GM = MB - GB where MB =

I0 Vol

and I0 is the waterline area moment of inertia of the box about its longitudinal axis through O. Therefore, 1 Lw3 12 h 1.2 GM = - a b Lw(1.2) 2 2 = 0.225 m

(Note: L = 4 m, w = 3 m, h = 2 m.) The specific gravity of seawater is 1.03; from Equation 2.19, the righting moment is M = WGM sin θ = [(9,790 N/m3)(1.03)5(4 m)(3 m)(1.2 m)6](0.225 m)(sin 8°)

= 4,550 N # m

Example 2.9 A 20 cm * 30 cm * 90 cm wood block has a specific weight of 6,000 N/m3. Determine whether the block floats in a stable condition if it is placed in water as shown in Figure 2.19 (a) and 2.19 (b). The specific weight of water is 9,810 N/m3. Solution (a) The center of gravity (point G) is 0.15 m above the lower edge of the block. The weight of the block is calculated as W = (6,000 N/m3)(0.20 m)(0.30 m)(0.90 m) = 324 N. The buoyancy force is equal to [y (0.20 m)(0.90 m)(9,810 N/m3)].

Problems

37 0.20 m

0.30 m G y

0.30 m

(a)

G B

0.20 m

(b) Figure 2.19

For equilibrium, the weight should be equal to the buoyancy force. Then y is determined as y = 324 N/[(0.20 m)(0.90 m)(9,810 N/m3)] = 0.183 m. Accordingly, the center of buoyancy (point B) is 0.0915 m above the lower edge of the block. The distance between G and B is then 0.15 - 0.0915 m = 0.0585 m. Let point M denote the metacenter. The location of M is unknown. By using Equation 2.17, the distance between M and B is calculated as MB =

(0.90 m)(0.20 m)3/12 = 0.0182 m (0.90 m)(0.20 m)(0.183 m)

Then point M is 0.0182 m above point B but 0.0585 - 0.0182 m = 0.0403 m below point G. Since the center of gravity is above the metacenter, the equilibrium is unstable and the block will tilt over due to a disturbance. (b) In this case, the center of gravity (point G) is 0.10 m above the lower edge of the block. Once again for equilibrium, the weight should be equal to the buoyancy force. Then y is determined as y = 324 N/[(0.30 m)(0.90 m)(9,810 N/m3)] = 0.122 m. Therefore, the center of buoyancy (point B) is 0.061 m above the lower edge of the block. Accordingly, point G is 0.10 - 0.061 m = 0.039 m above point B. The distance between M and B is calculated as MB =

(0.90 m)(0.30 m)3/12 = 0.0615 m (0.90 m)(0.30 m)(0.122 m)

Then point M is 0.0615 m above point B and 0.0615 - 0.039 m = 0.0225 m above point G. Since the center of gravity is below the metacenter, the equilibrium is stable.

Problems (secTion 2.2) 2.2.1. A cylindrical water tank is suspended as shown in Figure P2.2.1. The tank has a 10-ft diameter and contains 20°C water that weighs 14,700 pounds. Determine the depth of water in the tank and the pressure (in lb/ft2) on the bottom of the tank by two different methods (using the weight of the water and the depth of the water).

Water Pressure and Pressure Forces

Chap. 2

Figure P2.2.1

2.2.2. The collapse (crush) depth of a certain diving bell is an absolute pressure greater of 5 atm. How deep (in meters and feet) can the diving bell go in seawater (S.G. = 1.03) before it is in danger of being crushed? 2.2.3. A simple barometer to measure atmospheric pressure is depicted in Figure P2.2.3. Atmospheric pressure on the water surface in the cup causes the water to rise in the inverted test tube. Determine the magnitude of the atmospheric pressure (in kN/m2) assuming that there is some vapor pressure (based on the water temperature; 30°C) in the closed end of the test tube but negligible surface tension effects. Also determine the percentage error introduced if the vapor pressure was ignored.

8.7 m

Figure P2.2.3

2.2.4. The gauge pressure at the bottom of a water tank reads 30 mm of mercury (S.G. = 13.6). The tank is open to the atmosphere. Determine the water depth (in cm) above the gauge. Find the equivalency in N/m2 of absolute pressure at 20°C. 2.2.5. A cube-shaped storage tank, measuring 10 ft in length, width, and height, is filled with water. Determine the force on the tank bottom and sides. Hint: The average pressure on the sides of the tank can be determined. 2.2.6. The two containers of water shown in Figure P2.2.6 have the same bottom areas (2 m * 2 m), the same depth of water (10 m), and are both open to the atmosphere. However, the L-shaped container on the right holds less water. Determine hydrostatic force (in kN), not the pressure, on the bottom of each container.

Figure P2.2.6

Problems

2.2.7. An underwater storage tank was constructed to store natural gas offshore. Determine the gas pressure in the tank (in psi, lb/in2 and Pascals, N/m2) when the water elevation in the tank is 18 ft below the sea level (Figure P2.2.7). The specific gravity of seawater is 1.03.

18 ft Figure P2.2.7

2.2.8. A closed tank contains a liquid (S.G. = 0.85) under pressure. The pressure gauge, depicted in Figure P2.2.8, registers a pressure of 3.55 * 104 N/m2 (Pascals). Determine the pressure at the bottom of the tank and the height of the liquid column that will rise in the vertical tube.

1.4 m

Figure P2.2.8

2.2.9. A closed tank contains oil with a specific gravity 0.85. If the gauge pressure at a point 12 ft below the oil surface is 25.7 psi (lb/in.2), determine the absolute pressure and gauge pressure (in psi) in the air space at the top of the oil surface? 2.2.10. An incrementally small triangle is submerged beneath the surface of a fluid (Figure P2.2.10). Three pressures (Px, Py, and Ps) act on the three tiny surfaces of length (∆y, ∆x, and ∆s). Prove that Px = Ps and Py = Ps (i.e., pressure is omnidirectional) using the principles of statics. (Note that Px acts on ∆y, Py acts on ∆x, and Ps acts on ∆s. Also, the angle between the horizontal leg of the triangle and the hypotenuse is θ.) Ps Px Py

Figure P2.2.10

2.2.11. A multiple-piston hydraulic jack has two output pistons, each with an area of 250 cm2. The input piston, whose area is 25 cm2, is connected to a lever that has a mechanical advantage of 9:1. If a 50 N force is exerted on the lever, how much pressure (kN/m2 or kPa) is developed in the system? How much force (kN) will be exerted by each output piston?

(secTion 2.4) 2.4.1. A layer of oil sits on top of a layer of water in an open tank in a petroleum company lab. The water height is 5 times the oil height (h). The oil has a specific gravity of 0.80. If the gauge pressure at the bottom of the tank measures 1.43 in. of mercury, determine the oil height h.

Water Pressure and Pressure Forces

Chap. 2

2.4.2. Referring to Figure 2.4 (c), if the height of oil (S.G. = 0.85) above point "8" is 61.5 cm, determine the height of the water (at 4°C) above point "7." (Note: Point "10" is 29.2 cm above point "8.") 2.4.3. A significant amount of mercury is poured into a U-tube with both ends open to the atmosphere. Then water is poured into one leg of the U-tube until the water column is 1 m above the mercury– water interface. Finally, oil (S.G. = 0.79) is poured into the other leg until the oil column is 60 cm above the mercury–oil interface. What is the elevation difference between the mercury–water interface and the mercury–oil interface? 2.4.4. A mercury manometer is used to measure pressure in the pipe depicted in Figure P2.4.4. Determine the pressure in the pipe in psi and in inches of mercury for a manometry reading of h = 3 ft.

h 2 ft H2O

Hg Figure P2.4.4

2.4.5. Figure 2.5 (a) depicts a mercury manometer used to measure water pressure in a pipe. Determine the pressure in the pipe (in lb/in.2, psi) if the value of y is 1.34 in. and the value of h is 1.02 in. The specific gravity of mercury is 13.6. 2.4.6. In Figure P2.4.6, a single reading mercury manometer is used to measure water pressure in the pipe. What is the pressure (kilo-Pascals, kPa) in the pipe if h1 = 18.0 cm and h2 = 60.0 cm.

Figure P2.4.6

Problems

2.4.7. An open manometer, shown in Figure P2.4.7, is installed to measure pressure in a pipeline carrying oil (S.G. = 0.82). If the monometer liquid has a specific gravity of 0.85, determine the pressure in the pipe (in kPa and meters of water).

Air 66 cm

58 cm Manometer liquid

Oil

Figure P2.4.7

2.4.8. In Figure P2.4.6, a single-reading mercury manometer is used to measure water pressure in the pipe. What is the pressure (in psi) if h1 = 6.0 in. and h2 = 18.0 in.? Also determine the change in liquid height h1 for a 4 in. change in h2 if the diameter of the manometer tube is 1 in. and the diameter of the manometer fluid reservoir is 5 in. 2.4.9. In Figure P2.4.9, oil (S.G. = 0.82) is flowing in pipe A and water is flowing in pipe B. If carbon tetrachloride (S.G. = 1.6) is used as the manometer liquid, determine the pressure difference between A and B in psi.

A 32 in.

12 in.

20 in.

Manometer liquid

Figure P2.4.9

2.4.10. A micro-manometer consists of two reservoirs and a U-tube, as shown in Figure P2.4.10. If the density of the two liquids is ρ1 and ρ2, determine an expression for the pressure difference (P1 - P2) in terms of ρ1, ρ2, h, d1, and d2.

Water Pressure and Pressure Forces

Chap. 2

P2 ∆h

Diameter d1 Diameter d2

Figure P2.4.10

2.4.11. Determine the elevation at point A (EA) in Figure P2.4.11 if the air pressure in the sealed left tank is - 29.0 kPa (kN/m2).

40 m

Air

20 kN/m2

Air 37 m

Oil S.G. = 0.8 Water 35 m

S.G. = 1.6 EA A

Figure P2.4.11

Problems

2.4.12. For the system of manometers shown in Figure P2.4.12, determine the differential reading h.

Water A

60 cm

Water B

46 cm

20 cm

23 cm

20 cm

16 cm

S.G. = 0.8

S.G. = 13.6

S.G. = 0.8

Figure P2.4.12

(secTion 2.5) 2.5.1. A vertical, semicircular gate (Figure P2.5.1) that is hinged at the top keeps water from flowing in a semicircular channel that has a 2-ft radius (r). Determine the magnitude of the hydrostatic force on the gate and its location when water rises to the full 2-ft depth. Is the center of hydrostatic pressure deeper than the centroid of the gate?

r Figure P2.5.1

2.5.2. A concrete dam with a triangular cross section (Figure P2.5.2) is built to hold 9 m of water. Determine the hydrostatic force on a unit length (i.e., 1 m) of the dam and its location. Also, if the specific gravity of concrete is 2.78, determine if the dam is safe. That is, determine the moment generated with respect to the toe of the dam, A. Note that the hydrostatic pressure tends to overturn the dam and the weight acts to stabilize the dam.

A 4.5 m

Figure P2.5.2

Water Pressure and Pressure Forces

Chap. 2

2.5.3. A 1-m-diameter viewing window is mounted into the inclined side (45°) of a dolphin pool at a public aquarium. The center of the flat window is 5 m below the water's surface measured along the incline. Determine the magnitude (in kN) and location of the hydrostatic force acting on the window. 2.5.4. A circular gate is installed on a vertical wall as shown in Figure P2.5.4. Determine the horizontal force, P, necessary to hold the gate closed (in terms of the gate diameter, D, and depth, h). Neglect friction at the pivot.

h Pivot D P

Figure P2.5.4

2.5.5. A square gate 3 ft * 3 ft lies in a vertical plane. Determine the total pressure force on the gate and the distance between the center of pressure and the centroid when the upper edge of the gate is at the water surface. Compare these values to those that would occur if the upper edge of the gate is 10 ft below the water surface. 2.5.6. A vertical, rectangular gate 3 ft high and 2 ft wide is located on the side of a water tank. The tank is filled with water to a depth 5 ft above the upper edge of the gate. Locate a horizontal line that divides the gate area into two parts so that (a) the forces on the upper and lower parts are the same and (b) the moments of the forces about the line are the same. 2.5.7. The rectangular gate in Figure P2.5.7 is hinged at the top and separates water in the reservoir from the tail water tunnel. If the 2 m * 3 m gate weighs 32.3 kN, determine the maximum depth h for which the gate will stay closed. (Hint: Assume that the depth h is below the hinge.)

Hinged 45°

Figure P2.5.7

2.5.8. A gate is designed to open automatically when the water in a canal reaches a given level. The vertical, rectangular gate is depicted in Figure P2.5.8. The gate is 8-ft high (H) and will be required to open when h = 2 ft. Determine the location of the horizontal axis of rotation [email protected]′ (measured from the bottom of the gate) that will cause this to occur.

Problems

0´

Figure P2.5.8

2.5.9. Calculate the magnitude and the location of the resultant pressure force on the annular gate shown in Figure P2.5.9.

Hub

d = 3m

Figure P2.5.9

2.5.10. The circular gate of diameter "d " shown in Figure P2.5.10 is hinged at the horizontal diameter. If it is in equilibrium, determine the relationship between hA and hB as a function of γA, γB, and d.

Figure P2.5.10

Water Pressure and Pressure Forces

Chap. 2

2.5.11. A sliding gate 10 ft wide and 6 ft high is installed in a vertical plane and has a coefficient of friction against the guides of 0.2. If the gate weighs 6,040 lb and its upper edge is 17 ft below the water surface, calculate the vertical force required to lift the gate. 2.5.12. Calculate the minimum weight of the container cover necessary to keep it closed when the container is filled with water (Figure P2.5.12). The cover dimensions are 5 m * 10 m. Hinge

Cover

3m 4m Water

Q Figure P2.5.12

2.5.13. Determine the depth of the water (d) in Figure P2.5.13 that will cause the gate to open (begin to lay down). The gate is rectangular and is 8-ft wide. Neglect the weight of the gate in your computations. At what depth will it close?

5,000 lb

15 ft 60˚

Hinge

Figure P2.5.13

2.5.14. Neglecting the weight of the hinged gate, determine the depth h at which the gate will open in Figure P2.5.14.

h Hinge 1m

Figure P2.5.14

Problems

(secTion 2.6) 2.6.1. The hemispherical viewing window shown in Figure P2.6.1 is located on the bottom of an elevated fish tank in a marine science center. The viewing window has a 1-ft radius and is 5 ft below the surface of the water (h). Determine the hydrostatic force components (horizontal and vertical) on the viewing window (but not their locations). Salt water has a specific gravity of 1.03.

Figure P2.6.1

2.6.2. An 8-m-long curved gate depicted in Figure P2.6.2 is retaining a 6-m depth of oil (S.G. = 0.82) in a storage tank. Determine the magnitude and direction of the total hydrostatic force on the gate. Does the force pass through point A? Explain.

6m A 2m Gate

Figure P2.6.2

2.6.3. A hemispherical viewing port in a museum aquarium (Figure P2.6.3) has a 1-ft radius. The top of the viewing port is 3 ft below the water surface (h). Determine the magnitude of the horizontal component of the hydrostatic force acting on the viewing port. Also, determine the resultant hydrostatic force and its direction if the vertical force component equals 261 lb acting upwards. Note that the resultant force will pass through the center of the sphere since all pressures are normal to the surface and will pass through this point.

Figure P2.6.3

Water Pressure and Pressure Forces

Chap. 2

2.6.4. A hemispherical viewing port in a museum aquarium (Figure P2.6.3) has a 1-m radius. The top of the viewing port is 2 m below the water surface (h). Determine the magnitude of the vertical component of the hydrostatic force acting on the viewing port. 2.6.5. The bottom plate of a barge's hull (Figure P2.6.5) is curved with the radius of 1.75 m. When the barge is submerged in sea water (S.G. = 1.03), determine if the vertical force component is greater than the horizontal component on plate AB per unit length of hull.

3m 1.75 m

Figure P2.6.5

2.6.6. Calculate the horizontal and vertical forces acting on the curved surface ABC in Figure P2.6.6 if R = 2 ft.

A R

B R C

Figure P2.6.6

2.6.7. The cylindrical dome in Figure P2.6.7 is 8 m long and is secured to the top of an oil tank by bolts. If the oil has a specific gravity of 0.90 and the pressure gauge reads 2.75 * 105 N/m2, determine the total tension force in the bolts. Neglect the weight of the cover.

100 cm

75 cm

Figure P2.6.7

Problems

2.6.8. The tainter gate section shown in Figure P2.6.8 has a cylindrical surface with a 40-ft radius and is supported by a structural frame hinged at O. The gate is 33 ft long (in the direction perpendicular to the page). Determine the magnitude, direction, and location of the total hydrostatic force on the gate.

45˚ 0 Figure P2.6.8

2.6.9. Calculate the magnitude, direction, and location of the total hydrostatic pressure force (per unit length) on the gate shown in Figure P2.6.9.

11 ft r = 10 ft

8 ft

Figure P2.6.9

2.6.10. A 4-m-long, 2-m-diameter cylindrical tank is depicted in Figure P2.6.10. A 0.5-m-diameter pipe extends vertically upward from the middle of the tank. Water fills the tank and pipe to a level 3 m above the top of the tank (h). What is the hydrostatic force on one end of the tank? What is the total hydrostatic pressure force on one side (semicircle) of the tank?

Figure P2.6.10

Water Pressure and Pressure Forces

Chap. 2

2.6.11. Calculate the magnitude and location of the vertical and horizontal components of the hydrostatic force on the surface shown in Figure P2.6.11 (quadrant on top of the triangle, both with a unit width). The liquid is water and the radius R = 4.4 ft.

6 ft R

8 ft

Figure P2.6.11

2.6.12. Two reservoirs are interconnected as depicted in Figure P2.6.12. A homogeneous cone plugs a 0.1-m-diameter orifice between reservoir A that contains water and reservoir B that contains oil (S.G. = 0.8). Determine the specific weight of the cone if it unplugs when h0 reaches 1.5 m.

0.3 m 0.5 m

0.3 m

0.1 m

B Figure P2.6.12

2.6.13. What would be the specific weight and specific gravity of the cone if reservoir B in Figure P2.6.12 contains air at a pressure of 8,500 N/m2 instead of oil? 2.6.14. The homogeneous cylinder (S.G. = 2.0) in Figure P2.6.14 is 1 m long and 22 m in diameter and blocks a [email protected] opening between reservoirs A and B (S.G.A = 0.9, S.G.B = 1.5). Determine the magnitude of the horizontal and vertical components of the hydrostatic force on the cylinder.

Problems

A B 6m

2m √‾ u

45˚

Figure P2.6.14

(secTion 2.8) 2.8.1. A 1-m length of a certain standard steel pipe has a mass of 25.3 kg and has an outside diameter of 158 mm. Will the pipe sink in glycerin (S.G. = 1.26) if its ends are sealed? 2.8.2. A "youth" bowling ball is placed in water and found to be neutrally buoyant (i.e., when placed below the surface, it neither sinks to the bottom nor floats to the top). If the diameter of the bowling ball is 8.6 in., determine its weight, mass, and specific gravity. 2.8.3. Three people are in a boat with an anchor. If the anchor is thrown overboard, will the lake level rise, fall or stay the same theoretically? Explain. 2.8.4. A solid brass sphere of 28-cm diameter is used to hold a cylindrical buoy in place (Figure P2.8.4) in lake water (S.G. = 1.0). The buoy has a height of 1.5 m and is tied to the sphere at one end. Determine the specific gravity of the buoy if a rise in tide of h = 1.03 m lifts the sphere off the bottom?

50 cm

30 cm

sp. gr. = 13.5 Figure P2.8.4

2.8.5. The solid floating prism shown in Figure P2.8.5 has two components. Determine γA and γB in terms of γ if γB = 2γA.

Water Pressure and Pressure Forces

1.5 H

Chap. 2

2H Figure P2.8.5

2.8.6. A cylindrical anchor (h = 1.2 ft and D = 1.5 ft) is made of concrete. A force of 244 pounds pulling on the anchor line is required before the anchor is lifted from the bottom. If the anchor line is at an angle of 60° with respect to the lake bottom, determine the specific gravity of the concrete. 2.8.7. In Figure P2.8.7, the spherical buoy of radius R opens the square gate AB when water rises to the half-buoy height. Determine R if the weight of the gate is 6 kN and the weight of the buoy is negligible.

1m B 1m A Hinge 1m

Figure P2.8.7

2.8.8. A floating rod (6 in. * 6 in. * 12 ft) is hinged as shown in Figure P2.8.8 and weighs 165 lb. The surface of the water is 7 ft above the hinge. Calculate the angle θ assuming a uniform weight distribution in the rod.

6'' × 6'' × 12'

Hinge

Figure P2.8.8

Problems

2.8.9. A 40-ft long, 30-ft diameter cylindrical caisson floats upright in the ocean (S.G. = 1.03) with 12 ft of the caisson length above the water. The center of gravity measure 6 ft from the bottom of the caisson. Determine the metacentric height and the righting moment when the caisson is tipped through an angle of 5°, 10°, and 15°. 2.8.10. Figure P2.8.10 shows a buoy that consists of a wooden pole 25 cm in diameter and 2 m long, with a spherical weight at the bottom. The specific gravity of the wood is 0.62 and the specific gravity of the bottom weight is 1.40. Determine (a) how much of the wooden pole is submerged in the water, (b) the distance to the center of buoyancy from the water level, (c) the distance to the center of gravity from the water level, and (d) the metacentric height.

2m 0.25 m

0.5 m Figure P2.8.10

2.8.11. A wooden block is 2 m long, 1 m wide, and 1 m deep. Is the floating block stable if the metacenter is at the same point as the center of gravity? Explain. 2.8.12. A subway tunnel is being constructed across the bottom of a harbor. The process involves tugboats that pull floating cylindrical sections (or tubes as they are often called) across the harbor and sink them in place, where they are welded to the adjacent section already on the harbor bottom. The cylindrical tubes are 50 ft long with a diameter of 36 ft. When in place for the tugboats, the tubes are submerged vertically to a depth of 42 ft and, 8 ft of the tube is above the water (S.G. = 1.02). To accomplish this, the tubes are flooded with 34 ft of water on the inside. Determine the metacentric height and estimate the righting moment when the tubes are tipped through a heel (list) angle of 4° by the tugboats. (Hint: Assume that the location of the center of gravity can be determined based on the water contained inside the tubes and the container weight is not that significant.) 2.8.13. A 12-m-long, 4.8-m-wide, and 4.2-m-deep rectangular pontoon has a draft of 2.8 m in sea water (S.G. = 1.03). Assuming that the load is uniformly distributed on the bottom of the pontoon to a depth of 3.4 m, and the maximum design angle of list is 15°, determine the distance that the center of gravity can be moved from the center line toward the edge of the pontoon.

Water Flow in Pipes

3.1 Description of Pipe Flow In hydraulics, the term pressure pipe flow refers to full water flow in closed conduits of circular cross sections under a certain pressure gradient. For a given discharge (Q), pipe flow at any location can be described by the pipe cross section, the pipe elevation, the pressure, and the flow velocity in the pipe. The elevation (h) of a particular section in the pipe is usually measured with respect to a horizontal reference datum such as mean sea level (MSL). The pressure in a pipe generally varies from one point to another, but a mean value is normally used at a given cross section. In other words, the regional pressure variation in a given cross section is commonly neglected unless otherwise specified. In most engineering computations, the section mean velocity (V) is defined as the discharge (Q) divided by the cross-sectional area (A): V =

Q A

(3.1)

The velocity distribution within a cross section in a pipe, however, has special meaning in hydraulics. Its significance and importance in hydraulics are discussed next. 54

Sec. 3.2

The Reynolds Number

3.2 The Reynolds Number Near the end of the nineteenth century, British engineer Osborne Reynolds performed a very carefully prepared pipe flow experiment. Figure 3.1 shows the schematics of a typical setup for the Reynolds experiment. A long, straight, glass pipe of small bore was installed in a large tank with glass sides. A control valve (C) was installed at the outlet end of the glass pipe to regulate the outflow. A small bottle (B) filled with colored water and a regulating valve at the bottle's neck were used to introduce a fine stream of colored water into the entrance of the glass pipe when the flow was initiated. Water in the large tank was allowed to settle very quietly in a room for several hours so that water in every part of the tank became totally stationary. Valve C was then partially opened to allow a very slow flow in the pipe. At this time, the colored water appeared as a straight line extending to the downstream end, indicating laminar flow in the pipe. The valve was opened up slowly to allow the pipe flow rate to increase gradually until a certain velocity was reached; at that time, the thread of color suddenly broke up and mixed with the surrounding water, which showed that the pipe flow became turbulent at this point. Reynolds found that the transition from laminar to turbulent flow in a pipe actually depends not only on the velocity but also on the pipe diameter and the viscosity of the fluid. Furthermore, he postulated that the onset of turbulence was related to a particular index number. This dimensionless ratio is commonly known as the Reynolds number (NR) (see also Chapter 10) and it can be expressed as DV NR = (3.2) v When expressing Reynolds number for pipe flow, D is the pipe diameter, V is the mean velocity, and v is the kinematic viscosity of the fluid, defined by the ratio of absolute viscosity (μ) and the fluid density (ρ). μ v = (3.3) ρ B

Figure 3.1 Reynolds apparatus

Water Flow in Pipes

Chap. 3

It has been found and verified by many carefully prepared experiments that for flows in circular pipes the critical Reynolds number is approximately 2,000. At this point, the laminar pipe flow changes to a turbulent one. The transition from laminar to turbulent flow does not happen at exactly NR = 2,000 but varies from approximately 2,000 to 4,000 based on differences in experimental conditions. This range of Reynolds number between laminar and turbulent flow is commonly known as the critical zone, which will be discussed more fully later. Laminar flow occurs in a circular pipe when fluid flows in orderly laminae; this is analogous to the telescoping of a large number of thin-walled concentric tubes. The outer tube adheres to the pipe wall while the tube next to it moves with a very slight velocity. The velocity of each successive tube increases gradually and reaches a maximum velocity near the center of the pipe. In this case, the velocity distribution takes the form of a paraboloid of revolution with the mean velocity V equal to one-half of the maximum center line velocity, as shown in Figure 3.2. In turbulent flow, the turbulent motion causes the slower water particles adjacent to the pipe wall to mix continuously with the high-speed particles in the midstream. As a result, the lowspeed particles near the pipe wall are accelerated because of momentum transfer. For this reason, the velocity distribution in turbulent flow is more uniform than laminar flow. The velocity profiles in turbulent pipe flows have been shown to take the general form of a logarithmic curve in revolution. Turbulent mixing activities increase with the Reynolds number; hence, the velocity distribution becomes flatter as the Reynolds number increases. Under ordinary circumstances, water loses energy as it flows through a pipe. A major portion of the energy loss is caused by 1. friction against the pipe walls and 2. viscous dissipation occurring throughout the flow. Wall friction on a moving column of water depends on the roughness of the wall material (e) and the velocity gradient [(dV/dr) # r = D/2] at the wall (see Equation 1.2). For the same flow rate, it is evident in Figure 3.2 that turbulent flow has a higher wall velocity gradient than that of laminar flow; hence, a higher friction loss may be expected as the Reynolds number increases. At the same time, momentum transfer of water molecules between layers is intensified as the flow becomes more turbulent, which indicates an increasing rate of viscous dissipation in the flows. As a consequence, the rate of energy loss in pipe flow varies as a function of the Reynolds

V(r) r Vmax

Paraboloid in revolution (laminar)

V(r) r Vmax V

(a) Laminar

Figure 3.2

D Logarithmic curve in revolution (turbulent)

(b) Turbulent

Velocity profiles of laminar and turbulent flows in circular pipes

Sec. 3.3

Continuity and Momentum Equations in Pipe Flow

number and the roughness of the pipe wall. The ramifications of this in engineering applications will be discussed later in the chapter. Example 3.1 A 40-mm-diameter circular pipe carries water at 20°C. Calculate the largest flow rate for which laminar flow can be expected. Solution The kinematic viscosity of water at 20°C is v = 1.00 * 10-6 m2 >s (Table 1.3 or book jacket). Taking NR = 2,000 as the conservative upper limit for laminar flow, the velocity can be determined as NR =

(0.04 m)V DV = 2,000 = v 1.00 * 10-6 m2 >s

V = 2000(1.00 * 10-6 >0.04) = 0.05 m>s

The flow rate is

Q = AV =

π (0.04)2(0.05) = 6.28 * 10-5 m3 >s 4

3.3 Continuity and Momentum Equations in Pipe Flow In Figure 3.3, a fixed control volume is considered between sections 1-1 and 2-2. We will derive the continuity and momentum equations with reference to this control volume. The mass of an object, m, is the amount of matter contained in the object, and it is equal to the density multiplied by the volume. In a moving liquid (recalling that Q = discharge = volumetric flow rate), the rate of mass transfer is expressed as ρQ = ρAV. In Figure 3.3, the rate of mass transfer into the control volume at section 1-1 is ρA1V1 and the rate of mass transfer out of the control volume at section 2-2 is ρA2V2 for incompressible flow. Hence, the net rate of mass transfer into the control volume is (ρA1V1 - ρA2V2). From the conservation 1 F1 = P1A1 V1

y Wx Wy

x W

F2 = P2A2

2 h2

Figure 3.3

General description of flow in pipes

Datum

Water Flow in Pipes

Chap. 3

of mass principle, this must be equal to the rate of change of mass of water in the control volume. For a fixed control volume in steady, incompressible flow the mass of water in the control volume remains constant and therefore the rate of change is zero. Then (ρA1V1 - ρA2V2) = 0 or A1V1 = A2V2 = Q

(3.4)

The momentum of an object is a vector > quantity and is equal to the product of its mass and the velocity vector and expressed as mV . The rate of momentum transfer in a moving fluid > can then be expressed as ρQV. With reference to Figure 3.3, for steady flow, the > incompressible > momentum transfer into the control volume at section 1-1 is ρA V V = ρQV and the momen1 > 1 1 1 > tum transfer out of the control volume at section 2-2 is ρA> 2V2V2 = > ρQV2. Therefore, the net rate of momentum transfer into the control volume is (ρQV1 - ρQV2). According to the conservation principle, the net rate of momentum transfer into the control volume, > of momentum > > (ρQV1 - ρQV2), plus the sum of the external forces, (ΣF), must be equal to the rate of increase of momentum accumulated in the control volume. However, for a fixed control volume under steady, incompressible flow conditions, of momentum accumulated in the > the rate > of increase > control volume is zero. Therefore, ΣF + ρQV1 - ρQV2 = 0 or > > > ΣF = ρQ(V2 - V1) (3.5) In this equation, both the forces and velocities are vector quantities. They must be balanced in every direction considered. Along the axial direction of the flow, the external forces exerted on the control volume may be expressed as ΣFx = P1A1 - P2A2 - Fx + Wx

(3.6)

where V1, V2, P1, and P2 are the velocities and pressures at sections 1-1 and 2-2, respectively. Fx is the axial direction force exerted on the control volume by the wall of the pipe. Wx is the axial component of the weight of the liquid in the control volume. By recognizing (ρQ) in Equation 3.5 as mass flow rate, the principle of conservation of momentum (or the impulse-momentum equation) may be expressed, for the axial direction, as ΣFx = ρQ(Vx2 - Vx1)

(3.7a)

Similarly, for the other directions, ΣFy = ρQ(Vy2 - Vy1)

(3.7b)

ΣFz = ρQ(Vz2 - Vz1)

(3.7c)

In general, we may write in vector quantities > > > ΣF = ρQ(V2 - V1)

(3.7)

Example 3.2 A horizontal nozzle (Figure 3.4) discharges 0.01 m3>s of water at 4°C into the air. The supply pipe's diameter (dA = 40 mm) is twice as large as the nozzle diameter (dB = 20 mm). The nozzle is held in place by a hinge mechanism. Determine the magnitude and direction of the reaction force at the hinge, if the gauge pressure at A is 500,000 N>m2. (Assume the weight supported by the hinge is negligible.)

Sec. 3.3

Continuity and Momentum Equations in Pipe Flow

A FA

dA u = 60°

B VB FB

Figure 3.4 Flow through a horizontal nozzle

Solution The hinge force resists the pressure and momentum change in the system. This force can be computed from the conservation of momentum equation. The hydrostatic forces are Fx, A = PAA = (500,000 N>m2)[(π>4)(0.04 m)2] = 628 N Fy, A = 0 (all flow in the [email protected]); Fx,B = Fy,B = 0 (atmospheric pressure) VA = Q>AA = (0.01 m3 >s)>[(π>4)(0.04 m)2] = 7.96 m>s = Vx, A, Vy, A = 0 VB = Q>AB = (0.01 m3 >s)>[(π>4)(0.02 m)2] = 31.8 m>s

Vx, B = (31.8 m>s)(cos 60°) = 15.9 m>s Vy, B = (31.8 m>s)(sin 60°) = 27.5 m>s Now

ΣFx = ρQ(VxB - VxA) with the sign convention ( S +). Assuming Fx is negative and substituting yields 628 N - Fx = (998 kg>m3)(0.01 m3 >s)[(15.9 - 7.96) m>s]; Fx = 549 N d

Both forces and velocities are vector quantities that must adhere to the sign convention. Because the sign of Fx ended up positive, our assumed direction was correct. Likewise, ΣFy = ρQ(VyB - VyA) with the sign convention ( c + ). Assuming Fy is negative and substituting yields - Fy = (998 kg>m3)(0.01 m3 >s)[( - 27.5 - 0) m>s]; Fy = 274 N T

The resultant force is F = [(549 N)2 + (274 N)2]1>2 = 614 N

Water Flow in Pipes

Chap. 3

and its direction is θ = tan-1(Fy >Fx) = 26.5°

3.4 Energy in Pipe Flow

Water flowing in pipes may contain energy in various forms. The major portion of the energy is contained in three basic forms: 1. potential energy, 2. kinetic energy, and 3. pressure energy. The three forms of energy may be demonstrated by examining the flow in a general section of pipe, as shown in Figure 3.3. This section of pipe flow can represent the concept of a stream tube that is a cylindrical passage with its surface everywhere parallel to the flow velocity; therefore, the flow cannot cross its surface. Relative to a horizontal datum, the potential energy of an object of mass m is expressed as mgh, where h is the vertical distance between the centroid of the object and the datum. For a moving fluid, the rate of potential energy transfer can be expressed as ρQgh. In Figure 3.3, the net rate of potential energy transfer into the control volume for incompressible flow is (ρA1V1gh1 - ρA2V2gh2)

(3.8)

The kinetic energy of an object of mass m and speed V is defined as mV 2 >2. For a moving fluid, the rate of kinetic energy transfer can be expressed as ρQV 2 >2. Accordingly, in Figure 3.3, the net rate of kinetic energy transfer into the control volume is a ρA1V1

V 21 V 22 - ρA2V2 b 2 2

(3.9)

The pressure energy is due to the work performed by the pressure forces. Work is expressed as a product of a force and distance, and the rate of work is a product of a force and speed (magnitude of velocity). The rate of work performed by the surroundings on the control volume due to the hydrostatic pressure at section 1-1 is P1A1V1. The rate of work performed at section 2-2 due to the pressure force opposing the flow is P2A2V2. Therefore, the net rate of pressure energy transfer into the flow in the control volume is (P1A1V1 - P2A2V2)

(3.10)

The net rate of energy transfer into the control volume must be equal to the rate of change of total energy stored in the control volume. However, the energy stored in a fixed control volume under steady flow conditions remains constant. Therefore (ρA1V1gh1 - ρA2V2gh2) + (P1A1V1 - P2A2V2) + a ρA1V1

Dividing all the terms by ρgQ where Q = V1A1 = V2A2 h1 +

P1 V 21 P2 V 22 + = h2 + + γ γ 2g 2g

V 21 V 22 - ρA2V2 b = 0 (3.11) 2 2 (3.12)

Sec. 3.4

Energy in Pipe Flow

61 a′′

a 2

V1 2g

EGL

hL a′

HGL

P1 g

V22 2g P2 g V2

h1 h2 b

b′ Figure 3.5

Total energy and head loss in pipe flow

Therefore, the algebraic sum of the velocity head, the pressure head, and the elevation head accounts for nearly all the energy contained in a unit weight of water flowing through a particular section of pipe. In reality, however, a certain amount of energy loss occurs when the water mass flows from one section to another. Accounting for this loss in engineering applications is discussed next. Figure 3.5 depicts schematically the heads at two locations along a pipeline. At section 1, the upstream section, the three heads are V 21 >2g, P1 >γ, and h1. (Note that the energy per unit weight of water results in a length or height dimensionally.) The algebraic sum of these three heads gives the point a above the energy datum. The distance measured between points a and b represents the total head, or the total energy contained in each unit weight of water that passes through section 1. P1 V 21 H1 = h1 + + γ 2g

(3.13)

During the journey between the upstream and downstream sections, a certain amount of hydraulic energy is lost because of friction (i.e., primarily converted to heat). The remaining energy in each unit weight of water at section 2 is represented by the distance between points a′ and b′ in Figure 3.5. Once again, this is the total head and is the sum of the velocity head, the pressure head, and the elevation head. H2 = h2 +

P2 V 22 + γ 2g

(3.14)

The elevation difference between points a′ and a″ represents the head loss (hL), between sections 1 and 2. The energy relationship between the two sections can be written in the following form: h1 +

P1 V 21 P2 V 22 + = h2 + + + hL γ γ 2g 2g

(3.15)

Water Flow in Pipes

Chap. 3

This relationship is known as the energy equation, but occasionally it is mistakenly called Bernoulli's equation (which does not account for losses or assumes they are negligible). For a horizontal pipe of uniform size, it can be shown that the head loss results in a pressure drop in the pipe because the velocity heads and the elevation heads are equal P1 - P2 = hL γ

(3.15a)

For a pipe with a uniform size but different elevations h1 and h2 at the upstream and downstream ends P1 - P2 = hL + h2 - h1 γ

(3.15b)

Figure 3.5 displays a few other noteworthy hydraulic engineering concepts. For example, a line may be drawn through all of the points that represent total energy along the pipe. This is called the energy grade line (EGL). The slope of the EGL represents the rate at which energy is being lost along the pipe. A distance V 2>2g below the EGL is the hydraulic grade line (HGL). These concepts will be discussed in later sections. Example 3.3 A 25-cm circular pipe carries 0.16 m3>s of water under a pressure of 200 Pa. The pipe is laid at an elevation of 10.7 m above mean sea level in Freeport, Texas. What is the total head measured with respect to MSL? Solution The continuity condition (Equation 3.4), requires that

Q = AV Hence, V =

0.16 m3 >s Q = = 3.26 m>s A (π>4)(0.25 m)2

The total head measured with respect to mean sea level is (3.26 m>s)2 200 N>m2 P V2 + h = + + + 10.7 m = 11.3 m γ 2g 2(9.81 m>s2) 9,790 N>m3

Example 3.4 The elevated water tank shown in Figure 3.6 is being drained to an underground storage location through a 12-in. diameter pipe. The flow rate is 3,200 gallons per minutes (gpm), and the total head loss is 11.5 ft. Determine the water surface elevation in the tank. Solution An energy relationship (Equation 3.15) can be established between section 1 at the reservoir surface and section 2 at the end of the pipe. h1 +

P1 V 21 P2 V 22 + = h2 + + + hL γ γ 2g 2g

Sec. 3.5

Loss of Head from Pipe Friction

Datum 2

Figure 3.6

5 ft

Flow from an elevated water tank

The water velocity in the reservoir, being small compared with that in the pipe, can be neglected. Furthermore, both sections are exposed to atmospheric pressure so that P1 = P2 = 0 The mean velocity is 1 ft3 >s 3,200 gpm 3,200 gpm Q d = 9.07 ft>s = = c dc V = A πr 2 π(0.5 ft)2 449 gpm

Now solving the energy relationship by setting the datum at the ground elevation yields h = h1 =

(9.07 ft>s)2 V 22 + h2 + hL = - 5 ft + 11.5 ft = 7.78 ft 2g 2(32.2 ft>s2)

3.5 Loss of Head from Pipe Friction Energy loss resulting from friction in a pipeline is commonly termed the friction head loss (hf). This is the loss of head caused by pipe wall friction and the viscous dissipation in flowing water. Friction loss is sometimes referred to as the major loss because of its magnitude, and all other losses are referred to as minor losses. Several studies have been performed during the past century on laws that govern the loss of head by pipe friction. It has been learned from these studies that resistance to flow in a pipe is 1. 2. 3. 4. 5.

independent of the pressure under which the water flows, linearly proportional to the pipe length (L), inversely proportional to some power of the pipe diameter (D), proportional to some power of the mean velocity (V), and related to the roughness of the pipe, if the flow is turbulent.

Several experimental equations have been developed in the past. Some of these equations have been used faithfully in various hydraulic engineering practices.

Water Flow in Pipes

Chap. 3

The most popular pipe flow equation was derived by Henri Darcy (1803 to 1858), Julius Weisbach (1806 to 1871), and others about the middle of the nineteenth century. The equation takes the following form: hf = f a

L V2 b D 2g

(3.16)

This equation is commonly known as the Darcy–Weisbach equation. It is conveniently expressed in terms of the velocity head in the pipe. Moreover, it is dimensionally uniform since in engineering practice the friction factor ( f ) is treated as a dimensionless numerical factor; hf and V 2 >2g are both in units of length. 3.5.1 Friction Factor for Laminar Flow

In laminar flow, f can be determined by balancing the viscous force and the pressure force at the two end sections of a horizontal pipe separated by a distance L. In a cylindrical pipe section of radius r (Figure 3.7), the difference in pressure force between the two ends of the cylinder is (P1 - P2)πr 2, and the viscous force on the cylinder is equal to (2πrL)τ. The values of shear stress τ have been shown, in Equation 1.2, to be μ(dv/dr). Under equilibrium conditions, when the pressure force and the viscous force on the cylinder of water are balanced, the following expression results: -2πrL a μ

dv b = (P1 - P2)πr 2 dr

The minus sign is used because the velocity decreases as the radial position (r) increases (i.e., dv>dr is always negative in pipe flow). This equation can be integrated to give the general expression of flow velocity in terms of r: v = a

P1 - P2 b (r 20 - r 2) 4μL

(3.17)

where r0 is the inner radius of the pipe, and the equation shows that the velocity distribution in laminar pipe flow is a parabolic function of radius r. The total discharge through the pipe can be obtained by integrating the discharge through the elemental area, (2πr) dr. Q =

dQ =

r = r0

vdA =

Lr = 0

P1 - P2 2 (r 0 - r 2)(2πr) dr 4μL

dr r

L (a) Figure 3.7

(b) Geometry of a circular pipe

Sec. 3.5

Loss of Head from Pipe Friction

πr 40(P1 - P2) πD4(P1 - P2) = 8μL 128μL

(3.18)

This relationship is also known as the Hagen–Poiseuille law* of laminar flow. The mean velocity is V =

πD4(P1 - P2) Q 1 = a b A 128μL (π>4)D2

(P1 - P2)D2 V = 32μL

(3.19)

For a horizontal uniform pipe, the energy equation (3.15) leads to hf =

P1 - P2 γ

Thus, the Darcy–Weisbach equation may be written as P1 - P2 L V2 = fa b γ D 2g

(3.20)

Combining Equations 3.19 and 3.20, we have

64 μg γ VD

(3.20a)

64μ 64 = ρVD NR

(3.21)

f = Because γ = ρg, f =

which indicates a direct relationship between the friction factor (f) and the Reynolds number (NR) for laminar pipe flow. It is independent of the surface roughness of the pipe. 3.5.2 Friction Factor for Turbulent Flow When the Reynolds number approaches a higher value—say, NR W 2,000—the flow in the pipe becomes practically turbulent and the value of f then becomes less dependent on the Reynolds number but more dependent on the relative roughness (e/D) of the pipe. The quantity e is a measure of the average roughness height of the pipe wall irregularities, and D is the pipe diameter. The roughness height of commercial pipes is commonly described by providing a value of e for the pipe material. It means that the selected pipe has the same value of f at high Reynolds numbers as would be obtained if a smooth pipe were coated with sand grains of a uniform size e. The roughness height for certain common commercial pipe materials is provided in Table 3.1.

*Experimentally derived by G. W. Hagen (1839) and later independently obtained by J. L. M. Poiseuille (1840).

Water Flow in Pipes

Chap. 3

Table 3.1 Roughness Heights, e, for Certain Common Pipe Materials

Pipe Material

e (mm)

Brass 0.0015 Concrete Steel forms, smooth 0.18 Good joints, average 0.36 Rough, visible form marks 0.60 Copper 0.0015 Corrugated metal (CMP) 45 Iron (common in older water lines, except ductile or DIP, which is widely used today) Asphalt lined 0.12 Cast 0.26 Ductile; DIP—cement mortar lined 0.12 Galvanized 0.15 Wrought 0.045 Polyvinyl chloride (PVC) 0.0015 Polyethylene, high density (HDPE) 0.0015 Steel Enamel coated 0.0048 Riveted 0.9 ∼ 9.0 Seamless 0.004 Commercial 0.045

e (ft) 0.000005 0.0006 0.0012 0.002 0.000005 0.15 0.0004 0.00085 0.0004 0.0005 0.00015 0.000005 0.000005 0.000016 0.003–0.03 0.000013 0.00015

It has been determined that immediately next to the pipe wall there exists a very thin layer of flow commonly referred to as the laminar sublayer even if the pipe flow is turbulent. The thickness of the laminar sublayer δ′ decreases with an increase in the pipe's Reynolds number. A pipe is said to be hydraulically smooth if the average roughness height is less than the thickness of the laminar sublayer. In hydraulically smooth pipe flow, the friction factor is not affected by the surface roughness of the pipe. Based on laboratory experimental data, it has been found that if δ′ 7 1.7e, then the effect of surface roughness is completely submerged by the laminar sublayer and the pipe flow is hydraulically smooth. In this case, Theodore Von Kármán* developed an equation for the friction factor: 1 2f

= 2 loga

NR 2f b 2.51

(3.22)

At high Reynolds numbers, δ′ becomes very small. If δ′ 6 0.08e, it has been found that f becomes independent of the Reynolds number and depends only on the relative roughness height. In this case, the pipe behaves as a hydraulically rough pipe, and von Kármán found that f can be expressed as 1 2f

= 2 loga 3.7

D b e

(3.23)

*Theodore Von Kármán, "Mechanische Ähnlichkeit und Turbulenz" (Mechanical similitude and turbulence), Proc. 3rd International Congress for Applied Mechanics, Stockholm, Vol. I (1930).

Sec. 3.5

Loss of Head from Pipe Friction

Between these two extreme cases, if 0.08e 6 δ′ 6 1.7e, the pipe behaves neither completely smooth nor rough. C. F. Colebrook* devised an approximate relationship for this intermediate range: e 1 D 2.51 = -log§ + ¥ 3.7 2f NR 2f

(3.24)

In the early 1940s, it was somewhat cumbersome to use any of these implicit equations in engineering practice. A convenient chart was prepared by Lewis F. Moody† (Figure 3.8); it is commonly called the Moody diagram of friction factors for pipe flow. The chart clearly shows the four zones of pipe flow: 1. a laminar flow zone where the friction factor is a simple linear function of the Reynolds number, 2. a critical zone where values are uncertain because the flow might be neither laminar nor truly turbulent, 3. a transitional zone where f is a function of both the Reynolds number and the relative roughness of the pipe, and 4. a zone of fully developed turbulence where the value of f depends solely on the relative roughness and is independent of the Reynolds number. Figure 3.8 may be used together with Table 3.1 to obtain the friction factor f for circular pipes. Subsequent to the development of the Moody diagram, the Swamee–Jain equation‡ was proposed to solve for the friction factor once NR is known. f =

0.25 2 e>D 5.74 Jloga + 0.9 b R 3.7 NR

(3.24a)

This explicit expression is supposed to provide a very accurate estimate (within 1 percent) of the implicit Colebrook–White equation for 10-6 6 e>D 6 10-2 and 5,000 6 NR 6 108.

*C. F. Colebrook, "Turbulent flow in pipes, with particular reference to the transition region between smooth and rough pipe laws," Jour. 1st. Civil Engrs., London (Feb. 1939). †L. ‡P.

F. Moody, "Friction factors for pipe flow," Trans. ASME, 66 (1944).

K. Swamee and A. K. Jain, "Explicit equations for pipe-flow problems," Journal of the Hydraulics Division, ASCE 102(5): 657–664, 1976.

0.01 0.009 0.008

0.015

0.02

0.025

0.03

0.04

7 9 103

2(103) 104

2(105)

2(106)

3 4 56 7 9 3 4 56 7 9 106 105 Reynolds number NR

3 4 56 7 9 107

2(107)

Figure 3.8 Friction factors for flow in pipes: the Moody diagram. Source: From L. F. Moody, "Friction factors for pipe flow," Trans. ASME, vol. 66, 1944.

3 4 56 7 9

2(104)

0.000001 0.000005

3 4 56 7 9 108

0.00001

0.00005

0.0001

0.0002

0.001 0.0008 0.0006 0.0004

0.002

0.004

0.01 0.008 0.006

0.015

0.02

0.05 0.04

0.05

Smooth pipes

Complete turbulence, rough pipes

0.03

64 NR Laminar Critical Transition zone flow zone

Laminar flow f =

0.06

0.07

0.09 0.08

0.1

Relative roughness e D

Sec. 3.5

Loss of Head from Pipe Friction

Example 3.5 Compute the discharge capacity of 3-m concrete (rough) pipe carrying water at 10°C. It is allowed to have a head loss of 2 m/km of pipe length. Solution From Equation 3.16, the friction head loss in the pipe is hf = f a

Hence,

2m = f a V2 =

L V2 b D 2g

1000 m V2 b 3m 2(9.81 m>s2)

0.118 m2 >s2 f

(1)

From Table 3.1, taking e = 0.60 mm, we obtain for the 3-m cast-iron pipe: e = 2.00 * 10-4 = 0.0002 D At 10°C, the kinematic viscosity of water is v = 1.31 * 10-6 m2 >s. Therefore, NR =

DV 3V = = (2.29 * 106)V -6 v 1.31 * 10

(2)

By using Figure 3.8, Equations (1) and (2) are solved by iteration until both conditions are satisfied. The iteration procedure is demonstrated as follows. The Moody diagram (Figure 3.8) is used to find f. However, V is not available, so NR cannot be solved. But the e/D value and the Moody diagram can be used to obtain a trial f value by assuming flow is in the complete turbulence regime. This is generally a good assumption for water-transmission systems because the viscosity of water is low and the velocities are high, yielding high NR values. Thus, with e>D = 0.0002, an f = 0.014 is obtained from the Moody diagram assuming complete turbulence (i.e., reading directly across the Moody diagram from the relative roughness on the right to the associated f value on the left). Using this friction factor in Equation (1) we obtain V = 2.90 m>s; using this value in Equation (2), we get NR = 6.64 * 106. This Reynolds number and the e/D value are then taken to the Moody diagram to obtain a new friction factor, f = 0.014, which is unchanged from the assumed friction factor. (If the friction factor was different, then additional iterations would be performed until the trial f and the computed f are essentially equal.) Now using the final flow velocity, the discharge is calculated as Q = AV = [(π>4)(3 m)2](2.90 m>s) = 20.5 m3 >s

Note: The Darcy–Weisbach equation for friction head loss (Equation 3.16), Reynolds number (Equation 3.2), and Colebrook's friction factor relationship (Equation 3.24) or the Swamee–Jain relationship (Equation 3.24a) can be solved simultaneously by a computer algebra software system (e.g., Mathcad, Maple, or Mathematica) and should yield the same result. However, the relationships are highly nonlinear, and a good initial estimate may be required to avoid numerical instability.

Water Flow in Pipes

Chap. 3

Example 3.6 Estimate the size of a uniform, horizontal welded-steel pipe installed to carry 14.0 ft3 >s of water at 70°F (approximately 20°C). The allowable pressure loss resulting from friction is 17 ft/mi of pipe length. Solution The energy equation can be applied to two pipe sections 1 mile apart: h1 +

P1 V 21 P2 V 22 + = h2 + + + hL γ γ 2g 2g

For a uniformly sized, horizontal pipe with no localized (minor) head losses, V1 = V2; h1 = h2; hL = hf and the energy equation reduces to P1 P2 = hf = 17 ft γ γ From Equation 3.16, hf = f Therefore,

8 f LQ2 Q2 L V2 L = f = D 2g D 2g(πD2 >4)2 gπ2D5 5

D =

8 f LQ2 gπ2hf

= 1,530 f

(a)

where L = 5,280 ft, and hf = 17 ft. At 20°C, v = 1.08 * 10-5 ft2 >s. Assuming welded-steel roughness to be in the lower range of riveted steel, e = 0.003 ft, the diameter can then be found using the Moody chart (Figure 3.8) by means of an iteration procedure as follows. Let D = 2.5 ft, then V =

14 ft3 >s Q = = 2.85 ft>s A π(1.25 ft)2

and NR =

(2.85 ft>s)(2.5 ft) VD = 6.60 * 105 = v 1.08 * 10-5 ft2 >s e>D =

0.003 ft = 0.0012 2.5 ft

Entering these values into the Moody chart, we get f = 0.021. A better estimate of D can now be obtained by substituting this friction factor into Equation (a), which gives D = [(1,530)(0.021)]1>5 = 2.0 ft

Sec. 3.6

Empirical Equations for Friction Head Loss

A second iteration yields V = 4.46 ft>s, NR = 8.26 * 105, e>D = 0.0015, f = 0.022, and D = 2.02 ft ≈ 2.00 ft. More iterations will produce the same result. Once again, a computer algebra system (e.g., Mathcad, Maple, or Mathematica) should yield the same result.

3.6 Empirical Equations for Friction Head Loss Throughout the history of civilization, hydraulic engineers have built systems to deliver water for people to use. The hydraulic design of these systems was significantly aided in the twentieth century by empirical equations, which are actually formulas in the strict sense. Generally speaking, these design equations were developed from experimental measurements of fluid flow under a certain range of conditions. Some do not have a sound analytical basis. For this reason, the empirical equations may not be dimensionally correct; if so, they can only be applicable to the conditions and ranges specified. The two equations discussed below contain empirical roughness coefficients that depend on the roughness of the tested pipes and not relative roughness, further limiting their usefulness. One of the best examples is the Hazen–Williams equation, which was developed for water flow in larger pipes (D Ú 5 cm, approximately 2 in.) within a moderate range of water velocity (V … 3 m>s, approximately 10 ft/s). This equation has been used extensively for the designing of water-supply systems in the United States. The Hazen–Williams equation, originally developed for the British measurement system, has been written in the form 0.54 V = 1.318CHWR0.63 h S

(3.25)

where S is the slope of the energy grade line, or the head loss per unit length of the pipe (S = hf >L), and Rh is the hydraulic radius, defined as the water cross-sectional area (A) divided by the wetted perimeter (P). For a circular pipe, with A = πD2 >4 and P = πD, the hydraulic radius is πD2 >4 A D Rh = = = P πD 4

(3.26)

The Hazen–Williams coefficient, CHW, is not a function of the flow conditions (i.e., Reynolds number). Its values range from 140 for very smooth, straight pipes down to 90 or even 80 for old, unlined tuberculated pipes. Generally, the value of 100 is taken for average conditions. The values of CHW for commonly used water-carrying conduits are listed in Table 3.2. Note that the coefficient in the Hazen–Williams equation shown in Equation 3.25, 1.318, has units of ft0.37 >s. Therefore, Equation 3.25 is applicable only for the British units in which the velocity is measured in feet per second and the hydraulic radius (Rh) is measured in feet. Because 1.318 ft0.37 >s = 0.849 m0.37 >s, the Hazen–Williams equation in SI units may be written in the following form: 0.54 V = 0.849CHWR0.63 h S

where the velocity is measured in meters per second and Rh is measured in meters.

(3.27)

Water Flow in Pipes

Chap. 3

Table 3.2 Hazen–Williams Coefficient, CHW , for Different Types of Pipes

Pipe Materials

CHW

Brass Cast iron (common in older water lines) New, unlined 10-year-old 20-year-old 30-year-old 40-year-old Concrete or concrete lined Smooth Average Rough Copper Ductile iron (cement mortar lined) Glass High-density polyethylene (HDPE) Plastic Polyvinyl chloride (PVC) Steel Commercial Riveted Welded (seamless) Vitrified clay

130–140 130 107–113 89–100 75–90 64–83 140 120 100 130–140 140 140 150 130–150 150 140–150 90–110 100 110

Example 3.7 A 100-m-long pipe with D = 20 cm and CHW = 120 carries a discharge of 30 L/s. Determine the head loss in the pipe. Solution Area:

A =

πD2 π = (0.2)2 = 0.0314 m2 4 4

Wetted perimeter:

P = πD = 0.2π = 0.628 m

Hydraulic radius:

Rh = A>P =

0.0314 = 0.0500 m 0.628

Applying Equation 3.27, Q 0.54 = 0.849CHWR0.63 h S A hf 0.54 0.03 = 0.849(120)(0.05)0.63 a b 0.0314 100 V =

hf = 0.579 m

Sec. 3.6

Empirical Equations for Friction Head Loss

Another popular empirical equation is the Manning equation, which was originally developed in metric units. The Manning equation has been used extensively for open-channel designs (discussed in detail in Chapter 6). It is also quite commonly used for pipe flows. The Manning equation may be expressed in the following form: V =

1 2>3 1>2 R S n h

(3.28)

where the velocity is measured in meters per second and the hydraulic radius is measured in meters. The n is Manning's coefficient of roughness, specifically known to hydraulic engineers as Manning's n. In British units, the Manning equation is written as V =

1.486 2>3 1>2 Rh S n

(3.29)

where Rh is measured in feet and the velocity is measured in units of feet per second. The coefficient in Equation 3.29 serves as a unit conversion factor because 1 m1>3 >s = 1.486 ft1>3 >s. Table 3.3 contains typical values of n for water flow in common pipe materials. Table 3.3 Manning's Roughness Coefficient, n, for Pipe Flows

Type of Pipe

Manning's n

Brass Cast iron Cement mortar surfaces Cement rubble surfaces Clay drainage tile Concrete, precast Copper Corrugated metal (CMP) Ductile iron (cement mortar lined) Glass High-density polyethylene (HDPE) Polyvinyl chloride (PVC) Steel, commercial Steel, riveted Vitrified sewer pipe Wrought iron

Min.

Max.

0.009 0.011 0.011 0.017 0.011 0.011 0.009 0.020 0.011 0.009 0.009 0.009 0.010 0.017 0.010 0.012

0.013 0.015 0.015 0.030 0.017 0.015 0.013 0.024 0.013 0.013 0.011 0.011 0.012 0.020 0.017 0.017

Example 3.8 A horizontal pipe (old cast iron) with a 10-cm uniform diameter is 200 m long. If the measured pressure drop is 24.6 m of water, what is the discharge? Solution Area: Wetted perimeter:

A =

π 2 π D = (0.1)2 = 0.00785 m2 4 4

P = πD = 0.1π = 0.314 m

Water Flow in Pipes

Hydraulic radius:

Rh = A>P =

Energy slope:

S = hf >L =

Manning>s roughness coefficient:

Chap. 3

0.00785 = 0.0250 m 0.314 24.6 m = 0.123 200 m

n = 0.015 (Table 3.3)

Substituting the above quantities into the Manning equation, Equation 3.28, we have V = Q =

Q 1 = R2>3 S1>2 n h A 1 (0.00785)(0.025)2>3(0.123)1>2 = 0.0157 m3 >s 0.015

3.7 Friction Head Loss–Discharge Relationships Many engineering problems involve determination of the friction head loss in a pipe given the discharge. Therefore, expressions relating the friction head loss to discharge are convenient. Noting that A = πD2 >4 and g = 32.2 ft>s2 for the British unit system, we can rearrange the Darcy–Weisbach equation, Equation 3.16, as hf = f L

0.0252 Q2 D5

(3.30)

As discussed previously, the friction factor generally depends on the pipe size, roughness, and the Reynolds number. However, an inspection of the Moody diagram (Figure 3.8) reveals that the graphed lines become horizontal at high Reynolds numbers where the friction factor depends only on the e/D ratio for fully turbulent flow. In other words, for a given pipe size and material, f is constant for fully turbulent flow. This is generally the case for most water-transmission systems because the viscosity of water is low and the velocities are high, which yields high NR values. For practical purposes, Equation 3.30 is written as hf = KQm

(3.31)

where K = (0.0252 # f # L)>D5 and m = 2.0. Other friction head loss equations can also be expressed in the form of Equation 3.31 as summarized in Table 3.4. Note that in Equation 3.31 m is dimensionless, and the dimension of K depends on the friction equation and the unit system chosen. There is an abundance of computer software available to solve the pipe flow equations that have been discussed (i.e., Darcy–Weisbach, Hazen–Williams, and Manning). Some of these computer programs are free and readily available on the Internet as pipe flow calculators. Others are proprietary [e.g., FlowMaster, PIPE FLO, and Pipe Flow Wizard (Great Britain)], tend to be more robust, and have better print options for technical reports. For most of these computer programs, any four of the five variables (L, D, Q, hf, and the loss coefficient) are required to solve for the desired variable. Spreadsheet programs can easily be written to accomplish the same thing.

Sec. 3.8

Loss of Head in Pipe Contractions

Table 3.4 Friction Equations Expressed in the Form of hf = KQ m

Equation

K (BG System)

K (SI System)

Darcy–Weisbach

0.0252 f L

0.0826 f L

D Hazen–Williams

1.85

Manning

4.73 L

10.7 L

D4.87C 1.85 HW

4.64n2 L

10.3n2 L

D5.33

3.8 Loss of Head in Pipe Contractions A sudden contraction in a pipe usually causes a marked drop in pressure in the pipe because of both the increase in velocity and the loss of energy to turbulence. The phenomenon of a sudden contraction is schematically represented in Figure 3.9. The vertical distance measured between the energy grade line and the pipe centerline represents the total head relative to the pipe centroid at any particular location along the pipe. The vertical distance measured between the hydraulic grade line and the pipe centerline represents the pressure head (P>γ), and the distance between the EGL and HGL is the velocity head (V 2 >2g) at that location. After point B, the HGL begins to drop as the stream picks up speed, and a region of stagnant water appears at the corner of contraction C. Immediately downstream from the contraction, the streamlines separate from the pipe wall and form a high-speed jet that reattaches to the wall at point E. The phenomenon that takes place between C and E is known to hydraulic engineers as the vena contracta, which will be discussed in detail in Chapter 9. Most of the energy loss in a pipe contraction takes place between C and D where the jet stream velocity EGL 2

HGL

V1 2g

hc EGL 2

V2 2g

HGL A

C D

Figure 3.9

E V2

Head loss and pressure variation resulting from sudden contraction

Water Flow in Pipes

Chap. 3

is high and the pressure is low. A certain amount of pressure recovers between D and E as the jet stream gradually dissipates and normal pipe flow resumes. Downstream from point E, the EGL and HGL lines become parallel to each other again, but they take a steeper slope (than in the pipe upstream of the contraction) where a higher rate of energy dissipation (resulting from friction) in the smaller pipe is expected. The loss of head in a sudden contraction may be represented in terms of velocity head in the smaller pipe as hc = Kc a

V 22 b 2g

(3.32)

Here Kc is the coefficient of contraction. Its value varies with the ratio of contraction, D2 >D1, and the pipe velocity as shown in Table 3.5. Head loss from pipe contraction may be greatly reduced by introducing a gradual pipe transition known as a confusor as shown in Figure 3.10. The head loss in this case may be expressed as hc′ = K′c a

V 22 b 2g

(3.33)

The values of K′c vary with the transition angle α and the area ratio A2 >A1, as shown in Figure 3.11. The loss of head at the entrance of a pipe from a large reservoir is a special case of loss of head resulting from contraction. Because the water cross-sectional area in the reservoir is very large compared with that of the pipe, a ratio of contraction of zero may be taken. For a square-edged entrance, where the entrance of the pipe is flush with the reservoir wall as shown in Figure 3.12 (a), the Kc values shown for D2 >D1 = 0.0 in Table 3.5 can be used. The general equation for an entrance head loss is also expressed in terms of the velocity head of the pipe: he = Ke a

V2 b 2g

(3.34)

The approximate values for the entrance loss coefficient (Ke) for different entrance conditions are shown in Figure 3.12(a–d).

Table 3.5

Values of the Coefficient Kc for Sudden Contraction

Velocity in Smaller Pipe (m/s)

1 2 3 6 12

Sudden Contraction Coefficients, Kc (Ratio of Smaller to Larger Pipe Diameters, D2/D1) 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.49 0.48 0.47 0.44 0.38

0.49 0.48 0.46 0.43 0.36

0.48 0.47 0.45 0.42 0.35

0.45 0.44 0.43 0.40 0.33

0.42 0.41 0.40 0.37 0.31

0.38 0.37 0.36 0.33 0.29

0.28 0.28 0.28 0.27 0.25

0.18 0.18 0.18 0.19 0.20

0.07 0.09 0.10 0.11 0.13

0.03 0.04 0.04 0.05 0.06

Sec. 3.8

Loss of Head in Pipe Contractions

V2 A2

Figure 3.10

Pipe confusor

0.05

0.04 A2/A1 = 0.5 0.6

0.03 K′c

0.7 0.02 0.8 0.01

0.9

5º

10º

15º

20º

25º

30º a

35º

40º

45º

50º

55º

60º

Figure 3.11 Coefficient K′c for pipe confusors. Source: From Chigong Wu et al., Hydraulics (Chengdu, Sichuan, China: The Chengdu University of Science and Technology Press, 1979).

a r Ke = 0.5

Ke = 0.04

Ke = 1.0

(a)

(b) Figure 3.12

(c)

Coefficient Ke for pipe entrances

Ke = 0.5 + 0.3 cos a + 0.2 cos2 a

(d)

Water Flow in Pipes

Chap. 3

3.9 Loss of Head in Pipe Expansions The behavior of the energy grade line and the hydraulic grade line in the vicinity of a sudden pipe expansion is schematically depicted in Figure 3.13. At the corner of the sudden expansion A, the stream line separates from the wall of the large pipe and leaves an area of relative stagnation between A and B, in which a large recirculating eddy is formed to fill the space. Most of the energy loss in a sudden expansion takes place between A and B where the stream lines reattach to the wall. A recovery of pressure may take place here as a result of the decrease of velocity in the pipe. The high-speed jet stream gradually slows down and reaches equilibrium at point C. From this point downstream, the normal pipe flow conditions resume and the energy grade line takes on a smaller slope than that of the approaching pipe, as expected. The loss of head from a sudden expansion in a pipe can be derived from the momentum considerations (e.g., Daugherty and Franzini, 1977*). The magnitude of the head loss may be expressed as (V1 - V2)2 hE = 2g

(3.35)

Physically, this equation shows that the change in velocities expressed as a velocity head is the head loss in the sudden expansion. The head loss resulting from pipe expansions may be reduced by introducing a gradual pipe transition known as a diffusor, which is depicted in Figure 3.14 (Finnemore and Franzini, 2001**). The head loss in this pipe transition case may be expressed as hE=

K E=

(V1 - V2)2 2g

(3.36)

EGL

V 21 2g

V 22 2g

At centerline

EGL

HGL

HGL At wall A

Figure 3.13

Lead loss from sudden expansion

*R. L. Daugherty and J. B. Franzini, Fluid Mechanics with Engineering Applications (New York: McGraw-Hill Book Company, 1977). ** E. J. Finnemore and J. B. Franzini, Fluid Mechanics with Engineering Applications, 10th edition (New York: McGraw Hill Book Company, 2001).

Sec. 3.10

Loss of Head in Pipe Bends

V2 Figure 3.14

Pipe diffusor

EGL

HGL

V2 2g

V≈ 0

Figure 3.15

Exit (discharge) head loss

The values of K′E vary with the diffusor angle (α): α

10°

20°

30°

40°

50°

60°

75°

K′E

0.16

0.40

0.64

0.95

1.13

1.17

1.08

Note that for diffuser angles greater than 40°, the values of K′E are higher than 1.0. This is caused by the increased vortex motion due to turbulence set up induced currents in a diffuser. A submerged pipe discharging into a large reservoir is a special case of head loss from expansion. The flow velocity (V) in the pipe is discharged from the end of a pipe into a reservoir that is so large that the velocity within it is negligible. From Equation 3.35 we see that the entire velocity head of the pipe flow is dissipated and that the exit (discharge) head loss is hd = Kd

V2 2g

(3.37)

where the exit (discharge) loss coefficient Kd = 1.0. The phenomenon of exit loss is shown in Figure 3.15. 3.10 Loss of Head in Pipe Bends Pipe flow around a bend experiences an increase of pressure along the outer wall and a decrease of pressure along the inner wall. A certain distance downstream from the bend, the velocity and pressure resume normal distributions. To achieve this, the inner wall pressure must rise back to the normal value. The velocity near the inner wall of the pipe is lower than that at the outer wall, and it must also increase to the normal value. The simultaneous demand of energy may

Water Flow in Pipes

Chap. 3

Outer wall (high pressure)

Separation Inner wall (low pressure) Inner wall (low pressure)

(a)

Figure 3.16

(b)

Head loss at a bend: (a) flow separation in a bend and (b) secondary flow at a bend

cause separation of the stream from the inner wall as shown in Figure 3.16 (a). In addition, the unbalanced pressure at the bend causes a secondary current as in Figure 3.16 (b). This transverse current and the axial velocity form a pair of spiral flows that persist as far as 100 diameters downstream from the bend. Thus, the head loss at the bend is combined with the distorted flow conditions downstream from the bend until the spiral flows are dissipated by viscous friction. The head loss produced at a bend was found to be dependent on the ratio of the radius of curvature of the bend (R) to the diameter of the pipe (D) (Figure 3.16). Because the spiral flow produced by a bend extends some distance downstream from the bend, the head loss produced by different pipe bends placed close together cannot be treated by simply adding the losses of each one separately. The total loss of a series of bends placed close together depends not only on the spacing between the bends but also on the direction of the bends. Detailed analysis of head loss produced by a series of bends is a rather complex matter, and it can only be analyzed on an individual case-by-case basis. In hydraulic design the loss of head due to a bend, in excess of that which would occur in a straight pipe of equal length, may be expressed in terms of the velocity head as hb = Kb

V2 2g

(3.38)

For a smooth pipe bend of 90°, the values of Kb for various values of R/D as determined by Beij* are listed in the following table. The bend loss has also been found nearly proportional to the angle of the bend (α) for pipe bends other than 90° in steel pipes and drawn tubings. Pipe manufacturers are more than willing to supply prospective buyers with loss coefficients for bends, contractions, confusors, expansions, and diffusors. R/D

0.35

0.19

0.17

0.22

0.32

0.38

0.42

*K. H. Beij, "Pressure Losses for Fluid Flow in 90° Pipe Bends," Jour. Research Natl. Bur. Standards, 21 (1938).

Sec. 3.11

Loss of Head in Pipe Valves

3.11 Loss of Head in Pipe Valves Valves are installed in pipelines to control flow by imposing high head losses. Depending on how the particular valve is designed, a certain amount of energy loss usually takes place even when the valve is fully open. As with other losses in pipes, the head loss through valves may also be expressed in terms of velocity head in the pipe: hv = Kv

V2 2g

(3.39)

The values of Kv vary with the type and design of the valves. When designing hydraulic systems, it is necessary to determine the head losses through any valves that are present. Valve manufacturers are more than willing to supply prospective buyers with loss coefficients. The values of Kv for common valves are listed in Table 3.6. Table 3.6

Values of Kν for Common Hydraulic Valves

A. Gate valves Closed Kv = 0.15 (fully open) Open

B. Globe valves Closed Kv = 10.0 (fully open) Open C. Check valves Closed Hinge (swing type)

Swing type: Kv = 2.5 (fully open) Ball type: Kv = 70.0 (fully open) Lift type: Kv = 12.0 (fully open)

Open D. Rotary valves Closed Kv = 10.0 (fully open)

Open

Water Flow in Pipes

Chap. 3

Example 3.9 Figure 3.17 shows two sections of cast-iron pipe connected in series that transport water from a reservoir and discharge it into air through a rotary valve at a location 100 m below the water surface elevation. If the water temperature is 10°C, and square-edged connections are used, what is the discharge? Solution The energy equation can be written for section 1 at the reservoir surface and section 3 at the discharge end as V 33 P3 V 21 P1 + + h1 = + + h3 + hL γ γ 2g 2g Selecting the reference datum at section 3 yields h3 = 0. Because the reservoir and the discharge end are both exposed to atmospheric pressure and the velocity head at the reservoir can be neglected, we have h1 = 100 =

V 23 + hL 2g

The total available energy, 100 m of water column, is equal to the velocity head at the discharge end plus all the head losses incurred in the pipeline system. This relationship, as shown in Figure 3.17, may be expressed as (noting that V3 = V2): he + hf1 + hc + hf2 + hv +

V 22 = 100 2g

where he is the entrance head loss. For a square-edged entrance, Equation 3.34 and Figure 3.12 yield he = (0.5)

V 21 2g

The head loss from friction in pipe section 1–2 is hf1. From Equation 3.16, hf1 = f1

1,000 V 21 0.40 2g

The head loss from the sudden contraction at section 2 is hc. From Table 3.5 and Equation 3.32 (assume Kc = 0.33 for the first trial), hc = Kc

V 22 V 22 = 0.33 2g 2g

V 21 2g V 22 2g 100 m 1

40 cm f L = 1,000 m

20 cm f L = 1,200 m

Figure 3.17 Flow through a pipeline

Sec. 3.11

Loss of Head in Pipe Valves

The head loss from friction in pipe section 2–3 is hf2 : hf2 = f2

1,200 V 22 0.20 2g

The head loss at the valve is hv. From Table 3.6 and Equation 3.39, hv = Kv

V 22 V 22 = (10) 2g 2g

Therefore, 100 = a1 + 10 + f2

V 22 V 21 1,200 1,000 + 0.33b + af1 + 0.5b 0.20 2g 0.40 2g

From the continuity equation (Equation 3.4), we have

A1V1 = A2V2 π π (0.4)2V1 = (0.2)2V2 4 4 V1 = 0.25V2 Substituting V1 in the above relationship gives V 22 =

1,960 11.4 + 156 f1 + 6,000 f2

To evaluate f1 and f2 we have: D1V1 0.4 = V1 = (3.05 * 105)V1 v 1.31 * 10-6 D2V2 0.2 = = V2 = (1.53 * 105)V2 v 1.31 * 10-6

NR1 = NR2

where v = 1.31 * 10-6 at 10°C. For the 40-cm pipe, e>D = 0.00065, which yields f1 ≈ 0.018 (assuming complete turbulence). For the 20-cm pipe, e>D = 0.0013, so f2 ≈ 0.021. Solving the above equation for V2 yields the following: V 22 =

1,960 11.4 + 156(0.018) + 6,000(0.021)

V2 = 3.74 m>s and V1 = 0.25(3.74 m>s) = 0.935 m>s Hence, NR1 = 3.05 * 105 (0.935) = 2.85 * 105;

f1 = 0.0192

NR2 = 1.53 * 105 (3.74) = 5.72 * 105;

f2 = 0.0215

These f values do not agree with the assumed values, so a second trial must be made. For the second trial, we assume that Kc = 0.35, f1 = 0.0192, and f2 = 0.0215. Repeating the above calculation, we obtain V2 = 3.71 m>s, V1 = 0.928 m>s, NR1 = 2.83 * 105, NR2 = 5.66 * 105. Now from Figure 3.8, f1 = 0.0192; f2 = 0.0215.

Water Flow in Pipes

Chap. 3

Therefore, the discharge is Q = A2V2 =

π (0.2 m)2(3.71 m>s) = 0.116 m3 >s 4

Note: The energy equation, the Reynolds number expressions, Colebrook's friction factor relationship, or the Swamee–Jain equation and the continuity equation can be solved simultaneously by a computer algebra software system (e.g., Mathcad, Maple, or Mathematica) and should yield the same result. However, the relationships are highly nonlinear, and a good initial estimate may be required to avoid numerical instability. 3.12 Method of Equivalent Pipes The method of equivalent pipes is used to facilitate the analysis of pipe systems containing several pipes in series or in parallel. An equivalent pipe is a hypothetical pipe that produces the same head loss as two or more pipes in series or parallel for the same discharge. The expressions presented for equivalent pipes account for the losses from friction only. 3.12.1 Pipes in Series The method of equivalent pipes saves very little computation time when applied to pipes in series. However, the method is included herein for the sake of completeness. Consider pipes 1 and 2 shown in Figure 3.18. Suppose the pipes' diameters, lengths, and friction factors are known. We want to find a single pipe, E, that is hydraulically equivalent to 1 and 2 in series. For the two systems shown in Figure 3.18 to be equivalent, neglecting the head loss from pipe expansions and contractions, we must have Q1 = Q2 = QE

(3.39)

hfE = hf1 + hf2

(3.40)

and Suppose we employ the Darcy–Weisbach equation to solve the series pipe problem. In terms of the discharge (Q), Equation 3.16 becomes hf = f

8LQ2

(3.41)

gπ2D5

Then writing Equation 3.41 for pipes 1, 2, and E, substituting into Equation 3.40, and simplifying with QE = Q1 = Q2, we obtain fE

LE D5E

= f1

L1 D51

+ f2

(3.42)

D52

Q2 E QE

Figure 3.18 Pipes in series

Sec. 3.12

Method of Equivalent Pipes

Any combination of fE, L E, and DE that satisfies Equation 3.42 is acceptable. To find the characteristics of the hypothetical equivalent pipe, we pick two of the three unknowns— fE, L E, and DE—arbitrarily and calculate the third from Equation 3.42. In other words, an infinite number of single hypothetical pipes are hydraulically equivalent to two pipes in series. For N pipes in series, an equivalent pipe can be found using fE

N Li = fi 5 a 5 DE i = 1 Di

(3.43)

We should note that Equations 3.42 and 3.43 are valid for the Darcy–Weisbach equation only. The equivalent pipe relationships for the Hazen–Williams and Manning equations are given in Table 3.7. These relationships are valid both for the British and the SI unit systems. 3.12.2 Pipes in Parallel The method of equivalent pipes is a very powerful tool for analyzing pipe systems containing pipes in parallel. Consider the system with parallel pipes 1 and 2 shown in Figure 3.19. Suppose we want to determine a single pipe that is equivalent to pipes 1 and 2 in parallel. The two systems will be equivalent if hf1 = hf2 = hfE

(3.44)

QE = Q1 + Q2

(3.45)

and In considering the two requirements for parallel pipe flow (Equations 3.44 and 3.45), the flow equation is the most intuitive. However, the friction loss equality is the most critical to the solution process. Basically, the friction loss equality states that flow from one junction to another produces equal head losses regardless of the path taken. This concept is important in the solution of pipe network problems that will be covered in the next chapter. To solve the parallel pipe problem, we can rearrange Equation 3.41 as Q = b

gπ2 D5 hf 8f L

1>2

(3.46)

Table 3.7 Equivalent Pipe Equations

Equation Darcy–Weisbach

Manning

Pipes in Series fE

LE D5E

L En2E D5.33 E

Hazen–Williams

LE 4.87 C 1.85 HWEDE

= a fi N

i=1

D5i

= a N

Pipes in Parallel N D5i D5E = a B fEL E i = 1B fiL i

L in2i

D5.33 E

5.33 i = 1 Di

= a N

Li 1.85 4.87 i = 1 C HWiDi

B n2EL E

N D5.33 i = a 2 i = 1 B ni L i

4.87 4.87 N C 1.85 C 1.85 HWEDE HWiDi 1.85 = a Li B LE i=1 B

1.85

Water Flow in Pipes

Chap. 3

1 Q1 Q2 2

E QE

Figure 3.19

Pipes in parallel

Writing Equation 3.46 for pipes 1, 2, and E, substituting into Equation 3.45, and simplifying with hf1 = hf2 = hfE, we obtain D5E D51 D52 = + A fE L E A f1L 1 A f2L 2

(3.47)

For N pipes in parallel, Equation 3.47 can be generalized to obtain D5i D5E = a A fE L E i = 1 A fi L i N

(3.48)

Again, two of the three unknowns— fE, L E, and DE—can be chosen arbitrarily, and the third one is obtained from Equation 3.48. Also note that Equations 3.47 and 3.48 are applicable for the Darcy–Weisbach equation. Refer to Table 3.7 for the Hazen–Williams and Manning equations. Example 3.10 Pipes AB and CF in Figure 3.20 have a diameter of 4 ft, possess a Darcy–Weisbach friction factor of 0.02, and carry a discharge of 120 cubic feet per second (cfs). The length of AB is 1,800 ft and that of CF is 1,500 ft. Branch 1 is 1,800 ft long and has a diameter of 3 ft and a friction factor of 0.018. Branch 2 has a length of 1,500 ft, a diameter of 2 ft, and a friction factor of 0.015. (a) Determine the total head loss resulting from friction between points A and F, and (b) determine the discharge in each of the two branches (1 and 2). Branch 1 Q A

Q1 B

Branch 2 Figure 3.20

Flow through parallel pipes

Solution (a) We will first determine a hypothetical pipe that is hydraulically equivalent to the two pipes in parallel, branches 1 and 2. Let us arbitrarily pick a diameter of 4 ft and a friction factor of 0.02 for the equivalent pipe. By using Equation 3.47, 45 35 25 = + A 0.02 L E A (0.018)(1,800) A (0.015)(1,500)

Problems

Solving for L E, we obtain L E = 3,310 ft. Then hfAF = hfAB + hfBC + hfCF and by using Equation 3.30, hfAF = (0.02)(1,800) + (0.02)(1,500)

0.0252 (120)2 45

+ (0.02)(3,310)

0.0252 (120)2 45

hfAF = 12.8 + 23.5 + 10.6 = 46.9 ft (b) For branch 1, hfBC = 23.5 = (0.018)(1,800)

0.0252 Q21 35

Solving for Q1, we obtain Q1 = 83.6 cfs. Likewise, for branch 2 hfBC = 23.5 = (0.015)(1,500)

0.0252 Q22 25

and we obtain, Q2 = 36.4 cfs. Note that Q1 + Q2 = 120 cfs, which satisfies mass balance.

Problems (secTion 3.3) 3.3.1. A strength versus intelligence competition is introduced at a firefighter's convention between two contestants. Each is armed with a fire hose and a shield. The goal is to push your opponent backward a certain distance with the spray from the fire hose. A choice of shields is offered. One shield is a flat lid and the other is a hemispherical lid. Which shield would you choose (Figure P3.3.1)? Why? (a)

(b)

Vin Vout

Vout Vin

Figure P3.3.1

3.3.2. A jet of water exits a nozzle heading in the negative x-direction and strikes a flat plate at a 90° angle. The water sprays through a 360° arc (y- and z-direction) exiting the plate (Figure P3.3.1a). If the nozzle has an 8-in. diameter and the flow has an 11.3 ft/s velocity, determine the force exerted on the plate by the water. 3.3.3. A 1,040 N force is recorded on a hemispherical vane (Figure P3.3.1b) as it redirects a 2.5 cmdiameter water jet through a 180° angle. Determine the velocity of the flowing water jet if the blade is assumed to be frictionless. 3.3.4. Water flows through a horizontal, 0.5-m-diameter pipe at a rate of 0.9 m3 >s (Figure P3.3.4). It is ejected from the pipe through a 0.25-m-diameter nozzle. Determine the force (F) that holds the nozzle in place if the water pressure (FP1) in the pipe just upstream of the nozzle is 283 kN>m2 (kPa).

Water Flow in Pipes

Chap. 3

Fp1

Vout

Vin

Figure P3.3.4

3.3.5. The force (F) holding a nozzle connection at the end of a 0.6-m-diameter pipe is 63.5 kN. The pipe is connected to a 0.3-m-diameter nozzle. If the flow rate is 1.1 m3 >s in the positive x-direction (Figure P3.3.4), determine the water pressure (FP1) in the pipe just upstream of the nozzle. 3.3.6. Water flowing in a positive x-direction passes through a 90° elbow in a 6-in.-diameter pipeline and heads in a positive y-direction (Figure P3.3.6). If the flow rate is 3.05 ft3 >s, compute the magnitude and direction of the reaction force (F). The pressure upstream of the elbow is 15.1 psi; just downstream it is 14.8 psi.

Vout Vin Fp1

Fp2

F Figure P3.3.6

3.3.7. A 1-m-diameter pipe is carrying 1 m3 >s of water in a positive x-direction and passes through a 90° bend into the positive y-direction (Figure P3.3.6). Entrance and exit pressures in the bend are measured in water column heights of 42 and 41 m. Determine the magnitude and direction of the bend's reaction force (F). 3.3.8. Water flows through a reducing pipe bend and is deflected 30° in the horizontal plane. The velocity is 4 m/s entering the bend (15-cm diameter) with a pressure of 250 kPa. The pressure leaving the bend is 130 kPa (7.5-cm diameter). Determine the anchoring force required to hold the bend in place. (Assume that the water if flowing in a positive x-direction entering the bend and continues in a positive x-direction and a positive y-direction after the bend.)

(secTion 3.5) 3.5.1. Water at 4°C flows at the rate of 3.5 cms (m3 >s) through a corrugated metal pipe (CMP). If the diameter of the pipe is 2.25 m, determine the friction factor and flow type (i.e., laminar, critical zone, turbulent—transitional zone, turbulent—smooth pipe, or turbulent—rough pipe). 3.5.2. A wrought iron pipe, 1.50 ft in diameter and 100 ft long, carries 12 cfs (ft3 >s) of water at 68°F. Determine the friction factor and the type of flow that exists in the pipeline (i.e., laminar, critical zone, turbulent—transitional zone, turbulent—smooth pipe, or turbulent—rough pipe). 3.5.3. A horizontal, commercial steel pipe, 1.5 m in diameter, carries 3.5 m3 >s of water at 20°C. Calculate the pressure drop in the pipe per kilometer length. Assume that minor losses are negligible. 3.5.4. A 15-in. galvanized iron pipe is installed on a 1/50 slope (uphill) and carries water at 68°F (20°C). What is the pressure drop in the 65-ft-long pipe when the discharge is 18 cfs (ft3 >s)? Assume that minor losses are negligible. 3.5.5. The commercial steel pipeline depicted in Figure P3.5.5 is 200 m long and has a diameter of 0.45 m. Determine the height of the water tower (h) if the flow rate is 0.85 m3 >s. Assume that minor losses are negligible and a water temperature is 4°C.

Problems

89 1

Figure P3.5.5

3.5.6. Determine the flow rate of water (20°C) that will cause a pressure drop of 17,250 N>m2 in 350 m of horizontal, cast-iron pipe (D = 60 cm). Ignore minor losses. 3.5.7. The pressure heads are measured at two sections in a pipeline and are found to be 8.3 m at point A and 76.7 m at point B. The two sections are 5.5 km apart along a 4.5-m-diameter riveted-steel pipe (best condition). A is 100 m higher than B. If the water temperature is 20°C, what is the flow rate? Minor losses are negligible. 3.5.8. A smooth concrete pipe (1.5-ft diameter) carries water from a reservoir to an industrial treatment plant 1 mile away and discharges it into the air over a holding tank. The pipe leaving the reservoir is 3 ft below the water surface and runs downhill on a 1:100 slope. Determine the flow rate (in cfs, ft3 >s) if the water temperature is 40°F (4°C) and minor losses are negligible. 3.5.9. Two pressure gauges measure a pressure drop of 16.3 psi (lb>in.2) at the entrance and exit of an old buried pipeline. The original drawings have been lost. If the 6-in. galvanized iron pipe carries water at 68°F with a flow rate of 1.64 cfs (ft3 >s), determine the length of the horizontal underground line ignoring minor losses. 3.5.10. Determine the diameter of a 400-m-long wrought iron pipe required to convey water (15°C) at a flow rate of 45 L/s with a head loss not to exceed 9.8 m. 3.5.11. A 2,500-ft long pipeline is required to carry 21.5 cfs (ft3 >s) of water to an industrial client. The limiting pressure drop mandated by the client is 40 psi (lb>in.2). Determine the pipe size required if the material available is polyvinyl chloride (PVC) and the pipeline is level (horizontal). Assume that minor losses are negligible and the water temperature is 68°F. 3.5.12. City officials want to transport 1,800 m3 of water per day to a water treatment plant from a reservoir 8 km away. The water surface elevation at the reservoir is 6 m above the entrance of the pipe, and the water surface in the receiving tank is 1 m above the exit of the pipe. The pipe will be laid on a 1/500 slope. What is the minimum required diameter of a concrete pipe (good joints) if the water temperature varies between 4°C and 20°C? Assume that minor losses are negligible. 3.5.13. Equation (3.19) defines the mean velocity for laminar flow using the Hagen–Poiseuille law. Equation (3.20) gives the Darcy–Weisbach equation applied to a horizontal uniform pipe. Derive Equation (3.20a) showing all steps in the process. 3.5.14. A cast-iron pipeline was installed 20 years ago with a friction factor (measured) of 0.0195 and a roughness height (e) of 0.26 mm. The horizontal pipeline is 2,000 m long and has a diameter of 30 cm. Significant tuberculation has occurred since it was installed, and field tests are run

Water Flow in Pipes

Chap. 3

to determine the existing friction factor. A pressure drop of 366,000 Pascals is measured over the pipeline length for a flowrate (at 20°C) of 0.136 m3 >s. Determine the existing friction factor (effective) and the existing roughness height. Note: The friction factor is called "effective" since the loss of flow area due to tuberculation contributes to the reduced flow rate. Assume that minor losses are negligible. 3.5.15. A water supply pipeline contains a long segment that is horizontal and 30 cm in diameter (cast iron). A leak along an inaccessible portion of the buried pipeline is highly likely. A pair of pressure gauges located upstream of the leak indicate a pressure drop of 23,000 N>m2. Another pair of pressure gauges located downstream of the leak indicate a pressure drop of 20,900 N>m2. The distance between the gages in each pair is 100 m. Determine the magnitude of the leak. Assume that minor losses are negligible and that the water temperature is 20°C.

(secTion 3.7) 3.7.1. A flow rate 450 />s (water at 20°C) is carried in a 50-cm-diameter pipeline (ductile iron—new) for a distance of 1.5 km. Compute the friction head loss from (a) the Hazen–Williams, (b) Manning, and (c) Darcy–Weisbach equations and discuss the differences. Use computer software to verify your results. 3.7.2. A 3.5-ft-diameter commercial steel pipe (new, very smooth) carries water (39°F) from reservoir A to reservoir B (Figure P3.7.2). The pipeline is 2.5 miles long and the elevation difference between the surfaces of the two reservoirs is 395 ft. The discharge computed using the Darcy–Weisbach equation is 181 cfs (ft3 >s). Determine the discharge using the Hazen–Williams equation and the Manning equation. Ignore minor losses and verify your results with computer software. 1

Res. A 2 Res. B

Figure P3.7.2

3.7.3. Use the (a) Hazen–Williams equation, (b) Manning equation, and (c) the Darcy–Weisbach equation to calculate the flow rate in a smooth concrete pipe that carries water (20°C) between reservoirs A and B (Figure P3.7.2). The 4.5-m-diameter pipeline is 5.5 km long. There is a 60-m difference in the two water surface elevations. Compare the results and discuss the differences. Assume that minor losses are negligible. Verify your results with computer software. 3.7.4. Do some research to find two or three additional empirical equations involving head loss in pipelines. List the author(s) and limitations of each equation. 3.7.5. The pressure heads are measured at two sections in a pipeline and are found to be 8.3 m at point A and 76.7 m at point B. The two sections are 5.5 km apart along a 4.5-m-diameter riveted-steel pipe (best condition). Point A is 100 m higher than point B. The flow rate using the Darcy–Weisbach equation was found to be 95.6 m3 >s (ignoring minor losses). How closely does this compare with

Problems

3.7.6.

3.7.7.

3.7.8.

3.7.9.

the flow rates found using the Hazen–Williams equation and the Manning equation. Verify your results using computer software. The pressure head drop along a lengthy segment of a horizontal, buried pipeline is 29.9 ft. Unfortunately, the original plans for the pipeline route have been lost. Two pipeline routes of very different lengths are suspected. If the flow rate is 30.0 cfs (ft3 >s), determine the length of the 2-ft-diameter concrete pipe (n = 0.012). How much would the answer change (in %) if the n value was difficult to determine and could be assigned a value of 0.013 instead of 0.012? Assume that minor losses are negligible and verify your results with computer software. A 20-year-old, cast-iron pipeline is 3,800 m long with a diameter of 40 cm. The original Hazen– Williams coefficient was 130. A lot of tuberculation has occurred since it was installed. Flow tests are run to determine the existing CHW. A pressure drop of 232 kPa is measured over the horizontal pipeline length for a flow rate of 0.176 m3 >s. Determine the existing Hazen–Williams coefficient. Assume that minor losses are negligible and verify your results with computer software. A semicircular, precast concrete culvert with a radius of 3 ft flows full with a discharge of 150 cfs (ft3 >s). What is the friction head loss in 200 ft using Manning's equation? Can you verify your results with computer software? The water surface elevation difference between two reservoirs 2,000 m apart is 26 m. Compute the flow rate if (a) a 30-cm commercial steel (CHW = 140) pipeline connects the reservoirs, and (b) two 20-cm commercial steel pipelines are used instead. Ignore minor losses and verify your results with computer software.

(secTion 3.11) 3.11.1. What produces a greater head loss; a sudden contraction or a sudden expansion? Prove your answer by calculating the contraction loss for a flow rate of 106 />s in a 20-cm diameter pipe that suddenly reduces to 15-cm diameter pipe. Compare this with the head loss incurred when the 15-cm pipe suddenly expands to 20 cm. 3.11.2. In Problem 3.11.1, the head loss due to the abrupt contraction and expansion were calculated to be 0.275 and 0.353 m, respectively. With the same flow rate and geometry, determine the corresponding head losses if a 30° confusor and a 30° diffuser are used to reduce the head losses. 3.11.3. Valve manufacturers want to supply prospective buyers with the loss coefficients of their products. Hence, they perform lab tests to determine these coefficients. Determine the loss coefficient for a new valve if water flows through the 3-in. diameter valve at the rate of 1.40 ft3 >s and produces a pressure drop of 14.5 psi. 3.11.4. The pressure on the upstream side of a sudden contraction (D = 60 cm) is 285 and 265 kPa on the downstream side (D = 30 cm). Determine the flow rate in the horizontal pipeline that contains this sudden contraction. 3.11.5. Water flows through a 4-cm, wrought iron, horizontal pipeline from point A to point B. The pipeline is 50 meters long and contains a fully open gate valve and two elbows (R>D = 4). If the pressure at B (downstream) is 192 kPa and the flow rate is 0.006 m3 >s, determine the pressure at point A. 3.11.6. The pressure head drop across a short section of an 8-in.-pipeline (PVC) is 12 ft. The pipeline section contains a globe valve and another valve of some kind that is open but not labeled. Determine what kind of valve it is if the flow rate is 2.74 ft3 >s. Assume that the friction loss is negligible in the short pipe segment. 3.11.7. Determine the maximum discharge obtainable in a 3.5-ft-diameter commercial steel penstock that carries water from a mountain reservoir to a hydroelectric power plant. The penstock entrance is squared-edged and is located 100-ft below the reservoir's water surface. It is 1,500 ft long, contains a globe valve, and discharges water into the atmosphere at an elevation 750 ft below the entrance.

Water Flow in Pipes

Chap. 3

3.11.8. The sudden contraction headloss equation may be expressed as: hc = Kc[(VS)2 >2g]; and the sudden expansion head loss equation as: hE = [(VS - VL)2 >2g]; where VS is the velocity in the smaller pipe and VL is the velocity in the larger pipe. Prove that hE = 0.563 [(VS)2 >2g] for a diameter reduction of 50% (i.e., DS >DL = 0.5). Note that this shows that a sudden expansion loss is always greater than a sudden contraction loss by examining the Kc values in Table 3.5 for D2 >D1 = 0.5. 3.11.9. A 34-m-high water tower supplies drinking water (20°C) to a residential area with a 20-cm-diameter, 800-m-long (horizontal) commercial steel pipe. To increase the pressure head at the delivery point, engineers are considering replacing 94% of the pipe length with a larger (30-cm-diameter) steel pipe and a 30° confusor that connects to remainder of the smaller pipe. If the peak water demand is 0.10 m3 >s, determine the pressure head that would be gained by this strategy. 3.11.10. A 16-ft-diameter, cylindrical tank contains 10 ft of water depth ("h" in Figure P3.11.10). A short horizontal pipe with an 8-in. diameter and a rotary value is used to drain the tank (square-edged entrance) from the bottom. How long does it take to drain 50% of the tank? Assume that friction losses are negligible.

Figure P3.11.10

3.11.11. An oil flow rate of 0.012 m3 >s is required in an industrial process. The flow system includes a pressure tank pushing the oil through 200 m of new ductile iron pipe (DIP; 15 cm diameter, square-edged entrance) to point "2" (atmospheric pressure) as shown in Figure P3.11.11. The surface of the fluid in the tank (point "1") is at elevation 100 m and the end of the pipe (point "2") is at elevation 106 m. What air pressure will be needed over the fluid to produce the requisite flow? (S.G. (oil) = 0.84, ν = 2.03 * 10-6 m2 >s, and ε = 0.00012 m.) 2

Air 1

Figure P3.11.11

Problems

(secTion 3.12) 3.12.1. Derive an expression for N pipes in parallel using the Manning equation. 3.12.2. Derive an expression for N pipes in parallel using the Hazen–Williams equation. 3.12.3. Redo Example 3.10 using the Hazen–Williams equation assuming that CHW = 120 for all the pipes. 3.12.4. In the pipe system depicted in Figure P3.12.4, the discharge in pipe AB is 100 m3 >s. Branch 1 is 500 m long, and it has a diameter of 2 m and a friction factor of 0.018. Branch 2 has a length of 400 m, diameter of 3 m, and a friction factor of 0.02. Determine the length of an equivalent pipe to replace branches 1 and 2 assuming the pipe diameter is 3 m and f = 0.02. Also, determine the flow in branch 1. Branch 1 Q

Branch 2 Figure P3.12.4

3.12.5. In the pipe system depicted in Figure P3.12.4, the discharge in pipe AB is 100 m3 >s. Branches 1 and 2 can be replaced by a 222-m pipe (D = 3 m, f = 0.02). Pipe AB is 500 m long, and it has a diameter of 2 m and a friction factor of 0.018. Pipe CF has a length of 400 m, diameter of 3 m, and a friction factor of 0.02. Determine the length and head loss of an equivalent pipe (D = 3 m and f = 0.02) to replace the system of pipes. 3.12.6. Pipes AB and CF in the Figure P3.12.6 have a diameter of 3 m and Darcy–Weisbach friction factor of 0.02. The length of AB is 1,000 m and that of CF is 900 m. The discharge in pipe AB is 60 m3 >s. Branch 1 is 1,000 m long, has a diameter of 2 m, and a friction factor of 0.018. Branch 2 has a length of 800 m, a diameter of 3 m, and a friction factor of 0.02. A discharge of 20 m3 >s is added to the flow at point B and 10 m3 >s is taken out at point C as shown in the figure. (a) Determine the total head loss due to friction between section A and F and (b) determine the discharge in each of branches 1 and 2. 20 m3/s 60 m3/s A

Branch 1

10 m3/s

Q1 B

Branch 2 Figure P3.12.6

3.12.7. Can the method of equivalent pipes be used to find a single hypothetical pipe that is equivalent to the pipe system of Problem 3.12.6? If your answer is yes, determine an equivalent pipe. If your answer is no, explain your answer.

Robert Houghtalen

Pipelines and Pipe Networks

In general, when a number of pipes are connected together to transport water for a given project, they perform as a system that may include series pipes, parallel pipes, branching pipes, elbows, valves, meters, and other appurtenances. The arrangement is known as a pipeline if all elements are connected in series. Otherwise, the arrangement is a pipe network. Although the basic knowledge of pipe flow discussed in Chapter 3 is applicable to each individual pipe in the system, the design and analysis of a pipeline or pipe network does create certain complex problems unique to the system. This is particularly true if the system consists of a large number of pipes like those that frequently occur in the water-distribution networks of large metropolitan areas. The physical phenomena and problems that are pertinent to pipelines and pipe networks, as well as the special techniques developed for the analysis and design of such systems, are discussed in the following sections. 4.1 Pipelines Connecting Two Reservoirs A pipeline is a system of one or more pipes connected in series and designed to transport water from one location (often a reservoir) to another. There are three principal types of pipeline problems. 94

Sec. 4.1

Pipelines Connecting Two Reservoirs

1. Given the flow rate and the pipe combinations, determine the total head loss. 2. Given the allowable total head loss and the pipe combinations, determine the flow rate. 3. Given the flow rate and the allowable total head loss, determine the pipe diameter. The first type of problem can be solved by a direct approach, but the second and third types involve iterative procedures as shown in the following examples. Example 4.1 Two cast-iron pipes in series connect two reservoirs (Figure 4.1). Both pipes are 300 m long and have diameters of 0.6 m and 0.4 m, respectively. The elevation of the water surface (WS) in reservoir A is 80 m. The discharge of 10°C water from reservoir A to reservoir B is 0.5 m3/s. Find the elevation of the surface of reservoir B. Assume a sudden contraction at the junction and a square-edged entrance. 80 m EGL

hL HGL 1 2

Figure 4.1

Solution Applying the energy equation (Equation 3.15) between the reservoir surfaces A and B, hA +

PA V 2A PB V 2B + = hB + + + hL γ γ 2g 2g

Because PA = PB = 0, and the velocity heads can be neglected in a reservoir, hB = hA - hL Designating subscript 1 for the upstream pipe and subscript 2 for the downstream pipe to compute the head loss, we may write

From Table 3.1, we have

V1 =

Q 0.5 = = 1.77 m/s A1 (π/4)(0.6)2

V2 =

Q 0.5 = = 3.98 m/s A2 (π/4)(0.4)2

NR1 =

V1D1 1.77(0.6) = = 8.11 * 105 v 1.31 * 10-6

NR2 =

V2D2 3.98(0.4) = = 1.22 * 106 v 1.31 * 10-6 e 0.26 = = 0.00043 D1 600

Pipelines and Pipe Networks

Chap. 4

e 0.26 = = 0.00065 D2 400 From the Moody chart (Figure 3.8), we have f1 = 0.017 and f2 = 0.018 For the total head loss, hL = he + hf1 + hc + hf2 + hd. From Equations 3.16, 3.32, 3.34, and 3.37, we may write hL = a0.5 + f1

L 1 V 21 L2 V 22 b + a0.21 + f2 + 1b = 13.3 m D1 2g D2 2g

The elevation of the surface of reservoir B is

hB = hA - hL = 80 - 13.3 = 66.7 m

Example 4.2 Pipeline AB connects two reservoirs. The difference in elevation between the two reservoirs is 33 ft. The pipeline consists of an upstream section, D1 = 30 in. and L 1 = 5,000 ft; and a downstream section, D2 = 21 in. and L 2 = 3,500 ft. The pipes are concrete (smooth) and are connected end to end with a sudden reduction of area. Assume the water temperature is 68°F. Compute the discharge capacity. Solution The energy equation can be written as follows between the reservoir surfaces: hA +

PA V 2A PB V 2B + = hB + + + hL γ γ 2g 2g

By eliminating zero and small terms, we can rewrite this as hL = hA - hB = 33 ft Because the discharge is not yet known, the velocity in each pipe can only be assumed to be V1 and V2, respectively. The total energy equation, as above, will contain these two assumed quantities. It cannot be solved directly, so an iteration procedure is used. For water temperature at 68°F, v = 1.08 * 10-5 ft2/s. The corresponding Reynolds numbers may be expressed as NR1 =

V1D1 V1(2.5) = = (2.31 * 105)V1 -5 ν 1.08 * 10

(1)

NR2 =

V2D2 V2(1.75) = = (1.62 * 105)V2 ν 1.08 * 10-5

(2)

From the continuity condition, A1V1 = A2V2, we have π π (2.5)2(V1) = (1.75)2(V2) 4 4 V2 = 2.04 V1 (3) Substituting Equation (3) into Equation (2) we get NR2 = (1.62 * 105)(2.04 V1) = (3.30 * 105)V1

Sec. 4.1

Pipelines Connecting Two Reservoirs

The energy equation may be written as 33 = J0.5 + f1 a

V 21 V 22 5,000 3,500 bR + J0.18 + f2 a b + 1R 2.5 2g 1.75 2g

33 = (0.084 + 31.1f1 + 129f2)V 21

From Table 3.1, e/D1 = 0.00024 and e/D2 = 0.00034. As a first trial, try f1 = 0.014 and f2 = 0.015 (assuming complete turbulence, generally a good assumption for water-transmission systems because the viscosity of water is low and the velocities or pipe diameters are large, which yields high NR values). Thus 33 = [0.084 + 31.1(0.014) + 129(0.015)]V 21 V1 = 3.67 ft/s NR1 = 2.31 * 105(3.67) = 8.48 * 105 NR2 = 3.30 * 105(3.67) = 1.21 * 106 From Figure 3.8, f1 = 0.0155 and f2 = 0.016. These values do not agree with the values assumed previously. For the second trial, assume that f1 = 0.0155 and f2 = 0.016. V1 = 3.54 ft/s NR1 = 8.17 * 105 NR2 = 1.17 * 106 We obtain, from Figure 3.8, f1 = 0.0155 and f2 = 0.016. These values are the same as the assumed values, suggesting that V1 = 3.54 ft/s is the actual velocity in the upstream pipe. Therefore, the discharge is π Q = A1V1 = (2.5)2(3.54) = 17.4 cfs (ft3/s) 4

Example 4.3 A concrete pipeline is installed to deliver 6 m3/s of water (10°C) between two reservoirs 17 km apart. If the elevation difference between the two reservoirs is 12 m, what is the required pipe size? Solution As in the previous examples, the energy relationship between the two reservoirs is hA +

PA V 2A PB V 2B + = hB + + + hL γ γ 2g 2g

Thus, hL = hA - hB = 12 m The mean velocity may be obtained by using the continuity condition, Equation 3.4: V = and

Q 6 7.64 = = 2 A (π/4)D D2 Da

7.64

b DV 5.83 * 106 D2 = = NR = v D 1.31 * 10-6 Neglect the minor losses (for long pipes of L/D8 1,000, minor losses may be neglected). Therefore, the energy loss contains only the friction loss term. Equation 3.16 gives 17,000 L V2 L Q2 62 12 = f a b = fa b = f a b a b D 2g D 2gA2 D 2(9.81)(π/4)2D4

Pipelines and Pipe Networks

Chap. 4

Simplified, 0.000237 =

f D5

(a)

For concrete pipes, e = 0.36 mm (average), and assume that D = 2.5 m for the first trial, e 0.36 = = 0.00014 D 2500 and NR = 2.33 * 106. From Figure 3.8 we obtain f = 0.0135. Substituting these values into Equation (a), we see that D = a

0.0135 1/5 b = 2.24 0.000237

Hence, a different pipe diameter must be used for the second iteration. Use D = 2.24 m. Hence, e/D = 0.00016 and NR = 2.6 * 106. From Figure 3.8, we obtain f = 0.0136. From Equation (a), we now have D = a

0.0136 1/5 b = 2.25 m 0.000237

The value on the right-hand side is considered close enough to that on the previous iteration, and the pipe diameter of 2.25 m is selected.

Note: For all of these pipeline problems, simultaneous equations can be established and solved using computer algebra software (e.g., Mathcad, Maple, or Mathematica). The equations that are required include the energy equation, the Darcy–Weisbach equation for friction head loss, minor loss equations, the continuity equation, Reynolds number, and Colebrook's friction factor relationship or the Swamee–Jain equation. However, the relationships are highly nonlinear, and a good initial estimate may be required to avoid numerical instability. 4.2 Negative Pressure Scenarios (Pipelines and Pumps) Pipelines used to transport water from one location to another over a long distance usually follow the natural contours of the land. Occasionally, a section of the pipeline may be raised to an elevation that is above the local hydraulic grade line (HGL), as shown in Figure 4.2. As discussed in Chapter 3, the vertical distance measured between the energy grade line (EGL) and the hydraulic grade line at any location along the pipeline is the velocity head V2/2g at that location. The vertical distance measured between the hydraulic grade line and the pipeline is the local pressure head (P/γ). In the vicinity of the pipeline summit (S in Figure 4.2), the pressure head may take on a negative value. V2 2g

EGL HGL

–Ps g

P g

P H

Datum line Figure 4.2 Elevated section in a pipeline

Sec. 4.2

Negative Pressure Scenarios (Pipelines and Pumps)

Negative pressure scenarios in pipelines are not difficult to understand in light of the principles of hydraulic head. Recall that the total head H = a

V2 P + hb + γ 2g

must equal the vertical distance between the datum and the energy grade line at any location in the pipeline. At the summit, for example, the elevation head (hs) is the vertical distance from the datum to the centerline of the pipe. The velocity head (V 2/2g) is also a fixed positive value. The sum V 2/2g + hs may become larger than Hs, the total head at the summit. If this occurs, the pressure head (Ps/γ) must take on a negative value. Negative gauge pressure (in reference to atmospheric pressure as zero, Patm = 0) exists in a pipeline wherever the pipeline is raised above the hydraulic grade line (between P and Q in Figure 4.2). This negative pressure reaches a maximum value at the summit, -(Ps/γ). Water flow from S to R must flow against the pressure gradient. In other words, it flows from a point of lower pressure toward a point of higher pressure. This is possible because water always flows toward lower-energy locations, and in the closed conduit the elevation head decrease more than compensates for the pressure head increase. For example, if a unit weight of water flowing from S to R experiences a pressure increase equal to 3 m of water column, then the elevation of S must be at least 3 m higher than the elevation of R. In fact, the difference in elevation between S and R must equal 3 m plus the loss of head between S and R. Or, more generally, the elevation difference between any two points 1 and 2 in a pipeline is ∆h1 - 2 = a

P2 P1 b + hf γ γ

(4.1)

For design purposes, it is important to maintain pressure at all points in a pipeline above the vapor pressure of water. As we discussed in Chapter 1, the vapor pressure (gauge) of water is approximately equal to a negative water column height of 10 m at 20°C. When the pressure in a pipe drops below this value, water will be vaporized locally to form vapor pockets that separate the water in the pipe. These vapor pockets collapse in regions of higher pressure downstream. The action of vapor collapse is very violent, causing vibrations and sound that can greatly damage the pipeline. The entire process is often referred to as cavitation. It is important to note that the vaporization (boiling) of water in pipelines occurs even at normal atmospheric temperatures if the pressure drop is of sufficient magnitude. Theoretically, a pipeline may be designed to allow the pressure to fall to the level of vapor pressure at certain sections in the pipeline. For example, the vapor pressure of water at 20°C is 2,335 N/m2 (Table 1.1). This is an absolute pressure; gauge pressure is found by subtracting atmospheric pressure (1.014 * 105 N/m2) or, in terms of pressure head, (Pvapor - Patm)/γ = (2,335 - 101,400)/9,790 = -10.1 m. However, in practice it is generally not acceptable to allow the pressure to fall to vapor-pressure levels. Water usually contains dissolved gases that will vaporize well before the vapor-pressure point is reached. Such gases return to the liquid phase very slowly. They usually move with water in the form of large bubbles that reduce the effective flow area and thus disrupt the flow. For this reason, negative pressure should not be allowed to exceed approximately two-thirds of standard atmosphericpressure head (10.3 m of H2O) in any section of the pipeline (i.e., about -7.0 m or -23 ft of H2O, gauge pressure).

100

Pipelines and Pipe Networks

Chap. 4

Example 4.4 A PVC pipeline, 40 cm in diameter and 2,000 m long, carries water at 10°C between two reservoirs as shown in Figure 4.2. The two reservoirs have a water surface elevation difference of 30 m. At midlength, the pipeline must be raised to carry the water over a small hill. Determine the maximum height the pipeline may be raised (at the summit, S) above the lower reservoir's water surface elevation in order to prevent cavitation. Solution Pipeline problems are most often solved by using the energy equation. However, two points must be chosen to balance energy. In choosing the points, you generally identify where information is needed and where the most information is available in terms of energy. In this problem, the elevation of the summit is requested, and the most information is available at the reservoir water surfaces (i.e., where velocity head and pressure heads are negligible). Because the elevation of the summit is requested with respect to the lower reservoir (B), start by balancing energy between these two points. This yields Ps PB V 2B V2 hs + + = hB + + + hL γ γ 2g 2g Setting the datum at the water surface of reservoir B, all terms on the right-hand side of the equation disappear except the head loss term. The head loss includes the friction loss (hf) and the exit loss (hd). Rewriting the equation and substituting known values results in hs - 7.0 m +

1,000 m V2 V2 = af + 1.0b 2g 0.4 m 2g

(1)

where an allowable vapor-pressure head of -7.0 m (gauge) is used. Note that an allowable vapor pressure head of -7.0 m is the gauge pressure at which dissolved gases begin to vaporize. It is obvious from Equation (1) that the flow rate in the pipeline, which allows us to determine the pipeline velocity, will have to be determined in order to find the allowable height of the summit, hs. In the previous section, pipeline flow rates between reservoirs were determined by balancing energy between the reservoir surfaces. In this case, the total head loss between the two reservoirs is 30 m, which includes the entrance loss (he), the friction loss in the pipeline (hf), and the discharge loss (hd). Therefore, we may write hA - hB = 30 = aKe + f

L V2 + Kd b D 2g

Assuming a square-edged entrance and complete turbulence, e/D = 0.0015 mm/400 mm = 0.00000375, basically a smooth pipe that requires NR. Thus, try f = 0.015, which is in the middle of the Moody diagram (Figure 3.8) for smooth pipes. Substituting into the energy equation yields 30 = a0.5 + (0.015)

2,000 V2 + 1b 0.4 2(9.81)

Hence, V = 2.77 m/s. We must check the assumed friction factor (f) using this velocity. The relative roughness remains the same, and the Reynolds number is NR =

(2.77)(0.4) VD = = 8.46 * 105 -6 v 1.31 * 10

Sec. 4.2

Negative Pressure Scenarios (Pipelines and Pumps)

101

The Moody diagram now yields f = 0.012, which is different from the assumed value. The energy relationship is rewritten as follows: 30 = a0.5 + (0.012)

2,000 V2 + 1b 0.4 2(9.81)

from which V = 3.09 m/s. Now, NR = 9.44 * 105, and the Moody chart gives f = 0.012, which matches our previous value. Rearranging Equation (1) and substituting for V and f yields the maximum height (hs) the pipeline may be raised (at the summit) above the surface elevation of the lower reservoir to prevent cavitation: hs = 7.0 + a0.012

1,000 V2 V2 + 1.0b 0.4 2g 2g

= 7.0 + c (0.012)

1,000 (3.09)2 d = 21.6 m 0.4 2(9.81)

Therefore, the maximum height the pipeline may be raised above the lower reservoir is 21.6 m to prevent cavitation.

Pumps may be needed in a pipeline to lift water from a lower elevation or simply to boost the rate of flow. Pumps add energy to water in pipelines by increasing the pressure head. The details of pump design and selection will be discussed in Chapter 5, but the analysis of pressure head and energy (in the form of head) provided by a pump to a pipeline system are discussed here. The computations for pump installation in a pipeline are usually carried out by separating the pipeline system into two sequential parts: the suction side and the discharge side. Figure 4.3 shows a typical pump installation in a pipeline and the associated EGL and HGL. The head provided by the pump to the system (HP) is represented by the vertical distance between the low point (L) and the high point (M) on the energy grade line (at the inlet and outlet of the pump). The elevation of M represents the total head at the outlet of the pump that delivers the water to the receiving reservoir (R). An energy equation can be written between the supply reservoir (S) and the receiving reservoir as HS + HP = HR + hL

(4.2)

where HS and HR are the position heads in the supply and receiving reservoirs, respectively (i.e., generally the water surface elevations). HP is the head added by the pump, and hL is the total head loss in the system. Additional information is evident from the EGL and HGL in Figure 4.3. The suction side of the system from the supply reservoir (section 1–1) to the inlet of the pump (section 2–2) is subjected to negative pressure, whereas the discharge side from the outlet of the pump (section 3–3) to the receiving reservoir (section 4–4) is subjected to positive pressure. The change from negative pressure to positive pressure is the result of the pump adding energy to the water, mainly in the form of pressure head. Also note that the EGL drops significantly and quickly coming out of the supply reservoir. This is because of the friction loss in the vertical portion of the suction pipe, the entrance loss, the strainer loss, and the 90° bend. Also note that the 135° bends in the discharge line are assumed to have negligible losses as noted by the absence of discontinuities in the EGL. It is left to the reader to comment on whether the slope of the EGL in the discharge line should be uniform based on the fact that a portion of the pipe is not horizontal.

102

Pipelines and Pipe Networks

Chap. 4

V2 2g M

EGL HGL

4 HR R

4 HP

3 P

hp 2

HS S

3 L

1 Figure 4.3

Energy grade line and hydraulic grade line of a pumping station

Example 4.5 A pump is necessary to lift water from the clear well (reservoir) at a water-treatment plant to a storage tower 50-ft high and some distance away. A flow rate of 15 cubic feet per second (cfs) is required (68°F). The 15-in. pipeline (e/D = 0.00008) between the two reservoirs is 1,500-ft long and contains minor losses that amount to 15 times the velocity head. Determine the pressure head required from the pump. Also determine the pressure head on the suction side of the pump if it is 10 ft above the water surface of the clear well and 100 ft down the pipeline. Solution From Equation 4.2, Hp = HR - HS + hL = 50 ft + hL where hL = af

L V2 + 15b D 2g

For the flow rate required, the velocity and Reynolds number are Q (15) = = 12.2 ft/s A (π/4)(1.25)2 12.2(1.25) VD NR = = = 1.41 * 106 v 1.08 * 10-5 V =

And with e/D = 0.00008, the Moody chart gives f = 0.013. The energy loss in the pipeline is hL = a(0.013)

(12.2)2 1,500 + 15b = 70.7 ft 1.25 2(32.2)

Sec. 4.3

Branching Pipe Systems

103

The minimum pressure head the pump must provide is Hp = 70.7 + 50 = 120.7 ft (Note: A certain amount of pressure head must be added to compensate for the energy loss that occurs at the pump when it is in operation.) Balancing energy between the clear well and the suction side of the pump (sections 1–1 and 2–2 in Figure 4.3) gives Hs = h2 +

P2 V 22 + + hL γ 2g

where HS = 0 (datum) and the energy loss is hL = aKe + f

L V2 b D 2g

Assuming Ke = 4.0 (entrance loss with strainer), we have hL = a4.0 + 0.013

100 (12.2)2 b = 11.6 ft 1.25 2(32.2)

Therefore, the pressure head on the suction side of the pump is

P2 (12.2)2 = 0.0 - 10 - 11.6 = -23.9 ft γ 2(32.2) This is above the vapor pressure of water of 0.344 lb/in.2 (at 68°F, found in front jacket of book and Table 1.1) which equates to a gauge pressure of - 14.4 lb/in.2 (Pvapor - Patm = 0.344 lb/in.2 - 14.7 lb/in.2) and converts to a pressure head of - 33.3 ft of water [(Pvapor - Patm)/γ = ( - 14.4 lb/in.2)(144 in.2/ft2)/ 62.3 lb/ft3]. Therefore, the water in the pipeline will not vaporize. However, it is on the threshold of what is allowed in practice ( -7 m = -23.0 ft) based on other dissolved gases in the water that could vaporize.

4.3 Branching Pipe Systems Branching pipe systems are the result of more than two pipelines converging at a junction. These systems must simultaneously satisfy two basic conditions. First, the total amount of water brought by pipes to a junction must always be equal to that carried away from the junction by the other pipes (conservation of mass). Second, all pipes that meet at the junction must share the same energy level at the junction (conservation of energy). The hydraulics of branching pipe systems at a junction can be best demonstrated by the classical three-reservoir problem in which three reservoirs of different elevation are connected to a common junction J, as shown in Figure 4.4. Given the lengths, diameters, and material of all pipes involved, as well as the water elevation in each of the three reservoirs, the discharges to or from each reservoir (Q1, Q2, and Q3) can be determined. If an open-ended vertical tube (piezometer) is installed at the junction, the water elevation in the tube will rise to the elevation HJ. The elevation of HJ is the total energy level (position head plus pressure head) if the velocity head is assumed to be negligible at the junction where all of the flows are coming together. Therefore, the difference in elevation between the water surface at reservoir A and elevation HJ represents the friction losses in transporting water from A to J, as indicated by hf1. (Minor losses are neglected; friction losses often dominate in large water-conveyance

104

Pipelines and Pipe Networks

Chap. 4

A hf 1

B hf 2 hf 3

Q2 J Q3

Figure 4.4

Branching pipes connecting three reservoirs

systems.) Likewise, the elevation difference, hf2, between reservoir B and HJ represents the friction losses in transporting water from B to J; hf3 represents the friction losses in transporting water from J to C. Because the mass of water brought to the junction must equal the mass of water taken from the junction, we may simply write Q3 = Q1 + Q2

(4.3)

or ΣQ = 0 at the junction (assuming that density remains constant). These types of problems can be solved iteratively. Not knowing the discharge in each pipe, we may first assume a total energy elevation, HJ, at the junction. This assumed elevation establishes the friction head losses hf1, hf2, and hf3 for each of the three pipes. From this set of head losses and the given pipe diameters, lengths, and materials, friction loss equations yield a set of values for the discharges Q1, Q2, and Q3. If the assumed total energy elevation HJ is correct, then the computed Q's should satisfy the mass balance condition stated above—that is, ΣQ = Q1 + Q2 - Q3 = 0

(4.4)

Otherwise, a new elevation HJ is assumed for the second iteration. The computation of another set of Q's is performed until the above condition is satisfied. The correct values of discharge in each pipe are thus obtained. Note that if the assumed HJ is higher than the elevation in reservoir B, then Q1 should be equal to Q2 + Q3; if it is lower, then Q1 + Q2 = Q3. The error in the trial Q's indicates the direction in which the assumed energy elevation HJ should be set for the next trial. Numerical techniques such as the bisection method can be programmed and applied to the error to quickly obtain the correct result. To understand the concept, it may be helpful to plot the computed trial values of HJ against ΣQ. The resulting flow residual (ΣQ) may be positive or negative for each trial. However, with values obtained from three trials, a curve may be plotted as shown in Example 4.6. The correct discharge is indicated by the intersection of the curve with the vertical axis. The computation procedure is demonstrated in the following example.

Sec. 4.3

Branching Pipe Systems

105

Example 4.6 In Figure 4.5 (a), the three reservoirs A, B, and C are connected by pipes to a common junction J. Pipe AJ is 1,000 m long and 30 cm in diameter; pipe BJ is 4,000 m long and 50 cm in diameter; and pipe CJ is 2,000 m long and 40 cm in diameter. The pipes are made of concrete for which e = 0.6 mm may be assumed. Determine the discharge in each pipe if the water temperature is 20°C (v = 1.003 * 10-6 m2/s). Neglect the minor losses. Elevation 120 m

Elevation 100 m B

A 30 cm ϕ of 1000 m

50 cm ϕ of 4000 m Q2

J Elevation 40 m

Q1 40 cm ϕ of 2000 m

Elevation 80 m C

Figure 4.5 (a) Three-reservoir problem

Solution Let subscripts 1, 2, and 3 represent, respectively, pipes AJ, BJ, and CJ. At this point, we do not know the flow direction in BC since the total energy head, HJ, at J is unknown. As a first trial, we assume HJ = 100 m (same as the total energy head in reservoir B). Therefore Q2 = 0, and Q1 and Q3 may be calculated as follows. For reservoir A, hf1 = 120 - 100 = f1 a

Assuming fully developed turbulent flow and with

L 1 V 21 V 21 1000 b = f1 a b D1 2g 0.3 2(9.81)

e1 0.6 = = 0.002 D1 300 we obtain f1 = 0.024 from the Moody diagram, which yields V1 = 2.21 m/s. Hence, NR = V1D1/ v = 6.61 * 105, and returning to the Moody diagram f1 = 0.024 is verified. Therefore, π Q1 = V1A1 = 2.21 (0.3)2 = 0.156 m3/s 4 Similarly for reservoir C, hf3 = 100 - 80 = f3 a

e3 0.6 = = 0.0015 D3 400

L 3 V 23 V 23 2000 b = f3 a b D3 2g 0.4 2(9.81)

and from the Moody diagram f3 = 0.022, which yields V3 = 1.89 m/s and NR = 7.54 * 105. Again, no further approximations are needed for f3, so π Q3 = V3A3 = 1.89 (0.4)2 = 0.238 m3/s 4

106

Pipelines and Pipe Networks

Chap. 4

Hence, the summation of flows into the joint J with Q 2 = 0 is ΣQ = Q1 - Q3 = 0.156 - 0.238 = -0.082 m3/s The continuity equation at the junction is not satisfied, and another trial is needed. However, if we carefully review the results we notice that HJ = 100 m resulted in a Q3 value larger than Q1. To achieve a balance we should increase Q1 and decrease Q3. This is possible if we lower HJ below 100 m, which also implies that the flow in pipe BJ is from B to J. At the second trial we assume HJ = 90 m. Similar computations are repeated to obtain Q1 = 0.193 m3/s Q2 = 0.212 m3/s Q3 = 0.168 m3/s Hence, ΣQ = (Q1 + Q2) - Q3 = 0.237 m3/s Now inflow to the junction exceeds outflow. Therefore, the computations are repeated again for HJ = 95 m to obtain Q1 = 0.176 m3/s Q2 = 0.149 m3/s Q3 = 0.205 m3/s Hence, ΣQ = (Q1 + Q2) - Q3 = 0.120 m3/s With the values computed above, a small graph may be constructed with HJ plotted against the corresponding values of ΣQ, as shown in Figure 4.5 (b). The curve intersects the ΣQ = 0 line at HJ = 99.0 m, which is used to compute the final set of discharges. We obtain

HJ (m) 100

HJ = 99 m

–0.08

Figure 4.5 (b)

0.120 ΣQ at junction (m3/s)

0.237

Three-reservoir problem solution graph

Sec. 4.3

Branching Pipe Systems

107

Q1 = 0.161 m3/s Q2 = 0.065 m3/s Q3 = 0.231 m3/s Hence, the condition ΣQ = (Q1 + Q2) - Q3 = 0 is satisfied within (0.005 m3/s). Note: The energy equation for each pipe, the Darcy–Weisbach equation for friction head loss (or, alternatively, the Hazen–Williams equation or the Manning equation), the Reynolds number expression, the Von Kármán equation for the friction factor assuming complete turbulence, Colebrook's implicit friction factor equation when NR is available (or, alternatively, the explicit Swamee–Jain equation), and mass balance at the junction can be solved simultaneously by a computer algebra software system (e.g., Mathcad, Maple, or Mathematica) and should yield the same result. Likewise, a simple spreadsheet program can be formulated to quickly perform the iterations. (See Problem 4.3.1.)

Example 4.7 A horizontal, galvanized iron pipe system consists of a 10-in-diameter, 12-ft-long main pipe between the two junctions 1 and 2, as depicted in Figure 4.6. A gate valve is installed at the downstream end immediately before junction 2. The branch pipe has a 6-in. diameter and is 20 ft long. It consists of two 90° elbows (R/D = 2.0) and a globe valve. The system carries a total discharge of 10 cfs of water at 40°F. Determine the discharge in each of the pipes when the valves are both fully opened. (Note that, due to the different local losses along the two branches, the equivalent pipe formulas derived in Chapter 3 cannot be used in this example.) Gate valve

b Globe valve Figure 4.6

Solution The cross-sectional area of pipes a and b are, respectively, Aa = Mass balance requires that

π 10 2 a b = 0.545 ft2 4 12

Ab =

π 6 2 a b = 0.196 ft2 4 12

10 cfs = AaVa + AbVb = 0.545Va + 0.196Vb

(a)

where Va and Vb are the velocities in pipes a and b, respectively. The head loss between junctions 1 and 2 along the main pipe is V 2a L a V 2a ha = fa a b + 0.15 Da 2g 2g

108

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Chap. 4

The second term accounts for the fully open gate valve (Table 3.6). The head loss between junctions 1 and 2 along the branch pipe is hb = fb a

V 2b V 2b L b V 2b b + 2(0.19) + 10 Db 2g 2g 2g

The second term accounts for the elbow losses; the third term accounts for the fully open globe valve (Table 3.6). Since the head losses through both pipes must be the same, ha = hb, we have

Jfa a

V 2a V 2b 12 20 b + 0.15 R = Jfb a b + 0.38 + 10 R 0.833 2g 0.5 2g (14.4fa + 0.15)V 2a = (40fb + 10.4)V 2b

(b)

Equations (a) and (b) can be solved simultaneously for Va and Vb once friction factors have been established. For galvanized iron, Table 3.1 gives a

and

e 0.0005 b = = 0.00060 D a 0.833

e 0.0005 b = = 0.0010 D b 0.50

Assuming complete turbulence, the Moody diagram yields the following f values: fa = 0.0175

and

fb = 0.020

as a first approximation. Substituting the above values into Equation (b), we have 314.4(0.0175) + 0.154V 2a = 340(0.020) + 10.44V 2b 0.402V 2a = 11.2V 2b Va = Substituting Va into Equation (a), we find

11.2 V = 5.28Vb A 0.402 b

10 = 0.545(5.28Vb) + 0.196Vb = 3.07Vb Vb =

10 = 3.26 ft/s 3.07

Hence, Va = 5.28, Vb = 17.2 ft/s. The corresponding Reynolds numbers are calculated to verify the assumed friction factors. For pipe a, NRa =

VaDa 17.2(0.833) = = 8.48 * 105 v 1.69 * 10-5

The Moody chart gives f = 0.0175, which matches the original assumption. For pipe b, NRb =

VbDb 3.26(0.5) = = 9.65 * 104 -5 v 1.69 * 10

Sec. 4.3

Branching Pipe Systems

109

The Moody chart gives fb = 0.0225 ≠ 0.020. Equations (a) and (b) are solved again using the new value of fb : 314.4(0.0175) + 0.154V 2a = 340(0.0225) + 10.44V 2b Va = 5.30Vb

Substituting Va into Equation (a), 10 = 0.545(5.30Vb) + 0.196Vb Vb = 3.24 ft/s

and

Va = 5.30 (3.24) = 17.2 ft/s

Therefore, the discharges are Qa = AaVa = 0.545(17.2) = 9.37 cfs and Qb = AbVb = 0.196(3.24) = 0.635 cfs

Branching pipes connecting more than three reservoirs to a junction (Figure 4.7) are not common in hydraulic engineering. However, problems of multiple (more than three) reservoirs can be solved by following the same principles. Assume that the total energy head is HJ at junction J. The differences in water surface elevations between reservoirs A and B, A and C, and A and D are, respectively, ∆H1, ∆H2, and ∆H3. The head losses between reservoirs A, B, C, and D and the junction are, respectively, hf1, hf2, hf3, and hf4, as shown in Figure 4.7. A set of four independent equations may be written in the following general form for the four reservoirs: ∆H1 = hf1 - hf2

(4.5)

∆H2 = hf1 + hf3

(4.6)

∆H3 = hf1 + hf4

(4.7)

ΣQj = 0

(4.8)

∆H1 B

hf 2

∆H3

A hf 1

∆H2

HJ Q1

hf 4 J

Q4 D

Figure 4.7

hf 3

Multiple reservoirs connected at a junction

110

Pipelines and Pipe Networks

Chap. 4

For each of the pipe branches, the head loss may be expressed in the form of the Darcy–Weisbach equation,* Equation 3.16, as follows: hf1 = f1 a hf2 = f2 a hf3 = f3 a hf4 = f4 a

L 1 V 21 L 1 Q12 b = f1 a b D1 2g D1 2gA12 L 2 V 22 L 2 Q22 b = f2 a b D2 2g D2 2gA22 L 3 V 23 L 3 Q32 b = f3 a b D3 2g D3 2gA32 L 4 V 24 L 4 Q42 b = f4 a b D4 2g D4 2gA42

Substituting these relationships into Equations 4.5, 4.6, and 4.7 yields ∆H1 = ∆H2 = ∆H3 = and

L 1 Q21 L 2 Q22 1 a f1 f b 2 2g D1 A21 D2 A22 L 3 Q32 L 1 Q21 1 a f1 + f b 3 2g D1 A21 D3 A32 L 1 Q21 L 4 Q42 1 a f1 + f b 4 2g D1 A21 D4 A42 ΣQj = 0

(4.9)

(4.10)

(4.11)

(4.8)

Equations 4.8 through 4.11 can then be solved simultaneously for the four unknowns: Q1, Q2, Q3, and Q4. These values are the flow rates for each of the pipe branches shown. This procedure could be applied to any number of reservoirs connected to a common junction. 4.4 Pipe Networks Water-supply distribution systems in municipal districts are usually composed of a large number of pipes interconnected to form loops and branches. Although the calculations of flow in a pipe network involve a large number of pipes and may become tedious, the solution scenario is based on the same principles that govern flow in pipelines and branching pipes previously discussed. In general, a series of simultaneous equations can be written for the network. These equations are written to satisfy the following conditions: 1. At any junction, ΣQ = 0 based on the conservation of mass (junction equation). 2. Between any two junctions, the total head loss is independent of the path taken based on the conservation of energy (loop equation). *Empirical equations for friction head loss, such as the Hazen–Williams equation and the Manning equation, may be applied using analogous expressions.

Sec. 4.4

Pipe Networks

111

Q1 a

Loop A c

d Q3

i h

Figure 4.8

Schematic of a pipe network

Depending on the number of unknowns, it is usually possible to set up a sufficient number of independent equations to solve the problem. A typical problem would be to determine the flow distribution in each pipe of a network (Figure 4.8) when the inflows (Q1 and Q2) and outflows (Q3 and Q4) to the network are known. These equations may then be solved simultaneously. For the simple network shown in Figure 4.8, a set of 12 independent equations (eight junction equations and four loop equations) is needed to solve for the flow distribution in the 12 pipes. As a general rule, a network with m loops and n junctions provides a total of m + (n - 1) independent equations. For more complex networks, the number of equations increases proportionally. At a certain point it will be obvious that algebraic solution of the network equations becomes impractical. For most engineering applications, pipe-network solutions are obtained using computer software designed specifically for the task. Two algorithms commonly used for pipe network analysis are outlined below. 4.4.1 The Hardy–Cross Method The Hardy–Cross method utilizes successive flow approximations based on the two conditions stated previously for each junction and loop in the pipe network. In loop A shown in Figure 4.8, the arrowheads indicate the presumed flow direction. This loop must satisfy the conditions of mass and energy balance. 1. At each junction b, c, d, and e, the total inflow must equal the total outflow. 2. The loss of head from flow in the counterclockwise direction along pipes bc and cd must equal the loss of head from flow in the clockwise direction along pipes be and ed. To initiate the process, a distribution of flows in each pipe is estimated such that the total inflow equals the total outflow at each junction throughout the pipe network. For a network with n junctions, (n - 1) junction equations can be established to determine the flow rates in the system. Once flows are established for the first (n - 1) junctions, the flows to and from the last junction are fixed and thus dependent. The estimated flows, along with the pipe diameters, lengths, friction coefficients, and other network data (e.g., connectivity, junction elevations) are required for the Hardy–Cross method. The loss of head resulting from the estimated flow rates in all pipes is then computed for the network.

112

Pipelines and Pipe Networks

Chap. 4

The possibility of the assumed flow distribution satisfying the m loop equations is small. Invariably, the estimated pipe flows need to be adjusted until head losses in the clockwise direction equal head losses in the counterclockwise direction within each loop. The successive computational procedure uses the loop equations, one at a time, to correct the assumed pipe flows and thus equalizes the head losses in the loop. Because the flow balance at each junction must be maintained, a given correction to flow in any pipe (e.g., pipe be) in the clockwise direction requires a corresponding flow correction of the same magnitude in the clockwise direction in the other pipes (pipes bc, cd, and ed). The successive flow corrections to equalize the head loss are discussed next. Knowing the diameter, length, and roughness of a pipe, we see that the head loss in the pipe is a function of the flow rate, Q. Applying the Darcy–Weisbach equation (3.16), we may write hf = f a

L V2 L 1 b = Jf a b R Q2 = KQ2 D 2g D 2gA2

(4.12)

In any network loop, such as a loop A, the total head loss in the clockwise direction (hereafter designated with the subscript c) is the sum of the head losses in all pipes that carry flow in the clockwise direction around the loop: Σhfc = ΣKcQ2c

(4.13)

Similarly, the loss of head in the counterclockwise direction (subscript cc) is Σhfcc = ΣKccQ2cc

(4.14)

Using the assumed flow rates, Q's, it is not expected that these two values will be equal during the first trial, as mentioned previously. The difference, ΣKcQ2c - ΣKccQ2cc is the closure error of the first trial. We need to determine a flow correction ∆Q that, when subtracted from Qc and added to Qcc, will equalize the two head losses. Thus, the correction ∆Q must satisfy the following equation: ΣKc(Qc - ∆Q)2 = ΣKcc(Qcc + ∆Q)2 Expanding the terms in the parentheses on both sides of the equation, we have ΣKc(Q2c - 2Qc ∆Q + ∆Q2) = ΣKcc(Q2cc + 2Qcc ∆Q + ∆Q2) Assuming that the correction term is small compared with both Qc and Qcc, we may simplify the previous expression by dropping the last term on each side of the equation and write ΣKc(Q2c - 2Qc ∆Q) = ΣKcc(Q2cc + 2Qcc ∆Q) From this relationship, we may solve for ∆Q: ΣKcQ2c - ΣKccQ2cc ∆Q = 2(ΣKcQc + ΣKccQcc)

(4.15)

Sec. 4.4

Pipe Networks

113

If we take Equation 4.12 and divide it by Q on both sides, we have KQ =

(4.16)

Equations 4.13, 4.14, and 4.16 can be substituted into Equation 4.15 to obtain ∆Q =

(Σh fc - Σh fcc) h fc h fcc 2a Σ + Σ b Qc Qcc

(4.17a)

Equation 4.17(a) is appropriate when the Manning equation (3.28) is used to determine friction head losses instead of the Darcy–Weisbach equation. However, when the Hazen–Williams Equation (3.27) is used, the equation should be ∆Q =

Σh fc - Σh fcc 1.85a Σ

h fc Qc

+ Σ

h fcc Qcc

(4.17b) b

Once the error magnitude is established, a second iteration uses this correction to determine a new flow distribution. The computation results from the second iteration are expected to give a closer match of the two head losses along the c and cc directions in loop A. Note that pipes bc, cd, and ed in loop A are each common to two loops and, therefore, need to be subjected to double corrections, one from each loop. The successive computational procedure is repeated until each loop in the entire network is balanced (mass and energy) and the corrections become negligibly small. The Hardy–Cross method can best be described using an example problem. Even though commercial computer software is available to solve these laborious calculations, the student should go through the procedure a few times with small networks. By becoming familiar with the algorithms, a more judicious and appreciative use of the computer software can be expected. Example 4.8 A water-supply distribution system for an industrial park is schematically shown in Figure 4.9 (a). The demands on the system are currently at junctions C, G, and F with flow rates given in liters per second. Water enters the system at junction A from a water storage tank on a hill. The water surface elevation in the tank is 50 m above the elevation of point A in the industrial park. All the junctions have the same elevation as point A. All pipes are aged ductile iron (e = 0.26 mm) with lengths and diameters provided in the table below. Calculate the flow rate in each pipe. Also determine if the pressure at junction F will be high enough to satisfy the customer there. The required pressure is 185 kPa. A table of pipe and system geometry is a convenient way to organize the available information and make some preliminary calculations. The table below has been set up for that purpose. The first column identifies all of the pipes in the network. Column 2 contains flow rates for each pipe, which were estimated to initiate the Hardy–Cross algorithm. These estimated flow rates and directions are shown on the system schematic in Figure 4.9(a). Note that mass balance was maintained at each junction. Friction factors (column 6) are found assuming complete turbulence and read from the Moody diagram using e/D or, alternatively, from Equation 3.23. The "K" coefficient (column 7) is used later in the procedure to obtain the head loss in each pipe according to Equation 4.12.

114

Pipelines and Pipe Networks

Chap. 4

Q = 300 A

200

Q = 50 H

0 D

120 100

Q = 100

E 100

100 Q = 150 (a)

Q = 300 A

205

B 125

8 E

2 33

Q = 100

Q = 50 H F

87 Q = 150 (b) Figure 4.9

Pipe

Flow (m3/s)

Length (m)

Diameter (m)

e/D

K (s2/m5)

AB AD BC BG GH CH DE GE EF HF

0.20 0.10 0.08 0.12 0.02 0.03 0.10 0.00 0.10 0.05

300 250 350 125 350 125 300 125 350 125

0.30 0.25 0.20 0.20 0.20 0.20 0.20 0.15 0.20 0.15

0.00087 0.00104 0.00130 0.00130 0.00130 0.00130 0.00130 0.00173 0.00130 0.00173

0.019 0.020 0.021 0.021 0.021 0.021 0.021 0.022 0.021 0.022

194 423 1,900 678 1,900 678 1,630 2,990 1,900 2,990

Solution The Hardy–Cross method utilizes a relaxation technique (method of successive approximations). Using the estimated flow rates, head losses are found in each pipe by using Equation 4.12, one loop at a time. Equation 4.17 (a) is then used to determine a flow correction and thus improve the flow estimate. The same procedure is applied to all the remaining loops, and then the cycle repeats itself. The process is ended when the flow corrections become acceptably small. At this point, conservation of mass is satisfied for each junction, and the loss of head around each loop is the same for counterclockwise and clockwise flow (conservation of energy). We will proceed through the calculations by using a series of tables; explanations will follow each table. Having said all that, let us commence with loop 1.

Sec. 4.4

Pipe Networks

115

Loop

Pipe

Q (m3/s)

K (s2/m5)

hf (m)

hf /Q (s/m2)

New Q (m3/s)

AB BG GE AD DE

0.200 0.120 0.000 (0.100) (0.100)

194 678 2,990 423 1,630

7.76 9.76 0.00 (4.23) (16.3)

38.8 81.3 0.0 (42.3) (163.0)

0.205 0.125 0.005 (0.095) (0.095)

The flows listed in column 3 are the original estimates. Flows in loop 1 that are counterclockwise are placed in parentheses. Head losses are computed from: hf = KQ2 The flow correction is found using Equation 4.17(a). ∆Q =

Σhfc - Σhfcc 2 3 Σ(hfc/Qc) + Σ(hfcc/Qcc) 4

(7.76 + 9.76) - (4.23 + 16.3) 2 3 (38.8 + 81.3) + (42.3 + 163.0) 4

= - 0.005 m3/s

The negative sign on the flow adjustment indicates that counterclockwise head losses dominate (Σhfcc 7 Σhfc). Therefore, the flow correction of 0.005 m3/s is applied in the clockwise direction [column 7 ("New Q")]. This will help to equalize the losses in the next iteration. Now we continue with loop 2. Loop

Pipe

Q (m3/s)

K (s2/m5)

BC CH BG GH

0.080 0.030 (0.125) (0.020)

1,900 678 678 1,900

hf (m) 12.2 0.61 (10.6) (0.76)

hf /Q (s/m2)

New Q (m3/s)

152.5 20.3 (84.8) (38.0)

0.078 0.028 (0.127) (0.022)

Because pipe BG is shared by loops 1 and 2, the revised flow from the calculation in loop 1 is used here. Note that in loop 1 the flow in BG is clockwise; in loop 2, the flow is counterclockwise. The flow correction is found to be ∆Q =

Σhfc - Σhfcc 2 3 Σ(hfc/Qc) + Σ(hfcc/Qcc) 4

(12.2 + 0.61) - (10.6 + 0.76) 2 3 (152.5 + 20.3) + (84.8 + 38.0) 4

= + 0.002 m3/s

Clockwise flow dominates the loss so the correction of 0.002 m3/s is added in the counterclockwise direction. We complete the first iteration by correcting flows in loop 3. Loop

Pipe

Q (m3/s)

K (s2/m5)

hf (m)

hf /Q (s/m2)

New Q (m3/s)

GH HF GE EF

0.022 0.050 (0.005) (0.100)

1,900 2,990 2,990 1,900

0.92 7.48 (0.07) (19.0)

41.8 149.6 (14.0) (190.0)

0.035 0.063 0.008 (0.087)

∆Q =

Σhfc - Σhfcc 2 3 Σ(hfc/Qc) + Σ(hfcc/Qcc) 4

(0.92 + 7.48) - (0.07 + 19.0) 2 3 (41.8 + 149.6) + (14.0 + 190.0) 4

= -0.013 m3/s

116

Pipelines and Pipe Networks

Chap. 4

Counterclockwise head losses dominate so the flow correction is added in the clockwise direction. Note that this is a large enough correction to reverse the flow direction in GE; it will be labeled EG the next time. We now begin the second iteration with loop 1. Loop

Pipe

Q (m3/s)

K (s2/m5)

hf (m)

AB BG AD DE EG

0.205 0.127 (0.095) (0.095) (0.008)

194 678 423 1,630 2,990

8.15 10.9 (3.82) (14.7) (0.19)

∆Q =

Σhfc - Σhfcc 2[Σ(hfc/Qc) + Σ(hfcc/Qcc)]

hf /Q (s/m2)

New Q (m3/s)

39.8 85.8 (40.2) (154.7) (23.8)

0.205 0.127 (0.095) (0.095) (0.008)

(8.15 + 10.9) - (3.82 + 14.7 + 0.19) 2[(39.8 + 85.8) + (40.2 + 154.7 + 23.8)]

= + 0.000 m3/s The correction is very small ( 6 0.0005 m3/s). We continue on to loop 2. Loop

Pipe

Q (m3/s)

K (s2/m5)

hf (m)

hf /Q (s/m2)

New Q (m3/s)

BC CH BG GH

0.078 0.028 (0.127) (0.035)

1,900 678 678 1,900

11.6 0.53 (10.9) (2.33)

148.7 18.9 (85.8) (66.6)

0.080 0.030 (0.125) (0.033)

∆Q =

Σhfc - Σhfcc 2[Σ(hfc/Qc) + Σ(hfcc/Qcc)]

(11.6 + 0.53) - (10.9 + 2.33) = -0.002 m3/s 2[(148.7 + 18.9) + (85.8 + 66.6)]

Once again the correction appears to be acceptably small. Finally, we check loop 3. Loop

Pipe

Q (m3/s)

K (s2/m5)

hf (m)

GH HF EG EF

0.033 0.063 0.008 (0.087)

1,900 2,990 2,990 1,900

2.07 11.9 0.19 (14.4)

∆Q =

Σhfc - Σhfcc 2[Σ(hfc/Qc) + Σ(hfcc/Qcc)]

hf/Q (s/m2)

New Q (m3/s)

62.7 188.9 23.8 (165.5)

0.033 0.063 0.008 (0.087)

(2.07 + 11.9 + 0.19) - (14.4) 2[(62.7 + 188.9 + 23.8) + (165.5)]

= - 0.000 m3/s Because the correction is very small on all three loops, the flow rates are accepted and the process is ended. The final flows appear in Figure 4.9 (b). The following table summarizes information about the pipe system. Final head losses are determined by using final flows and Equation 4.12. Noting that all the pipes are horizontal in this example and by using Equation 3.15a the head loss is converted to a pressure drop in the last column (∆P = γhf). Strictly speaking this approach is approximate since it neglects the differences in velocity heads in the different pipes of the system.

Sec. 4.4

Pipe Networks

117

Pipe

Q (L/s)

Length (m)

Diameter (cm)

hf (m)

∆P (kPa)

AB AD BC BG GH CH DE EG EF HF

205 95 80 125 33 30 95 8 87 63

300 250 350 125 350 125 300 125 350 125

30 25 20 20 20 20 20 15 20 15

8.2 3.8 12.2 10.6 2.1 0.6 14.7 0.2 14.4 11.9

80.3 37.2 119.4 103.8 20.6 5.9 143.9 2.0 141.0 116.5

Now we determine whether or not the pressure at junction F is large enough to satisfy the customer at that location. We will use an energy balance to do this. But first, because the water surface elevation in the tank is 50 m above junction A, the pressure there is P = γh = (9790 N/m3)(50 m) = 489.5 kPa The pressure at junction F may now be determined by subtracting the pressure drops in pipes AD, DE, and EF or any alternative route from A to F. (In this case, all the junctions are at the same elevation and the variations in the velocity heads are negligible, so an energy balance only involves pressure heads and friction head losses.) Therefore, PF = PA - ∆PAD - ∆PDE - ∆PEF = 489.5 - 37.2 - 143.9 - 141.0 = 167.4 kPa Because the pressure is less than 185 kPa, the industrial customer is not likely to be satisfied. Minor losses in the system were not accounted for, so the pressure is likely to be even lower when the system is running at the full demand specified. It is left to the student to suggest modifications to the system to accommodate the customer (Problem 4.4.3). Also note that the pressure at F could have been determined by subtracting head losses from the total head at A and converting the pressure head to pressure.

The procedure outlined above for the Hardy–Cross method is valid if all the inflows entering the network are known. In practice, this occurs when there is only one source of inflow. In this case, the inflow rate is equal to the sum of the known withdrawal rates at all the junctions. However, if the flow is supplied to the network from two or more sources, as shown in Figure 4.10(a), the inflow rates entering the network will not be known a priori. Therefore, we need to add inflow path calculations to the procedure. The number of inflow paths to be considered is equal to the number of inflow sources minus one. In Figure 4.10(a), there are two reservoirs supplying flow to the network. Therefore, only one inflow path needs to be considered. This can be any path connecting the two reservoirs. For example, one can choose inflow path ABCDG in Figure 4.10(a). There are several other possibilities like ABFEDG, GDCFBA, and so on. The results will not be affected by the choice of the inflow path. Once an inflow path is selected, the path calculations are carried out in a manner similar to the loop calculations. Using the subscript p (path) to denote the flows in the same direction as the inflow path followed and cp (counterpath) to denote the flows in the opposite direction, the discharge correction ∆Q is calculated as ∆Q =

(Σhfp - Σhfcp) + Hd - Hu 2a Σ

hfp Qp

+ Σ

hfcp Qcp

(4.18a)

118

Pipelines and Pipe Networks

Chap. 4

HG HA G QC

QF Figure 4.10(a)

Pipe network with two reservoir sources

where Hu and Hd are the total heads at the beginning (upstream) point of the inflow path and at the end (downstream) point of the path. For path ABCDG in Figure 4.10(a), Hu = HA and Hd = HG. A positive value of ∆Q indicates that losses in the direction of the path are dominating. Therefore, the correction would be applied in the counterpath direction. In other words, the flows along the inflow path direction would be decreased and those in the opposite direction would be increased by ∆Q. Equation 4.18(a) can be used in conjunction with the Manning equation (3.28), as well as the Darcy–Weisbach equation (3.16). However, when the Hazen–Williams equation (3.27) is used, the equation is reformulated as ∆Q =

(Σhfp - Σhfcp) + Hd - Hu hfp hfcp 1.85a Σ + Σ b Qp Qcp

(4.18b)

Example 4.9 Consider the pipe network shown in Figure 4.10(a), which contains two reservoir sources. Suppose HA = 85 m, HG = 102 m, Qc = 0.10 m3/s, QF = 0.25 m3/s, and QE = 0.10 m3/s. The pipe and junction characteristics are tabulated below. Also tabulated are the initial estimates of the flow rates in all the pipes (cast iron; e = 0.26 mm). The flow directions are shown in Figure 4.10(a). Determine the discharge in each pipe and the pressure head at each junction. Pipe

Length (m)

Diameter (m)

e/D

K (s 2/m5)

Q (m3/s)

Junction

Elev. (m)

AB BC BF CF DC EF DE GD

300 350 350 125 300 300 125 250

0.30 0.20 0.20 0.20 0.20 0.20 0.20 0.25

0.00087 0.00130 0.00130 0.00130 0.00130 0.00130 0.00130 0.00104

0.019 0.021 0.021 0.021 0.021 0.021 0.021 0.020

194 1,900 1,900 678 1,630 1,630 678 423

0.200 0.100 0.100 0.050 0.050 0.100 0.200 0.250

A B C D E F G

48 46 43 48 44 48 60

Sec. 4.4

Pipe Networks

119

Solution Using the estimated flow rates, head losses are found in each pipe by using Equation 4.12, one loop at a time. Equation 4.17(a) is then used to determine a flow correction and thus improve the flow estimate. The same procedure is applied to all the remaining loops, and Equation 4.18(a) is applied to inflow path ABCDG connecting reservoir A to G. Then the cycle repeats itself. The process is ended when the flow corrections become acceptably small. At this point, conservation of mass is satisfied for each junction and the loss of head around each loop is the same for counterclockwise and clockwise flow (conservation of energy). We will proceed through the calculations by using a series of tables; explanations will follow each table. Having said all that, let us commence with loop 1 (L 1). Loop

Pipe

Q (m3/s)

K (s2/m5)

hf (m)

hf /Q (s/m2)

New Q (m3/s)

BC CF BF

0.100 0.050 (0.100)

1,900 678 1,900

19.00 1.70 (19.00)

190.00 33.90 (190.00)

0.098 0.048 (0.102)

Σhfc - Σhfcc 2[Σ(hfc/Qc) + Σ(hfcc/Qcc)]

(19.00 + 1.70) - (19.00) = 0.002 m3/s 2[(190.0 + 33.9) + (190.0)]

The positive sign on the flow adjustment indicates that clockwise head losses dominate (Σhfcc 6 Σhfc). Therefore, the flow correction of 0.002 m3/s is applied in the counterclockwise direction [column 7 ("New Q")]. This will help to equalize the losses in the next iteration. Now we continue with loop 2 (L 2). Loop

Pipe

Q (m3/s)

K (s2/m5)

hf (m)

hf /Q (s/m2)

New Q (m3/s)

DE EF DC CF

0.200 0.100 (0.050) (0.048)

678 1,630 1,630 678

27.12 16.30 (4.08) (1.56)

135.60 163.00 (81.50) (32.54)

0.154 0.054 (0.096) (0.094)

Because pipe CF is shared by loops 1 and 2, the revised flow from the calculation in loop 1 is used here. Note that in loop 1 the flow in CF is clockwise; in loop 2 the flow in CF is counterclockwise. The flow correction is found to be ∆Q =

Σhfc - Σhfcc [Σ(hfc/Qc) + Σ(hfcc/Qcc)]

(27.12 + 16.30) - (4.08 + 1.56) [(135.60 + 163.00) + (81.50 + 32.54)]

= + 0.046 m3/s Clockwise flow dominates the loss so the correction of 0.046 m3/s should be added in the counterclockwise direction. We complete the first iteration by correcting flows along inflow path ABCDG. Note that the flows in DC and GD are in a direction opposite to the direction of the inflow path chosen and are placed in parentheses.

120

Pipelines and Pipe Networks

Chap. 4

Inflow Path

Pipe

Q (m3/s)

K (s2/m5)

hf (m)

hf /Q (s/m2)

New Q (m3/s)

ABCDG

AB BC DC GD

0.200 0.098 (0.096) (0.250)

194 1,900 1,630 423

7.76 18.25 (15.02) (26.44)

38.80 186.20 (156.48) (105.75)

0.198 0.096 (0.098) (0.252)

By using Equation 4.18(a), ∆Q =

Σhfp - Σhfcp + HG - HA

2[Σ(hfp/Qp) + Σ(hfcp/Qcp)]

(7.76 + 18.25) - (15.02 + 26.44) + 102 - 85 2[(38.80 + 186.20) + (156.48 + 105.75)]

= 0.002 m3/s We find that head losses dominate along the designated inflow path, so the flow correction is added in the counterpath direction. The first iteration cycle is now complete. We will proceed to the second iteration cycle using the most recent flow rate calculated for each pipe. The calculations for loops 1 and 2 and the path ABCDG for the second iteration cycle are as follows: Loop

Pipe

Q (m3/s)

K (s2/m5)

hf (m)

hf /Q (s/m2)

New Q (m3/s)

BC CF BF

0.096 0.094 (0.102)

1,900 678 1,900

17.51 5.99 (19.77)

182.40 63.73 (193.80)

0.092 0.090 (0.106)

∆Q =

Σhfc - Σhfcc 2[Σ(hfc/Qc) + Σ(hfcc/Qcc)]

(17.51 + 5.99) - (19.77) = 0.004 m3/s 2[(182.4 + 63.73) + (193.8)]

Loop

Pipe

Q (m3/s)

K (s2/m5)

hf (m)

DE EF DC CF

0.154 0.054 (0.098) (0.090)

678 1,630 1,630 678

16.08 4.75 (15.65) (5.49)

∆Q =

Σhfc - Σhfcc [Σ(hfc/Qc) + Σ(hfcc/Qcc)]

hf /Q (s/m2) 104.41 88.02 (159.74) (61.02)

New Q (m3/s) 0.154 0.054 (0.098) (0.090)

(16.08 + 4.75) - (15.65 + 5.49) [(104.41 + 88.02) + (159.74 + 61.02)]

= 0.000 m3/s Inflow Path

Pipe

Q (m3/s)

K (s2/m5)

hf (m)

hf /Q (s/m2)

New Q (m3/s)

ABCDG

AB BC DC GD

0.198 0.092 (0.098) (0.252)

194 1,900 1,630 423

7.61 16.08 (15.65) (26.86)

38.41 174.80 (159.74) (106.60)

0.200 0.094 (0.096) (0.250)

∆Q =

Σhfp - Σhfcp + HG - HA 2 3 Σ(hfp/Qp) + Σ(hfcp/Qcp) 4

(7.61 + 16.08) - (15.65 + 26.86) + 102 - 85 2 3 (38.41 + 174.80) + (159.74 + 106.60) 4

= - 0.002 m3/s

Sec. 4.4

Pipe Networks

121

Similar calculations are carried out for one more iteration cycle, at which point all of the corrections become negligible. The final results are tabulated below and shown in Figure 4.10(b). Also tabulated are the total head and the pressure head values calculated for all of the nodes. Once the pipe discharges are found, the energy equation is used to calculate the total heads. For example, HB = HA - hfAB = 85.00 - 7.76 = 77.24 m and HC = HB - hfbc = 77.24 - 16.79 = 60.45 m

102 m

All flow rates are in m3/s 85 m

G 0.10

A 0.200

0.250

0.096

0.094

0.154

0.090 0.106 F

0.054

0.10

0.25 Figure 4.10(b)

Results of Example 4.9

Note that various paths can be used to determine the total head at a given node. For example, HC can also be calculated as HC = HA - hf AB - hf BF + hfCF. The results obtained from the different paths should be the same except for round-off errors. The junction pressure heads can be determined using a similar approach as in Example 4.8, but employing Equation 3.15(b). However, in practice, velocity heads at junctions are neglected. With this assumption the pressure head at a node is equal to the total head minus the elevation of the node. Pipe

Q (m3/s)

hf (m)

Junction

Elevation (m)

Total Head (m)

Pressure Head (m)

AB BC BF CF DC EF DE GD

0.200 0.094 0.106 0.090 0.096 0.054 0.154 0.250

7.76 16.79 21.34 5.49 15.02 4.75 16.08 26.44

A B C D E F G

48.00 46.00 43.00 48.00 44.00 48.00 60.00

85.00 77.24 60.45 75.56 59.48 55.90 102.00

37.00 31.24 17.45 27.56 15.48 7.90 42.00

122

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Chap. 4

4.4.2 The Newton Method The Newton method is an appropriate and convenient procedure for analyzing pipe networks containing a large number of pipes and loops. In general the Newton iteration method was developed to solve a set of N simultaneous equations, Fi, written as Fi[Q1, Q2, p , Qi, p , QN] = 0 where i = 1 to N, and Qi are the N unknowns. Computation for the iterative procedure begins by assigning a set of trial values to the unknowns Qi for i = 1 to N. Substitution of these trial values into the N equations will yield the residuals F1, F2, p FN. These residuals are likely to be different from zero because the trial values assigned to the unknowns are probably not the actual solutions. New values for Qi for i = 1 to N for the next iteration are estimated to make the residuals approach zero. We accomplish this by calculating corrections ∆Qi for i = 1 to N such that the total differentials of the functions Fi are equal to the negative of the calculated residuals. In matrix form, 0F1 0Q1 0F2 0Q1 0F3 0Q1

# #

0FN - 1 0Q1 0FN 0Q1

0F1 0Q2 0F2 0Q2 0F3 0Q2

# #

0FN - 1 0Q2 0FN 0Q2

0F1 0Q3 0F2 0Q3 0F3 0Q3

# #

0FN - 1 0Q3 0FN 0Q3

..... ..... ..... ..... ..... ..... .....

0F1 0QN - 2 0F2 0QN - 2 0F3 0QN - 2

0F1 0QN - 1 0F2 0QN - 1 0F3 0QN - 1

0F1 0QN 0F2 0QN 0F3 0QN

0FN - 1 0QN - 2 0FN 0QN - 2

0FN - 1 0QN - 1 0FN 0QN - 1

0FN - 1 0QN 0FN 0QN

# #

∆Q1 ∆Q2 ∆Q3 G

# #

-F1 -F2 -F3 W = G

∆QN - 1 ∆QN

# #

(4.19)

-FN - 1 -FN

The solution of Equation 4.19 by any matrix inversion method provides the corrections to the trial values of Qi for the next iteration. Thus, in equation form (Qi)k + 1 = (Qi)k + (∆Qi)k where k and (k + 1) indicate consecutive iteration numbers. This procedure is repeated until the corrections are reduced to acceptable magnitudes. The number of iterations required to achieve the correct solution depends on how close the initial trial values are to the correct solution. If the initial guesses are quite different from the actual results, then the procedure may not converge. The initial trial values of Qi do not need to satisfy mass balance at the junctions in the Newton method. This is a major advantage over the Hardy–Cross method, especially when large pipe networks are considered. Also, the equations are formulated based on the flow directions initially chosen. A positive result for a flow rate will indicate that the direction initially chosen is correct. A negative value will indicate that the flow in that particular pipe is in the direction opposite to the direction initially guessed. Because the flow rates can take positive and negative values in this formulation, the friction loss is expressed as hf = KQ ' Q ' m - 1 to ensure that changes in the head are consistent with the flow directions.

Sec. 4.4

Pipe Networks

123

The application of the Newton method to a pipe network analysis problem can best be shown through an example. Example 4.10 Analyze the pipe network of Example 4.9 using the Newton method. Solution We will assign numbers to the junctions as shown in Figure 4.10(c). For instance junction B of Example 4.9 is designated as J1. In Newton method applications, a junction means points where two or more pipes join. There are eight pipes in the network. Let us designate the flow rates in these pipes by Qi with i = 1, 2, p , 8. For example the flow rate in pipe AB is designated Q1 with flow direction A to B. The solution is formulated using the flow directions chosen. HG HA Q8

A Q

C Q2

J1 L1

Q3 QF

L2 J4

J5 F

Figure 4.10(c)

We will first write the junction equations. The correct values of the flow rates are those that will make the right-hand sides of these equations equal to zero since the sum of all the flow rates entering and leaving a junction must be zero. F1 = -Q1 + Q2 + Q3 F2 = -Q2 - Q4 + Q5 + QC F3 = Q4 + Q7 - Q8 F4 = Q6 - Q7 + QE F5 = -Q3 - Q5 - Q6 + QF Next, noting that the sum of friction losses around a closed loop must be zero, we will write the loop equations for loops 1 and 2, respectively, as F6 = K2Q2 ' Q2 ' - K3Q3 ' Q3 ' + K5Q5 ' Q5 ' F7 = -K4Q4 ' Q4 ' - K5Q5 ' Q5 ' + K6Q6 ' Q6 ' + K7Q7 ' Q7 ' Again the correct values of the flow rates will make the right-hand sides of these equations zero. Finally, the inflow path equation between reservoirs A and G is written as F8 = HA - K1Q1 ' Q1 ' - K2Q2 ' Q2 ' + K4Q4 ' Q4 ' + K8Q8 ' Q8 ' - HG

124

Pipelines and Pipe Networks

Chap. 4

Many members of the coefficient matrix are zero because only a few Q's appear in each equation. The nonzero values are evaluated as 0F1 0Q1 0F2 0Q2 0F3 0Q4 0F4 0Q6 0F5 0Q3 0F6 0Q2 0F7 0Q4 0F8 0Q1

= -1 = -1 = 1 = 1 = -1 = 2K2Q2 = - 2K4Q4 = - 2K1Q1

0F1 0Q2 0F2 0Q4 0F3 0Q7 0F4 0Q7 0F5 0Q5 0F6 0Q3 0F7 0Q5 0F8 0Q2

= 1 = -1 = 1

0F1 = 1 0Q3 0F2 = 1 0Q5 0F3 = -1 0Q8

= -1 = -1 = - 2K3Q3 = - 2K5Q5 = - 2K2Q2

0F5 0Q6 0F6 0Q5 0F7 0Q6 0F8 0Q4

= -1 = 2K5Q5 = 2K6Q6 = 2K4Q4

0F7 = 2K7Q7 0Q7 0F8 = 2K8Q8 0Q8

The initial (trial) flow rates selected for the pipes are Q1 = 0.20 m3/s, Q2 = 0.50 m3/s, Q3 = 0.10 m3/s, Q4 = 0.05 m3/s, Q5 = 0.50 m3/s, Q6 = 0.10 m3/s, Q7 = 0.30 m3/s, and Q8 = 0.25 m3/s. The flow directions are designated in Figure 4.10(c). Substituting these values into the equations formulated above for this example, we obtain

-1.0 0.0 0.0 0.0 H 0.0 0.0 0.0 - 77.6

1.0 - 1.0 0.0 0.0 0.0 1900.0 0.0 - 1900.0

1.0 0.0 0.0 0.0 -1.0 -380.0 0.0 0.0

0.0 -1.0 1.0 0.0 0.0 0.0 -163.0 163.0

0.0 1.0 0.0 0.0 - 1.0 678.0 -678.0 0.0

0.0 0.0 0.0 1.0 -1.0 0.0 326.0 0.0

0.0 0.0 1.0 -1.0 0.0 0.0 406.8 0.0

∆Q1 0.0 -0.4000 ∆Q2 0.0 -0.0500 - 1.0 ∆Q3 -0.1000 0.0 ∆Q4 0.1000 XH X = H X 0.0 ∆Q5 0.4500 0.0 ∆Q6 -625.5000 0.0 ∆Q7 96.2550 211.5 ∆Q8 469.2475

Solving this matrix equation using a computer program, we obtain the discharge corrections as ∆Q1 = 0.0419 m3/s, ∆Q2 = -0.2513 m3/s, ∆Q3 = -0.1068 m3/s, ∆Q4 = 0.0233 m3/s, ∆Q5 = -0.2780 m3/s, ∆Q6 = - 0.0652 m3/s, ∆Q7 = -0.1652 m3/s, and ∆Q8 = -0.0419 m3/s. Thus, for the second iteration we will use Q1 = 0.2419 m3/s, Q2 = 0.2487 m3/s, Q3 = -0.0068 m3/s, Q4 = 0.0733 m3/s, Q5 = 0.2220 m3/s, Q6 = 0.0348 m3/s, Q7 = 0.1348 m3/s, and Q8 = 0.2081 m3/s. The same procedure will be repeated until all the corrections become negligible. The table below summarizes the Q values obtained by the iteration process.

Sec. 4.5

Water Hammer Phenomenon in Pipelines

125

Flow Rates (m3/s) Iteration Number

Initial 1 2 3 4 5 6

0.2000 0.2419 0.2511 0.1989 0.2007 0.2008 0.2008

0.5000 0.2487 0.1226 0.0938 0.0932 0.0933 0.0933

Q3 0.1000 - 0.0068 0.1286 0.1051 0.1075 0.1075 0.1075

0.0500 0.0733 0.0821 0.0925 0.0961 0.0961 0.0961

0.5000 0.2220 0.1047 0.0863 0.0894 0.0894 0.0894

0.1000 0.0348 0.0168 0.0585 0.0532 0.0531 0.0531

0.3000 0.1348 0.1168 0.1585 0.1532 0.1531 0.1531

0.2500 0.2081 0.1989 0.2511 0.2493 0.2492 0.2492

The results are obtained in six iterations. These results are essentially the same as those of Example 4.9 except for rounding off. The resulting total heads are HA = 85 m, HB = 77.18 m, HC = 60.65 m, HD = 75.72 m, HE = 59.83 m, HF = 55.23 m, and HG = 102.00 m. The resulting pressure heads at nodes A, B, C, D, E, F, and G, respectively are 37.00 m, 31.18 m, 17.65 m, 27.72 m, 15.83 m, 7.23 m, and 42.00 m. Again, these results are practically the same as Example 4.9. The discrepancies are from round-off errors.

4.5 Water Hammer Phenomenon in Pipelines A sudden change of flow rate in a large pipeline (caused by valve closure, pump shutoff, etc.) may affect a large mass of water moving inside the pipe. The force resulting from changing the speed of the water mass could cause a pressure rise in the pipe with a magnitude several times greater than the normal static pressure in the pipe. This phenomenon is commonly known as the water hammer phenomenon. The excessive pressure may fracture the pipe walls or cause other damage to the pipeline system. The possible occurrence of water hammer, its magnitude, and the propagation of the pressure wave must be carefully investigated in connection with the pipeline design. The sudden change of pressure from valve closure may be viewed as the consequence of the force developed in the pipe necessary to stop the flowing water column. The column has a total mass m and is changing its velocity at the rate of dV/dt. According to Newton's second law of motion, F = m

dV dt

(4.20)

If the velocity of the entire water column could be reduced to zero instantly, Equation 4.20 would become F =

m(V0 - 0) mV0 = = ∞ 0 0

The resulting force (hence, pressure) would be infinite. Fortunately, such an instantaneous change is impossible because a mechanical valve requires a certain amount of time to complete a closure operation. In addition, neither the pipe walls nor the water column involved are perfectly rigid under large pressure. The elasticity of both the pipe walls and the water column play very important roles in the water hammer phenomenon.

126

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Chap. 4

To examine the water hammer phenomenon more thoroughly, consider a pipe of length L with an inside diameter D, a wall thickness e, and a modulus of elasticity Ep. Furthermore, assume that water is flowing from a reservoir through the pipe and a valve is at the end of the pipe as depicted in Figure 4.11(a). Assuming losses (including friction) are negligible, the energy grade line is depicted as a horizontal line. Immediately following valve closure, the water in close proximity to the valve is brought to rest. The sudden change of velocity in the water mass causes a local pressure increase. As a result of this pressure increase, the water column in this section is somewhat compressed, and the pipe walls expand slightly from the corresponding increase of stress in the walls. Both of these phenomena help provide a little extra volume, allowing water to enter the section continuously until it comes to a complete stop. The next section immediately upstream is involved in the same procedure an instant later. In this manner, a wave of increased pressure propagates up the pipe toward the reservoir as shown in Figure 4.11(b). When this pressure wave reaches the upstream reservoir, the entire pipe is expanded, and the water column within is compressed by the increased pressure. At this very instant, the entire water column within the pipe comes to a complete halt. This transient state cannot be maintained because the EGL in the pipe is much higher than the EGL of the open reservoir. Because energy differences create flow, the halted water in the pipe flows back into the reservoir as soon as the pressure wave reaches the reservoir. This process starts at the reservoir end of the pipe, and a decreased pressure wave travels downstream toward the valve, as shown in Figure 4.11(d). During this period, the water behind the wave front moves in the upstream direction as the pipe continuously contracts and the column decompresses. The time required for the pressure wave to return to the valve is 2L/C, where C is the speed of the wave traveling through the pipe. It is also known as celerity. The speed of pressure wave travel in a pipe depends on the modulus of elasticity of water Eb, and the modulus of elasticity of the pipe wall material Ep. The relationship may be expressed as C =

Ec Aρ

(4.21)

where Ec is the composite modulus of elasticity of the water-pipe system and ρ is the density of water. Ec is a function of the elasticity of the pipe walls and the elasticity of the fluid within. It may be calculated by the following relationship: 1 1 Dk = + Ec Eb Ep e

(4.22a)

The modulus of elasticity of water, Eb, and the density of water are given in Chapter 1 (and the front jacket of the book). The modulus of elasticity of various common pipe materials is listed in Table 4.1; k is a constant depending on the method of pipeline anchoring, and e is the thickness of the pipe walls. Typical k values are k = (1 - ε2) for pipes anchored at both ends against longitudinal movement, k = a

5 - εb for pipes free to move longitudinally (negligible stresses), and 4

k = (1 - 0.5ε) for pipes with expansion joints,

Sec. 4.5

Water Hammer Phenomenon in Pipelines

EGL

127

WS C Region of decreased pressure

Valve V0

V0 V=0

Wave front (a)

(f)

∆H

∆H Wave front

H0 Wave front

V=0 V=0 (g)

(b)

∆H

WS C

Wave front

V=0 V=0 (c)

(h)

Wave front V0

Wave front

V=0

V0 L (i)

(d) WS

Wave front

V0 (e)

Figure 4.11 Propagation of water hammer pressure waves (friction in pipe neglected). (a) Steady-state condition prior to valve movement. (b) Transient conditions at t 6 L/C. (c) Transient conditions at t = L/C. (d) Transient conditions at L/C 6 t 6 2L/C. (e) Transient conditions at t = 2L/C. (f) Transient conditions at 2L/C 6 t 6 3L/C. (g) Transient conditions at t = 3L/C. (h) Transient conditions at t = 3L/C 6 t 6 4L/C. (i) Transient conditions at t = 4L/C. Note: After t = 4L/C, the cycle repeats and continues indefinitely if the friction in the pipe is zero. The symbol or is used to denote the direction of reflection of the wave front.

128

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Chap. 4

Table 4.1 Modulus of Elasticity (Ep) of Common Pipe Materials

Pipe Material

Ep (N/m2)

Aluminum Brass, bronze Concrete, reinforced Copper Glass Iron, cast Iron, ductile Lead Lucite Rubber, vulcanized Steel

7.0 9.0 1.6 9.7 7.0 1.1 1.6 3.1 2.8 1.4 1.9

* * * * * * * * * * *

1010 1010 1011 1010 1010 1011 1011 108 108 1010 1011

Ep (psi) 1.0 1.3 2.5 1.4 1.0 1.6 2.3 4.5 4.0 2.0 2.8

* * * * * * * * * * *

107 107 107 107 107 107 107 104 104 106 107

where ε is the Poisson's ratio of the pipe wall material. For common pipe materials, ε = 0.25 is often used. 5 If the longitudinal stress in a pipe can be neglected—that is, k = a - εb = 1.0, 4 Equation 4.22(a) can be simplified to 1 1 D = + Ec Eb Ep e

(4.22b)

Figure 4.11(e) shows that by the time the decreased pressure wave arrives at the valve, the entire column of water within the pipe is in motion in the upstream direction. This motion cannot pull any more water from beyond the already closed valve and is stopped when the wave arrives at the valve. The inertia of this moving water mass causes the pressure at the valve to drop below the normal static pressure. A third oscillation period begins as a wave of negative pressure propagates up the pipe toward the reservoir as shown in Figure 4.11(f). At the instant the negative pressure reaches the reservoir, the water column within the pipe again comes to a complete standstill, and the EGL of the pipe is less than the EGL at the reservoir [Figure 4.11(g)]. Because of this energy difference, water flows into the pipe starting a fourth period of oscillation. The fourth period is marked by a wave of normal static pressure moving downstream toward the valve as in Figure 4.11(h). The water mass behind the wave front also moves in the downstream direction. This fourth-period wave arrives at the valve at time 4L/C, the entire pipe returns to the original EGL, and the water in the pipe is moving in the downstream direction. For an instant, the conditions throughout the pipe are somewhat similar to the conditions at the time of valve closure (the beginning of the first-period wave), except that the water velocity in the pipe has been reduced. This is a result of energy losses to heat from friction and the viscoelastic behavior of the pipe walls and the water column. Another cycle begins instantly. The four sequential waves travel up and down the pipe in exactly the same manner as the first cycle described above, except that the corresponding pressure waves are smaller in magnitude. The pressure-wave oscillation continues with each set of waves successively diminishing until finally the waves die out completely.

Sec. 4.5

Water Hammer Phenomenon in Pipelines

129

As mentioned previously, the closure of a valve usually requires a certain period of time t to complete. If t is less than 2L/C (valve closure is completed before the first pressure wave returns to the valve), the resulting rise in pressure should be the same as that of instantaneous closure. However, if t is greater than 2L/C, then the first pressure wave returns to the valve before the valve is completely closed. The returned negative pressure wave can offset the pressure rise resulting from the final closure of the valve. Knowing the maximum pressure rise created by the water hammer phenomenon is critical for the safe and reliable design of many pipeline systems. The appropriate design equations are based on fundamental principles and are derived as follows. Consider a pipe with a rapidly closing valve (t … 2L/C); the extra volume of water (∆Vol) that enters the pipe during the first period (t = L/C) [Figure 4.11(c)] is L ∆Vol = V0Aa b C

(4.23)

where V0 is the initial velocity of water flowing in the pipe and A is the pipe cross-sectional area. The resulting pressure rise ∆P is related to this extra volume by ∆P = Ec a

Ec (∆Vol) ∆Vol b = Vol AL

(4.24)

where Vol is the original volume of the water column in the pipe and Ec is the composite modulus of elasticity as defined by Equation 4.22(a). Substituting Equation 4.23 into Equation 4.24, we may write ∆P =

EcV0 Ec L c V0Aa b d = AL C C

(4.25a)

As the pressure wave propagates upstream along the pipe at speed C, the water behind the wave front is immediately brought to a stop from the initial velocity of V0. The total mass of water involved in this sudden change of speed from V0 to zero in time ∆t is m = ρAC∆t. Applying Newton's second law to this mass, we have ∆P(A) = m

(V0 - 0) ∆V = ρAC∆t = ρACV0 ∆t ∆t

or ∆P = ρCV0

(4.25b)

(Note: Equation 4.21 can be derived from Equation 4.25(b); see Problem 4.5.1.) Solving Equation 4.25(b) for C and substituting for C in Equation 4.25(a), we have ∆P = EcV0

ρV0 ∆P

or ∆P = V0 2ρEc

Also, ∆H =

V0 Ec V0 ∆P = = C ρg g Aρ g

(4.25c)

(4.26)

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where ∆H is the pressure head rise caused by water hammer. These equations are only applicable for rapid valve closure (t … 2L/C). For valve closures that are not rapid (i.e., t 7 2L/C), the pressure rise previously discussed (∆P) will not develop fully because the reflected negative wave arriving at the valve will reduce the pressure rise. For these slow valve closures, the maximum water hammer pressure may be calculated by the Allievi equation,* which is expressed as ∆P = P0 a

N N2 + + Nb 2 A 4

(4.27)

where P0 is the static-state pressure in the pipe, and N = a

ρLV0 2 b P0 t

Before applying the water hammer equations to pipe flow problems, the energy grade line and the hydraulic grade line for the pipe system under steady flow conditions must be determined, as shown in Figure 4.12. As the pressure wave travels up the pipeline, energy is being stored in the form of pressure in the pipe behind the wave front. Maximum pressure is reached when the wave front arrives at the reservoir, Pmax = γH0 + ∆P

(4.28)

where H0 is the total head before the valve closure, as indicated by the water surface elevation in the reservoir. The location immediately downstream of the reservoir is generally the most vulnerable to pipe and joint damage because the initial pressure is larger here than in the rest of the pipeline. Pressure wave celerity Water hammer pressure head C

(After valve closing)

EGL HGL

∆P g C (befo

re va

(befo

re va

lve c

losin

∆P g

V=0 Valve Figure 4.12

Water hammer pressure in a pipeline

*L. Allievi, "The Theory of Water Hammer" (translated by E. E. Halmos), Trans. ASME (1929).

Sec. 4.5

Water Hammer Phenomenon in Pipelines

131

Example 4.11 A steel pipe 5,000 ft long laid on a uniform slope has an 18-in. diameter and a 2-in. wall thickness. The pipe carries water from a reservoir and discharges it into the air at an elevation 150 ft below the reservoir free surface. A valve installed at the downstream end of the pipe permits a flow rate of 25 cfs. If the valve is completely closed in 1.4 s, calculate the maximum water hammer pressure at the valve. Assume the longitudinal stresses in the pipeline are negligible. Solution From Equation 4.22(b), 1 1 D = + Ec Eb E pe where Eb = 3.2 * 105 psi (from Chapter 1 or the front jacket of the book) and Ep = 2.8 * 107 psi (Table 4.1). The above equation may thus be written as 1 1 18 = + 5 Ec (2.8 * 107)2.0 3.2 * 10 Hence, Ec = 2.90 * 105 psi From Equation 4.21 we may obtain the speed of wave propagation along the pipe as C =

Ec 2.90 * 105(144) = 4,640 ft/s = A Aρ 1.94

The time required for the wave to return to the valve is t =

2(5,000) 2L = = 2.16 s C 4,640

Because the valve closes in 1.4 s ( 6 2.16 s), rapid-valve-closure equations may be applied. Therefore, water velocity in the pipe before valve closure is V0 =

25 = 14.1 ft/s π 2 (1.5) 4

and the maximum water hammer pressure at the valve can be calculated using Equation 4.25(b) as ∆P = ρCV0 = 1.94(4,640)(14.1) = 1.27 * 105 lb/ft2 (881 psi)

Example 4.12 A ductile-iron pipe with a 20-cm diameter and 15-mm thick walls is carrying water when the outlet is suddenly closed. If the design discharge is 40 L/s, calculate the pressure head rise caused by water hammer if (a) the pipe wall is rigid, (b) the pipe is free to move longitudinally (negligible stresses), and (c) the pipeline has expansion joints throughout its length. Solution A =

π (0.2)2 = 0.0314 m2 4

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hence, V0 =

Q 0.04 = = 1.27 m/s A 0.0314

(a) For rigid pipe walls, Dk/Epe = 0, Equation 4.22(a) gives the following relation: 1 1 = Ec Eb

Ec = Eb = 2.2 * 109 N/m2

From Equation 4.21, we can calculate the speed of pressure wave: C =

Ec 2.2 * 109 = 1,480 m/s = A 998 Aρ

From Equation 4.26, we can calculate the pressure head rise caused by water hammer as ∆H =

V0C 1.27(1,480) = = 192 m (H20) g 9.81

(b) For pipes with no longitudinal stress, k = 1, we may use Equation 4.22(b): 1

Ec = a

and

1 D + b Eb E pe

= a

1 9

2.2 * 10

C =

0.2 (1.6 * 1011)(0.015)

= 1.86 * 109 b

Ec = 1,370 m/s Aρ

Hence, the pressure head rise caused by the water hammer can be calculated as ∆H =

V0C 1.27(1,370) = = 177 m (H20) g 9.81

Note that once the pipe is free to move longitudinally [as opposed to the rigid pipe considered in part (a)], some of the pressure energy is absorbed by the expanding pipe, and the wave speed is reduced. This in turn reduces the magnitude of the head and pressure rise associated with the water hammer. (c) For pipes with expansion joints, k = (1 - 0.5 * 0.25) = 0.875. From Equation 4.22(a), Ec =

and

1 1 Dk a + b Eb E pe

= a

1 2.2 * 109

C =

(0.2)(0.875) (1.6 * 1011)(0.015)

= 1.90 * 109 b

Ec = 1,380 m/s Aρ

Again, we can calculate the pressure head rise caused by water hammer as ∆H =

V0C 1.27 # 1,380 = = 179 m (H20) g 9.81

which is essentially the same as case (b) (a pipe free to move longitudinally).

Water Hammer Phenomenon in Pipelines

133

Pressure

Sec. 4.5

Time Figure 4.13

Friction effects on water hammer's pressure–time pattern

In water hammer analysis, the time history of pressure oscillation in a pipeline is instructive. Because of the friction between the oscillatory water mass and the pipe wall, the pressure– time pattern is modified, and the oscillation gradually dies out. A typical pressure oscillation is shown in Figure 4.13. In reality, a valve cannot be closed instantaneously. The time required for closure of a valve is a certain period, tc. The water hammer pressure increases gradually with the rate of closure of the valve. The typical valve closure curve is shown in Figure 4.14. If tc is smaller than the time required for the wave front to make a round trip along the pipeline and return to the valve site (tc 6 2L/C), the operation is defined as rapid closure. The water hammer (or shock) pressure will reach its maximum value. The computation of a rapid closure operation is the same as that of an instantaneous closure. To keep water hammer pressure within acceptable limits, valves are commonly designed with closure times considerably greater than 2L/C. For slow closure operation (tc 7 2L/C), the pressure wave returns to the valve site before the closure is completed. A certain amount of water continuously passes through the valve when the pressure wave returns. As a result, the pressure wave pattern will be altered. A complete treatment of the water hammer phenomenon, with consideration of friction and slow valve closure operation, may be found in Chaudhry (1987)* and Popescu et al. (2003).†

Stepwise gate movement (used for computations)

Effective gate opening (%)

100

Actual gate movement ∆t

Figure 4.14

5 6 Time (s)

Typical valve closure curve

*M. Hanif Chaudhry, Applied Hydraulic Transients, 2nd ed. (New York: Van Nostrand Reinhold, 1987). Popescu, D. Arsenie, and P. Vlase, Applied Hydraulic Transients: For Hydropower Plants and Pumping Stations (London: Taylor & Francis, 2003).

†M.

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4.6 Surge Tanks There are many ways to eliminate the detrimental impacts of water hammer in pipelines. One method is slow valve closure, which was discussed in the last section. Other methods that are effective include relief valves (or diverters) and surge tanks. Relief valves rely on the water hammer pressure to open a valve and divert a large portion of the flow for a short period of time. Although relief valves may be a simple solution to the problem, it results in a waste of water. The inclusion of a surge tank near the control station (Figure 4.15) in a pipeline will harness the forces produced when a mass of water is slowed or stopped. A surge tank is defined as a stand pipe or storage reservoir placed at the downstream end of a long pipeline to prevent sudden pressure increases (from rapid valve closure) or sudden pressure decreases (from rapid valve opening). When a valve is being closed, the large mass of water moving in a long pipeline takes time to adjust accordingly. The difference in flow between the pipeline and that allowed to pass the closing valve causes a rise of water level in the surge tank. As the water rises above the level of the reservoir, an energy imbalance is produced so that the water in the pipeline flows back toward the reservoir and the water level in the surge tank drops. The cycle is repeated with mass oscillation of water in the pipeline and the surge tank until it is gradually damped out by friction. Newton's second law can be applied to analyze the effect of the surge tank on the water column AB, between the two ends of the pipeline. At any time during the closure or opening of the valve, the acceleration of water mass is always equal to the forces acting on it; that is, ρLA

dV = (the pressure force on the column at A) dt + (the weight component of the column in the direction of the pipeline) - (the pressure force acting on the column at B) { (friction losses)

The pressure force at A is the result of the elevation difference between the water surface in the reservoir and the pipeline inlet, modified by the entrance loss. The pressure force acting on Area As

Reservoir level HA

EGL for steady flo w Q0 V L

ymax hL HB B

Area A Q Turbine

Figure 4.15

Surge tank

Sec. 4.6

Surge Tanks

135

the column at B depends on the elevation of the water surface in the surge tank, also modified by the losses occurring at the entry (may be a restrictive throttle) to the tank. Hence, ρLA

dV = ρgA 3 (HA { entrance loss) + (HB - HA) dt

- (HB + y { throttle loss) { (pipeline losses)]

(4.29)

The sign of the pipeline losses depends on the direction of the flow. The losses always occur in the direction of the flow. If we introduce the modulus form, hL = KfV ' V ' and HT = KTU ' U ' , where U =

dy dt

(4.30)

and represents the upward velocity of the water surface in the tank, the sign of the losses would always be correct. Here Kf is the pipeline friction factor Kf = fL/(2gD) and hL is the total head loss in the pipeline between A and B. HT is the throttle loss. Substituting these values into Equation 4.29 and simplifying, we have the dynamic equation for the surge tank: L dV + y + KfV ' V ' + KTU ' U ' = 0 g dt

(4.31)

In addition, the continuity condition at B must be satisfied VA = UAs + Q

(4.32)

where Q is the discharge allowed to pass the closing valve at any given time t. The combination of Equations 4.30 through 4.32 produces second-order differential equations that can only be solved explicitly for special cases. A special solution may be obtained by what is frequently called the logarithmic method.* The method provides simple theoretical analysis of surge heights that are close to those observed in practice if the cross-sectional area As remains constant. The solution for a simple (unrestricted) constant-area surge tank (Figure 4.15) may be expressed as β ymax + hL = lna b β β - ymax

(4.33)

where β is the damping factor, which is defined as β =

LA 2gKfAs

(4.34)

Equation 4.33 is an implicit equation and may be solved for the surge height (y) by successive approximations or by computer algebra software (e.g., Mathcad, Maple, or Mathematica) as demonstrated in the following example. *John Pickford, Analysis of Water Surge (New York: Gordon and Beach, 1969), pp. 111–124.

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Example 4.13 A simple surge tank 8.00 m in diameter is located at the downstream end of a 1,500-m-long pipe, 2.20 m in diameter. The head loss between the upstream reservoir and the surge tank is 15.1 m when the flow rate is 20.0 m3/s. Determine the maximum elevation of the water in the surge tank if a valve downstream suddenly closes. Solution For a smooth entrance where the head loss may be neglected, we may write hL ≅ hf = KfV 2 or Kf =

hL V

15.1 (5.26)2

= 0.546 s2/m

and the damping factor, from Equation 4.34, is β =

(1,500)(3.80) LA = = 10.6 m 2gKf As 2(9.81)(0.546)(50.3)

By applying Equation 4.33, ymax + 15.1 10.6 = ln a b 10.6 10.6 - ymax

The solution is obtained by an iterative process. ymax

LHS

RHS

9.50 9.60 9.57

2.32 2.33 2.33

2.27 2.36 2.33

The maximum elevation of the water in the surge tank is 9.57 m over the reservoir level. The same solution is obtained using calculators or software that solve implicit equations.

4.7 Pipe Network Modeling There are many hydraulic computer models available that will quickly perform the pipe network calculations discussed in this chapter. Some of these models are proprietary and costly, but others are freely available on the Internet. Development of some of them started in the 1970s. They continued to be improved through the decades to a point where they are now quite versatile and user-friendly. Taken collectively, these pipe network models have a broad range of capabilities. They are able to: š š š š š

Determine flows in pressure pipe systems of almost unlimited size. Calculate pressures at pipe junctions (nodes) throughout the system. Incorporate water tanks, reservoirs, pumps, and valves into the solution process. Compute friction loss using Darcy–Weisbach, Manning, or Hazen–William methods. Determine losses due to bends, fittings, and other appurtenances.

Sec. 4.7

Pipe Network Modeling

137

š Evaluate extended period simulations including pumping energy and cost. š Accept multiple demand and supply patterns at designated nodes. š Track the concentration of chemical constituents throughout the network. In addition, the model set-up and data input is fast and intuitive with graphical user interfaces (GUIs). The model output is flexible and report-ready with accompanying tables, graphs, and schematics. In this section, we discuss a particular hydraulic model that is available from the U.S. Environmental Protection Agency (EPA). EPA's pressure pipe network model is called EPANET. This model was selected for three reasons: 1. It is nonproprietary and freely available on the Internet. 2. It is fundamentally sound in handling a large variety of applications. 3. It is widely-used and accepted in the engineering and regulatory community. 4.7.1 The EPANET Model The EPANET model was initially developed to compute flow rates and pressures in municipal water distribution systems. Even though the model has a multitude of capabilities, this remains its primary use and the one we will discuss. The order of the tasks performed in the model is as follows: define the distribution system, enter the element (object) properties, describe the system operation, select analysis options, and perform the desired computations. The process sequence, model structure, and data requirements are described in the following paragraphs. Define Distribution System S After the new project is named and default options are selected, a schematic of the distribution system to be modeled is drawn in the network map window. A toolbar is available to select system objects (e.g., pipes, junctions, reservoirs, tanks, and pumps) and position them on the map. An exact scale is not required, but the general orientation and location of system objects should resemble what you would see on a utility map. The network solved in Example 4.9 (eight pipes, five junctions, and two supply reservoirs) is displayed in Figure 4.16. Enter Element Properties S The properties for all of the elements (objects) in the network are entered next. To bring up the property editor, simply double click on the object. The property editor for Junction C and Pipe BC from Figure 4.16 are displayed in Figure 4.17. Essential information for the junctions includes the elevation and external water demand removed at that location. Once the model is executed, the total head and pressure head will be computed. Essential information for pipes includes length, diameter (mm), and roughness. In this case, we were solving for friction loss using the Darcy–Weisbach method (roughness, e is given in millimeters). Once the model is executed, the flow (L/s), velocity, and unit head loss (hL per 100 m) are computed. In like manner, property editors for all of the network elements are activated and the required data entered. Describe the System Operation S For networks that contain pumps, a pump characteristic curve is required (covered in Chapter 5 of your textbook). For extended period simulations, the demand is likely to vary over time (e.g., daily fluctuations). These additional data requirements are entered. Select Analysis Options S There are a variety of hydraulics options available for analyzing a pressure pipe network. For example, the user must choose the flow units (L/s, gal/min, ft3/s, etc.) and the head loss formula (Darch–Weisbach, Manning, or Hazen–Williams). In addition,

138

Pipelines and Pipe Networks

Chap. 4

Res-G GD

Res-A C

AB B

EF F

Figure 4.16

Figure 4.17

Pipe network from Example 4.9 as displayed in the map window

Property editors for Junction C and Pipe BC from Example 4.9

liquid properties need to be defined (specific gravity and viscosity). Finally, numerical accuracy limits need to be defined for the finite difference methods employed. Many other data items are required for simulations involving extended time periods, energy and cost requirements, and water quality analyses. Figure 4.18 displays the hydraulics options editor used to solve Example 4.9. Perform the Desired Computations S After the requisite data and options are entered, EPANET computations can be performed. The "Run" button is selected from the toolbar (or from the menu) and the analysis if performed. A status report is available if the run was unsuccessful indicating the problem. For example, negative pressures may prevent the model from obtaining a solution. EPANET Model Output S Model output is displayed in tabular and graphical form. Tabular output is available for pipes and junctions (Figure 4.19) listing user-defined variables.

Sec. 4.7

Pipe Network Modeling

139

Figure 4.18

Figure 4.19

Hydraulics options editor (single period simulation) for Example 4.9

Pipe (link) and junction (node) output tables for Example 4.9

Many different types of graphs are available: system flows, time series, profile plots, and contour plots. For example, Figure 4.20 depicts chlorine concentrations (color coded) throughout the network. In summary, the output possibilities are extensive, flexible, and report-ready.

140

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Tank

Source Pump

Chlorine 0.20 0.40 0.60 0.80 mg/L

Figure 4.20

Example of a graphical plot of chlorine concentrations

Source: EPANET 2 User's Manual, Lewis A. Rossman, Office of Research & Development, EPA, Sept. 2000.

(Note: The solution differs slightly from Example 4.9 because it goes through more iterations. Also, negative pipe flows indicate that it is in the opposite direction from how the pipe was entered.) Classroom Computer Exercise—Pipe Networks Obtain or write computer software that is appropriate for solving pipe networks. Examples are EPANET (public domain from the U.S. Environmental Protection Agency), WaterCAD and WaterGEMS (proprietary from Bentley), and KYPipe (proprietary from KYPipe LLC). Or you can write your own spreadsheet program. Answer the following questions by performing a computer analysis of the pipe network described in Example 4.8 and its modifications. (a) Before using the computer software, what data do you anticipate the software will need to analyze the pipe network in Example 4.8? (b) Now use the computer software to analyze Example 4.8. Enter the data requested by the software and perform a network of analysis. Compare flow rates generated by the computer model to those in the example problem. Why are the solutions not exactly the same? (Note: Some computer models require a pipe material and then assign an "f " value based on the pipe material and the Reynolds number. You may have to "manipulate" the model to get it to match the "f " values in Example 4.8.) (c) What would happen to the flow rate in pipe EF if the friction factor was reduced? What would happen to the pressure at F? Write down your answers, and then reduce "f " in EF from 0.021 to 0.014 and perform a new network analysis. List the original and new flow rate in pipe EF and the original and new pressure at node F. (Hint: The program may not let you change the friction factor directly but may allow a change in the pipe material or roughness value. You may have to assume complete turbulence on the Moody diagram and back into the roughness value or pipe material you need to reduce the friction factor in pipe EF to 0.014.) Once you have completed the analysis, restore pipe EF to its original friction factor and proceed to the next question.

Problems

141

(d) What would happen to the flow rate in pipe HF if the diameter was doubled? What would happen to the pressure at F? Estimate the magnitude of these changes and write them down. Now double the diameter and analyze the network. Did you reason correctly? List the original and new flow rate in pipe HF and the original and new pressure at node F. Now restore HF to its original size and proceed to the next question. (e) What would happen to the pressure at F if the demand for water at that point increased by 50 liters/s? Estimate the magnitude of these changes and write them down. Now increase the demand for water at F and perform a new network analysis. Did you reason correctly? List the original and new pressure at node F. Now restore the demand at node F to its original value and proceed to the next question. (f) What would happen to the flow rate in pipe EF if a new pipe were added to the system from node G to halfway between nodes A and D? What would happen to the pressure at F? Add this new pipe with the same characteristics as pipe DE and perform a new network analysis. Did you reason correctly? Now restore the network to its original configuration. (g) Perform any other changes that your instructor requests. Problems (secTion 4.1) 4.1.1. Examine the EGL and HGL in Figure 4.1 and explain the following: (a) The location of the EGL at the water surface of the two reservoirs. (b) The drop in the EGL moving from reservoir A into pipe 1. (c) The slope of the EGL in pipe 1. (d) The separation distance between the EGL and the HGL. (e) The drop in the EGL moving from pipe 1 to pipe 2. (f) The slope of the EGL in pipe 2 (note that it has a greater slope than pipe 1). (g) The drop in the EGL moving from pipe 2 to reservoir B. 4.1.2. Sketch the energy grade line (EGL) and the hydraulic grade line (HGL) for the pipeline shown in Figure P4.1.2. Consider all the losses and changes of velocity and pressure heads.

2 3

Figure P4.1.2

4.1.3. Often, the iterative procedure encountered in pipeline problems can be shortened by assuming complete turbulence in the pipe (if e/D is available) to obtain a preliminary friction factor. This assumption is often valid for water transmission systems since the viscosity of water is low and the velocities and/or pipe diameters are large which yields high NR values. Refer to the Moody diagram

142

Pipelines and Pipe Networks

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and determine three to five combinations of velocity and pipe size that would produce complete turbulence at a friction factor of 0.018 (assuming water at 20°C). 4.1.4. The flow rate (water; 20°C) through a smooth concrete pipeline (Figure P4.1.2) is 1.2 m3/s. Determine the water surface elevation of the upstream reservoir (A) if the water surface elevation of the downstream reservoir (B) is 600 m and given: Valve 1-2: fully open globe valve Pipe 3: 1,000 m long (D = 1.0 m)

Pipe 1 & 2: each 500 m long (D = 0.5 m) Pipe 5 & 6: each 500 m long (D = 0.5 m)

Ignore the confusor losses (at point "4") and all other minor losses except entrance (square edged), exit, and the fully open globe valve. 4.1.5. Water (68°F) flows at the rate of 22.5 cfs (ft3/s) from reservoir A to reservoir B through two cast iron (e = 0.001 ft) pipes connected in series. If L 1 = 2,500 ft, D1 = 2.0 ft., L 2 = 500 ft, and D2 = 1.5 ft, determine the difference in water surface elevations. The coefficient of sudden contraction from pipe 1 to pipe 2 is 0.36 and assume fully turbulent flow (i.e., f depends on e/D only). A

B 2

Figure P4.1.5

4.1.6. The 40-m-long, 4-in. commercial steel pipe connects reservoirs A and B as shown in Figure P4.1.6. Determine the pressure at each point designated in the figure (points "1" through "6") if the water flow is 10.1 L /s (20°C), reservoir A is subjected to a pressure of 9.79 kPa (gauge), all valves are fully open, and bend losses are negligible. P0

Note: The pipe is 4-in. commercial steel (ignore bend losses)

10 m

4 1 9.5 m

Lift-type check-valve

2 2m

10 m

5 Globe valve

7.5 m

B 7m

Figure P4.1.6

4.1.7. Water (at 10°C with a ν = 1.31 * 10-6 m2/s) flows from reservoir A (surface elevation 100 m, square-edged entrance) through a 2.25-m-diameter, concrete (good joints, average) pipe to reservoir

Problems

143

B (surface elevation 88 m). If the two reservoirs are 17 km apart, determine the flow rate taking into account minor losses. How much would the flow rate change if minor losses were neglected? 4.1.8. The water surface in reservoir A is 12 m higher than the surface of reservoir B. A 40-cm-diameter, 0.7 km long pipe carries water at 20°C from A to B. Determine the discharge for the following pipes: (a) commercial steel, (b) cast iron, and (c) smooth concrete. Determine flow increase in percent if the highest capacity pipe material were chosen instead of the lowest capacity pipe material. 4.1.9. Water flows from tank A to tank B (Figure P4.1.9). The water surface elevation difference is 60 ft. The water temperature is 68°F, the pipe material is cast iron, and the pipeline (square-edged entrance) has the following characteristics: Expansion: sudden Bends: Four (R/D = 4; in large pipe)

Small pipe: 1,000 ft long (D = 8 in.) Large pipe: 1,000 ft long (D = 16 in.)

Determine the system's existing flow rate. What percentage increase in the flow would occur if the 8-in. line were completely replaced with the 16-in. pipeline?

A B

Figure P4.1.9

4.1.10. The elevation difference between a supply tank and a cooling pond is 6.6 ft. A 75-ft long pipe must transport 2.5 cfs of 40°F water from the tank to the pond. Determine the diameter of the commercial steel pipe that is required. Assume square-edged connections and include a globe valve in the pipe. 4.1.11. The surface elevation difference between two reservoirs, 25 km apart, is 80 m. Ductile iron pipes (DIPs) are used to transport water between the two reservoirs. (a) Determine the pipe diameter required to transport 200 L/s. (b) Using the computed pipe diameter in part (a), determine the required water surface elevation difference if the flow must be increased to 250 L/s. Assume a water temperature of 20°C and include minor losses. 4.1.12. Determine the pressure P0 in tank A (Figure P4.1.6) if the pressure at point "1" is measured to be 97.9 kPa. Tank A and reservoir B are connected by a 40-m-long, 4-in. (0.102 m) commercial steel pipe. Assume that the water is at 20°C, all valves are fully open, and bend losses are negligible. 4.1.13. An industrial process requires a 7.5 L/min flow rate for a water–glycerol solution (specific gravity = 1.1; v = 1.03 * 10-5 m2/s) in a 2.5-m long tube under a pressure head of 60 mmHg. Determine the diameter of a glass tube (e = 0.003 mm) to fulfill these requirements. Assume that the pressure head (60 mmHg) is needed to overcome the friction loss in the horizontal tube; no other losses are considered. 4.1.14. All the pipes in Figure P4.1.14 have a Hazen–Williams coefficient of 100. Pipe AB is 3,000 ft long and has a diameter of 2.0 ft. Pipe BC1 is 2,800 ft long with a diameter of 1.0 ft, while pipe BC2 has a length of 3,000 ft and a diameter of 1.5 ft. Pipe CD is 2,500 ft long with a diameter of 2.0 ft. The water surface elevation of reservoir A is 230 ft (HA) and reservoir D (HD) is 100 ft. Determine the discharge in each pipe and the total head at points B and C if QB = 0 and QC = 0. Ignore minor losses.

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HA A B

QB HD

2 QC

D Figure P4.1.14

4.1.15. Redo Problem 4.1.14 with QB = 8 cfs and QC = 8 cfs. Should the discharge in AB in this case be greater or less than that of Problem 4.1.14? Why?

(secTion 4.2) 4.2.1. A 14-cm-diameter siphon tube is used to empty the reservoir and discharge it into the air (Figure P4.2.1). If the total head loss between the tube's intake and the summit (S) is 1.0 m and between S and the discharge end is 2.0 m, determine the discharge in the 16-m-long tube and the water pressure at the summit. S 2m 7m Figure P4.2.1

4.2.2. Do all siphons encounter negative pressure at their summits? Prove your answer by drawing an energy grade line (EGL) and hydraulic grade line (HGL) sketch on Figure P4.2.1. 4.2.3. A siphon spillway with a square cross-sectional area of 12 ft2 is used to discharge water to a downstream reservoir 60 ft below its crest or summit (i.e., the highest elevation in the siphon as shown in Figure P4.2.3). If the upstream water surface level is 8 ft below the summit, determine the danger of cavitation (i.e., a pressure less than two-thirds of atmospheric pressure or -9.8 lb/in.2) in the siphon at its summit. Assume that the frictional head loss is equal to 2.5 times the velocity head and is evenly distributed throughout its length. The entrance is square-edged and the distance from the siphon entrance to the summit is one-fifth the total siphon length.

15 ft

60 ft

Figure P4.2.3

Problems

145

4.2.4. A horizontal, 40-cm-diameter pipe conveys water at 440 L/s and contains a confusor (negligible head loss). The pressure upstream of the confusor is 104,000 N/m2. Cavitation concerns require a pressure head above -4.0 m just downstream of the confusor. Determine the minimum diameter of the downstream pipe. 4.2.5. A 20-cm, 300-m-long smooth concrete pipe carries water (20°C) from reservoir A to B, as shown in Figure P4.2.5. The pipe is elevated at S, which is 150 m downstream from reservoir A. The water surface in reservoir B is 25 m below the water surface in reservoir A. If ∆s = 7.0 m, is cavitation a concern?

∆s A

Gate valve B

Figure P4.2.5

4.2.6. A PVC pipeline with a 40 cm diameter and 2,000-m long carries water at the rate of 0.388 m3/s (10°C) from reservoir A to reservoir B. The water surface elevation of reservoir A is 30 m higher than that of reservoir B. A pump is needed to boost the flow rate when needed. Determine the pressure head the pump must add to the pipeline to double the flow rate. 4.2.7. A pump will be used to remove 50 cfs of ground water from an aquifer (underground reservoir) and transport it to a water treatment plant. A 24-in. pipe is used and made of high-density polyethylene (HDPE) with an entrance screen (K = 2.5). Determine the maximum height above the aquifer that the pump can be placed without encountering cavitation. Assume that friction losses on the suction side of the pump are negligible. 4.2.8. Water flows in a new 8-in., 1,000-ft-long galvanized iron pipe between reservoirs A and B (Figure P4.2.5). The pipe is elevated at S, which is 500 ft downstream from reservoir A. The water surface in reservoir B is 82 ft below the water surface in reservoir A. A booster pump is installed 150 ft downstream from reservoir A and provides additional energy to the system (98 ft of head). Determine the flow rate and the likelihood of cavitation at the summit of the pipe if ∆s is 1.3 ft. 4.2.9. A pump is installed in a 100-m pipeline to lift water at 20°C from reservoir A to reservoir B (see Figure P4.2.9). The pipe is rough concrete with a diameter of 80 cm. The design discharge is 4.5 m3/s. Determine the maximum distance from reservoir A that the pump could be installed without encountering cavitation problems.

50 m

Pump Figure P4.2.9

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Pipelines and Pipe Networks

Chap. 4

4.2.10. A pump is installed in a 10-cm pipeline and transports water from reservoir A to a higher reservoir (B) as shown in Figure P4.2.9. The head loss from A to the pump is four times the velocity head, and the head loss from the pump to B is seven times the velocity head. The pressure head measured at the inlet (suction side) of the pump is - 6 m. Calculate the pressure head the pump is delivering under these operating conditions. Note that the pressure head the pump delivers must be greater than 50 m, the elevation difference between the surfaces of the two reservoirs. Why? Draw the EGL and HGL. 4.2.11. A pump is needed to convey 8 cfs of water at 68°F through a 130-ft-long horizontal, ductile iron pipeline to a pressurized tank. The pump and pipeline are installed at an elevation of 10 ft, the water surface elevation in the receiving tank is 20 ft, and the pressure at the top of the tank is 32.3 psi. The pipe has a 15-in.-diameter on the suction side of the pump and a 12-in.-diameter on the discharge side of the pump. If the pump delivers 111 ft of head, determine the pressure head on the discharge side of the pump (in psi) and whether cavitation is a concern on the suction side of the pump? Sketch the system and draw the HGL and EGL. 4.2.12. A 40-m-long, 4-in. commercial steel pipe connects reservoirs A and B as shown in Figure P4.1.6. Determine the minimum pressure P0 that would keep the pressure head throughout the pipe positive. Assume that all valves are fully open, bend losses are negligible, and the water temperature is 20°C.

(secTion 4.3) 4.3.1. When solving a three-reservoir problem, it is advantageous to set the total energy elevation, HJ, at the junction to match exactly with the elevation of the middle reservoir for the first iteration. (a) What is the advantage of doing this? If the answer is not immediately apparent, trace back through the solution of Example 4.6 where this assumption was used. (b) Use a computer algebra software system or a spreadsheet program to formulate the classic three reservoir problem and verify its accuracy using Example 4.6. [Hint: If using a spreadsheet, use the von Karman equation (Equation 3.23) for the initial estimate of the Darcy–Weisbach friction factor assuming complete turbulence. Then use the Swamee–Jain equation* (Equation 3.24a; shown below) to solve for the friction factor once the NR value is known.] 0.25

f = Jloga

2 e 5.74 + 0.9 b R 3.7D NR

4.3.2. Determine flow rates in the branching pipe system depicted in Figure P4.3.2 given the following water surface (WS) elevation and pipe data (lengths and diameters): WS1 = 5,200 ft WS2 = 5,170 ft WS3 = 5,100 ft

L1 = 4,000 ft L2 = 2,000 ft L3 = 3,000 ft

D1 = 3 ft D2 = 5 ft D3 = 4 ft

All of the pipes are lined ductile iron (DIP, e = 0.0004 ft) and the temperature of the water is 68°F. Also determine the elevation of the junction (J) if the pressure head (PJ/γ) at the junction measured by a piezometer (height from J to HJ) is 30 ft.

Problems

147

WS1 Reservoir 1

WS2 HJ Pipe 1

WS3 Reservoir 3

Reservoir 2

Pipe 2

Pipe 3

Figure P4.3.2

4.3.3. Determine flow rates in the branching pipe system depicted in Figure P4.3.2 given the following water surface (WS) elevation and pipe data (lengths and diameters): WS1 = 1,600 m WS2 = 1,580 m WS3 = 1,550 m

L1 = 1,200 m L2 = 600 m L3 = 900 m

D1 = 1.2 m D2 = 1.0 m D3 = 1.5 m

All of the pipes are concrete (smooth, e = 0.18 mm) and the temperature of the water is 20°C. Also determine the pressure head (PJ/γ) at the junction (height from J to HJ) if the elevation of the junction (J) is 1,550 m. 4.3.4. Solve Problem 4.3.2 using the Hazen–Williams equation (CHW = 140 for lined ductile iron) for friction losses instead of the Darcy–Weisbach equation. 4.3.5. Solve Problem 4.3.2 using the Manning equation (n = 0.011 for new lined ductile iron) for friction losses instead of the Darcy–Weisbach equation. 4.3.6. The highest reservoir in a three-reservoir branching pipe system (Figure P4.3.2) is inaccessible after a mountain storm. Determine the surface elevation of this reservoir given the following water surface (WS) elevation and pipe data (lengths and diameters): WS1 = ? WS2 = 2,090 m WS3 = 2,020 m

L1 = 1,900 m L2 = 1,200 m L3 = 2,400 m

D1 = 0.30 m D2 = 0.40 m D3 = 0.50 m

All of the pipes are rough concrete (e = 0.6 mm), so complete turbulence is assumed. The actual elevation of the junction (J) is 2,049 m and the pressure at the junction is 97.9 kPa. 4.3.7. Two roof-top cisterns supply a tropical bungalow with shower water. The water surface in the uppermost cistern (A) is 8 m above the ground, and the water surface in the lower cistern (B) is 7 m above the ground. They both supply water to a junction below the lowest cistern's water surface through 3-cm-diameter PVC tubes (n = 0.011). Each tube is 2 m long. A 5-m long supply line leads from the junction through another 3-cm-diameter PVC tube to a pan shower,

148

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Chap. 4

essentially a bucket with holes in the bottom and a water surface elevation 3 m above the ground. What flow rate can be expected in the shower in liters per second? 4.3.8. Three reservoirs are connected to a junction by branching pipes (Figure P4.3.2). An 8,000-ft-long, 3-ft-diameter pipeline carries 75.0 cfs of water from reservoir 1 (elevation 3,200 ft) to junction J. Flow from the junction travels through a 2,000-ft-long, 2.5-ft-diameter pipeline to reservoir 3 (elevation 3,130 ft). Additional flow from the junction travels through a 3,000-ft-long, 2-ft-diameter pipeline to reservoir 2. Determine the water surface elevation of reservoir 2. All of the pipes are made of PVC and have a Hazen–Williams coefficient of 150.

(secTion 4.4) 4.4.1. The total discharge from A to B in Figure P4.4.1 is 0.936 m3/s. Pipe 1 is 100 m long with a diameter of 40 cm, and pipe 2 is 100 m long with a diameter of 50 cm. Determine the head loss between A and B (junctions) and the flow rate in each pipe using (a) Hardy–Cross principles and (b) the method of equivalent pipes. Assume e1 = 2.0 mm and e2 = 1.25 mm for the highly tuberculated pipes, 20°C water, and minor losses are negligible. Hint: A direct solution is possible using the conservation of mass and energy principles.

1 Q1

B Q2 2

Figure P4.4.1

4.4.2. The total discharge from A to B in Figure P4.4.2 is 50.0 cfs (ft3/s). Pipe 1 is 400 ft long with a diameter of 1.5 ft, and pipe 2 is 300 ft long with a diameter of 2 ft. Using (a) Hardy–Cross principles, and (b) the method of equivalent pipes, determine the head loss between A and B and the flow rate in each concrete (good joints) pipe. Assume water at 39°F and account for the two bend losses in each pipe (Kb = 0.2). Hint: A direct solution is possible using the conservation of mass and energy principles.

1 A

B 2

Figure P4.4.2

4.4.3. Refer to Example 4.8 to answer the following questions. (a) Determine the pressure at junction F by accounting for pressure drops in pipes AB, BC, CH, and HF. (Recall that in Example 4.8 we arrived at the pressure at F by a different sequence of pressure drops.) Comment on your answer. (b) Where is the lowest total energy in the system? How can you determine this by inspection (without calculations)? (c) In Example 4.8, it was determined that the pressure at F would be too low to satisfy the customer. What system changes could be made to raise the pressure at junction F? (Hint: Examine the components of the Darcy–Weisbach equation, which was used to compute the head loss.)

Problems

149

(d) Obtain or write computer software appropriate for solving pipe network problems (e.g., EPANET—public domain, US-EPA; WaterCAD/WaterGEM—proprietary, Bentley Systems, Inc.; KYPipe—proprietary, KYPipe LLC; or write a spreadsheet program). Then determine whether your suggestions for increasing the pressure at F work. Check flows and pressures for the existing system first to determine if you are entering the data correctly. 4.4.4. The three-loop water distribution system in Example 4.8 is not functioning effectively. The demand for water at junction F is being met, but not at the pressure required by the industrial customer. The water company has decided to increase the diameter of one pipe in the network by 5 cm. Determine which pipe should be replaced to have the greatest impact on the pressure in the system, specifically the pressure at node F. (Hint: Examine the output table of Example 4.8 for head losses, flow rates, and pipe sizes. One pipe stands out as the best choice, although there is a second pipe that is only slightly worse.) Replace the pipe of your choice (increase the diameter by 5 cm) and determine the pressure increase at junction F. 4.4.5. The water distribution system for an industrial park is schematically depicted in Figure P4.4.5. Water enters the system at junction A from a storage tank (water surface elevation of 355.0 m). The demands on the system are currently at junctions D (0.550 m3/s) and E (0.450 m3/s). All pipes are concrete (e = 0.36 mm) with lengths and diameters provided in the table below. In addition, the elevations of the junctions are given in the table below. Calculate the flow rate in each pipe (initial estimated flows are provided). Also determine if the pressure at each junction exceeds 170 kPa, a requirement of the water company by the industrial park. Assume complete turbulence for friction factors (f) and go through two solution (flow correction) iterations for each loop.

QD A

L2 QE C

Figure P4.4.5

Pipe

Flow (m3/s)

Length (m)

Diameter (m)

e/D

Junction

Elevation (m)

AB AC BD CE CB ED

0.450 0.550 0.500 0.500 0.050 0.050

300 300 400 400 300 300

0.40 0.45 0.40 0.40 0.20 0.20

0.00090 0.00080 0.00090 0.00090 0.00180 0.00180

A B C D E

355.0 315.5 313.8 312.3 314.1

4.4.6. Solve Problem 4.4.5 using the Hazen–Williams equation instead of the Darcy–Weisbach for friction losses. Let CHW = 120 for the concrete pipes. 4.4.7. A three-loop water distribution system is depicted in Figure P4.4.7. The demands on the system are currently at junctions C (6.00 cfs), D (10.0 cfs), and E (12.0 cfs). Water enters the system at junction A from a storage tank with a pressure of 40 psi. Using the pipe network data in the table below, calculate the flow rate in each pipe (initial estimated flows are provided). Also determine if the pressure at any junction drops below 30 psi, the pressure required by the customers. Pipe network analysis software (e.g., EPANET) or a spreadsheet program would be useful.

150

Pipelines and Pipe Networks

D QD

2 1

G QE

Chap. 4

Figure P4.4.7

Pipe

Flow (ft3/s)

Length (ft)

Diameter (ft)

CHW

Junction

Elevation (ft)

AB AC BD CE BF CF FG GD GE

12.00 16.00 8.00 8.00 4.00 2.00 6.00 2.00 4.00

600 600 800 800 400 400 800 400 400

1.50 1.50 1.25 1.25 1.00 1.00 1.25 1.00 1.00

120 120 120 120 120 120 120 120 120

A B C D E F G

325.0 328.5 325.8 336.2 330.2 332.7 333.4

4.4.8. Water companies are often required to deliver water at a minimum pressure. The two-loop water distribution system in Example 4.9 is not functioning effectively. The demand for water at junction F is being met, but not at the pressure required by the industrial customer. (The industrial customer requires water delivery at a pressure head of 20 m.) The water company has decided to increase the diameter of two pipes in the network by 5 cm (pipes BF and GD). Determine the pressure increase at junction F. Pipe network analysis software (e.g., EPANET) or a spreadsheet program would be useful. 4.4.9. Verify that Equation (4.17b) is the proper flow correction equation when the Hazen–Williams formula is used for friction head loss instead of the Darcy–Weisbach formula [i.e., derive Equation (4.17b)]. 4.4.10. Using computer software (e.g., EPANET or your own spreadsheet program), determine the flow rate and head loss in each cast-iron pipe in the network shown in Figure P4.4.10. The demands on the system are at junctions C (0.030 m3/s), D (0.250 m3/s), and H (0.120 m3/s). Water enters at junctions A (0.100 m3/s) and F (0.300 m3/s). The lengths and diameters of the pipes are provided in the table below. As with all computer programs, the results should be checked for accuracy. Spot check a few junctions to see if there is mass balance. Check one or two loops to see if the energy balances. Pipe

Length (m)

Diameter (m)

AB FA BC BD DE EC FG GD GH EH

1,200 1,800 1,200 900 1,200 900 1,200 900 1,200 900

0.25 0.35 0.20 0.35 0.40 0.20 0.35 0.35 0.20 0.25

Problems

151

QC E

D 3

Figure P4.4.10

4.4.11. For Problem 4.4.10, use computer software (e.g., EPANET or your own spreadsheet program) and the Hazen–Williams equation (CHW = 130) to solve for the flows instead of the Darcy–Weisbach equation. As with all computer programs, the results should be checked for accuracy. Spot check a few junctions to see if there is mass balance. Check one or two loops to see if the energy balances. 4.4.12. A planning study is being proposed for the pipe network of Example 4.10. In particular, the water company wants to determine the impact of increasing the outflow at junction F from 0.25 to 0.30 m3/s. Although there is an adequate supply of water to meet this demand, there is a concern that the resulting pressure heads may become too low. Using the Newton method and appropriate computer software, determine if the pipe network can accommodate this increase satisfactorily if a minimum of 10.5 m of pressure head is required at every node. 4.4.13. Using the Newton method and appropriate computer software, analyze the pipe network in the Figure P4.4.13 if HA = 190 ft, HE = 160 ft, HG = 200 ft, QB = 6.0 cfs, QC = 6.0 cfs, QD = 6 cfs, and QF = 12.0 cfs. Suppose the K = 1.0 s2/ft5 for pipes 1,7, and 8 and 3.0 s2/ft5 for the other pipes. Use initial estimates of Q1 = 10 cfs, Q2 = 1.0 cfs, Q3 = 2.0 cfs, Q4 = 1.0 cfs, Q5 = 10 cfs, Q6 = 4.0 cfs, Q7 = 10.0 cfs, and Q8 = 10.0 cfs. HG HE HA E

A QC Q1

L1 Q3

J3 L2

Q2 B

J4 QF

F Figure P4.4.13

(secTion 4.5) 4.5.1. Equation (4.25b) provides a solution for the pressure rise in a pipeline due to a rapid valve closure based on Newton's second law. Derive Equation (4.21), the speed of a pressure wave in a pipe, from this equation [Equation (4.25b)].

152

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Chap. 4

4.5.2. The pressure head rise caused by water hammer can be evaluated by using Equation (4.26). Review the derivation and answer the following questions. (a) What concepts (fundamental principles) are used in the derivation? (b) What limitations are placed on the use of the equation? 4.5.3. The valve in an oil pipeline must be closed rapidly when a leak is detected, but not so fast as to produce a large pressure rise. A 1,250-m-long pipeline carries oil (S.G. = 0.85, bulk modulus of 1.5 * 109 N/m2) through a 0.5-m-diameter steel pipe with expansion joints and a wall thickness of 2.5 cm. The normal discharge rate is 1.65 m3/s. Determine the minimum valve closure time that would prevent a large pressure rise (i.e., rapid closure). 4.5.4. At a mining operation, a 2.0-ft.-diameter, 1,000-ft.-long, concrete pipeline conveys water at a flow rate of 10 cfs from a reservoir to a holding tank. The elevation difference between the two water surfaces is 25.5 ft. A valve at the downstream end of the pipeline (rigid pipe wall, 2 in. thick) controls the flow rate. Determine the pressure head rise in the pipeline (in feet of water) if the valve were closed instantaneously. Since the water comes from a slurry pond, would the pressure head rise change if the sediment laden water had a specific gravity of 1.2? No calculations are required for the last part of the question, just an explanation. 4.5.5. Water flows in a 1,200-m-long, ductile iron pipeline connecting two reservoirs. The flow is controlled by a gate valve just upstream of the lower reservoir. When the valve is fully open in the 0.5-m-diameter pipeline (2.0 cm wall thickness and free to move longitudinally), the flow rate is 1.35 m3/s. If the valve is closed in 1.5 s, determine the maximum water hammer pressure. 4.5.6. A 700-m-long, 2-m-diameter, commercial steel pipeline conveys water from a reservoir to a turbine. The reservoir surface is 150 m above the turbine (where the pressure is assumed to be nearly atmospheric). A gate valve is installed at the downstream end of the pipe. If the gate valve is closed suddenly, determine the total (maximum) pressure the pipeline will be exposed to during the water hammer phenomenon (i.e., operational plus water hammer pressure). The pipeline has a thickness of 10 cm and the longitudinal stresses in the pipe are negligible. 4.5.7. All well-engineered reservoirs have emergency drawdown structures which can be subject to water hammer. Consider a reinforced, smooth concrete drawdown pipe that is 500-ft long with a 2.0-ftdiameter and a wall thickness of 2.0 in. If the gate valve at the end of the pipe is closed suddenly, determine the maximum water pressure (i.e., operational plus water hammer) that will develop in the pipeline. The pipe is anchored at both ends, the water level in the reservoir is 98.4 ft above the outlet, and water discharges into the air prior to valve closure. 4.5.8. Determine the required pipe thickness of an 8-in.-diameter, ductile iron pipeline designed to withstand a water hammer pressure of 250 psi. The operational head is assumed to be negligible, the pipeline is anchored at both ends, and the design flow rate is 1.45 cfs. 4.5.9. A 700-m-long, 2.0-m-diameter steel penstock conveys water from a reservoir to a turbine. The reservoir water surface is 150 m above the turbine and the flow rate is 77.9 m3/s. A gate valve is installed at the downstream end of the pipe. Determine the wall thickness to avoid damage to the pipeline if the gate valve is closed rapidly. Use the equation (P # D = 2τ # thickness) to determine the allowable pressure the pipe can withstand based on hoop stress theory with τ = 1.1 * 108 N/m2. Neglect longitudinal stresses and assume that the operational pressure is minimal compared to the maximum water hammer pressure. 4.5.10. Determine the wall thickness in Problem 4.5.9 if the valve closes in 60 s and the pipe walls are assumed to be rigid.

Problems

153

(secTion 4.6) 4.6.1. By using logic, sketches, and review of the relevant design equations, answer the following questions about surge tanks: (a) Does a surge tank eliminate elevated pressures due to water hammer in the entire pipeline? If not, what portions of the pipeline will still be subject to some increase in pressure? Refer to Figure 4.15. (b) What concepts (fundamental principles) are used in the derivation of Equation (4.31)? (c) What limitations are placed on the use of Equation (4.31)? 4.6.2. A surge tank is installed in a pipeline to protect an electric generator. The circular concrete (good joints) tunnel between the reservoir and the surge tank is 1,600 m long and 1.5 m in diameter. If the maximum flow is 6 m3/s, compute the maximum water rise if the surge tank is 6 m in diameter. Neglect minor losses. 4.6.3. A surge tank is being designed to retard the water hammer in a 3,450-ft-long pipeline with a diameter of 6.50 ft. The design discharge is 460 cfs and the pipe material is smooth concrete. Determine the diameter of the surge tank if the water in the tank is allowed to rise to an elevation 16.5 ft above the feeding reservoir after the flow is suddenly stopped. Neglect minor losses. 4.6.4. A flow of 100 cfs is carried 3-ft-diameter pipeline between a reservoir and a distribution junction. A simple surge tank needs to be installed in the 1,400-ft-long, commercial steel pipeline just upstream from the control valve to protect it from water hammer damage. Compute the maximum water rise if the surge tank is 6.5 ft in diameter. Ignore minor losses. 4.6.5. A 1,500-m-long pipeline requires a surge tank to reduce water hammer pressures. The 2.2-m-diameter pipeline conveys 20.0 m3/s. If the head loss between the reservoir and the surge tank is 15.1 m, determine the size of the surge tank if the allowable surface rise is 8.5 m. 4.6.6. A court case involving pipeline damage hinges on knowing the flow rate at the time of valve closure. A simple surge tank was operational in a 1,500-m-long pipeline to protect a turbine, but the flow gauge had malfunctioned. The pipeline is 2 m in diameter and made of rough concrete. If a 5-m rise was measured in a 10-m-diameter surge tank, what was the flow rate in the pipeline when the water was suddenly stopped? (Hint: Assume complete turbulence in the pipeline.)

(secTion 4.7) 4.7.1 Obtain or write computer software appropriate for solving pipe network problems (e.g., EPANET—public domain, US-EPA; WaterCAD/WaterGEM—proprietary, Bentley Systems, Inc.; KYPipe—proprietary, KYPipe LLC; or write a spreadsheet program). Then solve any of the problems from Section 4.4. 4.7.2 From a reservoir 1,200 m away, a 15-cm-diameter main pipe supplies water to six multistory buildings in an industrial park. The reservoir is 80 m above the datum elevation. The positions of the buildings are shown in Figure P4.7.2. The height and the water demand of each building are as follows: Buildings

Height (m)

9.4

8.1

3.2

6.0

9.6

4.5

Water demand (L/s)

5.0

6.0

3.5

8.8

8.0

10.0

154

Pipelines and Pipe Networks

Gate valve

100 m

70 m

1200 m, 15 cmf

Chap. 4

80 m J

100 m

120 m

Main pipe

80 m

90 m

80 m

130 m A

70 m

C Figure P4.7.2

If commercial steel pipes are used for the network (downstream from the junction J), determine the size of each pipeline to maintain a pressure of 210 kPa at each demand point. A gate valve (K = 0.15 when fully opened) is installed in the main pipe immediately upstream from the junction J. Determine the material to be used for the main pipe. Determine the water hammer pressure if the valve is suddenly closed. What must be the minimum wall thickness of the pipe in order to withstand the pressure?

Siavash Beik/Christopher B. Burke Engineering, LLC

Water Pumps

Water pumps are devices designed to convert mechanical energy to hydraulic energy. In general, water pumps can be classified into two basic categories: 1. turbo-hydraulic pumps, 2. positive-displacement pumps. Turbo-hydraulic pumps move fluids with a rotating vane or another moving fluid. Analysis of turbo-hydraulic pumps involves fundamental principles of hydraulics. The most common types of turbo-hydraulic pumps are centrifugal pumps, propeller pumps, and jet pumps. Positivedisplacement pumps move fluids strictly by precise machine displacements such as a gear system rotating within a closed housing (screw pumps) or a piston moving in a sealed cylinder (reciprocal pumps). Analysis of positive-displacement pumps involves purely mechanical concepts and does not require detailed knowledge of hydraulics. This chapter will only treat the first category, which constitutes most of the water pumps used in modern hydraulic engineering systems. 5.1 Centrifugal (Radial Flow) Pumps The fundamental principle of the centrifugal pump was first demonstrated by Demour in 1730. The concept involves a simple "pump" consisting of two straight pipes joined to form a tee, as shown in Figure 5.1. The tee is primed (filled with water), and the lower end of the tee is 155

156

Water Pumps

Chap. 5

Figure 5.1 Demour's centrifugal pump

submerged. Then the horizontal arms are rotated with sufficient velocity to propel the water from the ends of the tee (i.e., normal acceleration). The exiting water reduces the pressure at the ends of the tee (i.e., creating suction); enough to overcome the friction head loss of the moving water and the position head difference between the ends of the tee and the supply reservoir. Modern centrifugal pumps are constructed with this same hydraulic principle but with new configurations designed to improve the efficiency. Modern centrifugal pumps basically consist of two parts: 1. the rotating element, which is commonly called the impeller; and 2. the housing, which encloses the rotating element and seals the pressurized liquid inside. The power required by the pump is supplied by a motor connected to the shaft of the impeller. The rotary motion of the impeller creates a centrifugal force that enables the liquid to enter the pump at the low-pressure region near the center (eye) of the impeller and to move along the direction of the impeller vanes toward the higher-pressure region near the outside of the housing surrounding the impeller, as shown in Figure 5.2 (a). The housing is designed with a gradually expanding spiral shape so that the entering liquid is led toward the discharge pipe with minimum loss [Figure 5.2 (b)]. In essence, the mechanical energy of the pump is converted into pressure energy in the liquid. The theory of centrifugal pumps is based on the principle of angular momentum conservation. Physically, the term momentum, which usually refers to linear momentum, is defined as the product of a mass and its velocity, or momentum = (mass) (velocity) The angular momentum (or moment of momentum) with respect to a fixed axis of rotation can thus be defined as the moment of the linear momentum with respect to the axis: angular momentum = (radius) (momentum) = (radius) (mass) (velocity) The principle of conservation of angular momentum requires that the time rate of change of angular momentum in a body of fluid be equal to the torque resulting from the external force acting on the body. This relationship may be expressed as torque =

(radius) (mass) (velocity) volume = (radius) ρ a b (velocity) time time

The diagram in Figure 5.3 can be used to analyze this relationship. The angular momentum (or moment of momentum) for a small fluid mass per unit time (ρdQ) is (ρdQ)(V cos α)(r)

Sec. 5.1

Centrifugal (Radial Flow) Pumps

157

Housing Impeller

Vanes Eye

(a) Vd 2 2g Pressure head distribution Hp Pd g

Shaft V i2 2g Pi g

(b) Figure 5.2 (a, b) Cross sections of a centrifugal pump

where V cos α is the tangential component of the absolute velocity depicted in Figure 5.3. For the total fluid mass that enters the pump per unit time, the angular momentum can be evaluated by the following integral: ρ

rV cos α dQ

The torque applied to a pump impeller must equal the difference of angular momentum at the inlet and outlet of the impeller. It may be expressed as follows: ρ

roVo cos αo dQ - ρ

riVi cos αi dQ

(5.1)

158

Water Pumps

Chap. 5

Vo bo ao

uo = rov

Vro yro

bo ao yi bi

Vto

uo= rov

yto

Eye

ui = riv

V ti

Vri ui yti

yri yi bi

Figure 5.3 Velocity vector diagram; inlet side on the bottom and outlet side on the top. (Note: u is the speed of the impeller vane (u = rω); υ is the relative velocity of the liquid with respect to the vane; V is the absolute velocity of the liquid, a vector sum of u and υ. β0 is the vane angle at the exit, βi is the vane angle at the entrance, r = ri is the radius of the impeller eye at the entrance, and r = r0 is the radius of the impeller at the exit.)

For steady flow and uniform conditions around the pump impeller, roVo cos αodQ and riVi cosαi dQ have constant values. Equation 5.1 may be simplified to ρQ(roVo cos αo - riVi cos αi)

(5.2)

Let ω be the angular velocity of the impeller. The power input to the pump (Pi in bold to differentiate it from pressure, P) can be computed as Pi = ωT = ρQω(roVo cos αo - riVi cos αi)

(5.3)

The output power of a pump is usually expressed in terms of the pump discharge and the total energy head that the pump imparts to the liquid (Hp). As previously discussed, the energy head of a fluid can be usually expressed as the sum of the three forms of hydraulic energy head: 1. kinetic (V 2/2g), 2. pressure (P/γ), and 3. elevation (h). Referring to Figure 5.2 (b), we can see that the total energy head that the pump imparts to the liquid is Hp =

V 2d - V 2i Pd - Pi + + hd γ 2g

Sec. 5.1

Centrifugal (Radial Flow) Pumps

159

(Refer to Section 4.2 for an alternative way of determining Hp when a pump operates between two reservoirs.) The pump output power may be expressed as Po = γQHp

(5.4)

The polar vector diagram (Figure 5.3) is generally used in analyzing the vane geometry and its relationship to the flow. As designated previously, the subscripts i and o are used, respectively, for the inlet and exit flow conditions; u represents the peripheral or tangential velocity of the impeller or vane speed; υ represents the water velocity relative to the vane blade (hence, in the direction of the blade); and V is the absolute water velocity. Vt is the tangential component of the absolute velocity, and Vr is the radial component. Theoretically, the energy loss at the inlet reaches its minimum value when water enters the impeller without whirl. This is achieved when the impeller is operated at such a speed that the absolute water velocity at the inlet is in the radial direction. The efficiency of a centrifugal pump depends largely on the particular design of the vane blades and the pump housing. It also depends on the conditions under which the pump operates. The efficiency of a pump is defined by the ratio of the output power to the input power of the pump: ep = Po/Pi = (γQHp)/(ωT)

(5.5)

A hydraulic pump is usually driven by a motor. The efficiency of the motor is defined as the ratio of the power applied to the pump by the motor (Pi) to the power input to the motor (Pm): em = Pi/Pm

(5.6)

The overall efficiency of the pump system is thus e = epem = (Po/Pi) (Pi/Pm) = Po/Pm

(5.7)

Po = ePm

(5.8)

or Efficiency values are always less than unity because of friction and other energy losses that occur in the system. In Figure 5.2, the total energy head at the entrance to the pump is represented by Hi =

Pi V 2i + γ 2g

and the total energy head at the discharge location is Hd = hd +

Pd V 2d + γ 2g

The difference between the two is the amount of energy that the pump imparts to the liquid: Hp = Hd - Hi = a hd +

Pd V 2d Pi V 2i + b - a + b γ γ 2g 2g

(5.9)

160

Water Pumps

Chap. 5

Example 5.1 A centrifugal pump has the following characteristics: ri = 12 cm, ro = 40 cm, βi = 118°, βo = 140°. The width of the impeller vanes is 10 cm and is uniform throughout. At the angular speed of 550 rpm, the pump delivers 0.98 m3/s of water between two reservoirs with a 25-m elevation difference. If a 500-kW motor is used to drive the centrifugal pump, what are the efficiency of the pump and the overall efficiency of the system at this stage of operation? Solution The peripheral (tangential) speeds of the vanes at the entrance and at the exit of the impeller are, respectively, ui = ωri = 2π

550 (0.12 m) = 6.91 m/s 60

uo = ωro = 2π

550 (0.40 m) = 23.0 m/s 60

and the radial velocity of the water may be obtained by applying the continuity equation: Q = AiVri = AoVro, where Ai = 2πriB and Ao = 2πroB. (Note: B is the width of the impeller vane which directly impacts the flow rate.) Therefore, Vri =

Q Q 0.98 = = = 13.0 m/s Ai 2πriB 2π (0.12) (0.1)

Vro =

Q Q 0.98 = = = 3.90 m/s Ao 2πroB 2π (0.4) (0.1)

and from Figure 5.3, we see that Vri = υri and Vro = υro. Given the information now available, we can construct vector diagrams specific to this pump. The vector diagrams shown in Figure 5.3 represent the inlet side (bottom) and the outlet side (top) of the impeller. Each of the three major vectors (u , υ, and V ) is composed of components. The components of υ and V are radial and tangential components and form right triangles. Utilizing the computed values for ui, uo, Vri, Vro, υri, and υro, and the vector diagrams, the remainder of the vectors and angles can be computed: υri 13.0 = = -6.91 m/s tan βi tan 118° υro 3.90 υto = = = - 4.65 m/s tan βo tan 140° υti =

and Vi = 2V 2ri + (ui + υti)2 = 2(13.0)2 + (0.00)2 = 13.0 m/s αi = tan-1

Vri

(ui + υti)

= tan-1 a

13.0 b = 90° 0.00

Thus, cos αi = 0 (note that the absolute water velocity is completely in the radial direction, which minimizes the energy loss at the inlet). Continuing with the vector analysis, Vo = 2V 2ro + (uo + υto)2 = 2(3.90)2 + (18.4)2 = 18.8 m/s αo = tan-1 Thus, cos αo = 0.978.

Vro

(uo + υto)

= tan-1 a

3.90 b = 12.0° 18.4

Sec. 5.2

Propeller (Axial Flow) Pumps

161

Applying Equation 5.3, we get Pi = ρQω(roVo cos αo - riVi cos αi) Pi = (998) (0.98) a2π

550 b[(0.40) (18.8) (0.978) - 0] = 414,000 watts 60

Pi = 414 kW (Note: 1 N # m/s = 1 watt)

Applying Equation 5.4 and assuming the only energy head added by the pump is the elevation (neglect losses in Equation 4.2), we get Hp = HR - HS Po = γQHp = (9.79 kN/m3)(0.98 m3/s)(25 m) = 240 kW From Equation 5.5, the efficiency of the pump is ep = Po/Pi = (240)/(414) = 0.580 (58.0,) From Equation 5.7, the overall efficiency of the system is e = ep em = (Po/Pi) (Pi/Pm) = (0.580)(414/500) = 0.480 (48.0,)

5.2 Propeller (Axial Flow) Pumps A rigorous mathematical analysis for designing propellers based strictly on the energy–momentum relationship is not available. However, the application of the basic principle of impulse momentum provides a simple means of describing their operation. Linear impulse is defined as the integral of the product of the force and the time, dt from t′ to t″, during which the force acts on the body: t″

I =

Lt′

F dt

If a constant force is involved during the time period, T, then the impulse may be simplified to (impulse) = (force) (time) The principle of impulse momentum requires that the linear impulse of a force (or force system) acting on a body during a time interval be equal to the change in linear momentum in the body during that time. (force)(time) = (mass) (velocity change) or (force) =

(mass) (velocity change) (time)

(5.10)

The relationship may be applied to a body of fluid in steady motion by taking a control volume between any two sections as shown in Figure 5.4. The force represents all forces acting on the control volume. The factor, (mass)/(time), can be expressed as the mass involved per unit time (i.e., mass flow rate) or (density) (volume) (mass) = = (density) (discharge) = ρQ (time) (time)

162

Water Pumps

P3 g P1 g

P4 g

P2 g

2 1

Chap. 5

Propeller Figure 5.4 Propeller pump

and the velocity change is therefore the change of the fluid velocity between the two ends of the control volume: (velocity change) = Vf - Vi Substituting the above relationships into Equation 5.10, we have ΣF = ρQ(Vf - Vi)

(5.11)

Figure 5.4 schematically shows a propeller pump installed in a horizontal position. Four sections are selected along the pump's flow conduit to demonstrate how the system can be readily analyzed by the principle of impulse momentum. As the fluid moves from section 1 to section 2, the velocity increases and the pressure drops according to the Bernoulli principle of energy balance: P1 V 21 P2 V 22 + = + γ γ 2g 2g Between sections 2 and 3, energy is added to the fluid by the propeller. The energy is added to the fluid in the form of pressure head, which results in a higher pressure immediately downstream from the propeller. Farther downstream at the exit end of the pump (section 4), the flow condition is more stable, and a slight drop in pressure head may result from both the head loss between sections 3 and 4 and a slight increase in mean stream velocity. Applying the impulse momentum relationship, Equation 5.11, between sections 1 and 4, we may write the following equation: P1A1 + F - P4A4 = ρQ(V4 - V1)

(5.12)

where F is the force exerted on the fluid by the propeller. The right-hand side of Equation 5.12 drops out when the pump is installed in a flow conduit of uniform diameter, yielding F = (P4 - P1)A In this case, the force imparted by the pump is totally used to generate pressure. Ignoring losses and applying the Bernoulli principle between sections 1 and 2 results in

Sec. 5.2

Propeller (Axial Flow) Pumps

163

P1 V 21 P2 V 22 + = + γ γ 2g 2g

(5.13)

and between sections 3 and 4, we may write P3 V 23 P4 V 24 + = + γ γ 2g 2g

(5.14)

Subtracting Equation 5.13 from Equation 5.14 and noting that V2 = V3 for the same crosssectional area, we have P3 - P2 P4 V 24 P1 V 21 = a + b - a + b = Hp γ γ γ 2g 2g

(5.15)

where Hp is the total energy head imparted to the fluid by the pump. The total power output from the pump may be expressed as Po = γQHp = Q(P3 - P2)

(5.16)

The efficiency of the pump may be computed by the ratio of the output power of the pump to the input power from the motor. Propeller pumps are generally used for low-head (under 12 m), high-capacity (above 20 L/s) applications. However, more than one set of propeller blades may be mounted on the same axis of rotation in a common housing to form a multistage propeller pump, as shown in Figure 5.5. In this configuration, propeller pumps are capable of delivering a large quantity of

Figure 5.5 Multistage propeller pump

164

Water Pumps

Chap. 5

water over a great elevation difference. They are usually designed for self-priming operations and are used most frequently for pumping deepwater wells. Example 5.2 A 10-ft-diameter propeller pump is installed to deliver a large quantity of water between two reservoirs with a water surface elevation difference of 8.5 ft. The shaft power supplied to the pump is 2,000 hp. The pump operates at 80% efficiency. Determine the discharge rate and the pressure just upstream of the pump if the pressure just downstream is 12 psi. Assume the pipe size remains uniform throughout. Solution The energy imparted to the flow by the pump is given in Equation 5.5: Po = ep Pi = 0.8 (2,000 hp) = 1,600 hp = 8.80 * 105 [email protected]/s Assuming that friction losses are negligible for this short pipe results in Po = γQHp = γQ Jh + ΣK a

V2 bR 2g

For Ke = 0.5 (entrance coefficient) and Kd = 1.0 (exit coefficient), we have Po = γQHp = γQ Jh + 1.5a

Q2 2gA2

and 8.80 * 105 ft # lb/s = 62.3Q J8.5 + 1.5a

bR Q2

2g(25π)2

Solving the above equation yields

Q = 1,090 cfs Using Equation 5.16, the pressure just upstream of the pump is found to be Po = Q(P3 - P2)

8.80 * 10

ft # lb/s

= 1,090 ft3/s (P3 - 12)144

P3 = 17.6 psi

5.3 Jet (Mixed-Flow) Pumps Jet pumps capitalize on the energy contained in a high-pressure stream of fluid. The pressurized fluid ejects from a nozzle at high speed into a pipeline, transferring its energy to the fluid requiring delivery, as shown in Figure 5.6. Jet pumps are usually used in combination with a centrifugal pump, which supplies the high-pressure stream, and can be used to lift liquid in deep wells. The pumps are usually compact in size and light in weight. They are sometimes used in construction for dewatering the work site. Because the energy loss during the mixing procedure is significant, the efficiency of a jet pump is normally very low (rarely more than 25%). A jet pump can also be installed as a booster pump in series with a centrifugal pump. The jet pump may be built into the casing of the centrifugal pump suction line to boost the water surface elevation at the inlet of the centrifugal pump as shown schematically in Figure 5.7. This arrangement avoids any unnecessary installation of moving parts in the well casing, which is usually buried deep below the ground surface.

Sec. 5.4

Centrifugal Pump Characteristic Curves

165

∆Q Q

Q + ∆Q Throat Nozzle

∆Q Q

Q2 Figure 5.6

Figure 5.7 booster

Jet pump

Jet pump as a

5.4 Centrifugal Pump Characteristic Curves Pump characteristic curves (or performance curves), which are produced and supplied by reputable manufacturers, are graphical representations of a pump's expected operational performance. These manufacturers will test their pumps in the laboratory, and even field verify the results, to ascertain pump operational performance. Different formats are used by different pump manufacturers. However, these curves generally display the variation of the pump head, the brake horsepower, and the efficiency with the pump's generated flow rate. Figure 5.8 depicts typical pump characteristic curves for a centrifugal (radial flow) pump. Similar curves are available for axial flow and mixed-flow pumps, although the shapes of their curves are generally different. The pump head is the energy head added to the flow by the pump. The brake horsepower is the power input required by the pump in power units, and the efficiency is the ratio of the power output to the power input. The pump head at zero discharge is called the shutoff head. The discharge corresponding to the maximum efficiency is called the rated capacity. For variable speed pumps, some manufacturers display the characteristics at various speeds in the same figure. The characteristics of a given pump vary with the rotational speed. However, if the characteristics are known for one rotational speed, then the characteristics for any other rotational speed with the same impeller size can be obtained using the affinity laws (Section 5.10): Q2 Nr2 = Q1 Nr1

(5.17a)

166

Water Pumps

Chap. 5

350 Pump Head

300

250 Eff (%) 86

Head (ft)

Efficiency 200

84 82 80

150

BHP

100

800 600

Brake horsepower

400

200 0 0

Figure 5.8

15 Discharge (cfs)

Typical pump characteristic curves

Hp2 Hp1

= a

Nr2 2 b Nr1

BHP2 Nr2 3 = a b BHP1 Nr1

(5.17b) (5.17c)

in which Q = discharge, Hp = pump head, BHP = brake horsepower, and Nr = rotational speed. The efficiency curve is not affected significantly by the rotational speed. 5.5 Single Pump and Pipeline Analysis A single pump placed in a pipeline to move water from one reservoir to another reservoir or to a demand point represents the most common pump-application scenario. Determining the flow rate that is produced in these pump-pipeline systems requires knowledge of both pump operation and pipeline hydraulics. Consider the pump-pipeline system shown in Figure 5.9. Suppose pipe characteristics, pump characteristics, and the upstream and downstream water surface elevations are given, but the flow rate is unknown. To analyze this system for the flow rate, neglecting the minor losses, we can write the energy equation as HA + Hp = HB + hf

(5.18a)

Sec. 5.5

Single Pump and Pipeline Analysis

167 HB

Hs P Figure 5.9

Single pump and pipeline

or Hp = (HB - HA) + hf

(5.18b)

where Hp is the required pump head and HA and HB are the water surface elevations of the two reservoirs. This expression can be explained as follows. Part of the energy added to the flow by the pump is expended in raising water from elevation HA to HB, and part of it is expended to overcome the flow resistance. With Hs = HB - HA = elevation rise (static head), Hp = Hs + hf

(5.19)

Equation 5.19 can be analyzed further to determine the pump flow rate. Note that in this equation, Hs is constant, whereas hf depends on Q. If more flow is pushed through the system, then more friction losses result and a greater pump head is required. Thus, the right-hand side of this equation can be computed for various flow rates; in so doing, it is referred to as the system head, denoted by HSH. Substituting any friction loss equation (e.g., Darcy–Weisbach, Hazen–Williams, Manning, etc.) for hf will yield a relationship between HSH and Q. A plot of this relationship is called the system head curve. However, our objective is to determine the actual flow rate in the pipeline with the existing pump, not to determine hypothetical pump heads required for different flow rates. To determine the actual flow rate, we can superimpose a plot of the existing pump's characteristic curve (Hp and Q) on the system head curve. The intersection of the two curves (often called the match point) represents the flow rate of that particular pump operating in that particular pipe system. In essence, two equations in two unknowns are being solved graphically.* Once the discharge is determined, we can calculate the velocity and other flow characteristics as well as the energy grade line. The following example problem will help to clarify the solution process. Example 5.3 Consider the pump-pipeline system shown in Figure 5.9. The reservoir water surface elevations are known: HA = 100 ft and HB = 220 ft. The 2.0-ft-diameter pipe connecting the two reservoirs has a length of 12,800 ft and a Hazen–Williams coefficient (CHW) of 100. (a) The pump characteristics are known (columns 1 and 2 in the following table) and are plotted in Figure 5.10 (a). Determine the discharge in the pipeline, the velocity of flow, and the energy grade line. (b) Suppose the pump characteristics given in part (a) are at a rotational speed of 2,000 rpm. Determine the discharge in the pipeline and the pump head if the pump runs at 2,200 rpm. * A graphical solution is required because the Hp versus Q relationship for the pump is generally available in graphical form. If it is available or can be cast in equation form, then the pump characteristic equation is solved simultaneously with the system head equation to produce an identical solution.

168

Water Pumps

Q (cfs)

Hp (ft)

hf (ft)

Hs (ft)

HSH (ft)

0 5 10 15 20 25 30 35

300.0 295.5 282.0 259.5 225.5 187.5 138.0 79.5

0.0 8.1 29.2 61.9 105.4 159.3 223.2 296.8

120.0 120.0 120.0 120.0 120.0 120.0 120.0 120.0

120.0 128.1 149.2 181.9 225.4 279.3 343.2 416.8

Chap. 5

350 System head curve

300

Head (ft)

250 225 200

150 100 Hs

Pump characteristics

50 0 0

Discharge (cfs) Figure 5.10 (a) Single pump and pipeline analysis

Solution (a) For this system, Hs = HB - HA = 220 - 100 = 120 ft. We will use the Hazen–Williams friction formula to calculate the losses resulting from friction. With reference to Table 3.4, the Hazen–Williams formula in the British unit system can be expressed as hf = KQ1.85 where K =

4.73L D4.87C 1.85 HW

4.73(12,800) (2)4.87(100)1.85

= 0.413 s1.85/ft4.55

The friction loss and the system head are calculated for various values of Q as summarized in the preceding table. The system head is then plotted as in Figure 5.10 (a). The intersection of the system head curve and the pump characteristics curve yields Q = 20 cfs and Hp ≈ 225 ft. We can also read from the preceding table that the friction loss is about 105 ft. The velocity is found as V =

Q Q 20 = = = 6.37 ft/s 2 A πD /4 π(2.0)2/4

Sec. 5.6

Pumps in Parallel or in Series

169

The energy head just before the pump is HA = 100 ft (ignoring losses on the suction side of the pump) and just after the pump is HA + Hp = 100 + 225 = 325 ft. The energy head decreases linearly along the pipe to HB = 220 ft at reservoir B. (Note: These problems can be solved quickly and accurately with spreadsheets.) (b) To obtain the pump characteristics at 2,200 rpm, we use Equations 5.17a and 5.17b with Nr1 = 2,000 rpm, Nr2 = 2,200 rpm, Nr2 >Nr1 = 1.10, and (Nr2 >Nr1)2 = 1.21. The calculations are summarized in the following table. Nr1 = 2,000 rpm

Nr2 = 2,200 rpm

Q1 (cfs)

Hp1 (ft)

Q2 (cfs)

Hp2 (ft)

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0

300.0 295.5 282.0 259.5 225.5 187.5 138.0 79.5

0.0 5.5 11.0 16.5 22.0 27.5 33.0 38.5

363.0 357.6 341.2 314.0 272.9 226.9 167.0 96.2

The values of Q2 are obtained by multiplying the Q1 values by 1.10. Likewise, the values of Hp2 are obtained by multiplying the Hp1 values by 1.21. A plot of Q2 versus Hp2 will give the pump characteristics curve at 2,200 rpm as displayed in Figure 5.10 (b). The system head curve will be the same as in part (a). The point of intersection of the two curves yields Q = 23.4 cfs and Hp = 261 ft. 400 System head curve

350 300 261

Head (ft)

250 200

Pump characteristics

150 100 50 0 0

23.4

Discharge (cfs) Figure 5.10

(b) Single pump and pipeline analysis at a different rotational speed

5.6 Pumps in Parallel or in Series As discussed in Section 5.1, the efficiency of a pump varies with the discharge rate of the pump and the total head overcome by the pump. The optimum efficiency of a pump can be obtained only over a limited range of operation (i.e., discharges and total heads). Therefore, it is often

170

Water Pumps

Chap. 5

Pump Head

Pump 1 a

Pumps 1 and 2 in parallel

a Pump 2

Discharge Figure 5.11

Pump characteristics for two pumps in parallel

advantageous to install several pumps in parallel or series configurations in pumping stations to efficiently operate over a broad range of expected flow rates and required system heads. When two pumps are installed in parallel, neglecting the minor losses in their resident branch lines, the energy head added to the flow by the two pumps must be the same to satisfy the energy equation of the resident pipeline system. The discharge through the two pumps will be different, however, unless the two pumps are identical. The total discharge will split between the two pumps such that Hp1 = Hp2 where Hp1 and Hp2 denote the pump heads for the first and second pumps, respectively. To obtain the pump characteristic curves for two-pump parallel configurations, we add the flow rates (abscissa) of the characteristic curves of the individual pumps for each value of pump head as illustrated in Figure 5.11. If the two pumps are identical, we simply double the discharge value for each value of pump head. However, this does not mean that the actual discharge will be doubled when two pumps are operated in parallel (see Example 5.4). The same principles are applied to determine pump system characteristic curves when more than two pumps are installed in parallel. When two pumps are installed in series, the discharge through the two pumps must be the same. However, the pump heads will be different unless the pumps are identical. To obtain the pump characteristic curves for two-pump series configurations, we add the pump heads (ordinates) of the individual characteristic curves for each value of pump flow as shown in Figure 5.12. If the two pumps are identical, we simply double the head values for each value of pump flow. However, this does not mean that the actual head will be doubled when two pumps are operated in series (see Example 5.4). The same principles are applied to determine pump system characteristic curves when more than two pumps are installed in series. Pump combinations add flow and head flexibility to pump-pipeline systems while maintaining high operational efficiency. For example, when flow requirements are highly variable, several pumps can be connected in parallel and switched on and off to meet the variable demand. Again note that two identical pumps operating in parallel may not double the discharge in a pipeline because the total head loss in a pipeline is proportional to the second power of discharge: HP ∝ Q2. The additional resistance in the pipeline will cause a reduction in the total discharge.

Sec. 5.6

Pumps in Parallel or in Series

171

Pumps 1 and 2 in series

Pump Head

b Pump 2

Pump 1 a b

Discharge Figure 5.12

Pump characteristics for two pumps in series

Curve B in Figure 5.13 schematically shows the operation of two identical pumps in parallel. The joint discharge of the two pumps is always less than twice the discharge of a single pump. Pump combinations can also add flexibility when pump head requirements change. For example, in pipeline installations where pipeline losses or the elevation rise are variable, pumps can be connected in series and switched on and off to meet the variable head demand. Curve C in Figure 5.13 schematically shows the operation of two identical pumps connected in series.

Q vs. H

pum ps

System curve Q vs. Hp (two pumps in parallel)

Q vs. H

A 20 10

c ie ffi

ncy

100

E (one

pum p) C d n ) a A rve (cu

100 80

B) urve c ( y ienc Effic

200

300

400 500 Discharge (L/s)

Efficiency (%)

40 30

se ri

) es

Pumb Head (m)

(two

40 20 600

700

800

Figure 5.13 Typical performance curves of two pumps connected in parallel B and in series C

172

Water Pumps

Pumps A

Intake

Discharge

Pumps

Figure 5.14

Chap. 5

Schematic of pump operation either in series or in parallel

The efficiency of two (or more) identical pumps operating in parallel or in series is essentially the same as that of the single pump based on discharge. The installation can be arranged with a separate motor for each pump or with one motor operating two (or more) pumps. Multipump installations can be designed to perform in either series or parallel operations with the same set of pumps. Figure 5.14 is a typical schematic of such an installation. For series operations, valve A is opened and valves B and C are closed; for parallel operations, valve A is closed and valves B and C are open. Example 5.4 Two reservoirs are connected by a 300-m-long, asphalt-lined, cast-iron pipeline, 40 cm in diameter. The minor losses include the entrance, the exit, and a gate valve. The elevation difference between the reservoirs is 10 m, and the water temperature is 10°C. Determine the discharge, head, and the efficiency using (a) one pump, (b) two pumps in series, and (c) two pumps in parallel. Use the pump with characteristics depicted in Figure 5.13. Solution To deliver the water, the pump system must provide a total energy head (HSH) of HSH = Hs + af

L V2 V2 + ΣK b = 10 + (750f + 1.65) D 2g 2g

based on Equation 5.19 with minor losses considered. Using ν = 1.31 * 10-6 m2/s (Table 1.3) and e/D = 0.0003, the following values are computed for a range of discharges within which each pump system may expect to operate. A Q versus HSH curve is constructed based on the values computed in the following table (curve E, Figure 5.13). Q (L/s)

V (m/s)

0 100 300 500 700

0 0.80 2.39 3.98 5.57

NR 2.44 7.30 1.22 1.70

— * * * *

105 105 106 106

HSH (m)

— 0.0175 0.0160 0.0155 0.0155

10.0 10.5 14.0 20.7 31.0

From curve E in Figure 5.13 (using a more refined grid), we obtain the following. (a) For one pump: Q ≈ 420 L/s, Hp ≈ 18 m, and ep ≈ 40, (b) For two pumps in series: Q ≈ 470 L/s, Hp ≈ 20 m, and ep ≈ 15,

Sec. 5.7

Pumps and Branching Pipes

173

(c) For two pumps in parallel: Q ≈ 590 L/s, Hp ≈ 26 m, and ep ≈ 62, Note: Based on the system requirements, two pumps in parallel are the best choice for operation at high efficiency.

5.7 Pumps and Branching Pipes Consider the simple branching pipe system shown in Figure 5.15 in which a single pump delivers flow from reservoir A to two reservoirs (B and C) through pipes 1 and 2. The system heads, HSH1 and HSH2, respectively, for pipes 1 and 2 can be expressed as HSH1 = Hs1 + hf1 HSH2 = Hs2 + hf2 where Hs1 = HB - HA and Hs2 = HC - HA. The total discharge will be split between the two pipes such that HSH1 = HSH2 = Hp. These three heads must be equal to satisfy the energy equation for both pipes. When the two pipes are considered together as a single system, the system head curve of the combined system can be obtained by adding the flow rates (abscissa) of the individual system head curves for each value of head. To analyze a branching pipe system for flow rates, we plot the individual and the combined system head curves as well as the pump characteristics curve. The point of intersection of the pump characteristic curve and the combined system head curve will yield the total discharge and the pump head. The discharge in each pipe is obtained from the individual system head curves at the value of the pump head. As a check, the sum of the two discharges should be equal to the total system discharge already determined. The following example problems will clarify the procedure. HC

Pipe 2 HA

HB Hs2 Pipe 1

H s1

P Figure 5.15

Single pump and two pipes

Example 5.5 In Figure 5.15, HA = 110 ft, HB = 120 ft, and HC = 140 ft. Both pipes have a Darcy–Weisbach friction factor of 0.02. Pipe 1 is 10,000 ft long and has a diameter of 2.5 ft. Pipe 2 is 15,000 ft long and has a diameter of 2.5 ft. The pump characteristics are given in the following table and plotted in Figure 5.16. Determine the discharge in each pipe.

174

Water Pumps

Pipe 2 system head curve

Chap. 5

Combined system head curve

Pipe 1 system head curve

Head (ft)

60 50 40 30 20

Pump Curve

10 0

16.2

28 30 40 Discharge (cfs)

Figure 5.16

44.2

Graphical solution for Example 5.5

Q (cfs)

Hp (ft)

78.5

66.5

42.5

Solution For this system, Hs1 = 120 - 110 = 10 ft and Hs2 = 140 - 110 = 30 ft. With reference to Table 3.4, the Darcy–Weisbach friction formula can be written as hf = KQ2, where K =

0.025fL D5

For pipe 1, K =

0.025(0.02)(10,000) 2.55

= 0.0512 s2/ft5

and for pipe 2 K =

0.025(0.02)(15,000) 2.55

= 0.0768 s2/ft5

Then the system head curves for pipes 1 and 2, respectively, are calculated as HSH1 = 10 + 0.0512Q21 HSH2 = 30 + 0.0768Q22 by assigning various values to Q, solving the above equations for HSH, and plotting them in Figure 5.16. The combined system head curve is obtained by adding the discharges on these curves for the same head. The point intersection of the combined system head curve with the pump characteristics curve yields a total discharge of 44.2 cfs and a pump head of 50.1 ft. For this pump head, we read Q1 = 28.0 cfs and

Sec. 5.7

Pumps and Branching Pipes

175

Q2 = 16.2 cfs from the respective system head curves. The reader should verify these results by checking if the energy equation is satisfied separately for pipes 1 and 2.

Example 5.6 Consider the pump and the branching pipe system shown in Figure 5.17. The reservoir water elevations and the characteristics of pipes 1 and 2 are the same as in Example 5.5. Pipe 3 has a Darcy–Weisbach friction factor of 0.02, a diameter of 3 ft, and a length of 5,000 ft. Determine the discharge in each pipe. HC

Pip HA

HB Hs2 Pipe 1

Pip

Hs1

P Figure 5.17 Branching pipe system of Example 5.6

Solution We can solve this example in the same way as Example 5.5 if we can incorporate the friction losses in pipe 3. The easiest way is to subtract the losses in pipe 3 from the heads of the pump curve for respective discharges. For pipe 3, 0.025(0.02)(5,000)

K =

= 0.0103

Then the head loss pipe 3 is calculated as hf = 0.0103Q2 Calculating the head loss for the tabulated values of Q and subtracting from the respective Hp values will yield the net Hp values. Q (cfs)

Hp (ft) hf (ft) net Hp (ft)

80.0 0.0 80.0

78.5 1.0 77.5

74.0 4.1 69.9

66.5 9.3 57.2

56.0 16.5 39.5

42.5 25.8 16.7

We can now solve the problem just like in Example 5.5, except that we now plot the net pump heads as shown in Figure 5.18. The intersection of the combined system head curve with the net pump characteristics curve (match point) yields a total discharge of 38.2 cfs and a net pump head of 42.8 ft. For this net pump head, we read Q1 = 25.3 cfs and Q2 = 12.9 cfs from the respective system head curves. The actual

176

Water Pumps

Pipe 2 system head curve

Chap. 5

Combined system head curve

Pipe 1 system head curve

Head (ft)

60 50 42.8 40

30 20

Net pump curve

10 0 0

12.9

25.3

38.2 40

Discharge (cfs)

Figure 5.18

Graphical solution for Example 5.6

pump head is 42.8 + 0.0103 (38.2)2 = 57.8 ft. The reader should verify these results by checking if the energy equation is satisfied separately for the flow paths along pipes 1 and 2.

5.8 Pumps and Pipe Networks Pumps are an integral part of many pipe networks. The Hardy–Cross and Newton methods, introduced in Chapter 4, can be modified easily to analyze pipe networks that contain pumps. The modification is applied to the energy equation for the path (or loop) containing the pump. It now must include the head added to the flow by the pump. To facilitate this, the pump characteristics are expressed in a polynomial form such as Hp = a - bQ / Q / - cQ The coefficients a, b, and c are fitting parameters. They can be determined by using three data points from the pump characteristics curve. Example 5.7 The pipe system shown in Figure 5.19 is identical to that of Examples 4.9 and 4.10. However, the demand has increased at junction F (QF = 0.30 m3/s, not 0.25 m3/s), which necessitates adding a pump to the system just downstream of reservoir A. The pump characteristics can be expressed as Hp = 30 - 50Q2 - 5Q

Sec. 5.9

Cavitation in Water Pumps

177

where Hp is in m and Q is in m3/s. Determine the discharge in each pipe using the same initial values as in Example 4.10. HG HA Q8 A

P Q1

J5 QF

Figure 5.19

Q2 J1

J4 Q6

Pipe network for Example 5.7

Solution We will use the same equations as in Example 4.10 except F8 = HA + a - bQ1 / Q1 / - cQ1 - K1Q1 / Q1 / - K2Q2 / Q2 / + K4Q4 / Q4 / + K8Q8 / Q8 / - HG and 0F8 = - 2bQ1 - c - 2K1Q1 0Q1 where a = 30 m, b = 50 s2/m5, and c = 5 s/m2. The results are obtained in five iterations as summarized in the following table. (m3/s) Iteration Number

Initial 1 2 3 4 5

0.2000 0.2994 0.2955 0.2647 0.2650 0.2650

0.5000 0.2577 0.1464 0.1253 0.1245 0.1245

0.1000 0.0417 0.1492 0.1394 0.1405 0.1405

0.0500 0.0659 0.0760 0.0829 0.0843 0.0843

0.5000 0.2236 0.1223 0.1081 0.1088 0.1088

0.1000 0.0347 0.0285 0.0524 0.0507 0.0507

0.3000 0.1347 0.1285 0.1524 0.1707 0.1507

0.2500 0.2006 0.2045 0.2353 0.2350 0.2350

The resulting energy heads are HA = 85.0 m, HB = 96.5 m, Hc = 67.1 m, HD = 78.6 m, HE = 63.2 m, HF = 59.0 m, and HG = 102.0 m. Also, the energy head added to flow by the pump is 25.2 m.

5.9 Cavitation in Water Pumps One of the important considerations in pump installation design is the relative elevation between the pump and the water surface in the supply reservoir. Whenever a pump is positioned above the supply reservoir, the water in the suction line is under pressure lower than atmospheric. The

178

Water Pumps

Chap. 5

phenomenon of cavitation becomes a potential danger whenever the water pressure at any location in the pumping system drops substantially below atmospheric pressure. To make matters worse, water enters into the suction line through a strainer that is designed to keep out trash. This additional energy loss at the entrance reduces pressure even further. A common site of cavitation is near the tips of the impeller vanes where the velocity is very high. In regions of high velocities much of the pressure energy is converted to kinetic energy. This is added to the elevation difference between the pump and the supply reservoir, hp and to the inevitable energy loss in the pipeline between the reservoir and the pump, hL. Those three items all contribute to the total suction head, HS, in a pumping installation as shown schematically in Figure 5.20. The value of HS must be kept within a limit so that the pressure at every location in the pump is always above the vapor pressure of water; otherwise, the water will be vaporized and cavitation will occur. The vaporized water forms small vapor bubbles in the flow. These bubbles collapse when they reach the region of higher pressure in the pump. Violent vibrations may result from the collapse of vapor bubbles in water. Successive bubble breakup with considerable impact force may cause high local stresses on the metal surface of the vane blades and the housing. These stresses can cause surface pitting and will rapidly damage the pump. To prevent cavitation, the pump should be installed at an elevation so that the total suction head (HS) is less than the difference between the atmospheric head and the water vapor pressure head, or HS 6 a

Pvapor Patm b γ γ

The maximum velocity near the tip of the impeller vanes is not assessable by users. Pump manufacturers usually provide a value known commercially as the net positive suction head (NPSH), or H S′ . (The NPSH can be displayed as part of the pump characteristic curves, as shown in Figure 5.24 in the section on pump selection.) NPSH represents the pressure drop between the eye of the pump and the tip of the impeller vanes. With the value of NPSH given, the maximum pump elevation above the supply reservoir can be easily determined by accounting for all of the energy components that make up HS in Figure 5.20. The resulting expression is hp … a

Pvapor Patm V2 b - a H S′ + + hL b γ γ 2g

(5.20)

where hL is the total loss of energy in the suction side of the pump. It usually includes the entrance loss at the strainer, the friction loss in the pipe, and other minor losses. Another parameter commonly used for expressing cavitation potential in a pump is the cavitation parameter, σ, which is defined as σ =

H S′ Hp

(5.21)

where Hp is the total head developed by the pump and the numerator is the NPSH. The increase in velocity through the impeller vanes is accounted for in the parameter σ. The value of σ for each type of pump is usually furnished by the manufacturer and is based on pump test data. Applying Equation 5.21 to the relationships depicted in Figure 5.20, we may write H S′

Pvapor Patm V 2i = σHp = - a + hp + hL b γ γ 2g

(5.22)

Cavitation in Water Pumps

179

V2 2g

V Impeller Eye

Housing

Patm – Pvapor g

HS hc hf

V2 2g

Patm 10 m ≡ g

Sec. 5.9

H'S

Screen

Vapor pressure Vacuum Figure 5.20

Energy and pressure relationship in a centrifugal pump

180

Water Pumps

Chap. 5

where Vi is the speed of the water at the entrance to the impeller. Rearranging Equation 5.22, we get Pvapor Patm V 2i hp = - hL - σHp γ γ 2g

(5.23)

which defines the maximum allowable elevation of the pump intake (entrance to the impeller) above the surface of the supply reservoir. The losses (hL) are on the suction side of the pump. If the value determined by Equation 5.23 is negative, then the pump must be placed at an elevation below the water surface elevation in the supply reservoir. Example 5.8 A pump is installed in a 15-cm, 300-m-long pipeline to pump 0.060 m3/s of water at 20°C. The elevation difference between the supply reservoir and the receiving reservoir is 25 m. The pump has an 18-cm impeller intake diameter, a cavitation parameter of σ = 0.12, and experiences a total head loss of 1.3 m on the suction side. Determine the maximum allowable distance between the pump intake and the water surface elevation in the supply tank. Assume the pipeline has CHW = 120. Solution The friction loss in the pipeline can be determined from Equation 3.31 and Table 3.4 as hf = KQm = [(10.7L)/(D4.87C 1.85)]Q1.85 hf = [10.7 (300)/5(0.15)4.87 (120)1.856](0.06)1.85 = 25.8 m

The only minor loss in the discharge line is the exit (discharge) head loss, where Kd = 1.0. The velocities in the intake pipe and in the main pipeline are, respectively, Vi = Vd =

Q 0.06 = = 2.36 m/s Ai π(0.09)2 Q 0.06 = = 3.40 m/s Ad π(0.075)2

The total pump head can be determined by applying the energy equation as V 21 P1 V 22 P2 + + h1 + Hp = + + h2 + hL γ γ 2g 2g where subscripts 1 and 2 refer to the reservoirs at the supply end and delivery end, respectively. At the surface of the reservoirs, we may write V1 ≅ V2 ≅ 0 and P1 = P2 = Patm, therefore (3.40)2 Hp = (h2 - h1) + hL = 25 + a1.3 + 25.8 + (1) b = 52.7 m 2g

The vapor pressure is found in Table 1.1:

Pvapor = 0.02304 bar = 2,335 N/m2 whereas Patm = 1 bar = 101,400 N/m2 Finally, by applying Equation 5.23, we obtain the maximum allowable height of the pump above the supply reservoir: hp =

Pvapor Patm V 2i - ΣhLs - σHp γ γ 2g

Sec. 5.10

Specific Speed and Pump Similarity

181

(2.36)2 101,400 2,335 - 1.3 - (0.12)(52.7) = 2.21 m 9,790 9,790 2(9.81)

5.10 Specific Speed and Pump Similarity The selection of a pump for a particular purpose is based on the required discharge rate and the head against which the discharge is delivered. To raise a large quantity of water over a relatively small elevation (e.g., removing water from an irrigation canal onto a crop field), a high-capacity, low-stage pump is required. To pump a relatively small quantity of water against great heights (e.g., supplying water to a high-rise building), a low-capacity, high-stage pump is required. The designs of these two pumps are very different. Generally speaking, impellers of a relatively large radius and narrow flow passages transfer more kinetic energy from the pump into pressure head in the flow stream than impellers of smaller radius and large flow passages. Pumps designed with geometry that allows water to exit the impeller in a radial direction impart more centrifugal acceleration to the flow than those that allow water to exit axially or at an angle. Thus, the relative geometry of the impeller and the pump housing determine the performance and the field application of a specific pump. Dimensional analysis, a computational procedure that is described in Chapter 10, shows that centrifugal pumps built with identical proportions but different sizes have similar dynamic performance characteristics that are consolidated into one number called a shape number. The shape number of a particular pump design is a dimensionless number defined as S =

ω2Q (gHp)3/4

(5.24)

where ω is the angular velocity of the impeller in radians per second, Q is discharge of the pump in cubic meters per second, g is the gravitational acceleration in meters per second squared, and Hp is the total pump head in meters. In engineering practice, however, the dimensionless shape number is not commonly used. Instead, most commercial pumps are specified by the term specific speed. The specific speed of a specific pump design (i.e., impeller type and geometry) can be defined in two different ways. Some manufacturers define the specific speed of a specific pump design as the speed an impeller would turn if reduced enough in size to deliver a unit discharge at unit head. This way, the specific speed may be expressed as Ns =

ω2Q H 3/4 p

(5.25)

Other manufacturers define the specific speed of a specific pump design as the speed an impeller would turn if reduced enough in size to produce a unit of power at unit head. This way, the specific speed is expressed as Ns =

ω2Pi H 5/4 p

(5.26)

Most of the commercial pumps manufactured in the United States are currently specified with U.S. conventional units: gallons per minute (gpm), brake horsepower (bhp), feet (ft), and revolutions per minute (rpm). In SI units, cubic meter per second, kilowatts, meters, and radians per second are usually used in the computations. The conversions of specific speed among the U.S., English, metric, and SI units are provided in Table 5.1.

182

Water Pumps Table 5.1

Units United States English Metric SI

Chap. 5

Conversion of Specific Speed

Discharge Units

Head Units

Pump Speed

Equation

Symbol

U.S. gal/min

rev/min

(5.25)

Ns1

Ns1 = 45.6 S

Ns1 = 51.6 Ns3

Imp. gal/min m3/s m3/s

ft m m

rev/min rev/min rad/s

(5.25) (5.25) (5.24)

Ns2 Ns3 S

Ns2 = 37.9 S Ns3 = 0.882 S S = 0.022 Ns1

Ns2 = 43.0 Ns3 Ns3 = 0.019 Ns1 S = 1.134 Ns3

Conversion

Note: g = 9.81 m/s2 = 32.2 ft/s2

S ≈ 11

S ≈ 22

S ≈ 75

S ≈ 140

S ≈ 220

Figure 5.21

Relative impeller shapes and the approximate values of shape numbers, S, as defined in Table 5.1

Normally, the specific speed is defined as the optimum point of operational efficiency. In practice, pumps with high specific speeds are generally used for large discharges at low-pressure heads, whereas pumps with low specific speeds are used to deliver small discharges at highpressure heads. Centrifugal pumps with identical geometric proportions but different sizes have the same specific speed. Specific speed varies with impeller type. Its relationship to discharge and pump efficiency is shown in Figure 5.21.

Sec. 5.11

Selection of a Pump

183

Example 5.9 A centrifugal water pump operating at its optimum efficiency delivers 2.5 m3/s over a height of 20 m. The pump has a 36-cm diameter impeller and rotates at 300 rev/min. Compute the specific speed of the pump in terms of (a) discharge and (b) power if the maximum efficiency of the pump is 80%. Solution The given conditions are Q = 2.5 m3/s, Hp = 20 m, and ω = 300 rev/min. Applying Equation 5.25, we get 30022.5

Ns =

(20)3/4

= 50

At 80% efficiency, the shaft power is Pi = (γQHp)/ep = [(9,790)(2.5)(20)]/0.80 = 6.12 * 105 W (612 kW) Applying Equation 5.26, we get Ns =

3002612 (20)5/4

= 175

Example 5.10 The impeller of the pump in Example 5.9 has a diameter of 0.36 m. What diameter should the impeller of a geometrically similar pump be for it to deliver one-half of the water discharge at the same head? What is the speed of the pump? Solution Applying Equation 5.25, from Example 5.9, we have 1 ω (2.5) A2 Ns = = 50 (20)3/4 ω =

50(20)3/4 (1.25)1/2

= 423 rev/min

By definition, two pumps are geometrically similar if they have the same ratio of water discharge velocity to vane tip peripheral speed. When this is the case, the following relationship will be satisfied: Q1 Q2 = (a) 3 ω1 D 1 ω2D32 Hence, 2.5 3

300 (0.36)

1.25 423 (D2)3

The diameter: D2 = 0.255 m = 25.5 cm.

5.11 Selection of a Pump There are many different types of water pumps, and hydraulic engineers are faced with the task of choosing the proper pump for a particular application. However, certain types of pumps are more suited than others depending on the discharge, head, and power requirements of the pump. For the major pump types discussed in this chapter, the approximate ranges of application are indicated in Figure 5.22.

184

Water Pumps

Chap. 5

300 Positivedisplacement pumps

er ) ep ow

o (h

0 10

or s

Centrifugal pumps

40 20

20 15

Jet (mixed-flow) pumps

10 5 3

5 1

3.0 2.0 1.5

Propeller pumps

1.0 0.5 0.3

1 1.5 2

10 1.5 2

102 1.5 2

103 1.5 2

Discharge (L/s or 10–3 m3/s) Figure 5.22

Discharge, head, and power requirements of different types of pumps

50 Pump Head (m)

Pump Head (m)

00 20 00 10 0 60 0 40 0 20

200 150 100

40 I

III

100

150

200

Q, Flow (L/s) Figure 5.23

Pump model selection chart

250

300

104 1.5

Sec. 5.11

Selection of a Pump

185

The total head that a pump must produce to deliver the requisite flow rate is the sum of the required elevation rise and the head losses incurred in the system. Because friction loss and minor losses in the pipeline depend on the velocity of water in the pipe (Chapter 3), the total head loss is a function of the flow rate. For a given pipeline system (including the pump), a unique system head curve can be plotted by computing the head losses for a range of discharges. The process was discussed in detail in Section 5.5. In selecting a particular pump for a given application, the design conditions are specified and the appropriate pump model is chosen (e.g., Figure 5.23; Pump I, II, III, or IV). The system head curve is then plotted on the pump performance curve (e.g., Figure 5.24) provided by the manufacturer. The intersection of the two curves, called the match point (M), indicates the actual operating conditions. The selection process is demonstrated in the following example. Pump, I

4350 rpm 4050 rpm 3850 rpm 3550 rpm 3250 rpm

3250 rpm 0

4050 rpm 3850 rpm 3550 rpm

30 40 Flow (L/s)

60%

3250 rpm

4050 rpm 3850 rpm 3550 rpm

3250 rpm 80 120 160 Flow (L/s)

Pump, II

4350 rpm

30% 40% 50% 3250 rpm

10 0

4050 rpm 3850 rpm 3550 rpm 4350 rpm 4050 rpm 3850 rpm 3250 rpm 3550 rpm 60 80 Flow (L/s)

100

Figure 5.24

92 69 46 23

Horsepower input

40% 30

re requi NPSH

54%

50 Pump Head (m)

uired

200

35 0 @4

10 5 0

r pm

62% 62%

40 4350 rpm 4050 rpm 3850 rpm 3550 rpm

30 20 10

40% 50% 55%

q NPSH re

4 2 0

Pump Head (m)

50 r @ 43

184 138 92 46

Pump, IV NPSH (m)

55% 4350 rpm 4050 rpm 3850 rpm 3550 rpm 4350 rpm

30 20

9 6 3 0

NPSH (m)

4350 rpm

69 46 23

rpm

60% rpm 4350

3250 rpm

4050 rpm 3850 rpm

3250 rpm 3550 rpm 0

100

150 200 Flow (L/s)

Characteristic curves for several pump models

250

228 171 114 57

Horsepower input

40%

Horsepower input

40%

Pump Head (m)

43%

20% 30%

Pump Head (m)

43% 50

ired requ H S P N

350 @4

NPSH (m)

Horsepower input

4 2 0

50 rpm

40% 50% 55%

3 NPSH required @ 4

NPSH (m)

Pump, III

186

Water Pumps

Chap. 5

Example 5.11 A pump will be used to deliver a discharge of 70 L/s of water between two reservoirs 1,000 m apart with an elevation difference of 20 m. Commercial steel pipes 20-cm in diameter are used for the project. Select the proper pump and determine the operating conditions for the pump based on the pump selection chart (Figure 5.23) and the pump characteristics curves (Figure 5.24), both provided by the manufacturer. Solution For commercial steel pipe, the roughness height, e = 0.045 mm (Table 3.1). The flow velocity in the pipe is V =

0.070 m3 >s Q = = 2.23 m>s A π 2 (0.2 m) 4

and the corresponding Reynolds number at 20°C is NR = and

(2.23 m>s) (0.2 m) VD = = 4.5 * 105 ν 1 * 10-6 m2 >s

e>D = 0.045 mm>200 mm = 2.3 * 10-4 = 0.00023 The friction coefficient can be obtained from the Moody diagram (Figure 3.8) ( f = 0.016). The pipe friction losses are then hf = f a

(2.23)2 L V2 1000 b = 0.016 a b¢ ≤ = 20.3 m D 2g 0.2 2(9.81)

The total head (neglecting minor losses) that the pump must work against is HSH = (∆ elevation) + (friction loss) = 20.0 m + 20.3 m = 40.3 m

From the pump-selection chart provided by the manufacturers (e.g., Figure 5.23), pumps II and III may be used for the project. The system head curve should be determined before making a selection. Q (L/s)

V (m/s)

HSH

50 60 80

1.59 1.91 2.55

3.2 * 105 3.8 * 105 5.1 * 105

0.0165 0.0160 0.0155

10.7 14.9 25.6

30.7 34.9 45.6

Values of HSH versus Q (system head curve) are graphed on the pump characteristic (performance) curves shown in Figure 5.24. Superimposing this system curve over the characteristics of pumps II and III as provided by the manufacturer (Figure 5.24), we find that the possibilities are as follows. First selection: Pump II at 4,350 rpm, Q = 70 L>s, Hp = 40.3 m hence, Pi = 71 hp and efficiency = 52, Second selection: Pump III at 3,850 rpm, Q = 68 L/s, Hp = 39 m

Problems

187

hence, Pi = 61 hp and efficiency = 58, Third selection: Pump III at 4,050 rpm, Q = 73 L/s,

Hp = 42 m

hence, Pi = 70 hp and efficiency = 59, Hydraulically, the final selection should be pump II at 4,350 rpm because it best fits the given conditions. However, one may notice that the second selection (pump III at 3,850 rpm) and the third selection (pump III at 4,050 rpm) also fit the conditions rather closely with higher pump efficiencies. In this case, the selection would best be made based on the considerations of the cost of the pump versus the cost of electricity.

Problems (secTion 5.1) 5.1.1. Define the physical meaning of Vo, βo, and υo depicted in the centrifugal pump vector diagram in Figure 5.3. Also, describe the difference between a centrifugal pump and a propeller (axialflow) pump with regards to flow direction. Is the impulse-momentum principle used to develop flow equations for both pumps? 5.1.2. Determine the flow rate of a water pump that overcomes an energy head of 65 ft. A 750-kW motor drives the pump with an efficiency of 77% and the pump operates with an efficiency of 85%. (Note: 1 kW = 1.34 horsepower (hp) and 1 hp = 550 [email protected]/s) 5.1.3. An earth dam is in danger of failing. A pump is needed to quickly drain the small lake behind the dam. The total energy head required to move water over the top of the dam is 2.43 m. The only pump available is an old 10-cm diameter propeller pump. The power requirement for the motor is 1,000 W and the pump–motor combination has a low efficiency of 50%. Estimate the drawdown (how many centimeters the lake goes down) in the first 2-hour period if the lake has a surface area of 5,000 m2. (Note: Assume that the energy head the pump must overcome does not change much as the lake is lowered in the first 2 hours.) 5.1.4. Answer the following questions about pumps. (a) Referring to Figure 4.3, balance energy between positions 1 and 4 (i.e., the water surfaces of the two reservoirs). Solve for Hp and describe what the energy head added by the pump accomplishes physically in the system. (b) Referring to Figure 4.3, balance energy between positions 2 and 3. Solve for Hp and describe what the energy head added by the pump accomplishes. Assume that the pipe diameter at 2 and 3 are the same. (c) What concepts (first principles) are used in deriving the power input to a centrifugal pump [Equation (5.3)]? 5.1.5. Determine the power requirement (kW) for a motor that is needed to drive the pump installed in a pipeline that moves 2.05 m3/s from reservoir A to reservoir B. The 100-m-long, 80-cm-diameter pipe is made of rough concrete. The water surface of reservoir B is 20 m higher than the water surface of reservoir A. The pump efficiency is 80% and the motor efficiency is 74%. 5.1.6. A pump impellor has an outside radius of 50 cm, an inside radius of 15 cm, and vanes with a uniform opening (width) of 20 cm. When the impellor is rotated at an angular speed of 450 rpm, water exits from the impellor with an absolute velocity of 45 m/s. The angle of the exiting water (αo) is 35°. Determine the torque generated by the exiting flow.

188

Water Pumps

Chap. 5

5.1.7. The absolute velocity of water exiting a centrifugal pump impeller is 120 ft/s. The angle of the exiting water is 55° measured from a radial line originating at the center of the pump impeller. If the impeller radius at the exit is 1.5 ft and the width of the vane is 0.5 ft, determine the angular momentum (torque) of the flow leaving the pump. 5.1.8. A centrifugal pump delivers a flow rate of 70 cfs (ft3/s) in overcoming a head of 33 ft. It has the following specifications: a uniform impeller thickness of 4 in., an inlet radius of 1 ft, an outlet radius of 2.5 ft, βi = 120°, and βo = 135°. If the pump rotates at such a speed that no tangential velocity component of the water exists at the inlet (i.e., shockless entry, αi = 90°), what is the rotational speed (in rpm) of the pump? Also calculate the power input (in hp) to the pump. 5.1.9. A centrifugal pump impeller has an inlet diameter of 50 cm and outlet diameter of 150 cm. With βi = 135° and βo = 150°, the pump is rotating at an angular velocity of 100 rad/s. The impeller has uniform thickness of 30 cm. If the radial velocity component υri is the same magnitude as the tangential velocity component Vti, calculate the discharge of the pump and the power input to the pump.

(secTion 5.5) 5.5.1. A flow throttling valve is installed just upstream of the pump in the pump-pipeline system of Example 5.3. Determine the discharge and pump head if the head loss due to this valve in feet can be expressed as 0.125 Q2 where Q is in cfs. (Note: A spreadsheet will prove useful in filling out the following solution table.) Q (cfs)

Hp (ft)

0 5 10 15 20 25 30 35

300.0 295.5 282.0 259.5 225.5 187.5 138.0 79.5

hf (ft)

Hs (ft)

hvalve (ft)

HSH (ft)

5.5.2. A flow throttling valve is installed just upstream of the pump in the pump-pipeline system of Example 5.3. What is the required head loss due to this valve if the desired discharge in the pipe is 15 cfs? Would this be an efficient system? 5.5.3. A pump delivers water from reservoir A (HA = 47.5 m) to reservoir B (HB = 55.9 m). The pipeline has a length of L = 1,650 m, diameter of D = 0.50 m, and a Darcy–Weisbach friction factor of f = 0.018. Ignoring minor losses, determine the flow rate and the velocity in the pipeline. The pump characteristics are tabulated below. Q (m3/s) Hp (m)

0.00 91.4

0.15 89.8

0.30 85.1

0.45 77.2

0.60 65.9

0.75 52.6

0.90 36.3

1.05 15.7

5.5.4. A 1.75-ft-diameter pipeline conveys water from reservoir A to B. The pipeline is 13,800 ft long and has a Darcy–Weisbach friction factor of 0.016. The water surface in reservoir A is 14.7 ft higher than that of B. Accounting for minor losses (square-edged entrance and exit, swing-type check valve, and two rotary valves), determine the flow rate, pump head, and flow velocity. The pump characteristics are tabulated below. Q (cfs) Hp (ft)

0 300

5 289

10 256

15 201

20 124

25 25

Problems

189

5.5.5. A centrifugal pump delivers water through a 50-cm-diameter, 1,250-m-long commercial steel pipe from reservoir A to B with HA = 120.5 m and HB = 131.4 m. The friction factor varies according to the Moody diagram [alternatively, Swamee–Jain; Equation (3.24a)]. Neglecting minor losses, determine the discharge and pump head in the pump-pipeline system using the pump performance characteristics tabulated below. Q (L/s) Hp (m)

0 30.0

100 29.5

200 28.0

300 25.0

400 18.0

500 6.0

5.5.6. A centrifugal pump is installed in a pipeline to raise water 14.9 m into an elevated holding tank. The length of the ductile iron pipeline (f = 0.019) connecting the reservoirs is 22.4 m. The pipe is 5.0 cm in diameter and the performance curve of the pump is given by Hp = 23.9 - 7.59 Q2 where Hp is in meters, Q is in liters per second, and the equation is valid for flows up to 1.5 L/s. Using this pump, what flow do you expect in the pipeline if minor losses are ignored? What pump head is required for this flow? 5.5.7. In the transmission system shown in Figure P5.5.7, HA = 100 ft and HD = 150 ft. The Darcy– Weisbach friction factor 0.02 for all the pipes. Pipe AB is 4,000 ft long with a diameter of 3 ft, and pipe CD is 1,140 ft long with a diameter of 3 ft. The branch BC1 has a length of 100 ft and a diameter of 2 ft, and branch BC2 is 500 ft long with a diameter of 1 ft. The pump characteristics are tabulated below. Determine the system flow rate, pump head, and the discharge in branches 1 and 2. Q (cfs) Hp (ft)

0 60

10 55

20 47

30 37

40 23

50 7

D 1

B P A

Figure P5.5.7

(secTion 5.6) 5.6.1. The table below provides the results of a pump performance test. Discharge (gpm)* Dynamic head (ft)

0 100

200 96

400 88

600 72

800 46

1,000 6

*A common flow unit for pumps in the U.S. is gallons per minute (gpm).

190

Water Pumps

Chap. 5

(a) (b) (c) (d)

Plot the pump characteristic (performance) curve. Plot the characteristic curve for two pumps in series. Plot the characteristic curve for two pumps in parallel. What pump configuration would work for a required flow of 600 gpm, which must overcome a head of 140 ft? (e) What pump configuration would work for a required flow of 1,200 gpm, which must overcome a head of 140 ft? 5.6.2. A 1.5-ft-diameter pipeline conveys water from reservoir A to B. The pipeline is 13,800 ft long and has a Darcy–Weisbach friction factor of 0.02. The water surface in reservoir B is 196 ft higher than that of A. Neglecting minor losses, determine the flow rate and energy head added to the flow by each pump if two identical pumps are used in series. The pump characteristics are tabulated below. Q (cfs) Hp (ft)

0 300

5 289

10 256

15 201

20 124

25 25

5.6.3. Two identical pumps are connected in series and deliver water through a 50-cm-diameter, 1,000-m-long, commercial steel pipe from reservoir A (HA = 52.1 m) to reservoir B (HB = 98.7 m). Neglecting minor losses, compute the discharge, the total head added by the pumps, and the total power output (in hp) of each pump. The characteristics for the two pumps are tabulated below. Assume that the water is at 20°C and use the Swamee–Jain Equation (3.24a) to determine the friction factor. Q (L/s) Hp (m)

0 30.0

100 29.5

200 28.0

300 25.0

400 19.0

500 4.0

5.6.4. A 2.0-ft-diameter, 3,000-ft-long pipeline (CHW = 100) connects two reservoirs. Two identical pumps are used in parallel to move water through the pipeline from reservoir A to reservoir B where HA = 840 ft and HB = 921 ft. Ignoring minor losses, determine the discharge in the pipeline and the pump head. The pump characteristics are provided in the table below. Q (cfs) Hp (ft)

0 300

5 296

10 286

15 264

20 230

25 180

30 110

35 30

5.6.5. A pump-pipeline system delivers water from reservoir A to B with HA = 45.5 m and HB = 92.9 m. The pipe has a length of L = 1,860 m, diameter of D = 0.50 m, and a Darcy–Weisbach friction factor of f = 0.02. Minor losses include an inlet, exit, and swing-type check valve. The pump characteristics are tabulated below. When a single pump is used in the pipeline, the flow rate is 0.50 m3/s with a pump head of 73 m. Determine the flow rate in the pipeline if two identical pumps are used in a parallel combination. Also, determine the flow rate in the pipeline if two identical pumps are used in a series combination. Q (m3/s) Hp (m)

0.00 91.4

0.15 89.8

0.30 85.1

0.45 77.2

0.60 65.9

0.75 52.6

0.90 36.3

1.05 15.7

(secTion 5.7) 5.7.1. A branching pipeline-pump system is configured like Figure 5.15 with HA = 8 m, HB = 16 m, and HC = 20 m. Both pipes have a Hazen–Williams coefficient of 100. Pipe 1 is 1,000 m long and has a diameter of 1.0 m. Pipe 2 is 2,000 m long and has a diameter of 1.0 m. The pump characteristics are tabulated below. Determine the discharge in each pipe.

Problems

191

0 30.0

Q (m3/s) Hp (m)

1 29.5

2 28.0

3 25.5

4 22.0

5 17.5

6 12.0

7 5.0

5.7.2. A branching pipeline-pump system is shown in Figure P5.7.2. The water surface elevation at D is HD = 100 ft, and at C it is HC = 124 ft. There is a pump between B and C, and the pump characteristics are tabulated below. Every pipe in the system has a length of 1,000 ft, a diameter 2 ft, and a friction factor of 0.02. The discharge in pipe DB is 50 cfs from D to B. Also, it is known that flow direction in pipe BC is from B to C. Determine (a) the discharge in pipe BC, (b) the discharge and the flow direction in pipe AB, and (c) the water surface elevation HA. HC

Figure P5.7.2 Q (cfs) Hp (ft)

0 125

10 120

20 108

30 85

40 55

50 15

5.7.3. A branching pipeline-pump system is shown in Figure P5.7.3. Each pipe has the following properties; D = 1 m, L = 1,600 m, and f = 0.02. The two pumps are identical and the pump characteristics are tabulated below. The water surface elevation in reservoir A is 26.2 m and that in reservoir D is 33.5 m. The discharge in pipe BC is 500 L/s in the direction from B to C. Determine (a) the discharge in AB and BD, (b) the water surface elevation in reservoir C, and (c) the head added to the flow by each pump. Q (L/s) Hp (m)

0 21.5

250 20.5

500 19.0

750 16.5

1,000 13.0

1,250 8.5

1,500 3.0

P A

Figure P5.7.3

192

Water Pumps

Chap. 5

5.7.4. Determine the discharge in each pipe in Figure P5.7.4 if HA = 100 ft, HB = 80 ft, and HC = 120 ft. The pipe and pump characteristics are tabulated below. Pipe Characteristics Pipe

L (ft)

D (ft)

8,000

0.02

9,000

0.02

15,000

2.5

0.02

Pump Characteristics Q (cfs)

Hp1 (ft)

Hp2 (ft)

0.0

200.0

150.0

10.0

195.0

148.0

15.0

188.8

145.5

20.0

180.0

142.0

25.0

168.8

137.5

30.0

155.0

132.0

40.0

120.0

118.0

50.0

75.0

100.0

HB P

Figure P5.7.4

(secTion 5.9) 5.9.1. Refer to Figure 5.20 in answering the following questions concerning cavitation: (a) What types of energy losses are encountered on the suction side of the pump? (b) The design engineer must avoid cavitation problems when designing a pipeline containing a pump. What parameter does the designer have the most control over to eliminate the likelihood of cavitation? (c) Where is cavitation most likely to occur in a pump installation?

Problems

193

5.9.2. A pump delivers water at 20°C between a reservoir and a water tank 20 m higher. The suction side contains an entrance strainer (Ks = 2.5), three 90° bends (R/D = 2), and 10 m of ductile-iron pipe, 25 cm in diameter. The discharge side includes a 160-m-long, ductile-iron pipe, 20 cm in diameter, and a gate valve. The friction factor for the pipeline is 0.02, the net positive suction head is 7.5 m, and the design discharge is 170 L/s. Determine the allowable elevation difference between the pump and the reservoir water surface to avoid cavitation. 5.9.3. A pump delivers 6.0 cfs of 68°F water to a holding tank 65 ft above the supply reservoir. The suction side possesses a strainer (Ks = 2.5), a foot valve (Kv = 0.1), and 35 ft of cast-iron pipe, 10 in. in diameter. Determine the allowable height the pump can be placed above the supply reservoir to avoid cavitation if the NPSH is 15 ft. (Note: The strainer incorporates the entrance loss.) 5.9.4. A pump-pipeline system in a desert environment transports 40°C water at a flow rate of 1.40 cfs to a farm for irrigation purposes. The pump pushes the water through a 6-in.-diameter, 1,250-ft-long pipeline from the supply reservoir (water surface elevation of 280 ft) to a holding tank (water surface elevation of 198 ft). The pump's cavitation parameter is 0.08, and the total head loss between the inlet and the suction side of the pump is 1.6 ft. Determine the maximum height of the pump intake relative to the supply reservoir water surface elevation to avoid cavitation. Assume that the pipeline has a Hazen–Williams coefficient of 130. 5.9.5. A flow rate of 2.19 m3/s is required to fill an elevated tank from a supply reservoir with a maximum water surface elevation difference of 55 m. This is accomplished by installing a pump in an 80-cmdiameter, rough-concrete pipeline that is 250 m long. The pump is placed outside the supply reservoir (0.9 m below the water surface of the supply reservoir) and has a cavitation parameter of 0.15. Determine the maximum distance in meters that the pump could be installed away from the supply reservoir (i.e. allowable length of the suction line) without encountering cavitation problems. The only minor losses are a square-edged entrance loss and an exit loss. The water temperature is assumed to be 20°C. 5.9.6. A pump is tested by a manufacturer to establish the NPSH. At the onset of cavitation, the pump is positioned 10.2 ft above the supply reservoir. The suction line is 10 ft long (f = 0.014) with a 1-ft diameter. Minor loss coefficients in the suction line add up to 3.5. If the pumping rate is 3,000 gpm (gallons per minute), determine the NPSH. Assume that the water temperature is 68°F (20°C). 5.9.7. The efficiency of a pump will drop suddenly if cavitation takes place in the pump. This phenomenon is observed in a particular pump (with σ = 0.08) operating at sea level when the pump is delivering 0.42 m3/s of water at 40°C. Determine the sum of the gauge pressure head and the velocity head at the inlet of the pump (i.e., the sum, not the individual components). The total head delivered by the pump is 85 m and the suction pipe diameter is 30 cm.

(secTion 5.10) 5.10.1. A tested pump operating at optimum efficiency has a specific speed based on unit discharge of 2160. A geometrically similar pump produces a flow rate of 1,500 gal/min (U.S.) while operating at 185 rad/s and an efficiency of 75%. Determine the power output from the pump in units of horsepower. 5.10.2. Answer the following questions about specific speed and pump similarity. (a) Using both U.S. and SI units, show that the shape number given in Equation (5.24) is dimensionless. (b) Is specific speed a dimensionless number [Equations (5.25) and (5.26)]? (c) Is it possible to derive Equation (5.25) from Equation (5.26) based on the relationship between power and flow rate? Will the specific speed defined by the two different relationships be identical?

194

Water Pumps

Chap. 5

(d) In Example 5.10, Equation (a) was used to determine the impeller diameter. Derive this equation based on the premise that geometrically similar pumps have the same ratio of water discharge velocity to vane tip peripheral speed. 5.10.3. Determine the efficiency and shaft power of a pump that operates at a speed of 1,760 rpm with a flow rate of 0.21 m3/s. The pump is geometrically similar to another pump that has a specific speed of 77 (based on unit discharge) and 280 (based on unit power). Note: The specific speeds are based on ω in rpm. 5.10.4. A pump is required for a U.S. field application with the following specifications: a flow rate of 12.5 cfs (ft3/s) against a head of 95 ft. To design the pump, a model is built with a 6-in. diameter impeller and tested under optimum conditions. The test results show that at a speed of 1,150 rpm the pump requires 3.1 hp to discharge 1 cfs against a head of 18 ft at 65.6% efficiency. Determine the power requirement, diameter, and speed of a geometrically similar pump for the field. 5.10.5. The design of a centrifugal water pump is studied by a 1/10 scale model in a hydraulic laboratory. At the optimum efficiency of 89% the model delivers 75.3 L/s of water against a 10-m head at 4,500 rpm. If the prototype pump has a rotational speed of 2,250 rpm, determine the discharge and efficiency required to operate the pump under this condition.

(secTion 5.11) 5.11.1. Referring to Example 5.11, plot the system head curve with a spreadsheet program. On the same graph, plot the pump characteristic curve for pump III at a speed of 4,050 rpm. Now answer the following questions. (a) What is the shape of the system curve? Why does it take this shape? (b) What is the shape of the characteristic curve? Why does it take this shape? (c) What flow rate produces the highest total head on the pump characteristic curve? What physical meaning does this have? (d) What is the intersection point of the two curves? Does it meet the design conditions? 5.11.2. A 150-m-long, 35-cm-diameter, concrete (smooth) pipeline raises water 40 m from a water supply reservoir to an elevated tank. A design flow rate is 120 L/s. Select the best pump (based on highest efficiency) from Figures 5.23 and 5.24 and determine its operating conditions (ω, Q, Hp, and e). Assume complete turbulence and minor loss coefficients that add up to 4.7. 5.11.3. Select the pump with the highest efficiency from Figures 5.23 and 5.24 in your book based on the following conditions. A 1,500 m-long, 15-cm diameter pipeline ( f = 0.02) connects two reservoirs (elevation difference of 35 m). Minor losses include entrance, exit, and a globe valve. The design flow is 20 L/s. Determine the speed, discharge, head, and efficiency for the selected pump. 5.11.4. Determine the working conditions (Q, Hp, e, and Pi) of a pump capable of moving water (68°F) from reservoir A to reservoir B, a water surface rise of 100 ft. The old 1.0-ft-diameter pipe connecting the two reservoirs has a length of 7,800 ft and a Hazen–Williams coefficient of 90. The pump characteristics are available in Figure P5.11.4. Also verify that the pump efficiency obtained from the pump characteristic curves corresponds with the pump efficiency obtained from efficiency equations.

195 200

100

180

160

120 100

He ad Effi cie nc y

Pump Head (ft)

140

cap

ac it y

50 40

80 60

ut ower inp Horsep

40 20 0

Tested at 1,450 rpms 200

400 600 Flow (gpm)

800

Efficiency(per cent); Horsepower

Problems

10 0 1000

Figure P5.11.4

5.11.5. Select two different pumps (model and operating conditions) from Figure 5.24 that are capable of supplying water (20°C) to a reservoir at a flow rate of 40 L/s. The water must be raised 27.9 m from a supply reservoir to an elevated tank using a 100-m-long, 15-cm-diameter, galvanized iron pipeline that contains a lift-type check valve. Determine the working conditions and the pump efficiencies. Use the Swamee–Jain Equation (3.24a) to determine friction factors and a spreadsheet to determine the system curve at flows of 0, 20, 40, and 60 L/s. 5.11.6. A project requires a pump that will operate with a minimum discharge of 20 L/s against an elevation head of 39 m. The distance between the supply and delivery points is 150 m. A ball check valve will be used in the system that consists of commercial steel pipes. Determine the most economical pipe diameter for the pipeline and select a single pump from Figure 5.24 (including its operating conditions) if the total cost can be expressed as C = D1.75 + 0.75P + 18 where D is the pipe diameter in centimeters and P is the pump horsepower input. 5.11.7. A pump-pipeline system is designed to transport water (20°C) from a reservoir to an elevated storage tank at a minimum discharge of 300 L/s. The difference in water surface elevations is 15 m, and a 1,500-m-long, wrought-iron pipe that is 40 cm in diameter is used. Select the pump(s) from the set given in Figure 5.24. Determine the number of pumps, the configuration (series or parallel), discharge, total head, and efficiency at which the pumps operate. Ignore minor losses.

196

Water Pumps

Chap. 5

ProjecT Problem 5.11.8. A pumping system is designed to pump water from a 6-m-deep supply reservoir to a water tower 40 m above the ground. The system consists of a pump (or a combination of pumps), a 20-m long pipeline with one elbow on the suction side of the pump, a 60-m long pipeline, a gate valve, a check valve, and two elbows on the delivery side of the pump. The system is designed to pump 420 L/s of water, operating 350 days a year. Select a pump (or pumps) based on the characteristics provided in Figures 5.23 and 5.24, and a pipe size for optimum economy. Hazen–Williams coefficient 100 can be assumed for all pipe sizes listed below, and all elbows are 90° (R/D = 2.0). Power cost is $0.04/kW-hr. Motor efficiency is 85% for all sizes listed. Pumps

Cost ($)

Motor (hp)

Cost ($)

700

200

800

250

III

900

180

300

1020

250

340

Appurtenances

Size/Cost 20 cm

25 cm

30 cm

($)

Pipe (10 m)

120

150

180

Elbow

Gate valve

120

Check valve

105

130

Siavash Beik/Christopher B. Burke Engineering, LLC

Water Flow in Open Channels

Open-channel flow differs from pipe flow in one important aspect. Pipe flow fills the entire conduit, so the flow boundaries are fixed by the conduit geometry. In addition, pipe flow possesses hydraulic pressure that varies from one section to another along the pipeline. Open-channel flow has a free surface that adjusts itself depending on the flow conditions. The free surface is subject to atmospheric pressure, which remains relatively constant throughout the entire length of the channel. Therefore, open-channel flow is driven by the component of the gravitational force along the channel slope. Notice that the channel slope will appear in all the open-channel flow equations, whereas the pipe flow equations include only the slope of the energy grade line. In Figure 6.1, open-channel flow is schematically compared to pipe flow. Figure 6.1 (a) shows a pipe flow segment with two open-ended vertical tubes (piezometers) installed through the pipe wall at an upstream section, 1, and a downstream section, 2. The water level in each tube represents the pressure head (P/γ) in the pipe at the section. A line connecting the water levels in the two tubes represents the hydraulic grade line (HGL) between these sections. The velocity head at each section is represented in the familiar form, V 2/2g, where V is the mean velocity, V = Q/A, at the section. The total energy head at any section is equal to the sum of the elevation (potential) head (h), the pressure head (P/γ), and the velocity head (V 2/2g). A line connecting the total energy head at the two sections is called the energy grade line (EGL). The amount of energy lost when water flows from section 1 to section 2 is indicated by hL. 197

198

Water Flow in Open Channels 1

Chap. 6

V 21 2g

EGL

V 22 2g

HGL P1 g

P2 g V

h2 Datum line (a)

V 12 2g

2 EGL

WSL

P g V

y y'

z1 Datum line

V 22 2g

(b) Figure 6.1

Comparison of: (a) pipe flow and (b) open-channel flow

Figure 6.1 (b) shows an open-channel flow segment. The free water surface is subjected to only atmospheric pressure, which is commonly referred to as the zero pressure reference in hydraulic engineering practice. The pressure distribution at any section is directly proportional to the depth measured from the free water surface. In this case, the water surface line corresponds to the hydraulic grade line in pipe flow. To solve open-channel flow problems, we must seek the interdependent relationships between the slope of the channel bottom, the discharge, the water depth, and other channel characteristics. The basic geometric and hydraulic definitions used to describe open-channel flow through a channel section are: Discharge (Q) Flow area (A) Average velocity (V) Flow depth (y) Top width (T)

Volume of water passing through a flow section per unit time Cross-sectional area of the flow Discharge divided by the flow area: V = Q>A Vertical distance from the channel bottom to the free surface Width of the channel section at the free surface

Sec. 6.1

Open-Channel Flow Classifications

Wetted perimeter (P) Hydraulic depth (D) Hydraulic radius (Rh) Bottom slope (S0) Side slope (m) Bottom width (b)

199

Contact length of the water and the channel at a cross section Flow area divided by the top width: D = A>T Flow area divided by the wetted perimeter: Rh = A>P Longitudinal slope of the channel bottom Slope of channel sides defined as 1 vertical over m horizontal Width of the channel section at the bottom

Table 6.1 depicts the cross-sectional characteristics for various types of channel sections and their geometric and hydraulic relationships. 6.1 Open-Channel Flow Classifications Open-channel flow may be classified by space and time criteria. Based on the space criterion, an open channel characterizes uniform flow if the water depth remains the same throughout a length of channel reach at a given time. Uniform flow is more likely to occur in prismatic channels, channels where the cross-sectional area and bottom slope do not change over the channel reach. An open channel characterizes varied flow if the water depth or the discharge change along the length of the channel. Varied flow can further be classified as gradually varied flow or rapidly varied flow, depending on whether the changes in the flow depth are gradual or abrupt. Examples of uniform, gradually varied, and rapidly varied flows are shown in Figure 6.2 (a), where flow enters a channel with a mild slope from underneath a sluice gate. The flow will reach its minimum depth immediately after the sluice gate and will become gradually varied further downstream. The depth will then change rapidly through a hydraulic jump and will remain constant afterward. Based on the time criterion, open-channel flow may be classified into two categories: steady flow and unsteady flow. In steady flow, the discharge and water depth at any section in the Sluice gate

Gradually varied flow

Rapidly varied flow

Uniform flow

Hydraulic jump

(a)

(b)

(c)

Figure 6.2 Classifications of open-channel flow: (a) gradually varied flow (GVF), rapidly varied flow (RVF), and uniform flow (UF); (b) unsteady varied flow; (c) unsteady varied flow

Section Type

200

Table 6.1

Cross-Sectional Relationships for Open-Channel Flow

Area (A)

Wetted perimeter (P)

Hydraulic Radius (Rh)

Top Width (T)

Hydraulic Depth (D)

b + 2y

by b + 2y

(b + my)y

b + 2y21 + m2

(b + my)y

b + 2my

(b + my)y b + 2my

2my

y 2

Rectangular T y b

Trapezoidal T 1

b + 2y21 + m

Triangular 1

my 2

2y21 + m2

my 2

221 + m

Circular (θ is in radians) T

1 (2θ - sin 2θ)d 20 8

θd0

(sin θ)d0 or 22y(d0 - y)

1 2θ - sin 2θ a bd0 8 sin θ

Chap. 6

Source: Based on V. T. Chow, Open Channel Hydraulics (New York: McGraw-Hill, 1959).

1 sin 2θ a1 bd0 4 2θ

Water Flow in Open Channels

Sec. 6.2

Uniform Flow in Open Channels

201

reach do not change with time during the period of interest. In unsteady flow, the discharge and the water depth at any section in the reach change with time. Uniform flows in open channels are mostly steady; unsteady uniform flows are very rare in nature. Varied open-channel flow may be either steady or unsteady. A flood wave [Figure 6.2 (b)] and a tidal surge [Figure 6.2 (c)] are examples of varied unsteady flows. 6.2 Uniform Flow in Open Channels Uniform flow in an open channel must satisfy the following conditions: 1. The water depth, flow area, discharge, and the velocity distribution must remain unchanged in all sections of the entire channel reach. 2. The EGL, the water surface, and the channel bottom must be parallel to each other. Based on the second condition, the slopes of these lines are the same, Se = SW.S. = S0 as shown in Figure 6.3. Water in an open channel can reach the state of uniform flow only if no acceleration (or deceleration) takes place between sections. This is only possible when the gravity force component and the resistance to the flow are equal and opposite in direction in the reach. Therefore, for a uniform open channel, a free-body diagram can be taken between two adjacent sections (the control volume) to show the balance of the force components of gravity and resistance (Figure 6.3). The forces acting on the free body in the direction of the flow include: 1. the hydrostatic pressure forces, F1 and F2, acting on the control volume; 2. the weight of the water body in the reach, W, which has a component, W sin θ, in the direction of the flow; and 3. the resistance force, Ff, exerted by the channel (bottom and sides) on the flow.

EGL

SW.S.

Water surfac e

W sin u

W u

F2 Channel bottom

Ff L Datum line 1 Figure 6.3

2 Force components in uniform open-channel flow

202

Water Flow in Open Channels

Chap. 6

A summation of all these force components in the direction of the channel yields F1 + W sin θ - F2 - Ff = 0

(6.1a)

This equation can be further simplified because there is no change in water depth in uniform flow. Therefore, the hydrostatic forces at the two ends of the control volume must be equal, F1 = F2. The total weight of the water body is W = γAL where γ is the unit weight of water, A is the cross-sectional area normal to the flow, and L is the length of the reach. In most open channels, the channel slopes are small and the approximation, sin θ = tan θ = S0, is made. The gravity force component may thus be expressed as W sin θ = γALS0

(6.1b)

The resistance force exerted by the channel boundaries may be expressed in terms of resisting force per unit area (i.e., shear stress) multiplied by the total channel bed area that is in contact with the flowing water. This channel contact area is the product of the wetted perimeter (P) and the length of the channel reach (L). In 1769, French engineer Antoine Chezy assumed that the resisting force per unit area of the channel bed is proportional to the square of the mean velocity, KV 2, where K is a constant of proportionality. The total resistance force may thus be written as Ff = τ0PL = KV 2PL

(6.1c)

where τ0 is the resisting force per unit area of the channel bed, also known as the wall shear stress. Substituting Equation 6.1b , and Equation 6.1c into Equation 6.1a , we have γALS0 = KV 2PL or V =

γ A a b a b S0 A K P

In this equation, A/P = Rh, and 2γ/K may be represented by a constant, C. For uniform flow, S0 = Se, the above equation may thus be simplified to V = C2RhSe

(6.2)

in which Rh is the hydraulic radius of the channel cross section. The hydraulic radius is defined as the water area divided by the wetted perimeter for all shapes of open-channel cross sections. Equation 6.2 is the well-known Chezy's formula for open-channel flow. Chezy's formula is probably the first formula derived for uniform flow. The constant C is commonly known as Chezy's resistance factor; it was found to vary in relation to both the conditions of the channel and the flow. Over the past two and a half centuries, many attempts have been made to determine the value of Chezy's C. The simplest relationship, and the one most widely applied in the United States, derives from the work of an Irish engineer, Robert Manning (1891 and 1895).* Using the

* Robert Manning, "On the Flow of Water in Open Channels and Pipes," Transactions, Institution of Civil Engineering of Ireland, 10 (1891), 161–207; 24 (1895), 179–207.

Sec. 6.2

Uniform Flow in Open Channels

203

analysis performed on his own experimental data and on those of others, Manning derived the following empirical relation: C =

1 1/6 R n h

(6.3)

in which n is known as Manning's coefficient of channel roughness. Some typical values of Manning's coefficients are given in Table 6.2. Substituting Equation 6.3 into Equation 6.2 , we have Manning's equation: V =

kM 2/3 1/2 R S n h e

(6.4)

where kM = 1.00 m1/3/s = 1.49 ft1/3/s is a unit conversion factor. This will allow the use of the same n-values in different unit systems. Manning's equation may be used for gradually varied flow using the EGL slope (Se) and uniform flow using the bottom slope (S0 = Se for uniform flow). In terms of the discharge (Q) and the flow area (A), the equation is written as Q = AV =

kM 2/3 1/2 ARh Se n

(6.5)

Setting kM = 1 in the SI unit system, these equations become V =

1 2/3 1/2 R S n h e

(6.4a)

and Q = AV =

1 2/3 1/2 AR S n h e

(6.5a)

Table 6.2 Typical Values of Manning's n

Channel Surface Glass, PVC, HDPE Smooth steel, metals Concrete Asphalt Corrugated metal Earth excavation, clean Earth excavation, gravel and cobbles Earth excavation, some weeds Natural channels, clean and straight Natural channels, stones or weeds Riprap lined channel Natural channels, clean and winding Natural channels, winding, pools, shoals Natural channels, weeds, debris, deep pools Mountain streams, gravel and cobbles Mountain streams, cobbles and boulders

n 0.010 0.012 0.013 0.015 0.024 0.022–0.026 0.025–0.035 0.025–0.035 0.025–0.035 0.030–0.040 0.035–0.045 0.035–0.045 0.045–0.055 0.050–0.080 0.030–0.050 0.050–0.070

204

Water Flow in Open Channels

Chap. 6

where V has units of m/s, Rh is given in m, Se in m/m, A is given in m2, and Q is given in m3/s. On the right-hand side of this equation, the water area (A) and the hydraulic radius (Rh) are both functions of water depth (y), which is known as the uniform depth or normal depth (yn) when the flow is uniform. Setting kM = 1.49 in the BG system, Manning's equation is written as V =

1.49 2/3 1/2 R S n h e

(6.4b)

1.49 2/3 1/2 ARh Se n

(6.5b)

or Q =

where V is in ft/s, Q is in ft3/s (or cfs), A in ft2, Rh in ft, and Se in ft/ft. The computation of uniform flow may be performed by the use of either Equation 6.4 or Equation 6.5 and basically involves seven variables: 1. 2. 3. 4. 5. 6. 7.

the roughness coefficient (n); the channel slope (S0) (because S0 = Se in uniform flow); the channel geometry that includes the water area (A) and the hydraulic radius (Rh); the normal depth (yn); the normal discharge (Q); and the mean velocity (V).

A successive substitution procedure is generally required when the normal depth is sought. Alternatively, Figure 6.4 (a) can be used to determine the normal depth in trapezoidal and rectangular channels. Likewise, Figure 6.4 (b) can be used to determine the normal depth in circular channels. Example 6.1 A 3-m-wide rectangular irrigation channel carries a discharge of 25.3 m3/s at a uniform depth of 1.2 m. Determine the slope of the channel if Manning's coefficient is n = 0.022. Solution For a rectangular channel, the wetted perimeter and the hydraulic radius are A = by = (3)(1.2) = 3.6 m2 P = b + 2y = 5.4 m Equation 6.5a can be rewritten as

Rh =

A 3.6 2 = = = 0.667 m P 5.4 3

S0 = Se = a

b = 0.041 2/3

ARh

Sec. 6.2

Uniform Flow in Open Channels

(a)

205 Rectangular

m 1 2 3 4

yn / b

0.1

0.01 0.001

0.1

0.01

100

1/2 8/3

(nQ)/(kMS0 b ) (b) 1.0 0.9 0.8 0.7 yn /d0

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00

0.05

0.10

0.15

0.20 1/2

0.25

0.30

0.35

8/3

(nQ)/(kMS0 d0 ) Figure 6.4 Normal depth solution procedure: (a) trapezoidal channels (m = side slope) and (b) circular channels (d0 = diameter)

Example 6.2 A 6-ft-diameter, concrete pipe is flowing with a free surface (i.e., not under pressure). If the pipe is laid on a slope of 0.001 and carries a uniform flow at a depth of 4 feet (y in Table 6.1), what is the discharge? Solution Based on Table 6.1 , the value of θ = 90° + α (in degrees) where α = sin-1(1 ft/3 ft) = 19.5°. Thus, θ = 90° + 19.5° = 109.5°; and in radians, θ = (109.5°/360°)(2π) = 0.608 π radians. The area of the circular section is A =

1 (2θ - sin 2θ)d 20 = 1/8[2(0.608 π) - sin 2(0.608 π)](6 ft)2 = 20.0 ft2 8

206

Water Flow in Open Channels

Chap. 6

The wetted perimeter is P = θdo = (0.608 π)(6 ft) = 11.5 ft Then the hydraulic radius is Rh = A/P = (20.0 ft2)>(11.5 ft) = 1.74 ft Substituting the above values into Equation 6.5b with S0 = Se (uniform flow) and n = 0.013 (Table 6.2) yields 1.49 2/3 1/2 1.49 Q = ARh So = (20)(1.74)2/3(0.001)1/2 n 0.013 Q = 105 ft3 >s (or cfs) Example 6.3 If the discharge in the channel in Example 6.1 is increased to 40 m3/s, what is the normal depth of the flow? Solution The geometric parameters are Area:

A = by = 3y

P = b + 2y = 3 + 2y 3y A Hydraulic Radius: Rh = = P 3 + 2y Wetted Perimeter:

Substituting these values in Equation 6.5 with S0 = Se (uniform flow) yields Q = 40 = or

1 2/3 1/2 AR S n h 0 2/3 3y 1 (3y) a b (0.041)1/2 0.022 3 + 2y

AR2/3 h = (3y) a

2/3 3y (0.022)(40) b = = 4.346 3 + 2y (0.041)1/2

Solving by successive substitution, we find that

y = yn = 1.69 m Alternatively, we can use Figure 6.4 (a): nQ 8/3 (1.0)S1/2 0 b

(0.022)(40) (1.0)(0.041)1/2(3)8/3

= 0.23

Then, from the figure, yn/b = 0.56 and yn = (0.56)(3.0) = 1.68 m. Note: Uniform depth computations that involve implicit equations can be solved by some programmable calculators, computer algebra software (e.g., Mathcad, Maple, or Mathematica), spreadsheet programs, and computer software designed specifically for the task (both proprietary and freeware; try a Google search on the Internet).

Sec. 6.3

Hydraulic Efficiency of Open-Channel Sections

207

Classroom Computer Exercise—Normal Depth in Open Channels Review Example 6.3. Obtain or write software appropriate for solving normal depth in open channels. (See the note at the end of Example 6.3 and the book's preface for suggestions.) Answer the following questions by performing a computer analysis of the open channel described in Example 6.3 and its modifications. (a) Before using the software, what data do you anticipate it will need to compute normal depth for the channel in Example 6.3? (b) Now enter the data requested by the software and perform a normal depth analysis. Compare your computer result with the answer listed for Example 6.3. Is there any discrepancy? Comments? (c) What would happen to the flow rate in the channel if the depth was unchanged but the bottom slope was doubled? Estimate the magnitude of the change. Double the slope and perform a new analysis on the computer. Did you reason correctly? Now restore the slope and flow rate to their original values. (d) What would happen to the flow depth if the flow rate remained unchanged but the channel was lined with concrete (n = 0.013)? Estimate the magnitude of the depth change. Change the roughness value and perform a new normal depth analysis on your computer. Did you reason correctly? Now restore the roughness and flow depth to their original value. (e) Trapezoidal geometry is necessary if a channel is unlined. [For bank stability, the side slopes may be limited to 1(V):3(H).] Because of easement requirements, the unlined channel (n = 0.022) will be cheaper if the top width does not exceed 6 m. Using the computer software, determine the top width of the channel assuming the bottom width remains 3 m and the discharge remains 40 m3/s. Also, determine the side slope that would produce a top width that is exactly 6 m. (f) Is it possible to design a triangular channel with the software? Explain. (g) Perform any other changes your instructor requests. 6.3 Hydraulic Efficiency of Open-Channel Sections The Manning uniform flow equations (6.4 and 6.5) show that for the same cross-sectional area (A) and channel slope (S0), the channel section with a larger hydraulic radius (Rh) delivers a larger discharge. It is a section of higher hydraulic efficiency. Because the hydraulic radius is equal to the water cross-sectional area divided by the wetted perimeter, for a given cross-sectional area, the channel section with the least wetted perimeter is the best hydraulic section. Among all open-channel shapes, the semicircle has the least perimeter for a given area, so it is the most hydraulically efficient of all sections. A channel with a semicircular cross section, however, has sides that are curved and almost vertical at the water surface level, which makes the channel expensive to construct (excavation and forming) and difficult to maintain (bank stability). In practice, semicircular sections are only used when pipes are appropriate or in smaller flumes of prefabricated materials.

208

Water Flow in Open Channels

Chap. 6

60°

Figure 6.5

Hydraulically efficient sections

For large channels, trapezoidal sections are most commonly used. The most efficient trapezoidal section is a half hexagon, which can be inscribed into a semicircle with its center at the free water surface and 60° angles on the sides. Another commonly used channel section is the rectangular section. The most efficient rectangular section is the half-square section, which can also be inscribed into a semicircle with the center of the circle at the free water surface. The hydraulically efficient semicircular, half-hexagon, and half-square sections are shown in Figure 6.5. The concept of hydraulically efficient sections is only valid when the channel is lined with stabilized, nonerodible materials. Ideally, a channel should be designed for the best hydraulic efficiency, but it should be modified for practicality and construction cost. It should be noted that although the best hydraulic section offers the least water area for a given discharge, it does not necessarily have the lowest excavation cost. A half-hexagon section, for example, is a best hydraulic section only when the water surface reaches the level of the bank top. This section is not suitable for general applications because a sufficient distance above the water surface must be provided to prevent waves or fluctuations of water surface from overflowing the sides. The vertical distance from the designed water surface to the top of the channel banks is known as the freeboard of the channel. Freeboard and other channel design issues are further discussed in Section 6.9. Example 6.4 Prove that the best hydraulic trapezoidal section is a half-hexagon. Solution The water cross-sectional area (A) and the wetted perimeter (P) of a trapezoidal section are A = by + my 2

(1)

and P = b + 2y21 + m2

(2)

From Equation (1), b = A/y - my. This relationship is substituted into Equation (2): P =

A - my + 2y21 + m2 y

Now consider both A and m constant and let the first derivative of P with respect to y equal zero to obtain the minimum value of P: dP A = - 2 - m + 221 + m2 = 0 dy y Substituting A from Equation (1) , we get by + my 2 y2

= 221 + m2 - m

Sec. 6.3

Hydraulic Efficiency of Open-Channel Sections

209

or b = 2y( 21 + m2 - m)

(3)

Note that this equation provides a relationship between the flow depth and the channel bottom for an efficient section if the side slope m is fixed at a predetermined value. The most efficient section is obtained as follows if m can vary. By definition, the hydraulic radius, Rh, may be expressed as by + my 2 A = Rh = P b + 2y21 + m2 Substituting the value of b from Equation (3) into the above equation and simplifying, we have Rh =

y 2

It shows that the best hydraulic trapezoidal section has a hydraulic radius equal to one-half of the water depth. Substituting Equation (3) into Equation (2) and solving for P, we have P = 2y(221 + m2 - m)

(4)

To determine the value of m that makes P the least, the first derivative of P is taken with respect to m. Equating it to zero and simplifying, we have m =

23 = cot 60° 3

(5)

and thus, b = 2ya

1 +

1 23 23 y or b = 2 3 3 3

y =

23 b = b sin 60° 2

This means that the section is a half-hexagon, as depicted in Figure 6.6.

m y

y = b sin 60° =

b 60°

b Figure 6.6

3 √‾ 1 = 3 tan 60°

Best hydraulic trapezoidal section

3 √‾ b 3

210

Water Flow in Open Channels

Chap. 6

6.4 Energy Principles in Open-Channel Flow The energy principles derived from pressure flow in pipes are generally applicable to openchannel flow. The energy contained in a unit weight of water flowing in an open channel may also be measured in the three basic forms: 1. kinetic energy, 2. pressure energy, and 3. elevation (potential) energy above a certain energy datum line. Kinetic energy in any section of an open channel is expressed in the form of V 2/2g, where V is the mean velocity defined by the discharge divided by the water area (i.e., V = Q/A) in the section. The actual velocity of water flowing in an open-channel section varies in different parts of the section. The velocities near the channel bed are retarded because of friction, and reach a maximum near the water surface in the center part of the channel. The distribution of velocities in a cross section results in a different value of kinetic energy for different parts of the cross section. An average value of the kinetic energy in a cross section of open channel may be expressed in terms of the mean velocity as α(V 2/2g), where α is known as the energy coefficient. The value of α depends on the actual velocity distribution in a particular channel section. Its value is always greater than unity. The ordinary range of α lies between 1.05 for uniformly distributed velocities and 1.20 for highly varied velocities in a section. In simple analysis, however, the velocity heads (kinetic energy heads) in an open channel are taken as V 2/2g by assuming α equal to unity as an approximation. Because open-channel flow always has a free surface that is exposed to the atmosphere, the pressure on the free surface is constant and commonly taken as a zero pressure reference. Pressure energy in open-channel flow is usually computed with reference to the free surface. If the free surface in a channel approximates a straight-line slope, the pressure head at any submerged point A is equal to the vertical distance measured from the free surface to the point. Therefore, the water depth (y) at a given cross section is commonly used to represent the pressure head: p/γ = y. However, if the water is flowing over a vertical curve, such as a spillway or a weir, the centrifugal force produced by the fluid mass flowing over the curved path may cause a marked difference in pressure from a depth measurement alone. When water flows over a convex path [Figure 6.7 (a)], the centrifugal force acts in the direction opposite to the gravity yv gr

yv gr

r y y

(a) Figure 6.7

(b)

Flow over curved surfaces: (a) convex surface and (b) concave surface

Sec. 6.4

Energy Principles in Open-Channel Flow

211

force and the pressure is less than that of the water depth, by mυ2/r, where m is the mass of water column immediately above a unit area and υ2/r is the centrifugal acceleration of the water mass flowing along a path with radius of curvature (r). The resulting pressure head is p yυ2 = y gr γ

(6.6a)

When water flows over a concave path [Figure 6.7 (b)], the centrifugal force is in the same direction as the gravity force, and the pressure is greater than that represented by the water depth. The resulting pressure head is p yυ2 = y + gr γ

(6.6b)

where γ is the unit weight of water, y is the depth measured from the free water surface to the point of interest, υ is the velocity at the point, and r is the radius of curvature of the curved flow path. The elevation (potential) energy head in open-channel flow is measured with respect to a selected horizontal datum line. The vertical distance measured from the datum to the channel bottom (z) is commonly taken as the elevation energy head at the section. Therefore, the total energy head at any section in an open channel is generally expressed as H = z + y +

V2 2g

(6.7)

Specific energy in a channel section is defined as the energy head measured with respect to the channel bottom at the section. According to Equation 6.7 , the specific energy at any section is E = y +

V2 2g

(6.8)

or the specific energy at any section in an open channel is equal to the sum of the velocity head and the water depth at the section. Given the water area (A) and the discharge (Q) at a particular section, Equation 6.8 may be rewritten as E = y +

Q2 2gA2

(6.9)

Thus, for a given discharge Q, the specific energy at any section is a function of the depth of the flow only. When the depth of the flow, y, is plotted against the specific energy for a given discharge at a given section, a specific energy curve is obtained (Figure 6.8). The specific energy curve has two limbs: AC and CB. The lower limb always approaches the horizontal axis toward the right and the upper limb approaches (asymptotically) a 45°-line that passes through the origin. At any point on the specific energy curve, the ordinate represents the depth of the flow at the section, and the abscissa represents the corresponding specific energy. Usually, the same scales are used for both the ordinate and the abscissa. In general, a family of similar curves may be plotted for various values of discharge at a given section. For higher discharge, the curve moves to the right: A′C′B′. For lower discharge, the curve moves to the left: A″C″B″.

212

Water Flow in Open Channels

Chap. 6

y B B″

B′

s al

it Cr

tat

y c′ C′

yc yc ″ y1

Q+∆ Q

C″

A′

Q–∆Q Ec ″ Ec Figure 6.8

Ec′

A″ E1

A E

Specific energy curves of different discharges at a given channel section

The vertex C on a specific energy curve represents the depth (yc) at which the discharge Q may be delivered through the section at minimum energy (Ec). This depth is commonly known as the critical depth for the discharge Q at the given section. The corresponding flow in the section is known as the critical flow. At a smaller depth, the same discharge can be delivered only by a higher velocity and a higher specific energy. The state of rapid and shallow flow through a section is known as supercritical flow or rapid flow. At a larger depth, the same discharge may be delivered through the section with a smaller velocity and a higher specific energy than at critical depth. This tranquil, high-stage flow is known as subcritical flow. For a given value of specific energy, say E1, the discharge may pass through the channel section at either depth y1 (supercritical flow) or y2 (subcritical flow), as shown in Figure 6.8. These two depths, y1 and y2, are commonly known as alternate depths. At the critical state the specific energy of the flow takes a minimum value. This value can be computed by equating the first derivative of the specific energy with respect to the water depth to zero: Q2 dA dE d Q2 = a + yb = + 1 = 0 dy dy 2gA2 gA3 dy

The differential water area (dA/dy) near the free surface is dA/dy = T, where T is the top width of the channel section. Hence, -

Q 2T gA3

+ 1 = 0

(6.10a)

Sec. 6.4

Energy Principles in Open-Channel Flow

213

An important parameter for open-channel flow is defined by A/T = D, which is known as the hydraulic depth of the section. For rectangular cross sections, the hydraulic depth is equal to the depth of the flow. The above equation may thus be simplified to Q2 dE V2 = 1 = 1 = 0 dy gD gDA2

(6.10b)

or V 2gD

= 1

(6.11)

The quantity V> 2gD is dimensionless. It can be derived as the ratio of the inertial force in the flow to the gravity force in the flow (see Chapter 10 for detailed discussion). This ratio may be interpreted physically as the ratio between the mean flow velocity (V) and the speed of a small gravity (disturbance) wave traveling over the water surface. It is known as the Froude number (NF): NF =

V 2gD

(6.12)

When the Froude number is equal to unity, as indicated by Equation 6.11, V = 2gD, the speed of the surface (disturbance) wave and that of the flow is the same. The flow is in the critical state. When the Froude number is less than unity, V 6 2gD, the flow velocity is smaller than the speed of a disturbance wave traveling on the water surface. The flow is classified as subcritical. When the Froude number is greater than unity, V 7 2gD, the flow is classified as supercritical. From Equation 6.10, we may also write (for critical flow), Q2 A3 = = DA2 g T

(6.13)

In a rectangular channel, D = y and A = by. Therefore, Q2 = y 3b2 g Because this relation is derived from the critical flow conditions stated above, y = yc, which is the critical depth, and yc =

2 Q2 3 q = A gb2 Ag 3

(6.14)

where q = Q>b, is the discharge per unit width of the channel. For trapezoidal and circular channels, an explicit equation such as Equation 6.14 is not available, and a successive substitution procedure is required to solve Equation 6.13 for critical depth. Alternatively, Figures 6.9 (a) and 6.9 (b) can be used to determine critical depth in trapezoidal and circular channels, respectively. For an open channel of any sectional shape, the critical depth is always a function of the channel discharge and does not vary with the channel slope.

214

Water Flow in Open Channels 10

myc / b

(a)

Chap. 6

0.1 0.01

0.1

3/2

100

1/2 5/2

(Qm )/(g b ) (b)

1 0.9 0.8 0.7

yc /d0

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

5 1/2

Q/(gd0 ) Figure 6.9

Critical depth solution procedure: (a) trapezoidal channels and (b) circular channels

Example 6.5 A hydraulic transition is designed to connect two rectangular channels of the same width by a sloped floor, as shown in Figure 6.10 (a). Assume the channel is 3 m wide and is carrying a discharge of 15 m3/s at 3.6 m depth. Also assume a 0.1 m energy loss uniformly distributed through the transition. Determine the water surface profile in the transition. Solution The specific energy curve can be constructed based on the given discharge and the sectional geometry by using the following relationship from Equation 6.9: E = y +

Q2 2

2gA

= y +

(15)2 2(9.81)(3y)

= y +

1.27 y2

Sec. 6.4

Energy Principles in Open-Channel Flow

215

EGL hL = 0.1 m

y1 = 3.6 m

Datum line 20 m

z1 = 0.4 m

(a) y

yc =

52 = 1.37 m g

Ec(m) = 2.05 m

3 2 yc 1 0.5

y (m)

E (m)

0.5 1.0 2.0 3.0 3.0 4.0

5.60 2.27 2.32 2.32 3.14 4.07

45° 1

3 (b)

Figure 6.10 Hydraulic transition

At the inlet to the transition, the velocity is Vi : Vi =

Q 15 = = 1.39 m>s Ai (3.6)(3)

where Ai is the water area at the inlet and the velocity head is V 2i (1.39)2 = = 0.10 m 2g 2(9.81) The total energy head at the inlet as measured with respect to the datum line is Hi = zi + yi +

V 2i = 0.40 + 3.60 + 0.10 = 4.10 m 2g

The top horizontal line in Figure 6.10 (a) shows this energy level. At the exit of the transition, the total energy available is reduced by 0.1 m, as indicated by the EGL in Figure 6.10 (a): He = ze + ye +

V 2e = Hi - 0.1 = 4.00 m 2g

Ee is the specific energy measured with respect to the channel bottom: Ee = He = 4.00 m

216

Water Flow in Open Channels

Chap. 6

This value is applied to the specific energy curve shown in Figure 6.10 (b) to obtain the water depth at the exit section. Water surface elevations at four other sections (4.00 m, 8.00 m, 12.00 m, and 16.00 m from the entrance section) are computed by using the same method. The results for all six sections are shown in the following table.

Section

Inlet

4.00 m

8.00 m

12.00 m

16.00 m

Exit

Specific Energy, E (m) Water Depth, y (m)

3.70 3.60

3.76 3.67

3.82 3.73

3.88 3.79

3.94 3.86

4.00 3.92

Example 6.6 A trapezoidal channel has a bottom width of 5 m and side slopes m = 2. If the flow rate is 20 m3/s, what is the critical depth? Solution By using Equation 6.13 and Table 6.1: [(b + my)y]3 Q2 A3 2 = DA = = g T b + 2my or [(5 + 2y)y]3 202 = 40.8 = 9.81 5 + 2(2)y By successive substitution, we obtain y = yc = 1.02 m. Alternatively, using Figure 6.9: Qm3>2 g1>2b5>2

(20) (2)3>2 (9.81)1>2 (5)5>2

= 0.323

From Figure 6.9 (a), we obtain myc /b = 0.41. Therefore, yc = (0.41)(5)/2 = 1.03 m. Note: Critical depth computations that involve implicit equations can be solved by some programmable calculators, computer algebra software (e.g., Mathcad, Maple, or Mathematica), spreadsheet programs, and computer software designed specifically for the task (both proprietary and freeware; try a topical search on the Internet).

6.5 Hydraulic Jumps Hydraulic jumps can occur naturally in open channels but are more common in constructed structures such as energy dissipation (or hydraulic jump) basins. They are the result of an abrupt reduction in flow velocity by means of a sudden increase in water depth in the downstream direction. Most energy dissipation basins are rectangular in cross section, so this book limits the discussion of hydraulic jumps to rectangular channels. Hydraulic jumps convert a high-velocity supercritical flow (upstream) into a low-velocity subcritical flow (downstream). Correspondingly, a low-stage supercritical depth (y1) is changed to a high-stage subcritical depth (y2); these are known, respectively, as the initial depth and the sequent depth of a hydraulic jump (Figure 6.11). In the region of the hydraulic jump, the

Sec. 6.5

Hydraulic Jumps

217

Specific force curve

Specific energy curve y2

∆E E2

y2 yc y1

Fs1 = Fs2 Figure 6.11

Hydraulic jump

characteristic rolling water surface and turbulence of water can be seen. These violent motions are accompanied by a significant loss of energy head through the jump. Given the discharge in a particular channel, the amount of energy head loss through a jump (∆E) can be determined by simply measuring the initial and sequent depths and using the specific energy curve shown in Figure 6.11. Predicting the sequent depth by estimating the energy loss, however, is impractical because it is difficult to determine the energy loss through a jump. The relationship between the initial depth and the sequent depth in a hydraulic jump may be determined by considering the balance of forces and momentum immediately before and after the jump. Consider a control volume in the vicinity of a jump, as shown in Figure 6.11. The balance between the hydrostatic forces and the momentum flux through section 1 and section 2, per unit width of the channel, may be expressed as F1 - F2 = ρq(V2 - V1)

(6.15)

where q is discharge per unit width of the channel. Substituting the following quantities F1 =

γ 2 y 2 1

F2 =

γ 2 y 2 2

V1 =

q y1

V2 =

q y2

into Equation 6.15 and simplifying, we get q2 y1 + y2 = y1y2 a b g 2

(6.16)

This equation may be rearranged into a more convenient form as follows: y2 1 = 1 21 + 8N 2F1 - 1 2 y1 2

(6.17)

where NF1 is the Froude number of the approaching flow: NF1 =

V1 2gy1

(6.18)

Example 6.7 A 10-ft-wide rectangular channel carries 500 cfs of water at a 2-ft depth before entering a jump. Compute the downstream water depth and the critical depth.

218

Water Flow in Open Channels

Chap. 6

Solution The discharge per unit width is q = Using Equation 6.14, the critical depth is

500 = 50 ft3 >s # ft 10 502 = 4.27 ft A 32.2 3

yc = The approaching velocity is V1 =

q 50 = = 25 ft>s y1 2

The Froude number for the approaching flow can be computed by using this velocity and the initial depth y1 = 2.0: NF1 = Substituting this value into Equation 6.17 gives

V1 2gy1

= 3.12

y2 1 = 1 21 + 8(3.12)2 - 1 2 2.0 2

and solving for the sequent depth yields:

y2 = 7.88 ft

Equation 6.15 may also be arranged as F1 + ρqV1 = F2 + ρqV2 where Fs = F + ρqV

(6.19)

The quantity Fs is known as the specific force per unit width of the channel. For a given discharge, the specific force is a function of the water depth at a given section. When Fs is plotted against the water depth, the resulting curve is similar to a specific energy curve with a vertex that appears at the critical depth. A typical specific force curve is shown in Figure 6.11. A hydraulic jump usually takes place in a rather short reach in a channel. Therefore, it is reasonable to assume that through a hydraulic jump the specific forces immediately before and after a jump are approximately the same. The value of Fs can be computed from the given conditions of the approaching flow. If we apply this value to the specific force curve in Figure 6.11, we can draw a vertical line that gives both the initial and sequent depths of a jump. The energy head loss through the hydraulic jump (∆E) may then be estimated by applying the definition ∆E = a =

V 21 V 22 + y1 b - a + y2 b 2g 2g

q2 1 1 2 1 (V 1 - V 22) + (y1 - y2) = a 2 - 2 b + (y1 - y2) 2g 2g y 1 y2

Sec. 6.6

Gradually Varied Flow

219

Substituting Equation 6.16 into the above equation and simplifying, we get (y2 - y1)3 ∆E = 4y1y2

(6.20)

Example 6.8 A long, rectangular open channel 3 m wide carries a discharge of 15 m3/s. The channel slope is 0.004, and the Manning's coefficient is 0.01. At a certain point in the channel, flow reaches normal depth. (a) Determine the flow classification at normal depth. Is it supercritical or subcritical? (b) If a hydraulic jump takes place at normal depth, what is the sequent depth? (c) Estimate the energy head loss through the jump. Solution (a) The critical depth is calculated using Equation 6.14 and yc = 1.37 m. The normal depth of this channel can be determined by the Manning equation (Equation 6.5): Q =

1 A R2>3S1>2 n 1 h1

where A = y1b, Rh =

y1b A1 = , b = 3m P1 2y1 + b

We have 15 =

2>3 3y1 1 (3y1) a b (0.004)1>2 0.01 2y1 + 3

Solving the equation for y1, we obtain

y1 = 1.08 m,

V1 =

15 = 4.63 m>s 3y1

and NF1 = Because NF1 7 1, the flow is supercritical. (b) Applying Equation 6.17, we get y2 =

V1 2gy1

= 1.42

y1 1 21 + 8N 2F1 - 1 2 = 1.57y1 = 1.70 m 2

(y2 - y1)3 (0.62)3 = = 0.032 m 4y1y2 4(1.70)(1.08)

6.6 Gradually Varied Flow Gradually varied flow in open channels differs from rapidly varied flow (hydraulic jumps, flow through a streamlined transition, etc.) in that the change in water depth in the channel takes place very gradually with distance.

220

Water Flow in Open Channels

Chap. 6

In uniform flow, the water depth remains a constant value known as the normal depth (or uniform depth). The energy grade line is parallel to the water surface and the channel bottom. The velocity distribution also remains unchanged throughout the reach. Thus, the computation of only one water depth is sufficient for the entire reach. In rapidly varied flow such as a hydraulic jump, rapid changes in water depth take place in a short distance. A significant change in water velocities is associated with the rapid variation of water cross-sectional area. At this high rate of flow deceleration, the energy loss is inevitably high. The computation of water depths using the energy principles is not reliable. In this case, computations can only be carried out by applying the momentum principles (i.e., Equation 6.15). In gradually varied flow, velocity changes take place very gradually with distance so that the effects of acceleration on the flow between two adjacent sections are negligible. Thus, computation of the water surface profile, defined as depth changes along the channel length, can be carried out strictly on energy considerations. The total energy head at any section in an open channel, as defined in Equation 6.7, is restated here as Q2 V2 H = z + y + = z + y + 2g 2gA2 To compute the water surface profile, we must first obtain the variation of the total energy head along the channel. Differentiating H with respect to the channel distance x, we obtain the energy gradient in the direction of the flow: dy dy -Q2 dA Q2T dy dH dz dz = + + = + + 3 3 dx dx dx dx dx gA dx gA dx where dA = T(dy). Rearranging the equation gives dy = dx

dH dz dx dx 1 -

Q 2T

(6.21)

gA3

The term dH/dx is the slope of the energy grade line. It is always a negative quantity because the total energy head reduces in the direction of the flow, or Se = -dH/dx. Similarly, the term dz/dx is the slope of the channel bed. It is negative when the elevation of the channel bed reduces in the direction of the flow; it is positive when the elevation of the channel bed increases in the direction of the flow. In general, we may write S0 = -dz/dx. The energy slope in gradually varied flow between two adjacent sections may also be approximated by using a uniform flow formula. For simplicity, the derivation will be demonstrated with a wide rectangular channel section where A = by, Q = bq, and Rh = A/P = by/(b + 2y) ≅ y (for wide rectangular channels because b 7 7 y). Using the Manning formula (Equation 6.5), we get n2Q2 n2Q2 dH Se = = 4/3 2 = 2 10/3 dx Rh A by

(6.22)

Sec. 6.7

Classifications of Gradually Varied Flow

221

The slope of the channel bed may also be expressed in similar terms if uniform flow were assumed to take place in the channel. Because the slope of the channel bed is equal to the energy slope in uniform flow, the hypothetical uniform flow conditions are designated with the subscript n. We have n2Q2 dz S0 = = ¢ 2 10/3 b dx by n

(6.23)

From Equation 6.14 for rectangular channels, yc = or

q2 = 3 Q 2 Ag A gb2 3

Q2 = gy 3c b2 =

gA3c b

(6.24)

Substituting Equations 6.22, 6.23, and 6.24 into Equation 6.21, we have dy = dx

S0 J1 - a

yn 10/3 b R y

yc 3 J1 - a b R y

(6.25a)

For nonrectangular channels, Equation 6.24a can be generalized as dy = dx

S0 J1 - a

yn N b R y

yc M J1 - a b R y

(6.25b)

where the exponents M and N depend on the cross-sectional shape and the flow conditions as given by Chow.* This form of the gradually varied flow equation is very useful for a qualitative analysis, which helps to understand the gradually varied flow classifications covered in the next sections. Other forms are often used to compute water surface profiles. Physically, the term dy/dx represents the slope of the water surface with respect to the bottom of the channel. For dy/dx = 0, the water depth remains constant throughout the reach or the special case of uniform flow. For dy/dx 6 0, the water depth decreases in the direction of the flow. For dy/dx 7 0, the water depth increases in the direction of the flow. Solutions of this equation under different conditions will yield the various water surface profiles that occur in open channels. 6.7 Classifications of Gradually Varied Flow In analyzing gradually varied flow, the role of critical depth, yc, is very important. When openchannel flow approaches critical depth (y = yc), the denominator of Equation 6.25 approaches zero and the value of dy/dx approaches infinity. The water surface becomes very steep. This is * V. T. Chow, Open Channel Hydraulics (New York: McGraw-Hill, 1959).

222

Water Flow in Open Channels

Chap. 6

seen at hydraulic jumps or at a water surface entering a steep channel from a mild channel or a lake. The latter case provides a unique one-to-one relationship between the discharge and the water depth in a channel and is known as a control section in open-channel flow. Depending on the channel slope, geometry, roughness, and discharge, open channels may be classified into five categories: 1. 2. 3. 4. 5.

steep channels, critical channels, mild channels, horizontal channels, and adverse channels.

The classification depends on the flow conditions in the channel as indicated by the relative positions of normal depth (yn) and critical depth (yc) calculated for each particular channel. The criteria are as follows: Steep channels: Critical channels: Mild channels: Horizontal channels: Adverse channels:

yn >yc yn >yc yn >yc S0 = S0 6

6 1.0 = 1.0 7 1.0 0 0

or or or

yn 6 yc yn = yc yn 7 yc

A further classification of water surface profile curves depends on the actual water depth and its relationship to the critical and normal depths. The ratios of y/yc and y/yn may be used in the analysis, where y is the actual water depth at any section of interest in the channel. If both y/yc and y/yn are greater than 1.0, then the water surface profile curve is above both the critical depth line and the normal depth line in the channel, as depicted in Figure 6.12. The curve is designated as a type 1 curve. There are S-1, C-1, and M-1 curves for steep, critical, and mild channels, respectively. If the water depth (y) is between the normal depth and the critical depth, the curves are designated as type 2 curves. There are S-2, M-2, H-2, and A-2 curves. The type 2 curve does not exist in critical channels. In critical channels, normal depth is equal to critical depth. Thus, no depth of flow can come between the two. If the water depth is less than both yc and yn, then the water surface profile curves are type 3. There are S-3, C-3, M-3, H-3, and A-3 curves. Each of these water surface profile curves is listed and shown schematically in Figure 6.12. Examples of physical occurrences in open channels are also given. Certain important characteristics of water surface profile curves can be demonstrated from direct analysis of the gradually varied flow equation (Equation 6.25). By making substitutions into Equation 6.25, we note the following: 1. For type 1 curves, y/yc 7 1 and y/yn 7 1. Thus, the value of dy/dx is positive, indicating that water depth increases in the direction of the flow. 2. For type 2 curves, the value of dy/dx is negative. The water depth decreases in the direction of the flow. 3. For type 3 curves, the value of dy/dx is again positive. The water depth increases in the direction of the flow.

Sec. 6.7

Classifications of Gradually Varied Flow

223

M-1 M-2 M1 M-3

Mild slope (a)

(b) S-1

S3 S-2

Jump

S-3

S1 S2

Steep slope (c)

(d)

C-1 C-3

Critical slope

(e)

Rare occurrences

H-2

Jump H-3 Horizontal slope (f)

(g)

A-2

Jump A-3

Adverse slope

(h)

(i) Normal depth line Critical depth line

Figure 6.12

Classifications of gradually varied flow

224

Water Flow in Open Channels

Chap. 6

Table 6.3 Characteristics of Water Surface Profile Curves

Channel Mild Mild Mild Critical Critical Steep Steep Steep Horizontal Horizontal Adverse Adverse

Symbol

Type

M M M C C S S S H H A A

1 2 3 1 3 1 2 3 2 3 2 3

Slope

Depth

Curve

7 7 7 7 7 7 7 7 = = 6 6

y 7 yn 7 yc yn 7 y 7 yc yn 7 yc 7 y y 7 yn = yc yn = yc 7 y y 7 yc 7 yn yc 7 y 7 yn yc 7 yn 7 y y 7 yc yc 7 y y 7 yc yc 7 y

M-1 M-2 M-3 C-1 C-3 S-1 S-2 S-3 H-2 H-3 A-2 A-3

S0 S0 S0 S0 S0 S0 S0 S0 S0 S0 S0 S0

0 0 0 0 0 0 0 0 0 0 0 0

4. When the actual water depth approaches critical depth, y = yc, Equation 6.25 yields dy/dx = ∞, indicating that the slope of the water surface profile curve is theoretically vertical. Likewise, as y approaches yn, dy/dx approaches zero, indicating that the water surface profile approaches the normal depth line asymptotically. 5. A few types of water surface profile curves never approach a horizontal line (S-2, S-3, M-2, M-3, C-3, H-3, and A-3). Others approach a horizontal line asymptotically, except for the C-1 curve, which is horizontal throughout the channel reach. Because yn = yc in a critical channel, Equation 6.25 yields dy/dx = S0, indicating the water depth increases at the same rate as the channel bed elevation decreases, which theoretically results in a horizontal water surface profile. In channels where y 6 yc, the velocity of water flow is greater than that of the disturbance wave. For this reason, flow conditions in the downstream channel will not affect those upstream. The change of water depth resulting from any channel disturbance propagates only in the downstream direction. Thus, computation of the water surface profile should be carried out in the downstream direction (M-3, S-2, S-3, C-3, H-3, and A-3). In channels where y 7 yc, the speed of wave propagation is greater than the velocity of water flow. Any disturbance in the downstream channel can travel upstream and affect the flow conditions upstream as well as downstream. Any change of water depth in the downstream channel propagates upstream and may also change the water depth in the upstream channel. Thus, computation of the water surface profile should be carried out in the upstream direction (M-1, M-2, S-1, C-1, H-2, and A-2). At the break of a channel from mild to steep slope or a significant drop of the channel bottom, critical depth is assumed to take place in the immediate vicinity of the brink. At this point, a definite depth–discharge relationship can be obtained (i.e., control section) and is frequently used as a starting point for water surface profile computations. Table 6.3 provides a summary of the water surface profile curves. 6.8 Computation of Water Surface Profiles Water surface profiles for gradually varied flow may be computed by using Equation 6.25. The computation normally begins at a section where the relationship between the water surface elevation (or flow depth) and the discharge is known. These sections are commonly

Sec. 6.8

Computation of Water Surface Profiles Reservoir height

Control section

yc Reservoir

225

Reservoir

Mild slope

Steep slo

Control section

Depth computed from the mild slope reach Control section

Control section y yn

yc Horizontal slope

Mild slope

Reservior

Figure 6.13 Control sections in open channels

known as control sections (or mathematically, boundary conditions). A few examples of common control sections in open channels are depicted in Figure 6.13. Locations where uniform flow occurs can also be viewed as a control section because the Manning equation describes a flow depth–discharge relationship. Uniform flow (i.e., flow at normal depth) tends to occur in the absence of or far away from other control sections and where the stream slope and cross section are relatively constant. A successive computational procedure based on an energy balance is used to obtain the water surface elevation at the next section, either upstream or downstream from the control section. The distance between sections is critical because the water surface will be represented by a straight line. Thus, if the depth of flow is changing quickly over short distances, adjacent sections should be closely spaced to represent accurately the water surface profile. The step-by-step procedure is carried out in the downstream direction for rapid (supercritical) flows and in the upstream direction for tranquil (subcritical) flows. 6.8.1

Standard Step Method

The standard step method is presented in this section to calculate gradually varied flow water surface profiles. The method employs a finite difference solution scheme to solve the differential, gradually varied flow equation (Equation 6.25). It is the most common algorithm used in computer software packages that solve gradually varied flow profiles. For example, it is the primary algorithm in the widely used HEC-RAS program developed by the U.S. Army Corps of Engineers. For less common methods of computation, the reader is referred to the classic textbook by Ven T. Chow.* The standard step method is derived directly from an energy balance between two adjacent cross sections (Figure 6.14) that are separated by a sufficiently short distance so that the water * V. T. Chow, Open Channel Hydraulics (New York: McGraw-Hill, 1959).

226

Water Flow in Open Channels

EGL

V22 2g

Chap. 6

hL = Se ∆L V12 2g

∆z = S0 ∆L ∆L

Figure 6.14

Energy relationships in a water surface profile

surface can be approximated by a straight line. The energy relation between the two sections may be written as ∆z + y2 +

V 22 V 21 = y1 + + hL 2g 2g

(6.26a)

where ∆z is the elevation difference in the channel bottom and hL is the energy head loss between the two sections, as shown in Figure 6.14. Equation 6.26a may be rewritten as

a z2 + y2 +

V 22 V 21 b = a z1 + y1 + b + Se ∆L 2g 2g H2 = H1 + losses

(6.26b)

(6.26c)

where z is the elevation head (channel bottom elevation with respect to some datum) and H is the total energy head (elevation head + depth + velocity head). It is important to note that, in Equation 6.26, the sections 1 and 2 represent downstream and upstream sections, respectively. If the sections are numbered differently, the losses should always be added to the downstream side. The computation procedure yields the correct depth at a cross section that is a distance ∆L away from a section with a known depth. Computations begin at a control section and progress upstream (subcritical flow) or downstream (supercritical flow). For subcritical flow, the water surface profile is occasionally called a backwater curve because the process moves from downstream to upstream. Likewise, the profile for supercritical flow is occasionally called a frontwater curve. Equation 6.26b cannot be solved directly for the unknown depth (e.g., y2) because V2 and Se depend on y2. Therefore, an iterative procedure is required using successive approximations of y2 until the downstream and upstream energies balance (or come within an acceptable range). The energy slope (Se) can be computed by applying the Manning equation, in either SI units Se =

n2V 2 R4/3 h

(6.27a)

Sec. 6.8

Computation of Water Surface Profiles

227

or BG units Se =

n2V 2

(6.27b)

2.22R4/3 h

where Se is the average of the energy (EGL) slopes at the upstream and downstream sections. A tabulated computation procedure is recommended as illustrated in the example problems to follow. The astute reader may ask why ∆L is not solved for in Equation 6.26b. By assigning a depth to the next section instead of assuming the depth, the equation could be used to determine the distance between sections and avoid the iterative process altogether. This is a legitimate solution procedure called the direct step method, but it only works for prismatic channels (channels of uniform slope and cross section). When water surface profiles are sought in natural stream channels that are nonprismatic, cross sections on these streams are field surveyed or obtained from geographic information system maps at predetermined locations, which establishes the distance between sections. Then the standard step method is used to assess depth of flow at these sections. Fortunately, water surface profiles are generally solved with the aid of computer software, which takes the drudgery out of the iterative process 6.8.2 Direct Step Method In the direct step method, the gradually varied flow equations are rearranged to determine the distance (∆L) explicitly between two selected flow depths. This method is applicable to prismatic channels only because the same cross-sectional geometric relationships are used for all the sections along the channel. Replacing sections 1 and 2 with D and U, respectively, and noting that S0 = (zU - zD)/∆L = ∆z/∆L Equation 6.26b is rearranged as

∆L =

V 2U V 2D a yD + b - a yU + b 2g 2g S0 - Se

ED - EU S0 - Se

(6.26d)

where E = y + V 2/2g is specific energy. In Equation 6.26d, U and D represent upstream and downstream sections, respectively. For subcritical flow, the computations begin at the downstream end and progress upstream. In this case yD and ED would be known. An appropriate value for yU is selected and the associated EU is calculated. Then ∆L is determined by using Equation 6.26d. For supercritical flow, the computations begin at the upstream end and progress downstream. In this case, yU and EU would be known. An appropriate value for yD is selected, and the associated ED is calculated. Then ∆L is determined by using Equation 6.26d. Example 6.9 A grouted-riprap, trapezoidal channel (n = 0.025) with a bottom width of 4 meters and side slopes of m = 1 carries a discharge 12.5 m3/s on a 0.001 slope. Compute the backwater curve (upstream water surface profile) created by a low dam that backs water up to a depth of 2 m immediately behind the dam. Specifically, water depths are required at critical diversion points that are located at distances of 188 m, 423 m, 748 m, and 1,675 m upstream of the dam.

228

Water Flow in Open Channels

Chap. 6

Solution Normal depth for this channel can be calculated by using Equation 6.5 (iterative solution), Figure 6.4 (a), or appropriate computer software. Using Figure 6.4 (a), nQ 8/3 kMS1/2 0 b

(0.025)(12.5) (1.00)(0.001)1/2(4)8/3

= 0.245

From Figure 6.4 (a), with m = 1, we obtain yn/b = 0.415 therefore, yn = (4 m)(0.415) = 1.66 m. Critical depth for this channel can be calculated by using Equation 6.13 (iterative solution), Figures 6.9 (a), or appropriate computer software. Using Figures 6.9 (a), Qm3/2 g1/2b5/2

(12.5)(1)3/2 (9.81)1/2 (4)5/2

= 0.125

From Figures 6.9 (a), we obtain myc >b = 0.230; therefore, yc = (4 m)(0.230)>1.0 = 0.92 m

We will first use the standard step method. Water surface profile computations require the use of the Manning equation (Equation 6.27a), which contains the variables Rh and V. Recall from earlier discussions that Rh = A/P, where A is the flow area and P is the wetted perimeter, and V = Q/A. The computation procedure displayed in Table 6.4 (a) is used to determine the water surface profile. The depth just upstream from the dam is the control section, designated as section 1. Energy balance computations begin here and progress upstream (backwater) because the flow is subcritical (yc 6 yn). The finite difference process is iterative; the depth of flow is assumed at section 2 until the energy at the first two sections match using Equation 6.26b. Once the water depth at section 2 is determined, the depth of flow at section 3 is assumed until the energies at sections 2 and 3 balance. This stepwise procedure continues upstream until the entire water surface profile is developed. Because the starting depth of 2.00 m is greater than the normal depth and normal depth exceeds critical depth, the profile has an M-1 classification (Figure 6.12). The flow depth will approach normal depth asymptotically as the computations progress upstream, as depicted in Figure 6.13 (c). Once the depth becomes normal, or relatively close, the computation procedure is ended. The first few standard step computations are displayed in Table 6.4 (a); completion of the problem is left to the student in Problem 6.8.7. Because the channel considered in this example is prismatic, we can also use the direct step method to calculate the water surface profile. Table 6.4 (b) is used to determine the profile by setting up and solving Equation 6.26d. The calculations in the table are self-explanatory. Like the standard step method, the computations begin at the downstream end and progress upstream. For the first channel reach considered, yD = 2.00 m is known, and yU = 1.91 m is a depth we select based on the water surface profile (M-1; depths go down) and to compare with the standard step method solution. Then we calculate the distance between the sections with these two depths. For the next reach, 1.91 m becomes the downstream depth, and we select yU = 1.82 m. The results are slightly different from those of the standard step method. The discrepancies result from the iterative nature of the standard step method in which the results depend on the tolerance limit selected.

2.00 1.94

0.000 0.188

12.00 11.52

(5) A (m2) 1.042 1.085

(6) V (m/s) 0.0553 0.0600

(7) V 2/2g (m) 9.657 9.487

(8) P (m) 1.243 1.215

(9) Rh (m) 0.000508 0.000567

(10) Se 0.000538 (∆L = 188 m)

2.00 1.91

0.000 0.188

12.00 11.29

1.042 1.107

0.0553 0.0625

9.657 9.402

1.243 1.201

0.000508 0.000600

1.91 1.80

0.188 0.423

11.29 10.44

1.107 1.197

0.0625 0.0731

9.402 9.091

1.201 1.148

0.000601 0.000745

0.000673 (∆L = 235 m)

0.000554 (∆L = 188 m)

1.91 1.82

0.188 0.423

11.29 10.59

1.107 1.180

0.0625 0.0710

9.402 9.148

1.201 1.158

0.000601 0.000716

Column (4)

Column (1) Column (2) Column (3)

0.000659 (∆L = 235 m)

0.1549

0.1582

0.1042

0.1011

2.315 2.314

2.319 2.296

2.159 2.160

2.156 2.188

(13) Total Energy (m)

Section numbers are arbitrarily designated from downstream to upstream. Sections are designated as either downstream (D) or upstream (U) to assist in the energy balance. Depth of flow (meters) is known at section 1 and assumed at section 2. Once the energies balance, the depth is now known at section 2, and the depth at section 3 is assumed until the energies at sections 2 and 3 balance. The channel bottom elevation (meters) above some datum (e.g., mean sea level) is given. In this case, the datum is taken as the channel bottom at section 1. The bottom slope and distance interval are used to determine subsequent bottom elevations. Water cross-sectional area (square meters) corresponds to the depth in the trapezoidal cross section. Mean velocity (meters per second) is obtained by dividing the discharge by the area in column 5. Velocity head (meters). Wetted perimeter (meters) of the trapezoidal cross section based on the depth of flow. Hydraulic radius (meters) equal to the area in column 5 divided by the wetted perimeter in column 8. Energy slope obtained from Manning equation (Equation 6.27a). Average energy grade line slope of the two sections being balanced. Energy loss (meters) from friction between the two sections found using hL = Se(avg)(∆L) from Equation 6.26b. Total energy (meters) must balance in adjacent sections (Equation 6.26b). Energy losses are always added to the downstream section. Also, the energy balance must be very close before proceeding to the next pair of adjacent sections or errors will accumulate in succeeding computations. Thus, even though depths were only required to the nearest 0.01 m, energy heads were calculated to the nearest 0.001 m.

Note: The trial depth of 1.82 m is correct. Now balance energy between sections 3 and 4.

D U

Note: The trial depth of 1.80 m is too low; the energy does not balance. Try a higher upstream depth.

D U

Note: The trial depth of 1.91 m is correct. Now balance energy between sections 2 and 3.

D U

(11) Se(avg)

(12) hL (m)

Computation of Water Surface Profiles

Column (5) Column (6) Column (7) Column (8) Column (9) Column (10) Column (11) Column (12) Column (13)

2 3

1 2

D U

1 2

(4) z (m)

Water Surface Profile (Backwater) Computations Using the Standard Step Method (Example 6.9)

Note: The trial depth of 1.94 m is too high; the energy does not balance. Try a lower upstream depth.

(2) U/D

(1) Section

(3) y (m)

Table 6.4 (a)

Sec. 6.8 229

U/D

D U

Section

1 2

2 3

Table 6.4 (b)

12.00 11.29

A (m2) 9.657 9.402

P (m) 1.243 1.201

Rh (m) 1.042 1.107

V (m/s) 0.0553 0.0625

V 2/2g (m)

11.29 10.59

9.402 9.148

1.201 1.158

1.107 1.180

0.0625 0.0710

A distance of 239 m separates the two flow depths (1.91 m and 1.82 m).

1.91 1.82

A distance of 186 m separates the two flow depths (2.00 m and 1.91 m).

2.00 1.91

y (m)

1.9725 1.8910

2.0553 1.9725

E (m)

0.000601 0.000716

0.000508 0.000601

Water Surface Profile (Backwater) Computations Using the Direct Step Method (Example 6.9)

239

186

∆L (m)

186 425

0 186

Distance to Dam (m)

230 Water Flow in Open Channels Chap. 6

Sec. 6.8

Computation of Water Surface Profiles

231

Example 6.10 A rough-concrete trapezoidal channel (n = 0.022) with a 3.5-ft bottom width, side slope m = 2, and bed slope of 0.012 discharges 185 cfs of fresh water from a reservoir. Determine the water surface profile in the discharge channel to within 2% of normal depth. Solution Normal depth and critical depth are calculated before solving water surface profiles in order to determine the gradually varied flow classification. Normal depth may be determined using the Manning equation in conjunction with Figure 6.4 (a): nQ 8/3 kMS1/2 0 b

(0.022)(185) (1.49)(0.012)1/2(3.5)8/3

= 0.883

Then, from the figure, yn/b = 0.685 and yn = (0.685)(3.5) = 2.40 ft. Alternatively, normal depth can be obtained with the Manning equation and Table 6.1 (using successive substitution) or by using appropriate computer software. Critical depth can be computed using Equation 6.13 and Table 6.1: Q 2T gA3

Q2(b + 2myc) g[(b + myc)yc]3

(185)2[3.5 + 2(2)yc] 32.2[(3.5 + 2yc)yc]3

= 1

By successive substitution (or, alternatively, appropriate computer software), we obtain yc = 2.76 ft Because critical depth exceeds normal depth, the channel is steep with an S-2 classification (Figure 6.12). Water from the reservoir will enter the channel and pass through critical depth as depicted in Figure 6.13 (a). Because the control section is at the entrance to the channel and flow is supercritical (yn 6 yc), computations will proceed in the downstream direction (front-water) starting from critical depth at the entrance section and approaching normal depth asymptotically. In an S-2 profile, the water surface elevation changes quickly at first and then approaches normal depth more gradually. Therefore, use five cross sections, including the control section, with separation distances (∆L) of 2, 5, 10, and 40 feet, respectively, moving downstream. The first few standard step computations are displayed in Table 6.5 (a); completion of the problem is left to the student in Problem 6.8.8. We can also use the direct step method to calculate the water surface profile in this problem because the channel is prismatic. The calculations are summarized in Table 6.5 (b). Again, the calculations begin from the upstream end and progress downstream. For the first channel reach considered, yU = 2.76 ft is known, and yD = 2.66 ft is a depth we select based on the type of the water surface profile (S-2; depths go down) and to compare with the standard step method solution. We then calculate the distance between the sections with these two depths. For the next reach, 2.66 ft is the upstream depth, and we select yD = 2.58 ft. The discrepancies between the standard step and the direct step method result from the iterative nature of the standard step method in which the results depend on the tolerance limit selected.

Hydraulic radius (ft) corresponding to the depth. Refer to Table 6.1 for appropriate equation.

Energy slope obtained from Manning equation (Equation 6.27b).

Average energy grade line slope of the two sections being balanced.

Column (7)

Column (8)

Column (9)

Column (11)

Total energy (ft) must balance in adjacent sections (Equation 6.26b). Energy losses are always added to the downstream section. Also, the energy balance must be very close before proceeding to the next pair of adjacent sections or errors will accumulate in succeeding computations. Thus, even though depths were only required to the nearest 0.01 ft, energy heads were calculated to the nearest 0.001 ft.

Column (10) Energy loss (ft) from friction between the two sections found using the average energy grade line slope.

Velocity head (ft).

13.602 13.602

Column (6)

(∆L = 5) 0.041

13.617 13.616

Mean velocity (ft/s) obtained by dividing the discharge by the area in column 4.

0.00827

(∆L = 2) 0.014

(11) Total Energy (ft)

Column (5)

0.00773 0.00882

0.00716

(10) hL = Se ∆L (ft)

Water cross-sectional area (ft2) corresponding to the depth. Refer to Table 6.1 for appropriate equation.

1.524 1.486

0.00659 0.00773

(9) Se

Column (4)

0.966 1.065

1.571 1.524

(8) Se

The channel bottom elevation (ft) above some datum is given. In this case, the datum is 10.0 ft below the channel bottom at section 1. The bottom slope and distance interval are used to determine subsequent bottom elevations.

7.885 8.280

0.857 0.966

(7) Rh (ft)

Column (3)

23.46 22.34

7.431 7.885

(6) V 2/2g (ft)

Depth of flow (ft) is known at section 1 and assumed at section 2. Once the energies balance, the depth is now known at section 2, and the depth at section 3 is assumed until the energies at sections 2 and 3 balance. In this table, only the final depths that result in an energy balance are shown.

9.976 9.916

24.90 23.46

(5) V (ft/s)

Column (2)

2.66 2.58

2 3

10.000 9.976

(4) A (ft2)

Section numbers are arbitrarily designated from upstream to downstream.

2.76 2.66

1 2

(3) z (ft)

Column (1)

(2) y (ft)

(1) Section

Table 6.5 (a) Water Surface Profile (Front-Water) Computations Using the Standard Step Method (Example 6.10)

232 Water Flow in Open Channels Chap. 6

2 3

1 2

Section

2.76 2.66

24.90 23.46

A (ft2) 15.843 15.396

P (ft) 1.571 1.524

Rh (ft) 7.431 7.885

2.66 2.58

23.46 22.34

15.396 15.038

1.524 1.486

7.885 8.280

A distance of 5.12 ft separates the two flow depths (2.66 ft and 2.58 ft).

U D

V (ft/s)

0.9655 1.0646

0.8575 0.9655

V 2/2g (ft)

3.6255 3.6446

3.6175 3.6255

E (ft)

0.00773 0.00882

0.00659 0.00773

5.12

1.66

∆L (ft)

Water Surface Profile (Front-Water) Computations Using the Direct Step Method (Example 6.10)

A distance of 1.66 ft separates the two flow depths (2.76 ft and 2.66 ft).

U D

U/D

y (ft)

Table 6.5 (b)

1.66 6.78

0 1.66

Distance to Dam (ft)

Sec. 6.8 Computation of Water Surface Profiles 233

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Classroom Computer Exercise—Water Surface Profiles Review Example 6.10. Obtain or write computer software appropriate for determining normal and critical depth and water surface profiles. Spreadsheets can be programmed quickly to perform the task, and the widely used water surface profile model called HEC-RAS discussed in Section 6.10 is freely available on the Internet. (See the book's preface for other suggestions.) Answer the following questions by performing a computer analysis of the open channel described in Example 6.10 and modifications thereof. (a) Before using the software, what data do you anticipate the software will need to evaluate the waters surface profile of Example 6.10? Is it necessary to solve for critical and normal depth? Why? (b) Using normal and critical depth software, enter the data requested to determine normal and critical depth. Compare your computer results with the answers listed for Example 6.10. Are there any discrepancies? Comments? (c) Using the water surface profile software, enter the data requested to evaluate the entire water surface profile. Compare your computer results with the answers for the first two sections given in Example 6.10. (The channel depths for the remaining two cross sections are not provided in Example 6.10 but are available in the back of the book for Problem 6.8.7.) Are there any discrepancies between the two results? Comments? (d) What would happen to the water surface profile if the channel slope were doubled? Will you need to compute a new normal depth? Critical depth? Double the slope and compute the new water surface profile. Did you reason correctly? Now restore the channel to its original slope. (e) What would happen to the water surface profile if the flow rate were doubled? Will you need to compute a new normal depth? Critical depth? Double the flow rate and compute the new water surface profile. Did you reason correctly? Now restore the flow rate to its original value. (f) Determine the channel slope required to make normal depth equal critical depth. Speculate on the water surface profile that would result if normal depth was greater than critical depth. Refer to Figure 6.13 (a) for assistance in reasoning through this. Perform any other changes your instructor requests of you. 6.9 Hydraulic Design of Open Channels Open channels are usually designed for uniform flow or normal conditions. Therefore, uniform flow equations are used in sizing these channels. Designing an open channel involves the selection of channel alignment, channel size and shape, longitudinal slope, and the type of lining material. Normally, we consider several hydraulically feasible alternatives and compare them to determine the most cost-effective alternative. This section focuses on the hydraulic considerations involved in channel design. The topography of the project site, the available width of right-of-way, and the existing and planned adjacent structures control the channel alignment. The topography also controls the bottom slope of the channel. Slope stability considerations often govern the selection of the

Sec. 6.9

Hydraulic Design of Open Channels

235

Table 6.6 Stable Side Slopes for Channels

Side Slopea (Horizontal:Vertical)

Material Rock Muck and peat soils Stiff clay or earth with concrete lining

Nearly Vertical 1

Earth with stone lining or earth for large channels

.4:1 :1 .2 to 1:1 1:1

Firm clay or earth for small ditches Loose, sandy earth

1 1.2:1 2:1 to 4:1

Sandy loam or porous clay

3:1

If channel slopes are to be mowed, a maximum side slope of 3:1 is recommended. Source: Based on V. T. Chow, Open Channel Hydraulics (New York: McGraw-Hill, 1959).

side slopes. The recommended side slopes for different types of channel materials are given in Table 6.6. There may also be limitations on channel depth because of a high water table in the underlying soil or underlying bedrock. Most open channels are designed for subcritical flow. It is important to keep the Froude number sufficiently lower than the critical value of 1.0 under design conditions. If the design Froude number is close to 1.0, there is a possibility that the flow will be unstable and fluctuate between subcritical and supercritical conditions because of variations in the actual discharge. Channels are often lined to prevent the sides and bottom of the channel from eroding because of the shear stresses caused by the flow. The types of channel linings available can be categorized into two broad groups: rigid and flexible. Rigid liners are inflexible—for example, concrete. Flexible liners are slightly pliable (with the underlying soil) and self-healing such as gravel, riprap, gabions, and grass liners. The discussion in this section is limited to unlined earthen channels and channels lined with rigid materials. Flexible linings are beyond the scope of this book but are discussed in the literature by Chen and Cotton* and Akan.† Freeboard is the vertical distance between the top of the channel and the water surface that prevails under the design flow conditions. This distance should be sufficient to allow variations in the water surface because of wind-driven waves, tidal action, occurrence of flows exceeding the design discharge, and other causes. There are no universally accepted rules to determine an acceptable freeboard. In practice, freeboard selection is often a matter of judgment, or it is stipulated as part of the prevailing design standards. For example, the U.S. Bureau of Reclamation recommends that unlined channel freeboard be computed as F = 2Cy

(6.28)

where F = freeboard, y = flow depth, and C = freeboard coefficient. If F and y are in ft, C varies from 1.5 for a channel capacity of 20 cfs to 2.5 for a channel capacity of 3,000 cfs or more. If metric units are used with F and y in meters, C varies from 0.5 for a flow capacity for 0.6 m3/s to 0.76 for a flow capacity of 85 m3/s or more. For lined channels, the Bureau recommends that the curves displayed in Figure 6.15 are used to estimate the height of the bank above the water surface (W.S.) and the height of the lining above the water surface.

* Y. H. Chen and G. K. Cotton, Design of Roadside Channels with Flexible Linings, Hydraulic Engineering Circular No. 15, Federal Highway Administration, 1988. † A. O. Akan, Open Channel Hydraulics (New York: Butterworth-Heinemann/Elsevier, 2006).

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7 6

Height (ft)

5 4

Height of bank above W.S.

Height of hard surface or buried membrane lining above W.S.

2 1

Height of earth lining above W.S.

0.5 10

20 30 40 60 100

200

400 600 1000 Capacity (cfs)

2000

4000 6000 10,000 20,000

Figure 6.15 Recommended freeboard and height of banks in lined channels. Source: U.S. Bureau of Reclamation, Linings for Irrigation Canals, 1976.

6.9.1 Unlined Channels The sides and bottoms of earthen channels are both erodible. The main criterion for earthen channel design is that the channel is not eroded under the design flow conditions. There are two approaches to erodible channel design, namely, the maximum permissible velocity method and the tractive force method. We will discuss the maximum permissible velocity approach because of its simplicity. This method is based on the assumption that a channel will not erode if the average crosssectional velocity in the channel does not exceed the maximum permissible velocity. Therefore, a channel cross-section is designed so that, under the design flow conditions, the average flow velocity remains below the maximum permissible value. The value of the maximum permissible velocity depends on the type of material into which the channel is excavated and the channel alignment. Table 6.7 presents the maximum permissible velocities for various types of soils. Following Lane,* the values given in Table 6.7 can be reduced by 13% for moderately sinuous channels and by 22% for very sinuous channels. In a typical channel design problem, the channel bottom slope (S0), the design discharge (Q), and the channel material would be given. The procedure to size the channel section would consist of the following steps: 1. For the specified channel material, determine the Manning roughness coefficient from Table 6.2, a stable side slope from Table 6.6, and the maximum permissible velocity from Table 6.7. * E. W. Lane, "Design of Stable Channels," Transactions of the American Society of Civil Engineers, Vol. 120, 1955.

Sec. 6.9

Hydraulic Design of Open Channels

237

Table 6.7 Suggested Maximum Permissible Channel Velocities

Channel Material

Vmax (ft/s)

Vmax (m/s)

2.0 4.0 6.0

0.6 1.2 1.8

2.0 3.5 6.0

0.6 1.0 1.8

Sand and Gravel Fine sand Coarse sand Fine gravela Earth Sandy silt Silt clay Clay

aApplies to particles with median diameter (D ) less than 0.75 in (20 mm). 50 Source: U.S. Army Corps of Engineers. "Hydraulic Design of Flood Control Channels," Engineer Manual, EM 1110-2-1601. Washington, DC: Department of the Army, 1991.

2. Compute the hydraulic radius (Rh) from the Manning equation rearranged as Rh = a

3. 4. 5. 6. 7.

n Vmax kM 2S0

3/2

(6.29)

where kM = 1.49 ft1/3/s for the conventional U.S. unit system and 1.0 m1/3/s for the metric system. Compute the required flow area from A = Q/Vmax. Compute the wetted perimeter from P = A/Rh. Using the expressions for A and P given in Table 6.1, solve for the flow depth (y) and the bottom width (b) simultaneously. Check the Froude number and ensure that it is not close to unity. Add a freeboard (Equation 6.28) and modify the section for practical purposes.

Example 6.11 An unlined channel to be excavated in stiff clay will convey a design discharge of Q = 9.0 m3/s on a slope of S0 = 0.0028. Design the channel dimensions using the maximum permissible velocity method. Solution From Table 6.6, m = 1.0 for stiff clay; from Table 6.2, use n = 0.022 (clean and smooth surface). Also, from Table 6.7, Vmax = 1.8 m/s. Using Equation 6.29 with kM = 1.00 Rh = c

0.022(1.8) 1.0020.0028

3/2

= 0.647 m

Also, A = Q/Vmax = 9.0/1.8 = 5.0 m2. Hence, P = A/Rh = 5.0/0.647 = 7.73 m. Now, from expressions given in Table 6.1 and using m = 1.0, A = (b + my)y = (b + y)y = 5 m2 and P = b + 2y21 + m2 = b + 2.83 y = 7.73 m

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We now have two equations with two unknowns, y and b. From the second equation, b = 7.73 - 2.83y. Substituting this into the first equation and simplifying yields 1.83y 2 - 7.73y + 5.00 = 0 This equation has two roots, y = 0.798 m and 3.43 m. The first root results in a channel width of b = 7.73 - 2.83(0.798) = 5.47 m. The second root results in a channel width of b = 7.73 - 2.83(3.43) = -1.98 m. Obviously, a negative channel width has no physical meaning. Therefore y = 0.798 m will be used. Next we will check to see if the Froude number is close to the critical value of 1.0. From the expression given for the top width in Table 6.1, T = b + 2my = 5.47 + 2(1)0.798 = 7.07 m Then the hydraulic depth becomes D = A/T = 5.0/7.07 = 0.707 m, and finally NF =

V 2gD

1.8 29.81(0.707)

= 0.683

This value indicates that, under the design flow conditions, the flow will not be near the critical state. Finally, we will determine a freeboard using Equation 6.28. It is known that C varies from 0.5 for a channel capacity of 0.6 m3/s 0 to 0.76 for a capacity of 85 m3/s. Assuming this variation is linear, we determine C as being 0.526 for Q = 9.0 m3/s by interpolation. Then, F = 20.526(0.798) = 0.648 m

The total depth for the channel is y + F = (0.798 + 0.64) = 1.45 m ≈ 1.5 m (for practicality in field construction). The bottom width of 5.47 m is increased to 5.5 m for the same reason. The top width of the excavated channel then becomes b + 2m(y) = 5.5 + 2(1)(1.5) = 8.5 m.

6.9.2 Rigid Boundary Channels Channels lined with materials such as concrete, asphaltic concrete, soil cement, and grouted riprap are considered to have rigid boundaries. These channels are nonerodible because of the high shear strength of the lining material. In general, there are not any design constraints on the maximum velocity. Therefore, the best hydraulic section concept may be used to size channels with rigid boundaries. The best hydraulic section concept was discussed in Section 6.3. In summary, the conveyance capacity of a channel section for a given flow area is maximized when the wetted perimeter is minimized. For trapezoidal channel shapes, the best hydraulic section for a fixed side slope (m) is represented by b = 2 1 21 + m2 - m 2 (6.30) y The procedure to size a trapezoidal section using the best hydraulic section approach is as follows: 1. Select m and determine n for the specified lining material. 2. Evaluate the ratio b/y from Equation 6.30. 3. Rearrange the Manning formula as y =

[(b/y) + 221 + m2]1/4 [(b/y) + m]5/8

Qn kM 2S0

3/8

and solve for y knowing all the terms on the right-hand side. Then find b.

(6.31)

Sec. 6.10

Open Channel Flow Modeling

239

4. Check the Froude number. 5. Determine the height of lining and the freeboard using Figure 6.15 and modify the section for practical purposes. Example 6.12 A trapezoidal, concrete-lined channel is required to convey a design discharge of 15 m3/s. The channel bottom slope is S0 = 0.00095, and the maximum side slope based on local ordinances is m = 2.0. Design the channel dimensions using the best hydraulic section approach. Solution From Table 6.2, n = 0.013 for concrete. Substituting m = 2 into Equation 6.30, we find b = 2 1 ( 21 + 22 - 2 2 = 0.472 y

Next, using Equation 6.31 with kM = 1.0 for the metric unit system, y =

[(0.472) + 221 + 22]1/4 (15.0)(0.013) 3/8 c d = 1.69 m [(0.472) + 2]5/8 1.020.00095

Then, b = 0.472(1.69) = 0.798 m. For this section, A

= (b + my)y = [0.798 + 2(1.69)]1.69 = 7.06 m2,

= b + 2my = 0.798 + 2(2)1.69 = 7.56 m,

= A/T = 7.06/7.56 = 0.934 m,

= Q/A = 15.0/7.06 = 2.12 m/s, and

NF = V/(gD)1/2 = 2.12/[9.81(0.933)]1/2 = 0.701. The Froude number is sufficiently lower than the critical value of 1.0. Finally, from Figure 6.15 (with Q = 15 m3/s = 530 cfs), the lining height above the free surface is 1.2 ft (0.37 m). Also, the freeboard (height of bank) above the free surface is 2.9 ft (0.88 m). Thus, the design channel depth is y + F = (1.69 + 0.88) = 2.57 m ≈ 2.6 m (for practicality in field construction). The bottom width of 0.798 m is increased to 0.8 m for the same reason. The top width of the channel is b + 2m(y) = 0.8 + 2(2)(2.6) = 11.2 m.

6.10 Open Channel Flow Modeling There are many hydraulic computer models available that will quickly perform the open channel flow calculations discussed in this chapter. Some of these models are proprietary and costly, but others are freely available on the Internet. Development of some of them started in the 1960s. They continued to be improved through the decades to a point where they are now quite versatile and user-friendly. Taken collectively, these hydraulic models have a broad range of capabilities. They are able to: š š š š

Determine normal depth and critical depth in open channels. Design prismatic channels to efficiently convey uniform flow. Evaluate energy levels and flow classifications in steady, open channel flow. Calculate water surface profiles (gradually varied flow) in natural or constructed channels. š Analyze hydraulic jumps (rapidly varied flow) and energy dissipation structures.

240

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Chap. 6

š Evaluate the stability of channel linings and design flexible (non-erodible) liners. š Model unsteady flow situations including dam break analyses. š Simulate sediment transport and perform movable boundary calculations. In addition, the model set-up and data input are fast and intuitive with graphical user interfaces (GUIs). The model output is flexible and report-ready with accompanying tables and graphs. In this section, we discuss a particular hydraulic model that is available from the U.S. Army Corps of Engineers (ACE). The Corp's model was developed by their Hydrologic Engineering Center and is called the River Analysis System (HEC-RAS). This model was selected for three reasons: 1. It is nonproprietary and freely available on the Internet. 2. It is fundamentally sound in handling a large variety of applications. 3. It is widely-used and accepted in the engineering and regulatory community. 6.10.1

The HEC-RAS Model

The HEC-RAS model was initially developed to compute water surface profiles for flow in natural and constructed channels. Even though the model has a multitude of capabilities, this remains its primary use and the one we will discuss. The order of the tasks performed in the model is as follows: define the river reach, enter the cross section data, specify all hydraulic structures, input the flow values and boundary conditions, and perform the desired hydraulic computations. The process sequence, model structure, and data requirements are described in the following paragraphs. Define River Reach The river system to be modeled is sketched in the geometric data window. An exact scale is not required, but the general orientation and flow direction should resemble what you would see on a map of the area. In addition, the river reaches are named along with any junctions as displayed in Figure 6.16. Enter Cross Section Data Cross sections along the various river reaches are entered into the cross section data editor (Figure 6.17). Additional information that is required includes the river station (the distance from the outlet or mouth of the river reach to this cross section), the left and right bank locations, Manning's roughness values for the channel and overbanks (left and right overbanks, LOB and ROB), the distances to the next downstream cross section, and expansion and contraction coefficients (i.e., minor loss coefficients). Once entered, the cross sections appear at the appropriate locations and scale (proportional to the sketched river length) in the geometric data window (Figure 6.16). In addition, there is an option that allows you to plot the cross section to visually check for mistakes. For example, the cross section data entered in Figure 6.17 is displayed graphically in Figure 6.18. The cross section plot also displays the Manning "n" values for the channel and the overbanks. Specify Hydraulic Structures Any hydraulic structures in the stream reach being modeled will likely affect the water surface profile. Hydraulic structures include bridges, culverts, weirs, dams, and spillways. Icons of these structures are found on the left margin of the geometric data window (Figure 6.16). You simply select the appropriate icon and the model will request the necessary data. Figure 6.19 depicts a bridge structure and shows the original valley cross section (channel and overbanks), the roadway fill (shaded area), the bridge opening, and the bridge piers.

Sec. 6.10

Open Channel Flow Modeling

Figure 6.16 Geometric data editor with a sketched river system (Note: The cross sections are proportionally drawn by the model once they are entered.) Source: HEC-RAS River Analysis System, User's Manual, U.S. Army Corps of Engineers, Version 4.1, January, 2010.

Figure 6.17 Cross section data editor Source: HEC-RAS River Analysis System, User's Manual, U.S. Army Corps of Engineers, Version 4.1, January, 2010.

241

242

Water Flow in Open Channels

Chap. 6

Figure 6.18 Cross section plot for the data in Figure 6.17 Source: HEC-RAS River Analysis System, User's Manual, U.S. Army Corps of Engineers, Version 4.1, January, 2010.

Figure 6.19 A portion of the bridge data editor showing the bridge cross section plot Source: HEC-RAS River Analysis System, User's Manual, U.S. Army Corps of Engineers, Version 4.0, December, 2013.

Input Flow Values & Boundary Conditions Flow values are required prior to performing the hydraulic calculations. HEC-RAS can evaluate steady and unsteady flow in open channels. The steady flow data editor is depicted in Figure 6.20. Multiple water surface profiles can be analyzed in one simulation. In this case, the 10-, 50-, and 100-year flood are being modeled. Flow values are given at the most upstream cross section and wherever there is a change in flow moving downstream. The most notable location for a change in flow is when a tributary stream joins

Sec. 6.10

Open Channel Flow Modeling

Figure 6.20

243

Steady (uniform or gradually varied) flow data editor

the main channel. Note that a data editor for boundary conditions can be accessed by pressing the appropriate button on the flow data editor. A number of choices exist for inputting a boundary condition including normal depth or a known water surface elevation. In subcritical flow, the boundary condition at the downstream end of the channel reach is required. Perform Hydraulic Computations After the requisite geometric data and flow data are entered, the HEC-RAS hydraulic computations can be performed. Five different choices exist: steady flow, unsteady flow, sediment transport, water quality analysis, and hydraulic design. The simulation of a water surface profile is our immediate interest. The steady flow analysis option is selected from the Run menu bar of the main window, the appropriate flow and geometry files are selected in the simulation manager (Figure 6.21), and the compute button is pressed. Note that gradually varied flow is processed under the steady flow option in HEC-RAS.

Figure 6.21

Steady (uniform or gradually varied) flow simulation manager

244

Water Flow in Open Channels

Figure 6.22

Chap. 6

Profile output table

HEC-RAS Model Output Model output is displayed in tabular and graphical form. Tabular output can display a limited number of hydraulic variables for many cross sections (Figure 6.22) or a large number of hydraulic variables at a single cross section. However, the graphical output is much more informative. For example, a profile plot displays the changing water level throughout the channel length (Figure 6.23). Cross section plots display the water surface elevation and extent of flooding in the channel and overbanks (Figure 6.24). There is even an option to produce a three-dimensional schematic of the channel and floodplain (Figure 6.25) that can be rotated in all directions. In summary, the output possibilities are extensive, flexible, and report-ready.

5305 5300

Elevation (ft)

5295 5290 5285 Legend 5280 EG PF 1 5275

WS PF 1 Crit PF 1

5270

Ground 5265

500

1000 Main channel distance (ft)

1500

Figure 6.23 Profile plot (Note: A bridge is located 700 ft upstream that causes the water surface to rise 5 ft.)

2000

Problems

245

0.05

0.045

5282

Elevation (ft)

5280 5278 5276

Legend EG PF 1

5274 5272 5270

WS PF 1 Crit PF 1 Ground Bank Sta

5268 300

400

500

600

700

800

Station (ft) Figure 6.24

Figure 6.25

Cross section plot

Three-dimensional perspective plot

(Note: There is a small bike path bridge in the foreground and a vehicular bridge upstream.)

Problems (secTion 6.1) 6.1.1. Using the time and space criteria, classify the following open channel flow scenarios (steady or unsteady and uniform or varied): (a) Constant flow in a long, prismatic channel with a mild slope. (b) Flow in the transition of the channel in part (a) to a channel with a steep slope. (c) Flow on a sloped parking lot during a uniform-intensity rainfall event.

246

Water Flow in Open Channels

Chap. 6

(d) Flow on a sloped parking lot during a rainfall event that decreases in intensity over time. (e) Flow in a prismatic channel from a rapidly opening sluice gate. (f) Flow during the dry season in an urban (natural) stream. 6.1.2. Why is uniform flow rare in natural channels? Is steady flow rare in natural channels (e.g., rivers, streams, etc.)? Explain.

(secTion 6.2) 6.2.1. A concrete channel with an unusual cross section (Figure P6.2.1) carries water at a flow rate of 30 m3/s. Determine the channel's slope.

4.0 m 1.6 m 3.6 m

2.0 m

Figure P6.2.1

6.2.2. A trapezoidal channel has a bottom width of 30 ft and side slopes of 2:1 (H:V). The channel is paved with a concrete surface. If the channel is laid on a slope of 0.0001 and carries a uniform flow at a depth of 6 ft, determine the discharge. 6.2.3. A rectangular, irrigation channel with slope 0.0004 needs to be extended and cut through a short section of rock outcropping (n = 0.035). If the width is designed to be twice the depth, determine its dimensions assuming uniform flow and a discharge of 50 m3/s. 6.2.4. A 3-m-wide rectangular irrigation channel carries a discharge of 50.0 m3/s. The channel has a slope of 0.041 and a Manning's coefficient is n = 0.022. Determine the normal depth using successive substitution and Figure 6.4. 6.2.5. A trapezoidal channel with an 18-ft-wide bottom and side slope m = 2 discharges 300 cfs. The natural channel (clean/straight) has a 0.04 ft/ft bottom slope. Compute the normal depth of flow using successive substitution and Figure 6.4. Check your solution using appropriate computer software. 6.2.6. A corrugated metal stormwater drain is discharging 55 cfs. Assuming uniform flow in the 4-ftdiameter pipe, determine the flow depth if the 150 ft long pipe goes through a 3-ft drop in elevation. Check your solution using Figure 6.4b and appropriate computer software. 6.2.7. A triangular highway gutter (Figure P6.2.7) is designed to carry a discharge of 52 m3/min on a channel slope of 0.0016. The concrete gutter is 0.8 m deep with one side vertical and one side sloped at m:1 (H:V). Determine the side slope m. Check your solution using appropriate computer software.

Problems

247

Figure P6.2.7

6.2.8. Uniform flow occurs in a 20-ft-wide rectangular channel with a discharge of 2,520 cfs. If the normal depth of the flow is 15 ft, what will be the new normal depth when the width of the channel expands to 30 ft? Assume that the slope and channel roughness remain constant in both channels. Check your solution using appropriate computer software. 6.2.9. Determine the diameter of a corrugated-metal, storm water pipe that is designed to carry a flow rate of 5.83 m3/s while flowing half full. The slope of the pipe is 0.02 m/m and uniform depth is assumed. Also, determine the pipe size required to carry the same flow rate if the pipe is to flow full. Check your solution using Figure 6.4 and appropriate computer software. 6.2.10. Using appropriate computer software, design a trapezoidal channel and a rectangular channel to convey 100 cfs on a slope of 0.002. Both channels are lined with concrete. Specify width, depth, and side slopes. In both cases, try to obtain channels where the depth is about 60% of the bottom width.

(secTion 6.3) 6.3.1. Derive the relationship between depth of flow and width for a rectangular channel that is the most hydraulically efficient (i.e., best hydraulic section). Use Example 6.4 as a guide and show all steps. 6.3.2. Design the best hydraulic (rectangular) section for a metal channel to carry a flow rate of 31.2 cfs on a slope of 0.04. Check your solution with appropriate computer software. 6.3.3. An open channel (n = 0.011) is to be designed to carry 7.14 m3/s on a slope of 0.0063. Find the diameter of the best hydraulic section (semicircle). 6.3.4. A concrete open channel is required to pass a flow rate of 42.5 m3/s. Design the best hydraulic (trapezoidal) section for the channel. Check your solution using appropriate computer software. 6.3.5. Using the information provided in Example 6.4, determine the channel depth and bottom width of the best hydraulic section (trapezoidal channel) that requires a flow (excavation) area that will not exceed 150 ft2. Check your answer by showing the side length is equal to the bottom width (halfhexagon; Figure 6.6). 6.3.6. Show all of the computation steps in progressing from Equation (4) to Equation (5) in Example 6.4. 6.3.7. Determine the side slopes of the best hydraulic (triangular) section.

(secTion 6.4) 6.4.1. Starting with the specific energy equation for open channel flow, E = V 2/2g + y, derive the equations for critical depth (i.e., minimum energy); Q2T/(gA3) = 1 that leads to; V 2/(gD) = 1. Show all steps and define the variables T and D.

248

Water Flow in Open Channels

Chap. 6

6.4.2. There are two ways of determining the flow classification (subcritical, critical, or supercritical) in an open channel: (a) compute the Froude number (if Nf 6 1, subcritical, if Nf = 1, critical, and if Nf 7 1, supercritical), or (b) compute critical depth (yc) and compare it with the depth of flow (y) (if y 7 yc, flow is subcritical, if y = yc, flow is critical, and if y 6 yc, flow is supercritical) From Example 6.1, compute the flow classification for the 3-m-wide, rectangular irrigation channel (Q = 25.3 m3/s, yn = 1.2 m) using both methods. 6.4.3. A 3-m-wide rectangular channel carries a discharge 15 m3/s at a uniform depth of 1.7 m. The Manning's coefficient is n = 0.022. Determine (a) the channel slope, (b) the critical depth, (c) the Froude number, and (d) flow classification (subcritical or supercritical). 6.4.4. A flow of 834 cfs is conveyed in a 10-ft-wide rectangular channel at a depth of 6.0 ft. What is the specific energy of the channel? Is this flow subcritical or supercritical? If n = 0.014, what channel slope must be provided to produce a uniform flow at this depth? Check your solution using appropriate computer software. 6.4.5. A flow of 100 m3/s occurs in a trapezoidal canal having a bottom width of 10 m, a side slope of 2:1 (H:V) and n = 0.017. Calculate the critical depth and critical slope (i.e., the slope required to maintain this flow at a normal depth equal to critical depth). Check your critical depth solution using Figure 6.9. 6.4.6. A corrugated metal pipe (n = 0.024) with a 2.25-ft diameter and slope of 0.005 ft/ft flows half full. Determine the specific energy and the flow classification (subcritical or supercritical). Check your solution using appropriate computer software. 6.4.7. A trapezoidal channel conveys 480 cfs at a depth of 8.3 ft. If the channel has a bottom width of 10 ft and 1:1 side slopes, what is the flow classification (subcritical, critical, or supercritical)? What is the specific energy of flow? Also determine the critical depth using Figure 6.9. Check your solution using appropriate computer software. 6.4.8. A 40-ft-wide rectangular channel with a bottom slope of S = 0.0025 and a Manning's coefficient of n = 0.035 is carrying a discharge of 1,750 cfs. Using appropriate computer software (or desktop methods), determine the normal and critical depths. Also construct a specific energy curve for this discharge. 6.4.9. A transition is constructed to connect two trapezoidal channels with bottom slopes of 0.001 and 0.0004, respectively. The channels have the same cross-sectional shape, with a bottom width of 3 m, a side slope of z = 2, and Manning's coefficient n = 0.02. The transition is 20 m long and is designed to carry a discharge of 20 m3/s. Assume that an energy loss of 0.02 m is uniformly distributed through the transitional length. Determine the change in the bottom elevations at the two ends of the transition. Assume uniform (normal) depth before and after the transition and use appropriate computer software to assist you. 6.4.10. A hydraulic transition 100 ft long is used to connect two rectangular channels, 12 ft and 6 ft wide, respectively. The design discharge is 500 cfs, n = 0.013, and the slopes of the channels are 0.0009 for both. Determine the change in the bottom elevation and the water surface profile in the transition if the energy loss in the transition if 1.5 ft and is uniformly distributed throughout the transition length. Assume uniform (normal) depth before and after the transition and use appropriate computer software to assist you.

(secTion 6.5) 6.5.1. When the specific energy for a given discharge is plotted against the depth of the flow at a given section, a specific energy curve is obtained. Explain why the specific energy curve approaches the abscissa asymptotically for low depths (supercritical flow) and approaches a 45° line asymptotically for high depths (subcritical flow) using the equation for specific energy.

Problems

249

6.5.2. What is the difference between alternate depths and initial/sequent depths? 6.5.3. A hydraulic jump occurs in a 5-ft-wide rectangular channel. The initial and sequent depths are 0.66 ft and 3.00 ft, respectively. Determine the energy loss and the Froude number in the channel prior to the jump. 6.5.4. A hydraulic jump occurs in a 15-m-wide rectangular, concrete channel. The initial (normal) depth comes from a spillway discharge of 100 m3/s with a slope of 10%. The sequent depth is 3.00 m. Determine the energy loss and the specific force per unit width of channel. 6.5.5. Construct the specific energy and specific force curves for a 5-ft-wide rectangular concrete channel that carries 40 cfs. Use 0.5 ft depth increments up to 3 ft. 6.5.6. Plot the specific energy and specific force curves for a 10-m rectangular channel carrying 15 m3/s discharge. Use 0.2 m depth increments up to 1.4 m. Also, determine the critical depth, the minimum specific energy, and discuss how a change in discharge would affect the specific force curve.

(secTion 6.8) 6.8.1. Identify all of the gradually varied flow classifications depicted in Figure 6.13. The complete list of possible classifications is depicted in Figure 6.12. 6.8.2. A 5-m-wide, rectangular channel (n = 0.035) flows into a reservoir with a water surface that is 1.93 m above the channel bottom at its end. If the flow rate is 20 m3/s and the channel slope is 1.9%, determine the channel and flow classification (e.g., M-3, S-2, etc.) and explain your supporting logic. 6.8.3. An obstruction is lodged in a 10-ft wide rectangular channel (concrete) that has a bottom slope of 0.0025. The depth of the channel just upstream of the obstruction is 5.0 ft. If the flow rate is 325 cfs, determine the channel and flow classification (e.g., S-3, M-2, etc.). Explain your supporting logic. 6.8.4. Water emerges from under a gate into a trapezoidal, concrete channel. The gate opening is 0.55 m, and the flow rate is 12.6 m3/s. The entry channel has a 1.5 m bottom width, side slopes of 0.5:1 (H:V), and a bottom slope of 0.018. Determine the channel and flow classification (e.g., M-1, S-2, etc.) and explain your supporting logic. 6.8.5. At a certain location, the depth of flow in a 40-ft-wide rectangular channel is 4.00 ft. The channel has a bottom slope of S = 0.0025, a Manning's coefficient of n = 0.035, and a discharge of 1,600 cfs. Determine the channel and flow classification (e.g., M-1, S-2, etc.) and explain your supporting logic. Will the depth increase or decrease going upstream? Using the standard step method (one step), determine the depth of flow 100 ft upstream from the location where the depth is 4.00 ft. (Note: Normal depth and critical depth are yn = 6.52 ft and yc = 3.68 ft.) 6.8.6. Water flows through a trapezoidal channel (n = 0.011) at the rate of 40 m3/s. The channel has a 5-m bottom width, side slopes of m = 1.0, and a bottom slope of 0.004. Determine the depth of flow 5 m upstream from a section that has a measured depth of 1.79 m. But first determine the channel and flow classification (e.g., M-1, S-2, etc.) and explain your supporting logic. Will the depth increase or decrease going upstream? Use the standard step method (one step) to determine the depth. (Note: Normal and critical depths are yn = 1.21 m and yc = 1.67 m.) 6.8.7. In example Problem 6.9, water depths were required at critical diversion points located at distances of 188, 423, 748, and 1,675 m upstream of the dam. Depths were only found at locations 188 and 423 m upstream. Determine the depths of flow at the other two diversion points using the standard step method and a spreadsheet program. Has normal depth been reached in the last cross section?

250

Water Flow in Open Channels

Chap. 6

6.8.8. In example Problem 6.10, water depths were required in the discharge channel until the water level was within 2% of normal depth. Depths were found at 2 and 7 ft downstream [separation distances (∆L) of 2 ft and 5 ft] but are still needed at distances of 17 and 57 ft downstream. Determine the depths of flow at these other two locations using the standard step method and a spreadsheet program. Has normal depth been reached (within 2%) in the last cross section? 6.8.9. In example Problem 6.9, water depths were required at critical diversion points located at distances of 188, 423, 748, and 1,675 m upstream of the dam. The standard step method solution (Problem 6.8.7) yielded depths of 1.91, 1.82, 1.74, and 1.66 m, respectively, at these locations. Use the direct step method and a spreadsheet program to solve for upstream distances using these depths. How closely do these distances compare with the standard step method distances prescribed in the original example problem? 6.8.10. In example Problem 6.10, water depths were required in the discharge channel at the following downstream distances; 2, 7, 17, and 57 ft (separation distances of 2, 5, 10, and 40 ft). The standard step method solution (Problem 6.8.8) yielded depths of 2.66, 2.58, 2.51, and 2.42 ft, respectively, at these locations. Use the direct step method and a spreadsheet program to solve for upstream distances using these depths. How closely do these distances compare with the standard step method distances prescribed in the original example problem? 6.8.11. In Figure P6.8.11, a 10-m-wide rectangular channel carries 16.0 m3/s and has a roughness coefficient of 0.015 and a slope of 0.0016. Normal and critical depths in this channel are yn = 0.780 m and yc = 0.639 m. If a 5-m high dam is placed across the channel raising the water depth to 5.70 m at a location 5.0 m upstream from the dam, determine the channel and flow classification (e.g., M-2, S-1, etc.) and explain your supporting logic. Then determine the water surface profile upstream from the control section (where the depth is 5.70 m) by finding the depth of flow at distances of 200, 600, 1,500, and 3,000 m upstream. Use a spreadsheet program or appropriate computer software to perform the computations. (See the book's preface for computer software suggestions.)

S0 = 0.0016 Figure P6.8.11

6.8.12. The downstream side of the dam in Problem 6.8.11 has a slope of 1/10. Water that goes over the dam is conveyed in this 10-m wide rectangular overflow channel until it rejoins the original channel at the bottom of the dam. This steep channel carries 16.0 m3/s and has a roughness coefficient of 0.015. Normal and critical depths are yn = 0.217 m and yc = 0.639 m. Determine the channel and flow classification (e.g., M-2, S-1, etc.) in this channel and explain your supporting logic. Also, determine the location and depth of the control section. (Hint: Refer to Figures 6.12 and 6.13 after you determine the channel and flow classification.) Then determine the water surface profile in the

Problems

251

overflow channel by finding the depth of flow at distances of 0.3, 2.0, 7.0, and 30.0 m downstream from the top of the dam. Use a spreadsheet or appropriate computer software to perform the computations. (See the book's preface for computer software suggestions.) 6.8.13. A diversion gate backs up flow in a 5-ft wide rectangular irrigation channel (S0 = 0.001, n = 0.015). The discharge in the channel is 50 cfs. If the depth of flow at the gate is 3.5 ft, determine the channel and flow classification (e.g., M-2, S-1, etc.) upstream of the gate and explain your supporting logic. Then determine the water surface profile by finding the depth of flow at distances of 100, 300, and 600 ft upstream of the gate. Use a spreadsheet or appropriate computer software to perform the computations. (See the book's preface for computer software suggestions.) 6.8.14. A long trapezoidal channel with a roughness coefficient of 0.015, bottom width measuring 3.6 m, and m = 2.0 carries a flow rate of 44 m3/s. An obstruction is encountered in the channel that raises the water depth to 5.8 m. If the channel slope is 0.001, determine the channel and flow classification (e.g., M-2, S-1, etc.) upstream of the obstruction and explain your supporting logic. Then compute the water surface profile upstream from the obstruction by finding depths at 250 m intervals until the depth is within 2% of normal depth. Use a spreadsheet or appropriate computer software to perform the computations. (See the book's preface for computer software suggestions.) 6.8.15. A very long trapezoidal canal with a bottom width of 18 ft, side slopes of m = 2.0, a bottom slope of 0.001, and n = 0.020, carries a flow rate of 800 cfs. The canal terminates at a free overfall (e.g., a water falls). (a) Determine the channel and flow classification upstream of the free overfall (e.g., M-2, S-1, etc.) and explain your supporting logic. (b) Will depths of 5.25 and 5.30 ft occur in this channel? If your answer is YES calculate how far from the downstream end. If your answer is NO explain why. (c) Will depths of 3.65 and 3.85 ft occur in this channel? If your answer is YES calculate how far from the downstream end. If your answer is NO explain why.

(secTion 6.9) 6.9.1. An earthen channel will be excavated into a silt clay soil (n = 0.024, m = 3). The channel will have a bottom slope of 0.002 and accommodate a design flow of 4.58 m3/s. Ignoring freeboard, design (size) the channel section. 6.9.2. Design (size) a channel section for the following conditions: earthen channel (sandy soil) with side slopes of 3:1, a design flow of 303 cfs, bottom slope of 0.0011, Vmax = 4.0 ft/s, and a roughness coefficient of 0.022. Account for freeboard in the design. 6.9.3. A rectangular, 100-m-long channel will be cut in rock (n = 0.035) and is required to deliver a flow of 25 m3/s on a slope of 0.003 m/m. Design the channel dimensions using the best hydraulic section approach and include freeboard. 6.9.4. A trapezoidal channel lined with concrete is required to carry 342 cfs. The channel will have a bottom slope of S0 = 0.001 and side slopes of m = 1.5. Design (size) the channel using the best hydraulic section approach and include freeboard.

252

Water Flow in Open Channels

Chap. 6

(secTion 6.10) 6.10.1. Review Section 6.10 on the capabilities and use of HEC-RAS, the U.S. Army Corps of Engineers' hydraulic modeling program. Then download the program from the website: http://www .hec.usace.army.mil/software/hec-ras/downloads.aspx. Examine the program's functions and then apply the program to Example 6.9. Verify that the solution determined using HEC-RAS matches the solution in the example problem. Note that slight variations in the depths and the energy levels are to be expected since HEC-RAS is capable of going through a large number of iterations to match energy levels. 6.10.2. Choose any of the problems that you solved in Section 6.8 using the standard step method. Now use HEC-RAS to solve the same problem. Verify that the solution determined using HEC-RAS matches the solution you obtained with the standard step method. Note that slight variations in the depths and the energy levels are to be expected since HEC-RAS is capable of going through a large number of iterations.

Robert Houghtalen

Groundwater Hydraulics

Groundwater is found in permeable, water-bearing geologic formations known as aquifers. There are basically two types of aquifers. 1. A confined aquifer is a relatively high-permeability, water-bearing formation (e.g., sand or gravels) confined below a layer of very low permeability (e.g., clay). 2. An unconfined aquifer is a relatively high-permeability, water-bearing formation with a definable water table, a free surface subjected to atmospheric pressure from above and below which the soil is completely saturated. Figure 7.1 schematically shows several examples of groundwater occurrence in both confined and unconfined aquifer formations. Movement of groundwater occurs because of a hydraulic gradient or gravitational slope in the same way water movement occurs in pipes or open channels. Hydraulic gradients in aquifers may occur naturally (e.g., sloping water tables) or as a result of artificial means (e.g., well pumps). The pressure level, or pressure head, in a confined aquifer is represented by the piezometric surface that usually originates from a distant source such as the water table elevation in a recharging location. An artesian spring is formed if the impermeable confining stratum is perforated at a location where the ground surface falls below the piezometric surface. It becomes an artesian well if the ground surface rises above the piezometric surface. The water table of an unconfined aquifer is usually unrelated to the piezometric surface of a confined aquifer in the 253

254

Groundwater Hydraulics

Piezometric surface

Water table

Im pe

Water table Artesian well well

Artesian spring

Pumping well in an unconfined aquifer

bl ec

Chap. 7

Pumping well in a confined aquifer

Water table

fin

in g

str at

Unconfined aquifer Confined aquifer

Figure 7.1

Groundwater occurrence in confined and unconfined aquifers

same region, as shown in Figure 7.1. This is because the confined and unconfined aquifers are hydraulically separated by the impermeable confining stratum. The capacity of a groundwater reservoir in a particular geological formation depends strictly on the percentage of void space in the formation and how the interstitial spaces are connected. However, the ability to remove the water economically depends on many other factors (e.g., the size and interconnectivity of the pore spaces, the direction of flow) that will be discussed later. Figure 7.2 schematically shows several types of rock formations and their relation to the interstitial space. The ratio of void space to the total volume of an aquifer (or a representative sample) is known as the porosity of the formation, defined as α =

Figure 7.2

Volv Vol

Examples of interstitial spaces in various rock formations

(7.1)

Sec. 7.1

Movement of Groundwater Table 7.1

255

Porosity Range in Common Water-Bearing Formations

Material Clay Silt Medium to coarse mixed sand Fine to medium mixed sand Uniform sand Gravel Gravel and sand mixed Sandstone Limestone

0.45–0.55 0.40–0.50 0.35–0.40 0.30–0.35 0.30–0.40 0.30–0.40 0.20–0.35 0.10–0.20 0.01–0.10

where Volv is the volume of the void space, and Vol is the total volume of the aquifer (or the sample). Table 7.1 lists the porosity range for common water-bearing formations. 7.1 Movement of Groundwater Groundwater velocity is proportional to the hydraulic gradient in the direction of the flow. The apparent velocity of groundwater movement in a porous medium is governed by Darcy's law* V = K

dh dL

(7.2)

where dh/dL is the hydraulic gradient in the direction of the flow path (dL) and K is a proportionality constant known as the coefficient of permeability, sometimes called the hydraulic conductivity. The apparent velocity is defined by the quotient of discharge divided by the total cross-sectional area of the aquifer stratum through which it flows. The coefficient of permeability depends on the aquifer properties and the properties of the fluid. By dimensional analysis, we may write K =

Cd 2γ μ

(7.3)

where Cd 2 is a property of the aquifer material, while γ is the specific weight and μ is the viscosity of the fluid, respectively. The constant C represents the various properties of the aquifer formation that affect the flow other than d, which is a representative dimension proportional to the size of the interstitial space in the aquifer material. The coefficient of permeability can be effectively determined by laboratory experiments or field tests by applying Darcy's equation (Equation 7.2), as will be discussed later. Table 7.2 gives a representative range of the coefficients of permeability for some natural soil formations. The seepage velocity (Vs) is the average speed that water moves between two locations ∆L apart in an aquifer over a time increment ∆t. It is different from the apparent velocity (V) defined in Darcy's equation because water can only move through the pore spaces. The appropriate equations are Vs = ∆L> ∆t

and

V = Q>A

* H. Darcy, Les Fontaines Publiques de la Ville de Dijon (Paris: V. Dalmont, 1856).

256

Groundwater Hydraulics Table 7.2

Typical Coefficient of Permeability Ranges for Some Natural Soil Formations

K (m>s)a

Soils Clays Sandy clays Peat Silt Very fine sands Fine sands Coarse sands Sand with gravels Gravels aFor

Chap. 7

610-9 10-9 to 10-8 10-9 to 10-7 10-8 to 10-7 10-6 to 10-5 10-5 to 10-4 10-4 to 10-2 10-3 to 10-2 710-2

K in (ft >s), multiply by 3.28; for K in (gpd>ft2), multiply by 2.12 * 106.

Since water can only travel through pore space, Vs = Q>(αA) = V>α where A is the total cross-sectional flow area and (αA) represents just the open (pore) area. Even the seepage velocity is not the actual velocity or speed of an individual water molecule as it travels through the pore spaces. The actual distance that water molecules travel between any two locations ∆L apart in the porous medium is not a straight line but a tortuous path, so it must be longer than ∆L. In this chapter, the movement of groundwater is treated from a hydraulic engineer's point of view; that is, we are primarily interested in the movement of water at the macro level, not the micro level. The fluid mechanical aspect of water particle movement in pores will not be included in this discussion. For an area A perpendicular to the direction of the water movement in an aquifer, the discharge may be expressed as Q = AV = KA

dh dL

(7.4)

Laboratory measurement of the coefficient of permeability can be demonstrated by the following example. Example 7.1 A small sample of an aquifer (uniform sand) is packed in a test cylinder (Figure 7.3) to form a column 30 cm long and 4 cm in diameter. At the outlet of the test cylinder, 21.3 cm3 of water is collected in 2 min. During the testing period, a constant piezometric head difference of ∆h = 14.1 cm is observed. Determine the coefficient of permeability of the aquifer sample. Solution The cross-sectional area of the sample volume is A =

π 2 (4) = 12.6 cm2 4

Sec. 7.1

Movement of Groundwater

257

∆h P1 g P2 g

Water entrance z2 Water exit Figure 7.3

Laboratory determination of coefficient of permeability

The hydraulic gradient is equal to the change of energy head (velocity head is negligible in groundwater flow) per unit length of the aquifer measured in the direction of the flow. dh 14.1 = = 0.470 dL 30 The discharge rate is Q = 21.3 cm3 >2 min = 10.7 cm3 >min

Applying Darcy's law (Equation 7.4), we may calculate K =

Q 1 10.7 # 1 = = 1.81 cm3 >min # cm2 = 3.02 * 10-4 m>s A dh 12.6 0.470 a b dL

Note that permeability units are actually stated in terms of flow rate (volume/time) per unit area of aquifer. However, this basic unit can be reduced to a form of velocity. Also note that the movement of water is very slow even in sand, which is why the velocity head component of energy head is typically ignored.

Based on Darcy's law, several different types of permeameters have been developed in the past for laboratory measurements of permeability in small samples of aquifers.* Although laboratory testing is conducted under controlled conditions, such measurements may not be representative of the field permeability because of the small sample size. In addition, when unconsolidated samples are taken from the field and repacked in laboratory permeameters, the texture, * L. K. Wenzel, Methods for Determining Permeability of Water-Bearing Materials with Special Reference to Discharging-Well Methods, U.S.G.S. Water Supply Paper 887 (1942).

258

Groundwater Hydraulics

Chap. 7

porosity, grain orientation, and packing may be markedly disturbed and changed; consequently, permeabilities are modified. Undisturbed samples taken with thin-walled tubes are better but still suffer from some disturbance, wall effects of the sample tube, entrained air, and flow direction (i.e., the flow direction in the field is likely to be different than the laboratory flow direction through the sample). For better reliability, field permeability of aquifers can be determined by well-pumping tests in the field. This method will be discussed in Section 7.4. 7.2 Steady Radial Flow to a Well Strictly speaking, groundwater flow is three dimensional. However, in most aquifer flow situations the vertical component is negligible. This results from the fact that the horizontal dimensions of aquifers are several orders of magnitude larger than the vertical dimension or the thickness of the aquifer. Therefore, flow in aquifers can be assumed to take place horizontally with velocity components in the x and y directions only. All the aquifer and flow characteristics are assumed to remain constant over the thickness of the aquifer. In accordance with this assumption of aquifertype flow, we can say that the piezometric head is constant at a given vertical section of an aquifer. All the equations in this section and the subsequent sections are derived with the assumption that the aquifer is homogeneous (aquifer properties are uniform) and isotropic (permeability is independent of flow direction) and that the well completely penetrates the aquifer. The removal of water from an aquifer by pumping a well results in radial flow in the aquifer to the well. This happens because the pumping action lowers the water table (or the piezometric surface) at the well and forms a region of pressure depression that surrounds the well. At any given distance from the well, the drawdown of the water table (or piezometric surface) is defined by the vertical distance measured from the original to the lowered water table (or piezometric surface). Figure 7.4 (a) shows the drawdown curve of the water table in an unconfined aquifer; Figure 7.4 (b) shows the drawdown curve of the piezometric surface in a confined aquifer. Q Q

r0 r

Original water table

Original piezometric surface

rw Actual water table (drawdown curve)

Actual piezometric surface

h hw

(a)

Figure 7.4

Impermeable stratum

Confined aquifer

(b)

Radial flow to a pumping well from (a) an unconfined aquifer and (b) a confined aquifer

Sec. 7.2

Steady Radial Flow to a Well

259

In a homogeneous, isotropic aquifer, the axisymmetric drawdown curve forms a conic geometry commonly known as the cone of depression. The outer limit of the cone of depression defines the area of influence of the well. The flow is said to be at steady state if the cone of depression does not change with respect to time. For a hydraulic system to be at steady state, the volume of water being added to storage per unit time must be equal to the water being taken from storage. When a discharge well pumps water from an aquifer, a steady state would be reached only if the aquifer is recharged at the same rate from a lake or river, rainfall infiltration, leakage from another aquifer, or some other source of water. In the absence of such recharge, a strict steady state is not possible. However, if pumping continues at a constant rate for some time, an approximate steady state will be reached as the changes in the cone of depression become unnoticeable. 7.2.1

Steady Radial Flow in Confined Aquifers

Darcy's law may be directly applied to derive the radial flow equation that relates the discharge to the drawdown of the piezometric head in a confined aquifer after a steady state of equilibrium is reached. By using plane polar coordinates with the well as the origin, we find that the discharge flowing through a cylindrical surface at a radius r from the center of the well equals Q = AV = 2πrb a K

dh b dr

(7.5)

In this equation, b is the thickness of the confined aquifer. Because the flow is at steady state, Q is also equal to the well discharge, the flow rate at which the well is pumped. The piezometric head (h) can be measured from any horizontal datum. However, usually it is measured from the bottom of the aquifer. Figure 7.4 (b) depicts all of the variables. Integrating Equation 7.5 between the boundary conditions at the well (r = rw, h = hw) and at the radius of influence (r = r0, h = h0) yields Q = 2πKb

h0 - hw r0 lna b rw

(7.6)

The previous equations apply only to steady flow where the discharge remains a constant value at any distance from the well. In other words, the radius of influence is neither expanding nor contracting. A more general equation for the discharge (steady flow) may be written for any distance (r) as Q = 2πKb

h - hw r lna b rw

(7.7)

Eliminating Q between Equations 7.6 and 7.7 results in r b rw h - hw = (h0 - hw) r0 lna b rw lna

(7.8)

260

Groundwater Hydraulics

Chap. 7

which shows that the piezometric head varies linearly with the logarithm of the distance from the well, regardless of the rate of discharge. Transmissivity (T, or transmissibility) is a confined aquifer characteristic defined as T = Kb. Often the confined flow equations are written in terms of T by replacing Kb with T. Rearranging Equation 7.7 and using the definition of transmissivity, we obtain h = hw +

Q r lna b rw 2π T

(7.9)

This equation is useful to determine the piezometric head (h) at any distance r if the head at the well (hw) is known. If the head is known at a location other than the pumped well—say, at an observation well—then a similar equation can be written as h = hob +

Q r lna b rob 2π T

(7.10)

where rob is the distance between the pumped well and the observation well and hob is the head at the observation well. In most groundwater problems, we are more interested in the drawdown rather than the head. The drawdown (s) is defined as s = h0 - h where h0 is the undisturbed head. In terms of drawdown, Equation 7.10 becomes s = sob +

rob Q lna b r 2π T

(7.11)

Equation 7.11 can be used to determine the drawdown caused by a single pumping well. Because the confined flow equations are linear, we can use the principle of superposition to find the drawdown resulting from multiple pumping wells. In other words, the drawdown produced by multiple wells at a particular location is the sum of the drawdowns produced by the individual wells. Suppose, for example, a well located at point A is being pumped at a constant rate QA and a well at point B is being pumped at a constant rate QB, as shown in Figure 7.5. Suppose there is an observation well at point O at which the observed drawdown is sob. The drawdown (s) at point C can be calculated using s = sob +

rAo rBo QA QB ln a b + ln a b rA rB 2π T 2π T

(7.12a)

where rAo is the distance between pumped well A and the observation well, rBo is the distance between pumped well B and the observation well, rA is the distance from pumped well A to the point where s is sought, and rB is the distance from the pumped well B to the point where s is sought. If there are M pumping wells and the drawdown (sob) is known at one observation well, then we can generalize Equation 7.12 to obtain M rio Qi s = sob + a ln a b ri i = 1 2π T

(7.12b)

where Qi is the constant pumping rate of well i, rio is the distance between the pumped well i and the observation well, and ri is the distance between the pumped well i and the point where the drawdown is sought.

Sec. 7.2

Steady Radial Flow to a Well QA

rAO

261

A rBO

QB rB C

Figure 7.5 Plan view of an aquifer with two pumping wells

Example 7.2 A discharge well that completely penetrates a confined aquifer is pumped at a constant rate of 2,500 m3/day. The aquifer transmissivity is known to be 1,000 m2/day. The steady-state drawdown measured at a distance 60 m from the pumped well is 0.80 m. Determine the drawdown at a distance 150 m from the pumped well. Solution By using Equation 7.11, s = sob +

rob Q 2,500 60 lna b = 0.80 + lna b = 0.435 m r 2πT 2π(1,000) 150

Example 7.3 Two discharge wells (1 and 2) penetrating a confined aquifer are pumped at constant rates of 3,140 m3/day and 942 m3/day, respectively. The aquifer transmissivity is 1,000 m2/day. The steady-state drawdown measured at an observation well is 1.20 m. The observation well is 60 m from well 1 and 100 m from well 2. Determine the drawdown at another point in the aquifer located 200 m from well 1 and 500 m from well 2. Solution From Equation 7.12, we can write s = sob +

r1o r2o Q1 Q2 ln a b + ln a b r1 r2 2π T 2π T

s = 1.20 +

7.2.2

3,140 60 942 100 ln a ln a b + b = 0.357 m 2π (1,000) 200 2π (1,000) 500

Steady Radial Flow in Unconfined Aquifers

Darcy's law may be directly applied to derive the radial flow equation that relates the discharge to the drawdown of the water table in an unconfined aquifer after a steady state of equilibrium is reached. By using plane polar coordinates with the well as the origin, we find that the discharge flowing through a cylindrical surface at a radius r from the center of the well equals Q = AV = 2π rha K

dh b dr

(7.13)

262

Groundwater Hydraulics

Chap. 7

In this equation, h is measured from the bottom of the aquifer to the water table. Because the flow is at steady state, Q is also equal to the well discharge, the flow rate at which the well is pumped. Figure 7.4 (a) depicts all of the variables. Integrating Equation 7.13 between the boundary conditions at the well (r = rw, h = hw) and at the radius of influence (r = r0, h = h0) yields Q = πK

h20 - h2w r0 a ln b rw

(7.14)

The selection of the radius of influence, r0, can be somewhat arbitrary. The variation of Q is rather small for a wide range of r0 because the influence on the well by the water table at great distances is small. In practice, approximate values of r0 may be taken between 100 m and 500 m, depending on the nature of the aquifer and the operation of the pump. We can rearrange Equation 7.14 as h2w = h20 -

r0 Q lna b rw πK

(7.15)

A more general equation for any distance r from the pumped well and an observation well at distance rob from the pumped well can be written as h2 = h2ob -

rob Q ln a b r πK

(7.16)

The unconfined flow equations are not linear in h. Therefore, superposition of h values is not allowed for multiple wells in an unconfined aquifer. However, the equations are linear in differences in h2. Therefore, to find the steady-state h at a point resulting from M pumping wells, we can use M Q rio i h2 = h2ob - a ln a b ri i=1 π K

(7.17)

where hob is the head measured at the observation well, Qi is the constant pumping rate of well i, rio is the distance between the pumped well i and the observation well, and ri is the distance between the pumped well i and the point where the drawdown is sought. Example 7.4 A 95-ft-thick, unconfined aquifer is penetrated by an 8-in. diameter well that pumps at a rate of 50 gal/min (gpm). The drawdown at the well is 3.5 ft and the radius of influence is 500 ft. Determine the drawdown 80 ft from the well. Solution First, we must determine the permeability of the aquifer. Substituting the given conditions into Equation 7.14 and noting that there are 449 gpm in 1 cfs yields (952 - 91.52) (50>449) = π K ln(500>0.333) K = 3.97 * 10-4 ft>s

Sec. 7.3

Unsteady Radial Flow to a Well

263

Now we can use Equation 7.16 and the known head at the well (or alternatively, at the radius of influence) as the observation well data. h2 = 91.52 -

(50>449) -4

π(3.97 * 10 )

lna

0.333 b; h = 94.1 ft 80

Therefore, the drawdown is: s = ho - h = 95.0 - 94.1 = 0.9 ft.

Example 7.5 Two discharge wells (1 and 2) penetrating an unconfined aquifer are pumped at constant rates of 3,000 and 500 m3/day, respectively. The steady-state head (i.e., water table height) measured at an observation well is 40.0 m. The observation well is 50 m from well 1 and 64 m from well 2. The water table height measured at a second observation well is 32.9 m, which is located 20 m from well 1 and 23 m from well 2. Determine the aquifer permeability (in m/day). Solution From Equation 7.17, h2 = h2ob 32.92 = 402 Therefore, K = 2.01 m/day.

r1o r2o Q1 Q2 ln a b ln a b r1 r2 πK πK

3,000 50 64 500 ln a b ln a b πK 20 πK 23

7.3 Unsteady Radial Flow to a Well Groundwater flow is said to be unsteady if the flow conditions at a given point, such as piezometric head and velocity, are changing with time. These changes are also associated with changes in the volume of water in storage. Storage coefficient (S, also referred to as storage constant or storativity) is an aquifer parameter linking the changes in the volume of water in storage to the changes in the piezometric head. It is a dimensionless number defined as the water yield from a column of aquifer (of unit area) that results from lowering the water table or piezometric surface by a unit height. The upper limit of S is the porosity, although it is impossible to remove all of the water stored in the pores because of capillary forces. Although S may approach the porosity in unconfined aquifers, its value is much lower in confined aquifers because the pores do not drain. Water is removed by compression of the saturated layer and expansion of the groundwater. The derivations of the unsteady groundwater equations and their solutions are beyond the scope of this book. Therefore, only the results will be presented herein. 7.3.1

Unsteady Radial Flow in Confined Aquifers

With reference to Figure 7.6, unsteady radial flow in a confined aquifer at time t is expressed as* 02h 1 0h S 0h + = 2 r 0r T 0t 0r * W. C. Walton, Groundwater Resources Evaluation (New York: McGraw-Hill, 1970).

(7.18)

264

Groundwater Hydraulics

Chap. 7

Qw Initial piezometric surface s Piezometric surface at time t

Figure 7.6

Definition sketch for unsteady confined flow

where S is the storage coefficient and T is the transmissivity. Both of these aquifer characteristics are assumed to remain constant with time. An analytical solution can be obtained to this equation using the initial condition h = h0 at t = 0 for all r, and the two boundary conditions h = h0 at r = ∞, and r

Qw 0h = 0r 2πT

for

r = rw

and

rw S 0

where Qw is a constant well discharge starting at t = 0 and rw is the radius of the well. The initial condition implies that the piezometric surface is level at the time pumping first begins. The first boundary condition implies that the aquifer is infinitely large—that is, there are no boundaries such as rivers or lakes affecting the flow. The second boundary condition assumes that the rate of flow entering the well from the aquifer equals the well discharge. This boundary condition also implies the well radius is negligibly small compared to other horizontal dimensions in the aquifer, and it neglects the changes in the volume of water within the well with time. Theis† first presented a solution to the unsteady confined equation as ∞ -u Qw e h0 - h = s = du u 4πT L u

(7.19)

where h0 - h = s is the drawdown in an observation well at a distance r from the pumping well and u is a dimensionless parameter given by u =

r 2S 4Tt

(7.20)

In Equation 7.20, t is the time since the pumping began.

†

C. V. Theis, "This relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage," Trans. Am. Geophys. Union, 16 (1935): 519–524.

Sec. 7.3

Unsteady Radial Flow to a Well

265

The integral in Equation 7.19 is commonly written as W(u)—known as the well function of u—and the equation becomes s =

Qw W(u) 4π T

(7.21)

The well function is not directly integratable but may be evaluated by the infinite series W(u) = -0.5772 - ln u + u -

u2 u3 + g 2 # 2! 3 # 3!

(7.22)

The values of the well function W(u) are given in Table 7.3 for a wide range of u. For a reasonably large value of t and a small value of r in Equation 7.20, u becomes small enough that the terms following "ln u" in Equation 7.22 become very small and may be neglected. Thus, when u 6 0.01, the Theis equation may be modified to the Jacob formulation,* which is written as Qw r 2S s = h0 - h = c -0.5772 - ln d 4π T 4Tt -2.30 Qw 0.445r 2S = log10 4π T Tt

Table 7.3

u *1 * 10-1 * 10-2 * 10-3 * 10-4 * 10-5 * 10-6 * 10-7 * 10-8 * 10-9 * 10-10 * 10-11 * 10-12 * 10-13 * 10-14

1.0

2.0

3.0

4.0

0.219 1.823 4.038 6.332 8.633 10.936 13.238 15.541 17.843 20.146 22.449 24.751 27.054 29.356 31.659

0.049 1.223 3.355 5.639 7.940 10.243 12.545 14.848 17.150 19.453 21.756 24.058 26.361 28.663 30.966

0.013 0.906 2.959 5.235 7.535 9.837 12.140 14.442 16.745 19.047 21.350 23.653 25.955 28.258 30.560

0.0038 0.702 2.681 4.948 7.247 9.549 11.852 14.155 16.457 18.760 21.062 23.365 25.668 27.970 30.273

(7.23)

Well Function, W(u)

5.0 0.0011 0.560 2.468 4.726 7.024 9.326 11.629 13.931 16.234 18.537 20.839 23.142 25.444 27.747 30.050

6.0 0.00036 0.454 2.295 4.545 6.842 9.144 11.447 13.749 16.052 18.354 20.657 22.959 25.262 27.565 29.867

7.0

8.0

9.0

0.00012 0.374 2.151 4.392 6.688 8.990 11.292 13.595 15.898 18.200 20.503 22.805 25.108 27.410 29.713

0.000038 0.311 2.027 4.259 6.554 8.856 11.159 13.461 15.764 18.067 20.369 22.672 24.974 27.277 29.580

0.000012 0.260 1.919 4.142 6.437 8.739 11.041 13.344 15.646 17.949 20.251 22.554 24.857 27.159 29.462

* C. E. Jacob, "Drawdown test to determine effective radius of artesian well," Trans. ASCE, 112 (1947): 1047–1070.

266

Groundwater Hydraulics

Chap. 7

The confined flow equation and its solution are linear with respect to s, Qw, and t. This property allows us to utilize the Theis solution with the concept of superposition in many situations beyond those for which the solution was first obtained. Suppose the well discharge is not constant but varies as shown in Figure 7.7. The drawdown s at time t where tN 7 t 7 tN - 1 can be found as s =

1 N (Qk - Qk - 1) # W(uk) 4π T ka =1

(7.24)

r 2S 4T(t - tk - 1)

(7.25)

where uk =

Note that if we are seeking the drawdown at some point in time after the pumping stopped, QN = 0 in this formulation. Also, Q0 = 0 and t0 = 0. Fortunately, the concept of superposition can be used for multiple well situations also. For example, if M wells start to pump simultaneously at t = 0 with constant discharges Qj, j = 1, 2, p , M in an infinite aquifer, the total drawdown at a particular location at time t is M 1 M s = a sj = Qj # W(uj) 4π T ja j=1 =1

(7.26)

where uj =

r 2j S

(7.27)

4 Tt

and rj is the distance between the pumped well j and point where "s" is sought. For M wells starting to pump at different times tj with constant discharges Qj in an infinite aquifer, the total drawdown can still be calculated using Equation 7.26. However, for this case uj =

r 2j S

(7.28)

4 T (t - tj)

Q Qk QN21

Qk21

Qk11

t0 t1

t2 t3

tk21 tk Figure 7.7

tk11

Variable pumping rate

tN22 tN21

Sec. 7.3

Unsteady Radial Flow to a Well

267

If M wells have variable pumping rates as shown in Figure 7.7, the total drawdown can still be calculated using the method of superposition s = a sj M

j=1

(7.29)

where sj is the drawdown from well j to be obtained by applying Equation 7.26 to each well separately. 7.3.2

Unsteady Radial Flow in Unconfined Aquifers

The governing mechanism for the release of water from storage in an unconfined aquifer is the gravity drainage of water that once occupied some of the pore space above the cone of depression. However, the gravity drainage does not occur instantaneously. When an unconfined aquifer is pumped, the initial response of the aquifer is similar to that of a confined aquifer. In other words, the release of water is mainly caused by the compressibility of the aquifer skeleton and water. Therefore, the early stages of pumping flow in an unconfined aquifer can be calculated like the flow in a confined aquifer using the Theis solution and Equations 7.20 and 7.21. The storage coefficient to be used would be comparable in magnitude to that of a confined aquifer. However, as gravity drainage commences, the drawdowns will be affected by this mechanism and deviations from the Theis solution will occur. As gravity drainage is fully established at later stages, the behavior of the aquifer will again resemble that of a confined aquifer, provided that the drawdowns are much smaller than the initial thickness of the aquifer. The Theis solution will still apply, but the storage coefficient to be used in the equation will be that of unconfined conditions. A procedure developed by Neuman* to calculate drawdowns takes into account the delayed gravity drainage. A graphical representation of the Neuman procedure, which was also presented and explained by Mays,† is given in Figure 7.8 where ua = uy =

r 2 Sa 4Tt r 2 Sy

4Tt r2 η = 2 h0

(7.30) (7.31) (7.32)

with Sa = efficient early time storage coefficient, Sy = unconfined storage coefficient, T = Kh0, K = permeability coefficient, h0 = initial water table elevation, and W (ua, uy, η) = well function for unconfined aquifers with delayed gravity drainage. Equation 7.30 is applicable for small values of time, and it corresponds to those line segments to the left of printed η values in Figure 7.8. Equation 7.31 is applicable for large time values, and it

* S. P. Neumann, "Analysis of pumping test data from anisotropic unconfined aquifers considering delayed gravity response," Water Resources Research, 11 (1975): 329–342. †

L. W. Mays, Water Resources Engineering (New York: John Wiley and Sons, Ltd., 2001).

268

Groundwater Hydraulics

Chap. 7

1/ua 1021 100

102

103

104

105

106

107

102

103

h values

Theis curve for 1/ua

W (ua,uy,h)

pe Ty

0.2

0.1

0.06

0.03

0.004

0.001

0.4 0.6 0.8 1.0 1.5 2.0 2.5 3.0 4.0 5.0 6.0 7.0

0.1

0.01 1025

es urv c a

0.01

1024

Theis curve for 1/uy

1023

1022

Figure 7.8

1021 1/uy

Well function for unconfined aquifers

corresponds to the line segments to the right of printed η values. Once the value of the well function is obtained from Figure 7.8, the drawdown is calculated using s =

Qw W (ua, uy, η) 4πT

(7.33)

Example 7.6 A high-capacity discharge well fully penetrating a confined aquifer is pumped at constant rate of 5,000 m3/day. The aquifer transmissivity is 1,000 m2/day, and storage coefficient is 0.0004. (a) Determine the drawdown at a distance of 1,500 m from the pumped well after 1.5 days of pumping. (b) Suppose the pump is shut off after 1.5 days. Determine the drawdown at a distance 1,500 m from the pumped well one day after pumping has stopped. (c) Explain the difference between the results of parts (a) and (b). Solution (a) By using Equation 7.20, (1,500)2(0.0004) r 2S u = = = 0.15 4 Tt 4(1,000)(1.5) From Table 7.3, we obtain W(u) = 1.523. Therefore, by using Equation 7.21, s =

Qw 5,000 W(u) = (1.523) = 0.61 m 4π T 4π(1,000)

(b) For this case, we will use the concept of superposition for unsteady well flow because the well discharge is not constant during the study period. This requires the use of Equation 7.24, with N = 2, Q0 = 0, Q1 = 5,000 m3/day, Q2 = 0, t0 = 0, and t1 = 1.5 days.

Sec. 7.3

Unsteady Radial Flow to a Well

269

By using Equation 7.25 for k = 1, u1 =

(1,500)2(0.0004) r2 S = = 0.090 4 T (t - t0) 4(1,000)(2.5 - 0)

and for k = 2, u2 =

(1,500)2(0.0004) r2 S = = 0.225 4 T (t - t1) 4(1,000)(2.5 - 1.5)

From Table 7.3, we then obtain W(u1) = 1.919 and W(u2) = 1.144. Now using Equation (7.24), we obtain 1 5[Q1 - Q0] W(u1) + [Q2 - Q1] W(u2)6 4πT 1 s = 5(5,000 - 0)(1.919) + (0 - 5,000(1.144)6 = 0.31 m 4π(1,000)

s =

(c) The drawdown obtained in part (b) is smaller than the drawdown of part (a). The explanation is that when the pump is shut off at t = 1.5 days, the drawdown is 0.61 m. However, at this time, there is a hydraulic gradient towards the well because of the cone of depression formed. Therefore, flow toward the well will continue although pumping has stopped. As a result, the piezometric surface will rise. This is called aquifer recovery.

Example 7.7 A discharge well penetrating a confined aquifer will be pumped at a constant rate of 60,000 ft3 >day. At what constant rate can a second well be pumped so that the drawdown at a critical location 300 ft from the first well and 400 ft from the second well will not exceed 5 ft after two days of pumping? The aquifer transmissivity is 10,000 ft2 >day, and the storage coefficient is 0.0004. Solution For this problem, we will use the concept of superposition for unsteady flow in an aquifer with multiple wells. For the first well, using Equation 7.27 yields u1 =

r 21S (300)2(0.0004) = = 0.00045 4Tt 4(10,000)(2.0)

and from Table 7.3, W(u1) = 7.136. For the second well, using the same equation yields r 22S (400)2(0.0004) u2 = = = 0.0008 4Tt 4(10,000)(2.0) and from Table 7.3, W(u2) = 6.554. Now applying Equation 7.26, 1 [Q # W(u1) + Q2 # W(u2)] 4π T 1 1 5.0 = [60,000(7.136) + Q2(6.554)] 4π(10,000) s =

Solving for Q2, we obtain Q2 = 3.05 * 104 ft3 >day.

270

Groundwater Hydraulics

Chap. 7

Example 7.8 An unconfined aquifer has an early time storage coefficient of Sa = 0.0002, unconfined storage coefficient of Sy = 0.20, initial thickness of h0 = 100 ft, and a transmissivity of T = K h0 = 100 ft2/day. What will the drawdown be at a location 200 ft from the pumped well if a well fully penetrating this aquifer is pumped at a constant rate of 15,000 ft3/day for 2 days? Solution By using Equations 7.30 to 7.32 we obtain

ua = uy = η =

r 2 Sa (200)2(0.0002) = = 0.01 4Tt 4(100)(2) r 2 Sy 4Tt r2 h2o

(200)2(0.2) = 10 4(100)(2)

(200)2 (100)2

= 4

Then 1/ua = 1/0.01 = 102 and 1/uy = 1/10 = 10-1. We first mark 1/ua = 102 on the upper horizontal scale of Figure 7.8 and draw a vertical line. This vertical line intersect the line labeled η = 4 to the right of the label 4. However, the line segment to the right of the label is not applicable to 1/ua. So, we should work with the parameter uy. Now, we mark 1/uy = 10-1 on the lower horizontal axis and draw a vertical line. This vertical line intersects the line labeled η = 4 to the right of the label 4. This line segment is applicable to 1/uy. Then, for 1/uy = 10-1 and η = 4 we obtain W(ua, uy, η) = 0.08 from Figure 7.8. Then from Equation 7.33 we obtain

s =

Qw 15,000 W(ua, uy, η) = (0.08) = 0.955 ft 4πT 4(3.14)(100)

7.4 Field Determination of Aquifer Characteristics Laboratory tests of soil permeability (Example 7.1) are performed on small samples of soil. Their value in solving engineering problems depends on how well they represent the entire aquifer in the field. When used with consideration of field conditions and careful handling of the samples, the laboratory test methods can be very valuable. Nevertheless, important groundwater projects often require pumping tests in the field to determine an aquifer's hydraulic parameters such as permeability, transmissivity, and storage coefficient. One well is pumped at a known rate and the resulting drawdowns are measured, either in the pumping well itself or in one or more observation wells. The aquifer parameters are then determined from analyses of drawdown data. The basic idea in analyzing the pumping test data is to fit the observed drawdowns to available analytical solutions. The values of the aquifer parameters sought are those providing the best fit between the theoretical and observed results. We sometimes refer to this procedure as the inverse problem. Aquifer permeability and transmissivity can be obtained from drawdown data collected under either steady or unsteady flow conditions. The storage coefficient, however, will require unsteady drawdown data.

Sec. 7.4

Field Determination of Aquifer Characteristics Q

271

Observation wells r2 r1

Impermeable stratum

Figure 7.9

7.4.1

Field determination of transmissivity coefficient in confined aquifers

Equilibrium Test in Confined Aquifers

Pumping tests under equilibrium (steady) conditions are used to determine the transmissivity of confined aquifers. If, as shown in Figure 7.9, the drawdowns, s1 and s2, are respectively measured at two observation wells located at distances r1 and r2 from a pumped well under steadystate conditions, we determine the transmissivity (T) by rewriting Equation 7.11 as T =

Qw ln(r1 >r2) 2π (s2 - s1)

(7.34)

Recall that s = ho - h, where ho is the piezometric head of the undisturbed aquifer. In most pumping tests, however, multiple observation wells are used to characterize the aquifer better. For this purpose, we can rewrite Equation 7.11 in terms of common logarithms as T = -

2.30 Qw ∆ (log r) 2π ∆s

(7.35)

where ∆(log r) = (log r2 - log r1) and ∆s = s2 - s1. When more than two observation wells are available, based on the form of Equation 7.11, a plot of s versus r on semilog paper (with s on the linear scale and r on the log scale) produces a straight line. In reality, all the points will not fall exactly on a straight line. We usually draw (estimate) a best-fit straight line, as shown in Figure 7.10, and use the slope of the line to find T. Note that for any log cycle of r, ∆(log r) = 1.0. Thus, defining ∆*s = drop in s per log cycle of r yields T =

2.30 Qw 2 π (∆*s)

(7.36)

272

Groundwater Hydraulics

Chap. 7

3 Best fit line Data

s (ft)

2.5

∆*s

1.5

1 0.1

Figure 7.10

10 r (ft)

100

1000

Analysis of steady pumping test data for confined aquifers

Example 7.9 A field test is conducted in a confined aquifer by pumping a constant discharge of 400 ft3/hr from an 8-inch-diameter well. After an approximate steady state is reached, a drawdown of 2.48 ft is measured in the pumped well, and a drawdown of 1.72 ft is measured at 150 ft from the pumped well. Determine the transmissivity for this aquifer. Solution Using Equation 7.34 with observation well 1 represented by the pumped well yields T =

Qw ln(r1 >r2) 400 ln(0.33>150) = = 513 ft2 >hr 2π s2 - s1 2π (1.72 - 2.48)

Example 7.10 Suppose in the case of Example 7.9, three additional observations are available. The drawdowns measured at r = 10, 300, and 450 ft are, respectively, s = 2.07, 1.64, and 1.58 ft. Determine the transmissivity for the aquifer. Solution A plot of s versus r is prepared on semilog paper as shown in Figure 7.10. From the best-fit line, we determine ∆*s = 0.29 ft. Then Equation 7.36 yields the transmissivity. T =

2.30 Qw 2.30 (400) = = 505 ft2 >hr 2 π (∆*s) 2 π (0.29)

Sec. 7.4

Field Determination of Aquifer Characteristics

7.4.2

273

Equilibrium Test in Unconfined Aquifers

The aquifer permeability coefficient, K, in unconfined aquifers can be effectively measured in the field by well-pumping tests. In addition to the pumped well, the pumping test requires two observation wells which penetrate the aquifer. The observation wells are located at two arbitrary distances r1 and r2 from the pumped well, as schematically represented in Figure 7.11. After pumping the well at a constant discharge Qw for a long period, the water levels in the observation wells, h1 and h2, will reach final equilibrium values. The equilibrium water levels in the observation wells are measured to calculate the aquifer's permeability coefficient. For unconfined aquifers, the coefficient can be calculated by integrating Equation 7.13 between the limits of the two observations wells to obtain Qw

K =

π(h22

h21)

r2 lna b r1

(7.37)

To determine the coefficient of permeability when multiple observation wells are available, we rewrite Equation 7.37 in terms of common logarithms as K =

2.30Qw ∆(log r) π ∆h2

(7.38)

where ∆(log r) = (log r2 - log r1) and ∆h2 = h22 - h21. When more than two observation wells are available, a plot of h2 versus r on semilog paper (with h2 on the linear scale and r on the log scale) produces a straight line. In reality, all the points will not fall exactly on a straight line. We usually draw (estimate) a best-fit straight line, as shown in Figure 7.12, and use the

Observation wells

r2 r1

h2 h1 h

Figure 7.11

Field determination of permeability coefficient in unconfined aquifers

274

Groundwater Hydraulics

Chap. 7

1,550 1,500

h2 (ft2)

1,450 1,400 1,350

∆*h2

Best fit line Data

1,300 1,250 1,200 0.1

Figure 7.12

1.0

10.0 r (ft)

100.0

1000.0

Analysis of steady pumping test data for unconfined aquifers

slope of the line to find K. Note that for any log cycle of r, ∆(log r) = 1.0. Therefore, defining ∆*h2 = increase in h2 per log cycle of r yields K =

2.30Qw π(∆*h2)

(7.39)

Example 7.11 A well 20 cm in diameter completely penetrates the undisturbed water table of a 30.0 m deep unconfined aquifer. After a long period of pumping at the constant rate of 0.1 m3/s, the drawdown at distances of 20 m and 50 m from the well are observed to be 4.0 m and 2.5 m, respectively. Determine the coefficient of permeability of the aquifer. What is the drawdown at the pumped well? Solution In reference to Figure 7.11, the conditions given are as follows: Q = 0.1 m3/s, r1 = 20 m, r2 = 50 m; also, h1 = 30.0 m - 4.0 m = 26.0 m and h2 = 30.0 m - 2.5 m = 27.5 m. Substituting these values into Equation 7.37, we have K =

0.1 2

π(27.5 - 26.0 )

lna

50 b = 3.63 * 10-4 m>s 20

The drawdown at the pumped well can be calculated by using the same equation with the calculated value of the coefficient of permeability and the well diameter. At the well, r = rw = 0.1 m and h = hw. Therefore, 3.63 * 10-4 =

Q π(h21

h2w)

lna

r1 0.1 20 b = lna b 2 2 rw 0.1 π(26 - hw)

Sec. 7.4

Field Determination of Aquifer Characteristics

275

from which we have hw = 14.5 m The drawdown at the well is (30 - 14.5) = 15.5 m.

Example 7.12 A field test is conducted in an unconfined aquifer by pumping a constant discharge of 1,300 ft3/hr from a 6-inch well penetrating the aquifer. The undisturbed aquifer thickness is 40 ft. The drawdowns measured at steady state at various locations are tabulated in the first two columns of the following table. Determine the coefficient of permeability. r (ft)

s (ft)

h = 40 - s (ft)

0.25 35.00 125.00 254.00

4.85 1.95 1.35 0.90

35.15 38.05 38.65 39.10

h2 (ft) 1,236 1,448 1,494 1,529

Solution We first calculate the values of h2 as shown in the preceding table. Then we prepare a plot of h2 versus r on semilog paper as displayed in Figure 7.12. From the best-fit line we obtain ∆*h2 = 95 ft2. Then, by using Equation 7.39, K =

7.4.3

2.30Qw π(∆*h2)

2.30(1,300) = 10.0 ft>hr π(95)

Nonequilibrium Test

Field data collected under steady-state conditions can be used to determine the aquifer permeability or transmissivity, as discussed in the previous sections. The storage coefficient can be determined only if unsteady-state drawdown data are available. As mentioned previously, the procedures for determining aquifer characteristics basically fit the field data to the analytical solutions developed for groundwater flow. Unfortunately, analytical solutions are not available for unsteady flow in unconfined aquifers. Therefore, the procedures discussed herein are limited to confined aquifers. However, for practical purposes, the same procedures can be applied to unconfined aquifers if the drawdowns are very small compared to the thickness of the aquifer. Although various procedures are available to analyze unsteady pumping test data, those based on the Jacobs solution are adopted here because of their simplicity. The Jacobs solution, previously introduced as Equation 7.23, can be rewritten in terms of common logarithms as s = or s =

2.30 Qw 2.25 Tt c log 2 d 4πT r S

2.30 Qw 2.30 Qw 2.25 T log 2 + log t 4π T 4π T r S

(7.40)

(7.41)

276

Groundwater Hydraulics

Chap. 7

3 2.5 ∆°s

s (m)

2 1.5

Best fit line Data

1 0.5 0 t 0.01 0

0.1

100

t (hr) Figure 7.13

Analysis of unsteady pumping test data from a single observation well

This equation implies that a plot of s versus t on semilog paper (with s on the linear axis and t on the log axis) should give a straight line as shown in Figure 7.13. The slope of this line (∆s/∆log t) is equal to (2.30 Qw/4πT). Therefore T =

2.30 Qw ∆log t 4π ∆s

For any log cycle of t, ∆log t = 1.0. Thus T =

2.30 Qw 4 π(∆°s)

(7.42)

where ∆°s = increase in s per log cycle of t. Also, we can show from (Equation 7.40) that S =

2.25 T to r2

(7.43)

where r = distance between the pumped well and the observation well, and to = the value of the time when the fitted straight line intersects the horizontal axis as shown in Figure 7.13. Equation 7.43 is obtained by simply setting t = to for s = 0 in Equation 7.40. When the Jacobs solution was initially introduced as Equation 7.23, it was noted that it was valid only for small values of u (or large values of t). Therefore, the duration of the pumping test should be long enough for the observed drawdowns to satisfy Equation 7.40 for the straight-line analysis to be applicable. Otherwise, the data points will not form a straight line. Sometimes when we plot the data points, we see that the points for larger t values form a straight line whereas the points for smaller t values do not fall on that line. In that event, we simply disregard the small t points as long as we have enough data points to obtain a straight line.

Sec. 7.4

Field Determination of Aquifer Characteristics

277

The interpretation is that the data points for small t values do not satisfy the Jacob solution, so they will not be used in the analysis. When drawdown data from more than one observation well are available, the procedure described above can be applied to each well individually. Then we can use the averages of the T and S values obtained from different wells. Alternatively, we can use all the data at once. For this purpose, Equation 7.40 is rewritten as 2.25 T>(r 2 >t) 2.30 Qw s = c log d 4πT S

(7.44)

2.30 Qw 2.30 Qw 2.25 T r2 s = log log 4π T S 4π T t

(7.45)

As implied by Equation 7.45, a plot of s versus r 2/t on a semilog graph paper (s on the linear axis and r 2/t on the log axis) forms a straight line as shown in Figure 7.14. The slope of this straight line is 2.30 Qw ∆s = 4π T ∆log(r 2 >t) 3

2.5 Best fit line Data for r = 20 m Data for r = 25 m

s (m)

2 ∆+s 1.5

0.5

0 10

100

1,000

10,000

(r 2/t)0 100,000

r2/ t (m2/hr) Figure 7.14

Analysis of unsteady pumping test data from multiple observation wells

278

Groundwater Hydraulics

Chap. 7

Solving for T, T = -

2.30 Qw ∆log(r 2 >t) 4π ∆s

For any log cycle of r 2/t, we have ∆log (r 2 >t) = 1.0. Then, defining ∆+s as the drop in s per log cycle of r 2/t, we obtain 2.30 Qw

T =

axis,

(7.46)

4 π (∆+s)

Also, defining (r 2/t)o as the value of r 2/t where the straight line intersects the horizontal

S =

2.25 T (r 2 >t)o

(7.47)

Example 7.13 A pumping test was conducted in a confined aquifer using a constant pump discharge of 8.5 m3/hr. The drawdowns are measured at an observation well located 20 m from the pumped well and are shown in the following table. Determine the aquifer transmissivity and storage coefficient. t (hr) s (m)

0.05 0.60

0.10 0.76

0.20 0.98

0.50 1.32

1.00 1.58

2.00 1.86

5.00 2.21

10.0 2.49

20.0 2.76

Solution The observed data are plotted and the best-fit line is drawn as shown in Figure 7.13. From the best-fit line, we obtain ∆°s = 0.90 m. Therefore, Equation 7.42 yields T =

2.30 Qw 2.30(8.5) = = 1.73 m2 >hr 4 π(∆°s) 4π(0.90)

Also from Figure 7.13, we obtain to = 0.017 hr. From Equation 7.43, we then obtain S =

2.25 T to r

2.25(1.73)(0.0170) (20)2

= 1.65 * 10-4

Example 7.14 A pumping test was conducted in a confined aquifer using a constant pump discharge of 8.5 m3/hr. The drawdowns measured at two observation wells located 20 m and 25 m from the pumped well are listed in the first three columns of the following table. Analyzing all the available data at once, determine the aquifer transmissivity and storage coefficient.

Sec. 7.5

Aquifer Boundaries

279

At r = 20 m

At r = 25 m

At r = 20 m

At r = 25 m

t (hr)

s (m)

0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 20.00

0.53 0.76 0.99 1.30 1.53 1.77 2.08 2.31 2.55

0.38 0.62 0.84 1.15 1.38 1.62 1.93 2.16 2.39

r 2/t (m2 >hr)

8,000 4,000 2,000 800 400 200 80 40 20

12,500 6,250 3,125 1,250 625 312.5 125 62.5 31.25

Solution The r 2/t values are first calculated for both observation wells as listed in the last two columns of the table above. Then all the available data are plotted as shown in Figure 7.14 and a best-fit line is drawn. From this line we obtain ∆+s = 0.78 m and (r 2 >t)o = 37,500 m2 >hr. Solving Equations 7.46 and 7.47 yields T =

S =

2.3 Qw +

4 π (∆ s) 2.25 T (r 2 >t)o

2.3(8.5) = 1.99 m2 >hr 4π(0.78)

2.25(1.99) = 1.19 * 10-4 37,500

7.5 Aquifer Boundaries Previous discussions on well hydraulics have assumed the impacted aquifers to be uniform (homogeneous and isotropic) and infinite in extent. This resulted in radially symmetrical drawdown patterns. Oftentimes, these drawdown patterns are impacted by aquifer boundaries such as nearby impermeable strata (no flow boundary) and water bodies such as lakes and rivers (constant head boundary). If the aquifer boundary is located within the radius of influence of the pumping well, then the shape of the drawdown curve may be significantly modified; this, in turn, affects the discharge rate as predicted by the radial flow equations. The solution of aquifer boundary problems can often be simplified by applying the method of images. Hydraulic image wells are imaginary sources or sinks, with the same strength (i.e., flow rate) as the original well, placed on the opposite side of the boundary to represent the effect of the boundary. Figure 7.15 depicts the effect of a fully penetrating impermeable boundary on a well located a short distance from it. Note that the impermeable boundary produces a greater drawdown and upsets the radially symmetric pattern. Figure 7.16 portrays the application of the method of images by placing an imaginary well of the same strength (Q) at the same distance from the boundary but on the opposite side. Thus, the original boundary condition is hydraulically replaced by the two-well system in the hypothetically uniform aquifer of infinite extent. Computations to determine the drawdown curve for wells impacted by fully penetrating boundaries depend on the aquifer type. For confined aquifers, the two wells (i.e., the real well and the image well) depicted in Figure 7.16 hydraulically affect each other in such a way that the drawdown curves can be linearly superimposed. Thus, using the principle of superposition, the resultant drawdown curve is found by adding the drawdown curves of the two wells together.

280

Groundwater Hydraulics

Chap. 7

Original water table or piezometric head

Drawdown without boundary

Impermeable surface

Drawdown with boundary

Figure 7.15

Pumping well near a fully penetrating impermeable boundary

Also note that the imaginary well and the real well offset one another at the boundary line creating a horizontal hydraulic gradient (dh>dr = 0); as a result, there is no flow across the boundary. Note that unconfined aquifers are not linear in drawdowns but linear in differences in h2. Various procedures are available to analyze unconfined aquifer boundaries, but we will concentrate on those for confined flow because of their simplicity. The presence of a lake, river, or other large body of water in the vicinity of a well increases flow to the well. The effect of a fully penetrating water body on the drawdown is exactly opposite to that of a fully penetrating impermeable boundary. The resulting drawdown, as depicted in Figure 7.17, is less than normal, but the symmetric pattern is still upset. Instead of an imaginary pumping well, the equivalent hydraulic system (Figure 7.18) involves an imaginary recharge well placed at an equal distance across the boundary. The recharge well infuses water at a discharge rate Q into the aquifer under positive pressure. Q

Real well

Imaginary well

Original water table or piezometric head

Drawdown component of real well

Drawdown component of imaginary well

Resultant drawdown curve a

Figure 7.16

Equivalent hydraulic system with imaginary well

Sec. 7.5

Aquifer Boundaries

281 Q Lake

Boundary

Well Original water table or piezometric head Drawdown with boundary Drawdown without boundary

Figure 7.17

Pumping well near a fully penetrating perennial water body

The drawdown curve in confined aquifers resulting from fully penetrating water bodies is obtained by linearly superimposing the drawdown component of the real well and the drawdown component of the image (recharge) well that replaces the water boundary, as shown in Figure 7.18. The resulting drawdown curve of the real well intersects the boundary line at the elevation of the free water surface. The steeper hydraulic gradient causes more water to flow across the boundary line. Thus, much of the water from the well is obtained from the water body rather than from the aquifer. Replacing aquifer boundaries by an equivalent hydraulic system of image wells can be applied to a variety of groundwater boundary conditions. Figure 7.19 (a) shows a discharge well pumping water from an aquifer bounded on two sides by impermeable boundaries. Three imaginary wells are required to provide the equivalent hydraulic system. The imaginary discharge well, I1 and I2, provide the required absence of flow across the boundaries; a third imaginary well, I3, is necessary to balance the system. The three image wells all have the same discharge rate (Q) as the real well. All wells are placed at equal distances from the physical boundaries.

ent on p m

Q Real discharge well

Draw

dow

Resultan t

n co

mpo

nent

o pc

d-u

il Bu

of im

nar y

Imaginary recharge well

drawdow

n curve

of re al w

ell a

agi

Figure 7.18 Equivalent hydraulic system with imaginary recharge well

282

Groundwater Hydraulics

Chap. 7

Permeable

Impermeable

Real well

(a)

Impermeable

Real well

(b)

Figure 7.19

Pumping wells near multiple boundaries

Figure 7.19 (b) represents the situation of a discharge well pumping water from an aquifer bounded on one side by an impermeable boundary and on the other side by a perennial stream. The equivalent hydraulic system includes three image wells of equal strength Q. I1 is a recharge well, and I2 is a discharge well. A third image well, I3, a recharge well like I1, is necessary to balance the system. Example 7.15 A factory located near a river bank needs to extract a discharge of 1.55 cfs from a confined aquifer (K = 0.0004 ft/s, and b = 66 ft). Local authorities require that, at a distance of 100 ft from the bank, the groundwater table may not be lower than one-third of a foot from the normal surface elevation of the river. Determine the minimum distance from the bank that the well can be located. Solution As shown in Figures 7.17 and 7.20, a pumping well located near a perennial water body may be hydraulically replaced by an imaginary recharge well of the same strength and at the same distance but on the opposite side of the boundary. The resulting drawdown curve may be obtained by superimposing the real and the imaginary wells and assuming an infinite extent of the aquifer without the boundary. However, this will only work for confined aquifers and fully penetrating water bodies. Assume P to be the distance between the pumping well and the river bank. Then the drawdown (100 ft from the bank or P - 100 ft from the real well and P + 100 ft from the imaginary recharge well) is the sum of the piezometric surfaces produced by the pumping well and the recharge well. Using Equation 7.11 for the real well yields sreal = sob +

rob Q 1.55 P ln a b = sP(real) + ln a b r 2π T 2π(0.0264) P - 100

where the observation well is placed at the river and noting that T = Kb = (0.0004)(66) = 0.0264 ft2/s. The same equation for the image well yields simage = sob +

rob Q -1.55 P lna b = sP(image) + lna b r 2π T 2π(0.0264) P + 100

Sec. 7.5

Aquifer Boundaries

283

where a negative sign is used for the recharge pump rate. Based on the principle of superposition, we will add drawdowns to obtain 0.333 ft at the location of interest (noting that sP(real) + sP(image) = 0; see Figure 7.18). Therefore, 1.55 P 1.55 P b b ln a ln a 2π(0.0264) P - 100 2π(0.0264) P + 100 1.55 P + 100 s = 0.333 = ln a b; 2π(0.0264) P - 100 P + 100 ln a b = 0.0356 P - 100

s = sreal + simage =

P + 100 = 1.036(P - 100); which yields P = 5,660 ft Alternate Solution To determine the drawdown created by two wells, we will apply Equation 7.12a: s = sob +

rAo rBo QA QB ln a b + ln a b rA rB 2π T 2π T

The observation well in this case is at the boundary where sob = 0.0 ft. In addition, the drawdown at the point of interest is s = 0.333 ft, based on a local mandate. The other variables are as follows: QA is the discharge well pumping rate, and QB is the recharge (image) well pumping rate. Making appropriate substitutions yields the minimum distance P: ( -1.55) 1.55 P P ln a b + lna b 2π(0.0264) P - 100 2π(0.0264) P + 100 1.55 P + 100 0.333 = lna b 2π(0.0264) P - 100

0.333 = 0.0 +

P + 100 = 1.036(P - 100); which yields P = 5,660 ft

P 100 ft

0.33 ft

66 ft

Aquifer

Figure 7.20

284

Groundwater Hydraulics

Chap. 7

Example 7.16 An otherwise infinitely large confined aquifer is bounded on one side by an impermeable barrier. The aquifer transmissivity is 5,000 ft2/day, and the storage coefficient is 0.0002. A well placed 200 ft from the boundary is pumped at a constant rate of 20,000 ft3/day. Determine the drawdown at half the distance between the well and the boundary three days after pumping has begun. Solution An image well, having the same characteristics as the real well, is placed on the other side of the boundary. The distance between the image well and the boundary is 200 ft. The distance from the image well to the point where the drawdown is to be calculated is 300 ft. We will calculate the drawdowns for the real well and the image well separately using Equations 7.20 and 7.21 and add them using the principle of superposition. For the real well, u =

(100)2(0.0002) r 2S = = 3.33 * 10-5 4 Tt 4(5,000)(3)

From Table 7.3, W(u) = 9.741. Then s =

Qw 20,000 W(u) = (9.741) = 3.10 ft 4π T 4π(5,000)

For the image well, u =

(300)2(0.0002) r 2S = = 3.00 * 10-4 4 Tt 4(5,000)(3)

From Table 7.3, W(u) = 7.535. Then s =

Qw 20,000 W(u) = (7.535) = 2.40 ft 4π T 4π(5,000)

The total drawdown is 3.10 + 2.40 = 5.50 ft.

Example 7.17 Suppose the boundary in Example 7.16 is a river rather than a barrier. Calculate the drawdown and explain why it is different than the drawdown obtained in Example 7.16. Solution In this case, the image well is a recharge well. The drawdowns caused by the real well and the image well are calculated in the same manner as in Example 7.16. However, in this case, the drawdown caused by the image well is negative ( - 2.40 ft.). So the resulting drawdown is 3.10 - 2.40 = 0.70 ft. This is smaller than the drawdown obtained in Example 7.16. The reason is that the river produces a constant head boundary that prevents the piezometric surface from dropping as much between the river and the well. Alternatively, if you look at the discharge well and the image well in the equivalent hydraulic system (Figure 7.18), the recharge well is building up the piezometric surface so it will not drop as much.

Sec. 7.6

Surface Investigations of Groundwater

285

7.6 Surface Investigations of Groundwater Locating subsurface groundwater using information obtained on the Earth's surface is an ancient art known as divination. Some people still practice the "art" by using Y-shaped sticks or metal rods known as divining rods. (The senior author's grandmother was a practicing "water witch.") At the turn of the twentieth century, geophysical methods had been developed for petroleum and mineral explorations. A few of these methods have proven useful for locating and analyzing groundwater. More recently, various remote sensing methods have been developed. These methods are based on somewhat qualitative interpretation of satellite images to extract hydrogeological data. Remote sensing methods require expertise in earth sciences and digital image processing and, therefore, are beyond the scope of this book. Information obtained by surface methods can only provide indirect indications of groundwater. Correct interpretation of the data usually requires supplemental information that can only be obtained by subsurface investigations. Two of the most commonly used geophysical methods are described below. 7.6.1

The Electrical Resistivity Method

Electrical resistivity of rock formations varies over a wide range. The measured resistivity of a particular formation depends on a variety of physical and chemical factors such as the material and the structure of the formation; the size, shape, and distribution of pores; and the water content. The distinguishable difference between a dry rock formation and the same formation with large amounts of water filling the interstitial spaces is the key to detecting groundwater. The procedure involves measuring the electrical potential difference between two electrodes placed in the ground surface. When an electric current is applied through two other electrodes outside but along the same line with the potential electrodes, an electrical field penetrates the ground and forms a current flow network, as shown in Figure 7.21. A deeper penetration of the electrical field will occur by increasing the spacing between the electrodes. The variation in apparent resistivity is plotted against the electrode spacing from which a smooth curve can be drawn.* The interpretation of such a resistivity-spacing curve in terms of subsurface formations is frequently complex and often difficult. Nevertheless, with certain supplemental data from subsurface investigations to substantiate the surface measurement, correct predictions of the existence and depth of groundwater aquifers can often be made. 7.6.2

Seismic Wave Propagation Methods

By shocking the Earth's surface with a small explosion or the impact of a heavy weight, the time required for the sound or shock wave to reach a certain point at a known distance away can be measured. Seismic waves propagate through a transfer medium the same way that light waves do. They may be refracted or reflected at the interface of any two materials of different elastic properties. A change in propagation velocity takes place at the interface. The wave speed is greatest in solid igneous rock and the least in unconsolidated formations. The water content in a particular formation will significantly alter the wave speed in the formation. Because the seismic wave is traveling several hundred meters deep into the ground, subsurface information may be * H. M. Mooney and W. W. Wetzel, The Potentials About a Point Electrode and Apparent Resistivity Curves for a Two-, Three-, and Four-Layered Earth (Minneapolis: University of Minnesota Press, 1956).

286

Groundwater Hydraulics

Chap. 7

I = Current eletrodes E = Potential eletrodes V I

(a)

V I

Waterbearing layer (b) Figure 7.21 Arrangement of electrodes for resistivity determination: (a) current lines in a homogeneous medium and (b) current lines distorted by the presence of a water-bearing layer

obtained by placing several seismometers at various distances from the shock point along the same line. The wave travel time is plotted against the distance, as shown in Figure 7.22. A sudden change in the slope of the time-spacing curve can be interpreted to determine the depth of a groundwater table. 7.7 Seawater Intrusion in Coastal Areas Along the coastline, freshwater coastal aquifers are in contact with seawater. Under natural conditions, fresh groundwater is discharged into the sea under the water table, as shown in Figure 7.23. However, with an increased demand for groundwater in certain coastal regions, the seaward flow of fresh groundwater has been reduced or even reversed, causing saltwater from the

Sec. 7.7

Seawater Intrusion in Coastal Areas

Shock point 1

C 4

287 Seismic wave recorders

10 11 12 13 14 15 B

Layer 1

A Layer 2

Curve tangent at C used for reflection method

Time

Slope 1 V2 First arrival time used for refraction method

Slope 1 V1

2z /V1 S 0

B 1

10 11 12 13 14 15 16

Figure 7.22 Propagation of seismic waves in a two-layer medium. Waves propagate at speed V1 in the upper (dry) layer and at a higher speed (V2) in the lower (water-bearing) layer. For points to the lower right of line AB, the wave refracted at the point A through the lower layer (2) and reflected back to the surface arrives sooner than the wave propagating directly through the upper layer (1)

sea to enter and penetrate freshwater aquifers. This phenomenon is commonly known as seawater intrusion. If the seawater travels far enough inland and enters water supply wells, the groundwater supply becomes useless. Furthermore, once a coastal aquifer is contaminated by salt, it is very difficult to remove the salt from the formation, and the aquifer may be permanently damaged. Engineering prevention and control of seawater intrusion will be discussed in this section. Overdrafting of coastal aquifers results in lowering the water table in unconfined aquifers or the piezometric surface in confined aquifers. The natural gradient originally sloping toward the sea is reduced or reversed. Because of the difference in densities of saltwater and freshwater, an interface is formed when the two liquids are in contact. The shapes and movements of the interface are governed by the pressure balance of the saltwater on one side of the interface and freshwater on the other side. It has been found that the interface that occurs underground does not take place at sea level but at a depth below sea level that is approximately 40 times the height of the freshwater table above sea level, as shown in Figure 7.23. This distribution is caused by the equilibrium of hydrostatic pressure that exits between these two liquids, which have different densities.

288

Groundwater Hydraulics

Chap. 7

Ground surface h Water table

hs = 40 hr

Freshwater

Int ce

a erf

Saltwater x A Figure 7.23

x B

Schematic representation of freshwater and saltwater distribution in unconfined coastal aquifers

Figure 7.23 displays a cross section of a coastal aquifer. The total hydrostatic pressure at point A and at depth hs, below sea level is PA = ρsghs where ρs is the density of the saltwater and g is the gravitational acceleration. Similarly, the hydrostatic pressure at point B inland, at the same depth as A, and on the interface is PB = ρgh + ρghs where ρ is the density of freshwater. For a stationary interface, the pressure at A and B must be the same, and we may write ρsghs = ρgh + ρghs

(7.48)

Solving (Equation 7.48) for hs yields hs = c

ρ dh ρs - ρ

(7.49)

By taking ρs = 1.025 g>cm3 and ρ = 1.000 g>cm3, the above relationship yields hs = c

1.000 d h = 40h 1.025 - 1.000

(7.50)

This is commonly known as the Ghyben–Herzberg relation. This relation shows that a small depression in the water table near the coastline caused by well pumping could cause a big rise in the interface. Similarly, a buildup of the water table near

Sec. 7.7

Seawater Intrusion in Coastal Areas

289

the coastline caused by artificial recharge could drive the saltwater wedge deep into the ground, thus forcing it to move seaward. These phenomena are schematically demonstrated in Figure 7.24. Obviously, artificial recharge of an overdrafted coastal aquifer is an effective method of controlling seawater intrusion. By proper management, artificial recharge of the aquifer can eliminate the overdraft and maintain the proper water table level and gradient. Q Original water table

Discharge depression

Raised interface

rfa

nte li

na igi Or

40 a

(a)

Built up due to recharge

Original water table

rig

O al

in fa

r te in ce 40 a

(b) Figure 7.24 Seawater intrusion under the influence of (a) a discharge well and (b) a recharge well

290

Groundwater Hydraulics

Chap. 7

In addition to the artificial recharge, several other methods have been effectively applied for control of seawater intrusion. The most common methods are described as follows: 1. Pumping Trough: A pumping trough is a line of discharge wells situated along the coastline. By pumping the wells, a depression (trough line) is formed, as shown in Figure 7.25. Although saltwater is taken into the wells, a certain amount of freshwater in the aquifer is also removed. The freshwater motion is in the seaward direction toward the wells. This movement of fresh groundwater can stabilize the saltwater and freshwater interface. 2. Pressure Ridge: A pressure ridge is a series of recharge wells installed parallel to the coastline. Freshwater is pumped into the coastal aquifer to maintain a freshwater pressure ridge along the coastline to control the saltwater intrusion. The pressure ridge must be large enough to repel the seawater and must be located far enough inland; otherwise, the saltwater inland of the ridge will be driven farther inland, as demonstrated in Figure 7.26. Inevitably, a small amount of freshwater will be wasted to the sea; Q Water table

Saltwater Freshwater

Interface

Figure 7.25

Control of seawater by a pumping trough

40 a

Freshwater

Freshwater Saltwater

Saltwater (a) Figure 7.26

(b)

Control of seawater by a pressure ridge: (a) proper recharge arrangement and (b) improper recharge arrangement

Sec. 7.8

Seepage Through Dam Foundations

291

the remainder that moves landward can be used to supply part of the pumping draft. Reclaimed wastewater may be used to meet part of the recharge need. This method's advantage is that it does not deplete the usable groundwater capacity, but the disadvantages of high initial and operating costs and the need for supplemental freshwater often make small-scale operations impractical. 3. Subsurface Barriers: Subsurface barriers may be built along the coastline to reduce the coastal aquifer permeability. In relatively shallow-layer aquifers, subsurface dikes may be constructed with sheet piling, bentonite, or even concrete materials. An impermeable subsurface barrier may be formed by injecting flowable materials such as slurry, silicone gel, or cement grout into the aquifers through a line of holes. Subsurface barriers are best suited for certain locations such as narrow, alluvial canyons connected to large inland aquifers. Although the initial cost of installing the barriers may be very high, there is almost no operation or maintenance expense. 7.8 Seepage Through Dam Foundations In the generic sense, seepage is defined as the movement of water through soil. With regards to engineering, seepage is often undesirable and needs to be analyzed and controlled. For example, dams constructed to store water in a reservoir may continuously lose some of their water through seepage. Impermeable concrete dams constructed on an alluvial foundation may lose water through foundation seepage, whereas earth dams may lose water through the dam's embankment. Water movement caused by seepage is governed by Darcy's law in the same manner as groundwater flow. Seepage flow can be analyzed fairly quickly and accurately by applying the flow net technique. A flow net is a graphical representation of flow patterns expressed by a family of streamlines and their corresponding equipotential lines. Streamlines are always drawn in the direction of the flow, and they can be used to divide the flow field into a certain number of flow channels, each carrying the same discharge. Equipotential lines connect all points in the flow field that have equal velocity potential (or equal head). In a properly constructed flow net, the drop in head (∆h) between adjacent equipotential lines typically remains constant. The two sets of lines always meet at right angles and form an orthogonal net in the flow field. Figure 7.27 represents a portion of a flow net formed by a set of streamlines and equipotential lines. h 1 Dh h Dq

h 2 Dh Dn

Stream lines Equipotential lines

Figure 7.27

A flow net segment

292

Groundwater Hydraulics

Chap. 7

Flow nets are usually constructed so that the distance between a pair of adjacent streamlines (∆n) and the distance between a pair of equipotential lines (∆s) are equal in every cell. The square-net concept, as depicted in Figure 7.27, results in the equation ∆n = ∆s for each cell in the flow net. Because distances will be important in the following equations, flow nets are constructed on scale drawings. The velocity through the cell with the dimensions ∆n and ∆s noted in Figure 7.27 can be found using Darcy's law for steady flow through a porous media. Thus, V = K

dh ∆h = K ds ∆s

The volumetric flow rate through the corresponding flow channel per unit width of the dam is ∆q = AV = KA

∆h ∆s

(7.51)

where A is the flow area through the flow channel. Because the flow is per unit width of the dam, the flow area in the cell of interest (Figure 7.27) is A = ∆n

(7.52)

Substituting this area into (Equation 7.51) and noting ∆n = ∆s, we have ∆q = K(∆n)

∆h = K∆h ∆s

(7.53)

Because ∆h is a constant value of head drop (loss) between any two adjacent equipotential lines in Figure 7.27, we may write ∆h =

H n

(7.54)

H is the difference between the upstream reservoir water level and the tailwater level (as depicted in Figure 7.28), and n is the number of cells, or equipotential drops, in each flow channel of the flow net. Now Equation 7.53 may be rewritten as ∆q = K

H n

(7.55)

If there are m different flow channels in the flow net, then the total seepage flow rate per unit width of the dam is q = m∆q = Ka

m bH n

(7.56)

Therefore, the total seepage beneath the dam can be calculated by merely determining the m/n ratio from a graphically constructed flow net and determining the coefficient of permeability of the underlying soil.

Sec. 7.8

Seepage Through Dam Foundations

293

(a)

1 3

(b) Figure 7.28

Seepage through a previous dam foundation: (a) without a cutoff wall and (b) with a cutoff wall

Flow nets that depict seepage under a concrete dam, with and without a cutoff wall, are provided in Figure 7.28. A cutoff wall is a thin layer of impermeable material or sheet piling partially penetrating the aquifer under the dam. From the two depictions shown in Figure 7.28, it is obvious that the cutoff wall alters the seepage pattern by lengthening flow paths, thus increasing the resistance to flow. Hence, cutoff walls effectively decrease the amount of seepage and can significantly reduce the total uplift force on the base of the dam if they are strategically placed. (See Chapter 8.) The construction of flow nets by hand sketching is as much an art as a science. Sophisticated techniques exist for drafting complex flow nets. However, for our purposes, simple sketches will be relied on to help gain an understanding of the fundamental principles. Some useful guidelines in the construction of simple flow nets include: š Construct a scale drawing denoting all impervious boundaries (i.e., impermeable or lowpermeability natural strata or artificial boundaries such as sheet piling). š Sketch two to four streamlines entering and leaving the pervious boundaries at right angles and flowing essentially parallel to the impervious boundaries. š Equipotential lines are drawn perpendicular to the flow lines, forming a flow net that is made up of cells that are essentially square (equal median lines). š In regions of uniform flow, the cells are of equal size. In diverging flow, the cells increase in size; in converging flow, they decrease in size. It may be helpful to read these rules again while referring to the flow nets depicted in Figure 7.28.

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Example 7.18 A concrete gravity dam built on an alluvial channel bed, as shown in Figure 7.28, stores water at a depth of 50 m. If the coefficient of permeability is K = 2.14 m/day, estimate the seepage per meter width of dam (a) without a cutoff wall and (b) with a cutoff wall. Solution From Equation 7.56, q = Ka

m bH n

(a) For the dam without the cutoff wall [Figure 7.28 (a)], we count the number of flow channels (m = 5) and the number of cells along a flow channel (i.e., the number of equipotential drops) (n = 13). Applying Equation 7.56 yields q = (2.14) a

5 b(50) = 41.2 m3 >day per meter of dam width 13

(b) For the dam with the cutoff wall [Figure 7.28 (b)], the number of flow channels (m = 5) and the number of equipotential drops (n = 16) results in q = (2.14) a

5 b(50) = 33.4 m3 >day per meter dam width 16

Flow nets also allow you to determine energy heads, position heads, pressure heads, and seepage velocities at any location under the dam. Refer to Problem 7.8.1 for additional information about the valuable information that can be gleaned from flow nets.

7.9 Seepage Through Earth Dams Because an earth dam is built with pervious materials, it is of particular engineering concern. Excessive seepage through an earth dam may produce sloughing (slippage) of the downstream embankment and piping (removal of soil by exiting seepage water). Either scenario may lead to a complete failure of the dam. Therefore, seepage analysis should be performed for every earth dam by applying the method of flow nets. Seepage through an earth dam can be treated as flow through unconfined porous media. The upper surface of the flow, known as the surface of saturation or phreatic surface, is at atmospheric pressure. The typical shape of a phreatic line in a homogeneous earth dam is shown in Figure 7.29. The phreatic line is a streamline whose intersection with the equipotential lines is equally spaced vertically by the amount of ∆h = H/n, where H is the total head available and n is the number of equipotential drops counted in a graphic flow net. This line, which provides the upper boundary of the flow net, must be initially located by trial. An empirical rule for locating the phreatic line was suggested by Casagrande* and is shown in Figure 7.29. Section DF on the lower part of the downstream dam face must be protected against soil piping, which may eventually lead to dam failure. The seepage water may be removed * A. Casagrande, "Seepage through dams," J. New Eng. Water Works Assoc., 51 (1937): 139.

Sec. 7.9

Seepage Through Earth Dams

295

b 0.3 b B

1808 2 u8 x 400

Phreatic line

E F u Figure 7.29 Seepage flow net through a homogeneous earth dam. (A large portion of the phreatic line AD can be approximated by the parabola BCE with F as the focus and passing through the point B. Point A on the upstream face of the dam is the intersection of the water surface with the dam, Point D is the downstream transition where the seepage is exposed to the atmosphere.)

permanently from the downstream surface by a properly designed drainage system. For a nonstratified, homogeneous earth dam, a narrow, longitudinal drain will effectively intercept all the water seeping through the embankment. Figure 7.30 schematically shows the dimensions of a typical earth dam drainage blanket. The total discharge through an earth dam can be determined by using a graphically constructed flow net, as discussed in Section 7.8. Equation 7.56 gives the seepage discharge per unit width of the dam: q = Ka

m bH n

where m is the number of flow channels and n is the number of equipotential drops in the flow net.

L > 3H Figure 7.30

Drainage blanket in an earth dam

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Problems (secTion 7.1) 7.1.1. Equation (7.1) is the fundamental equation for determining porosity of an aquifer material. However, porosity can also be determined from α = 1 - (ρb /ρs) where ρb is the bulk density of the sample and ρs is the density of the solids in the sample. Derive this equation from Equation (7.1). Hint: Use the definition of density. 7.1.2. Determine the porosity of a limestone sample based on the following information. The sample is oven dried and weighed (145 N). It is then saturated with kerosene and weighed again (153 N). Finally, it is submerged in kerosene. The displaced kerosene is collected and weighed (80.0 N). 7.1.3. Estimate the porosity of an aquifer sample based on the following information. A cylindrical sample of relatively fine sand is extracted from a 4-in.-diameter well (borehole). When the sand from the 12-in. long sample is dried and poured into a graduated cylinder filled with water, it displaces 1,700 ml of water. 7.1.4. In a laboratory experiment a uniform sand sample is packed into a cylindrical sample space 4 cm in diameter and 20 cm long. Under a steady head of 30 cm, 100 cm3 of water is collected in 5 min. What is the apparent velocity of water flowing through the sample in cm/min? Determine the coefficient of permeability of the sample in m/s. Based on the permeability, does that sample behave like a typical fine sand or a typical course sand. What is the average time (in min) it takes water to travel 1.0 m through this sand under the same hydraulic gradient? 7.1.5. Referring to Example 7.1, answer the following questions. (a) Determine the average time it takes molecules of water to move from one end of the sample to the other end. (b) Estimate the actual velocity of an individual water molecule as it travels through the permeameter. (c) Why is the permeability measurement from the unconsolidated sample not likely to match the field (in situ) permeability? (d) If an undisturbed (in situ) sand sample was obtained instead of the unconsolidated sample from a test well (bore hole) using a thin tube sampler, would there still be reasons to doubt the accuracy of the laboratory determined permeability? Why or why not? 7.1.6. The water in Example 7.1 is assumed to be at room temperature (20°C). Applying Equation (7.3), determine the permeability coefficient if the water was at 10°C. Recall that in Example 7.1, 21.3 cm3 of water was collected in 2 min. How much 10°C water would be collected during the same test? 7.1.7. In a field test, a time of 84 h was required for a tracer to travel from one observation well to another. The wells are 100 ft apart, and the difference in their water surface elevations is 2 ft. Samples of the aquifer between the wells indicate the porosity is about 35%. Compute the coefficient of permeability (in ft/s) of the aquifer assuming it is homogeneous. 7.1.8. A rapid sand filter (course sand, α = 0.35; Figure P7.1.8) is 1 m deep and has a surface area of 4 m2. Determine the discharge through the filter (in m3/hr) and the average time (in min) it takes a drop of water to pass through the filter.

Problems

297

0.8 m Sand filter

Figure P7.1.8

7.1.9. Groundwater is moving through a narrow opening between impermeable rock outcroppings as depicted in Figure P7.1.9. The groundwater table elevations are displayed on the map based on observation wells in the vicinity. Given a typical cross section, estimate the flow-rate (in cfs) through the coarse sandy soil that is resident in the confined opening.

9 90 5,1 009 5,2

Rock outcrop

5,22

(Scale: 1 in. = 2,000 ft)

Water table elevation (5,2009)

109 5,2 09

409

(Not to Scale)

9 5,230

1,5009

Plan view Typical section view Figure P7.1.9

7.1.10. An unconfined aquifer (K = 12.2 m/day) is separated from an underlying confined aquifer (K = 15.2 m/day) by a semi-impervious layer that is 1.5 m thick as shown in Figure P7.1.10. Explain why there is flow from the unconfined aquifer to the confined aquifer. Also, determine the coefficient of permeability (in m/day) of the semi-impervious layer if the flow rate through it is 0.407 m3/day per square meter.

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Ground surface Water table

Wells 2.0 m

Piezometric surface

3.5 m Unconfined Aquifer Semipervious layer

1.5 m

Confined Aquifer Figure P7.1.10

(secTion 7.2) 7.2.1. Sketch the area described in Equation (7.5) through which the radial flow passes. Also verify Equations (7.6) and (7.14) by integrating Equations (7.5) and (7.13) using the boundary conditions given. 7.2.2. A comment is made in the paragraph which follows Equation (7.14) suggesting the selection of the radius of influence (r0) can be somewhat arbitrary (i.e., the discharge is not overly sensitive to this variable). Suppose the radius of influence for a 20-cm-diameter well is approximately 250 m (plus or minus 50 m). Determine the percent error in discharge that results from this {20, change in the radius of influence. Assume a 50-m thick, unconfined aquifer (course sand) with a drawdown at the well of 3.0 m. 7.2.3. A discharge well is located near the center of a circular island, as depicted in Figure P7.2.3. The island is approximately 800 m in diameter. The 30-cm-diameter well is pumped at the rate of 0.200 m3/s and produces a drawdown of 6.9 m (sw). Estimate the coefficient of permeability (in m/s) of the 50-m-thick, unconfined aquifer.

Q = 0.2 m3/s

Lake surface

sw = ?

0.30 m 800 m Figure P7.2.3

Problems

299

7.2.4. A large industry is concerned that the radius of influence of their production well extends beyond their property boundaries. Their 16-in. radius well draws water from a confined aquifer at the rate of 1570 gal/min (gpm). The confined aquifer is 100 ft thick with a piezometric surface (prior to pumping) that is 350 ft above the bottom of the aquifer. The drawdown at the well is 100 ft. Determine the radius of influence if the aquifer's hydraulic conductivity is 4.01 * 10-4 ft/s. 7.2.5. A confined aquifer (Figure P7.2.5) has average thickness of 10.0 m. When a 30-cm-diameter well is pumped at the rate of 30 m3/hr, the drawdown in the piezometric surface is 15 m at the well. If the drawdown 30 m from the well is 9.6 m, determine the aquifer's hydraulic conductivity (in m/s), transmissivity, and the drawdown 60 m away from the well. Q = 30 m3/hr

30 m

Original peizometric surface

s = 9.6 m

sw = 15 m

Impermeable stratum 10 m

Figure P7.2.5

7.2.6. A 40-cm-diameter well draws water from a 30.5-m-deep unconfined aquifer at a rate of 0.0151 m3/s. The drawdown at an observation well 50 m away is 1.5 m, and the drawdown at the well is 6.5 m. Determine the radius of influence of the well. Also, determine the drawdown at a distance of 100 m from the well. 7.2.7. A 16-in. radius well draws water from a confined aquifer at the rate of 3.5 cfs. The confined aquifer is 100 ft thick with a piezometric surface (prior to pumping) that is 350 ft above the bottom of the confined aquifer. The drawdown at an observation well 165 ft away is 33 ft, and the radius of influence is 1780 ft. Determine the drawdown at the well and the coefficient of permeability (in tt/s). 7.2.8. The water table at a construction site must be lowered to accomplish the required foundation work. A pumping well is installed in the unconfined sandy aquifer to lower the water table. The steady state drawdown requirements are as follows: at least 1.5 m of drawdown within a distance of 30 m from the well and 3.0 m of drawdown within a distance of 3.0 m from the well. The hydraulic conductivity of the sand is 1.00 * 10-4 m/s. An impermeable clay with a hydraulic conductivity 1.00 * 10-10 m/s forms the base of the aquifer. The depth of the water in the aquifer above the clay layer before pumping is 8.2 m. Calculate the required discharge from the well to meet the design conditions assuming the well has a radius of influence of 150 m. 7.2.9. A beverage industry owns a 12-in-diameter well that completely penetrates a confined aquifer and is pumped at the rate of 1.56 cfs. Steady flow has been achieved and the drawdown in the well is 37.4 ft. The transmissivity of the aquifer has been determined to be 0.0538 ft2/s. What is the drawdown impact (in feet) on a neighbor's domestic well 325 ft away? If the beverage industry installs

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a second identical well 650 ft away from the neighbor's well, what is the combined impact of both wells? (Hint: In confined aquifers, the drawdown produced by multiple wells at a particular location is the sum of the drawdowns produced by the individual wells.) 7.2.10. An observation well is located on an industrial site that contains two discharge wells. The water table depth in the observation well registers 131 ft in the unconfined aquifer (K = 6.56 ft/day). The two discharge wells (#1 and #2) are being pumped at the rate of 550 and 91.7 gpm (gal/min), respectively. The observation well is 164 ft from well #1 and 210 ft from well #2. Determine the water table depth at the location of an oil spill which is 65.6 ft from well #1 and 75.4 ft from well #2. Can you make a logical guess as to which well the oil will end up contaminating? Justify your answer. 7.2.11. Determine the transmissivity (in m2/day) of a confined aquifer that contains two discharge wells (#1 and #2). The wells completely penetrate the aquifer and are pumped at constant rates of 2,950 m3/day and 852 m3/day, respectively. The steady state drawdown measured at observation well A is 1.02 m (50 m from well #1 and 90 m from well #2). The steady state drawdown measured at observation well B is 0.242 m (180 m from well #1 and 440 m from well #2). 7.2.12. A field pump test is performed to determine the transmissivity of a high-capacity aquifer. The confined aquifer thickness is 20 ft and it has a porosity of 0.26. Unfortunately, the flow data for the well test was misplaced by the field crew. Can you still estimate the transmissivity (in ft2/s) given the following field data? š Two observation wells are located 500 ft and 1,000 ft from the pumped well along the same radial line. š The piezometric surface of the two wells at equilibrium differs by 42.8 ft. š The time it takes a conservative tracer to move from the outer observation well to the inner observation well is 49.5 hr.

(secTion 7.3) 7.3.1. An aquifer test will be conducted using a pumped well that completely penetrates a confined aquifer. An observation well is proposed as a distance 800 ft from the pumped well, but the depth of the well needs to be estimated to minimize drilling expenses. The approximate characteristics of the aquifer include a transmissivity of 5,000 ft2/day and a storage coefficient of 0.0005. Determine the approximate drawdown if the well test requires a pumping rate of 2.5 cfs for 48 hr. 7.3.2. A manufacturing plant has an occasional need for groundwater. The water comes from a well that completely penetrates a confined aquifer with a transmissivity is 1,000 m2/day and storage coefficient is 0.0004. However, the drawdown produced by the well cannot exceed 0.75 m at a distance of 1,500 m from the well without impacting a neighbor's well. Determine the maximum flow capacity (in m3/hr) that is obtainable from the well if the duration of pumping is limited to 36 hr? 7.3.3. Determine the drawdown at a distance 200 m from a pumped well at 50, 100, 150, 200, and 250 hr after pumping has begun. The pumped well fully penetrates a confined aquifer and is pumped at a constant rate of 300 m3/hr. The aquifer transmissivity is 25.0 m2/hr and the storage coefficient is 0.00025. 7.3.4. Determine the drawdown 50 ft from a pumped well (Q = 104 gpm) two days after pumping has begun. The pumped well completely penetrates a 500-ft-thick, unconfined aquifer that possesses an early-time storage coefficient of 0.0005, an unconfined storage coefficient of 0.10, and a permeability of 5 ft/day. 7.3.5. A confined aquifer has a transmissivity of 500 m2/day and a storage coefficient of 0.0005. Wells 1 and 2 completely penetrate this aquifer and will be pumped at constant rates of 600 m3/day and

Problems

301

1400 m3/day respectively. Pumping will start in both wells at the same time. An observation well is located 400 m from well 1 and 500 m from well 2. How long can the two wells be pumped without exceeding a 1.0 m drawdown limit in the observation well? At that limit, what is the drawdown contribution of each well? (Note: A spreadsheet will be helpful.) 7.3.6. Determine the drawdown at a critical location in a confined aquifer. The aquifer will be affected by two completely penetrating wells, both of which will be pumped at 207 gpm. However, pumping from the second well (279 ft from the point of interest) will start a day and a half after the first well (322 ft from the point of interest). Determine the drawdown at the point of interest three days after pumping begins in the first well if the aquifer transmissivity is 450 ft2/hr and the storage coefficient is 0.0005. 7.3.7. Two wells penetrating a confined aquifer are 600 ft apart. Each well will be pumped at 40,000 ft3/day. However, the pumping from the second well will start a day and a half after the first one. Determine the drawdown at a point halfway between the two wells 3 days after the pumping has begun in the first well. The aquifer transmissivity is 10,000 ft2/day, and the storage coefficient is 0.0005. 7.3.8. A discharge well penetrating a confined aquifer will be pumped at a rate of 800 m3/hr for two days, and then the pumping rate will be reduced to 500 m3/hr. The aquifer transmissivity is 40 m2/hr, and the storage coefficient is 0.00025. Determine the drawdown at a location 50 m from the pumped well three days after pumping has started.

(secTion 7.4) 7.4.1. Explain why a non-equilibrium test, rather than an equilibrium test, is needed to determine the storage coefficient of an aquifer. 7.4.2. Derive Equation (7.43) from (7.40) and Equation (7.47) from (7.44). 7.4.3. A confined aquifer of uniform thickness (18 m) is completely penetrated by a pumping well. After a long period of pumping at the constant rate of 0.3 m3/s, the water elevations in the observation wells (r1 = 20 m, r2 = 65 m) are stabilized. The drawdown measured at the observation wells are, respectively, 16.3 m and 3.4 m. Determine the coefficient of permeability, the drawdown at a distance of 40 m from the well, and the radius of influence. 7.4.4. A well located in the middle of a circular island is depicted in Figure P7.4.4. Sketch the surface of the water table that results from pumping the well. If the well yields 9.8 gpm, determine the coefficient of permeability. Also determine the drawdown at a distance of 150 ft from the well.

300 ft

300 ft 2 ft

Steady state

60 ft 30 ft

Figure P7.4.4

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7.4.5. A field test is conducted in a confined aquifer by pumping a constant discharge of 525 ft3/hr from a 8-in. diameter well. After steady state is reached, a drawdown of 3.2 ft is measured in the pumped well. Also measured are drawdowns of 2.55 ft, 2.1 ft, 1.9 ft, and 1.86 ft respectively at 10, 150, 300, and 450 ft from the pumped well. Determine the transmissivity for this aquifer and the drawdown at a location 1,500 ft from the well. 7.4.6. A pharmaceutical industry is doing a study at a proposed site to determine ground water potential. A field test is conducted in the confined aquifer by pumping the 20-cm-diameter test well at a rate of 13.2 m3/hr. After approximate steady state is reached, a drawdown of 1.20 m is measured in the pumped well. Also measured are drawdowns of 0.91 m, 0.67 m, 0.61 m, and 0.56 m respectively at 3.0 m, 45.0 m, 90.0 m, and 150 m from the pumped well. Determine the transmissivity of this aquifer and the distance from the well when the drawdown falls below 0.45 m. 7.4.7. The characteristics of a confined aquifer are required. A pump test is conducted at a flow rate of 5.4 m3/hr. Drawdowns are measured at an observation well located 18.0 m from the pumped well and are tabulated below. Determine the aquifer transmissivity, the storage coefficient, and the drawdown after 50 hr. Time (hr) s (m)

0.05 0.47

0.10 0.60

0.20 0.74

0.50 0.93

1.0 1.09

2.0 1.25

5.0 1.46

10.0 1.61

20.0 1.77

7.4.8. The following drawdown information was collected from an observation well 50 ft away from a 12-in.-diameter well that is pumped at a uniform rate of 1.55 cfs. Determine the permeability and storage coefficient of the 280-ft-thick unconfined aquifer. (Hint: Disregard small time drawdowns if they do not form a straight line on the semi-log plot.) Time (hr) s (ft)

1 1.1

2 1.8

3 2.5

4 3.2

5 3.8

6 4.4

8 5.5

10 6.5

12 7.2

18 8.9

24 10

7.4.9. A pumping test was conducted in a confined aquifer using a constant pump discharge of 10.0 m3/hr. The drawdowns measured at two observation wells located 20 m and 25 m from the pumped well are listed in the table below. Analyzing all the available data at once, determine the aquifer transmissivity, the storage coefficient, and the drawdown in the closest observation well after 10 days.

Time (hr)

at r = 20 m s (m)

at r = 25 m s (m)

0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 20.00

0.77 1.13 1.50 1.98 2.35 2.72 3.20 3.57 3.93

0.53 0.90 1.26 1.75 2.11 2.48 2.97 3.33 3.70

Problems

303

(secTion 7.5) 7.5.1. Locate imaginary wells on Figure P7.5.1, which will replace the actual boundaries with an equivalent hydraulic system.

Stream

Real well

Impermeable boundary

Figure P7.5.1

7.5.2. A large industry owns a 16-in. radius well that draws water from a confined aquifer at the rate of 1570 gal/min (gpm). The confined aquifer (T = 4.01 * 10-2 ft2/s) has a piezometric surface (prior to pumping) that is 350 ft above the bottom of the aquifer. The drawdown at the well is 100 ft. A fully penetrating impermeable boundary is 1,500 ft away from the well. Will the boundary impact the drawdown curve of the industrial well? Prove your answer. 7.5.3. A 40-cm-diameter well draws water from a confined aquifer at the rate of 1.33 m3/s. The aquifer is 30.5 m thick with a piezometric surface (prior to pumping) that is 107 m above the bottom of the confined aquifer. The drawdown at an observation well 50 m away is 10.1 m. If the same well is placed in the same aquifer, 400 m from a completely penetrating stream, determine the drawdown at the well and at the boundary. The aquifer permeability is 4.35 * 10-3 m/s. 7.5.4. A 10-m-thick confined aquifer (T = 1.30 * 10-3 m2/s) is located 60 m away from a fully penetrating impermeable boundary. When a 30-cm-diameter well is pumped at the rate of 30 m3/hr, the drawdown in the piezometric surface at the well is 15.0 m when it is not impacted by an aquifer boundary. Determine the drawdown at the well, the impermeable boundary, and halfway between the well and the boundary when the aquifer contains the impermeable boundary condition. 7.5.5. A factory is pumping 1.55 cfs from a confined aquifer, 5,660 ft from a fully penetrating river. An observation well, located between the river and the pumped well (100 ft from the river and 5,560 ft from the discharge well), registers a drawdown of 0.333 ft. Determine the transmissibility of the confined aquifer. 7.5.6. A 16-in.-diameter well is extracting a flow rate of 2.78 cfs from a confined aquifer. The aquifer has a transmissivity of 250 ft2/hr and a storage coefficient of 0.001. The well is located 300 ft away from a fully penetrating impermeable boundary. Determine the drawdown at the boundary after 100 hr of pumping. Also determine how long it would take for the drawdown at the boundary to reach 48.0 ft. 7.5.7. A confined aquifer has a transmissivity is 25.0 m2/hr and the storage coefficient of 0.00025. A discharge well fully penetrates the confined aquifer and is pumped at a constant rate of 400 m3/hr. A fully penetrating stream is located 500 m away. Determine the drawdown at a location between the well and the stream (100 m from the well and 400 m from the stream) 50 hr after pumping has begun.

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(secTion 7.8) 7.8.1. An abundance of useful information can be obtained from flow nets. In addition to seepage rates, which were calculated in Example 7.18, the total energy head at any location in the flow net can be estimated using equipotential lines. Recall that equal head reductions (drops) occur between any two adjacent equipotential lines. Therefore, if boundary conditions are known (i.e., the reservoir and tail water depths in Example 7.18), the total energy level can be determined at any location in the flow net. Once the total energy head is determined, the pressure head can be obtained by subtracting the position head. In addition, the Darcy velocity can be estimated at any flow net location using the total head differences between the nearest equipotential lines and measured distances from the scale drawing (Darcy's Law, V = K∆h/∆s). With this background, estimate the total energy head and the seepage velocity (magnitude and direction) at locations 1, 2, and 4 in Figure 7.28(b). Assume the upstream water depth is 80 ft (which also acts as a scale), K = 3.51 ft/day, and the porosity is 0.35. 7.8.2. Constructing a good flow is an artistic endeavor. Since the flow nets drawn by different people will vary, we may wonder about the accuracy of using it for seepage calculations. However, if carefully drawn there is surprising consistency in seepage predictions from engineer to engineer. To lend greater credibility to this premise, perform the following exercise: (a) With a pencil, sketch five more streamlines in Figure 7.28(a) by bisecting the existing flow channels. Why isn't the result a proper flow net? What additional step must be taken? With this additional step, recompute the seepage rate of Example 7.18 How do the answers compare? (b) Trace the outline of the dam and boundaries in Figure 7.28(a). Draw your own flow net using two streamlines instead of four. Estimate the seepage of Example 7.18 using your new flow net. How do the answers compare? 7.8.3. Sheetpiles are used to keep water out of a bridge pier construction site, as depicted in Figure P7.8.3 (drawn to scale). Determine the quantity of seepage (in gallons/min per unit length of sheetpile) that can be expected in order to design an appropriate pump to de-water the construction site. The drawing dimensions are as follows: d = 18.0 ft, b = 24.5 ft, and z = 7.0 ft. The soil permeability is 19.3 ft/day and the porosity is 0.40. Also determine the approximate exit velocity of the water (in ft/s) next to the sheetpile. (Note: If this velocity is high enough, it may transport the soil with it and eventually undermine the sheetpile.)

Impervious sheetpile d Construction site z b

Figure P7.8.3

Problems

305

7.8.4. Estimate the seepage rate (in m3/day) under the concrete dam drawn to scale in Figure P7.8.4. The 50-m-long dam rests on an alluvial foundation with a permeability of 2.23 * 10-7 m/s. The upstream water depth is 20 m. Also, estimate the energy head (using the dam bottom as the datum) and the seepage velocity immediately below the middle of the dam. The soil porosity is 0.40.

30 m

Impervious layer

Figure P7.8.4

7.8.5. Place a cut-off wall at the heel of the dam depicted in Figure P7.8.4 much like Example 7.18. The cut-off wall should extend downward one-third of the distance to the impervious layer. Estimate the seepage per unit meter width of dam. Assume the permeability is 2.23 * 10-7 m/s and the upstream water depth is 20 m. If the seepage per unit meter without the cutoff wall is 2.23 * 10-6 m/s per meter, determine the percent reduction in seepage using the cutoff wall.

(secTion 7.9) 7.9.1. A centrifugal pump is required to dispose of the water seeping through the earth levee depicted in Figure P7.9.1. Using the flow net provided, find the seepage rate (in gpm). The water level behind the levee (H) is 30 ft and it is 980 ft long. The permeability coefficient of the soil is 7.50 * 10-6 ft/s with a porosity of 0.40.

Figure P7.9.1

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Groundwater Hydraulics

Chap. 7

7.9.2. The homogeneous earth (silt) dam depicted in Figure 7.30 has an upstream water depth (H) of 5 m. Construct a proper flow net to determine the seepage rate through the 150-m-long dam in m3/day. The phreatic line depicted on the figure is the upper most streamline. (Hint: All streamlines terminate in the drain.) 7.9.3. Determine the seepage in ft3/day through the earth dam depicted in Figure 7.29. The permeability of the soil (silt) in the dam is 1.64 * 10-7 ft/s. Also determine the seepage velocity just before the water begins to surface on the downstream embankment at point "D." The upstream headwater is 25 ft and the dam is 260 ft long. Assume Figure 7.29 is a scale drawing with the upstream depth of 25 ft being the scale. 7.9.4. An earth dam, as schematically shown to scale in Figure P7.9.4, is constructed with a uniform material having a coefficient of permeability of 2.00 * 10-6 m/s on a relatively impervious foundation. The dam is 30 m high, which can be used as the drawing scale. Compute the seepage rate in units of cubic meters per day per unit width of the dam. Assume the phreatic surface emerges on the downstream slope at a distance of x = 30 m as defined in Figure 7.29.

18 m

25 m

30 m

Figure P7.9.4

7.9.5. Determine the seepage rate in m3/day per unit width of the dam in Problem 7.9.4 if a drainage blanket (as depicted in Figure 7.30) extends 30 m back from the toe of the dam. Begin the flow net by sketching a phraetic line (upper most flow line, as depicted in Figure 7.30) entering the drain 5 m from its upstream end. All other flow lines enter the drain between that point and the upstream end.

Siavash Beik/Christopher B. Burke Engineering, LLC

Hydraulic Structures

Water is more useful to people when it is properly controlled, conveyed, and contained. Hydraulic structures are designed and built to serve these purposes. Some of the most common hydraulic structures are pipes, pumps, open channels, wells, water-measuring devices, and stormwatercollection and stormwater-transport systems. These structures are covered in other chapters of this book. Dams, weirs, spillways, culverts, and stilling basins are also common, and they will be addressed in this chapter. 8.1 Functions of Hydraulic Structures Any classification of hydraulic structures will inevitably be discretionary because many of them can be built to serve more than one purpose. In addition, a general classification of hydraulic structures based on use is not satisfactory because many identical structures may be used to serve completely different purposes. For example, a low head dam could be built across a channel as a device to measure discharge, or it could be built to raise the water level at the entrance to an irrigation canal to permit diversion of water into the canal. Instead of classifying various hydraulic structures into arbitrary categories, we have listed the common functions of hydraulic structures and the basic design criteria.

307

308

Hydraulic Structures

Chap. 8

1. Storage structures are designed to hold water under hydrostatic conditions. A storage structure usually has a large capacity for a relatively small change in hydrostatic head (water elevation). 2. Conveyance structures are designed to transport water from one place to another. The design normally emphasizes delivery of a given discharge with a minimum consumption of energy. 3. Waterway and navigation structures are designed to support water transportation. Maintenance of a minimum water depth under various conditions is critical. 4. Coastline structures are constructed to protect beaches, inlets, harbors, and buildings. Wave action is a key consideration in the design of these structures. 5. Measurement or control structures are used to quantify the discharge in a particular conduit. Stable performance and a one-to-one relationship between the discharge and some indicator (usually elevation) are necessary. 6. Energy-conversion structures are designed to transform hydraulic energy into mechanical or electric energy (e.g., hydraulic turbine systems) or electrical or mechanical energy into hydraulic energy (e.g., hydraulic pumps). The design emphasis is on the system efficiency and the power consumed or produced. 7. Sediment- and fish-control structures are designed to direct or regulate the movement of the nonhydraulic elements in water. An understanding of the basic mechanisms and behaviors of the elements involved is an essential requirement for the design. 8. Energy-dissipation structures are used to control and disperse excess hydraulic energy to prevent channel erosion. 9. Collection structures are designed to gather and admit water to hydraulic systems. A typical example is a surface drainage inlet used to collect surface runoff and direct it into a stormwater conveyance system. Obviously, detailed consideration of all these functions and their design criteria are beyond the scope of this book. Only the most commonly encountered hydraulic structures are discussed here to demonstrate how fundamental considerations are used in their design. 8.2 Dams: Functions and Classifications A dam is a barrier structure placed across a watercourse to store water and modify normal stream flow. Dams vary in size from a few meters in height (farm pond dams) to massive structures more than 100 meters in height (large hydroelectric dams). Two of the largest dams in the United States are Hoover Dam (on the Colorado River and located on the border of Arizona and Nevada) and Grand Coulee Dam (on the Columbia River in Washington). Completed in 1936, Hoover Dam is 222 m high and 380 m long, and it can store 3.52 * 1010 m3 of water. Grand Coulee Dam, completed in 1942, is 168 m high and 1,592 m long, and it can store 1.17 * 1010 m3 of water. These two dams are dwarfed by the world's largest dams. The highest dam in the world is Jinping-I Dam in China; at 305 m high, it was completed in 2014. The largest dam in the world by reservoir size is Karibe Dam in Zimbabwe. It was completed in 1959 and stores 185 billion cubic meters of water. Dams fulfill many functions. Hoover and Grand Coulee dams provide an enormous amount of electricity to the western United States. However, like most large dams, they have multiple

Sec. 8.2

Dams: Functions and Classifications

309

purposes. They also provide downstream flood control (see storage routing in Chapter 11), irrigation water to vast amounts of farm land, and recreational opportunities. Dams are also built to provide industrial water, cooling water (for power plants), and municipal water. Locks and dams are built to support navigation on many large rivers. In the past, dams were built in conjunction with water wheels to supply the power for gristmills. Dams can be classified in a several ways. It may prove useful to classify them according to how they achieve stability and what materials are used in their construction. One such classification scheme can be seen in Table 8.1. Four types of dams are noted: gravity, arch, buttress, and earth. A typical gravity dam is a massive structure (Figure 8.1). The enormous weight of the dam body provides the necessary stability against overturning (about the toe of the dam) or Table 8.1

Classification of Dams

Type

Stability

Material

Gravity

Large mass

Concrete, rock, or masonry

Arch

Arch action on rock canyon

Concrete

Buttress

Mass of dam and water on upstream face

Concrete, steel, or timber

Earth

Mass of dam and water on upstream face

Earth or rock

Cross Section

Clay core

310

Hydraulic Structures

Chap. 8

(a) Gravity dam

(b) Arch dam

Figure 8.1 Top-view representation of (a) gravity dam and (b) arch dam

shear failure (along the bottom). Arch dams are normally built into solid rock foundations that provide resistance to the hydrostatic force by arch actions (Figure 8.1). Combining gravity action and arch action is a common practice. A typical buttress dam supports a slanted concrete slab (upstream face) at intervals with buttress supports. Because much of the cross section is empty space between buttresses, the stability comes from the water weight acting on the face slab. Because of their importance and the basic principles involved, the stability of gravity and arch dams is discussed in the next section. The most common type of dam encountered is the earth dam. It achieves stability from its mass and from the water on its upstream face. Because these dams are made of porous material, they will continually seep water. Controlling the amount and location of the seepage is an important design concern (Chapter 7). Few major dams are being built anymore, but many small dams are designed and built every year. Most are earth dams built for the purpose of urban stormwater management (Chapter 11) or for supplemental water supplies in rural areas of developing countries. Because of their importance and prevalence, Section 8.4 describes some of the key design considerations and a typical construction scenario.

Sec. 8.3

Stability of Gravity and Arch Dams

311

8.3 Stability of Gravity and Arch Dams 8.3.1 Gravity Dams The major forces acting on a gravity dam are represented in Figure 8.2. They are 1. 2. 3. 4. 5. 6.

the hydrostatic force (FHS), the weight of the dam (W), the uplifting force on the base of the dam (Fu), the sedimentation (silt deposit) pressure force (Fs), the earthquake force on the dam (FEQ), and the earthquake force caused by the water mass behind the dam (FEW).

Many gravity dams have a uniform cross section throughout their width that permits a force analysis per unit width of the dam. The binding forces between each unit width segment are neglected in the analysis because they can only add to the stability of the dam. The hydrostatic force acting on the upstream face of the dam may be resolved into a horizontal component and a vertical component. The horizontal component of the hydrostatic force acts along a horizontal line H/3 above the base of the dam. This horizontal force creates a clockwise moment about the toe of the dam (labeled in Figure 8.2), and it may cause the dam to fail by overturning. It may also cause dam failure by shearing along a horizontal plane at the base of the dam. The vertical component of the hydrostatic force is equal to the weight of the water mass directly above the upstream face of the dam. It acts along a vertical line that passes through the centroid of that mass. The vertical component of hydrostatic force always forms a counterclockwise moment about the toe. It is a stabilizing factor in gravity dams.

FEW H FEQ FHS W FS

Key

Toe

Heel Fu

Cutoff wall Figure 8.2 Cross-section of a gravity dam

312

Hydraulic Structures

Chap. 8

The largest stability force is the weight of the dam, which depends not only on the dimensions but also on the material used. The unit weight of most masonry or solid earth materials is approximately 2.4 to 2.6 times that of water. The importance of this stability force explains the name: gravity dam. The uplifting force on the base of a dam can be determined by foundation seepage analysis (Chapter 7). Acting in a direction opposite to the weight, this force should be minimized in every dam design if possible. It weakens the foundation and tends to overturn the dam. If the foundation soil is porous and homogeneous, then the uplifting pressure on the base varies linearly from full hydrostatic pressure at the heel of the dam (i.e., P = γH) to the full hydrostatic pressure at the toe. The total resulting uplifting force can be determined by integrating the resulting trapezoidal pressure distribution. The magnitude of the uplifting force as well as the overturning (clockwise) moment can be greatly reduced by installing an impermeable cutoff wall, as depicted in Figure 8.2. The cutoff wall alters the seepage course by lengthening the pathway, thus reducing the seepage and uplifting force downstream for the cutoff wall. The water velocity immediately behind the dam is very slow or nearly zero. Consequently, it loses its ability to carry sediments or other suspended matter. These heavier materials are deposited on the bottom of the reservoir, some being near the base of the dam. The silt–water mixture is approximately 50% heavier than water (sp. gr. = 1.5) and forms excess pressure force near the heel. Normally, the thickness of the silt layer will increase slowly with time. This force may contribute to dam failure by shear along the base. In earthquake zones, the forces generated by earthquake motion must be incorporated into dam design. Earthquake forces on dams result from acceleration associated with the earthquake motions. The magnitude of the earthquake force in the dam body (FEQ) is proportional to the acceleration and the mass of the dam body. The force may act in any direction through the centroid of the dam body. The earthquake force resulting from acceleration of the water body behind the dam is approximately equal to 5 aγ a bH2 9 g

where a is the earthquake acceleration, γ is the specific weight of water, and H is the hydrostatic head, or depth of water, immediately behind the dam. The earthquake force of the water body acts in a horizontal direction at a distance (4/3π)H above the base of the dam.* To ensure stability, the safety factors against sliding or overturning failures must be greater than 1.0 and are generally much higher. In addition, the maximum pressure exerted on the foundation must not exceed the bearing strength of the foundation. The force ratio against sliding (FRslide) is defined by the ratio of the total horizontal resistance force that the foundation can develop to the sum of all forces acting on the dam that tend to cause sliding. The ratio may be expressed as FRslide =

μ(ΣFV) + Asτs ΣH

(8.1)

* J. I. Bustamante, "Water pressure on dams subjected to earthquakes," J. Engr. Mech., Div., ASCE, 92 (Oct. 1966): 116–127.

Sec. 8.3

Stability of Gravity and Arch Dams

313

where μ is the coefficient of friction between the dam base and the foundation (ordinarily 0.4 6 μ 6 0.75), ΣFV is the summation of all vertical force components acting on the dam, τs is the shear stress strength of keys, and As is the total shear area provided by the keys. ΣFH is the summation of all horizontal force components acting on the dam. The keys (or keyways, depicted in Figure 8.2) are dam components built into the foundation to add resistance against dam sliding. Horizontal forces are transmitted to the foundation via the shear force in the keys. The total shear force provided by the keys, τsAs, must be larger than the difference between the total horizontal force acting on the dam, ΣFH, and the friction force provided by the base, μ(ΣFV). τsAs 7 [ΣFH - μ(ΣFV)] The force ratio against overturning (FRover) is defined by the ratio of the resisting moments (counterclockwise moments about the toe) to the overturning moments (clockwise moments about the toe): FRover =

Wlw + (FHS)vlv ΣFHYH + Fulu

(8.2)

where (FHS)v is the vertical component of the hydrostatic force and lw, lv, and lu are the horizontal distances from the toe to the lines of action of the weight (W), the vertical component of hydrostatic force, and the uplifting force (Fu), respectively. YH is the vertical distance measured from the toe to the lines of action of each respective horizontal force component (FH) acting on the upstream face of the dam. Oftentimes, we may assume that the vertical pressure on the foundation is a linear distribution between the toe and heel as shown in Figure 8.3. If we let RV represent the resultant of all vertical forces acting on the base of the dam and PT and PH represent the resulting foundation pressure at the toe and heel, respectively, we may write RV =

(PT + PH) (B) 2

and by equating moments (vertical forces only) about the center line RV (e) = c

(PT - PH) B (B) d a b 2 6 CL

B e

PH PT Figure 8.3

Pressure distribution on dam foundation

314

Hydraulic Structures

Chap. 8

Solving these two equations simultaneously, we have PT = a PH = a

RV 6e b a1 + b B B

RV 6e b a1 b B B

(8.3) (8.4)

The vertical resultant force normally acts through a point on the downstream side of the base centerline. Therefore, PT is usually the critical pressure in design. The value of PT must be kept less than the bearing strength of the foundation. The pressure at the heel (PH) is less important. Nevertheless, it is desirable to keep PH a positive value at all times to prevent tension cracks from developing in the heel region. Negative pressure indicates tension, and masonry materials have very low resistance to tension stress. A positive PH value can be ensured if the vertical resultant force (RV) is kept within the middle third of the base, or e 6

B 6

(8.5)

The value of e can be found by using the principle of moments; that is, the moment produced by the individual vertical force components about the center line is equal to the moment produced by RV. 8.3.2 Arch Dams The force load on an arch dam is essentially the same as on a gravity dam. To resist these forces, the dam foundation must provide a horizontal arch reaction. The large horizontal reactions can only be provided by strong, solid rock abutments at the two ends of the arch (Figure 8.1). Arch dams are usually high dams built in relatively narrow rock canyon sections. The efficiency of using material strength rather than bulk results in a very slender cross section compared to a gravity dam, making arch dams the best choice in many situations. Because arch dams combine the resistance of arch action with gravity, there is high stress in each segment of the dam, and detailed stress analysis is required. The stability analysis on an arch dam is usually carried out on each horizontal rib. Take a rib situated h meters below the designed reservoir water level. The forces acting in the direction of the dam centerline may be summed up as follows: θ θ = 2r(γh)sin 2 2 where R is the reaction from the abutment, θ is the central angle of the rib, r is the outer radius (extrados) of the arch, and γh is the hydrostatic pressure acting on the rib (Figure 8.4). The previous equation can be simplified for the abutment reaction 2R sin

R = r γh

(8.6)

This value is determined by considering only the arch reaction for resisting the hydrostatic load on the dam. In practice, however, several other resistive forces, as discussed with gravity dams,

Sec. 8.3

Stability of Gravity and Arch Dams

315

should also be included. The analysis should consider the combination resistance from both the arch and gravity actions. The volume of an arch dam is directly related to the thickness of each rib (t), the width (height) of the rib (B), and the center angle (θ). For the minimum volume of the dam, it can be shown that θ = 133°34′ (Figure 8.4). Other factors, such as topographic conditions, often prevent the use of this optimal value. Values within the range of 110° 6 θ 6 140° are commonly used in arch dam design. A simple approach to arch dam design is to keep the central angle constant while the radii vary from rib to rib as shown in Figure 8.4 (a). Another approach frequently used is to keep the radii of the ribs at a constant value and allow variations of the central angle, as shown in Figure 8.4 (b).

CL u = 133° 34 R

(a)

u = 133° 34 CL R r

u = 150° (b) Figure 8.4

Arch dams of (a) constant angle and (b) constant radius

316

Hydraulic Structures

Chap. 8

8.4 Small Earth Dams Small earth dams (or embankments) are designed and built for a variety of reasons. For example, small earth dams are frequently used in stormwater-management ponds, mining applications (tailings ponds), farm ponds (irrigation or stock water), constructed wetlands, and flood protection (levees). Because these structures are so common, a basic understanding of critical design features and construction considerations is important. The following paragraphs provide a rudimentary discussion of some of these items. A detailed examination can be found in the classic reference Design of Small Dams.* Small earth dams must be designed and constructed properly to fulfill their intended purpose. Special attention must be given to the following features. š Foundation: Placing an embankment on native material without any site preparation will only suffice for very small earth dams. Normally, site clearing, grading or scraping, and compaction are required to provide stability and minimize settling and seepage. Often a trench (also called a key or keyway) is excavated along the center line of the cross section (Figure 8.5). The trench is then backfilled and compacted, usually with clay, to minimize seepage. š Embankment: The height, slopes, top width, and materials all have to be specified. The height is normally dictated by storage requirements or site limitations such as elevation restrictions or land holdings. The slopes depend somewhat on the material used but are often between 2:1 and 3:1. The top width may have to accommodate maintenance equipment, otherwise 1 to 3 meters is common for small dams. Finally, the embankment materials need to be specified. Small dams and levees are often constructed by using a homogeneous material throughout (simple embankment). More commonly, a clay core is placed in the middle of the embankment with more pervious material on either side (zoned embankment; see Figure 8.5). The clay core reduces seepage, and the outer materials (silt or silt and sand) provide stability. Compaction testing (with a nuclear density gauge) is often required because the embankment is constructed and compacted in measured lifts. š Service Spillway: An outlet device is necessary to pass the normal stream flow coming into the reservoir downstream. A common outlet device for small earth dams is a riser and barrel assembly, often called a drop inlet (Figure 8.5). Water flows over the top of the riser pipe, drops into the barrel or pipe, and is passed through the earth embankment. The riser needs to be sized based on expected flows and some acceptable increase in water level. The riser acts as a weir; the design equations will be covered in the next section. The barrel likewise needs to be sized to accommodate the flow without allowing water to back up the riser and choke the flow. It acts much like a culvert; the design equations are covered in Section 8.9. Often the barrel will have a gated outlet on the bottom (low-level outlet) to drain the reservoir if necessary. Antiseep collars (Figure 8.5) minimize seepage along the barrel. In lieu of antiseep collars, the pipe may be encased in concrete for two-thirds of its length with a filter collar for the final third. š Emergency Spillway: Any earth dam whose failure will cause economic distress downstream or potential loss of life requires an emergency spillway. An emergency spillway is designed to accommodate the flows associated with rare storm events without * U.S. Department of the Interior (Bureau of Reclamation), Design of Small Dams (Washington, DC: U.S. Government Printing Office, 1977).

Sec. 8.4

Small Earth Dams

317

Low-level outlet Service spillway

Outlet of barrel Stilling basin

Dam crest

Riprap Reserivor

Emergency spillway

Plan view Service spillway (with trash rack) Riser

Riprap

Clay core

Dam crest Embankment Stilling basin (lined with riprap)

Antiseep collars

Low-level outlet Native material

Barrel Key

Section view Figure 8.5

Typical small earth dam

allowing the dam to be overtopped. (About one-third of all earth dam failures are caused by overtopping and subsequent embankment erosion.) Normally, emergency spillways are constructed by excavating a channel through the native (virgin) material around one end of the dam (Figure 8.5). The control elevation (which permits water to flow through the emergency spillway) is higher than the service spillway elevation, so it is rarely called into use. Design considerations include hydrologic analysis to determine the peak flow rate, channel sizing to pass the peak flow, and lining design to resist the erosive forces associated with the peak flow. Grass-lined emergency spillways are common and hold up well under infrequent use if the slopes are not too steep. At the end of the construction process, a survey is necessary to confirm final grades, elevations, and distances. Of paramount importance are the elevations for the top of the dam (crest), the service spillway, and the emergency spillway. The appropriate sizes of the service spillway pipes and the emergency spillway channel are also critical.

318

Hydraulic Structures

Chap. 8

8.5 Weirs A weir is a flow obstruction that causes water to rise to pass over it. Because weirs are usually installed in streams and channels that have free surfaces, the flow behavior over a weir is governed by gravity forces. One unique application of a weir is to help keep bridges from flooding, as schematically shown in Figure 8.6. By placing an obstruction of adequate dimensions in an otherwise subcritical stream flow, the water level is raised upstream from the weir. With the increased available head, the flow accelerates as it passes over the weir crest. This acceleration causes the depth of water to decrease and attain supercritical flow after passing through critical depth. Some distance downstream from the weir, the flow returns to a normal subcritical depth through a hydraulic jump. This arrangement protects the bridge structure from being overtopped. This concept is used in Australia to design "minimum energy" bridges. The cost of these bridges is reduced by minimizing the waterway opening. Flow acceleration over a weir provides a unique one-to-one relationship between the approaching water height (depth) and the discharge for each type of weir. Thus, weirs are commonly built to measure the discharge in open channels. Weirs are also used to raise stream-flow levels in order to divert water for irrigation and other purposes. The use of a weir as a flow-measuring device will be discussed in detail in Chapter 9. In this section the hydraulic characteristics of weirs will be presented. As previously mentioned, a weir increases the flow depth immediately upstream of the weir and reduces the cross-sectional flow area at the crest. The increase of water depth reduces the flow velocity upstream, but the sudden reduction of cross-sectional area causes the flow to speed up quickly as it passes over the crest. The occurrence of critical flow on weirs is the essential feature of weir structures. The hydraulics of an overflow weir may be examined by using an ideal frictionless weir (Figure 8.7). At the location where critical depth occurs, the discharge per unit width of the weir can be determined by using the critical flow equations (from Chapter 6). V 2gD

and yc =

= 1

(6.11)

q2 Q2 = 3 2 A gb Ag 3

(6.14)

Subcritical flow Weir

Figure 8.6

Supercritical flow

Acceleration of flow over a weir

Normal depth

Sec. 8.5

Weirs

319

V2 2g Energy line

yc H x

Figure 8.7

write

Flow over a frictionless weir

Rearranging Equation 6.11 and substituting yc for D for rectangular channels, we may V 2c yc = 2g 2 Therefore, the specific energy at the critical section is V 2c yc 3 E = yc + = yc + = yc 2g 2 2

(6.8)

If the approaching velocity head can be neglected, then the energy of the approaching flow is approximately equal to the water depth upstream of the weir, H. Therefore, for a frictionless weir, we may write an energy balance as E + x =

3 y + x = H 2 c

(8.7)

where x is the height of the weir, as depicted in Figure 8.7. Combining Equation 8.7 with Equation 6.14 and defining Hs = H - x, we have 2Hs 3 (8.8a) b 3 where q is the discharge per unit width of the weir. This is the basic form of the weir equation. In British units, the weir equation is q = 2gy 3c =

A ga

q = 3.09 H 3/2 s

(8.8b)

In the SI system, the equation becomes q = 1.70 H 3/2 s

(8.8c)

The stated discharge coefficients (3.09 ft0.5/s and 1.70 m0.5/s) are higher than the coefficients obtained in experiments because friction loss is neglected in the above analysis. Also note that Hs is defined as the vertical distance from the top of the weir to the upstream water level.

320

Hydraulic Structures

V2 2g

Chap. 8

V2 2g

Nappe

Figure 8.8 (a)

Flow over sharp-crested weir (free-falling nappe and submerged flow)

A sharp-crested weir is shown in Figure 8.8 (a). Upstream a short distance from the weir, all the velocity vectors are nearly uniform and parallel. However, as flow approaches the weir, the water near the bottom of the channel rises in order to pass over the crest. The vertical component of the flow near the upstream face of the weir causes the lower surface of the stream to separate from the weir and form a nappe after the flow passes over the weir [see the left panel in Figure 8.8 (a)]. The nappe usually traps a certain amount of air between its lower surface and the downstream side of the weir. If no means of restoring air is provided, a void will appear that represents a negative pressure on the structure. The nappe will also cling intermittently to the side of the weir and cause the flow to be unstable. The dynamic effect of this unstable flow may result in added negative pressure that may eventually damage the structure. When the downstream water level rises over the weir crest, the weir is said to be submerged [see the right panel in Figure 8.8 (a)]. In this case, the negative pressure no longer exists, and a new set of flow parameters may be considered in the determination of the discharge coefficient. A low head dam is a specific type of weir designed to span a stream or river, raising the upstream water level slightly as the flow passes over its entire length. This allows for a relatively constant diversion of water upstream for open-channel irrigation or power-plant cooling water, two common purposes of these hydraulic structures. Most low head dams are less than 3 m high. Depending on the downstream depth, different hydraulic conditions develop and are depicted in Figure 8.8 (b). Low head dams must be designed hydraulically to fulfill their purpose, but an additional concern is human safety. Water enthusiasts often underestimate the power of moving water, and these dams can pose formidable dangers. The downstream condition that should be avoided is Case 3 in Figure 8.8 (b). Although this condition may look harmless to the observer, it is responsible for many drowning deaths. This hydraulic condition, which includes a reverse roller, presents at least three dangers. The first danger is the reverse current that ensnares anyone who ventures too close to the backside of the dam. The second danger is the reduced "buoyancy" resulting from the large amounts of entrained air created from the plunging water. The third danger is the force of the water falling over the dam and striking someone who is not able to resist the reverse current. Hydraulic engineers should be aware of the dangers of these structures and avoid designs that are hazardous to the public. Appropriate designs are beyond the scope of this book but can be found in the engineering literature. The most common retrofits for existing structures include strategically placed riprap and altered downstream depths.

Sec. 8.5

Weirs

321

Overflow

Case 1

Low head dam

Low tailwater (swept-out jump) Case 2

Overflow

Low head dam

Increased tailwater (optimum jump)

Case 3

Overflow

Low head dam

High tailwater with reverse roller (drowned jump)

Case 4

Overflow

Low head dam

Very high tailwater (surface nappe) Figure 8.8 (b) Weir flow over low head dams (four hydraulic conditions). Source: Based on M. A. Robinson et al., "Dangerous dams," CE News (Feb. 2007): 24–29.

322

Hydraulic Structures

Chap. 8

Example 8.1 Uniform flow at a depth of 2 meters occurs in a long rectangular channel that is 4 meters wide. The channel is laid on a slope of 0.001, and the Manning coefficient is 0.025. Determine the minimum height of a low weir that can be built on the bottom of this channel to produce critical depth. Solution For a uniform flow condition, the Manning equation (6.5a) may be used to determine the channel discharge (Q): Q =

1 2/3 1/2 AR S n h 0

In this case, A = (2 m)(4 m) = 8 m2, P = 2(2 m) + 4 m = 8 m, and Rh = A/P = 1.0 m. Hence, Q =

1 (8)(1.0)2/3(0.001)1/2 = 10.1 m3/s 0.025

and Q 10.1 = = 1.26 m/s A 8

V = The specific energy is E = y +

(1.26)2 V2 = 2 + = 2.08 m 2g 2(9.81)

Flow over the weir passes through critical depth. Using Equation 6.14 yields yc = The corresponding velocity is

A gb2

Vc =

(10.1)2

A (9.81)(4)2

= 0.87 m/s

Q 10.1 = = 2.90 m/s 4yc 4(0.87)

and the critical velocity head is V 2c = 0.43 m 2g Now an energy balance is possible between two locations, at the weir and just upstream of the weir, as depicted in Figure 8.9. Assuming no energy loss at the weir, the minimum weir height (x) that can be built to produce critical flow is E = yc +

V 2c + x 2g

2.08 = 0.87 + 0.43 + x; x = 0.78 m

Sec. 8.6

Overflow Spillways

323 0.08 m

0.43

0.87

x Figure 8.9

8.6 Overflow Spillways An overflow spillway acts as a safety valve on a dam. Overflow spillways are designed to pass large amounts of water safely over the crest of a dam to maintain target water levels. They often act as emergency spillways or, in conjunction with emergency spillways, to keep the dam from being overtopped during storm events. Overflow spillways are common on arch, gravity, and buttress dams. Many earth dams have a concrete section to accommodate an overflow spillway. For small dams, their design shape is not that critical. On large dams, however, their effectiveness is highly dependent on their shape. In essence, an overflow spillway is a hydraulically efficient weir followed by a steep open channel that allows excess water to flow over a dam at supercritical velocities. The ideal longitudinal profile or shape of an overflow spillway should closely match the underside of the free-falling water nappe of a sharp-crested weir as depicted in Figure 8.10. This will minimize the pressure on the spillway surface. However, caution must be exercised to avoid any negative pressure on the surface. Negative pressure is caused by separation of the high-speed flow from the spillway surface, resulting in a pounding action that can cause significant damage to the spillway structure (e.g., pitting). The U.S. Waterways Experimental Station suggests a set of simple crest profiles that have been found to agree with actual prototype measurements. The geometry of the U.S. Waterways Experimental Station spillway crest profiles is shown in Figure 8.11. The discharge of a spillway may be calculated by an equation similar to that derived for flow over a weir (Equation 8.8), Q = CLH 3/2 a

(8.9)

where C is the coefficient of discharge, L is the width of the spillway crest, Ha is the sum of the static head (Hs) and the approaching velocity head (V 2a/2g), at the crest (Figure 8.10). Therefore, Ha = Hs +

V 2a 2g

(8.10)

The coefficient of discharge of a particular spillway crest is often determined by scaled model tests (Chapter 10) and accounts for the energy losses and the magnitude of the approaching

324

Hydraulic Structures

hv 5

V 2a 2g

Energy gradient

Chap. 8

Nappe

Sharp-crested weir

(a)

hv 5

V 2a 2g

(b) Figure 8.10 Ideal longitudinal profile or shape of an overflow spillway: (a) water nappe over a sharp-crested weir; (b) flow profile of an overflow spillway

velocity head. The value of the coefficient normally ranges between 1.66 to 2.26 m1/2/s (or 3 to 4.1 ft1/2/s in British units). A detailed discussion of weirs is presented in Chapter 9. Example 8.2 An overflow spillway 80 m wide carries a maximum (design) discharge of 400 m3/s. Compute the static (design) head and define the crest profile for the spillway. Consider a 3:1 upstream slope and a 2:1 downstream slope for the crest profile. Assume a discharge coefficient of 2.22 based on model studies and a negligible approach velocity based on the dam height. Solution Applying Equation 8.9 and assuming a minimal approach velocity, we obtain Q = 2.22LH 3/2 s

Sec. 8.6

Overflow Spillways

325

and Hs = a

2/3 2/3 Q 400 b b = a = 1.72 m 2.22L 2.22(80)

From the table in Figure 8.11, we have

a = 0.139 Hs = 0.239 m;

r1 = 0.68 Hs = 1.170

b = 0.237 Hs = 0.408 m;

r2 = 0.21 Hs = 0.361 m

K = 0.516;

P = 1.836

and from Figure 8.11, a

y x P x 1.836 b = - K a b = - 0.516a b Hs Hs Hs

The downstream end of the profile curve will be matched to a straight line with slope 2:1. The position of the point of tangency is determined by da da Hence,

y b Hs x b Hs

= - KP a

x P-1 x 0.836 b = -0.947a b = -2 Hs Hs

X = 2.45 XP.T. = 4.21 m Hs Y = -2.67 YP.T. = - 4.59 m Hs The crest profile curve of the spillway is shown in Figure 8.12.

y a x r2

Upstream slope (vertical /horizontal) 3/0

b P.T.

S y Hs

_ _

x 52K Hs

Figure 8.11

_ _

3/1

3/2

3/3

a / Hs

0.175 0.139 0.115

b / Hs

0.282 0.237 0.214 0.199

r 1 / Hs

0.50

0.68

0.48

0.45

r2 / Hs 0.20

0.21

0.22

–

0.500 0.516 0.515 0.534

1.850 1.836 1.810 1.776

Overflow spillway profile

326

Hydraulic Structures

Chap. 8

d x/Hs

y/Hs

Point

0.5

0.86

–0.145 –0.249

1.0

1.72

–0.516 –0.888

1.5

2.58

–1.086 –1.868

2.0

3.44

–1.842 –3.168

2.5

4.30

–2.775 –4.773

e P.T. 1

Figure 8.12

8.7 Side-Channel Spillways A side-channel spillway carries water away from an overflow spillway in a channel parallel to the spillway crest (Figure 8.13). A

x L A'

Section A–A' Figure 8.13

Side-channel spillway

Sec. 8.7

Side-Channel Spillways

327

The discharge over the entire width (L) of an overflow spillway can be determined by Equation 8.9, and the discharge through any section of the side channel at a distance x from the upstream end of the channel is Qx = xCH 3/2 a

(8.11)

The side-channel spillway must provide a slope steep enough to carry away the accumulating flow in the channel. However, a minimum slope and depth at each point along the channel is desired in order to minimize construction costs. For this reason, an accurate water surface profile for the maximum design discharge is important in the side-channel spillway design. The flow profile in the side channel cannot be analyzed by the energy principle (i.e., Chapter 6, gradually-varied flow profile) because of the highly turbulent flow conditions that cause excess energy loss in the channel. However, an analysis based on the momentum principle has been validated by both model and prototype measurements. The momentum principle considers the forces and change of momentum between two adjacent sections, a distance of ∆x apart, in the side channel, ΣF = ρ(Q + ∆Q)(V + ∆V) - ρQV

(8.12)

where ρ is the density of water, V is the average velocity, and Q is the discharge at the upstream section. The symbol ∆ signifies the incremental change at the adjacent downstream section. The forces represented on the left-hand side of Equation 8.12 usually include the weight component of the water body between the two sections in the direction of the flow [(ρgA∆x) sin θ], the unbalanced hydrostatic forces ρgAycos θ - ρg(A + ∆A)a y + ∆yb cos θ and a friction force, Ff, on the channel bottom. Here A is the water cross-sectional area, y is the distance between the centroid of the area and the water surface, and θ is the angle of the channel slope. The momentum equation may thus be written as ρgA∆x sin θ + [ρgAy - ρg(A + ∆A)(y + ∆y)]cos θ - Ff = ρ(Q + ∆Q)(V + ∆V) - ρQV

(8.13)

Let S0 = sin θ for a reasonably small angle, and Q = Q1, V + ∆V = V2, A = (Q1 + Q2)/(V1 + V2), and Ff = γASf ∆x; the above equation may be simplified to ∆y = -

Q1(V1 + V2) ∆Q a ∆V + V2 b + S0 ∆x - Sf ∆x g(Q1 + Q2) Q1

(8.14)

where ∆y is the change in water surface elevation between the two sections. This equation is used to compute the water surface profile in the side channel. The first term on the righthand side represents the change in water surface elevation between the two sections resulting from the impact loss caused by the water falling into the channel. The middle term represents the change from the bottom slope, and the last term represents the change caused by friction in the channel. Relating the water surface profile to a horizontal datum, we may write ∆z = ∆y - S0 ∆x = -

Q1(V1 + V2) ∆Q a ∆V + V2 b - Sf ∆x g(Q1 + Q2) Q1

(8.15)

328

Hydraulic Structures

Chap. 8

Note that when Q1 = Q2 or when ∆Q = 0, Equation 8.15 reduces to V 22 V 21 ∆z = a b - Sf ∆x 2g 2g

(8.16)

which is the energy equation for constant discharge in an open channel as derived in Chapter 6. Example 8.3 A 20-ft overflow spillway discharges water into a side-channel spillway with a horizontal bottom slope. If the overflow spillway (C = 3.7 ft1/2/s) is under a head of 4.2 feet, determine the depth change from the end of the side channel (after it has collected all of the water from the overflow spillway) to a point 5 feet upstream. The concrete (n = 0.013) side channel is rectangular with a 10-ft bottom. The water passes through critical depth at the end of the side channel. Solution The flow at the end of the side channel (Equation 8.9) is Q = CL(Ha)3/2 = (3.7)(20)(4.2)3/2 = 637 ft3/s The flow at a point 5 feet upstream (Equation 8.11) is Q = xC(Ha)3/2 = (15)(3.7)(4.2)3/2 = 478 ft3/s Solving for critical depth (Equation 6.14) at the end of the channel, we have yc = [Q2/gb2]1/3 = [(637)2/5(32.2)(10)26]1/3 = 5.01 ft

The solution method employs a finite-difference solution scheme (Equation 8.14) and an iterative process can be employed to compute the upstream depth. The upstream depth (or depth change, ∆y) is estimated, and Equation 8.14 is solved for a depth change. The two depth changes are compared, and a new estimate is made if they are not nearly equal. Table 8.2 displays the solution. Because side-channel spillway profile computations involve implicit equations, computer algebra software (e.g., Mathcad, Maple, and Mathematic) or spreadsheet programs will prove very helpful.

8.8 Siphon Spillways Water that is passing through a closed conduit will experience negative pressure when the conduit is elevated above the hydraulic grade line (pressure line) as described in Section 4.2. A spillway designed to discharge water in a closed conduit under negative pressure is known as a siphon spillway. A siphon spillway begins to discharge water under negative pressure when the reservoir level reaches a threshold elevation that primes the conduit. Before this, the water overflows the spillway crest in the same manner as that of the overflow spillway described in Section 8.6 [Figure 8.14 (a)]. However, if the water that flows into the reservoir exceeds the capacity of the spillway, the water level at the crest will rise until it reaches and passes the level of the crown (C). At this point, the conduit is primed and siphon action begins changing free surface flow into pressure flow. Theoretically, the discharge head is increased by the amount equal to H - Ha [Figure 8.14 (b)], and the discharge rate can thus be substantially increased. The large head allows rapid discharge of the excess water until it drops to the spillway entrance elevation. The portion of the spillway conduit rising above the hydraulic grade line (HGL) is under negative pressure. Because the hydraulic grade line represents zero atmospheric pressure, the vertical distance measured between the HGL and the conduit (immediately above the HGL)

Sec. 8.8

Siphon Spillways

329

Table 8.2 Side-Channel Profile Computations (Example 8.3)

(1) ∆x

(2) ∆y

(3) y

(4) A

(5) Q

(6) V

— -1.00 -2.48 -2.72

5.01 6.01 7.49 7.73

50.1 60.1 74.9 77.3

637 478 478 478

12.7 7.95 6.38 6.18

Column (1)

(7) (8) Q1 + Q2 V1 + V2 — 1,115 1,115 1,115

— 20.7 19.1 18.9

(9) ∆Q

(10) ∆V

(11) Rh

(12) (13) Sf ∆y

— 159 159 159

— 4.75 6.32 6.52

— — — 2.73 0.0013 -2.48 3.00 0.0007 -2.68 3.04 0.0007 -2.71

Distances upstream (ft) from the end of the channel.

Column (2) The assumed change in depth (ft) between sections. Column (3) Channel depth (ft) obtained from the assumed ∆y. Column (4) Channel cross-sectional area (ft2) corresponding to the depth. Column (5) Channel discharge (ft3/s) based on location along the spillway. Column (6) Column (7–10) Column (11)

Average channel velocity, V = Q/A (ft/s). Variables needed for Equation 8.14. Recall that subscript 1 refers to the upstream section and subscript 2 refers to the downstream section. Hydraulic radius (ft) found by dividing area by wetted perimeter.

Column (12) Friction slope found from Manning's equation [n2V 2/(2.22 Rh4/3)]. Average values of V and Rh should be used. However, because the friction loss is small, upstream values are used for convenience. Column (13) Change in channel depth between sections found using Equation 8.14. This value is compared to the assumed value in column (2), and another estimate is made if they do not correspond. Balance occurs when the depth 5 ft upstream is 7.73 ft.

indicates the negative pressure head (-P/γ) at that location. The crown of a siphon is the highest point in the conduit, so it is subjected to the maximum negative pressure. The maximum negative pressure at a spillway crown must not be allowed to decrease below the vapor pressure of water at the temperature measured. If the negative pressure at any section in the conduit drops below the water vapor pressure, the liquid vaporizes, and vast numbers of small vapor cavities form. These vapor bubbles are carried down the conduit with the flow. When a bubble reaches the region of higher pressure, the vapor condenses into liquid form and sudden collapse takes place. As the bubble collapses,

a1.2D

TWE

D (a) Submerged inlet and outlet HWE yu >1.2D

TWE (b) Full pipe flow with free outlet HWE

yu >1.2D

TWE

HWE

TWE

yu ft3) = P + (2 ft)(62.3 lb>ft3) P = 2,420 lb>ft2 = 16.8 psi Pressure can be expressed as the height of any fluid. For mercury, h = P>γHg = (2,420 lb>ft2)/[(13.6)(62.3 lb>ft2)] h = 2.86 ft of Hg (34.3 in.)

9.2 Velocity Measurements Water velocity in every conduit varies from near-zero values close to the stationary boundary to a maximum value near midstream. It is interesting to measure the velocity distribution in pipes and open channels. This is done by making local measurements at several positions in a cross

350

Water Pressure, Velocity, and Discharge Measurements

Chap. 9

section. The measurements should only be made with velocity probes of small size so that local flow patterns will not be disturbed by the presence of the probe in the flow field. Instruments commonly used for velocity measurement are Pitot tubes and current meters. Pitot tubes* are hollow tubes bent to measure pressure in the flowing stream. The probe usually consists of two tubes that are bent in such a way that the open end of one tube is perpendicular to the velocity vector, and the other is parallel to the flow as depicted in Figure 9.4 (a). To facilitate measurements, the two tubes are usually combined into one concentric construction— that is, a smaller tube is inside a large one [Figure 9.4 (b)]. At the probe tip (0), a stagnation point is produced where the velocity is zero. The pressure sensed at this opening is the stagnant point pressure, or stagnation pressure. At the side openings (1), the flow velocity (V) is practically undisturbed. These openings sense the static (or ambient) pressure at the site. Applying Bernoulli's equation between the two positions, 0 and 1, and neglecting the small vertical distance in between, we may write P0 P1 V2 + 0 = + γ γ 2g It is evident that the stagnation pressure head (P0 >γ) is a combination of the static pressure head (P1 >γ) and the dynamic pressure head (V 2 >2g) (i.e., the conversion of the velocity head to pressure head at the probe tip). From this expression, the flow velocity can be determined: V 2 = 2ga or

P0 - P1 ∆P b = 2ga b γ γ

(9.1a)

V = 22g(∆P>γ)

(9.1b)

0 V Dh 1 Dh V 0 (a)

(b)

Figure 9.4 Schematic representation of a Pitot tube: (a) separated tubes and (b) combined tubes

* Henri de Pitot (1695–1771) first used an open-ended glass tube with a 90° bend to measure velocity distribution in the Seine River. The rise of water elevation in the vertical part of the tube indicated stagnation pressure. Pitot did not apply Bernoulli's principle to obtain the correct velocity as discussed in this section.

Sec. 9.2

Velocity Measurements

351

The quantity ∆P>γ indicates the pressure head difference between the two openings in the probe. It is a function of the liquid column displacement (∆h) in the manometer shown in Figure 9.4. The pressure difference (∆P) is found using manometry principles that were discussed in Chapter 2 and are reviewed next in Example 9.2. Pitot tubes are widely used to measure the pressure and the velocity of flowing water. They are both reliable and accurate because they involve a simple physical principle and a simple setup. The Pitot tube is very useful for measuring water velocities under conditions in which the exact direction of the stream flow cannot be determined. In these cases, misalignments of the probe in the flow are likely to occur. The Prandtl Pitot tube shown in Figure 9.4 (b) has approximately a 1% error at an angle of 20° to the direction of the stream flow. The outer diameter of a Pitot tube is typically small—say, 5 mm. The two pressure tubes inside are much smaller. Because of the small diameters of these tubes, caution must be used to keep air bubbles from becoming trapped inside. Surface tension at the interface can produce a significant effect in the small tubes and yield unreliable readings. Example 9.2 A Pitot tube is used to measure velocity at a certain location in a water pipe. The manometer indicates a pressure difference (column height) of 14.6 cm. The indicator fluid has a specific gravity of 1.95. Compute the velocity. Solution Referring to Figure 9.4 (b), let x be the distance from position 1 to the interface between the water and manometry fluid (left side) and let γm be the specific weight of the manometry fluid. Applying manometry principles from Chapter 2 yields P1 - γx + γm ∆h - γ∆h + γx = Po or Po - P1 = ∆P = ∆h(γm - γ) Substituting this into Equation 9.1a, V 2 = 2g∆ha

γm - γ b = 2g∆h[(sp.gr.)m - 1] γ

14.6 b[1.95 - 1.0] 100 V = 1.65 m>s

V 2 = 2(9.81) a

Current meters are frequently used to measure the speed of water in open channels. There are two different types of mechanical current meters: cup and propeller. The cup type current (velocity) meter usually consists of four to six evenly shaped cups mounted radially about a vertical axis of rotation [Figure 9.5 (a)]. The moving water rotates the cups around the axis at a rate proportional to the water velocity. A mechanical or fiber-optic sensor relays each revolution to an electronic data-collection device. A conversion is made to flow velocity, and the data are stored or presented as a digital readout. Most cup-type current meters do not register velocity below a few centimeters per second because of starting friction. The propeller-type current meter has a horizontal axis of rotation [Figure 9.5 (b)]. It is more suitable for measuring higher velocity ranges in the midstream regions and is less susceptible to interference by weeds and debris. Depending on the design and construction of the current meter, the speed of axial rotation may not be linearly proportional to the speed of the water current. For this reason, each current

352

Water Pressure, Velocity, and Discharge Measurements

(a) Cup type Figure 9.5

Chap. 9

(b) Propeller type Current meters: (a) cup type and (b) propeller type

meter must be individually calibrated before it is used for field measurements. Calibrations may be carried out by towing the probe through still water at constant speeds. However, a calibration curve covering the range of applicable speeds is generally provided by the manufacturer. Acoustic Doppler (sonar) velocity meters can be used to measure the velocity of water in open channels. They rely on the Doppler principle by transmitting acoustic pulses along various paths. Because a sonic pulse moving with the current moves faster than one against the current, time-of-arrival differences can be used to determine flow velocity. Additional information can be obtained from the manufacturers of these devices. 9.3 Discharge Measurements in Pipes Although pipe flow measurements can be accomplished by several different methods, the simplest and most reliable measurement is the volumetric (or weight) method. This method requires only a stopwatch and an open tank to collect the water flowing from the pipe. The discharge rate can be determined by measuring the water volume (or weight) collected per unit time. Because of its absolute reliability, this method is frequently used for calibration of various types of flow meters. It is impractical for most operational applications because flowing water is totally diverted into a container when a measurement is made. However, in some cases there are water towers or ground level tanks within the pipe network that can be used for measurement. Flow diversion is not necessary to obtain accurate flow measurements in pressurized pipe flow. Pipe flow rates can be correlated to variations in energy (head) distribution associated with a sudden change in pipe cross-sectional geometry. This principle is utilized in Venturi meters, nozzle meters, and orifice meters. A Venturi meter is a precisely engineered section of pipe with a narrow throat. Two piezometric openings are installed at the entrance and at the throat as shown in Figure 9.6. Applying the Bernoulli equation at sections 1 and 2, and neglecting head loss, we get z1 +

P1 V 21 P2 V 22 + = z2 + + γ γ 2g 2g

(9.2)

The continuity equation between the two sections is A1V1 = A2V2

(9.3)

Sec. 9.3

Discharge Measurements in Pipes

353

Figure 9.6

Venturi meter

where A1 and A2 are the pipe and throat cross-sectional areas, respectively. Substituting Equation 9.3 into Equation 9.2 and rearranging, we obtain Q =

C A1 2 a b - 1 C A2

2ga

P1 - P2 + z1 - z2 b γ

(9.4)

The equation may be simplified to Q = CdA1

2gJ ∆ a

P + zb R γ

(9.5a)

where the dimensionless discharge coefficient Cd is evaluated as Cd =

1 A1 2 a b - 1 C A2

(9.5b)

For Venturi meters installed in a horizontal position, Q = CdA1

2ga

∆P b γ

(9.5c)

The coefficient Cd can be directly computed from the values of A1 and A2. The difference between this theoretically computed value and that obtained from experiments (which accounts for losses) should not exceed a few percentage points for well-manufactured Venturi meters. For satisfactory operation, the meter should be installed in a section of the pipe where the flow is relatively undisturbed before it enters the meter. To ensure this, a section of straight and uniform pipe that is free from fittings and at least 30 diameters in length must be provided upstream of the meter installation. Example 9.3 A 6-cm (throat) Venturi meter is installed in a 12-cm-diameter horizontal water pipe. A differential (mercury–water) manometer installed between the throat and the entry section registers a mercury (sp. gr. = 13.6) column reading of 15.2 cm. Calculate the discharge.

354

Water Pressure, Velocity, and Discharge Measurements

Chap. 9

Solution From manometry principles (as reviewed in Example 9.2), ∆P = ∆h(γHg - γ) or ∆P>γ = ∆ha

γHg - γ γ

b = ∆h 3 (sp.gr.)Hg - 1 4

= (15.2 cm)(13.6 - 1.0)

A1 = (π>4)(12)2 = 113 cm2 A2 = (π>4)(6)2 = 28.3 cm2 The dimensionless coefficient (Cd) can be calculated from the area ratio by using Equation 9.5b: 1 Cd = = 0.259 A1 2 a b - 1 C A2

To determine the discharge, apply Equation 9.5c for a horizontally installed Venturi meter: Q = 0.259a

113 15.2 b 2(9.81) J a b12.6 R = 0.0179 m3 >s 10,000 C 100

Nozzle meters and orifice meters (Figure 9.7) are also based on variations in energy (head) distribution associated with a sudden change in pipe cross-sectional geometry. In fact, the discharge equations for the nozzle meter and the orifice meter have the same form as that derived for the Venturi meter (Equation 9.5a). The main difference in application is that the value of the coefficient of discharge for the nozzle meters and orifice meters would be different from the theoretical value, Cd, calculated by using Equation 9.5b. This is primarily because of separation of the stream flow from the pipe wall boundary immediately downstream from the flow constriction (vena contracta). Nozzle meters and orifice meters produce a significant amount of head loss because most of the pressure energy that converts to kinetic energy (to speed the fluid through the narrow

(a)

(b)

Figure 9.7

(a) Nozzle meter and (b) orifice meter

Sec. 9.3

Discharge Measurements in Pipes

355

opening) cannot be recovered. The coefficient of discharge may vary significantly from one meter to another. The value depends not only on the status of flow in the pipe (the pipe's Reynolds number) but also on the area ratio between the nozzle (or orifice) and the pipe, the location of the pressure taps, and the upstream and downstream condition of the pipe flow. For this reason, on-site calibration is recommended for each meter installed. If installation of a pipe orifice meter is made without an on-site calibration, then reference should be made to the manufacturer's data, and the detailed installation requirements should be followed. As previously stated, the coefficient of discharge for nozzle meters and orifice meters cannot be computed directly from the area ratio, A1 >A2. The discharge equations (Equations 9.5a and 9.5c) must be modified by an experimental, dimensionless coefficient, Cv : Q = CvCdA1

2gJ ∆ a

P + zb R γ

(9.6a)

where z is the difference in elevations between the two pressure taps. For horizontal installations, Q = CvCdA1

2ga

∆P b γ

(9.6b)

Extensive research on nozzle meters has been sponsored by ASME and the International Standards Association to standardize the nozzle geometry, installation, specification, and experimental coefficients. One of the typical ASME flow nozzle installations commonly used in the United States with the corresponding experimental coefficients is shown in Figure 9.8. Compared with Venturi meters and nozzle meters, orifice meters are affected even more by flow conditions. For this reason, detailed instructions of installation and calibration curves must be provided by the manufacturer for each type and size. If a meter is not installed strictly according to the instructions, then it should be calibrated meticulously on site. Ellipse

2 3

d2 1.00

d2 d1

0.99

d2 Cv

0.6d2 0.5d1

0.4

0.2

0.98

0.6 0.8 d2/d1

0.97 0.96 0.95

105

p2 Figure 9.8

ASME nozzle dimensions and coefficients

5 V2d2 y

106

356

Water Pressure, Velocity, and Discharge Measurements

Chap. 9

Example 9.4 The Venturi meter in Example 9.3 is replaced with an ASME flow nozzle meter. During operation, the attached differential (mercury–water) manometer registers a mercury column reading of 15.2-cm. The water in the pipeline is 20 °C. Determine the discharge. Solution The diameter ratio is d2 >d1 = 6>12 = 0.5, and Cd = 0.259 (from Example 9.3). Assume that the experimental meter coefficient Cν = 0.99. The corresponding discharge can be calculated from Equation 9.6b as Q = (0.99)(0.259) a

113 15.2 b 2(9.81) J a b(12.6) R = 0.0178 m3 >s 10,000 C 100

This value must be verified by checking the corresponding Reynolds number of the nozzle. The NR value calculated based on the discharge is (0.0178) c (0.06) d V2d2 (π/4)(0.06)2 NR = = = 3.78 * 105 v 1.00 * 10-6 With this Reynolds number value, the chart in Figure 9.8 gives a better value of the experimental coefficient, Cν = 0.986. Hence, the correct discharge is Q = (0.98>0.99)(0.0178) = 0.0177 m3 >s

A bend meter (Figure 9.9) measures the pressure difference between the outer and inner sides of a bend in a pipeline. Centrifugal force developed at a pipe bend forces the main stream to flow closer to the outer wall of the pipe at the bend. A difference in pressure is developed between the inside and outside of the bend. The pressure difference increases as the flow rate increases. The relationship between the measured pressure difference and the discharge in the pipe can be calibrated for flow rate determinations. The discharge equation may be expressed as Q = CdA

2ga

Po Pi - b γ γ

(9.7)

where A is the pipe cross-sectional area and Pi and Po are the local pressure values registered at the inside and outside of the pipe bend, respectively. Cd is the dimensionless discharge coefficient, which can be determined by calibration on site. Note that Equation 9.7 is appropriate if the elevation difference between taps (∆z) is negligible. Otherwise, the elevation difference must be included with the pressure difference.

Q Po Pi

Figure 9.9 Bend meter

Sec. 9.4

Discharge Measurements in Open Channels

357

If the bend meter cannot be calibrated in place, then the pipe discharge may still be determined within an accuracy of approximately 10% if the pipe flow's Reynolds number is sufficiently large and if at least 30 diameters of straight pipe are provided upstream from the bend. In such a case, the discharge coefficient is approximated by Cd =

R 2D

(9.8)

where R is the center line radius at the bend and D is the pipe diameter. Bend meters are inexpensive and convenient. An elbow already in the pipeline may be used without additional installation cost or added head loss. 9.4 Discharge Measurements in Open Channels A weir is a simple overflow structure extending across a channel and normal to the direction of the flow. Various types of weirs exist, and they are generally classified by shape. Weirs may be either sharp-crested (useful for measuring flow) or broad-crested (incorporated into hydraulic structures with flow measurement as a secondary function). 9.4.1 Sharp-Crested Weirs Sharp crested weirs (Figure 9.10) include the following four basic types: 1. 2. 3. 4.

horizontal weirs without end contractions, horizontal weirs with end contractions, V-notch weirs, and trapezoidal weirs. 2

Vo 2g

H Vo

(a)

u L

(b) Figure 9.10

(c)

(d)

Common sharp-crested weirs: (a) uncontracted horizontal weir, (b) contracted horizontal weir, (c) V-notched weir, and (d) trapezoidal weir

358

Water Pressure, Velocity, and Discharge Measurements

Chap. 9

An uncontracted horizontal weir extends across the entire width of a uniform reach in the channel. A standard uncontracted, horizontal weir should meet the following requirements. 1. The crest of the weir should be horizontal, sharp-edged, and normal to the flow. 2. The weir plate should be vertical and have a smooth upstream surface. 3. The approach channel should be uniform and the water surface should be free from large surface waves. The basic discharge equation for a standard, uncontracted, horizontal weir (Figure 9.10) is Q = CLH 3/2

(9.9)

where L is the length of the crest, H is the head on the weir, and C is the discharge coefficient having units of length0.5 >time. Discharge coefficients are often derived from experimental data by governmental agencies such as the U.S. Bureau of Reclamation (USBR). Using British units (H, L, p in feet, and Q in cubic feet per second), the discharge coefficient in ft0.5 >s may generally be expressed as C = 3.22 + 0.40

H p

(9.10a)

where p is the weir height (Figure 9.10). In the SI unit system, the coefficient becomes C = 1.78 + 0.22

H p

(9.10b)

A contracted horizontal weir has a crest that is shorter than the width of the channel. Thus, water contracts both horizontally and vertically in order to flow over the crest. The weir may be contracted at either end or at both ends. The general discharge equation may be expressed as Q = Ca L -

nH b H 3/2 10

(9.11)

where n is the number of contractions at the end [n = 1 for a contraction at one end and n = 2 for contractions at both ends, as shown in Figure 9.10 (b)]. The coefficient of discharge, C, should be determined by calibration in place. Note that the value will depend on the unit system used because C has units of length0.5 >time. A standard contracted horizontal weir is one whose crest and sides are so far removed from the bottom and sides of the channel that full contraction is developed. The dimensions of a standard contracted horizontal weir are shown in Figure 9.11 (a), where the weir length (L) is denoted as b. The discharge equation of this standard weir is given by the USBR* as Q = 3.33 (L - 0.2H)H 3/2

(9.12a)

This expression was developed for the British measurement system with L and H in feet and Q in cubic feet per second. In the SI unit system, the equation becomes Q = 1.84(L - 0.2H)H 3/2

(9.12b)

* U.S. Bureau of Reclamation, Water Measurement Manual (Washington, DC: U.S. Government Printing Office, 1967).

Sec. 9.4

Discharge Measurements in Open Channels

a $ 3H

359

c $ 3H

s $ 2H

H b $ 3H

908

p $ 6H

(a) Figure 9.11

p $ 2H

(b)

USBR standard weirs: (a) contracted horizontal weir and (b) 90° V-notch weir

A V-notch weir is especially useful when accuracy in measurement is desired for a large range of water depths. The discharge equation for a V-notch weir takes the general form θ Q = Ca tan b H 5/2 2

(9.13)

where θ is the weir angle as shown in Figure 9.10 (c), and the discharge coefficient (C) is determined by calibration in place. Note that the value will depend on the unit system used because C has units of length0.5/time. A standard USBR 90° V-notch weir consists of a thin plate with each side of the notch inclined 45° from the vertical [Figure 9.11(b)]. The weir operates like a contracted horizontal weir, and all requirements stated for the standard, uncontracted, horizontal weir apply. The minimum distances of the sides of the weir from the channel banks should be at least twice the head on the weir, and the minimum distance from the weir crest to the channel bottom should be at least twice the head on the weir. The discharge equation of the standard 90°-V-notch weir is given by the USBR as Q = 2.49H 2.48

(9.14a)

This expression was developed for the British measurement system with H in feet and Q in cubic feet per second. In the SI unit system the equation becomes Q = 1.34H 2.48

(9.14b)

with Q in cubic meters per second and H in meters. The trapezoidal weir has hydraulic characteristics in between that of the contracted horizontal weir and the V-notch weir. The general discharge equation developed for the contracted horizontal weir may be applied to the trapezoidal weir with an individually calibrated discharge coefficient. The USBR standard trapezoidal weir (Figure 9.12) is also known as the Cipolletti weir. It has a horizontal crest, and the sides incline outwardly at a slope of 1:4 (horizontal to vertical). All requirements stated for the standard, uncontracted, horizontal weir apply. The height of the weir crest should be at least twice the head of the approach flow above the crest (H), and the distances from the sides of the notch to the sides of the channel should also be at least twice the head. The discharge equation for the Cipolletti weir is given by the USBR as Q = 3.367LH 3/2

(9.15a)

360

Water Pressure, Velocity, and Discharge Measurements

Chap. 9

4 H

s $ 2H p $ 2H

Figure 9.12 USBR standard trapezoidal weir

This expression was developed for the British measurement system with L and H in feet and Q in cubic feet per second. In the SI unit system the equation becomes Q = 1.858 LH 3/2

(9.15b)

Example 9.5 Laboratory measurements are made on a contracted (both sides) horizontal weir with a crest length of 1.56 m. The measured discharge is 0.25 m3 >s under a head of H = 0.2 m. Determine the discharge coefficient in the given (SI) units. Solution Applying Equation 9.11 for the contracted horizontal weir, we get nH Q = C aL bH 3/2 10

Here L = 1.56 m, H = 0.2 m, and n = 2 for contraction at both ends; 0.25 = C a1.56 -

2(0.2) b(0.2)3/2 10

C = 1.84 m0.5 >s

9.4.2 Broad-Crested Weirs

A broad-crested weir provides a stretch of elevated channel floor over which critical flow takes place (Figure 9.13). Depending on the height of the weir in relation to the depth of the approaching channel, the discharge equation may be derived from the balance of forces and momentum between the upstream approach section (1) and the section of minimum depth (2) on the crest of the weir. For a unit width of the weir, the following equation may be written as ρqa

q q 1 - b = γ c y 21 - y 22 - h(2y1 - h) d y2 y1 2

(9.16)

where q is the discharge per unit width, h is the weir height measured from the channel floor, and y1 and y2 are the upstream and downstream depths, respectively. The conditions provided above are not sufficient to simplify Equation 9.16 into a one-toone relationship between the approach water depth and the discharge. An additional equation was obtained from experimental measurements* for the average flow * H. A. Doeringsfeld and C. L. Barker, "Pressure-momentum theory applied to the broad-crested weir," Trans. ASCE, 106 (1941): 934–946.

Sec. 9.4

Discharge Measurements in Open Channels

361

y1 h

Figure 9.13 Broad-crested weir

y1 - h = 2y2

(9.17)

Substituting Equation 9.17 into Equation 9.16 and simplifying, we have q = 0.43322ga The total discharge over the weir is

1/2 y1 b H 3/2 y1 + h

Q = Lq = 0.43322ga

1/2 y1 b LH 3/2 y1 + h

(9.18)

(9.19)

where L is the overflow length of the weir crest (as seen in a plan view) and H is the height of the approach water above the weir crest. 9.4.3 Venturi Flumes The use of a weir is probably the simplest method for measuring discharge in open channels. However, there are disadvantages to using weirs, including the relatively high energy loss and the sedimentation deposited in the pool immediately upstream of the weir. These difficulties can be partially overcome by using a critical flow flume called a Venturi flume. A variety of Venturi flumes have been designed for field application. Most of the flumes operate with a submerged outflow condition and create critical depth at a contracted section (throat) followed by a hydraulic jump at the exit. The discharge through the flume can be calculated by reading the water depth from the observation wells located at the critical flow section and at another reference section. The most extensively used critical flow flume in the United States is the Parshall flume, developed by R. L. Parshall* in 1920. The flume was experimentally developed for the British measurement system. It has fixed dimensions as shown in Figure 9.14 and Table 9.1. Empirical discharge equations were developed to correspond to each flume size. These equations are listed in Table 9.2.

* R. L. Parshall and C. Rohwer, The Venturi flume. Colorado Agricultural Experimental Station Bulletin No. 265 (1921); R. L. Parshall, "The improved Venturi flume," Trans. ASCE, 89 (1926): 841–851.

362

Water Pressure, Velocity, and Discharge Measurements Diverging section

Converging section Throat section

Ha 2 3

Hb H Z

Crest A Alternate 458 wing wall

Chap. 9

W Hc

Plan M

Slope

1 4

Flow

Crest Y

131

1 38

angles

Water surface, S

Level floor

X 1 1 3 1 3 8 angles

Section L2L Figure 9.14 Parshall flume dimensions. Source: Courtesy of U.S. Bureau of Reclamation.

In Equations 9.20 through 9.24, Q is the discharge in cubic feet per second (cfs), W is the throat width in feet, and Ha is the water level reading from observation well a measured in feet. These equations are derived strictly for the units stated and for the dimensions specified in Figure 9.14 and Table 9.1. They do not have an equivalent version in the metric system. When the ratio of gauge reading Hb (from observation well b) to gauge reading Ha (from observation well a) exceeds the following values: 0.50 for flumes 1, 2, and 3 in. wide 0.60 for flumes 6 in. and 9 in. wide 0.70 for flumes 1 ft to 8 ft wide 0.80 for flumes 10 ft to 50 ft wide the flow is said to be submerged. The effect of the downstream submergence is to reduce the discharge through the flume. In this case, the discharge computed by the above equations should be corrected by considering both readings Ha and Hb. Figure 9.15 shows the flow rate corrections for submerged flow through a 1-ft Parshall flume. The diagram is made applicable to larger flumes (up to 8 ft) by multiplying the corrected discharge for the 1-ft flume by a factor given for the particular size selected. Figure 9.16 shows the flow rate corrections for submerged flow through a 10-ft Parshall flume. The diagram is also made applicable to larger size flumes (up to 50-ft) by multiplying the corrected discharge for the 10-ft flume by a factor given for the particular size used.

363

Source

917 32 1078 1214

1 ft

5 416

1 ft

9 232 5 416 638

2 ft

7 16

10 '8 11 '8

4 ft

5 ft

6 ft

7 ft

0 ft

1-in.

0 ft

6-in.

9-in.

1 ft

0-in.

6-in.

2 ft

0-in.

3 ft

0-in.

4 ft

0-in.

5 ft

0-in.

6 ft

0-in.

7 ft

0-in.

3-in.

2-in.

0 ft

in.

5 ft

4 ft

3 ft

in.

ft.

2 3A

0 6

4 '8

4 '4

8 ft

7 ft

10 '8

7 ft

6 ft

4 '2

6 ft

10 '8

5 ft

4 '4

4 ft

5 ft

10 '8

3 ft

4 ft

7 '8

2 ft

3 '2

1 ft

321 32 5 516

0 ft

in.

ft.

4 ft

2 ft

1 ft

in.

ft.

11 '8

9 ft

8 ft

3 ft

6 '8

3 ft

2 ft

12 to 18

6 to 10

6 to 9

0 ft

in.

ft.

7 ft

4 '4

6 ft

1 '8

5 ft

11 '2

3 ft

16 '8

9 '4

2 ft

10 '8

3 '8

1 ft

619 32 813 32 3 1016

0 ft

in.

ft.

2 ft

1 ft

0-in.

3 ft

0-in.

3 ft

0-in.

3 ft

0-in.

3 ft

0-in.

3 ft

0-in.

3 ft

0-in.

3 ft

6-in.

0-in.

2 ft

12-in.

–

0 ft

5 1232

4 '2

10-in. 10 '8

8 '8

0 ft

in.

ft.

8-in.

0 ft

in.

ft.

0 ft

in.

ft.

1 ft

0 ft

12 ft

3 '2

11 ft

1 '4

10 ft

10 '4

8 ft

3 '2

7 ft

6 ft

10 '4

4 ft

4 '2 18 '2

2 ft

–

in.

ft.

4 '2 11 '2

0 ft

2 '4

111 16

1 '8

0 ft

in.

ft.

in.

ft.

Parshall Flume Dimensions

0 ft

in.

ft.

Table 9.1

2 ft

1 ft

–

in.

ft.

0 ft

5 16

0 ft

in.

ft.

0 ft 1

0 ft

in.

ft.

min

3-in.

0 ft

3-in.

0 ft

3-in.

0 ft

3-in.

0 ft

3-in.

0 ft

3-in.

0 ft

3-in.

0 ft

3-in.

0 ft

1 '2

1-in.

–

85.6

67.9

50.4

33.1

24.6

16.1

8.9

3.9

1.13

0.47

0.19

(continued)

3.0 121.4

2.6 103.5

1.6

1.3

0.61

0.42

0.15

0.11

0.09

0.05

'2 in. 0.03

'4 in. 0.02 1

max

Flow (cfs)

'2 in. '8 in. 0.01

in.

ft.

364

–

10 ft

6 ft

5 ft

0-in.

19 ft

50 ft

–

0-in.

16 ft

40 ft

–

0-in.

12 ft

–

30 ft

11 ft

–

0-in.

25 ft

0-in

9 ft

20 ft

–

0-in.

7 ft

15 ft

–

0-in.

6 ft

0-in.

–

8 ft

in.

12 ft

in.

ft.

2 3A

0-in.

27 ft

26 ft

25 ft

16 ft

14 ft

56 ft

45 ft

34 ft

29 ft

24 ft

18 ft

14 ft

12 ft

10 '8

9 ft

in.

ft.

7 ft

in.

ft.

0 0

9 '2

7 ft

60 ft

9 '2

7 ft

50 ft

4 '4

7 ft

40 ft

35 ft

30 ft

25 ft

6 ft

4 '4

5 ft

18 ft

7 '4

4 ft

3 ft

in.

ft.

15 ft

1 '4

11 ft

in.

ft.

6 ft

4 ft

3 ft

2 ft

in.

ft.

0-in.

20 ft

0-in.

16 ft

0-in.

14 ft

0-in.

13 ft

0-in.

12 ft

0-in.

10 ft

0-in.

8 ft

0-in.

6 ft

0-in.

3 ft

in.

ft.

–

in.

ft.

1 ft

0 ft

in.

ft.

–

1 ft

in.

ft.

2 ft

1 ft

1 '2

1 ft

1 '2

1 ft

0 ft

in.

ft.

Parshall Flume Dimensions (continued)

–

8 '4

13 ft

in.

ft.

–

2 ft

in.

ft.

0 ft

in.

ft.

1 1 * Tolerance on the throat width (W) is { 64 inch; tolerance on other dimensions is { 32 inch. The throat sidewalls must be parallel and vertical. Source (1) Colorado State University Technical Bulletin No. 61. Source (2) U.S. Department of Agriculture: Soil Conservation Service Circular No. 843. Source (3) Colorado State University Technical Bulletin No. 426-A.

Source

Table 9.1

0-in.

1 ft

0-in.

1 ft

0-in.

1 ft

0-in.

1 ft

0-in.

1 ft

0-in.

1 ft

0-in.

1 ft

0-in.

1 ft

3-in.

0 ft

in.

ft.

–

in.

ft.

max

600

350

200

25 3,000

20 2,000

15 1,500

15 1,200

10 1,000

8.0

6.0

3.5 139.5

min

Flow (cfs)

Sec. 9.4

Discharge Measurements in Open Channels Table 9.2

Parshall Flume Discharge Equations

Throat Width

Discharge Equation

3 in. 6 in. 9 in.

Q = 0.992H 1.547 a Q = 2.06H 1.58 a Q = 3.07H 1.53 a 1.522 # W0.026 Q = 4WH a

1 to 8 ft

Q = (3.6875W +

10 to 50 ft

Flume size (ft) Correction factor 2.5

1.0

0.3

2.5)H 1.6 a 1.0 1.0

2.0 1.8

(9.20) (9.21) (9.22)

0.03 to 1.9 0.05 to 3.9 0.09 to 8.9

(9.23)

Up to 140

(9.24)

Up to 2,000

3.0 2.4

4.0 3.1

6.0 4.3

8.0 5.4

me ub s f e o 70 72 74 76 78 80 82 84 86 88 90 92 94 96

Pe rce nt

Upper head Ha (ft)

1.5

0.6 0.5 0.4

Free Flow Capacity (cfs)

/H a e, H b c n rge

2.0

0.8

365

0.06 0.1

0.2

0.3 0.4 0.5 1 Correction (cfs)

4 5 6

8 10

Figure 9.15 Flow-rate correction for a 1-ft submerged Parshall flume. Source: After R. L. Parshall, Measuring Water in Irrigation Channels with Parshall Flumes and Small Weirs, U.S. Soil Conservation Service, Circular 843 (1950); R. L. Parshall, Parshall Flumes of Large Size, Colorado Agricultural Experimental Station Bulletin No. 426A (1953). Flume size (ft) Correction factor 6.0 5.0

/H a Hb , ce en 80 1 g r 8 e bm 82 3 u 8 4 fs o 8 5 e g 8 a t n 87 ce

Upper head Ha (ft)

4.0 3.0 2.0

10 1.0

r Pe

1.5

15 1.5

20 2.0

30 3.0

40 4.0

50 5.0

89 91 3 9 5 9 7 9 9 9

1.0 0.8 0.7 0.6 0.5

0.5

3 4 5 6 8 10 20 30 40 60 80 100 Correction (cfs)

200

400

Figure 9.16 Flow-rate correction for a 10-ft submerged Parshall flume. Source: After R. L. Parshall, Measuring Water in Irrigation Channels with Parshall Flumes and Small Weirs, U.S. Soil Conservation Service, Circular 843 (1950); R. L. Parshall, Parshall Flumes of Large Size, Colorado Agricultural Experimental Station Bulletin No. 426A (1953).

366

Water Pressure, Velocity, and Discharge Measurements

Chap. 9

Example 9.6 A 4-ft Parshall flume is installed in an irrigation channel to monitor the rate of flow. The readings at gauges Ha and Hb are 2.5 ft and 2.0 ft, respectively. Determine the channel discharge. Solution Ha = 2.5 ft

Hb = 2.0 ft

Submergence, Hb >Ha = 80,. Equation (9.23) provides the value of the unsubmerged discharge

W Qu = 4WH 1.522 a

0.026

= 4 (4) (2.5)1.522(4)0.026 = 67.9 cfs

Under the given conditions, the flume is operating at 80% submergence, and the value should be corrected accordingly. From Figure 9.15, we find the flow rate correction for a 1-ft Parshall flume to be 1.8 cfs. For the 4-ft flume, the corrected flow rate is Qc = 3.1(1.8) = 5.6 cfs The corrected discharge of the channel is Q = Qu - Qc = 67.9 - 5.6 = 62.4 cfs

Problems (secTion 9.1) 9.1.1. Water is poured into an open-ended U-tube (Figure P9.1.1). Then oil is poured into one leg of the U-tube and causes the water surface in one leg to rise 6 in. above the oil-water interface in the other leg. The oil column measures 7.4 in. What is the specific gravity of the oil?

Oil A

B H2O

7.4 in.

6 in. 2

Figure P9.1.1

9.1.2. Determine the pressure in the pipe shown in Figure P9.1.2 if y = 134 cm, h = 112 cm, and the manometry fluid is mercury (S.G. = 13.6). Also determine the height the pressurized water would climb in a piezometer if it was used to measure pressure at the same location.

Problems

367 Atmosphere

Water A

h 1

Liquid M

Figure P9.1.2

9.1.3. Natural gas is flowing through a pipeline, and the pressure is being measured by an inclined manometer (Figure 9.2). Determine the gas pressure (in psi and kN/m2) if the angle of inclination is 30°, the manometry fluid has a specific gravity of 13.6, and ∆l = 6 in.

(secTion 9.2) 9.2.1. A plexiglass tube (piezometer) is mounted on a pipe as shown in Figure P9.2.1. Another plexiglass tube with a 90° bend (i.e., Pitot tube) is inserted into the center of the pipe and directed toward the current. For a given discharge, the Pitot tube registers a water height of 330 cm when the water pressure is 31.3 kPa. Determine the height of water in the piezometer (cm) and the flow velocity (m/s). Is the flow velocity the average pipe velocity? Explain.

Piezometer

Pitot tube

Figure P9.2.1

9.2.2. Show that Equation (9.1b) can be written in the following form V = 22g # ∆h(SG - 1)

if water is the pipe fluid, the manometer fluid has a specific gravity of SG, and water from the pipe extends all the way to the manometer fluid. Hint: Apply manometry principles to the sketch shown in Figure 9.4(b) and combine with Bernoulli principles. 9.2.3. The center line velocity in a 4-in.-diameter pipe is 24.4 ft/s based on Pitot tube readings. Determine the reading on the Pitot tube scale (∆h in Figure 9.4b) if the pipe fluid is water and the manometer fluid is mercury (SG = 13.6). Estimate the discharge. Why is it an estimate? 9.2.4. Referring to Figure 9.4(a), determine the maximum measurable velocity if the maximum Pitot tube scale length is 25 cm. The pipe fluid is water, which extends all the way to the manometer fluid, which is mercury (SG = 13.6). Estimate the maximum discharge in the 50-cm-diameter pipe. Is your estimate of discharge high or low? Explain.

368

Water Pressure, Velocity, and Discharge Measurements

Chap. 9

9.2.5. A Pitot tube and piezometer are dipped into a 50-cm-deep open channel with relatively straight, parallel stream lines as shown in Figure P9.2.5. If the Pitot tube is facing into the current (25 cm below the surface) and water rises in the tube 16 cm above the water surface, determine the flow velocity. Also determine the water rise in the piezometer. Piezometer

Pitot tube

Figure P9.2.5

9.2.6. Refer to Figure 9.4(b) and answer the following questions. (a) If the pipe fluid is water and the manometer fluid is mercury (SG = 13.6), determine the pipe velocity if ∆h = 4.8 in. (b) If the pipe fluid is oil (SG = 0.85) and the manometer fluid is mercury, determine the pipe velocity if ∆h = 4.8 in.

(secTion 9.3) 9.3.1. The maximum flow rate in an 8-in. diameter (horizontal) waterline is 4.3 cfs. If a 4-in. Venturi meter is installed in the pipe to measure flow, estimate the necessary length of a vertical U-tube scale for the differential water-mercury (SG = 13.6) manometer. 9.3.2. A 50-cm diameter (horizontal) pipe contains a 20-cm Venturi meter. Determine the flow rate if the pressure difference reading between the throat and entry section in a water-mercury (SG = 13.6) manometer is 1.46 m. 9.3.3. A 10-cm ASME flow nozzle is installed in a 20-cm (horizontal) waterline. The attached manometer contains mercury (SG = 13.6) and water and registers a pressure difference of 42-cm. Calculate the discharge in the pipe. 9.3.4. Determine the discharge in the 16-in. diameter waterline shown in Figure P9.3.4. The nozzle meter has a throat diameter of 6.5 in. and is installed according to ASME standards. The distance between pressure taps is 10 in. [SG (mercury) = 13.6].

Water

Mercury 8 in. Water

Figure P9.3.4

Problems

369

9.3.5. Flow measurement is needed in a 21-in.-diameter water line with a maximum flow rate of 12.4 cfs. A 12-in. orifice meter (Cv = 0.675) has been selected with a 9-in. vertical difference between manometer taps (upwards flow). Estimate the necessary scale length associated with a vertical U-tube for the meter's water-mercury (SG = 13.6) manometer. 9.3.6. A 30-cm orifice plate is installed in a 50-cm-diameter horizontal waterline. In a field calibration, 6.55 m3 of water is collected in 19.7 s. The water-mercury (SG = 13.6) manometer reads a mercury difference of 18.2 cm. What is the discharge coefficient? 9.3.7. A bend meter is installed in a 75-cm-diameter water pipe as shown in Figure 9.9. The installation delivered 51 m3 of water in 1 min. Determine the pressure difference (in cm) that will register in a mercury-water manometer when the water pipe and bend are in a horizontal position. The bend radius is 80 cm. 9.3.8. A bend meter is installed in a 30-in.-diameter water pipe with a 36 in. bend radius as shown in Figure 9.9. The flow rate in the pipe is 34.7 cfs and the pressure difference reading between the outside and inside taps registers 2.1 in. on a mercury–water manometer. Determine the coefficient of discharge for the bend meter if the pipe bend is in a vertical position and flow is downward. Note that in the vertical position the pressure taps are mounted along a 45° line from the horizontal datum.

(secTion 9.4) 9.4.1. Determine the discharge over an uncontracted horizontal weir when the upstream water depth is 7.0 ft. The sharp-crested weir is 9.0 ft long and 6.0 ft high. 9.4.2. Determine the discharge over an uncontracted horizontal weir that is 1.5 m high and 4.5 m long. The upstream depth is 2.2 m. Determine the weir height if the same discharge was desired without exceeding an upstream depth of 1.8 m? 9.4.3. Flow occurs over a sharp-crested, uncontracted horizontal weir 3.5 ft high under a head of 1.0 ft. If this weir replaced another uncontracted weir, which was one-half its height, what change in depth occurred in the upstream channel? (Note: The weir length is not needed to complete this analysis.) 9.4.4. Determine the discharge over a standard contracted weir when the upstream water depth is 2.1 m. The weir (one end contraction) is 1.8 m high and has a 1.8 m overflow length. Also determine the approach velocity head (see Figure 8.10a) in the 4-m-wide channel. Is it significant? 9.4.5. A 4-m-wide irrigation channel discharges water over a 1.7-m high contracted (both ends) horizontal weir (C = 1.86) with a 1-m-long horizontal crest. The upstream depth is 2.3 m. This weir is to be replaced by an uncontracted horizontal weir that will maintain the same upstream depth. Determine the height of the weir if the discharge coefficient is based on Equation (9.10b). 9.4.6. A mining company would like to replace an old flow monitoring weir. The contracted (one end), horizontal weir has a crest height of 23.5 ft, a crest length of 13 ft, and a discharge coefficient of 3.28 ft0.5/s. At design flow it operates with a head of 3.6 ft. Determine the crest height of a standard U.S.B.R. 90°-V-notch (replacement) weir that would discharge the same design flow without raising the upstream flow depth. 9.4.7. A 90°-V-notch weir (standard U.S.B.R.) is used in an irrigation channel to measure a flow rate of 0.295 m3/s. Determine the length of a contracted horizontal weir (standard U.S.B.R.) that would produce the same head at that discharge. 9.4.8. A broad-crested rectangular weir is 1 m high and has a crest (overflow) length of 3 m. The weir is constructed with a well-rounded upstream corner and a smooth surface. What is the discharge if the head is 0.4 m? 9.4.9. Determine the discharge (in m3/s) through a 6-ft Parshall flume if the gauge reading Ha is 0.701 m and Hb is 0.579 m.

370

Water Pressure, Velocity, and Discharge Measurements

Chap. 9

9.4.10. The flow through an 8-ft Parshall flume is 129 cfs when Ha is 2.50 ft. Determine the downstream water level Hb that would produce this flow rate. 9.4.11. Derive Equation (9.18) from the impulse-momentum Equation (9.16) showing all steps. 9.4.12. Equation (9.19) is the general expression for a broad-crested weir. Considering the limits of a weir from zero height (h = 0) to infinity (h S ∞ ), Equation (9.19) may vary from Q = 1.92LH 3/2 to Q = 1.36LH 3/2 with H in meters and Q in meters per seconds. Verify these expressions and find equivalent expressions for the BG system of units. 9.4.13. In Chapter 8, it was stated that an essential feature of weirs is that flow achieves critical depth passing over the weir. Since that is the case, derive Equation (9.9) using Equation (6.14), which relates critical depth to the flow rate. Hint: Critical depth will also have to be related to the head on the weir by an energy balance (ignoring losses). What does the C term represent? 9.4.14. Laboratory tests on a 60°-V-notch weir gave the following results: for H = 0.3 m, Q = 0.022 m3/s, and for H = 0.6 m, Q = 0.132 m3/s. Determine the discharge equation for this V-notch weir in both SI and BG forms.

Siavash Beik/Christopher B. Burke Engineering, LLC

Hydraulic Similitude and Model Studies

The use of small models for predicting the behavior of hydraulic structures dates back at least to Leonardo da Vinci.* But the methods developed for using the results of experiments conducted on a scaled model to predict quantitatively the performance of a full-size hydraulic structure (i.e., the prototype) were not realized fully until the beginning of the twentieth century. The principles on which the model studies are based constitute the theory of hydraulic similitude. In other words, are the hydraulic relationships in the prototype sufficiently similar to those in the model? Assessment of the appropriate physical quantities and fundamental hydraulic relationships (both static and dynamic) involved in the actual performance of the structure is known as dimensional analysis. All significant hydraulic structures are now designed and built after certain preliminary model studies have been completed. Such studies may be conducted for any one or more of the following purposes: 1. to determine the discharge coefficient of a large measurement structure, such as an overflow spillway or a weir; 2. to develop an effective method for energy dissipation at the outlet of a hydraulic structure; 3. to reduce energy loss at an intake structure or at a transition section; * Leonardo da Vinci (1452–1519), a genius, Renaissance scientist, engineer, architect, painter, sculptor, and musician. 371

372

Hydraulic Similitude and Model Studies

Chap. 10

4. to develop an efficient, economic spillway or other type of flood-releasing structure for a reservoir; 5. to determine an average time of travel in a temperature control structure, for example, in a cooling pond at a power plant; 6. to establish the best cross section, location, and dimensions of various structural components such as the breakwater, the docks, and the locks in harbor and waterway design; and 7. to determine the dynamic behaviors of the floating, semi-immersible, and bottom structures in transportation or offshore facilities. River models have also been extensively used in hydraulic engineering to determine 1. the pattern a flood wave travels through a river channel; 2. the effect of artificial structures such as bends, levees, dikes, jetties, and training walls on the sedimentation movements in the channel reach, as well as the impacts to upstream and downstream channel reaches; and 3. the direction and force of natural and anthropogenic currents in channels or harbors and their effect on navigation and marine life. 10.1 Dimensional Homogeneity When a physical phenomenon is described by an equation or a set of equations, all terms in each of the equations must be kept dimensionally homogeneous.* In other words, all terms in an equation must be expressed in the same units. In fact, to derive a relationship among several parameters involved in a physical phenomenon, one should always check the equation for homogeneity of units. If all terms in the relationship do not result in the same units on both sides of the equation, then one can be sure that pertinent parameters are missing or misplaced, or extraneous terms have been included. Based on the conceptual and physical understanding of the phenomenon and the principle of dimensional homogeneity, the solution of many hydraulic problems may be formulated. For example, we understand that the speed of surface wave propagation on water (C) is related to the gravitational acceleration (g) and the water depth (y). Generally, we may write C = f (g, y)

(10.1)

where f is used to express a function. The units of the physical quantities involved—length (L) and time (T)—are indicated in the brackets: C = [LT -1] g = [LT -2] y = [L] Because the left-hand side of Equation 10.1 has the units of [LT -1], those units must appear explicitly on the right-hand side as well. Thus, d and g must combine as a product and the function ( f) must be the square root. Therefore, as discussed in Chapter 6 (Equation 6.11).

C = 1gy

* There are a few exceptions such as empirical equations (e.g., Section 3.6).

Sec. 10.2

Principles of Hydraulic Similitude

Table 10.1

Quantity Length Area Volume Angle (radians) Time Discharge Linear velocity Angular velocity Acceleration Mass Moment of inertia Density Viscosity

373

Dimensions of Physical Quantities Commonly Used in Hydraulic Engineering

Dimension

Quantity

L L2 L3 None T L3T -1 LT -1 T -1 LT -2 M ML2 ML-3 ML-1T -1

Force Pressure Shear stress Specific weight Modulus of elasticity Coefficient of compressibility Surface tension Momentum Angular momentum Torque Energy Power Kinematic viscosity

Dimension MLT -2 ML-1T -2 ML-1T -2 ML-2T -2 ML-1T -2 M -1LT 2 MT -2 MLT -1 ML2T -1 ML2T -2 ML2T -2 ML2T -3 L2T -1

The dimensions of the physical quantities most commonly used in hydraulic engineering are listed in Table 10.1.

10.2 Principles of Hydraulic Similitude Similarity between hydraulic models and prototypes may be achieved in three basic forms: 1. geometric similarity, 2. kinematic similarity, and 3. dynamic similarity. Geometric similarity implies similarity of form. The model is a geometric reduction of the prototype and is accomplished by maintaining a fixed ratio for all homologous lengths between the model and the prototype. The physical quantities involved in geometric similarity are length (L), area (A), and volume (Vol). To keep the homologous lengths in the prototype (L p) and the model (L m), a constant ratio (L r) requires adherence to the following expression: Lp Lm

= Lr

(10.2)

An area (A) is the product of two homologous lengths; hence, the ratio of the homologous area is also a constant and can be expressed as Ap Am

L 2p L 2m

= L 2r

(10.3)

374

Hydraulic Similitude and Model Studies

Chap. 10

A volume (Vol) is the product of three homologous lengths. The ratio of the homologous volume can be expressed as Volp Volm

L 3p

L 3m

= L 3r

(10.4)

Example 10.1 A geometrically similar open-channel model is constructed with a 5:1 scale. If the model measures a discharge 7.07 cfs (ft3/s), what is the corresponding discharge in the prototype? Solution The discharge ratio can be found using the expression Q = AV, which requires the area and velocity ratios. The area ratio between prototype and model using Equation 10.3 is Ap Am

L 2p

L 2m

= L 2r = 25

The velocity ratio between the prototype and the model is Lp Vp Vm

Lp T = = Lr = 5 Lm Lm T

Note that, for geometric similarity, the time ratio from the prototype to the model remains unscaled. Accordingly, the discharge ratio is Qp ApVp = = (25)(5) = 125 Qm AmVm Thus, the corresponding discharge in the prototype is Qp = 125 Qm = 125(7.07) = 884 cfs

Kinematic similarity implies similarity in motion. Kinematic similarity between a model and the prototype is attained if the homologous moving particles have the same velocity ratio along geometrically similar paths. Thus, kinematic similarity involves the scale of time as well as length. The ratio of times required for homologous particles to travel homologous distances in a model and its prototype is Tp = Tr (10.5) Tm Velocity (V) is defined in terms of distance per unit time; thus, the ratio of velocities can be expressed as Lp Lp Vp Vm

Tp Lm Tm

Lm Lr = Tp Tr Tm

(10.6)

Sec. 10.2

Principles of Hydraulic Similitude

375

Acceleration (a) is defined in terms of length per unit time squared; thus, the ratio of homologous acceleration is Lp ap am

T 2p Lm

Lp =

T 2m

Lm T 2p T 2m

Lr T 2r

(10.7)

Discharge (Q) is expressed in terms of volume per unit time; thus, L 3p Qp Qm

Tp L 3m Tm

L 3p L 3m L 3r = = Tp Tr Tm

(10.8)

Kinematic models constructed for hydraulic machinery may frequently involve angular displacement (θ) expressed in radians, which is equal to the tangential displacement (L) divided by the length of the radius (R) of the curve at the point of tangency. The ratio of angular displacements may be expressed as Lp θp θm

Rp Lm Rm

Lp =

Lm Lm = = 1 Rp Lp Rm

(10.9)

Angular velocity (N) in revolutions per minute is defined as angular displacement per unit time; thus, the ratio θp Np Nm

Tp θm Tm

θp =

θm 1 = Tp Tr Tm

(10.10)

Angular acceleration (α) is defined as angular displacement per unit time squared; thus, θp αp αm

T 2p θm T 2m

θp =

θm T 2p T 2m

1 T 2r

(10.11)

Example 10.2 A 10:1 scale model is constructed to study the flow movement in a cooling pond. The designed discharge from the power plant is 200 m3/s, and the model can accommodate a maximum flow rate of 0.1 m3/s. What is the appropriate time ratio?

376

Hydraulic Similitude and Model Studies

Chap. 10

Solution The length ratio between the prototype and the model is Lr = The discharge ratio is Qr =

Lp Lm

= 10

200 = 2,000, and 0.1 L 3p Qr =

Qp Qm

= a

L 3m Tm

Lp Lm

b a

Tm b = L 3r T r-1 Tp

Substituting the length ratio into the discharge ratio gives the time ratio Tr =

Tp Tm

L 3r (10)3 = = 0.5 Qr 2,000

or Tm = 2Tp Therefore, a unit time period measured in the model is equivalent to two time periods in the prototype pond.

Dynamic similarity implies similarity in the forces involved in motion. Dynamic similarity between a model and its prototype is attained if the ratio of homologous forces (prototype to model) is kept at a constant value, or Fp Fm

= Fr

(10.12)

Many hydrodynamic phenomena may involve several different kinds of forces in action. Typically, models are built to simulate the prototype on a reduced scale and may not be capable of simulating all the forces simultaneously. In practice, a model is designed to study the effects of only a few dominant forces. Dynamic similarity requires that the ratios of these forces be kept the same between the model and the prototype. Hydraulic phenomena governed by various types of force ratios are discussed in Sections 10.3–10.6. Because force is equal to mass (M) multiplied by acceleration (a) and because mass is equal to density (ρ) multiplied by volume (Vol), the force ratio is expressed as Fp Fm

Mpap Mmam

ρpVolpap ρmVolmam

Lp =

ρp L 3p ρm L 3m

T 2p Lm T 2m

ρp L 4p 1 = = ρrL 4r T r-2 ρm L 4m T 2p T 2m

(10.13)

Sec. 10.2

Principles of Hydraulic Similitude

377

and the mass (force divided by acceleration) ratio may be expressed as Fp Mp

Fm am

Fp Fm = a = FrT 2r L r-1 p

(10.14)

Work is equal to force multiplied by distance, so the ratio of homologous work in dynamic similarity is Wp

FpL p FmL m

= FrL r

(10.15)

Power is the time rate of doing work; thus, the power ratio may be expressed as Wp Pp

Tp Wm Tm

Wp 1 FrL r = Tr Wm Tp Tm

(10.16)

Example 10.3 A 59,700-w (80-hp) pump is used to power a water-supply system. The model constructed to study the system has an 8:1 scale. If the velocity ratio is 2:1, what is the power needed for the model pump? Solution By substituting the length ratio into the velocity ratio, the time ratio is obtained: Vr =

Lr = 2; Tr

Tr =

Lr 8 = = 4 2 2

Lr = 8

Assume that the same fluid is used in the model and the prototype, because an alternative was not specified. Thus, ρr = 1, and the force ratio is calculated from Equation 10.13 as Fr =

ρrL 4r T r-2

(1)(8)4 (4)2

= 256

From Equation 10.16, the power ratio is Pr =

Fr L r (256)(8) = = 512 Tr (4)

and the power required for the model pump is Pm =

Pp Pr

59,700 = 117 W = 0.157 hp 512

378

Hydraulic Similitude and Model Studies

Chap. 10

Example 10.4 The model designed to study the prototype of a hydraulic machine must 1. be geometrically similar, 2. have the same discharge coefficient defined as Q/(A 22gH), and 3. have the same ratio of peripheral speed to the water-discharge velocity [ωD/(Q/A)]. Determine the scale ratios in terms of discharge (Q), head (H), diameter (D), and rotational angular velocity (ω). Solution It is important to recognize that although the energy head (H) is expressed in units of length, it is not necessarily modeled as a linear dimension. To have the same ratio of peripheral speed to the water discharge velocity, we have ωp Dp Qp/Ap

ωm Dm Qm /Am

or ωr Dr Ar T r-1L r L 2r = = 1 Qr L 3r T r-1 To have the same discharge coefficient, we have Qp/ ( Ap 22gHp )

Qm/ ( Am 22gHm )

L 3r T r-1 L 2r (gH)1/2 r

Qr Ar 2(gH)r

= 1

from which we get (gH)r =

L 2r T 2r

Because the gravitational acceleration (g) is the same for the model and prototype, we may write, Hr = L 2r T r-2 The other ratios asked for are Discharge ratio Qr : Diameter ratio: Angular speed ratio:

L 3r T r-1 Dr = L r, and ω = T r-1

10.3 Phenomena Governed by Viscous Forces: Reynolds Number Law Water in motion always involves inertial forces. When the inertial forces and the viscous forces can be considered to be the only forces that govern the motion, then the ratio of these forces acting on homologous particles in a model and its prototype is defined by the Reynolds number law: NR =

inertial force viscous force

(10.17)

Sec. 10.3

Phenomena Governed by Viscous Forces: Reynolds Number Law

379

The inertial forces defined by Newton's second law of motion, F = ma, can be expressed by the ratio in Equation 10.13: Fr = Mr

Lr T 2r

= ρr L 4r T r-2

(10.13)

The viscous force defined by Newton's law of viscosity, F = μa may be expressed by

Fr =

μp a

dV bA dL

dV b A dL p p

dV μm a b Am dL m

= μrL 2r T r-1

(10.18)

where μ is the viscosity and V denotes the velocity. Equating values of Fr from Equations 10.13 and 10.18, we get ρrL 4r T r-2 = μrL 2r T r-1 from which ρrL 4r T r-2 μrL 2r T r-1

ρrL 2r ρrL rVr = = 1 μr μrTr

(10.19)

Reformulating the above equation, we may write a

ρpL pVp μp

ρmL mVm a b μm ρpL pVp μp

= (NR)r = 1

ρmL mVm = NR μm

(10.20)

Equation 10.20 states that when the inertial force and the viscous force are considered to be the only forces governing the motion of the water, the Reynolds number of the model and the prototype must be kept at the same value. If the same fluid is used in both the model and the prototype, the scale ratios for many physical quantities can be derived based on the Reynolds number law. These quantities are listed in Table 10.2.

380

Hydraulic Similitude and Model Studies Table 10.2

Scale Ratios for the Reynolds Number Law (water used in both model and prototype, ρr = 1, μr = 1)

Geometric Similarity Length Area Volume

Chap. 10

Kinematic Similarity

Lr L 2r L 3r

Time Velocity Acceleration Discharge Angular velocity Angular acceleration

L 2r L r-1 L r-3 Lr L r-2

Dynamic Similarity Force Mass Work Power

1 L 3r Lr L r-1

L r-4

Example 10.5 To study a transient process, a model is constructed at a 10:1 scale. Water is used in the prototype, and it is known that viscous forces are the dominant ones. Compare the time, velocity, and force ratios if the model uses (a) water or (b) oil that is five times more viscous than water, with ρoil = 0.8ρwater. Solution (a) From Table 10.2, Tr = L 2r = (10)2 = 100 Vr = L r-1 = (10) -1 = 0.1 Fr = 1 (b) From the Reynolds number law, ρpL pVp μp

ρmL mVm μm

we have ρrL rVr = 1 μr Because the ratios of viscosity and density are, respectively, μr = ρr =

μp μm ρp ρm

μwater μwater = = 0.2 μoil 5μwater

ρwater ρwater = = 1.25 ρoil 0.8ρwater

From the Reynolds number law, Vr =

μr (0.2) = = 0.016 ρrL r (1.25)(10)

Sec. 10.4

Phenomena Governed by Gravity Forces: Froude Number Law

381

The time ratio is Tr =

Lr (10) = = 625 Vr (0.016)

Tr =

ρrL 2r Lr (1.25)(10)2 = = = 625 μr Vr (0.2)

The force ratio, then, is Fr = Fr =

ρrL 4r T 2r ρrL 4r T 2r

= =

(1.25)(10)4 (625)2 (ρrL 4r ) ¢

ρ2r L 4r μ2r

≤

= 0.032

μ2r (0.2)2 = = 0.032 ρr 1.25

The Tr and Fr equations that were solved first simplify the computations. However, the reformulated equations that rely on the Reynolds number law (given in terms of ρ and μ) demonstrate the importance of selecting the model fluid. The properties of the liquid used in the model, especially the viscosity, greatly affect the performance in the Reynolds number models.

10.4 Phenomena Governed by Gravity Forces: Froude Number Law In some flow situations, inertial forces and gravity forces are considered to be the only dominant forces. The ratio of the inertial forces acting on the homologous elements of the fluid in the model and prototype can be defined by Equation 10.13, restated as Fp Fm

= ρrL 4r T r-2

(10.13)

The ratio of gravity forces, represented by the weight of the homologous fluid elements involved, may be expressed as Fp Fm

Mpgp Mmgm

ρpL 3pgp ρmL 3mgm

= ρrL 3r gr

(10.21)

Equating the values from Equations 10.13 and 10.21, we obtain ρrL 4r T r-2 = ρrL 3r gr which can be rearranged as grL r =

L 2r T 2r

= V 2r

or Vr 1/2 g1/2 r Lr

= 1

(10.22)

382

Hydraulic Similitude and Model Studies Scale Ratios for the Froude Number Law (gr = 1, ρr = 1)

Table 10.3

Geometric Similarity Length Area Volume

Chap. 10

Kinematic Similarity Time Velocity Acceleration Discharge Angular velocity Angular acceleration

Lr L 2r L 3r

Dynamic Similarity L 1/2 r L 1/2 r 1 L 5/2 r L r-1/2 L r-1

Force Mass Work Power

L 3r L 3r L r4 L 7/2 r

Equation 10.22 may be expressed as a a

and hence, Vp 1/2 1/2 gp L p

Vp b 1/2 1/2 gp L p Vm

1/2 g1/2 m Lm

= (NF)r = 1

Vm 1/2 1/2 gm L m

= NF (Froude number)

(10.23)

In other words, when the inertial force and the gravity force are considered to be the only forces that dominate the fluid motion, the Froude number of the model and the Froude number of the prototype should be kept equal. If the same fluid is used for both the model and the prototype, and they are both subjected to the same gravitational force field, then many physical quantities can be derived based on the Froude number law. These quantities are listed in Table 10.3. Example 10.6 An open-channel model 30 m long is built to satisfy the Froude number law. What is the flow in the model for a prototype flood of 700 m3/s if the scale used is 20:1? Also determine the force ratio. Solution From Table 10.3, the discharge ratio is 2.5 Qr = L 5/2 = 1,790 r = (20)

Thus, the model flow should be Qm =

Qp Qr

700 m3/s = 0.391 m3/s = 391 L/s 1,790

The force ratio is Fr =

Fp Fm

= L 3r = (20)3 = 8,000

Sec. 10.5

Phenomena Governed by Surface Tension: Weber Number Law

383

10.5 Phenomena Governed by Surface Tension: Weber Number Law Surface tension is a measure of molecular energy on the surface of a liquid body. The resulting force can be significant in the motion of small surface waves or the control of evaporation from a large body of water such as a water storage tank or reservoir. Surface tension, denoted by σ, is measured in terms of force per unit length. Hence, the resulting force is F = σL. The ratio of analogous surface tension forces in prototype and model is Fr =

Fp Fm

σpL p σmL m

= σrL r

(10.24)

Equating the surface tension force ratio to the inertial force ratio (Equation 10.13) gives σrL r = ρr

L 4r T 2r

Rearranging gives Tr = a

ρr 1/2 3/2 b Lr σr

(10.25)

By substituting for Tr from the basic relationship of Vr = L r/Tr, Equation 10.25 may be rearranged to give Vr =

Lr ρr 1/2 3/2 a b Lr σr

= a

σr 1/2 b ρrL r

ρrV 2r L r = 1 σr

(10.26)

Hence, ρpV 2pL p σp

ρmV 2mL m = NW (Weber number) σm

(10.27)

In other words, the Weber number must be kept at the same value in the model and in the prototype for studying phenomena governed by inertial and surface tension forces. If the same liquid is used in the model and the prototype, then ρr = 1.0 and σr = 1.0, and Equation 10.26 can be simplified to V 2r L r = 1 or Vr =

1 L 1/2 r

(10.28)

384

Hydraulic Similitude and Model Studies

Chap. 10

Because Vr = L r/Tr, we may also write Lr 1 = 1/2 Tr Lr Thus, Tr = L 3/2 r

(10.29)

10.6 Phenomena Governed by Both Gravity and Viscous Forces Both gravity and viscous forces may be important in the study of surface vessels moving through water or shallow water waves propagated in open channels. These phenomena require that both the Froude number law and the Reynolds number law be satisfied simultaneously; that is, (NR)r = (NF)r = 1, or ρrL rVr Vr = μr (grL r)1/2 Assuming that both the model and the prototype are affected by the Earth's gravitational field (gr = 1) and because v = μ/ρ, the above relationship may be simplified to vr = L 3/2 r

(10.30)

This requirement can only be met by choosing a special model fluid with a kinematic viscosity ratio to water equal to the three-halves power of the scale ratio. In general, this requirement is difficult to meet. For example, a 1:10 scale model would require that the model fluid have a kinematic viscosity of 31.6 times less than that of water, which is obviously impossible. However, two courses of action may be available, depending on the relative importance of the two forces in the particular phenomenon. In the case of ship resistance, the ship model may be built according to the Reynolds model law and may operate in a towing tank in accordance with the Froude number law. In the case of shallow water waves in open channels, empirical relationships such as Manning's formula (Equation 6.4) may be used as an auxiliary condition for the wave measurements, according to the Froude number law. 10.7 Models for Floating and Submerged Bodies Model studies for floating and submerged bodies are performed to obtain information on 1. 2. 3. 4.

friction drag along the boundary of the moving vessel, form drag resulting from boundary flow separation caused by the body shape, forces expended in the generation of gravity waves, and stability of the body in withstanding waves and wave forces on the body.

The first two phenomena are strictly governed by viscous forces, and therefore the models should be designed according to the Reynolds number law. The third phenomenon is governed by the gravity force and must be analyzed by applying the Froude number law. All three measurements may be performed simultaneously in a towing tank filled with water. In analyzing the data, however, the friction forces and the form drag forces are first computed from the measurements by using known formulas and drag coefficients. The remaining force measured in towing

Sec. 10.7

Models for Floating and Submerged Bodies

385

the vessel through the water surface is the force expended in generating the gravity waves (wave resistance), and it is scaled up to the prototype values by the Froude number law. The analysis procedure is demonstrated in Example 10.7. For subsurface vessels such as submarines, the effect of surface waves on the vessels may be neglected. Hence, the Froude number model is not needed. To study stability and wave forces on stationary offshore structures, the effect of inertial force must be taken into consideration. The inertial force, defined as Fi = M′a, can be calculated directly from the prototype dimensions. Here, M′ is the mass of water displaced by the portion of the structure immersed below the waterline (also known as the virtual mass), and a is the acceleration of the water mass. Example 10.7 A ship model, with a maximum cross-sectional area of 0.780 m2 immersed below the waterline has a characteristic length of 0.9 m. The model is towed in a wave tank at the speed of 0.5 m/s. For the particular shape of the vessel, it is found that the drag coefficient can be approximated by CD = (0.06/N 0.25 R ) for 104 … NR … 106, and CD = 0.0018 for NR 7 106. The Froude number law is applied for the 1:50 model. During the experiment, a total force of 0.400 N is measured. Determine the total resistance force on the prototype vessel. Solution Based on the Froude number law (Table 10.3), we may determine the velocity ratio as 1/2 Vr = L 1/2 = 7.07 r = (50)

Hence, the corresponding velocity of the vessel is Vp = VmVr = 0.5(7.07) = 3.54 m/s The Reynolds number of the model is VmL m 0.5(0.9) = = 4.50 * 105 v 1.00 * 10-6

NR =

and the drag coefficient for the model is CDm =

0.06 (4.50 * 105)1/4

= 0.00232

1 The drag force on a vessel is defined as D = CD a ρAV 2 b, where ρ is the water density and A is the pro2 jected area of the immersed portion of the vessel on a plane normal to the direction of the motion. Thus, the model drag force can be calculated as

1 1 Dm = CDm a ρmAmV 2m b = (0.00232)(998)(0.78)[(0.5)2] = 0.226 N 2 2 The model wave resistance is the difference between the measured towing force and the drag force: Fw = 0.400 - 0.226 = 0.174 N For the prototype, the Reynolds number is NR =

Vp L p vp

VpL rL m vp

3.54(50)(0.9) 1.00 * 10-6

= 1.59 * 108

386

Hydraulic Similitude and Model Studies

Chap. 10

Thus, the drag coefficient of the prototype vessel is CDp = 0.0018, and the drag force is 1 1 Dp = CDp a ρp ApV 2p b = CDp a ρp AmL 2r V 2p b 2 2

1 Dp = 0.0018a (998)(0.780)(50)2(3.54)2 b = 21,900 N 2

The wave resistance on the prototype vessel is calculated by applying the Froude number law (Table 10.3): Fwp = FwrFwm = L 3r Fwm = (50)3(0.174) = 21,800 N Hence, the total resistance force on the prototype is F = Dp + Fwp = 21,900 + 21,800 = 43,700 N

10.8 Open-Channel Models Open-channel models may be used to study velocity–slope relationships and the effects of flow patterns on the changes in channel bed configuration. For the former applications, relatively long reaches of the river channel can be modeled. A special example is the U.S. Army Corps of Engineers—Waterways Experiment Station in Vicksburg, Mississippi, where the Mississippi River was once modeled on one site. In these applications, where changes in bed configuration are only of secondary concern, a fixed-bed model may be used. Basically, this model is used in studying the velocity–slope relationship in a particular channel; therefore, the effect of bed roughness is important. An empirical relationship, such as the Manning equation (Equation 6.4), may be used to assume the similarity between the prototype and the model:

Vr =

Vp Vm

1 2/3 1/2 R S np hp p 1 2/3 1/2 R S nm hm m

1 2/3 1/2 R S nr hr r

(10.31)

If the model is built with the same scale ratio for the horizontal dimensions (X) and vertical dimensions (Y), known as the undistorted model, then Rhr = Xr = Yr = L r

and

Sr = 1

and Vr =

1 2/3 L nr r

(10.32)

The Manning's roughness coefficient n ∝ R1/6 h (Equation 6.3), so we may write nr = L 1/6 r

(10.33)

This often results in a model velocity so small (or, conversely, the model roughness will be so large) that realistic measurements cannot be made. In addition, the model water depth may be so shallow that the physical characteristics of the flow may be altered. These issues may be

Sec. 10.8

Open-Channel Models

387

resolved by using a distorted model in which the vertical scale and the horizontal scale do not have the same value, usually, a smaller vertical scale ratio, Xr 7 Yr. This means that Sr =

Sp Sm

Yr 6 1 Xr

Hence, Sm 7 Sp, and the result is a larger slope for the model. The use of the Manning equation requires that the flow be fully turbulent in both model and prototype. Open-channel models involving problems of sediment transport, erosion, or deposit require movable bed models. A movable channel bed consists of sand or other loose material that can be moved in response to the forces of the current at the channel bed. Normally, it is impractical to scale the bed material down to the model scale. A vertical scale distortion is usually employed on a movable bed model in order to provide a sufficient tractive force to induce bed material movement. Quantitative similarity is difficult to attain in movable bed models. For any sedimentation studies performed, it is important that the movable bed model be quantitatively verified by a number of field measurements. Example 10.8 An open-channel model is built to study the effects of tidal waves on sedimentation movement in a 10-km river reach (the reach meanders in an area 7 km long). The mean depth and width of the reach are 4 m and 50 m, respectively, and the discharge is 850 m3/s. Manning's roughness value is np = 0.035. If the model is to be constructed in a laboratory room 18 m long, determine a convenient scale, the model discharge, and the model roughness coefficient. Verify that turbulent flow prevails in the model. Solution In surface wave phenomena the gravitational forces are dominant. The Froude number law will be used for the modeling. The laboratory length will limit the horizontal scale: Xr =

Lp Lm

7,000 = 389 18

We will use Xr = 400 for convenience. It is judged reasonable to use a vertical scale of Yr = 80 (enough to measure surface gradients). Recall that the hydraulic radius is the characteristic dimension in open-channel flow and that for a large width-to-depth ratio the hydraulic radius is roughly equal to the water depth. Thus, we can make the following approximation: Rhr = Yr = 80 Because NF =

Vr 1/2 1/2 gr Rhr

= 1

(10.22)

then 1/2 1/2 Vr = R1/2 hr = Y r = (80)

Using Manning's formula (or Equation 10.31), Vr =

Vp Vm

1 2/3 1/2 R S nr hr r

Sr =

Yr Xr

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we have

nr =

1/2 R2/3 hr S r

Y 2/3 r a

Yr Xr

Y 1/2 r

1/2

Y 2/3 r X 1/2 r

(80)2/3 (400)1/2

= 0.928

Hence, nm =

np nr

0.035 = 0.038 0.928

The discharge ratio is 3/2 Qr = ArVr = XrYrVr = XrY 3/2 = 2.86 * 105 r = (400)(80)

Thus, the model discharge required is Qm =

Qp Qr

850 2.86 * 105

= 0.00297 m3/s = 2.97 L/s

To use the Manning formula, turbulent flow must be ensured in the model. To verify the turbulent flow condition in the model, it is necessary to calculate the value of the model Reynolds number. The horizontal prototype velocity is Vp =

850 m3/s = 4.25 m/s (4 m)(50 m)

Hence, Vm =

Vp Vr

4.25 (80)1/2

= 0.475 m/s

The model depth is Ym =

Yp Yr

4 = 0.05 m 80

The model Reynolds number is NR =

VmYm (0.475)(0.05) = = 23,800 v 1.00 * 10-6

which is much greater than the critical Reynolds number (2,000). Hence, the flow is turbulent in the model.

10.9 The Pi Theorem Complex hydraulic engineering problems often involve many variables. Each variable usually contains one or more dimensions. In this section, the Pi theorem is introduced to reduce the complexity of these problems when coupled with experimental model studies. The Pi theorem relies on dimensional analysis to group several independent variables into dimensionless combinations, thereby reducing the number of control variables required in the experiment. In addition to variable reduction, dimensional analysis indicates those factors that significantly influence the phenomenon; thus it guides the direction in which the experimental (model) work should be conducted.

Sec. 10.9

The Pi Theorem

389

Physical quantities in hydraulic engineering may be expressed either in the force–length– time (FLT) system or in the mass–length–time (MLT) system. These two systems are connected by Newton's second law, which states that force equals mass times acceleration, or F = ma. Through this relationship, conversion can be made from one system to the other. The steps in dimensional analysis can be demonstrated by examining a simple flow phenomenon such as the drag force exerted on a sphere as it moves through a viscous fluid. Our general understanding of the phenomenon is that the drag force is related to the size (diameter) of the sphere (D), the velocity of the sphere (V), the viscosity of the fluid (μ), and the fluid density (ρ). Thus, we may express the drag force as a function of D, V, μ, and ρ, or we may write: Fd = f (D, V, ρ, μ) A generalized approach to dimensional analysis of the phenomenon may be made through the Buckingham Pi theorem. This theorem states that if a physical phenomenon involves n dimensional variables in a dimensionally homogenous equation described by m fundamental dimensions, then the variables may be combined into (n - m) dimensionless groups for analysis. For drag force on a moving sphere, a total of five dimensional variables are involved. The previous equation may thus be expressed as f′(Fd, D, V, ρ, μ) = 0 These five variables (n = 5) are described by the fundamental dimensions, M, L, and T (m = 3). Because n - m = 2, we can express the function using two Π groups: ∅(Π1, Π2) = 0 The next step is to arrange the five dimensional parameters into two dimensionless Π groups. This is accomplished by selecting m-repeating variables (for this problem, we need three repeating variables) that will appear in each of the dimensionless Π groups. The repeating variables must contain all m-dimensions, must be independent (i.e., cannot be combined to form a dimensionless variable of their own), and should be the most dimensionally simple of all the variables in the experiment. In this case, we will select the following three repeating variables: sphere diameter (simple dimensionally and contains the length dimension), velocity (simple dimensionally and contains the time dimension), and density (the simplest variable left that contains the mass dimension). At this point, the three repeating variables are combined with the two nonrepeating variables to form the two Π groups. Π1 = DaV bρcμd Π2 = DaV bρcF dD The values of the exponents are determined by noting the Π groups are dimensionless, and they can be replaced by M 0L0T 0. Because most hydraulic studies involve certain common dimensionless groups such as the Reynolds number, the Froude number, or the Weber number, as discussed in the previous sections in this chapter, one should always look for them in performing the dimensional analysis. To determine the Π1 group, we write the dimensional expression: L b M c M d M 0L0T 0 = (L)a a b a 3 b a b T LT L

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where the dimensions are found in Table 10.1. Based on this algebraic relationship: for M: for L: for T:

0 = c + d 0 = a + b - 3c - d 0 = -b - d

There are four unknowns in the three conditions given. We can always solve for three of them in terms of the fourth—say, d. From the equations above, we have: c = -d, b = -d, and, by substitution, a = -d Thus, Π1 = D -dV -dρ-dμd = a

μ d DVρ -d b = a b μ DVρ

where the dimensionless variable combination results in the Reynolds number (NR). Working in similar fashion with the Π2 group, we get Π2 =

FD ρD2V 2

Note that the dimensions for the drag force in the MLT system are (ML/T2) based on Newton's second law, F = ma as displayed in Table 10.1. Finally, returning to the original condition that ∅(Π1, Π2) = 0, we may write Π1 = ∅′(Π2) or Π2 = ∅″(Π1). Thus, FD ρD2V 2

= ∅″(NR)

where ∅″ is the undefined function we are seeking. In other words, the dimensionless grouping of variables on the left side of the equation, which includes the drag force, is a function of the Reynolds number. We have now reduced the original problem from five variables to two dimensionless variables that contain the original five variables. Now the full benefit of the Pi theorem should be evident to the reader. Based on the original problem formulation, a five-variable experiment would have been difficult to set up and tedious to analyze. However, the new two variable experiment is much less complex. To find the appropriate relationship or function, ∅″, an experiment can be devised whereby the drag force is measured as the Reynolds number is changed. The resulting data can be graphed [FD/(ρD2V 2) vs. NR] and analyzed by statistical software to define the functional relationship. Incidentally, the Reynolds number can be changed easily by altering the experimental velocity [because NR = (ρDV)/μ] without changing the experimental fluid (required to change ρ and μ), a needlessly tedious endeavor. It should be emphasized that dimensional analysis does not provide solutions to a problem; rather, it provides a guide to point out the relationships among the parameters that are applicable to the problem. If one omits an important parameter, the results are incomplete and thus can lead to incorrect conclusions. However, if one includes parameters that are unrelated to the problem, additional dimensionless groups that are irrelevant to the problem will result. Thus, the successful application of dimensional analysis depends, to a certain degree, on the engineer's

Sec. 10.9

The Pi Theorem

391

basic understanding of the hydraulic phenomenon involved. These considerations can be demonstrated by the following example problem.

Example 10.9 A broad crested weir model is designed to study the discharge per foot of the prototype. Derive an expression for this discharge using the Buckingham Pi theorem. Assume that the overflowing water sheet is relatively thick so that the surface tension and thus viscosity of the fluid can be neglected. Solution Based on our general understanding of the phenomenon, we may assume that the discharge (q) would be affected by the spillway head (H), the gravitational acceleration (g), and the spillway height (h). Thus, q = f (H, g, h) or f′(q, H, g, h) = 0. In this case, n = 4, m = 2 (not 3, because none of the terms involve mass). According to the Pi theorem, there are n - m = 2 dimensionless groups, and ∅(Π1, Π2) = 0 Based on the rules used to guide the selection of repeating variables, we will use spillway head (which is simple dimensionally and contains the length dimension) and spillway discharge (which is simple dimensionally and contains the time dimension). Note that we could not use spillway height as a repeating variable once spillway head was selected because they can be combined to form a dimensionless parameter (i.e., h/H). However, gravitational acceleration (g) could have been selected as the second repeating variable without affecting the problem solution. (The reader is encouraged to complete Problem 10.9.2 to verify that this statement is accurate.) Using q and H as the basic repeating variables, we have Π1 = qa1H b1gc1 Π2 = qa2H b2hc2 From the Π1 group, we have L0T 0 = a

Thus, L: T:

L3 a1 b1 L c1 b L a 2b TL T

0 = 2a1 + b1 + c1 0 = -a1 - 2c1

Hence, 1 c1 = - a1 2

3 b1 = - a1 2 3

From the Π2 group we have

Π1 = qa1H -2 a1g-2 a1 = a L0T 0 = a

q g1/2H

L3 a2 b2 c2 b L L TL

b 3/2

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Chap. 10

Thus, L: T:

0 = 2a2 + b2 + c2 0 = -a2

Hence, a2 = 0 b2 = -c2 Π2 = q0H -c2hc2 = a

h c2 b H

Note that this dimensionless variable would not have resulted if both h and H were used as repeating variables. Now that the two dimensionless groups are identified to be

we may return to

q 1/2

g H

3/2

and

∅(Π1, Π2) = ∅a

q 1/2

g H

3/2

or q 1/2

g H

3/2

= ∅′ a

h b H h b = 0 H

h b H

q = g1/2H 3/2 ∅′ a

h b H

The result indicates that the discharge per unit length of the spillway is proportional to 2g and H 3/2. The flow rate is also influenced by the ratio (h/H), as shown in Chapter 9 (Equation 9.19).

Problems (secTion 10.2) 10.2.1. A 650 m long channel carries irrigation water at an average velocity of 2 m/s. The trapezoidal channel has a 2 m depth, 4.25 m bottom width, and 1:1 side slopes. Determine the channel area, velocity, and discharge of a geometrically similar model constructed with a 10:1 scale. 10.2.2. A rectangular irrigation channel is 16 ft wide and flows at a uniform depth of 4.0 ft while on a slope of 0.001. The Manning coefficient is 0.025. Determine the maximum depth and width of a geometrically similar channel if the flow in the model must be limited to 1.07 cfs (ft3/s). 10.2.3. A model study is proposed for a gated outlet device. The outlet is governed by the orifice equation: Q = CdA(2gh)1/2, Equation (8.19) where Cd is the coefficient of discharge. The prototype is a flood control reservoir, and the gated outlet is used to regulate water levels. The model reservoir (1:100 scale) is drained in 18.3 minutes. How many hours should it take to drain the prototype? Assume identical discharge coefficients for the prototype and the model. 10.2.4. A 328-ft-long overflow spillway will discharge flood water from a reservoir under a permitted maximum head of 9.85 ft. The operation of the prototype spillway is studied on a 1:50 scale model in a hydraulic laboratory. The spillway discharge is governed by the equation: Q = CLH 3/2, where C is the coefficient of discharge, L is the spillway length, and H is the head on the spillway. If the

Problems

393

model discharge is 2.30 cfs (ft3/s), determine the discharge in the prototype. Assume the discharge coefficient ratio between prototype and model is one. 10.2.5. A model study is being proposed for a 3.20-mile-long, shipping channel that has experienced a large accumulation of sediment. However, the available model length is only 70 ft. The shipping channel has a design flow rate of 2,650 cfs (ft3/s). Select a reasonable length scale and determine the model discharge if a time ratio of 10 is used. Also determine the model velocity if the channel velocity is 2.76 ft/s. 10.2.6. An overflow spillway with a 100-m-long crest will convey a design discharge of 1,150 m3/s under a permitted maximum head of 3.00 m. The operation of the prototype spillway is studied on a 1:25 scale model in a hydraulic laboratory. The time ratio on the model is L r 1/2. The model velocity measured at the end (toe) of the spillway is 3.00 m/s. Determine the flow rate in the model and the Froude numbers (NF = V/(gd)1/2 where d is the flow depth) at the toe of the spillway in the model and the prototype. 10.2.7. A sea wall has been proposed to dissipate wave forces on a beach front. A 3-ft-long, 1:30 model is used to study the effects on the prototype. If the total force measured on the model is 0.510 lb and the velocity scale is 1:10, determine the force per unit length on the prototype. Also determine the discharge ratio between the model and the prototype. 10.2.8. A 1:25 scale model is being designed to study a prototype hydraulic structure. The velocity ratio between the model and the prototype is 1:5, and the measurement accuracy is required to be within 1% of the total force. Determine the model force and the accuracy of the force measurement in the model if the expected force on the prototype is 45,000 N. 10.2.9. A 1:50 scale model is used to study the power requirements of a prototype submarine. The model will be towed at a speed 50 times greater than the speed of the prototype in a tank filled with sea water. Determine the conversion ratios from the prototype to the model for the following quantities: (a) time, (b) force, (c) power, and (d) energy. 10.2.10. The moment exerted on a gate structure is studied in a laboratory water tank with a 1:125 scale model. If the moment measured on the model is 1.5 N # m on the 1-m long gate arm, determine the moment exerted on the prototype. (Hint: In this case, the acceleration term from F = ma is the gravitational acceleration and thus Equation (10.13) needs to be adjusted.)

(secTion 10.3) 10.3.1.

For a particular model study, the inertial forces and viscous forces are known to govern the motion. If the model and prototype both use water, verify the scale ratios in Table 10.2 for: (a) time (Tr = L r2), (b) acceleration (ar = L r-3), (c) angular velocity (Nr = L r-2), (d) force (Fr = 1), and (e) power (Pr = L r-1). 10.3.2. The motion of a submarine is being studied in a laboratory. The prototype speed of interest is 5 m/s in the ocean. Inertial forces and viscous forces govern the motion. At what speed must a 1:10 model be towed to establish similarity between the model and the prototype? Assume that the sea water and towing tank water are the same. 10.3.3. The moment exerted on a ship's rudder is studied with a 1:20 scale model in a water tunnel using the same temperature water as the river water. Inertial and viscous forces govern the fluid motion. If the torque measured on the model is 10 N # m for a water tunnel velocity of 22 m/s, determine the corresponding water velocity and torque (moment) for the prototype. 10.3.4. A coffer dam will be built in an ocean harbor to facilitate the construction of bridge pilings. This temporary dam must resist 16 ft/s harbor currents. A 1:20 model is proposed within a water tunnel using sea water that is the same density (SG = 1.03) and temperature (40 °F) as that measured

394

Hydraulic Similitude and Model Studies

Chap. 10

in the harbor. Inertial forces and viscous forces are dominant. What current speed must the water tunnel provide in order to study the force load on the coffer dam? If the required water tunnel velocity is judged to be impractical, can the study be performed in a wind tunnel using air at 40 °F? Determine the air speed in the wind tunnel. 10.3.5. A 4-ft diameter oil pipeline is being proposed for a remote location. The oil has a specific gravity of 0.8 and a dynamic viscosity of 9.93 * 10-5 [email protected]/ft2. A model will by used to study the pipeline flow conditions using a 0.5-ft diameter pipe and water at normal conditions (68.4 °F). If the design flow rate in the prototype is 125 cfs, determine the required flow in the model. 10.3.6. A scale model is used to study the flow of hot water in a large pipeline. The 1:5 scale model uses water at 20°C. The prototype discharges 11.5 m3/s of water at 90°C. Determine the model discharge.

(secTion 10.4) 10.4.1. Verify the scale ratios given in Table 10.3 (Froude Number Law) for (a) velocity, (b) time, (c) acceleration, (d) discharge, (e) force, and (f) power. 10.4.2. A tidal basin is to be studied where inertial and gravity forces dominate. If a 1:500 scale tidal basin model is used, what length of time (in hours) in the model represents a period of one week in the prototype? 10.4.3. A ship 300 ft long designed to travel at a top speed of 3 ft/s is to be studied in a towing tank with a 1:50 scale model. Determine what speed the model must be towed to adhere to (a) the Reynolds number law and (b) the Froude number law. 10.4.4. An ogee spillway has a design flow rate of 400 m3/s. An energy dissipater is being designed to force a hydraulic jump at the end of a spillway channel. The initial depth of flow in the 30-m wide prototype is calculated to be 0.80 m. Assuming inertial and gravity forces are dominant, determine the discharge of the 1:10 scale model and the velocity in the prototype and model. 10.4.5. A 120-m-long overflow spillway will discharge flood water from a reservoir with a permitted maximum head of 2.75 m. The operation of the prototype spillway is studied on a 1:50 scale model in a hydraulic laboratory. Assuming inertial and gravity forces dominate, determine the discharge and spillway crest depth (critical) of the prototype if the model discharge is 67.9 L/s. In addition, assume the spillway discharge is governed by the equation: Q = CLH 3/2, where C is the coefficient of discharge, L is the spillway length, and H is the spillway head. If the discharge coefficient of the model crest measures 2.19, what is the prototype crest discharge coefficient? 10.4.6. A 1:25 model is built to study a stilling basin at the outlet of a steep spillway chute assuming inertial and gravity forces are dominant. The stilling basin consists of a horizontal floor (apron) with U.S.B.R. Type II baffles installed to stabilize the location of the hydraulic jump. The prototype has a rectangular cross section 82.0 ft wide designed to carry a 2,650 cfs discharge. The velocity immediately before the jump is 32.8 ft/s. Determine the following: (a) the model discharge; (b) the velocity in the model immediately before the jump; (c) the Froude number of the prototype and the model at this location; and (d) the prototype depth downstream of the jump (see Section 8.10, Figure 8.21).

Problems

395

(secTion 10.5) 10.5.1. Determine the surface tension of a liquid in the prototype if a time ratio of 2 is established with a 1:5 scale model. The surface tension of the liquid in the model is 130 dyn/cm. Also determine the force ratio? Assume the densities of the fluids in the prototype and the model are approximately the same. 10.5.2. A model is built to study the surface tension phenomena in a reservoir. Determine the conversion ratios between the model and the prototype for the following quantities if the model is built with a 1:100 scale (a) rate of flow, (b) energy, (c) pressure, (d) power. The same fluid is used in the model and prototype. 10.5.3. A measuring device includes certain small glass tubes of a given geometry. To study the surface tension effect, a 3:1 scale model (larger than prototype) is built. Determine the discharge ratio and the force ratio assuming the same liquid is used in the model and the prototype.

(secTion 10.7) 10.7.1. A wave tank is built to study the forces on a floating caisson. The prototype is 200 ft wide, 400 ft long, and 40 ft high. It will be towed in sea water (20 °C) in the longitudinal direction to an offshore construction site where it will be sunk. The calculated floating depth of the caisson is 26 ft, with 14 ft remaining above the water surface. If a 1:100 model is towed in a wave tank using sea water, what is the model speed that corresponds to the prototype speed of 5 ft/s? The Froude number law is applied. Also, determine the Reynolds number of the model if it is built according to the Reynolds number law, 10.7.2. The wave resistance on a ship is desired. A 1:250 model is towed in a wave tank and a wave resistance of 2.5 pounds is measured. Determine the corresponding wave resistance on the prototype. 10.7.3. A small barge 10 m long moves at 1.5 m/s in freshwater at 20°C. A 1:5 scale model of the prototype barge is to be tested in a towing tank containing a liquid of specific gravity 0.90. What viscosity must this liquid have for both Reynolds and Froude number laws to be satisfied? 10.7.4. A model barge (1 m long) is tested in a towing tank at a speed of 1 m/s. Determine the prototype velocity if the prototype is 150 m in length. The model has a 2-cm draft and is 10-cm wide. The drag coefficient is CD = 0.25 for NR 7 5 * 104, and the towing force required to tow the model is 0.3 N. What force is required to tow the barge in waterways?

(secTion 10.8) 10.8.1. As you recall, the laboratory site used to build a model in Example 10.8 was limited in length to 18 m. A new laboratory site has no length limit, but the roughness coefficient of the material to be used in the movable bed is nm = 0.021. Determine the appropriate horizontal scale (using the same vertical scale) and the corresponding model velocity. 10.8.2. A model is built to study flow in a stream segment. The stream has an average depth of 1.2 ft and is roughly 20 ft wide with a flow rate of 94.6 cfs. An undistorted model with a 1:200 scale is constructed to study the velocity-slope relationship. If the reach has a Manning's coefficient of 0.045, determine the model values of roughness and velocity.

396

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10.8.3. A 1:100 scale model is constructed to study the pattern of flow in a river reach. If the river reach has a Manning's coefficient n = 0.025, determine (a) the corresponding value of n in the model, and (b) the value of n in the model if the vertical scale is exaggerated to 1:25. (Assume that since the river has a large width-to-depth ratio, the hydraulic radius is roughly equal to the water depth.) 10.8.4. Determine the model roughness coefficient, velocity, and flow rate for Example 10.8 if the vertical scale used was 400, just like the horizontal scale. Are the values obtained for the model reasonable? Will the model flow remain fully turbulent? 10.8.5. A ship channel model is built to study sedimentation issues in the prototype which is 460 ft wide and 25 ft deep and carries a discharge of 31,800 cfs. For a vertical scale of 1:100 and a roughness coefficient of np = 0.03, determine the appropriate horizontal scale if nm = 0.02. (Assume for the large width-to-depth ratio, the hydraulic radius equals the channel depth.)

(secTion 10.9)

10.9.1. Equation 5.3 (P = ω # T) relates the power developed by a motor to the torque and rotational speed. Derive the essence of this expression using the Buckingham Pi theorem. (Hint: Use the FLT system of units instead of the MLT system.) 10.9.2. Example 10.9 utilized the Buckingham Pi theorem to derive and expression for the discharge of water over a broad crested weir. Repeat the analysis using the weir head and gravitational acceleration as repeating variables. 10.9.3. Use the Buckingham Pi-theorem to derive an expression for the drag force (FD) exerted on a submarine. The drag force is impacted by the submarine length, B, the velocity of the submarine, V, and the viscosity, μ, and density, ρ, of sea water. 10.9.4. Steady flow of an incompressible, Newtonian fluid occurs through a long, smooth-walled horizontal pipe. Use the Buckingham Pi-theorem to derive an expression for the pressure drop that occurs per unit length of pipe (∆P1) using the pipe diameter (D), the pipe velocity (V), and the fluid density (ρ) as repeating variables. The fluid viscosity (μ) is the only other pertinent variable. (Hint: Use Newton's 2nd Law to determine the units for ∆P1 and μ in the MLT system.) 10.9.5. The mean speed V of a liquid moving down an open channel is believed to depend upon: depth d, gravitational acceleration g, roughness height e, liquid density ρ, viscosity μ, and channel slope (sin θ). Find the dimensionless parameters that may affect the coefficient k in the formula V = k2dg sin θ. Use depth, gravitational acceleration, and density as repeating variables in the MLT system. 10.9.6. Determine an expression for the velocity (V) of an air bubble rising through a stationary liquid. The pertinent variables are bubble diameter (D), gravitational acceleration (g), viscosity (μ), density (ρ), and surface tension (σ). Use bubble diameter, density, and viscosity as repeating variables and the MLT unit system.

Robert Houghtalen

Hydrology for Hydraulic Design

Despite their apparent similarity, the terms hydrology and hydraulics should not be confused. As previously stated in Chapter 1, hydraulics is a branch of engineering that applies fluid mechanics principles to problems dealing with the collection, storage, control, transport, regulation, measurement, and use of water. In contrast, hydrology is a science dealing with the properties, distribution, and circulation of the Earth's water. Thus, hydrology generally refers to natural processes, whereas hydraulics generally refers to processes designed, constructed, and controlled by humans. The hydrologic cycle, or the circulation of the Earth's water, is a complex process with many subcycles, so a brief overview may be helpful. Water from the oceans evaporates by absorbing energy from the sun. This increases the water vapor in the overlying air mass. Condensation and precipitation occur when this vapor-laden air mass cools, usually because it rises in the atmosphere. If the precipitation occurs over land, the water can take a number of paths. Some is caught by buildings, trees, or other vegetation (interception). Most of this water eventually evaporates back into the atmosphere. Rainfall that makes it to the ground is stored in depressions, infiltrates into the ground, or runs off the land driven by gravitational forces. The depression storage water either infiltrates or evaporates. Infiltration water is either held in the soil pores or moves downward to the water table. Water held in soil pores can be used by plants and released back to the atmosphere through the process of transpiration. Water that 397

398

Hydrology for Hydraulic Design

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drains down to the groundwater aquifer often ends up in rivers and eventually the ocean. This is also the ultimate destination of surface runoff water. Figure 11.1 is a simple depiction of the process. Even though hydrology and hydraulics represent distinct disciplines, they are inexorably linked in engineering practice. Many hydraulic projects require a hydrologic study to establish the design flow rate (Q). Indeed, the design flow rate is critical in establishing the appropriate size and design of many hydraulic structures. For example, rainfall events result in water flowing over the land surface to natural or constructed channels. The design of stormwater pipes, channels, ponds, and low-impact development devices is predicated on establishing appropriate design flow rates for these structures. Where runoff data are available, the statistical methods discussed in Chapter 12 can be used to determine a design flow rate. However, at most project sites, only rainfall data are available. In that event, we would have to use hydrologic methods to determine the design runoff rate using the available rainfall information. This chapter is not meant to be a comprehensive coverage of hydrology. Entire books are devoted to that purpose. However, to better understand earlier chapters on open-channel flow and hydraulic structures, an introduction to hydrologic concepts and design methods is appropriate. This will give the reader an appreciation for the level of effort required to arrive at the flow rate on which many hydraulic analyses and designs depend. In addition, some hydrologic methods are presented to equip the reader with design tools that are popular and effective in establishing design flow rates. The hydrologic methods presented are primarily appropriate for small urban watersheds. A majority of the hydraulic design activity done in the United States is for hydraulic infrastructure in the urban environment.

Atmosphere Condensation

Precipitation

Snow pack Melt Depression storage

Interception Lake

Infiltration

Surface Transpiration runoff Evaporation (rivers & lakes) Soil River

Water table Confining layer

Aquifer

Figure 11.1

Hydrologic cycle

Evaporation (ocean) Radient energy

Evaporation (land)

Ocean

Sec. 11.1

Watershed Delineation

399

Flood protection hydraulic structures, including those in urban areas, are designed to control stormwater runoff. Therefore, a runoff event should be selected as the basis for designing such a structure. It is presumed that a hydraulic structure will function properly if it can accommodate the design runoff at full capacity. However, the structure would fail to function as intended if the magnitude of the design event is exceeded. Due to the uncertainty inherent in hydrologic events, there is always a finite probability that a design runoff event will be exceeded during the service life of a structure. Statistical methods are used to determine the exceedance probabilities associated with the different design runoff magnitudes. (See Chapter 12 for a more thorough coverage.) Since historical runoff data are unavailable at most sites, rainfall data are used to generate the design runoff. Two fundamentally different approaches can be used for this purpose. In the continuous simulation approach, chronological records of rainfall are converted to a chronological record of runoff by use of a mathematical rainfall–runoff model. The resulting chronological runoff record is then analyzed statistically (Chapter 12) to determine a design runoff. In the single-event approach, a historical rainfall record is analyzed to select a design storm. This storm event is then used to generate the design runoff by use of a mathematical rainfall–runoff model. A key assumption in this approach is that a selected design storm and the runoff resulting from this storm have the same probability of exceedance. Because the continuous simulation approach is time-consuming and costly, the single-event design-storm method is widely used in the practice. The single-event approach is adopted in this chapter. The order of presentation of the material in this chapter follows the order of the tasks performed to determine a design runoff event: delineate the watershed of the hydraulic structure to be designed, select a design storm, calculate the losses from rainfall to determine the effective rainfall that will become runoff, and generate a design runoff hydrograph. Applications involving various urban structures are also included. 11.1 Watershed Delineation The land area that contributes to flow at a hydraulic structure is usually called the watershed, catchment, or the drainage basin of that structure. In other words, if the rainfall excess produced at some land surface location eventually contributes to the flow at a hydraulic structure; this land surface location is included in the watershed of that structure. The location of the structure is usually referred as the design point, the watershed outlet, or the basin outlet. An accurate topographic map is necessary to properly delineate a watershed boundary. Figure 11.2 depicts the delineation procedure for a stream named Nelson Brook. The watershed is delineated in two steps: 1. identify the design point with a circle on the topographic map and designate all peak elevations with X's close to the design point and upstream to the headwaters of the watershed [Figure 11.2 (a)], and 2. circumscribe the stream with a watershed boundary that passes through the peak points and the design point [Figure 11.2 (b)]. The watershed boundary should intersect contours in a perpendicular fashion. Note that any significant precipitation that occurs within the watershed boundary will produce surface runoff that eventually flows into Nelson Brook and through the design point. Any significant

400

Chap. 11

ls Ne

n lso Ne

Kimball Hill

O U G H

Hydrology for Hydraulic Design

Kimball Hill Manroo Hill (a)

Manroo Hill (b)

Figure 11.2 Watershed delineation: (a) peak point identification; (b) circumscribing boundary Source: Adapted from the Natural Resource Conservation Service (http://www.nrcs.usda.gov/Internet/ FSE_DOCUMENTS/nrcs144p2_014819.pdf)

precipitation outside the watershed boundary will produce surface runoff that does not end up passing through the design point. It is often helpful to draw arrows on the topographic map that represents the flow direction of surface runoff, passing through the contour lines in a perpendicular fashion from higher elevations to lower elevations. From these flow arrows, it is easy to see what area contributes flow to the design point. Watershed delineation is somewhat intuitive. It helps to visualize the topography from the elevation contours, and your ability to do this improves with practice. Modern geographic information systems will often perform the watershed delineation task automatically once the design point is designated, but visual checking and field verification help to avoid program malfunctions in flat or disturbed terrain. 11.2 Design Storms A design storm is characterized by a return period, storm duration, the total depth or average intensity of rainfall, time distribution of rainfall, and spatial distribution of rainfall. In most design problems, the spatial variation is neglected, especially in small watersheds. Return period is defined as the average number of years between occurrences of a hydrologic event with a certain magnitude or greater. The inverse of the return period represents the probability that this magnitude will be exceeded in any given year. For example, if a 25-year event (or an event that has a return period of 25 years) is selected as the design event, then there is a 1/25 = 0.04 = 4% probability that the design event will be exceeded in any given year. The rainfall intensity refers to the time rate of rainfall. The total depth of rainfall is the depth to which the rainwater would accumulate if it stayed where it fell on the ground. The average intensity is equal to the total depth of rainfall divided by the storm duration.

Sec. 11.2

Design Storms

11.2.1

401

Storm Hyetograph

Rainfall intensity varies over the storm duration, and a plot of rainfall intensity versus time is called a hyetograph. Although the variation of rainfall intensity can best be represented by a continuous function, we often assume that the intensity remains constant within fixed time intervals. We sometimes present the rainfall hyetographs in terms of the cumulative rainfall or in the form of a dimensionless curve. All three types of hyetographs are shown in Figure 11.3.

3.5 3.0 i (cm/hr)

2.5 2.0 1.5 1.0 0.5 0

0.5

1.0

1.5

2.0

2.5

1.5

2.0

2.5

0.75

1.00

1.25

t (hr) (a) 3.0 2.5 P (cm)

2.0 1.5 1.0 0.5 0

0.5

1.0 t (hr) (b)

1.0

P/PT

0.8 0.6 0.4 0.2 0

0.25

0.50 t/td (c)

Figure 11.3 Rainfall hyetographs

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Table 11.1 Example Calculations of a Storm Hyetograph

(1) t (hr)

(2) i (cm/hr)

(3) ∆P (cm)

0 0.6

0.3

3.0

1.5

1.8

0.9

0.6

0.3

0.5 1.0 1.5 2.0

(4) P (cm)

(5) t/td

(6) P/PT

0.3

0.25

0.1

1.8

0.50

0.6

2.7

0.75

0.9

3.0

1.0

Suppose the intensity of a 2-hr rain is 0.6, 3.0, 1.8, and 0.6 cm/hr during the first, second, third, and fourth half-hour periods, respectively, as tabulated in columns 1 and 2 of Table 11.1. The hyetograph shown in Figure 11.3a represents this rain in graphical form. The incremental rainfall, ∆P, produced over each half-hour period, ∆t, is given in column 3. For example, between t = 1.0 and 1.5 hr, the amount of rain falling on the ground is ∆P = i # ∆t = (1.8 cm/hr) (0.5 hr) = 0.9 cm. The entries in column 4 represent the cumulative rainfall, which is zero at t = 0 hr. At t = 0.5 hr, P = 0.0 + 0.3 = 0.3 cm. Likewise, at t = 1.0 hr, P = 0.3 + 1.5 = 1.8 cm. The cumulative rainfall of 3.0 cm when the rain stops at t = 2 hr is the total depth of rainfall. Note that this value is obtained by summing up all the ∆P values listed in column 3. The entries in column 5 are obtained by dividing the t values of column 1 by td = 2 hr, the storm duration. Similarly, column 6 entries are obtained by dividing the P values of column 4 by PT = 3.0 cm, the total depth of precipitation. Figure 11.3b is a graphical representation of column 1 versus column 4. Likewise, Figure 11.3c is a graphical representation of column 5 versus column 6. In this example, the average intensity is calculated as the total depth divided by the duration or (3.0 cm)/(2 hr) = 1.5 cm/hr. 11.2.2

Intensity–Duration–Return Period Relationships

Frequency analysis techniques are used to develop the relationships between the average intensity, storm duration, and the return period from local historical rainfall data. These relationships are readily available in chart or graphical form at most locations. Often these graphs are referred to as the intensity–duration–frequency (IDF) curves. Typical IDF curves are displayed in Figure 11.4. Where these curves are not available, the procedure described in Chapter 12 can be used to develop the average intensity–duration–return period relationships. Alternatively, one can use the regional IDF relationships such as those shown in Figure 11.5. 11.2.3 Design-Storm Selection An appropriate design return period should first be selected depending on the importance of the structure, the cost, the level of protection it provides, and the consequences of its failure. The design return periods are usually specified in local drainage manuals or regulations. Typical design return periods vary from 2 to 5 years for street gutters, 2 to 25 years for stormwater pipes, and 10 to 100 years for detention basins. Design return periods of 5 to 10, 10 to 25, and 25 to 50 years are used for culverts under streets carrying low, intermediate, and high traffic volumes, respectively. Major highway bridges are designed to pass the 50- or 100-year runoff event.

Sec. 11.2

Design Storms

403

8 7

Intensity (in/hr)

6 25-yr average frequency

10-yr average frequency 5-yr average frequency

2-yr average frequency

3 2 1 0

50 60 70 Duration (min)

100 110 120

Figure 11.4 Typical IDF curves

10-YEAR 24-HR RAINFALL (IN.) 3

3.5 3

3.5

6 7

AT LA NT I C

3 4 8 3 3.5 4

O C EA N

3.5

5 6

8 7

GULF OF MEXICO 0 100 200 300 400 miles

9 9

9 8

Figure 11.5 Typical rainfall frequency map (10-year, 24-hr storm) Source: Rainfall Frequency Atlas of the United States for Durations from 30 Minutes to 24 Hours and Return Periods from 1 to 100 Years (Technical Paper No. 40), National Weather Service.

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The design-storm duration depends on the type of project. Structures such as stormwater pipes and culverts are sized to accommodate peak flows. Therefore, the design-storm duration should be selected such that the resulting peak discharge will be the largest for a given return period. Structures such as detention basins are designed to temporarily store runoff. Therefore, the duration causing the largest detention volume is the most critical. Ideally, the critical storm duration is determined after trying several values and studying how the storm duration affects the peak discharge and/or the detention volume. The average design storm intensity (or the total depth) is determined from the IDF curves once the design return period and the duration are selected. However, rainfall also varies temporally over the duration of a storm. This variation may have a significant effect on the runoff rates produced by the rainfall. Therefore, the time pattern of rainfall intensity needs to be specified for a complete description of a design storm. Once again, local drainage manuals or regulations usually specify the design rainfall time distribution of rainfall. The standard distributions developed by the Soil Conservation Service (SCS) are commonly used in the United States. 11.2.4

Synthetic Block Design-Storm Hyetograph

A synthetic design storm can be constructed from the local intensity-duration-return period curves for a given return period and duration. Figure 11.6 displays a 60-min block design-storm hyetograph. Storm events with shorter durations and higher intensity will be nested within the design-storm hyetograph obtained by this method. For example, a 60-min storm hyetograph constructed by using a time increment of ∆t = 10 min will also contain the most intense 10-, 20-, 30-, 40-, and 50-min storms for the same return period. In the synthetic design-storm approach, smaller ∆ts produce higher peak intensities. Although the selection of ∆t is somewhat arbitrary, it should not be greater than the time of concentration (a characteristic response time that will be covered in Section 11.4.1) of the watershed being studied. Also, the arrangement of the sequence of blocks representing different intensities is somewhat arbitrary. Normally, the peak intensity is placed between 1/3 and 1/2 of the duration. The others are then alternated from one side of the peak to the other until all are placed. 7 Intensity, i (in./hr)

6 5 4 3 2 1 0

Figure 11.6

30 40 Time, t (min)

Example synthetic block design-storm hyetograph

Sec. 11.2

Design Storms

405

Example 11.1 Construct a 10-year, 60-min design-storm hyetograph by using the IDF curves given in Figure 11.4. Use a time increment of 10 min. Solution The calculations are presented in Table 11.2. Columns 1 and 2 are read off the 10-year curve in Figure 11.4. For instance, i = 4.20 in./hr for td = 20 min. An interpretation for this is that the average intensity over the most intense 20-min portion of the 60-min design storm is 4.20 in./hr. Likewise, the depth, P = 4.20 (20/60) = 1.40 in., given in column 3, represents the rainfall depth produced during the most intense 20-min time period. Column 4 lists the incremental depths produced during different time increments of ∆t = 10 min = (1/6) hr. For instance, the incremental depth of 0.40 in. listed between 10 and 20 min is obtained by subtracting 1.00 in. from 1.40 in. Column 5 lists the rates at which the depth is produced during these intervals (i.e., rainfall intensities). The sequence of intensities in an actual rain, however, is not necessarily the same as listed in columns 1 and 5. For instance, the most intense 20-min portion does not necessarily follow a most intense 10-min portion. Therefore, the sequence of rainfall intensities listed in column 5 needs to be rearranged to form a more realistic intensity distribution. Columns 6 and 7 show the rearranged sequence. The peak intensity is first placed between 20 and 30 min, that is between 1/3 and 1/2 of the duration. The others are then alternated from one side of the peak to the other. Figure 11.6 displays the resulting block hyetograph.

11.2.5

Soil Conservation Service Hyetographs

The Soil Conservation Service* (SCS) presented several dimensionless rainfall distributions for different regions of the United States as shown in Figure 11.7. The four distributions are displayed in Figure 11.8 and tabulated in Table 11.3, where t = time, PT = total depth of rainfall, Table 11.2 Synthetic Block Design-Storm Hyetograph Example

(1)

(2)

(3)

(4)

(5)

(7)

(8)

td (min)

i (in./hr)

P = i td (in.)

∆P (in.)

∆P/∆t (in./hr)

t (min)

i (in./hr)

—

0 1.00

6.00

1.00

4.20

1.40

3.46

1.73

3.00

2.00

2.70

2.25

2.40

0 1.62 10 0.40

2.40

2.40 20

0.33

1.98

6.00 30

0.27

1.62

1.98 40

0.25

1.50

1.50 50

0.15

0.90

0.90 60

* Urban Hydrology for Small Watersheds (Technical Release 55), Soil Conservation Service. Washington, DC: U.S. Department of Agriculture, Engineering Division, 1986. [Note: The Soil Conservation Service is now called the Natural Resources Conservation Service (NRCS), although the methodologies presented here still use the old name.]

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Rainfall distribution Type I Type IA Type II Type III

Figure 11.7 Location of the four SCS rainfall distributions Source: Urban Hydrology for Small Watersheds, Technical Release 55, Soil Conservation Service.

Fraction of 24-hr rainfall, P/PT

0.5 IA I

III II

12 Time (hr)

Figure 11.8 The four SCS 24-hr rainfall distributions Source: Urban Hydrology for Small Watersheds, Technical Release 55, Soil Conservation Service.

Sec. 11.2

Design Storms

407 Table 11.3

SCS 24-HR Rainfall Distributions

t (hr)

Type I P/PT

Type IA P/PT

Type II P/PT

Type III P/PT

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0

0.000 0.008 0.017 0.026 0.035 0.045 0.055 0.065 0.076 0.087 0.099 0.112 0.125 0.140 0.156 0.174 0.194 0.219 0.254 0.303 0.515 0.583 0.624 0.654 0.682 0.706 0.728 0.748 0.766 0.783 0.799 0.815 0.830 0.844 0.857 0.870 0.882 0.893 0.905 0.916 0.926 0.936 0.946 0.956 0.965 0.974 0.983 0.992 1.000

0.000 0.010 0.022 0.036 0.051 0.067 0.083 0.099 0.116 0.135 0.156 0.179 0.204 0.233 0.268 0.310 0.425 0.480 0.520 0.550 0.577 0.601 0.623 0.644 0.664 0.683 0.701 0.719 0.736 0.753 0.769 0.785 0.800 0.815 0.830 0.844 0.858 0.871 0.884 0.896 0.908 0.920 0.932 0.944 0.956 0.967 0.978 0.989 1.000

0.000 0.005 0.011 0.017 0.023 0.029 0.035 0.041 0.048 0.056 0.064 0.072 0.080 0.090 0.100 0.110 0.120 0.133 0.147 0.163 0.181 0.203 0.236 0.283 0.663 0.735 0.776 0.804 0.825 0.842 0.856 0.869 0.881 0.893 0.903 0.913 0.922 0.930 0.938 0.946 0.953 0.959 0.965 0.971 0.977 0.983 0.989 0.995 1.000

0.000 0.005 0.010 0.015 0.020 0.026 0.032 0.037 0.043 0.050 0.057 0.065 0.072 0.081 0.089 0.102 0.115 0.130 0.148 0.167 0.189 0.216 0.250 0.298 0.600 0.702 0.751 0.785 0.811 0.830 0.848 0.867 0.886 0.895 0.904 0.913 0.922 0.930 0.939 0.948 0.957 0.962 0.968 0.973 0.979 0.984 0.989 0.995 1.000

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and P = rainfall depth accumulated up to time t. Note that the design storm lasts for 24 hrs. However, shorter and more intense rainfalls are nested within the 24-hr duration, and therefore these distributions are appropriate for both small and large watersheds. Applying the SCS procedure results in a design storm hyetograph, a relationship between rainfall intensity and time for a given location and return period. Design-storm hyetographs are the primary input into rainfall–runoff models. Example 11.2 Determine a 10-year, 24-hr storm hyetograph for Virginia Beach, Virginia. Solution From Figure 11.5, we determine that the 10-year, 24-hr rainfall for Virginia Beach, Virginia is 6 in. Likewise Figure 11.7 indicates that SCS Type II hyetographs can be used for Virginia Beach. (Virginia Beach is on the border of Type II and III hyetographs, but the governmental review agency has dictated a Type II.) The calculations are performed in tabular form as shown in Table 11.4. Table 11.4 is set up to determine the rainfall intensity for the time increments between times t1 and t2 tabulated in columns 1 and 2. The corresponding cumulative rain to total rain ratios are tabulated in columns 3 and 4, respectively. These values are obtained from Table 11.3 for the Type II distribution. In this example, PT = 6 in. To find the rainfall intensity between, say, t = 9.5 hr and t = 10 hr, we first note the precipitation ratios are 0.163 and 0.181, respectively, for 9.5 and 10 hrs. Thus, the rainfall depth at t = 9.5 hr is P = (0.163)(6) = 0.978 in., and that at t = 10 hr is P = (0.181)(6) = 1.086 in. A depth of 1.086 - 0.978 = 0.108 in. is produced between t = 9.5 and 10 hr, and, therefore, the intensity is 0.108 in./0.5 hr = 0.216 in./hr. Similarly, for the period between 10 and 10.5 hr, the intensity is equal to [(6)(0.203) - 6(0.181)]/0.5 = 0.264 in./hr. The intensity distribution of the whole design-storm hyetograph can be determined by repeating the calculations for all the time intervals.

11.3 Losses from Rainfall and Rainfall Excess Abstractions or losses from rainfall refer collectively to that part of the rainfall that does not produce runoff. In general, the abstractions include interception by trees and vegetation, surface depression storage, evaporation, transpiration through plants, and infiltration into the underlying soil. Rainfall excess or effective rainfall is the part of rainfall that becomes runoff. It is calculated as the total rainfall minus abstractions. The rate of rainfall excess or rate of effective rainfall is the depth of rainfall excess produced per unit time. It is calculated as the rate of rainfall minus the rate of loss. Based on Hortonian hydrology, the rainfall excess is assumed to be solely responsible for storm runoff. Therefore, the total volume of rainfall excess is equal to the total volume of runoff produced. Under design-storm conditions, evaporation and transpiration are generally negligible. Interception and depression storage are usually calculated by simple empirical equations or are assigned constant values specified in drainage manuals. Infiltration is the dominant mechanism through which losses from rainfall occur. The infiltration capacity is the maximum rate at which water can infiltrate. The actual rate of infiltration will be equal to the rate of rainfall if the rainfall rate is less than the infiltration capacity. Otherwise, the actual rate of infiltration will equal the infiltration capacity, and the rainwater that does not infiltrate will flow over the ground surface after filling the surface depressions.

Sec. 11.3

Losses from Rainfall and Rainfall Excess Table 11.4

409

Design-Storm Hyetograph Calculations for Example 11.2

(1) t1 (hr)

(2) t2 (hr)

(3) P1/PT

(4) P2/PT

(5) P1 (in.)

(6) P2 (in.)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0

0.005 0.011 0.017 0.023 0.029 0.035 0.041 0.048 0.056 0.064 0.072 0.080 0.090 0.100 0.110 0.120 0.133 0.147 0.163 0.181 0.203 0.236 0.283 0.663 0.735 0.776 0.804 0.825 0.842 0.856 0.869 0.881 0.893 0.903 0.913 0.922 0.930 0.938 0.946 0.953 0.959 0.965 0.971 0.977 0.983 0.989 0.995 1.000

0.000 0.030 0.066 0.102 0.138 0.174 0.210 0.246 0.288 0.336 0.384 0.432 0.480 0.540 0.600 0.660 0.720 0.798 0.882 0.978 1.086 1.218 1.416 1.698 3.978 4.410 4.656 4.824 4.950 5.052 5.136 5.214 5.286 5.358 5.418 5.478 5.532 5.580 5.628 5.676 5.718 5.754 5.790 5.826 5.862 5.898 5.934 5.970

0.030 0.066 0.102 0.138 0.174 0.210 0.246 0.288 0.336 0.384 0.432 0.480 0.540 0.600 0.660 0.720 0.798 0.882 0.978 1.086 1.218 1.416 1.698 3.978 4.410 4.656 4.824 4.950 5.052 5.136 5.214 5.286 5.358 5.418 5.478 5.532 5.580 5.628 5.676 5.718 5.754 5.790 5.826 5.862 5.898 5.934 5.970 6.000

(7) (8) ∆P = P2 - P1 (in.) i = ∆P/∆t (in./hr) 0.030 0.036 0.036 0.036 0.036 0.036 0.036 0.042 0.048 0.048 0.048 0.048 0.060 0.060 0.060 0.060 0.078 0.084 0.096 0.108 0.132 0.198 0.282 2.280 0.432 0.246 0.168 0.126 0.102 0.084 0.078 0.072 0.072 0.060 0.060 0.054 0.048 0.048 0.048 0.042 0.036 0.036 0.036 0.036 0.036 0.036 0.036 0.030

0.060 0.072 0.072 0.072 0.072 0.072 0.072 0.084 0.096 0.096 0.096 0.096 0.120 0.120 0.120 0.120 0.156 0.168 0.192 0.216 0.264 0.396 0.564 4.560 0.864 0.492 0.336 0.252 0.204 0.168 0.156 0.144 0.144 0.120 0.120 0.108 0.096 0.096 0.096 0.084 0.072 0.072 0.072 0.072 0.072 0.072 0.072 0.060

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11.3.1

Chap. 11

Green and Ampt Infiltration Model

The Green and Ampt model can best be explained by using a definition sketch such as the one given in Figure 11.9. It is assumed that the soil has a uniform degree of saturation equal to Si (i.e., the fraction of the pore volume containing water) at the time the rain begins. As rainwater enters the soil, the degree of saturation will increase as represented by the typical moisture profile displayed in Figure 11.9. Naturally, the moisture content is higher near the surface and increases with depth over time as the storm progresses and more infiltration occurs. The Green and Ampt model approximates this moisture profile by a sharp (squared-off) wetting front that will develop adjacent to the surface shortly after the rain begins. Above the wetting front, the soil is assumed to be saturated. The initial degree of saturation is maintained below it. However, the depth of the wetting zone, Z, will increase as more water infiltrates into the soil. Applying Darcy's law to the saturated zone, the infiltration capacity is determined as fp =

K (Z + Pf)

(11.1)

where fp is the infiltration capacity, K the hydraulic conductivity of the soil, Pf a characteristic suction head of the soil, and Z the depth of the wetted zone measured from the soil surface. If the rate of rainfall, i, is less than fp, the actual rate of infiltration, f, is set equal to the rate of rainfall, i. Otherwise, f = fp. The rate of rainfall excess is then determined as ie = i - f if infiltration is the dominant loss from rainfall. We usually carry out the infiltration calculations over discrete time intervals, ∆t. The increase in the depth of the saturated zone, ∆Z, during a time interval is calculated by using ∆Z =

f ∆t ϕ(1 - Si)

(11.2)

where f = rate of infiltration during the time interval, ϕ = effective porosity, and Si = initial degree of saturation. Ground surface 0

1.0

Distance from surface

Saturated zone

Moisture profile

Degree of saturation Z

Wetting front

Dry zone Initial moisture profile

Figure 11.9

Definition sketch for Green and Ampt model

Sec. 11.3

Losses from Rainfall and Rainfall Excess Table 11.5

Soil Type Sand Loamy sand Sandy loam Loam Silt loam Sandy clay loam Clay loam Silty clay loam Sandy clay Silty clay Clay

411

Green and Ampt Soil Properties

Pf (cm)

Ks (cm/hr)

0.437 (0.374–0.500) 0.437 (0.363–0.506) 0.453 (0.351–0.555) 0.463 (0.375–0.551) 0.501 (0.420–0.572) 0.398 (0.332–0.464) 0.464 (0.409–0.519) 0.471 (0.418–0.524) 0.430 (0.370–0.490) 0.479 (0.425–0.533) 0.475 (0.427–0.523)

4.95 (0.97–25.36) 6.13 (1.35–27.94) 11.01 (2.67–45.47) 8.89 (1.33–59.38) 16.68 (2.92–95.39) 21.85 (4.42–108.0) 20.88 (4.79–91.10) 27.30 (5.67–131.50) 23.90 (4.08–140.2) 29.22 (6.13–139.4) 31.63 (6.39–156.5)

23.56 5.98 2.18 1.32 0.68 0.30 0.20 0.20 0.12 0.10 0.06

(Source: Rawls, W. J., Brakensiek, D. L., and Miller, N. (1983). "Green–Ampt Infiltration Parameters from Soils Data," Journal of Hydraulic Engineering, ASCE, 109:62–70).

Defining F as the cumulative depth of water that has infiltrated between time zero and t yields

F = a f ∆t for discrete time intervals. Also the relationship between Z and F is Z =

F ϕ(1 - Si)

(11.3)

(11.4)

All the equations of the Green and Ampt model are dimensionally homogeneous, and they can be used with any consistent unit system. The soil properties used in the Green and Ampt model are tabulated for a variety of soil textures in Table 11.5. It should be noted that the Green and Ampt method is a common option for removing losses in most rainfall–runoff computer models. Example 11.3 An open urban area is comprised of a soil that has ϕ = 0.40, K = 0.30 cm/hr, and Pf = 20 cm. The initial degree of saturation is Si = 0.60 (i.e., 60% saturated). Rainfall rates are tabulated in column 4 of Table 11.6. Determine the losses due to infiltration. Also determine the rates of effective rainfall assuming that the losses occur due to infiltration only.

412

Hydrology for Hydraulic Design Table 11.6

Chap. 11

Green and Ampt Calculations for Example 11.3

(1) Time step

(2) t1 (min)

(3) t2 (min)

(4) i (cm/hr)

(5) Z1 (cm)

(6) fp (cm/hr)

(7) f (cm/hr)

(8) ∆Z (cm)

(9) Z2 (cm)

(10) ie (cm/hr)

1 2 3 4 5 6 7 8 9 10 11 12

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00

5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00

2.00 2.00 2.00 4.00 4.00 4.00 3.00 3.00 3.00 2.00 2.00 2.00 Σ = 33.00

0.00 1.04 2.08 3.13 4.28 5.17 5.93 6.61 7.24 7.83 8.38 8.91

∞ 6.06 3.18 2.22 1.70 1.46 1.31 1.21 1.13 1.07 1.02 0.97

2.00 2.00 2.00 2.22 1.70 1.46 1.31 1.21 1.13 1.07 1.02 0.97 Σ = 18.09

1.04 1.04 1.04 1.16 0.89 0.76 0.68 0.63 0.59 0.56 0.53 0.51

1.04 2.08 3.13 4.28 5.17 5.93 6.61 7.24 7.83 8.38 8.91 9.42

0.00 0.00 0.00 1.78 2.30 2.54 1.69 1.79 1.87 0.93 0.98 1.03 Σ = 14.91

Rate of rainfall and infitration (cm/hr)

Solution Table 11.6 presents the solution using a time increment of ∆t = t2 - t1 = 5 min = (1/12) hr. Subscripts 1 and 2 refer to the beginning and the end of a time interval, respectively. For the first time step, t1 = 0 represents the initial condition, therefore Z1 = 0. The fp values in column 6 are obtained from Equation 11.1. For simplicity, Z1 is used in place of Z (or an average Z over the time increment) in this equation as an approximation. The entries for f in column 7 are equal to the smaller of fp and i. The entries in column 8 for ∆Z are found by using Equation 11.2. In column 9, Z2 is the depth of the wetting front at the end of the time interval, and it is evaluated as Z2 = Z1 + ∆Z. The entries for the rate of effective rainfall in column 10 are calculated as ie = i - f. The value of Z1 needed at each time step is carried over from the previous time step. The results are plotted in Figure 11.10. 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0

Rainfall excess Losses due to infiltration 0

Figure 11.10

30 40 Time (min)

Green and Ampt results for Example 11.3

Sec. 11.3

Losses from Rainfall and Rainfall Excess

11.3.2

413

Soil Conservation Service Method

The Soil Conservation Service (SCS) developed a procedure for determining rainfall excess based on runoff curve numbers and cumulative rainfall depths. The runoff curve number (CN) is a watershed parameter with a range from 0 to 100. The value of CN depends on the hydrologic soil group, the soil cover type (i.e., land use) and condition, and the percent of impervious areas in the watershed. The recommended CN values for various urban land use types and some agricultural land uses are given in Table 11.7. These CNs are for average moisture conditions preceding a storm. If a watershed is composed of several subareas with different CNs, then a weighted average (based on area) or composite CN can be obtained for the whole watershed. In Table 11.7, hydrologic soil group A includes soils with high infiltration rates such as sand and gravel, and group B includes soils of coarse to fine texture with moderate rates of infiltration. Soils of moderately fine to fine texture with slow infiltration rates belong to group C, and those with the lowest infiltration rates like clays belong to group D. Table 11.7

SCS Runoff Curve Numbers

Cover Description Cover Type and Hydrologic Condition Open space (parks, cemeteries, etc.): Poor condition (grass cover 6 50,) Fair condition (grass cover 50,- 75,) Good condition (grass cover 775,) Impervious areas (parking lots, etc.) Urban districts: Commercial and business Industrial Residential areas (by average lot size): 1/8 acre or less (town houses) 1/4 acre 1/3 acre 1/2 acre 1 acre 2 acre Newly graded areas (no vegetation) Agricultural land or open land (good condition) Fallow land (crop residue) Row crops (contoured) Small grain crops (contoured) Pasture, grassland, or range Meadow (mowed for hay) Woods–grass combination (orchards) Woods

% Impervious

Hydrologic Soil Groups A

100

68 49 39 98

79 69 61 98

86 79 74 98

89 84 80 98

85 72

89 81

92 88

94 91

95 93

65 38 30 25 20 12

77 61 57 54 51 46 77

85 75 72 70 68 65 86

90 83 81 80 79 77 91

92 87 86 85 84 82 94

76 65 61 39 30 32 30

85 75 73 61 58 58 55

90 82 81 74 71 72 70

93 86 84 80 78 79 77

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Once the curve number is selected, the cumulative runoff (R) corresponding to cumulative rainfall (P) is calculated by using (P - 0.2S)2 (P + 0.8S)

(11.5)

1,000 - 10(CN) (CN)

(11.6)

R = with S =

where R = cumulative runoff (or rainfall excess) in inches, P = cumulative rainfall in inches, and S = [email protected] storage deficit in inches at the time runoff begins. These equations are valid if P 7 0.2S, otherwise R = 0. Figure 11.11 graphically presents these equations. The runoff produced over a time increment is the difference between the cumulative runoff at the end and at the beginning of the time increment. If P, R, and S are in millimeters, the soil moisture deficit relationship is written as S =

25,400 - 254(CN) (CN)

(11.7)

10 0 95

um be

(P – 0.2S)2

P + 0.8S

Cu r

Direct runoff, R (in.)

ve n

Curves on this sheet are for the case Ia = 0.2S, so that

45 40

25 20

Rainfall, P (in.) Figure 11.11 The SCS rainfall–runoff relationship Source: Urban Hydrology for Small Watersheds, Technical Release 55, Soil Conservation Service.

Sec. 11.4

Design Runoff Hydrographs

415

Example 11.4 An urban residential district contains 1/3 acre lots. It is 30% impervious, and the hydrologic soil group for the district is identified as group B. Determine the rainfall excess (i.e., runoff) resulting from a 10-hr storm that produces the rainfall intensities tabulated in column 3 of Table 11.8. Table 11.8 SCS Rainfall Excess Calculations for Example 11.4

(1) t1 (hr)

(2) t2 (hr)

(3) i (in./hr)

(4) ∆P (in.)

(5) P1 (in.)

(6) P2 (in.)

(7) R1 (in.)

(8) R2 (in.)

(9) ∆R (in.)

0 2 4 6 8

2 4 6 8 10

0.05 0.20 1.00 0.50 0.25

0.10 0.40 2.00 1.00 0.50

0.00 0.10 0.50 2.50 3.50

0.10 0.50 2.50 3.50 4.00

0.00 0.00 0.00 0.53 1.12

0.00 0.00 0.53 1.12 1.46

0.00 0.00 0.53 0.59 0.34

Solution From Table 11.7, we obtain CN = 72 for this urban watershed. The calculations are presented in Table 11.8. The rainfall intensities (i) are tabulated in the 2-hr intervals in column 3, where t1 and t2 listed in columns 1 and 2 mark the beginning and end of the time intervals. The incremental rainfall depth accumulating over a time interval is given in column 4 (i # ∆t). The cumulative rainfall at t1 is given in column 5. Obviously, P1 = 0 when the rainfall first begins at t1 = 0. In column 6, P2 is the cumulative rainfall at t2, and it is obtained as P2 = P1 + ∆P. The R1 values listed in column 7 represent the cumulative runoff (or rainfall excess) at time t1, and they are obtained by using Equations 11.5 and 11.6 (or Figure 11.11) given CN and P1. In column 8, R2 is the cumulative runoff at t2, and it is obtained in the same manner but using P2 in Equations 11.5 and 11.6. In column 9, ∆R = R2 - R1 is the incremental depth of runoff accumulating over the time interval ∆t.

11.4 Design Runoff Hydrographs As mentioned in the earlier chapters, flood protection hydraulic structures are designed to accommodate a design runoff event. If historical runoff has not been measured at or near the design point, the first step in determining the design runoff event is the delineation of the contributing watershed as discussed in Section 11.1. Next comes the selection of a design storm as discussed in Section 11.2. The third step is to determine the quantity of runoff (or rainfall excess) resulting from the design rainfall. This step was addressed in Section 11.3. The objective of this section is to calculate the flow rates at the design point resulting from the transport of the rainfall excess to the location of the structure being designed. The discharge, or flow rate, at a stream location is the volume of water passing through the section per unit time. A plot of discharges versus time is called a streamflow hydrograph or total runoff hydrograph. The total runoff hydrograph (TRH) at the design point corresponding to the design storm represents a design runoff hydrograph. Total runoff at a stream section is comprised of direct runoff and base flow as shown in Figure 11.12. The rainfall excess from a storm flowing over the ground toward the stream produces the direct runoff. In small watersheds, the rainfall excess can reach the design point in a matter of hours. The base flow of a total runoff hydrograph is supplied by an adjacent groundwater aquifer. Part of the rainfall that has infiltrated into the soil may also eventually reach a stream section via an aquifer. However, it may take weeks or even months for rain water to reach a

Hydrology for Hydraulic Design

Rainfall

416

Chap. 11

Peak

Rising limb

Volume of direct surface runoff

sio ces Re

Discharge

Recorded discharge

Direct runoff Estimated base flow Time Figure 11.12

Elements of streamflow hydrograph

stream along this path since groundwater flow is very slow. Therefore, the base flow component of a total runoff hydrograph corresponding to a design storm is not generated by that particular storm. It is rather generated by various rainfall events from the past. Therefore, while the total volume of rainfall excess from a design storm (depth of rainfall excess times the watershed area) should be equal to the volume of direct runoff, no such relationship exists between the base flow and losses from the design storm. A plot of direct runoff rates versus time is called a direct runoff hydrograph (DRH). 11.4.1

Time of Concentration

Time of concentration is defined as the time it takes runoff to reach the design point from the hydrologically most remote point in the watershed. Although it is difficult to compute accurately, time of concentration is a key parameter in many hydrologic analyses and design procedures. Many design tools are available for determining the time of concentration. Most techniques distinguish between the overland flow phase and the channel flow phase. One popular procedure promoted by the NRCS (formerly SCS) breaks down the flow time into three components: (1) sheet (overland) flow, (2) shallow concentrated flow, and (3) open-channel flow. Sheet flow is defined as flow over plane surfaces at very shallow depths (∼0.1 ft). Sheet flow occurs throughout a watershed before concentrating in swales and gullies. Resistance to sheet flow (Manning's n values; listed in Table 11.9) incorporates raindrop impact, surface drag, resistance from obstacles (e.g., grass, stones, litter), erosion, and sediment transport. Based on a Manning's kinematic solution, Overton and Meadows* suggest sheet flow travel time (Tt1 in hours) is found using 0.4 Tt1 = [0.007(nL)0.8]/(P 0.5 2 s )

(11.8)

* D. E. Overton and M. E. Meadows, Storm Water Modeling (New York: Academic Press, 1976): 58–88.

Sec. 11.4

Design Runoff Hydrographs

417

Table 11.9 Manning's n Values for Sheet Flow a,b

Surface Description

Range of n Values

Concrete, bare soil Grass Short grass Dense grass Bermuda grass Range (natural) Fallow, no residue Cultivated soils Residue cover Residue cover Woods Light underbrush Dense underbrush

0.011 0.15 0.24 0.41 0.13 0.05 0.06 0.17 0.4 0.8

E. T. Engman, "Roughness coefficients for routing surface runoff," Journal of Irrigation and Drainage Engineering, 112 (1986). b Soil Conservation Service, Urban Hydrology for Small Watersheds (Technical Release 55) Washington, DC: Soil Conservation Service, 1986.

where n is Manning's sheet flow roughness, L is flow length in feet, P2 is the 2-year, 24-hr rainfall in inches, and s is the land slope (ft/ft). The National Weather Service and the National Oceanic and Atmospheric Administration evaluate and publish 2-year (return interval), 24-hr precipitation depths for the United States. The sheet flow length was originally limited to 300 ft or less by the NRCS, but currently in practice it is often limited to 100 ft. Shallow concentrated flow occurs when two separate planes of sheet flow converge to consolidate flow but fail to form a defined stream or channel. This type of flow often occurs in street gutters. Shallow concentrated flow velocity is estimated from the SCS equations: Unpaved: V = 16.1345 s0.5

(11.9)

Paved: V = 20.3282 s0.5

(11.10)

where V is the average velocity (ft/s) and s is the watercourse slope (ft/ft). The shallow concentrated flow travel time (Tt2) is found by dividing the flow length by the average velocity. Open-channel flow begins where shallow concentrated flow ends. This transition can be subjective, but it is often accompanied by well-defined stream banks. Field reconnaissance and contour maps are helpful. U.S. Geological Survey quadrangle maps depict channels (or streams) with blue lines. The average velocity for open-channel flow is defined by Manning's equation as V = (1.49/n)(Rh)2/3(Se)1/2

(11.11)

where V is the average velocity (ft/s), n Manning's channel roughness coefficient (Table 11.10), Rh the hydraulic radius (ft) described in Chapter 6 (Section 2), and Se the slope of the energy grade line (ft/ft) or channel bottom slope (So) if uniform flow is assumed. The channel flow time (Tt3) is found by dividing the channel length by the average velocity.

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If metric units are used with P2 in millimeters, L in meters, V in meters per second, and Rh in meters, Equations 11.8 through 11.11 are replaced with

# s0.4) Tt1 = [0.091 # (n # L)0.8]/(P 0.5 2

(11.12)

Unpaved: V = 4.91 # s0.5

(11.13)

V = 6.19 # s0.5

(11.14)

Paved: and

# S1/2 V = (1.0/n) # R2/3 h e

(11.15)

An alternative procedure for determining the time of concentration for small watersheds (less than 2,000 acres) is based on the runoff curve numbers discussed in the preceding section. The empirical equation is Tc = [L0.8 (S + 1)0.7]/(1,140 Y 0.5)

(11.16)

where Tc is the time of concentration (hours), L is the length of the longest flow path in the watershed (from the design point to the hydrologically most remote point in the watershed located on the drainage divide in feet and often called the hydraulic length), Y is the average watershed slope expressed as a percentage, and S is calculated using Equation 11.6. If the hydraulic length L is in meters and S is in millimeters, Equation 11.16 is replaced with Tc = 52.586L0.8 # [(S/25.4) +1]0.76/(1,140 # Y 0.5)

(11.17)

and S is evaluated using using Equation 11.7.

Table 11.10

Typical Manning's n Values for Channel Flow

Channel Surface Glass, polyvinyl chloride, high-density polyethylene Smooth steel, metals Concrete Asphalt Corrugated metal Earth excavation, clean Earth excavation, gravel or cobbles Earth excavation, some weeds Natural channels, clean and straight Natural channels, stones or weeds Riprap-lined channel Natural channels, clean and winding Natural channels, winding with pools or shoals Natural channels, weeds with debris or deep pools Mountain streams, gravel or cobbles Mountain streams, cobbles or boulders

n 0.010 0.012 0.013 0.015 0.024 0.022–0.026 0.025–0.035 0.025–0.035 0.025–0.035 0.030–0.040 0.035–0.045 0.035–0.045 0.045–0.055 0.050–0.080 0.030–0.050 0.050–0.070

Sec. 11.4

Design Runoff Hydrographs

419

Example 11.5 Determine the time of concentration (Tc) for a watershed with the following characteristics: sheet flow segment with n = 0.20 and L = 120 ft (and slope of 0.005); 2-year, 24-hr rainfall is 3.6 in; a shallow concentrated flow (unpaved) segment where L = 850 ft with a slope of 0.0125; and a channel flow segment with n = 0.025, A = 27 ft2, P = 13 ft, So = 0.005, and L = 6,800 ft. Solution For the sheet flow segment, using Equation 11.8: Tt1 = [0.0075(0.20)(120)60.8]/[(3.6)0.5(0.005)0.4] = 0.39 hr For the shallow concentrated flow, from Equation 11.9: V = 16.1345(0.0125)0.5 = 1.80 ft/s and Tt2 = 850/[(1.80 ft/s)(3,600 sec/hr)] = 0.13 hr For the channel flow segment, Rh = A/P = 27/13 = 2.08 ft. Then, from Equation 11.11: V = (1.49/0.025)(2.08)2/3 (0.005)1/2 = 6.87 ft/s and Tt3 = 6,800/[(6.87 ft/s)(3,600 s/hr)] = 0.27 hr Therefore, Tc = 0.39 + 0.13 + 0.27 = 0.79 hr It is interesting to note that the sheet flow length (120 ft) makes up only 1.5% of the total flow length (120 + 850 + 6,800 = 7,700 ft). However, 0.39 hr of the time of concentration of 0.79 hr (i.e., 49%) is attributed to the sheet flow segment of the flow path.

11.4.2

Unit Hydrograph

The unit hydrograph concept is commonly employed in rainfall–runoff modeling. A unit hydrograph is a conceptual direct runoff hydrograph resulting from a rainfall excess of unit depth and of a particular duration. It represents how a rainfall excess of unit depth contributes to runoff at the watershed outlet. It can be viewed as a lumped watershed characteristic. A unit depth of 1 cm is used in the SI unit system, and a depth of 1 in. is employed in the British unit system. We usually abbreviate a unit hydrograph as UH∆D, where the subscript ∆D indicates the duration of the rainfall excess. For instance, the direct runoff hydrograph produced by a rainfall excess that has a duration of 3 hrs and constant intensity of (1/3) in./hr is denoted by UH3. Note that the depth of the rainfall excess is (1/3 in./hr)(3 hr) = 1 in. We can obtain a unit hydrograph for a watershed from simultaneous rainfall and runoff records. For this purpose, a rainfall event that produces a relatively uniform rainfall excess over certain duration ∆D is required. The base flow rates are estimated and subtracted from the total runoff hydrograph (TRH) to determine the direct runoff hydrograph (DRH). By definition, the volume of direct runoff, which is the area under the DRH, is equal to the watershed area multiplied by the depth of rainfall excess. Then, based on the linearity assumption used in the unit hydrograph applications, the depth of rainfall excess, de, can be found by dividing the volume of

420

Hydrology for Hydraulic Design

Chap. 11

direct runoff by the known watershed area. Finally, the ordinates of UH∆D can be determined by dividing the ordinates of the DRH by de at respective times. Simultaneous rainfall and runoff data are unavailable at most sites. In the absence of such data, synthetic unit hydrographs are derived based on the watershed characteristics. The SCS synthetic unit hydrograph procedure is commonly used for this purpose. This procedure only requires two parameters: (1) the time to peak and (2) the peak discharge. With these parameters, a unit hydrograph can be developed at any ungauged stream location. The time to peak is the elapsed time from the beginning of the effective (runoff-producing) rainfall to the peak discharge as shown in Figure 11.13. The SCS computes the time to peak (Tp, in hours) using the equation Tp = ∆D/2 + TL

(11.18)

where ∆D is the duration of effective rainfall (hours) and TL is the lag time (hours). The lag time is the elapsed time from the centroid of effective rainfall to the peak discharge (Figure 11.13). Lag time and storm duration are interrelated parameters and change with each storm. Time of concentration (Tc), a watershed characteristic, remains relatively constant and is readily determined (Section 11.4.1). SCS has related Tc to the effective rainfall duration (∆D) and lag time (TL) empirically in order to compute an optimal time to peak for the synthetic unit hydrograph. SCS recommends that the effective rainfall duration be set to ∆D = 0.133 Tc

(11.19)

based on the characteristics of the SCS curvilinear unit hydrograph. Also, the lag time is related to time of concentration by TL = 0.6 Tc

(11.20)

Therefore, the time to peak is computed as DD Effective rainfall TL

TP Figure 11.13 SCS hydrograph parameters

Time

Sec. 11.4

Design Runoff Hydrographs

421

Tp = 0.67 Tc

(11.21)

The time of concentration is determined using the equations presented in Section 11.4.1. The peak discharge qp in cfs/in. for the SCS synthetic unit hydrograph is determined from qp = (Kp A)/Tp

(11.22)

where A is the drainage area (square miles), Tp is the time to peak (hours), and Kp is an empirical constant. Kp ranges from 300 in flat swampy areas to 600 in steep terrain, but it is often assigned a value of 484. Local NRCS offices in the United States will provide guidance in evaluating Kp. The peak discharge and time to peak combine with the dimensionless unit hydrograph coordinates in Table 11.11 (Kp = 484 only) to produce the SCS unit hydrograph. The example problem that follows demonstrates the procedure. The dimensionless unit hydrograph is plotted in Figure 11.14, which also displays mass accumulation. Recall that the area under a hydrograph represents runoff volume (or mass). A little more than one-third (37.5%) of the total runoff volume accumulates before the peak. It is important to remember that the shape and peak of a unit hydrograph depend on the effective (runoff-producing) storm duration. Therefore, there are an infinite number of unit hydrographs for a particular watershed. However, for the SCS synthetic unit hydrograph method, the effective storm duration (∆D) is found using Equation 11.19. To use the resulting unit hydrograph to produce design storms, runoff depths must be developed in units of time that are ∆D in length (or multiples thereof).

Table 11.11

Dimensionless Unit Hydrograph and Mass Curve Coordinates for the SCS Synthetic Unit Hydrograph

Time Ratios (t/tp)

Flow Ratios (q/qp)

Mass Ratios (Qa/Q)

Time Ratios (t/tp)

Flow Ratios (q/qp)

Mass Ratios (Qa/Q)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

0.00 0.03 0.10 0.19 0.31 0.47 0.66 0.82 0.93 0.99 1.00 0.99 0.93 0.86 0.78 0.68 0.56

0.000 0.001 0.006 0.012 0.035 0.065 0.107 0.163 0.228 0.300 0.375 0.450 0.522 0.589 0.650 0.700 0.751

1.7 1.8 1.9 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.5 5.0

0.46 0.39 0.33 0.28 0.21 0.15 0.11 0.08 0.06 0.04 0.03 0.02 0.02 0.01 0.01 0.00

0.790 0.822 0.849 0.871 0.908 0.934 0.953 0.967 0.977 0.984 0.989 0.993 0.995 0.997 0.999 1.000

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Hydrology for Hydraulic Design

Chap. 11

1.0 0.9 q = discharge at time t qp = peak discharge Qa = accululated volume at time t Q = total volume t = a selected time Tp = time from beginning of rise to the peak

0.8

q/qp or Qa/Q

0.7 0.6 0.5 0.4

q qp

0.3 Qa Q

0.2 0.1

t/Tp Figure 11.14 The SCS dimensionless unit hydrograph and mass curve Source: Part 630 Hydrology National Engineering Handbook, Soil Conservation Service.

Equations 11.18 through 11.22 can be used for the metric system as well if A is in square kilometers, Tp, Tc, and ∆D are in hours, and qp is in cubic meter per second per centimeter. However, Kp = 2.08 should be used in Equation 11.22. Example 11.6 A development is being planned for a large tract of land in Albemarle County, Virginia. The development will encompass a 250-acre watershed with a hydraulic length of 4,500 feet. The predominant soil is of group C, and land use is horse pasture with an average land slope of 8%. The land use in the watershed is scheduled to change over the next 5 years to single family residential (1/2 acre lots). Determine the SCS synthetic unit hydrograph before development. Solution The SCS synthetic unit hydrograph requires the time to peak and the peak discharge. Time to peak: The time to peak requires the time of concentration. Applying Equation 11.16, Tc = [L0.8(S + 1)0.7]/(1,140 Y 0.5) requires a determination of the soil moisture storage deficit (S). Thus, from Equation 11.6, S = (1,000/CN) - 10 = (1,000/74) - 10 = 3.51 in.

Sec. 11.4

Design Runoff Hydrographs

423

where the curve number is found in Table 11.7 (pasture, good condition, and C soils). Substituting the hydraulic length and watershed slope into Equation 11.16 yields Tc = [(4,500)0.8(3.51 + 1)0.7]/[1,140 (8)0.5] = 0.75 hrs (45 min) By applying Equation 11.21, an estimate of the time to peak is Tp = 0.67 Tc = 0.67 (0.75) = 0.50 hrs (30 min) Peak discharge: The peak discharge of the unit hydrograph is calculated from Equation 11.22 as qp = (KpA)/Tp = 484 [250 acres (1 sq.mi/640 acres)]/0.5 hrs = 378 cfs/in. SCS synthetic unit hydrograph: The coordinates of the unit hydrograph displayed in Table 11.12 are extracted from Table 11.11. The peak flow is 378 cfs/in. and occurs 30 min into the storm. The storm duration is found using Equation 11.19 and equals ∆D = 0.133 Tc = (0.133)(0.75) = 0.10 hrs (6 min) Thus, the unit hydrograph developed is a 6-min unit hydrograph.

A unit hydrograph can be used to predict a watershed's response to any storm if the principles of linearity and superposition are assumed to be valid for the rainfall–runoff process. Linearity suggests that the response from a watershed to runoff is linear in nature. In other words, if 1 in. of rainfall excess (i.e., runoff) over a given duration of say 2 hrs produces a 2-hr unit hydrograph UH2, then 2 in. of runoff over the same duration (2 hrs) would produce a flow rate twice as high at each point in time. Therefore, as depicted in Figure 11.15a, the DRH produced by a rainfall excess of 2-hr duration and 1.5-in. depth would be equal to 1.5 UH2. Superposition suggests that the effects of rainfall on the watershed can be separately accounted for and accumulated. For example, suppose a storm of 4-hr duration produces a rainfall excess of 0.6 and 0.8 in. during the first and second 2-hr periods, respectively. The resulting direct Table 11.12

SCS Synthetic Unit Hydrograph Calculations for Example 11.6

Time Ratios (t/tp)

Flow Ratios (q/qp)

Time (min)

Flow (cfs)

Time Ratios (t/tp)

Flow Ratios (q/qp)

Time (min)

Flow (cfs)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

0.00 0.03 0.10 0.19 0.31 0.47 0.66 0.82 0.93 0.99 1.00 0.99 0.93 0.86 0.78 0.68 0.56

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48

0 11 38 72 117 178 249 310 352 374 378 374 352 325 295 257 212

1.7 1.8 1.9 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.5 5.0

0.460 0.390 0.330 0.280 0.207 0.147 0.107 0.077 0.055 0.040 0.029 0.021 0.015 0.011 0.005 0.000

51 54 57 60 66 72 78 84 90 96 102 108 114 120 135 150

174 147 125 106 78 56 40 29 21 15 11 8 6 4 2 0

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Chap. 11

Superposition DRH = 0.6 UH2 + 0.8 UH2 (lagged 2 hrs)

UH2

0.8 UH2 (lagged 2 hrs)

DRH = 1.5 UH2

0.6 UH2

(a)

(b) Figure 11.15

Linearity and superposition principles

runoff hydrograph, DRH, will be equal to 0.6 UH2 + 0.8 UH2 (lagged 2 hrs) as depicted in Figure 11.15b. Extensive research has shown that the rainfall–runoff process is exceedingly complex and not completely subject to these principles. Nonetheless, unit hydrograph theory has proven to be a useful design tool when judiciously applied.

11.4.3

Total Runoff Hydrograph

The total runoff hydrograph (TRH) or streamflow hydrograph produced by a design storm will be the design runoff hydrograph. This is based on the assumption that a storm and the runoff resulting from this storm have the same return period. The unit hydrograph method is commonly used to calculate design runoff hydrographs. In this approach, the appropriate watershed is delineated (Section 11.1) and the drainage area measured. Next, a design storm is selected as discussed in Section 11.2. Then the losses from the design storm are calculated to obtain the rainfall excess as described in Section 11.3. From a time of concentration determination (Section 11.4.1), a unit hydrograph can be obtained (Section 11.4.2) for the watershed. Finally, the direct runoff rates resulting from the rainfall excess are determined as described at the end of Section 11.4.2. The estimated base flow rates are then added to the direct runoff rates to obtain the total runoff rates. A plot of the total runoff rates versus time will be the design runoff hydrograph. Example 11.7 The design-storm hyetograph for an urban stormwater structure is tabulated in columns 1 and 2 of Table 11.13. The losses from rainfall have already been calculated and tabulated in column 3 of the same table. Table 11.13 Rainfall Excess Calculations for Example 11.7

(1) t (min)

(2) i (in./hr)

(3) f (in./hr)

(4) ie (in./hr)

(5) de (in.)

0–10 10–20 20–30 30–40 40–50 50–60

0.5 2.5 4.6 2.5 1.7 1.0

0.5 1.3 1.0 0.7 0.5 0.4

0 1.2 3.6 1.8 1.2 0.6

0 0.2 0.6 0.3 0.2 0.1

Sec. 11.4

Design Runoff Hydrographs Table 11.14

Total (Design) Runoff Hydrograph Calculations for Example 11.7

(1) (2) (3) UH1/6 0.2UH1/6 t (min) (cfs/in.) lagged 10 min (cfs) 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120

0 65 160 340 480 400 315 240 180 120 80 43 20 8 0

0 13 32 68 96 80 63 48 36 24 16 8.6 4 1.6 0 0 0

425

(4) 0.6UH1/6 lagged 20 min (cfs)

0 39 96 204 288 240 189 144 108 72 48 25.8 12 4.8 0 0 0

(5) 0.3UH1/6 lagged 30 min (cfs)

0 19.5 48 102 144 120 94.5 72 54 36 24 12.9 6 2.4 0 0 0

(6) 0.2UH1/6 lagged 40 min (cfs)

0 13 32 68 96 80 63 48 36 24 16 8.6 4 1.6 0 0 0

(7) 0.1UH1/6 lagged 50 min (cfs)

0 6.5 16 34 48 40 31.5 24 18 12 8 4.3 2 0.8 0

(8) DRH (cfs)

(9) BF (cfs)

(10) TRH (cfs)

0 0 0 13 32 107 192 303.5 399 403 401 362.5 330.5 266.6 217 151.4 103.5 65.7 40 23 12 5.9 2 0.8 0

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

10 10 10 23 42 117 202 313.5 409 413 411 372.5 340.5 276.6 227 161.4 113.5 75.7 50 33 22 15.9 12 10.8 10

The ordinates of the 10-min (1/6-hr) unit hydrograph, UH1/6, are provided in column 2 of Table 11.14 for this urban watershed. The base flow (BF) is estimated to be constant at 10 cfs. Determine the direct runoff hydrograph (DRH) and the total runoff hydrograph (TRH; i.e., the design runoff hydrograph). Solution The rainfall excess calculations are summarized in Table 11.13. The rates of rainfall excess, ie, tabulated in column 4 are obtained by subtracting the rates of loss, f, from the rates of rainfall, i, at respective time periods. For example, during the time period between 20 and 30 min, rainfall and losses occur at the rates 4.6 and 1.0 in./hr, respectively, and the rate of rainfall excess is the difference, or 3.6 in./hr. The incremental excess rainfall depths, de, tabulated in column 5 are obtained by multiplying the rates of rainfall excess by the time increment of 10 min [(1/6) hr]. For example, for the time period from 20 to 30 min, de = (3.6 in./hr)(1/6 hr) = 0.6 in. The contribution to the runoff rates of the rainfall excess produced over this 10-min (1/6-hr) time period is 0.6 UH1/6. However, the runoff due to this rainfall excess is delayed by 20 min with respect to time zero when the storm starts. Accordingly the direct runoff hydrograph produced by the design storm is expressed as DRH = 0.2 UH1/6 (lagged 10 min) + 0.6 UH1/6 (lagged 20 min) + 0.3 UH1/6 (lagged 30 min) + 0.2 UH1/6 (lagged 40 min) + 0.1 UH1/6 (lagged 50 min)

426

Hydrology for Hydraulic Design

4.0

450

3.5

400 350 Total runoff rate

2.5

300 250

2.0

200 1.5

150

Rate of rainfall

1.0

Total runoff rate (cfs)

Rate of rainfall (in./hr)

3.0

100

0.5 0

Chap. 11

50 0

Figure 11.16

50 60 70 Time (min)

0 100 110 120

Rainfall hyetograph and total (design) runoff hydrograph for Example 11.7

and the calculations are presented in Table 11.14. It should be noted that the rainfall occurring during the first 10 min will not produce any runoff. The contributions to direct runoff of the rainfall excess during second through sixth 10-min periods are tabulated in columns 3 through 7 of Table 11.14. The entries in column 8 represent the direct runoff rates and are calculated as the sum of the entries in column 3 through 7 at respective times. The total runoff hydrograph is calculated as TRH = DRH + BF The total runoff rates tabulated in column 10 are calculated by adding the base flow (BF) rate in column 9 to the direct runoff rates in column 8 at respective times. The rainfall hyetograph and the total runoff hydrograph are plotted in Figure 11.16.

11.5 Storage Routing The design hydrographs developed in the last few sections can be viewed as flood waves. As these waves move downstream, their shape changes. If no additional inflow occurs between an upstream and downstream observation point, storage (in the channel and floodplain) reduces the flood peak and broadens the flood wave. To reduce the peak even more dramatically requires more storage, and flood-control reservoirs are constructed to take advantage of this benefit. Even though few major reservoirs are being planned or built in the United States, stormwatermanagement ponds are abundant in developing suburban areas. Storage routing is the process of evaluating the changes in a storm hydrograph as it passes through a pond or reservoir. In other words, an outflow hydrograph is developed using the inflow hydrograph, storage characteristics of the pond, and the hydraulic properties of the outlet device. A hypothetical experiment will help us understand the concept. In Figure 11.17, an empty barrel with a hole in its bottom is positioned under a spigot. The spigot is turned on (at t = 0) and held at a steady flow rate (Qin) until it is shut off (at t = t0). Initially, the inflow rate exceeds

Sec. 11.5

Storage Routing

427

Flow

Qin Qin

Qout

Storage

Qout Figure 11.17

Time

The hole-in-the-barrel experiment

the outflow rate of the barrel's hole and water begins to build up in the barrel (i.e., storage). The outflow rate increases over time as the depth (head) increases. It reaches a maximum when the spigot is turned off, because no additional inflow is available to increase the depth further. After the spigot is turned off, the barrel takes a while to empty. Figure 11.17 also depicts the inflow and outflow hydrographs for the experiment. Note the steady inflow and time-varying outflow. Also recall that the area under a hydrograph represents a volume of water. Thus, the area under the inflow hydrograph represents the volume of water that entered the barrel, and the area under the outflow hydrograph represents the volume of water that drained from the barrel. The area in between the inflow and outflow hydrographs represents the storage of water in the barrel. This storage accumulates over time until it reaches a maximum when the spigot is shut off (represented by the shaded area in Figure 11.17). From this time forward, the area under the outflow hydrograph represents the volume of water that drains from the barrel after time t0. This volume (area) must match the maximum storage volume previously defined. Also, the total area under the inflow hydrograph and the total area under the outflow hydrograph should be equal based on mass balance. Application of the conservation of mass (with constant density) is necessary to solve the storage routing problem mathematically. Simply stated, the change in storage is equal to inflow minus outflow. In differential form, the equation may be expressed as dS/dt = I - O

(11.23)

where dS/dt is the rate of change of storage with respect to time, I the instantaneous inflow, and O the instantaneous outflow. If average rates of inflow and outflow are used, then an acceptable solution can be obtained over a discrete time step (∆t) using ∆S/∆t = I - O

(11.24)

where ∆S is the storage change over the time step. Finally, by assuming linearity of flow across the time step, the mass balance equation may be expressed as ∆S = [(Ii + Ij)/2 - (Oi + Oj)/2]∆t

(11.25)

428

Hydrology for Hydraulic Design

Chap. 11

Flow Inflow

DS Dt Outflow Time Figure 11.18

Graphical representation of the storage equations

where subscripts i and j designate the flow at the beginning and end of the time step, respectively. Figure 11.18 depicts the variables in the equation. The linearity assumption improves by decreasing ∆t. The mass–balance relationship in Equation 11.25 contains two unknowns. Because the inflow hydrograph must be defined before the routing calculations are initiated, the inflow values (Ii and Ij) are known. Likewise, the time increment (∆t) is chosen and the outflow value at the beginning of the time step (Oi) was solved in the previous time-step calculations. That leaves the storage increment (∆S) and the outflow at the end of the time step (Oj) as unknowns. In fact, these two unknowns are related. As can be seen in Figure 11.18, as Oj increases, ∆S decreases. To solve the mass–balance equation requires another relationship between storage and outflow. Because both storage and outflow (for uncontrolled outlet devices) are related to the depth of water in the reservoir, they are related to one another. This relationship is employed to complete the solution. The data requirements to perform storage routing computations include the following: š inflow hydrograph (using SCS or other appropriate procedures), š elevation versus storage relationship for the reservoir, and š elevation versus discharge (outflow) relationship for the outlet device. Figure 11.19 displays these data requirements graphically. The procedure for obtaining the stage (elevation) versus storage curve is described in the figure. Also, the two major types of outlet devices are noted with typical stage-versus-discharge relationships. Reservoir routing is performed for a number of reasons including outlet device sizing, storage volume requirements, and evaluation of downstream flooding potential. Routing computations are often performed using the modified Puls method. The modified Puls (or level pool) routing method reformulates Equation 11.25 into (Ii + Ij) + [(2Si/∆t) - Oi] = [(2Sj/∆t) + Oj]

(11.26)

where the change in storage during the time step (∆S) was replaced by Sj - Si. The advantage of this expression is that all of the known variables are on the left side of the equation and all of the unknowns are grouped on the right. The solution procedure for the modified Puls routing method is as follows. 1. Determine the design inflow hydrograph for the reservoir. 2. Select a routing interval (∆t). Remember that linearity of inflows and outflows is assumed, so ∆t must be chosen accordingly. A generally good estimate is ∆t = Tp/10.

Storage Routing

429

Inflow hydrograph

Time

Stage-storage curve Procedure 1. Determine the area encompassed by the contours 2. Determine volume between contours (average end area formula or alternative) 3. Accumulate volumes

Stage (elevation)

Sec. 11.5

Storage (a)

Stage-discharge curve Weir

Stage

Orifice

Discharge Discharge Weirs (overflow spillways, risers, Orifices (culverts, submerged unsubmerged drop inlets: see gates and inlets: see Chapter 8, Chapter 9, Equation 9.9) Equation 8.19) Weir equation Orifice equation Q = CLH 3/2 Q = Cd A(2gh)1/2 (b)

Figure 11.19

Data requirements for storage routing

3. Determine the elevation–storage relationship for the reservoir site and the elevation– outflow relationships for the outlet device selected. 4. Establish the storage–outflow relationship using the following table: Elevation

Outflow (O)

Storage (S)

2S/∆t

(2S/∆t) + O

5. Graph the [(2S/∆t) + O] versus (O) relationship. 6. Perform storage routing computations using a table with the following headings. Time

Inflow (Ii)

Inflow (Ij)

(2S/∆t) - 0

(2S/∆t) + O

Outflow (O)

The solution procedure is explained in more detail using the following example problem.

430

Hydrology for Hydraulic Design

Chap. 11

Example 11.8 (adapted from Normann and Houghtalen*) A development is being planned in Albemarle County, Virginia, that encompasses a 365-acre watershed (undeveloped: CN = 61, Tc = 80 min). Virginia's stormwater management ordinance requires that the 2-year peak discharge after development (CN = 68, Tc = 70 min) be no greater than the 2-year peak discharge for the undeveloped conditions (170 cfs). A detention pond is proposed at the outlet of the watershed to meet the criterion. Determine the required storage and the outlet device size if the peak discharge after development is 241 cfs. Solution (a) Determine the appropriate inflow hydrograph for the reservoir. SCS procedures were used to determine the 2-year, 24-hr hydrograph after development. Table 11.15 lists the design flows coming into the proposed pond. (b) Select a routing interval (𝚫t). It appears from the inflow hydrograph that it takes about an hr for the hydrograph to peak. Based on ∆t = Tp/10, a 5-min time increment is chosen. (c) Determine the elevation–storage relationship for the reservoir site and the elevation–outflow relationships for the outlet device selected. The potential pond site is depicted on the contour map in Figure 11.20. It is located at the outlet of the basin and is currently being used as a farm pond. An earthen dam will be built, and corrugated metal pipes will be used to release pond outflows. The elevation–area-storage relationship is determined using the average end-area method and given in Table 11.16. Two 36-in. corrugated metal pipes (CMPs) will be used for the outlet device. Remember, the outlet device (size and number of pipes) must be designed to meet Virginia's stormwatermanagement criterion. At this point, they are trial sizes until the reservoir routing is completed to determine whether they meet design requirements. The CMPs will be placed on the existing stream bed (elevation 878 ft, MSL) with the earthen dam built over them. We will assume the CMPs operate as orifices (inlet control). Therefore, we will use Equation 8.19: Q = Outflow (O) = Cd A(2gh)1/2 where Cd is a discharge coefficient set equal to 0.6 (square edge), A the flow area of the two pipes, and h the driving head measured from the middle of the pipe openings to the water surface. Table 11.17 provides the information for the elevation–outflow relationship. (d) Establish the storage–outflow relationship. The storage–outflow relationship is established as summarized in Table 11.18. Table 11.15

Inflow Hydrograph for Example 11.8

Time

Flow (cfs)

Time

Flow (cfs)

Time

Flow (cfs)

11:30 11:35 11:40 11:45 11:50 11:55 12:00 12:05 12:10

8 9 11 12 13 15 18 27 43

12:15 12:20 12:25 12:30 12:35 12:40 12:45 12:50 12:55

72 119 164 210 240 241 227 202 181

13:00 13:05 13:10 13:15 13:20 13:25 13:30 13:35 13:40

156 126 114 100 85 79 67 59 52

*R. J. Houghtalen and J. M. Normann, Basic Stormwater Management in Virginia (Richmond, VA: Division of Soil and Water Conservation, Virginia Department of Conservation and Historic Resources, 1982).

Sec. 11.5

Storage Routing

431

882 ft

886 ft

884 ft

880 ft Dam 36-in. pipes Stream

Figure 11.20 Contour map of potential pond site

The first three columns in the table are obtained from the elevation–storage and the elevation–discharge information previously computed. The fourth column is obtained by doubling the storage in the third column and dividing by the routing interval we have chosen (∆t = 5 min = 300 s). Of course, the storage will have to be converted from acre-feet to cubic feet to get the desirable units (cfs) for 2S/∆t. Recall that there are 43,560 square feet in an acre. The last column is found by adding the second column (outflow, O) to the fourth column (2S/∆t). The second and third columns represent the relationship between outflow and storage

Table 11.16

Elevation–Area-Storage Relationship for Example 11.8

Elevation (ft, MSL)

Area (acres)

878

0.00

∆Storage (acre-ft)

Storage (acre-ft) 0.00

0.22 880

0.22

0.22 1.00

882

0.78

1.22 2.44

884

1.66

3.66 4.80

886

3.14

Table 11.17

8.46

Elevation–Outflow Relationship for Example 11.8

Elevation (ft, MSL)

Head (ft)

Outflow (cfs)

878 880 882 884 886

0.0 0.5 2.5 4.5 6.5

0 48 108 144 173

432

Hydrology for Hydraulic Design Table 11.18

Elevation (ft, MSL)

Storage–Outflow Relationship for Example 11.8

Outflow (O) (cfs)

878 880 882 884 886

Chap. 11

Storage (S) (acre-ft)

0 48 108 144 173

2S/∆t (cfs)

0 0.22 1.22 3.66 8.46

(2S/∆t) + O (cfs)

0 64 354 1,060 2,460

0 112 462 1,210 2,630

that is needed to solve a mass–balance equation that contains two unknowns. However, it is more convenient computationally to use the relationship between outflow and (2S/∆t) + O, as will be seen. (e) Graph the [(2S/𝚫t) + O] versus (O) relationship. Information from Table 11.18 is used to produce Figure 11.21. (f) Perform the storage (modified Puls) routing computations. The storage routing computations are summarized in Table 11.19. Column 1 (time) and column 2 (inflow) represent the design inflow hydrograph. Column 3 is column 2 moved up one time increment (i.e., Ij follows Ii by one time increment). The rest of the columns are unknown or blank when the routing process begins. The primary objective is to fill in the last column representing the outflow hydrograph or discharges from the detention pond. The footnotes below apply to the numbers in the table with superscripts. 1. To initiate the routing process when there is very little inflow, assume (2Si /∆t - Oi) equals zero because there is very little storage in the pond or outflow from the pond. 2. The first value of outflow is given as 6 cfs. If it was not given, then we could assume that the first value of outflow is equal to the first value of inflow. 3. The values in this column are found by applying the mass–balance equation stated in Equation 11.26 as (Ii + Ij) + [(2Si /∆t) - Oi] = [(2Sj /∆t) + Oj]

Outflow (O, cfs)

150

100

Figure 11.21

500

1000

1500 2S/Dt + O (cfs)

2000

2500

Routing relationship between [(2S/∆t) + O] and outflow (O)

3000

Sec. 11.5

Storage Routing

433

Table 11.19 Storage Routing Calculations for Example 11.8

Time

Inflow (Ii) (cfs)

11:30 11:35 11:40 11:45 11:50 11:55 12:00 12:05 12:10 12:15 12:20 12:25 12:30 12:35 12:40 12:45 12:50 12:55 13:00 13:05 13:10 13:15 13:20 13:25 13:30

8 9 11 12 13 15 18 27 43 72 119 164 210 240 241 227 202 181 156 126 114 100 85 79 67

(2S/∆t) - O (cfs)

Inflow (Ij) (cfs) +

9 11 12 13 15 18 27 43 72 119 164 210 240 241 227 202 181 156 126 114 100 85 79 67

01 35 5 4 5 5 4 9 13 26 71 166 308 496 699 879 1,014 1,101 1,140 1,126 1,072 996 897 783

(2S/∆t) + O (cfs) 3

17 23 28 29 33 38 49 79 128 217 354 540 758 977 1,167 1,308 1,397 1,438 1,422 1,366 1,286 1,181 1,061

Outflow (O) (cfs) 62 74 9 12 12 14 17 20 33 51 73 94 116 131 139 144 147 148 149 148 147 145 142 139

or, in this case, (8 + 9) + [0] = [17] for time 11:35

In words, we are determining the value of [(2Sj /∆t) + Oj] at time 11:35. The equation states that we need to add the current inflow to the previous inflow and the previous value of [(2Si /∆t) - Oi]. The plus signs in the table show which numbers are to be added. The outflow values, except for the first one mentioned in footnote (2), are obtained from the [(2S/∆t) + O] versus (O) relationship graphed in step "e" (Figure 11.21). For time 11:35, using the [(2S/∆t) + O] = 17 cfs value, we read a value of outflow approximately equal to 7 cfs using a more detailed graph than could be displayed in Figure 11.21. Note that you may interpolate Table 11.18 to obtain roughly the same answer. The values of [(2S/∆t) - O] are found using algebra. Simply double the outflow values in the last column and subtract it from [(2S/∆t) + O]; in this case, 17 - 2(7) = 3 cfs. After completing the routing for time 11:35, move on to time 11:40. Repeat the preceding steps 3 through 5 by determining 2S/∆t + O from the mass balance equation, then outflow (O) from the graph in step "e," and finally (2S/∆t - O) using algebra. Repeat for the rest of the inflow hydrograph.

434

Hydrology for Hydraulic Design

Chap. 11

Comments on the storage (reservoir) routing table: Note that the modified Puls routing procedure was stopped at time 13:25. At this point, the outflow has already begun to decrease. Because our primary concern was the peak outflow, the procedure was stopped. The inflow and the calculated outflow hydrographs are plotted in Figure 11.22. The attenuation and the time lagging of the peak discharge represent typical effects of detention basins on flood waves. Larger detention basins or smaller outlet devices cause more pronounced effects. The area between the two hydrographs represents the volume of water stored in the detention basin. The maximum storage occurs at the time the two hydrographs intersect (at around 13:00 in Figure 11.22). Before this time, the inflow rates are higher than the outflow rates, and therefore the detention basin is filling. Beyond this time, the outflow rates exceed the inflow rates, indicating that the detention basin is emptying. Also, it is not a coincidence that the peak outflow occurs when the two hydrographs intersect. The outlet devices are uncontrolled (i.e., not operator controlled gates), and thus the outflow increases with increasing storage. Therefore, the outflow rate is maximized at the same time the storage is maximized. The outflow peak of 149 cfs is less than our target outflow of 170 cfs. Thus, the design storm is slightly "overcontrolled" with the outlet device chosen. The routing procedure can be repeated using slightly larger pipes. Larger pipes will cost more, but the pond will not get as deep during the design storm. This will leave more usable land to develop and will probably offset the added pipe cost. The elevation of the pond at peak outflow (or any outflow) can be obtained using the elevation–outflow relationship developed in Table 11.17. By interpolation using the peak discharge of 149 cfs, the peak elevation is 884.3 ft MSL. (The peak storage can be determined from Table 11.16.) The peak elevation would normally establish the emergency spillway elevation. A different design storm would be used to size the emergency spillway (usually a weir) to keep the dam from overtopping during rare events (i.e., the 100-year storm). This stormwater-management pond is considered a "dry pond" that has water in it during and shortly after a storm. Eventually, it drains completely because the pipes have been placed on the stream bed. Dry ponds are often placed in the corner of a park or a ball field to make use of the land when it is dry. "Wet ponds" have water in them all the time. Therefore, the volume required for storage must be reserved above the normal pool elevation. (b) 250 I

Flow Rate (cfs)

200

150 O 100

11:50

11:45

12:00

Figure 11.22

12:15

12:30 Time

12:45

13:00

Inflow (I) and outflow (O) hydrographs

13:15

13:30

Sec. 11.6

Hydraulic Design: The Rational Method

435

11.6 Hydraulic Design: The Rational Method Peak flows resulting from rainfall events represent the major design requirement for many hydraulic structures (e.g., storm drainage inlets, stormwater pipes, drainage channels, and culverts). The statistical techniques covered in Chapter 12 are effective tools for obtaining peak flows and their associated probabilities on gauged streams. For ungauged streams, the design runoff hydrograph approach described in Section 11.4 can be used. A simpler procedure, the rational method, is available for peak flow design of structures such as culverts, storm drainage inlets, and stormwater pipes. The rational method is one of the oldest and most widely accepted hydrologic methods used to size such structures. The rational method is based on the equation Qp = C I A

(11.27)

where C is a dimensionless runoff coefficient, I the average rainfall intensity in in./hr of probability (P), and A the contributing drainage area in acres. Qp is the peak discharge in acre-in./hr or cfs (because 1 ac-in./hr is approximately equal to 1 cfs). Equation 11.27 can be used in the SI system with a consistent set of units (e.g., Qp in m3/s, I in m/s, and A in m2). The original determination of watershed runoff coefficients resulted from separate frequency analyses of rainfall and runoff (Chapter 12). In other words, rainfall and runoff with the same return periods were determined from gauge information, and the runoff coefficient was then found as the ratio of runoff to rainfall. The concept is now simplified, assuming that runoff always has the same return period as the rainfall that produces it. Thus the runoff coefficient, essentially a ratio of the rates of runoff to rainfall, varies from 0 (no runoff) to 1.0 (complete runoff). In practice, it is determined from a table based on the land use, soil type, and land slope (Table 11.20). From Table 11.20, it is evident and intuitive that runoff increases with increasing slope and imperviousness or decreasing vegetative cover and soil permeability. An areaweighted average C is used for watersheds with mixed land uses. Table 11.20

Typical Range of Values for Runoff Coefficients

Land Use (Soils and Slopes)

Runoff Coefficient (C)

Parking lots, roofs Commercial areas Residential: Single family Apartments Industrial Parks, open space Forest, woodlands Lawns: Sandy soil, flat ( 62,) Sandy soil, steep ( 77,) Clay soil, flat ( 62,) Clay soil, steep ( 77,) Crop lands: Sandy soil Loam soil Clay soil

0.85–0.95 0.75–0.95 0.30–0.50 0.60–0.80 0.50–0.90 0.15–0.35 0.20–0.40 0.10–0.20 0.15–0.25 0.25–0.35 0.35–0.45 0.25–0.35 0.35–0.45 0.45–0.55

436

Hydrology for Hydraulic Design

Chap. 11

The average rainfall intensity (I) is obtained from an intensity-duration-frequency (IDF) curve (Figure 11.4). As discussed in detail in Chapter 12, IDF curves are graphs of average rainfall intensity versus storm duration for a given geographic location. An array of different return intervals is typically plotted. Long-term rainfall records are used to develop these curves for hundreds of U.S. cities and many large cities throughout the world. Figure 11.4 suggests that the longer the storm, the less intense the rainfall, although the total rainfall amount increases. Again this matches our intuition: If it is raining intensely, then it is not likely to last long. To make use of an IDF curve for the rational method, we need to establish the design-storm duration. It can be shown that the peak discharge occurs when the entire watershed is contributing runoff to the design point. Therefore, the storm duration is set equal to the time of concentration. With the storm duration and the return interval, the rainfall intensity is obtained from the IDF curve. The intensity is assumed to be constant throughout the storm and is used to solve Equation 11.27. Because of this assumption and many others in the rational method, it is only applicable to small watersheds. 200 acres or 80 hectares is a commonly mentioned upper limit. Example 11.9 Estimate the 10-year peak discharge (Q10) before and after development of the 250-acre watershed described in Example 11.6. Assume the rational equation is appropriate even though the watershed is larger than 200 acres and the IDF curve in Figure 11.4 is applicable to the location of the watershed. Solution The time of concentration, Tc, was determined as 45 min for the predevelopment condition in Example 11.6. Setting the design storm duration equal to 45 min, the 10-year design-storm intensity, I, is found as 2.8 in./hr from Figure 11.4. Also a C value of 0.35 is selected from Table 11.20. The C value for open space is used, and a value from the high end of the range is selected because of the relatively high runoff rate of "C" soils and the steep land slope. Then, by using Equation 11.27. Qp = C I A = (0.35) (2.8 in./hr) (250 acres) = 245 cfs For postdevelopment conditions, the new time of concentration must be computed. Applying Equation 11.16, Tc = [L0.8(S + 1)0.7]/(1,140 Y 0.5) requires a determination of the soil moisture storage deficit (S). Thus, from Equation 11.6, S = (1,000/CN) - 10 = (1,000/80) - 10 = 2.50 in. where the curve number, CN = 80, is found in Table 11.7 (residential areas, 1([email protected] lots, C soil). Substituting the hydraulic length and watershed slope into Equation 11.16 yields Tc = [(4,500)0.8 (2.50 + 1)0.7]/[1,140(8)0.5] = 0.624 hrs (37.4 min) Applying Equation 11.27 again for postdevelopment conditions yields Qp = C I A = (0.45)(3.1 in./hr) (250 acres) = 349 cfs where C = 0.45 is found using Table 11.20 and I = 3.1 in./hr is obtained from Figure 11.4 (with a storm duration equal to the 37.4-min time of concentration). Judgment was used in selecting the C value for residential areas. The large lots tend to lower the C-value, but the tight soils and steep slopes increase the C value. Thus, a C value in the upper end of the range was selected. Note that the development increased the 10-year peak discharge from 245 to 349 cfs. This is the result of a combination of more runoff (higher C value) from the increased impervious area and a reduced

Sec. 11.6

Hydraulic Design: The Rational Method

437

time of concentration because runoff makes it receive streams more quickly (flowing through street gutters and storm pipes). These are typical consequences of urbanizing watersheds that have led to the establishment of stormwater-management ordinances and low-impact development initiatives to increase infiltration and decrease the speed of runoff.

11.6.1 Design of Stormwater-Collection Systems Stormwater-collection and transport systems represent one of the most costly and important components of our urban infrastructure. These systems collect stormwater runoff and convey it to a nearby stream, river, detention basin, lake, estuary, or the ocean in order to minimize the damage and inconvenience of urban flooding. Stormwater system components include street gutters, storm drainage inlets, stormwater pipes, manholes, and, in some cases, constructed channels, ponds, infiltration devices, and stormwater wetlands. The cost of stormwater systems depends greatly on the storm frequency (or recurrence interval) they are designed to convey. The optimal storm frequency is based purely on economics: when the cost of additional capacity exceeds the benefits. This is rarely done in practice because the analysis is costly and difficult. Instead, local governmental agencies set a design standard, typically a 10-year storm. Consequently, on the average of once every 10 years, the system will be overloaded, resulting in some minor street or low-land flooding. Problem areas may require a more stringent design standard. Storm drainage inlets remove water from the street. The most common types are grate, curb opening, and combinations (Figure 11.23). Locating inlets usually constitutes the initial phase of stormwater system design. The number of inlets, each of which must be served by a pipe, directly affects the cost of the system. To minimize cost, the flow in the street gutter must be maximized. In general, inlets are placed in the following locations: š in all sumps where water collects with no other outlet, š along the curb when the gutter capacity (curb height) is exceeded, š along the curb when the water has spread out into the street far enough to hinder traffic flow or safety (pavement encroachment or "spread" criteria are established by many local governments), š upgrade of all bridges (to prevent bridge icing in cold weather), and š along the curb before an intersection for traffic safety reasons. Inlet location design to limit street spread combines Manning's equation for gutter capacity and the rational equation for the rate of surface runoff. Given the street geometry (cross-section information and street slope), pavement encroachment criteria (or gutter height depending on

Grate

Curb opening Figure 11.23

Typical storm drainage inlets

Combination

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what limits the depth of flow at the curb), and Manning's roughness coefficient, the capacity of the gutter is established using 1/2 Q = AV = (1.49/n) A R2/3 h Se

(11.28)

with the variables defined in Equation 11.11. The drainage area contributing flow to the inlet is calculated by substituting the gutter flow found above into the rational equation, which is rearranged as A = Q/(C I)

(11.29)

Using a topographic map, the proposed inlet location is moved up or down the street until the area contributing surface flow to the inlet matches the drainage area computed above. Although this seems easy, the rational equation requires a rainfall intensity, which depends on a time of concentration. If the inlet has not been located, then the time of concentration cannot be established and an iterative process is necessary. The following example problem will clarify the procedure. Example 11.10 Curb opening inlets are needed on Barudi Street, which serve a single-family residential neighborhood (C = 0.35). A contour map of the area is shown in Figure 11.24 (approximate scale of 1 in. = 100 ft) along with the street cross-section geometry. The asphalt street (n = 0.015) has a 1/4-in./ft cross slope and a longitudinal street slope of 2.5%. The local government stipulates a 5-year design storm and allows 6 ft of pavement encroachment on each side of the 30-ft-wide street. How far west from the drainage divide should the first inlet be placed to adequately drain the north side of the street? The 2-year, 24-hr rainfall is 3.2 in. and the IDF curve in Figure 11.4 applies. Solution Based on the cross slope of 1/4 in./ft and the limiting spread of 6 ft, the depth of flow at the curb is 1.5 in. This results in a wetted perimeter of 6.13 ft and a flow area of 0.375 ft2. (Draw the triangular flow section and verify.) Applying Manning's equation (Equation 11.28) results in a gutter capacity of 1/2 Q = (1.49/n) A R2/3 h Se

Q = (1.49/0.015)(0.375 ft2)(0.375/6.13)2/3(0.025)1/2 Q = 0.914 cfs N

a 705' 704'

Drainage divide

702' 703'

Street cross section 701' Barudi Street

6 ft

Contour map Figure 11.24

Curb 30 ft Inlet location information for Example 11.10

Sec. 11.6

Hydraulic Design: The Rational Method

439

Next we will apply the rational equation to determine how much surface area can be drained before the gutter has reached its capacity. From the contour map, it appears that the time of concentration will involve primarily overland flow time from the drainage boundary to the street with very little gutter flow time. The greatest overland flow distance is about 100 ft (flow line a, b on the contour map; s ≈ 3,). Applying Equation 11.8 for sheet flow yields 0.4 Tt1 = [0.007(nL)0.8]/(P 0.5 2 s )

Tt1 = [0.0075(0.15)(100)60.8]/[(3.2)0.5(0.03)0.4] Tt1 = 0.139 hr = 8.3 min Table 11.9 provided the n value (short grass), and the contour map supplied the land slope and distance in the preceding equation. Adding a little gutter flow time to the sheet flow time above gives us a time of concentration of roughly 10 min, which is set equal to the storm duration for the rational method. It should be mentioned that some local governmental agencies will specify a time of concentration for stormwater system design (e.g., 5 or 10 min, depending on local spread criteria and topography). Using the IDF curve in Figure 11.4, a storm duration of 10 min produces an intensity of 5.2 in./hr for the 5-year storm. Substituting into the rational equation (Equation 11.29) results in A = Q/(C I) A = 0.914 cfs/[(0.35)(5.2 in./hr)] A = 0.502 acres (approximately 22,000 ft2) Because the drainage area on the north side of the street is roughly rectangular with a width of 100 ft, the inlet is placed about 220 ft down the street from the drainage divide. Unless located in a sump, an inlet rarely captures all of the gutter flow. The percentage of flow captured depends on the street slope, cross slope, flow rate, and type of inlet. Inlet manufacturers provide information on the capture efficiency of their inlets. A similar design procedure may be used to locate the second inlet. š The gutter capacity at the second inlet location is computed with Manning's equation. š The gutter capacity is reduced by an amount equal to the flow that passes by the first inlet. š With the rational equation, the area contributing flow to the second inlet is determined using the reduced flow due to the previous inlet bypass. A new time of concentration must be calculated. If all design factors remain the same (C value, street slope and geometry, pavement-encroachment criteria, time of concentration, and capture efficiency of the inlets) and the width to the drainage divide remains constant, then the inlet spacing from the first to the second inlet repeats itself down the street until a street intersection is reached.

11.6.2 Design of Stormwater Pipes The next phase of the design is stormwater-pipe sizing. The analysis begins at the first (highest) inlet and proceeds downhill to the outlet point. The rational equation is used to compute the design flow rate for each pipe. Manning's equation is used to obtain a pipe size capable of conveying the peak discharge while flowing just full (not under pressure). Previous calculations to determine the peak flow using the rational equation for inlet location cannot be used for stormwater-pipe sizing except at the first inlet. Inlet location design accounts for local surface water contributions only. Stormwater pipes must accommodate local

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surface water contributions at an inlet as well as flows from all upstream pipes. Therefore, the rational equation is applied using the entire upstream drainage area. In addition, an area-weighted runoff coefficient (C) may be necessary for the upstream drainage area. Finally, the time of concentration is computed using the longest combination of inlet flow time and pipe flow time to the design point (i.e., the entrance to the pipe being sized). Using this time of concentration, the rainfall intensity is obtained from the appropriate IDF curve. Stormwater-pipe design is not difficult but does require a large data-collection effort. Most of the necessary data can be gleaned from contour maps of the area. (A contour interval of 2 ft or less is often required.) A well-designed table or spreadsheet helps to assimilate the data and clarify the necessary calculations. The rational equation (Equation 11.27) is used to determine the peak flow (Qp, in cfs) that each stormwater pipe must convey. Manning's equation (Equation 11.28) is used to determine required pipe diameter (Dr, in ft) and can be written as Dr = c

nQp 0.4632So

3/8

(11.30)

where the pipe slope (So) is used in place of the energy grade line slope (Se). Because the required pipe diameter is not likely to be commercially available, the next standard pipe size larger is selected for design purposes. (Standard pipe sizes are usually available in 3-in. intervals from 12 to 24 in., 6-in. intervals from 24 to 48 in., and 1-ft intervals thereafter.) Because the flow time in each pipe is required to size downstream pipes (i.e., time of concentration calculations for the rational equation), velocities in pipes flowing partially full will be needed. Hence, the selected pipe diameter (D) is used to determine the full flow area (Af), the full flow hydraulic radius (Rf), and the full flow velocity (Vf). The appropriate equations are as follows: Af = πD2/4

(11.31)

Rf = D/4

(11.32)

1/2 Vf = (1.49/n) R2/3 f So

(11.33)

Once full flow conditions have been obtained, design aids like Figure 11.25 may be used to determine the actual flow depth (y), the flow velocity (V), and the flow time (t) for the design peak discharge (Qp) as it passes through the selected pipe. The example problem that follows describes the design procedure in detail. Stormwater-pipe design is often subject to certain design standards established by convention or local governmental agencies. Typical standards are as follows: š Because stormwater pipes are buried, a minimum cover over the crown of the pipe of 3 to 4 ft is required for structural and other reasons. š Pipe slopes match overlying ground slopes when possible to minimize excavation costs. š In flat topography, minimum slopes should produce velocities of 2 to 3 ft/s when flowing full to minimize sedimentation. š A minimum pipe diameter of 12 or 15 in. is required to reduce clogging problems. š Pipe sizes are never reduced downstream even if increased slopes provide adequate flow capacity. Again, clogging is the concern. š Manholes (or inlets) are provided at pipe junctions, changes of grade, and changes of alignment for constructability and maintenance reasons.

Sec. 11.6

Hydraulic Design: The Rational Method

441

Ratio of

depth of water in pipe y = diameter of pipe D

1.0

0.8 Q Qfull

0.6

0.4 V Vfull

0.2

0.4

0.6

0.8

1.0

1.2

Q/Qfull and V/Vfull

Figure 11.25 Hydraulic characteristics of pipes flowing partly full

š Maximum manhole spacing in straight reaches may be specified (e.g., 400 ft) because of cleaning equipment limitations. Longer distances are permitted for large pipe sizes. š Because flow through a manhole produces a small head loss, a drop in the invert elevation from the incoming to the outgoing pipe is recommended (say, 1 in. or 0.1 ft). Minor loss equations are available in some local design standards to evaluate this explicitly. Example 11.11 Design the stormwater-collection pipes that service a portion of a small town depicted in Figure 11.26. Inlet times (in min), drainage areas (in acres), and runoff coefficients for each inlet are displayed in the figure, along with ground elevations at each manhole (in ft, MSL). Stormwater pipe (concrete, n = 0.013) lengths are provided in the computation table (Table 11.21). Use the 5-year design storm (IDF curve in Figure 11.4) and a minimum pipe size of 15 in. Solution Table 11.21 facilitates the design. The process begins at the highest manhole (which collects flow from two inlets) and proceeds to the outlet point (Race River). Each column represents the computations for one pipe. Proceed from one column to the next until each pipe is designed. Description of Parameters Stormwater pipe: Length: Inlet time: Time of concentration:

Pipes are designated by upstream and downstream manhole numbers. Pipes lengths are obtained from appropriate maps. Inlet time of concentration includes sheet flow and gutter flow times. The time of concentration is the longest flow time to the entrance of the current pipe (design point) through any flow path. If there are no upstream pipes, then the Tc is the inlet time. Otherwise, compare the local inlet time to all other Tcs to upstream manholes plus the pipe flow time from that manhole to the design point. The largest time governs. For pipe 2 - 3, the inlet time of 17 min exceeds the Tc of pipe 1- 2 plus its travel time (15 + 1 = 16 min).

442

Hydrology for Hydraulic Design t(inlet) = 17 C = 0.6, A = 2.8 MH–2 (84.8)

t(inlet) = 15 C = 0.4, A = 2.8

t(inlet) = 12 C = 0.4, A = 2.5

MH–3 (78.8)

MH–3A (80.9)

t(inlet) = 12 C = 0.6, A = 2.5

t(inlet) = 10 C = 0.8, A = 2.2 MH–5 (68.1)

t(inlet) = 15 C = 0.4, A = 2.8 MH–1 (88.3)

t(inlet) = 12 C = 0.4, A = 2.4 MH–5A (73.1)

Chap. 11

t(inlet) = 14 C = 0.6, A = 2.5 MH–4A (72.7)

MH–4 (71.3)

MH–6 (66.6) Race River Figure 11.26

Table 11.21

Stormwater pipe design information for Example 11.11

Stormwater Pipe Design Calculations for Example 11.11

Stormwater Pipe

1–2

2–3

3A–3

3–4

Length (ft) Inlet time (Ti) (min) Time of concentration (Tc) (min) Runoff coefficient (C) R/F intensity (I) (in./hr) Drainage area (A) (acres) Peak discharge (Qp) (cfs) Slope (ft/ft) Required pipe diameter Dr (in.) Design pipe diameter(D) (in.) Full pipe area (Af) (ft2) Full pipe velocity (Vf) (ft/s) Full pipe flow (Qf) (cfs) Qp/Qf (or Q/Qf ) y/D V/Vf Flow depth (y) (in.) Pipe velocity (V) (ft/s) Pipe flow time (min)

350 15 15 0.4 4.3 2.8 4.8 0.01 13.4 15 1.23 5.28 6.48 0.74 0.63 1.11 9.45 5.86 1.0

300 17 17 0.5 4.1 5.6 11.5 0.02 16.3 18 1.77 8.43 14.9 0.77 0.65 1.13 11.7 9.52 0.5

350 15 15 0.4 4.3 2.8 4.8 0.006 14.8 15 1.23 4.09 5.02 0.96 0.78 1.17 11.7 4.78 1.2

250 12 17.5 0.45 4.0 10.9 19.6 0.03 18.5 24 3.14 12.5 39.3 0.50 0.50 1.02 12.0 12.8 0.3

Sec. 11.7

Hydrologic Modeling

443

Runoff coefficient:

The area-weighted runoff coefficient is calculated for the entire upstream drainage area. Information is obtained from Figure 11.26. For pipe 2 -3: C = [(2.8)(0.4) + (2.8)(0.6)]/5.6 = 0.5. Rainfall intensity: The rainfall intensity is found from the IDF curve (Figure 11.4). Drainage area: The total drainage area contributing flow to the pipe is determined. Peak discharge: The rational peak discharge using Equation 11.27: Qp = C I A. Slope: The pipe slope is found by dividing the pipe length by the surface elevation difference at the pipe ends (manholes). The pipe will be buried deep enough to meet minimum cover requirements. If a full flow velocity of 2 ft/s (or more stringent local criteria) is not met, then a greater slope will be required. Required pipe diameter: Use Equation 11.30 to determine Dr and convert to inches. Design pipe diameter: Based on Dr, use the next larger commercially available pipe size. (Minimum size = 15 in., 21 in. not available.) Full pipe area: Solve Equation 11.31 using the design pipe diameter, D (in ft). Full velocity: Solve Equation 11.33; Rf = D/4 (Equation 11.32 with D in ft). Full pipe flow: Solve the continuity equation: Qf = Af Vf. Qp /Qf (or Q/Qf ): Obtain the ratio of the rational peak (design) flow divided by the full pipe flow. y/D: Using the ratio of Q/Qf, obtain the ratio of depth of flow to pipe diameter from Figure 11.25. V/Vf : Using the y/D ratio and Figure 11.25, obtain the ratio of the actual (partially full) pipe flow velocity to the full pipe flow velocity. Flow depth: Using y/D and the pipe diameter, determine the flow depth (y). Pipe velocity: Using V/Vf and the full pipe velocity, determine the actual pipe velocity (V). Pipe flow time: The pipe flow time is found by dividing the pipe length by the pipe velocity and converting the answer to minutes.

11.7 Hydrologic Modeling There are many hydrologic computer models available that will quickly perform the sequence of calculations discussed in this chapter to obtain design runoff hydrographs. Some of these models are proprietary and costly, but others are freely available on the Internet. Development of some of these models started in the 1960s. They continued to be improved through the decades to a point where they are now quite versatile and user-friendly. Taken collectively, these hydrologic models have a broad range of capabilities. š š š š š š š š

Both rural and urban watersheds can be analyzed for flood flows and flow depths. Rainfall, infiltration, and other hydrologic processes can be modeled in various ways. Urban storm pipes, channels, and low-impact development devices can be designed. Reservoir routing can be performed to determine storage needs and design outlet devices. "What if" scenarios can be accomplished quickly by changing design parameters. Sensitivity studies can be performed on various input or calibration parameters. Model setup and data input are fast and intuitive by using graphical user interfaces. Model output is flexible and report-ready with accompanying tables and graphs.

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In this section, we will discuss two hydrologic models, one is available from the U.S. Army Corps of Engineers (ACE) and the other from the U.S. Environmental Protection Agency (EPA). The Corp's model was developed by their Hydrologic Engineering Center and is called the Hydrologic Modeling System (HEC-HMS). EPA's hydrologic model is called the Storm Water Management Model (EPA-SWMM). These models were selected for three reasons: 1. They are nonproprietary and freely available on the Internet. 2. They are fundamentally sound in handling a variety of applications. 3. They are widely used and accepted in the engineering and regulatory community. 11.7.1

The HEC-HMS Model

The HEC-HMS model was developed to determine design runoff hydrographs in rural and urban watersheds. The order of the tasks performed in the model corresponds to the rainfall–runoff processes described in this chapter: define the watershed, select a design storm, remove the losses to obtain the runoff, and generate a design runoff hydrograph. The process sequence, model structure, and model capabilities are described in the following paragraphs. Define the Watershed The watershed to be modeled is characterized by hydrologic elements such as sub-basins, stream reaches, junctions, and reservoirs. Once identified, these elements are interconnected and represent the total watershed as displayed in Figure 11.27. To complete the watershed definition process, each element requires input data, such as sub-basin areas, stream reach slopes, and outlet pipe and weir sizes for reservoirs (ponds). Rainfall–runoff computations proceed from upstream elements to downstream elements culminating with a design runoff hydrograph at the outlet of the watershed (design point). Select a Design Storm Two major types of design storms can be modeled in HEC-HMS: historical events and synthetic storms. Historical events require gauged rainfall data within or near the watershed. Various methods are available to distribute the rainfall over the watershed

Sub-basin 1

Junction 1

Sub-basin 2 Reach 1

Junction 2

Reach 2

Pond

Figure 11.27

Typical hydrologic elements in HEC-HMS watersheds

Sec. 11.7

Hydrologic Modeling

445

Figure 11.28

Data input interface for the frequency storm method (IDF data) Source: Hydrologic Modeling System HEC-HMS User's Manual, Version 4.0, U.S. Army Corps of Engineers, December, 2013.

when multiple gauges are used. Radar rainfall information may also be used if the watershed is divided up into gridded areas. Synthetic storm options include the frequency storm method (Figure 11.28, essentially a synthetic block hyetograph that was covered in Section 11.2.4) and the SCS hyetograph (covered in Section 11.2.5). Remove the Losses Infiltration usually produces the most significant loss of rainfall during design storms. In HEC-HMS, infiltration can be simulated using many different methods. One choice is the physically based Green and Ampt method (Figure 11.29, covered in Section 11.3.1). A commonly used procedure is the SCS loss method (covered in Section 11.3.2). There are many other infiltration methods that may be more appropriate in certain circumstances, such as doing continuous simulation of historical rainfall over months or years. Interception, depression storage, and evapo-transpiration are simulated with different algorithms.

Figure 11.29

Data input interface for the Green and Ampt infiltration method Source: Hydrologic Modeling System HEC-HMS User's Manual, Version 4.0, U.S. Army Corps of Engineers, December, 2013.

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Figure 11.30 Data input interface for the SCS unit hydrograph method Source: Hydrologic Modeling System HEC-HMS User's Manual, Version 4.0, U.S. Army Corps of Engineers, December, 2013.

Generate a Design Runoff Hydrograph Several methods are available to generate a design runoff hydrograph from the effective precipitation (i.e., runoff depth). This process is referred to as transformation in HEC-HMS. One transform option is the SCS unit hydrograph method (Figure 11.30, covered in Section 11.4.2). Other unit hydrograph methods (covered in Section 11.4.3) are available to choose from along with the more physically based, kinematic wave method. Base flows can be modeled in a variety of methods as well. Route Hydrograph through Reaches and Reservoirs The transformation methods in the previous computation step create hydrographs at the outlet of each sub-basin. However, they need to be routed downstream through channels and reservoirs and combined with other subbasin hydrographs (Figure 11.27). Once again, there are many methods to choose from including the modified Puls method (Figure 11.31, covered in Section 11.5). For reservoirs, the outlet device(s) may be simulated using an elevation–discharge relationship or by inputting the type and geometry of the outlet devices. Hydrologic Simulation and Output Review The structure of HEC-HMS requires three models to be formulated by the user: the meteorlogic model, the basin model, and control specifications (Figure 11.32). The primary input for the meteorlogic model is the rainfall information.

Figure 11.31 Data input interface for the modified Puls routing method Source: Hydrologic Modeling System HEC-HMS User's Manual, Version 4.0, U.S. Army Corps of Engineers, December, 2013

Sec. 11.7

Hydrologic Modeling

447

Project Requirements (Operational and Data Structure)

Meteorlogical Model (precipitation, snowmelt, and evapotranspiration) Figure 11.32

Basin Model (hydrologic elements are interconnected to represent watershed)

Control Specifications (start time, end time and time interval)

The HEC-HMS model structure

The basin model contains all of the data necessary to describe the watershed elements (i.e., subbasin, channels, junctions, reservoirs, etc.). The control specifications contain the starting time, ending time, and the modeling time interval. Once these models have been constructed, they are combined and the hydrologic simulation is initiated. Model output is displayed in tabular and graphical form. The primary output is a global summary table showing the peak discharge and volume for each basin element (Figure 11.33). Hydrographs and hyetographs for subbasin elements are also available (Figure 11.34). 11.7.2

The EPA-SWMM Model

The EPA-SWMM model was developed primarily to simulate the rainfall–runoff process in urban areas. Unlike most hydrologic models, it is capable of simulating pressure flow in pipes

Figure 11.33 Typical global summary table displaying HEC-HMS output Source: Hydrologic Modeling System HEC-HMS User's Manual, Version 4.0, U.S. Army Corps of Engineers, December, 2013.

448

Hydrology for Hydraulic Design

Depth (mm)

Chap. 11

Subbasin "Subbasin-1" Result for Run "Current"

0.4 0.8 1.2 1.6

Flow (cms)

3.0 2.5 2.0 1.5 1.0 0.5

03 :0 0 04 :0 0 05 :0 0 06 :0 0 07 :0 0 08 :0 0 09 :0 0 10 :0 0 11 :0 0 12 :0 0

Legend (Compute Time: 10 May 2012, 11:31:15)

16 Jan 1973

Run: Current Element: Subbasin-1 Result: Precipitation Run: Current Element: Subbasin-1 Result: Precipitation Loss Run: Current Element: Subbasin-1 Result: Outflow Figure 11.34 Typical hydrograph and hyetograph for a subbasin element Source: Hydrologic Modeling System HEC-HMS User's Manual, Version 4.0, U.S. Army Corps of Engineers, December, 2013.

and forecasting runoff quality as well as quantity. The runoff component of SWMM is very similar to HEC-HMS. Rainfall is generated over subcatchments, losses are removed, and the runoff moves over the land surface to the collection system. The runoff is then picked up by the transport component and moves through the collection and treatment system (i.e., pipes, channels, storage/treatment devices, and pumps) resulting in hydrographs and pollutant loads. To initiate the modeling process, the drainage system is characterized by hydrologic and hydraulic objects such as subcatchments, pipes (conduits) or channels, junctions, pumps, and reservoirs (storage units). Once identified, these objects are interconnected and represent the study area as displayed in Figure 11.35. To completely define the system, each object requires input data, such as sub-catchment areas, pipe lengths and slopes, and storage device volumes. Rainfall–runoff computations proceed from the rain gauges to the subcatchments and on to the collection system. The transport system is much more versatile than in HEC-HMS. It can accommodate pressure flow, account for backwater, and model pumps, reverse flows, and even loop flows prior to discharging at the outlet. Because of this, the data requirements are more extensive, including the bottom elevations and depths of junctions (usually manholes). SWMM has a variety of hydrologic modeling options. It is capable of simulating: š Historical or synthetic storms, including synthetic block and SCS hyetographs š Rainfall interception and depression storage

Sec. 11.7

Hydrologic Modeling

449

Raingauge Subcatchment

Junction Conduit Divider

Outfall Regulator

Storage unit

Pump

Figure 11.35 Typical hydrologic and hydraulic objects in a SWMM study area Source: Storm Water Management Model User's Manual, Version 5.0, U.S. Environmental Protection Agency, July 2010.

š Infiltration, including the SCS Curve Number and the Green and Ampt methods š Overland flow routing by the nonlinear reservoir method š Capture and retention by low-impact development (LID) practices. In addition, SWMM has a variety of hydraulic modeling options including: š š š š

A huge variety of standard open and closed conduit shapes as well as natural channels Special elements such as pumps, weirs, orifices, and storage/treatment devices Kinematic wave or full dynamic wave routing methods Dry weather sanitary sewer flow.

Finally, SWMM is able to simulate the build-up and wash-off of non-point source pollutants for different land uses. These pollutant loads can be routed through the drainage system and reduced using various best management practices (BMPs). Model output is displayed in tabular and graphical form. A "status report" table provides a continuity (mass balance) check. For the subcatchments (Figure 11.36), the sum of the losses

Figure 11.36 Portion of a typical status report (continuity check on subcatchments) Source: Storm Water Management Model User's Manual, Version 5.0, U.S. Environmental Protection Agency, July 2010.

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Chap. 11

Figure 11.37 Typical hydrographs for pipes or channels (called links in SWMM) Source: Storm Water Management Model User's Manual, Version 5.0, U.S. Environmental Protection Agency, July 2010.

Figure 11.38 Typical profile plot of stormwater pipes (shows surcharges and flow depths) Source: Storm Water Management Model User's Manual, Version 5.0, U.S. Environmental Protection Agency, July 2010.

and the runoff (outflow) must closely correspond to the rainfall (inflow). The status report also indicates if there is flooding at any of the nodes (manholes) and which pipes are surcharging (under pressure flow). Many of the results are also displayed in graphical form. For example, hydrographs can be requested for any of the pipes or channels (Figure 11.37). Another very useful graph is a profile plot for any of the pipes or channels (Figure 11.38). For stormwater collection systems, this graph indicates flow depths in the pipes and surcharging of manholes.

Problems

451

Problems (InTroducTIon and secTIon 11.1) 11.1.1. Components of the hydrologic cycle may be classified as follows: (a) water-holding elements, (b) liquid transport phases, and (c) vapor transport phases. Using Figure 11.1, apply one of these three descriptors to each of the components of the hydrologic cycle. Can you think of any other components of the hydrologic cycle which are not shown? Which of the three descriptors applies to them? 11.1.2. As water moves through the hydrologic cycle, water quality changes are common due to natural phenomena or anthropogenic pollution. Using Figure 11.1, describe how water-quality changes occur during each phase of the hydrologic cycle. For example, when lake water is evaporated, trace elements and salts are left behind producing a water-quality change. 11.1.3. Delineate a watershed using a topographic map and design point provided by your course instructor. Alternatively, obtain a sample topographic map from a USGS (or alternative) website, designate an arbitrary design point along a stream, and delineate the contributing watershed.

(secTIon 11.2) 11.2.1. Identify and define the components of a design storm. 11.2.2. A 24-hr rainfall depth of 5 in. in New York City has a 10% chance of being equaled or exceeded in any given year. Determine the return period for this rainfall event. Also, determine the probability that this event will be equaled or exceeded in both of the next 2 years. 11.2.3. A 3.0-cm rainfall occurs in a 6-hr duration. If this represents a 5-year storm, what is the probability it will be equaled or exceeded in any given year? What is the probability it will be equaled or exceeded each year for the next 3 years? What is the probability it will not be equaled or exceeded in the next 3 years? 11.2.4. The 20-year, 2-hr storm has a rainfall intensity of 1.8 in. per hr. Determine the probability of this storm being equaled or exceeded next year. Also, determine the probability that it will be equaled or exceeded in both of the next 2 years, at least once in the next 2 years, and not at all in the next 2 years. 11.2.5. Based on the design storm hyetograph given in Table 11.1, determine the percentage of rainfall that occurred in the first hr (i.e., 50% of the storm duration). Note that most of the rainfall occurred in the second quartile of the storm (time 0.5 to 1.0 hrs). Reproduce the hyetograph table to produce a first quartile storm. That is, rearrange the four intensities from largest to smallest. Now determine the percentage of the rainfall that occurs in the first hr of the storm. A spreadsheet program will be helpful. 11.2.6. An 8-hr storm possesses 2-hr storm intensities of 1.0, 1.8, 1.2, and 0.6 in./hr. Construct a storm hyetograph table that identifies the t/td and P/PT ratios. A spreadsheet program will be helpful. 11.2.7. A 1-hr storm possesses 15-min storm intensities of 4.4, 10.4, 5.6, and 2.8 cm/hr. Construct a storm hyetograph table that identifies the t/td and P/PT ratios. A spreadsheet program will be helpful. 11.2.8. Note that in Figure 11.4, rainfall intensity decreases with increasing storm length. Based on that fact, which storm do you think produces more rainfall; a 10-year, 30-min storm or a 10-year, 60-min storm? Verify your answer using Figure 11.4. 11.2.9. Construct a 25-year, 60-min design storm (synthetic block) hyetograph by using the IDF curves given in Figure 11.4. Use a time increment of 10 min. A spreadsheet program will be helpful. 11.2.10. Construct a 10-year, 120-min design storm (synthetic block) hyetograph. The average intensity for the 10-year design storm can be determined from the equation: iavg = 181.5/(td + 20.1) where iavg is in inch per hr and td is in minutes. Use a time increment of 20 min. A spreadsheet program will be helpful.

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11.2.11. Construct a 10-year, 60-min design storm (synthetic block) hyetograph. The average intensity for the 10-year design storm can be determined from the equation: iavg = 461/(td + 20.1) where iavg is in centimeters per hr and td is in minutes. Use a time increment of 10 minutes. Also, identify the peak intensity of the storm. Would the peak intensity change if the time increment was changed to 5 minutes? Why? A spreadsheet program will be helpful. 11.2.12. The SCS 10-year, 24-hr storm hyetograph for Virginia Beach, Virginia, was computed in Example 11.2. The storm hyetograph was based on a Type II rainfall distribution, even though the city was on the border of the Type III distribution. How would the results change from time 9 to 12 hrs if the Type III distribution was used? Also compare the peak intensity using the Type III distribution to the peak intensity from the Type II distribution. A spreadsheet program may prove helpful. 11.2.13. Determine the SCS 10-year, 24-hr storm hyetograph for Miami, Florida, for the time period from 9 to 13 hrs. Also determine the peak intensity for the storm. A spreadsheet program may prove helpful. 11.2.14. Determine the SCS 10-year, 24-hr storm hyetograph for Chicago, Illinois, for the time period from 10 to 13 hrs. Also determine the peak intensity for the storm. A spreadsheet program may prove helpful. 11.2.15. The total volume of rainfall in a watershed for any portion of a storm can be extracted from SCS storm hyetographs, or any hyetograph for that matter. For example, the following incremental depths of rainfall (cm) are recorded during a 45-min storm. Determine the total depth of rainfall for the 10 hectare urban watershed, the maximum intensity (cm/hr), and the total rainfall volume (m3). Time (min)

0 to 6

6 to 18

18 to 21

21 to 30

30 to 36

36 to 45

Rainfall depth (cm)

0.15

0.71

0.46

1.27

0.76

0.46

11.2.16. The SCS 10-year, 24-hr storm hyetograph for Virginia Beach, Virginia, was determined in Example 11.2. From this data, determine the 10-year, 6-hr storm hyetograph at the same location.

(secTIon 11.3) 11.3.1.

11.3.2.

11.3.3.

Most hydrologic (rainfall–runoff) computer models include the Green and Ampt procedures for determining infiltration rates in a watershed. It is a physically based model that has gained widespread acceptance. Examine the procedure described in Section 11.3.1 and answer the following questions: (a) Is there a relationship between apparent velocity in groundwater flow found using Darcy's Law (Equation 7.2) and fp in Equation 11.1. (b) Determine the lower and upper limits of fp in Equation 11.1. (c) What is the basis (underlying principle) for Equation 11.2? Most hydrologic (rainfall–runoff) computer models include the Green and Ampt procedures for determining infiltration rates in a watershed. Example 11.3 demonstrates how it is applied to find the infiltration losses from a design storm for a sandy, clay-loam soil. (Table 11.5 can be used to match the values of the soil properties used with the soil type.) Reproduce the results of Table 11.6 using a spreadsheet. Also, determine the total amount of rain that fell during the storm and the total amount of water that was infiltrated during the storm. Determine the infiltration losses from a 1.5-hr storm having quarter-hr intensities of 1.2, 2.4, 4.8, 3.6, 1.4, and 0.6 in./hr. Use the Green and Ampt infiltration procedures for the watershed that contains silt-loam soil with an initial degree of saturation of 40%. Also, determine the total amount of rain that fell during the storm and the total amount of water that was infiltrated during

Problems

453

the storm. (Note: Remember to change units when obtaining values from Table 11.5. Also, a spreadsheet program may be helpful.)

11.3.4.

11.3.5.

11.3.6.

11.3.7.

11.3.8.

11.3.9.

Time (hrs)

0–0.25

0.25–0.5

0.5–0.75

0.75–1.0

1.0–1.25

1.25–1.5

Intensity (in./hr)

1.2

2.4

4.8

3.6

1.4

0.6

Estimating infiltration parameters is challenging because soils are notoriously heterogeneous (spatially variable). The soil in Example 11.3 is a sandy, clay-loam based on the infiltration parameters used (see Table 11.5). Slight deviations in the parameters could easily produce anything from a silty clay to a sandy loam. Use a spreadsheet to reproduce the Green and Ampt solutions presented in Table 11.6. Then, do a sensitivity analysis on the Green and Ampt parameters to see which one has the greatest impact on the total infiltration. (a) Vary ϕ from 0.4 to 0.5 while holding other parameters at their original values. (b) Vary Pf from 9 to 29 while holding other parameters at their original values. (c) Vary K from 0.1 to 2.2 while holding other parameters at their original values. These Green and Ampt parameters represent the range of values given in Table 11.5 for soils that vary from silty clay to sandy loam. Determine the SCS runoff depth from a watershed located in Miami, Florida (United States), for a 10-year 24-hr storm. The watershed contains 25 acres of commercial development and 75 acres of townhouses. The soils in the region are sand and gravel. Also, determine the volume of runoff for the watershed in acre-feet. A 200-hectare watershed is scheduled to be developed into an industrial area. Currently, the watershed is an open space (fair condition) with 40% B-soils and 60% D-soils. Determine the increase in runoff depth from a 10-cm rainfall event due to developing the watershed. A 100-hectare watershed is composed of three different land uses: 20 hectares of golf course (40% in Drexel soils and the rest in Bremer soils), 30 hectares of commercial area (Bremer soils), and 50 hectares of residential area (1/2-acre lots and Donica soils). Determine the runoff volume (cubic meters) from a 15 cm storm. (Note: Drexel soils are course to fine textured loams, Bremer are moderately fine to fine textured, and Donica are sandy soils.) An agricultural watershed containing row crops possesses two different soil groups; 18 hectares of B soil and 42 hectares of A soils. Determine the hourly runoff depth (cm) from the rainfall event tabulated below. A spreadsheet program may prove helpful. Time (hrs)

0 to 1

1 to 2

2 to 3

3 to 4

4 to 5

Intensity (cm/hr)

1.3

10.2

3.8

2.5

1.3

The owner of a golf course (150 acres, good condition, C soils) has decided to sell it to a developer. The plans are to convert it into residential housing (one-half acre lots). Determine the runoff depths (inches) in half-hour time increments from the 3-hr design rainfall event tabulated below. A spreadsheet program may prove helpful. Also determine how much the total runoff depth and runoff volume increases from this change in land use. (Note: Increases in runoff volume and peak flows from development are the reason stormwater management ordinances are passed and low impact development devices are installed.) Time (hrs) Intensity (in./hr)

0 to 0.5

0.5 to 1

1 to 1.5

1.5 to 2

2 to 2.5

2.5 to 3

0.7

1.4

2.3

1.8

1.0

0.4

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(secTIon 11.4) 11.4.1.

11.4.2.

11.4.3.

11.4.4. 11.4.5. 11.4.6.

11.4.7.

11.4.8.

11.4.9.

Determine the time of concentration for the runoff from a concrete parking lot. The overland flow segment is 25 m long with a slope of 0.5%. The overland flow segment moves water into a shallow V-channel (dropping 2 meter in elevation over 100 meters in length). Finally, the storm water is picked up in a 400-m-long, 2-m-wide, rectangular channel flowing 0.5 m deep. The concrete channel has a slope of 1.0%. Determine the time of concentration in minutes. The 2-year, 24-hr rainfall depth is 4 cm. Also determine what percent of the time of concentration is attributed to the 25 m of overland flow. The hydraulic length (longest flow path) for a watershed is 2,800 ft. Along this path, runoff initially travels over the land surface (dropping 3 ft in elevation over 150 ft in length through short grass) and then moves into shallow concentrated flow for a distance of 450 ft with an average velocity of 2.0 ft/s. The remainder of the travel path is through a 2-ft diameter (concrete) pipe flowing half full on a slope of 1%. Determine the time of concentration in minutes. The 2-year, 24-hr rainfall depth is 3.4 in. Also determine what percent of the time of concentration is attributed to the 150 ft of overland flow. A 200-acre watershed in Vermont is currently being used to grow and harvest hay (meadow). The watershed contains B soils with an average land slope of 7%. However, next year the land use is scheduled to change to row crops (contoured). The hydraulic length of the watershed is 3,200 ft. Determine the time of concentration before and after the land use change occurs. What are unit hydrographs and where is the concept used? What are synthetic unit hydrographs and why are they important? Referring to Example 11.6, plot the unit hydrograph using a spreadsheet program. Determine the volume of runoff in acre-feet and the depth of runoff in inches. Referring to Example 11.6, determine the SCS unit hydrograph after development if half the watershed is developed commercially and the rest with townhouses. Find flow rates in the unit hydrograph for t/tp ratios of 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.4, 2.8, 3.2, 3.6, and 4.0. A spreadsheet program may prove helpful. A 162-hectare watershed is being developed into an industrial park. The hydraulic length of the watershed is 1,610 m, the average land slope is 2%, and the soils are primarily clay. Determine the SCS synthetic unit hydrograph after development (i.e., flow rates for t/tp ratios of 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.4, 2.8, 3.2, 3.6, and 4.0). A spreadsheet program may prove helpful. A 300-acre watershed just south of Chicago is currently undeveloped (woods). The soils are primarily clay, the hydraulic length of the watershed is 3,280 ft, and the average land slope is 1%. Determine the SCS synthetic unit hydrograph before development (i.e., flow rates for t/tp ratios of 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.4, 2.8, 3.2, 3.6, and 4.0). A spreadsheet program may prove helpful. Example 11.7 provided information on losses from a design rainfall event. Low impact development (LID) devices have been installed in the watershed to enhance infiltration. The new loss rates are listed below. Assuming the unit hydrograph from Example 11.7 is still appropriate, determine the new total (design) runoff hydrograph. Also determine the percent decrease in runoff depth and peak flow that can be attributed to the LID devices? A spreadsheet program may prove helpful. Time (min)

0 to 10

10 to 20

20 to 30

30 to 40

40 to 50

50 to 60

f (in./hr)

0.5

1.9

1.6

1.3

1.1

1.0

11.4.10. The 1-hr unit hydrograph (UH1) for the Ty River watershed at the Chamberlain Avenue crossing (design point) is provided in the table below. Determine the total runoff hydrograph (TRH) at that location if a 4-hr storm produces 2.5 cm of rain in the first hr, 4.5 cm of rain in the second

Problems

455

hour, 3.5 cm of rain in the third hour, and 2 cm of rain in the final hour. Assume losses to rainfall are constant at 0.5 cm/hr, and the base flow is constant at 2 m3/s. A spreadsheet program may prove helpful. Time (hrs) 3

Q (m /s)

0.5

1.5

2.5

3.5

11.4.11. A 2-hr unit hydrograph (UH2) for the Wright Water Creek watershed is given below. Determine the watershed's response to a 6-hr design storm (i.e., the TRH) in which 3 in. of rain falls in the first 2 hrs, 4.5 in. in the next 2 hrs, and 1.5 in. in the last 2 hrs. Assume runoff depth is two-thirds of the rainfall and base flow is 50 cfs. A spreadsheet program may prove helpful. Time (hrs)

Q (cfs)

300

800

1,200

1,000

700

400

200

11.4.12. A 30-min unit hydrograph (UH1/2) for the No Name Creek watershed (922 acres) is given below. Determine the stream flow (TRH) that would result from a rainfall event on the watershed if the excess rainfall (runoff) in a 1-hr storm in 15-min increments is 0.75, 0.75, 0.25, and 0.25 in. Assume a base flow of 20 cfs. Also determine the runoff volume in acre feet. A spreadsheet program may prove helpful. Time (min)

105

120

Q (cfs)

240

640

1,000

840

560

320

120

11.4.13. The 30-min unit hydrograph (UH1/2) for an urban watershed is provided in the table below. Using the UH1/2, compute a 1-hr unit hydrograph (UH1). Hint: A 1-hr unit hydrograph is the result of 1 in. of runoff, a half inch of the runoff coming in the first half hour and a half inch coming in the second half hour. Once this is done, determine whether the 15-min unit hydrograph be computed in a similar way? A spreadsheet program may prove helpful. Time (hrs) 3

Q (m /s)

0.25

0.5

0.75

1.0

1.25

1.5

1.75

2.0

2.25

11.4.14. The flows on Wolf Creek tabulated below are the result of a 1-hr storm that produces 2 cm of runoff of uniform intensity. Determine the peak flow and time to peak of a 4-hr design storm that generates 2 cm of runoff in the first hour, 3 cm of runoff in the second hour, 2.5 cm of runoff in the third hour, and 1.5 cm of runoff in the last hour. Assume the base flow is negligible. A spreadsheet program may prove helpful. Time (hrs)

0400

0500

0600

0700

0800

0900

1000

1100

1200

Q (m3/s)

(secTIon 11.5) 11.5.1. Referring to Example 11.8, continue computations in the routing table through time 13:45. Additional inflows are provided in the table below. Also graph the inflow and outflow hydrographs on the same set of axes. Finally, determine the maximum elevation and storage attained in the pond during the design storm. A spreadsheet program may prove helpful.

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Time

13:30

13:35

13:40

13:45

13:50

Inflow (Ii), cfs

11.5.2. Build a 2S/∆t + O versus O relationship in 0.3-m increments for a pond that is expected to reach a depth of 1.5 m over the top of the service spillway (normal pond level). The elevation–storage relationship for the pond is described by S = 650 h1.2 where S = storage in the pond in m3 and h = water level in meters above the service spillway. The discharge over the spillway is described by O = kwL(2g)0.5 h1.5 where O = discharge in m3/s, kw = 0.45 (discharge coefficient), L = 0.50 m (spillway crest length), and g = gravitational acceleration. Initially, h = 0 and S = 0 and the routing interval (∆t) is 8 min. A spreadsheet program may prove helpful. 11.5.3. Build a 2S/∆t + O versus O relationship in half-foot increments for an underground cistern that is 2 ft deep, 6 ft long, and 3 ft wide. A 6-in-diameter drain on the side of the cistern acts as an orifice with a discharge coefficient of 0.6. The invert of the drain is at the bottom of the cistern. Assume a routing interval of 10 s. A spreadsheet program may prove helpful. 11.5.4. The [2S/∆t + O] versus Outflow (O) table for a large flood control reservoir is shown below. In addition, a storage routing table is provided for a 5-day flood wave that passes through the reservoir. Fill in the blank spaces (?) of both tables and determine the peak outflow and the peak elevation that occurred during the flood. Note that the routing interval is 12 hrs and 1 acre = 43,560 ft2. Elevation (ft, MSL)

Outflow (O) (cfs)

Storage (S) (acre-ft)

2S/∆t (cfs)

(2S/∆t) + O (cfs)

865 870 875 880 885

0 20 50 130 ?

0 140 280 ? 1,220

? 282 ? 1331 2,460

0 ? 615 1461 2,780

Day/Time

Inflow (Ii) (cfs)

Inflow (Ij) (cfs)

1—noon midnight 2—noon midnight 3—noon midnight 4—noon midnight 5—noon midnight 6—noon

2 58 118 212 312 466 366 302 248 202 122

58 118 212 312 466 366 302 248 202 122 68

(2S/∆t) - O (cfs) (2S/∆t) + O (cfs) Outflow, O (cfs) ? ? 198 444 802 1,286 1,668 1,824 1,850 1,798 1,672

? 228 528 968 1,580 2,118 2,336 2,374 2,300 2,122

? ? 15 42 83 147 225 256 262 251 225

11.5.5. A portion of a reservoir routing table is given below (90 min into the storm). Fill in the blanks (?) in the table. Also determine the outflow, stage (H), and volume in the reservoir at time 120 min. Note: The routing interval is 10 min. Stage (H) (m)

Outflow (O) (m3/s)

Storage (S) (m3)

(2S/∆t) + O (m3/s)

0.00 0.15 0.30 0.45 0.60

0.000 0.085 ? 0.481 0.860

0 1,230 2,470 ? 4,950

? ? 8.46 12.8 17.4

Problems

457

Time (min)

Inflow (Ii) (m3/s)

Inflow (Ij) (m3/s)

(2S/∆t) - O (m3/s)

(2S/∆t) + O (m3/s)

Outflow, O (m3/s)

90 100 110 120

1.42 1.27 1.22 0.85

1.27 1.22 0.85 0.74

13.3 ? ? 15.7

14.6 ? ? 17.4

0.62 ? ? 0.85

11.5.6. Referring to Example 11.8, perform the storage routing computations again using a 10-min routing increment starting with the flow at time 11:30 and progressing to 11:40, then 11:50 etc. and dropping the flows in between. Will this require a new [(2S/∆t) + O] versus (O) relationship? If so, revise this prior to performing the storage routing computations. Compare the new peak outflow with the peak outflow from Example 11.8 (149 cfs). A spreadsheet program may prove helpful. 11.5.7. The stage–storage relationship at a pond site is described by S = 450 h1.2 where S = storage in the pond in m3 and h = water level in meters above the spillway crest. The discharge over the spillway is described by O = kwL(2g)0.5 h1.5 where the discharge (O) is in cubic meter per second, kw = 0.40 (discharge coefficient), L = 1.0 m (spillway crest length), and g = 9.81 m/s2 (gravitational acceleration). Initially, h = 0 and S = 0. Build a [2S/∆t) + O] versus (O) relationship for pond stages in 0.2-m increments up to 0.6 m over the spillway crest. Perform storage routing computations for the inflow hydrograph provided below to determine the peak outflow and stage. A spreadsheet program may prove helpful. Time (min)

Inflow (Ii) (m3/s)

Time (min)

Inflow (Ii) (m3/s)

0 8 16 24 32

0.00 0.20 0.48 0.85 0.74

40 48 56 64 72

0.62 0.49 0.37 0.22 0.10

11.5.8. A stormwater detention pond has the storage characteristics listed in the table below. Outflow is over a weir with a crest elevation of 5.5 m according to the equation Q = 1.83 # H 3/2 where H is the depth of water above the weir crest elevation. Given the inflow hydrograph (also in a table below), perform the reservoir routing computations. Also determine the peak flow, peak elevation, and peak storage? A spreadsheet program may prove helpful. Pond Elev. (m) 3

Storage, S (m ) Time (min) 3

Q (m /s)

5.0

5.5

6.0

6.5

7.0

690

1520

2510

3650

120

150

180

210

240

270

0.4

1.2

2.4

3.6

3.1

2.1

1.6

1.1

0.5

11.5.9. An empty water storage tank (cylindrical shape), with a diameter of 66 ft and a height of 5 ft, is being filled at a rate of 88 cfs. However, it is also losing water (due to a hole in the bottom of the tank) at the rate of O = 14.5 (h)0.5 where h is the depth of water in feet and "O" is the outflow rate in cfs. Determine the depth of water in the tank after 4 min. Use a depth interval of 1 ft for the 2S/∆t + O versus O relationship and a routing interval of 30 s.

(secTIon 11.6) 11.6.1. Determine the 25-year peak discharge for a 20-acre watershed that is 40% single family residential and 60% apartments. The longest flow path is 50 ft of sheet flow (through short grass on a slope of 0.8%; the 2-year, 24-hr rainfall depth is 3.3 in.) and 1,460 ft of shallow concentrated flow in paved channels on a slope of 1.0%. Use Figure 11.4 for the rainfall intensity.

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11.6.2. A hilly, 20-acre watershed is composed of pasture land with dense grass and clay soils. The peak discharge is required for a culvert installation. The hydraulic length of the watershed is 1,200 ft, and the average watershed slope is 7%. The watershed is slated for commercial development in the next 2 years. Using the IDF curve in Figure 11.4, determine the 10-year peak discharge before and after development. 11.6.3. Determine the 10-year peak discharge (in m3/s) required to design a pipe that will carry stormwater away from a parking lot. The large, circular parking lot with a diameter of 140 m is being constructed as overflow parking for a sports center. All stormwater will drain as sheet flow to the center, flow into a drop inlet, and travel through a pipe system to the nearest stream. The parking lot will be short grass (clay soil) with a slope of 2% toward the center. The 2-year, 24-hr rainfall is 73 mm. The average intensity for the 10-year design storm can be determined from I = 438/(td + 20.1) where I is in centimeter per hour and the storm duration (td) is in minutes. 11.6.4. A 150-acre forested watershed with a time of concentration of 95 min is partially developed with the following land uses: 40 acres of single family homes, 60 acres of apartments, and the rest in native forest. After development, the longest flow path is 4,920 ft. A 100-ft sheet flow segment passes through woods with light underbrush on a 2% slope. An 800-ft segment of shallow concentrated flow occurs next over an unpaved area on a 1% slope. Channel flow occurs over the remaining distance on a slope of 0.5% in a stony natural stream. A typical cross-section of the stream has 18 ft2 of flow area and 24 ft of wetted perimeter. The 2-year, 24-hr precipitation is 3.8 in. Using the IDF curve in Figure 11.4, determine the 2-year peak discharge before and after development. Also determine the percent increase in peak flow due to the development in the watershed. 11.6.5. The paved parking lot depicted in Figure P11.6.5 drains as sheet flow into a concrete, drainage channel. The rectangular channel discharges into a stormwater management pond. Determine the 2-year peak discharge. Assume the 2-year, 24-hr rainfall is 1.1 in. Use Figure 11.4 to obtain the rainfall intensity. (Hint: Assume the channel flows at a depth of 1 ft to determine channel flow time.) 270´ 1´ Drainage channel (0.5% Slope)

2´

0.5% Slope

600´

Channel cross section

Figure P11.6.5

11.6.6. Stream flow data from a gauged watershed is used to determine the 2-year and the 10-year peak discharge, 44 and 72 cfs, respectively. The 50-acre watershed is the site of an industrial park with 55% of the watershed left in native woodlands. If the IDF curve in Figure 11.4 is applicable to the region, estimate the time of concentration of the watershed. 11.6.7. Referring to Example 11.10, describe subjectively (no analysis required) how the location of the inlet would change (one change at a time, not collectively) if (a) the pavement encroachment criterion is 5 ft. (b) the longitudinal street slope is 2%. (c) the street cross slope is 3/8 in./ft. (d) the 2-year design storm is the standard. (e) the land use was apartments. Also, are inlets required for the south side of the street? Why or why not?

Problems

459

11.6.8. Referring to Example 11.10, locate the first inlet if all of the following changes in the design conditions are taken collectively: (a) the 2-year design storm is the standard. (b) the pavement encroachment criteria is 5 ft. (c) the lawns are dense grass. (d) the street cross slope is 3/8 in./ft. All other design conditions remain the same. 11.6.9. Referring to Example 11.10, locate the next downstream inlet. Assume the street slope has increased from 2.5% to 3.5% and the first inlet intercepts 70% of the gutter flow. All other data can be obtained from Example 11.11. 11.6.10. Determine the pipe size (concrete) that would be required to convey the design flow (3.8 cfs) away from first inlet in a stormwater collection system. The street slope is 1.5% and the minimum pipe size specified by community standards is 15 in. Also, determine the depth of flow in the pipe at peak flow and the travel time through the 150-ft-long pipe. 11.6.11. Determine the pipe size (concrete) that would be required to convey the design flow (0.15 m3/s) away from first inlet in a stormwater collection system. The street slope is 2.0% and the minimum pipe size specified by community standards is 40 cm. Also, determine the depth of flow in the pipe at peak flow and the travel time through the 50-m-long pipe. 11.6.12. Referring to Example 11.11, repeat the design up to MH-4 given the following changes in the design conditions (taken collectively, not one at a time): (a) the minimum pipe size is 18 in. with next larger size 24 in. (b) all inlet times are 10 min. (c) the runoff coefficient into MH-3A is 0.5. All other design conditions remain the same. 11.6.13. Referring to Example 11.11, finish the problem by designing the stormwater collection pipes from MH-4A to MH-6. The pipe lengths are as follows: MH4A to 4 (350 ft); MH4 to 5 (320 ft); MH5A to 5 (250 ft); MH-5 to 6 (100 ft). 11.6.14. Design the stormwater collection pipes (concrete; minimum size of 12 in.) for the housing subdivision shown in Figure P11.6.14. The data for each of the drainage areas (basins) contributing flow to the manholes is provided in the table below, including the inlet time (Ti) and the runoff coefficient (C). In addition, stormwater pipe information is given, including the ground elevation at each manhole. The rainfall intensity–duration relationship for the design storm can be described by i = 18/(td)0.5, where i = intensity in inch per hour and td = storm duration = time of concentration in minutes. A spreadsheet program may prove useful.

Basin 1 A Inlet/manhole Basin 3

Basin 2

Storm pipe

Basin 4

Surface runoff C

B Figure P11.6.14

River

460

Hydrology for Hydraulic Design

Chap. 11

Basin

Area (acres)

Ti (min)

Stormwater Pipe

Length (ft)

Upstream Elev. (ft)

Downstream Elev. (ft)

1 2 3 4

2.2 1.8 2.2 1.2

14 10 10 10

0.3 0.4 0.3 0.5

AB CB BD DR

200 300 300 200

23.9 24.4 22.9 22.0

22.9 22.9 22.0 21.6

11.6.15. Design the stormwater collection pipes in Problem 11.6.14 given the following data changes: (a) the upstream elevation for pipe AB is 24.9 ft instead of 23.9 ft, (b) the inlet time for Basin 1 is 12 min instead of 14 min, and (c) the inlet time for Basin 3 is 13 min instead of 10 min. The changes are to be taken collectively, not one at a time. A spreadsheet program may prove useful. 11.6.16. Design the stormwater collection pipes in Problem 11.6.14 given the following data changes: (a) the minimum full flow velocity is greater than 2.5 ft/s, (b) upstream elevation for pipe AB is 24.9 ft instead of 23.9 ft, (c) the upstream elevation for pipe CB is 23.5 ft instead of 24.4 ft, (d) the inlet time for Basin 1 is 12 min instead of 14 min, (e) the inlet time for Basin 2 is 14 min instead of 10 min, (f) the inlet time for Basin 3 is 13 min instead of 10 min, and (g) the downstream elevation for pipe DR is 22.0 ft instead of 21.6 ft. The changes are to be taken collectively, not one at a time. A spreadsheet program may prove useful.

(secTIon 11.7) 11.7.1.

11.7.2. 11.7.3.

Review Section 11.7.1 on the capabilities and use of HEC-HMS, the U.S. Army Corps of Engineers' hydrologic modeling program. Then download the program from the website: http:// www.hec.usace.army.mil/software/hec-hms/downloads.aspx Examine the program's functions, and then apply the program to Example 11.8. There will only be two hydrologic elements in your basin model, a subbasin and a reservoir. Use the SCS CN method for losses (initial abstraction is 0.1 in. post development) and the SCS UH transform method. Recall that lag time is 60% of the time of concentration. The meteorological model is the SCS storm, Type 2 with a 2-year, 24-hr R/F depth of 3.5 in. Make sure you link the precipitation to the subbasin and your control specifications last for 24 hrs (with an arbitrary future date). Verify that the solution determined using HEC-HMS closely corresponds to the solution in the example problem. Slight variations in the two solutions are to be expected since HEC-HMS is capable of using very small time steps and a large number of computational iterations. Note that you could input the storm hydrograph directly using a source element in your basin model and eliminate the subbasin element. Use HEC-HMS to solve a watershed problem assigned to you by your professor. Review Section 11.7.2 on the capabilities and use of EPA-SWMM, the U.S. Environmental Protection Agency's stormwater modeling program. Then download the program from: http:// www2.epa.gov/water-research/storm-water-management-model-swmm. Examine the program's functions, and then apply the program to Example 11.11. Verify that the solution determined using EPA-SWMM matches the solution in the example problem. Note that the pipe system can be represented by the model exactly (i.e., the hydraulics). However, the model does not have the rational method capability (i.e., the hydrology). Use a rainfall gauge with a constant intensity storm for a duration equal to the time of concentration (the rational storm). Then, use the SCS method for infiltration and the kinematic wave model for the transform. Adjust parameters until the peak flows out of the subbasins matches the example problem. Once you are satisfied with

Problems

11.7.4.

11.7.5.

461

the hydrology, run a simulation to determine if the pipe sizes from the example problem pass the design flow without surcharging. Use EPA-SWMM to solve any of the chapter problems from 11.6.12 through 11.6.16. Verify that the solution determined using the model matches the solution you obtained using a solution table or Excel. Note that the model is not able to run the rational method to determine peak flows from the subbasins. Therefore, an alternative hydrologic simulation must be performed and the parameters adjusted until the peak flows are matched. Once that is accomplished, the designed pipe sizes should be able to accommodate the design storm without surcharging. Use EPA-SWMM to solve an urban stormwater problem assigned to you by your professor.

(bonus Problems—WaTer budgeTIng) Water budgeting is a widely used principle in hydrologic design. It forms the basis for the hydrologic cycle and is the key to watershed studies and sustainable water supply assessments. Water budgeting is based on the principle of mass balance. When constant density is assumed, it reduces to a volume balance. It was previously used in this chapter in the development of the reservoir routing technique. In differential form, the equation may be expressed as dS/dt = I - O

(11.23)

where dS/dt is the rate of change of storage with respect to time in a control volume, I the instantaneous inflow, and O the instantaneous outflow. The control volume (bounded by a control surface) may be a watershed, a reservoir, a storage tank, or a groundwater aquifer. If all inflows and outflows can be quantified over a discrete time step ∆t, the equation becomes ∆S = I∆t - O∆t

(11.23a)

where ∆S is the storage change over the time step. The principle is quite easy to understand and is essential to the solution of many problems in hydrology and water resources. However, the identification and quantification of all concievable inflows and outflows across the control surface are essential for accurate solutions. The following bonus problems are useful in displaying its many applications. B1. An in-ground swimming pool may have a leak. The 30-ft * 10-ft * 5-ft (depth) pool is filled on June 1. On June 13, a hose, flowing at a rate of 10 gpm (gallons per minute), is used to add water to the pool. The hose is turned off after 1 hr. Four inches of rain fell during the month. Evaporation from a nearby lake was 8 in. during June. It is estimated that 25% more will evaporate from the pool than the lake. On July 1, the pool is 5 in. below the top. Does the pool have a leak? If so, what is the rate of leakage in gallons per day? Are you confident enough in your assessment to testify as an expert witness in court? B2. A water supply reservoir has been lined with clay to limit leakage (infiltration) through the bottom. An assessment of the effectiveness of the liner is required to fulfill the terms of the construction contract. The following data have been collected for a test week: Parameter/Day Stream Inflow (m3/s) Stream Outflow (m3/s) City Use (m3/s) Precipitation (cm)

0.0 0.0 0.3 0.0

0.2 0.1 0.2 1.5

0.4 0.3 0.3 7.0

0.5 0.2 0.2 2.5

0.5 0.1 0.3 0.0

0.2 0.1 0.3 0.0

0.1 0.0 0.3 0.0

462

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Chap. 11

The surface area of the lake is 40 hectares. The reservoir surface elevation fell 15.5 cm during the week. If the evaporation for the week is estimated to be 3 cm, determine the rate of leakage (L) in cubic meters for the week. B3. A rainfall event occurs over a 150 square mile watershed. The following incremental depths of precipitation, interception, and infiltration are estimated during a 30-min storm in 5-min increments. Determine the total volume of rainfall (acre-feet). Also determine the total volume of runoff (acrefeet) that will drain into a reservoir at the outlet of the watershed. Parameter/Time Increment Precipitation (in.) Interception (in.) Infiltration (in.)

0.05 0.05 0.00

0.10 0.03 0.07

0.45 0.02 0.23

0.30 0.00 0.20

0.15 0.00 0.10

0.05 0.00 0.05

B4. An annual water budget for a critical water supply basin is needed. During the year, the following hydrologic data were collected for the 6,200 km2 basin: precipitation (P) = 740 mm, evaporation and transpiration (E + T) = 350 mm, average annual streamflow leaving the basin (outflow, Qo1) = 75.5 m3/s, ground water outflow (Qo2) = 0.200 km3, and ground water inflow (infiltration, I) = 0.560 km3. Determine the change of storage in the watershed (surface reservoirs) and the change in storage in the water supply basin (surface reservoirs and groundwater together) in cubic kilometer. Hint: Sketch the control volumes; a different one is needed for each solution.

Robert Houghtalen

Statistical Methods in Hydrology

Statistical methods are indispensable tools in hydrology. Most hydrologic processes, such as rainfall, are not amenable to purely deterministic analysis because of inherent uncertainties. These uncertainties arise from the randomness of natural processes, lack of data in sufficient quantity and quality, and a lack of understanding of all the causal relationships in complex hydrologic processes. Statistical methods account for these uncertainties, and competent predictions made using these methods are always accompanied by some probability of occurrence or likelihood. In applying statistical methods, we presume that the natural processes are governed by some mathematical rules rather than the physical laws underlying these processes. In allowing this presumption, statistical methods can be used to analyze diverse hydrologic processes. The coverage of statistical methods herein emphasizes frequency analysis techniques and the concepts used in these techniques. The purpose of a frequency analysis is to extract meaningful information from observed hydrologic data to make decisions concerning future events. As an example, suppose a hydrologic data series contains the instantaneous discharges at a stream section observed during the past 20 years and that a highway bridge is being planned on this stream. The bridge must be designed to pass a design discharge without being flooded. What discharge should be used in the design? What are the chances that the bridge will be flooded during its life span if a particular design discharge is used? A plot of the peak discharges from the past 20 years would look rather erratic, and it would provide no answers to these questions. Only after a frequency analysis of the streamflow data can we answer these types of questions intelligently. 463

464

Statistical Methods in Hydrology

Chap. 12

The hydrologic data used in a frequency analysis must represent the situation being studied; that is, the data set must be homogeneous. For example, the future surface runoff from a developed, urbanized area cannot be determined using historical runoff data observed under undeveloped conditions. Other changes that affect the data set include relocated gauges, streamflow diversions, and construction of dams and reservoirs during the period the observations were made. The available hydrologic data may contain more information than needed for frequency analysis. In that event, the data should be reduced into a useful form. For example, suppose we have a daily record of streamflow at a gauge site for the past N years. Such a record would be called a complete duration series. For the most part, however, we are interested in extremes, particularly the high streamflows for flood studies. We can form an annual exceedence series by considering the highest N values on the record. Alternatively, we can obtain an annual maximum series by using the largest N values occurring in each of the N years. Either the annual exceedence or the annual maximum series can be used in a frequency analysis. The results from the two approaches become very similar when extreme events of rare occurrence are investigated. Usually, however, we use the annual maximum series because the values included in this series are more likely to be statistically independent, as assumed in frequency analysis methods. 12.1 Concepts of Probability Understanding probability concepts requires the definition of some key terms: random variable, sample, population, and probability distribution. A random variable is a numerical variable that cannot be precisely predicted. In probabilistic methods, we treat all hydrologic variables as random variables. These include, but are not limited to, rates of rainfall, streamflow, evaporation, wind velocity, and reservoir storage. A set of observations of any random variable is called a sample. For example, the annual maximum streamflows observed at a specific gauge site during the past N years form a sample. Likewise, the annual maximum streamflows that will occur over some specific period in the future form another sample. We assume that samples are drawn from an infinite hypothetical population, which is defined as the complete assemblage of all the values representing the random variable being investigated. In simpler terms, the population of annual maximum streamflows at a specified site would contain the maximum yearly values observed over an infinite number of years. A probability distribution is a mathematical expression that describes the probabilistic characteristics of a population. A probability distribution is useful in calculating the chances that a random variable drawn from this population will fall in a specified range of numerical values. For example, the probability distribution of annual maximum streamflows will enable us to estimate the chances that the maximum streamflow will exceed a specified magnitude in any one year in the future. 12.2 Statistical Parameters Most theoretical probability distributions are expressed in terms of statistical parameters that characterize the population, such as the mean, standard deviation, and skewness. We cannot determine these parameters precisely because we do not know all the values included in the entire population. However, we can estimate these statistical parameters from a sample.

Sec. 12.2

Statistical Parameters

465

Let a sample contain N observed values of a random variable, xi, with i = 1, 2, p , N. For an annual maximum streamflow series, xi would denote the maximum streamflow observed during the ith year. The sample estimate of the mean (m) is 1 N m = a xi N i=1

(12.1)

Simply stated, m is the average of all the observed values included in the sample. The variance is a measure of the variability of the data. The square root of the variance is called the standard deviation. A sample estimate of the standard deviation (s) is N 1/2 1 2 s = c (xi - m) d N - 1 ia =1

(12.2)

Skewness, or skew, is a measure of the symmetry of a probability distribution about the mean. We can estimate the skew coefficient from the data as N a (xi - m)3 N

G =

i=1

(N - 1) (N - 2) s3

(12.3)

When performing frequency analyses on data measured at a gauging station, G is frequently referred to as the station skew of the sample. Experience indicates that logarithms of the observed values for many hydrologic variables are more apt to follow certain probability distributions. Hence, the aforementioned statistical parameters are calculated as ml =

1 N log xi N ia =1

sl = c

N 1/2 1 2 (log x m ) d i l N - 1 ia =1

(12.4)

(12.5)

N a (log xi - ml)3

Gl =

i=1

(N - 1) (N - 2) s3l

(12.6)

where ml, sl, and Gl are the mean, standard deviation, and skew coefficient of the logarithms (base 10) of the observed data values. Throughout this text, log refers to the common logarithm (base 10) of the operand, whereas ln is used to refer to the natural logarithm (base e = 2.718). Example 12.1 Annual peak discharges (Qi) of the Meherrin River at Emporia, Virginia, are tabulated in column 2 of Table 12.1 for the years 1952 to 1990. Determine the mean, the standard deviation, and the skew coefficient of these data. Solution Equations 12.1–12.3 will be used, substituting Qi for xi. The calculations are best performed in tabular (or spreadsheet) form as shown in Table 12.1. Column sums needed for the computations are given in the

9,410 11,200 5,860 12,600 7,520 7,580 12,100 9,400 8,710 6,700 12,900 8,450 4,210 7,030 7,470 5,200 6,200 5,800 5,400 7,800 19,400 21,100 10,000 16,200 8,100

Year

1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976

- 4.08E+ 02 1.38E+ 03 - 3.96E+ 03 2.78E+ 03 - 2.30E+ 03 - 2.24E+ 03 2.28E+ 03 - 4.18E+ 02 - 1.11E+ 03 - 3.12E+ 03 3.08E+ 03 - 1.37E+ 03 - 5.61E+ 03 - 2.79E+ 03 - 2.35E+ 03 - 4.62E+ 03 - 3.62E+ 03 - 4.02E+ 03 - 4.42E+ 03 - 2.02E+ 03 9.58E+ 03 1.13E+ 04 1.82E+ 02 6.38E+ 03 - 1.72E+ 03

Qi - m

Table 12.1

1.66E+ 05 1.91E+ 06 1.57E+ 07 7.74E+ 06 5.28E+ 06 5.01E+ 06 5.21E+ 06 1.74E+ 05 1.23E+ 06 9.72E+ 06 9.50E+ 06 1.87E+ 06 3.14E+ 07 7.77E+ 06 5.51E+ 06 2.13E+ 07 1.31E+ 07 1.61E+ 07 1.95E+ 07 4.07E+ 06 9.18E+ 07 1.27E+ 08 3.32E+ 04 4.07E+ 07 2.95E+ 06

(Qi - m)2 - 6.78E+ 07 2.64E+ 09 - 6.20E+ 10 2.15E+ 10 - 1.21E+ 10 1.12E+ 10 1.19E+ 10 - 7.29E+ 07 - 1.36E+ 09 - 3.03E+ 10 2.93E+ 10 - 2.56E+ 09 - 1.76E+ 11 - 2.17E+ 10 - 1.29E+ 10 - 9.85E+ 10 - 4.73E+ 10 - 6.49E+ 10 - 8.62E+ 10 - 8.21E+ 09 8.80E+ 11 1.44E+ 12 6.06E+ 06 2.60E+ 11 - 5.07E+ 09

(Qi - m)3 3.97E+ 00 4.05E+ 00 3.77E+ 00 4.10E+ 00 3.88E+ 00 3.88E+ 00 4.08E+ 00 3.97E+ 00 3.94E+ 00 3.83E+ 00 4.11E+ 00 3.93E+ 00 3.62E+ 00 3.85E+ 00 3.87E+ 00 3.72E+ 00 3.79E+ 00 3.76E+ 00 3.73E+ 00 3.89E+ 00 4.29E+ 00 4.32E+ 00 4.00E+ 00 4.21E+ 00 3.91E+ 00

log Qi 2.59E- 02 1.02E- 01 - 1.80E- 01 1.53E- 01 - 7.15E- 02 - 6.80E- 02 1.35E- 01 2.54E- 02 - 7.69E- 03 - 1.22E- 01 1.63E- 01 - 2.08E- 02 - 3.23E- 01 - 1.01E- 01 - 7.44E- 02 - 2.32E- 01 - 1.55E- 01 - 1.84E- 01 - 2.15E- 01 - 5.56E- 02 3.40E- 01 3.77E- 01 5.23E- 02 2.62E- 01 - 3.92E- 02

(log Qi - ml) 6.70E- 04 1.03E- 02 3.23E- 02 2.33E- 02 5.11E- 03 4.63E- 03 1.82E- 02 6.46E- 04 5.91E- 05 1.48E- 02 2.65E- 02 4.35E- 04 1.05E- 01 1.02E- 02 5.53E- 03 5.37E- 02 2.41E- 02 3.40E- 02 4.64E- 02 3.09E- 03 1.16E- 01 1.42E- 01 2.73E- 03 6.85E- 02 1.54E- 03

(log Qi - ml)2

Mean, Standard Deviation, and Skew Calculations (Meherrin River Peak Flows)

1.73E- 05 1.05E- 03 - 5.81E- 03 3.56E- 03 - 3.65E- 04 - 3.15E- 04 2.46E- 03 1.64E- 05 - 4.54E- 07 - 1.80E- 03 4.32E- 03 - 9.06E- 06 - 3.38E- 02 - 1.02E- 03 - 4.12E- 04 - 1.24E- 02 - 3.75E- 03 - 6.26E- 03 - 9.98E- 03 - 1.72E- 04 3.93E- 02 5.34E- 02 1.43E- 04 1.79E- 02 - 6.03E- 05

(log Qi - ml)3

466 Statistical Methods in Hydrology Chap. 12

5,640 19,400 16,600 11,100 4,790 4,940 9,360 13,800 8,570 17,500 16,600 3,800 7,390 7,060 3.83E+ 05

Year

1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 Sum

- 4.18E+ 03 9.58E+ 03 6.78E+ 03 1.28E+ 03 - 5.03E+ 03 - 4.88E+ 03 - 4.58E+ 02 3.98E+ 03 - 1.25E+ 03 7.68E+ 03 6.78E+ 03 - 6.02E+ 03 - 2.43E+ 03 - 2.76E+ 03 - 3.27E- 11

Qi - m 1.75E+ 07 9.18E+ 07 4.60E+ 07 1.64E+ 06 2.53E+ 07 2.38E+ 07 2.09E+ 05 1.59E+ 07 1.56E+ 06 5.90E+ 07 4.60E+ 07 3.62E+ 07 5.89E+ 06 7.60E+ 06 8.24E+ 08

(Qi - m)2 - 7.29E+ 10 8.80E+ 11 3.12E+ 11 2.11E+ 09 - 1.27E+ 11 - 1.16E+ 11 - 9.59E+ 07 6.32E+ 10 - 1.94E+ 09 4.53E+ 11 3.12E+ 11 - 2.18E+ 11 - 1.43E+ 10 - 2.10E+ 10 3.45E+ 12

(Qi - m)3

Table 12.1

3.75E+ 00 4.29E+ 00 4.22E+ 00 4.05E+ 00 3.68E+ 00 3.69E+ 00 3.97E+ 00 4.14E+ 00 3.93E+ 00 4.24E+ 00 4.22E+ 00 3.58E+ 00 3.87E+ 00 3.85E+ 00 1.54E+ 02

log Qi

(Continued )

- 1.96E- 01 3.40E- 01 2.72E- 01 9.76E- 02 - 2.67E- 01 - 2.54E- 01 2.36E- 02 1.92E- 01 - 1.47E- 02 2.95E- 01 2.72E- 01 - 3.68E- 01 - 7.91E- 02 - 9.89E- 02 3.11E- 14

(log Qi - ml) 3.86E- 02 1.16E- 01 7.42E- 02 9.53E- 03 7.15E- 02 6.45E- 02 5.56E- 04 3.69E- 02 2.17E- 04 8.72E- 02 7.42E- 02 1.35E- 01 6.25E- 03 9.78E- 03 1.47E+ 00

(log Qi - ml)2

- 7.58E- 03 3.93E- 02 2.02E- 02 9.30E- 04 - 1.91E- 02 - 1.64E- 02 1.31E- 05 7.10E- 03 - 3.19E- 06 2.58E- 02 2.02E- 02 - 4.98E- 02 - 4.94E- 04 - 9.67E- 04 6.53E- 02

(log Qi - ml)3

Sec. 12.2 Statistical Parameters 467

468

Statistical Methods in Hydrology

Chap. 12

bottom row of the table. All values are given in three significant figures to match the original flows. The mean is calculated from Equation 12.1 with N = 39: m =

1 N Qi = (3.83 * 105)/39 = 9,820 cfs N ia =1

Next, the standard deviation is computed from Equation 12.2: s = c

N 1/2 1 2 (Q m) d = [(8.24 * 108)/38]1/2 = 4,660 cfs i N - 1 ia =1

Finally, the skew coefficient is calculated using Equation 12.3: N a (Qi - m)3 N

G =

i=1

(N - 1) (N - 2) s

= (39)(3.45 * 1012)/[(38)(37)(4,660)3] = 0.946

Example 12.2 Consider the Meherrin River discharges given in Example 12.1. Compute the mean, standard deviation, and skew coefficient of the logarithms of the observed discharges. Solution First, the logarithms of the Q's are obtained and tabulated in column 6 of Table 12.1. Then calculations are performed in tabular form as in Example 12.1. However, in this case, Equations 12.4, 12.5, and 12.6 are used where log Qi has been substituted for the log xi terms. Thus, ml =

1 N log Qi = (1.54 * 102)/39 = 3.95 N ia =1

sl = c

and

N 1/2 1 (log Qi - ml)2 d = [(1.47)/38]1/2 = 0.197 a N - 1 i=1

N a (log Qi - ml)3 N

Gl =

i=1

(N - 1) (N - 2) s3l

= (39)(6.53 * 10-2)/[(38)(37)(0.197)3] = 0.237

Note that the mean flow for the log-transformed data set can be found by taking the antilog (sometimes referred to as the inverse log): Q (log mean) = 103.95 = 8,910 cfs which is much lower than the arithmetic mean.

12.3 Probability Distributions Among the many theoretical probability distributions available, the normal, log-normal, Gumbel, and log-Pearson type III distributions are used most often in hydrology. 12.3.1

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is probably the most common model of probability. However, it is rarely used in hydrology. The main limitation of the normal distribution is that it allows random variables to assume values from -∞ to ∞, while most

Sec. 12.3

Probability Distributions

469

hydrologic variables, such as stream discharge, are non-negative. In other words, in reality there is no such thing as a "negative" discharge, as may be computed using the normal distribution. The normal distribution is expressed in terms of a probability density function, fX (x), as fX(x) =

exp c -

s 22π

(x - m)2 2s2

(12.7)

where "exp" is the base of Naperian logarithms (i.e., 2.71828 . . .) raised to the power of the value in the brackets, and the mean (m) and standard deviation (s) of the sample are used as estimates of the mean and standard deviation of the population. To explain the meaning of the probability density function, let the random variable X represent the annual maximum discharges of a stream. Suppose the parameters m and s have been obtained by analyzing the annual maximum discharge series of this stream. Further suppose we want to determine the probability that the maximum discharge in any one year in the future will be between two specified numerical values x1 and x2. This probability can be computed as x2

P[x1 … X … x2] =

L x

fX(x)dx

(12.8)

Or, in words, the probability that a particular value X will fall between two values, x1 and x2, can be calculated as the definite integral from x1 to x2 of the probability density function. 12.3.2

Log-Normal Distribution

The log transformation of hydrologic random variables is more likely to follow the normal distribution than the original values. In such cases, the random variable is said to be log-normally distributed. The probability density function for log-normal distribution is fX(x) =

1 (x)sl 22π

exp c

-(log x - ml)2 2s2l

(12.9)

where "exp" is the base of Naperian logarithms (i.e., 2.71828 . . .) raised to the power of the value in the brackets. Note that the log function in the brackets is the common log (base 10) of x. 12.3.3

Gumbel Distribution

The Gumbel distribution, also known as the extreme value type I distribution, is commonly used for frequency analysis of floods and maximum rainfall. The probability density function for this distribution is fX (x) = (y) 5exp[-y(x - u) - exp[-y(x - u)]]6

(12.10)

in which y and u are intermediate parameters, defined as y = and

π s 26

u = m - 0.45s

(12.11)

(12.12)

where m and s represent the mean and standard deviation, respectively, of the sample, as defined previously.

470

Statistical Methods in Hydrology

12.3.4

Chap. 12

Log-Pearson Type III Distribution

The log-Pearson type III distribution was recommended by the U.S. Water Resources Council* to model annual maximum streamflow series. The council's duties on guidelines for flood-frequency studies were taken over by the Interagency Advisory Committee on Water Data of the U.S. Geological Survey in 1981. The probability density function for the log-Pearson type III distribution is fX(x) =

νb (log x - r)b - 1 exp[-ν (log x - r)] x Γ (b)

(12.13)

where Γ is the gamma function. Values of the gamma function can be found in standard mathematical tables. The parameters b, v, and r are related to the sample statistical parameters through the expressions b = ν = and

4 G 2l sl 2b

r = ml - sl 2b

(12.14) (12.15) (12.16)

The sample statistical parameters ml, sl, and Gl are obtained from Equations 12.4–12.6, respectively. The skew coefficient used in the log-Pearson type III distribution is sensitive to the sample size. For samples having less than 100 data values, the Water Resources Council recommended using a weighted average of the skewness coefficient obtained from the sample and the generalized map skews given in Figure 12.1. Following the weighting procedure described by the Interagency Advisory Committee on Water Data,†† the weighted skew coefficient (for samples having less than 100 data points) is expressed as g =

0.3025 Gl + VGGm 0.3025 + VG

(12.17)

In this expression Gl is the skew coefficient obtained from the sample using Equation 12.6, Gm is the generalized map skew obtained from Figure 12.1 depending on the geographical location, and VG is the mean square error of the sample skew. Based on the studies of Wallis et al.,† VG can be approximated as VG ≈ 10A - B log (N/10)

(12.18)

* Water Resources Council, "A uniform technique for determining flood flow frequencies." Water Resources Council Bulletin 15. Washington, DC: U.S. Water Resources Council, 1967. † Interagency Advisory Committee on Water Data, "Guidelines for Determining Flood Flow Frequency." Bul-

letin 17B. Reston, VA: U.S. Department of the Interior, U.S. Geological Survey, Office of Water Data Coordination, 1982. †† Wallis, J. R., Matalas, N. C., and Slack, J. R., "Just a moment!" Water Resources Research, 10(2): 211–221, 1974.

Sec. 12.3

Probability Distributions .2 0

02.1 2.3

.3 .6

.6 .1 2.1 2.4

2.4

2.1

2.3

2.1

3 2

471

0 .3

.1 0

2.4

2 2.

2.42.3

.5 .7

2 .3 2.1

.7 .5 .3

2.1

0 2.1

2.2

2.3 .0

2.2

.70 2.1

0 100 200 miles

.33

2.1 2.2 0

2.2

2.3 2.3

2.05 0 50 100 miles 0 25 50 miles

2.1

50 100 150 250 miles

Figure 12.1 Water Resources Council generalized map skews Source: Interagency Advisory committee on Water Data, U.S. Geological Survey, 1982.

where N is the number of data values in the sample, and A = -0.33 + 0.08 & Gl &

if & Gl & … 0.90

(12.19a)

or A = -0.52 + 0.30 & Gl &

if & Gl & 7 0.90

(12.19b)

B = 0.94 - 0.26 & Gl &

if & Gl & … 1.50

(12.19c)

if & Gl & 7 1.50

(12.19d)

or B = 0.55

If the sample has more than 100 points, then the skew coefficient to be used in the log-Pearson type III distribution is simply Gl (also referred to as the "station skew" in this case), which is computed using Equation 12.6. Example 12.3 The skew coefficient for the log-transformed annual peak discharge series of the Meherrin River was computed as Gl = 0.237 in Example 12.2. Calculate the weighted skew coefficient for this river. Solution The Meherrin River gauge used in Example 12.2 is located in Emporia, Virginia. From Figure 12.1, the generalized map skew for this location is 0.7. From Equation 2.19a with Gl = 0.237, A = -0.33 + 0.08 & 0.237& = -0.311 and from Equation 2.19c with Gl = 0.237, B = 0.94 - 0.26 & 0.237& = 0.878

472

Statistical Methods in Hydrology

Chap. 12

Then using Equation 12.18 with N = 39, VG ≈ 10-0.311 - 0.878 log (39/10) = 10-0.830 = 0.148 and finally, from Equation 12.17, the weighted skew coefficient is g = [0.3025(0.237) + 0.148(0.7)]/(0.3025 + 0.148) = 0.389

12.4 Return Period and Hydrologic Risk The probability density functions for different probability distributions were summarized in Section 12.3. The cumulative density functions are more useful from a practical viewpoint. Given the probability density function fX(x), the cumulative density function for any distribution can be expressed as x

FX(x) =

fX(u) du

(12.20)

-∞

where u is a dummy variable of integration. The lower limit of integration should be changed to zero if the distribution allows only positive values. The numerical value of FX(x) represents the probability that the random variable being modeled will take a value smaller than x. Suppose we hypothesize that one of the probability distributions discussed in Section 12.3 can be used to describe an annual maximum discharge series of a stream. Using the sample mean, standard deviation, skew coefficient, and the chosen probability density function, we evaluate Equation 12.20 with an upper limit of integration of x = 3,000 cfs. Suppose further that the resultant cumulative density, using Equation 12.20, computes to be 0.80. Then we can say that the annual maximum discharge of the stream being studied will be smaller than 3,000 cfs with a probability of 0.80, or 80%, in any single year in the future. It should be noted that the numerical value of FX(x) will always be between zero and unity. Occasionally, FX(x) is referred to as the non-exceedence probability. Hydrologists who deal with flood studies are usually more interested in the exceedence probability (p), which is expressed as p = 1 - FX (x)

(12.21)

Obviously, p will take values between zero and 1. In the previous example, p = 1 - 0.80 = 0.20. This means that in any given year in the future, the maximum discharge will exceed x = 3,000 cfs with a probability of 0.20, or 20%. Sometimes hydrologists state that 3,000 cfs has an exceedence probability of 0.20, or 20%, to express the same outcome. The return period, also called the recurrence interval, is defined as the average number of years between occurrences of a hydrologic event with a certain magnitude or greater. Denoting the return period by T, T =

1 p

(12.22)

by definition. For example, in the previous example, the return period of 3,000 cfs will be 1/0.20 = 5 years. In other words, the annual maximum discharge of the stream being considered will exceed 3,000 cfs once every 5 years on average. We can express the same outcome by stating that the 5-year discharge (or 5-year flood) is 3,000 cfs.

Sec. 12.5

Frequency Analysis

473

Hydraulic structures are usually sized to accommodate, at full capacity, a design discharge having a specified return period. Generally, the structure will fail to function as intended if the design discharge is exceeded. The hydrologic risk is the probability that the design discharge will be exceeded one time or more during the service life of the project. Denoting the risk by R and the service life of the project in years by n, R = 1 - (1 - p)n

(12.23)

Example 12.4 A highway culvert is required to accommodate a design discharge that has a return period of 50 years. The service life of the culvert is 25 years. Determine the hydrologic risk associated with this design. In other words, what is the probability that the culvert's capacity will be exceeded during its 25-year service life? Solution From Equation 12.22, p = 1/50 = 0.02. Then, from Equation 12.23, R = 1 - (1 - 0.02)25 = 0.397 = 39.7,

12.5 Frequency Analysis The purpose of a frequency analysis of a series of observed values of a hydrologic variable is to determine the future values of this variable corresponding to different return periods of interest. To achieve this, we need to determine the probability distribution that fits the available hydrologic data using statistical means. Only after we identify a probability distribution that adequately represents the data series can we interpolate and extrapolate from the observed data values in an intelligent manner. Frequency factors as well as special probability graph papers are useful for this purpose. 12.5.1

Frequency Factors

For most theoretical distributions used in hydrology, closed form analytical expressions are not available for cumulative density functions. However, Chow* showed that Equation 12.21 can be written in a more convenient form as xT = m + KTs

(12.24)

where m and s are the sample mean and standard deviation, respectively; xT is the magnitude of a hydrologic variable corresponding to a specified return period T; and KT is the frequency factor for that return period. When log-transformed variables are used, as in the case of log-normal and log-Pearson type III distributions, log xT = ml + KTsl

(12.25)

The value of the frequency factor depends on the probability distribution being considered. An explicit analytical expression for KT is available only for the Gumbel distribution: KT =

- 26 T ¢ 0.5772 + lnJlna bR≤ π T - 1

(12.26)

The Gumbel frequency factors for various return periods are tabulated in Table 12.2. * Chow, V. T., "A general formula for hydrologic frequency analysis." Transactions of the American Geophysical Union, 32(2): 231–237, 1952.

474

Statistical Methods in Hydrology Table 12.2

Chap. 12

Frequency Factors for Gumbel, Normal, and log–Normal Distributions

T (years)

KT (Gumbel)

KT (Normal)

KT (log-Normal)

1.11 1.25 1.67 2 2.5 4 5 10 20 25 40 50 100 200

0.9 0.8 0.6 0.5 0.4 0.25 0.2 0.1 0.05 0.04 0.025 0.02 0.01 0.005

-1.100 -0.821 -0.382 -0.164 0.074 0.521 0.719 1.305 1.866 2.044 2.416 2.592 3.137 3.679

-1.282 -0.841 -0.253 0 0.253 0.674 0.841 1.282 1.645 1.751 1.960 2.054 2.327 2.576

- 1.282 - 0.841 - 0.253 0 0.253 0.674 0.841 1.282 1.645 1.751 1.960 2.054 2.327 2.576

For normal and log-normal distributions, following Abramowitz and Stegun,* we can approximate the frequency factor by KT = z (12.27) where 2.515517 + 0.802853w + 0.010328w2 z = w (12.28) 1 + 1.432788w + 0.189269w2 + 0.001308w3 and w = [ln (T 2)]1/2

(12.29a)

w = [ln (1/p2)]1/2

(12.29b)

or Equations 12.27 to 12.29 are valid for p values of 0.5 or smaller (that is, T values of 2 years or greater). For p 7 0.5, 1 - p is substituted for p in Equation 12.29b, and a negative sign is inserted in front of z in Equation 12.28. The frequency factors listed for various return periods in Table 12.2 were obtained by using these equations. We can approximate the log-Pearson type III frequency factors, for return periods between 2 and 200 years, using a relationship developed by Kite† and expressed as KT = z + (z2 - 1)k + (z3 - 6z)

k2 k5 - (z2 - 1)k 3 + zk 4 + 3 3

(12.30)

where k =

Gl 6

(12.31a)

and z is as obtained from Equation 12.28. If the Water Resources Council's weighted skew coefficient concept is used, g k = (12.31b) 6 * Abramowitz, M., and Stegun, I. A., Handbook of Mathematical Functions. New York: Dover Publications, 1965. † Kite, G. W., Frequency and Risk Analysis in Hydrology. Fort Collins, CO: Water Resources Publications, 1977.

Sec. 12.5

Frequency Analysis Table 12.3

475

Frequency Factors (KT) for log-Pearson Type III Distribution

Exceedence Probability, p 0.5

0.1

0.04

0.02

0.01

Return Period, T (years) Skew Coefficient (g)

100

2.0 1.5 1.0 0.8 0.6 0.4 0.2 0.1 0.0 -0.1 -0.2 -0.4 -0.6 -0.8 -1.0 -1.5 -2.0

- 0.307 - 0.240 - 0.164 - 0.132 - 0.099 - 0.066 - 0.033 - 0.017 0.000 0.017 0.033 0.066 0.099 0.132 0.164 0.240 0.307

1.302 1.333 1.340 1.336 1.328 1.317 1.301 1.292 1.282 1.270 1.258 1.231 1.200 1.166 1.128 1.018 0.895

2.219 2.146 2.043 1.993 1.939 1.880 1.818 1.785 1.751 1.716 1.680 1.606 1.528 1.448 1.366 1.157 0.959

2.912 2.743 2.542 2.453 2.359 2.261 2.159 2.107 2.054 2.000 1.945 1.834 1.720 1.606 1.492 1.218 0.980

3.605 3.330 3.022 2.891 2.755 2.615 2.472 2.400 2.326 2.252 2.178 2.029 1.880 1.733 1.588 1.257 0.990

The values of KT for various values of g are given in Table 12.3. Linear interpolation of these values for g's not listed in the table is not recommended. Instead, Equations 12.27 through 12.31 should be used. Note that for g = 0, the frequency factors for the log-Pearson type III distribution are the same as those for the log-normal distributions. It is important to note that the frequency factors presented in these sections are useful to estimate the magnitudes of future events only if the probability distribution is specified. The methods for testing the goodness of fit of data to a probability distribution will be discussed in the next section. Example 12.5 The statistical parameters of the annual maximum discharge series of the Meherrin River have been computed as m = 9,820 cfs, s = 4,660 cfs, ml = 3.95, and sl = 0.197 in Examples 12.1 and 12.2. Also, we determined the weighted skew coefficient to be g = 0.389 in Example 12.3. Determine the magnitude of the 25-year discharge on the Meherrin River if the data fits (a) the normal distribution, (b) the log-normal distribution, (c) the Gumbel distribution, and (d) the log-Pearson type III distribution. Solution (a) To solve part (a), we first obtain K25 = 1.751 from Table 12.2 for p = 0.04 (i.e., T = 25 years). Then, from Equation 12.24, Q25 = m + K25(s) = 9,820 + 1.751(4,660) = 18,000 cfs (normal) (b) We can use the same frequency factor, K25 = 1.751, to solve part (b) of the problem. From Equation 12.25, log Q25 = ml + KT(sl) = 3.95 + 1.751(0.197) = 4.29 Then, taking the antilog of 4.29, we obtain Q25 = 19,500 cfs (log normal).

476

Statistical Methods in Hydrology

Chap. 12

(c) For the Gumbel distribution, we first obtain K25 = 2.044 from Table 12.2 for p = 0.04 (i.e., T = 25 years). Then, from Equation 12.24, Q25 = m + K25(s) = 9,820 + 2.044(4,660) = 19,300 cfs (Gumbel) (d) We will use the Equations 12.28 through 12.31 to solve part (d), the log-Pearson type III distribution. The sequence is Equations 12.31b, 12.29a, 12.28, and 12.30: k = g/6 = 0.389/6 = 0.0648 w = [ln T 2]1/2 = [ln (25)2]1/2 = 2.54 z = w z = 2.54 -

2.515517 + 0.802853w + 0.010328w2 1 + 1.432788w + 0.189269w2 + 0.001308w3

2.515517 + 0.802853(2.54) + 0.010328(2.54)2 1 + 1.432788(2.54) + 0.189269(2.54)2 + 0.001308(2.54)3

= 1.75

KT = z + (z2 - 1)k + (z3 - 6z)(k 2/3) - (z2 - 1)k 3 + zk 4 + k 5/3 KT = 1.75 + (1.752 - 1) 0.0648 + [1.753 - 6(1.75)] (0.06482/3) (1.752 - 1)0.06483 + 1.75(0.0648)4 + 0.06485/3 KT = 1.877 Now, from Equation 12.25, log Q25 = ml + KT (sl) = 3.95 + 1.877(0.197) = 4.32 Taking the antilog of 4.32, we obtain Q25 = 20,900 cfs (log-Pearson type III).

12.5.2

Testing Goodness of Fit

The chi-square test is a statistical procedure to determine the goodness of fit of data to a probability distribution. In this test, we divide the entire range of possible values of the hydrologic variable into k class intervals. We then compare the actual number of data values falling in these intervals to the number of data values expected according to the probability distribution being tested. The number of the class intervals, k, is selected so that the expected number of data values in each class interval is at least 3. The limits of the class intervals are determined so that the number of expected data values is the same in each interval. The frequency factors discussed in Section 12.5.1 can be used to determine the class limits. To perform a chi-square test, it is necessary to first choose a significance level, α. Commonly, α = 0.10 is used in hydrology. The meaning of α can be explained as follows: If we use α = 0.10, and as a result of the chi-square test we reject the probability distribution being considered, then there is a 10% chance that we have rejected a satisfactory distribution. The test statistic is calculated using (Oi - Ei)2 χ = a Ei i=1 2

(12.32)

where Oi and Ei are the observed and expected number of data values in the ith interval. Then we accept the distribution being tested if χ2 6 χ2α

Sec. 12.5

Frequency Analysis

477

Table 12.4

Values of χ2α

Significance Level v

α = 0.05

α = 0.10

1 2 3 4 5 6 7 8 9 10 15 20 25 30 40 50 60 70 80 90 100

3.84 5.99 7.81 9.49 11.1 12.6 14.1 15.5 16.9 18.3 25.0 31.4 37.7 43.8 55.8 67.5 79.1 90.5 101.9 113.1 124.3

2.71 4.61 6.25 7.78 9.24 10.6 12.0 13.4 14.7 16.0 22.3 28.4 34.4 40.3 51.8 63.2 74.4 85.5 96.6 107.6 118.5

α = 0.50 0.455 1.39 2.37 3.36 4.35 5.35 6.35 7.34 8.34 9.34 14.3 19.3 24.3 29.3 39.3 49.3 59.3 69.3 79.3 89.3 99.3

and reject it otherwise, where χ2α is the critical value of χ2 at significance level α. The values of χ2α for α = 0.05, 0.10, and 0.50 are given in Table 12.4 as a function of v, where ν = k - kk - 1

(12.33)

and kk = the number of sample statistics, such as the mean, standard deviation, and skew coefficient used to describe the probability distribution being tested. For the normal, log-normal, and Gumbel distributions, kk = 2; and for the log-Pearson type III distribution, kk = 3. Once again, k is the number of the class intervals used in the test. It should be noted that the result of a chi-square test is sensitive to the value of k being used. Therefore, this test must be used cautiously. Example 12.6 Consider the annual maximum discharges (log transformed) on the Meherrin River given in Example 12.2 with sample statistics ml = 3.95 and sl = 0.197. Test the goodness of fit of these data to the log-normal distribution at a significance level of α = 0.10. Use five class intervals—that is, k = 5. Solution Because the probability ranges from 0 to 1.0, the exceedence probability increment is (1.0 - 0.0)/5 = 0.20 for each class interval. Accordingly, the upper and lower probability limits of the five class intervals are determined as listed in Table 12.5. The exceedence probabilities of p = 0.8, 0.6, 0.4, and 0.2 correspond to return periods of T = 1.25, 1.67, 2.50, and 5.00 years. The frequency factors for these return periods for the log-normal distribution are obtained from Table 12.2 as KT = - 0.841, -0.253, 0.253, and 0.841. The corresponding discharges are determined, by using Equation 12.25 with ml = 3.95 and sl = 0.197, as being Q = 6,090, 7,940, 9,990, and 13,100 cfs, respectively. The upper and lower discharge limits listed

478

Statistical Methods in Hydrology Table 12.5

Class Interval

Exceedence Probability Limits

Chap. 12

Chi-Square Test for Example 12.6

Discharge Limits (cfs)

Higher

Lower

Upper

(Oi - Ei)2/Ei

1 2 3 4 5

1.0 0.8 0.6 0.4 0.2

0.8 0.6 0.4 0.2 0.0

0 6,090 7,940 9,990 13,100

6,090 7,940 9,990 13,100 Infinity Totals

7.8 7.8 7.8 7.8 7.8 39

9 9 7 6 8 39

0.185 0.185 0.082 0.415 0.005 0.872

in Table 12.5 are based on these values. Note that the upper limit in terms of the exceedence probability corresponds to the lower limit in terms of the discharge for each class interval. The expected number of data values is Ei = N/5 = 39/5 = 7.8 for each interval. The observed number of values (Oi) is obtained from Table 12.1. For instance, Table 12.1 contains only nine values between 0 and 6,090 cfs for class 1. The value in column 8 for class 1 is computed as χ2 =

(9 - 7.8)2 = 0.185 7.8

Other entries in column 8 are computed likewise. The test statistic is calculated by summing the values in column 8 of Table 12.5 to be 0.872. In this example, k = 5 and kk = 2 (because we are using the log-normal distribution). Therefore v = 5 - 2 - 1 = 2. Then, from Table 12.4 for α = 0.10, we obtain χ2α = 4.61. Because χ2 6 χ2α (i.e., because 0.872 6 4.61), we conclude that the log-normal distribution does indeed adequately fit the annual maximum discharge data series (log transformed) of the Meherrin River.

12.5.3

Confidence Limits

There are uncertainties associated with the estimates made using frequency factors. Usually, we present these estimates within a range called a confidence interval. The upper and lower limits of a confidence interval are called the confidence limits. The width of the confidence interval depends on the size of the sample and the confidence level. An interval is said to have a confidence level of 90% if the true value of the estimated hydrologic variable is expected to fall in this range with a probability of 0.90, or 90%. The upper and lower confidence limits are expressed, respectively, as UT = m + KTU(s) (12.34) L T = m + KTL(s) (12.35) where KTU and KTL are modified frequency factors developed by Chow et al.* For log-transformed samples, the corresponding equations are log UT = ml + KTU(sl) (12.36) log L T = ml + KTL(sl)

(12.37)

* Chow, V. T, Maidment, D. R., and Mays, L. W., Applied Hydrology. New York: McGraw-Hill, 1988.

Sec. 12.5

Frequency Analysis

479

The approximate expressions for the modified frequency factors are KTU

KT + 2K 2T - ab = a

(12.38)

KT - 2K 2T - ab KTL = (12.39) a where KT is the frequency factor appearing in Equations 12.24 and 12.25, and for a sample size of N: a = 1 b =

K 2T

z2 2(N - 1) z2 N

(12.40) (12.41)

The value of the parameter z depends on the confidence level. In practice, a 90% confidence level is most commonly used; at this level, z = 1.645. For other confidence levels, Equation 12.28 can be used to obtain z with w = Jln a

2 1/2 2 b R 1 - β

(12.42)

where β is the confidence level expressed as a fraction.

Example 12.7 The statistical parameters of the annual maximum discharge series of the Meherrin River have been computed as m = 9,820 cfs, s = 4,660 cfs, ml = 3.95, and sl = 0.197 in Examples 12.1 and 12.2. Also, we determined the weighted skew coefficient to be g = 0.389 in Example 12.3. Determine the 90% confidence limits for the 25-year discharge, assuming the data fit (a) the normal distribution, (b) the log-normal distribution, (c) the Gumbel distribution, and (d) the log-Pearson type III distribution. Solution In Example 12.5, we determined that K25 = 1.751 for the normal and log-normal distributions, K25 = 2.044 for the Gumbel distribution, and K25 = 1.877 for the log-Pearson type III distribution. Also, when a 90% confidence level is used, z = 1.645 for β = 0.90. From Equation 12.40, a = 1 -

(1.645)2 z2 = 1 = 0.9644 2 (N - 1) 2 (39 - 1)

For parts (a) and (b) of the problem, we first find the parameter b using Equation 12.41 as b = K 2T -

(1.645)2 z2 = (1.751)2 = 2.997 N 39

Next, from Equations 12.38 and 12.39, K25U =

KT + 2K 2T - ab 1.751 + 2(1.751)2 - (0.9644)(2.997) = = 2.25 a 0.9644

K25L =

KT - 2K 2T - ab 1.751 - 2(1.751)2 - (0.9644)(2.997) = = 1.38 a 0.9644

480

Statistical Methods in Hydrology

Chap. 12

Then, for the normal distribution using Equations 12.34 and 12.35, the confidence limits are U25 = m + K25U(s) = 9,820 + 2.25(4,660) = 20,300 cfs L 25 = m + K25L(s) = 9,820 + 1.38(4,660) = 16,300 cfs For the log-normal distribution, the confidence limits using Equations 12.36 and 12.37 are log U25 = ml + K25U(sl) = 3.95 + 2.25(0.197) = 4.39 (U25 = 24,500 cfs) log L 25 = ml + K25L(sl) = 3.95 + 1.38(0.197) = 4.22 (L 25 = 16,600 cfs) Parts (c) and (d) can be solved in a similar manner. For part (c), from Equation 12.41, b = K 2T -

(1.645)2 z2 = (2.044)2 = 4.109 N 39

and from Equations 12.38 and 12.39, K25U =

KT + 2K 2T - ab 2.044 + 2(2.044)2 - (0.9644)(4.109) = = 2.60 a 0.9644

K25L =

KT - 2K 2T - ab 2.044 - 2(2.044)2 - (0.9644)(4.109) = = 1.64 a 0.9644

Then, for the Gumbel distribution using Equations 12.34 and 12.35, the confidence limits are U25 = m + K25U(s) = 9,820 + 2.60(4,660) = 21,900 cfs L 25 = m + K25L(s) = 9,820 + 1.64(4,660) = 17,500 cfs To solve part (d) of this problem, with K25 = 1.877 for the log-Pearson type III distribution, from Equation 12.41, b = K 2T -

(1.645)2 z2 = (1.877)2 = 3.454 N 39

and from Equations 12.38 and 12.39, K25U =

KT + 2K 2T - ab 1.877 + 2(1.877)2 - (0.9644)(3.454) = = 2.40 a 0.9644

K25L =

KT - 2K 2T - ab 1.877 - 2(1.877)2 - (0.9644)(3.454) = = 1.49 a 0.9644

For the log-Pearson type III distribution, the confidence limits using Equations 12.36 and 12.37 are log U25 = ml + K25U(sl) = 3.95 + 2.40(0.197) = 4.42 (U25 = 26,300 cfs) log L 25 = ml + K25L(sl) = 3.95 + 1.49(0.197) = 4.24 (L 25 = 17,400 cfs)

Example 12.8 The statistical parameters of the annual maximum discharge series of the Meherrin River have been computed as m = 9,820 cfs, s = 4,660 cfs, ml = 3.95, and sl = 0.197 in Examples 12.1 and 12.2. Also, in Example 12.6 we determined that the Meherrin River data fit the log-normal distribution. Determine the 1.25-, 2-, 10-, 25-, 50-, 100-, and 200-year peak discharges and the 90% confidence limits for the Meherrin River.

Sec. 12.6

Frequency Analysis Using Probability Graphs

481

Solution The solution is presented in Table 12.6. The values in column 2 are obtained from Table 12.2, and Equation 12.25 is used to determine the entries in column 3. The antilogs of the values in column 3 become the discharges listed in column 4. Equations 12.40 and 12.41 are used to calculate the values in columns 5 and 6, respectively. Table 12.6 Peak Discharges and 90% Confidence Limits for the Meherrin River

1 T

2 KT

3 log QT

4 QT

1.25 2 10 25 50 100 200

- 0.841 0 1.282 1.751 2.054 2.327 2.576

3.78 3.95 4.20 4.29 4.35 4.41 4.46

6.09E+ 03 8.91E+ 03 1.59E+ 04 1.97E+ 04 2.26E+ 04 2.56E+ 04 2.87E+ 04

5 a

6 b

7 KTU

8 KTL

0.964 0.638 - 0.557 - 1.187 0.964 - 0.069 0.268 - 0.268 0.964 1.574 1.697 0.962 0.964 2.997 2.251 1.381 0.964 4.150 2.613 1.647 0.964 5.346 2.941 1.884 0.964 6.566 3.242 2.100

9 10 log UT log L T 3.840 4.003 4.284 4.393 4.465 4.529 4.589

3.716 3.897 4.140 4.222 4.274 4.321 4.364

11 UT

12 LT

6.92E+ 03 1.01E+ 04 1.92E+ 04 2.47E+ 04 2.92E+ 04 3.38E+ 04 3.88E+ 04

5.20E+ 03 7.89E+ 03 1.38E+ 04 1.67E+ 04 1.88E+ 04 2.10E+ 04 2.31E+ 04

Likewise, Equations 12.38 and 12.39 are used to determine the entries in columns 7 and 8. The logarithms of the upper and lower confidence limits listed in columns 9 and 10 are obtained by using Equations 12.36 and 12.37, respectively. The antilogs of these yield the upper and lower limits listed in columns 11 and 12, respectively.

12.6 Frequency Analysis Using Probability Graphs 12.6.1

Probability Graphs

Graphical representation of hydrologic data is an important tool for statistical analysis. Usually, we plot the data on specially designed probability paper. The ordinate usually represents the value of the hydrologic variable, and the abscissa represents the return period (T) or the exceedence probability (p). The ordinate scale can be linear or logarithmic, depending on the probability distribution being used. The abscissa scale is designed such that Equation 12.24, or Equation 12.25, will plot as a theoretical straight line. When plotted, the data points should fall on or near this straight line if the probability distribution being used represents the data series adequately. With this linear relationship, we can easily interpolate and extrapolate the plotted data. Normal, log-normal, and Gumbel distribution graph papers are available commercially. Figure 12.2 provides an example of normal distribution (probability) paper. For the log-Pearson type III distribution, there would have to be a different graph paper for each different value of the skew coefficient. Commercial log-Pearson type III papers are not available because of this impracticality. A log-normal probability paper can be used for log-Pearson type III distribution, but Equation 12.25 will plot as a smooth theoretical curve rather than a straight line, and extrapolation of the plotted data will be relatively difficult. 12.6.2 Plotting Positions Plotting position refers to the return period T (or the exceedence probability p = 1/T) assigned to each data value that will be plotted on a probability paper. Among the many methods available in the literature, most of which are empirical, the Weibull method is adopted herein. In this method, the data values are listed in an order of decreasing magnitude and a rank (r) is assigned to each data value. In other words, if there are N data values in the series, r = 1 for the largest

482

Statistical Methods in Hydrology

99.99

0.01

99.9 99.8

0.05 0.1 0.2

0.5

Figure 12.2

0.5

Chap. 12

0.2 0.1 0.05

99.8 99.9

0.01

99.99

Probability (normal distribution) graph paper

value in the series and r = N for the smallest. Then the exceedence probability assigned to each data value for plotting purposes is found as r (12.43) N + 1 which is equivalent to the plotting position formula adopted by the Water Resources Council,* and N + 1 T = (12.44) r where T is the assigned return period for plotting purposes. p =

Example 12.9 The annual maximum discharge series of the Meherrin River were tabulated in chronological order in Table 12.1. Determine the return periods assigned to these data values for plotting purposes. Solution This problem can be solved in tabular form. As shown in Table 12.7, the observed values of annual peak discharges are listed in decreasing order. Then a rank, r = 1 to 39, is entered in column 1. Subsequently,

Interagency Advisory Committee on Water Data, "Guidelines for determining flood flow frequency." Bulletin 17B. Reston, VA: U.S. Department of the Interior, U.S. Geological Survey, Office of Water Data Coordination, 1982.

Sec. 12.6

Frequency Analysis Using Probability Graphs Table 12.7

483

Plotting Positions for Example 12.9

Rank (r)

Q (cfs)

Plotting Position (p)

Plotting Position (T) years

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

21,100 19,400 19,400 17,500 16,600 16,600 16,200 13,800 12,900 12,600 12,100 11,200 11,100 10,000 9,410 9,400 9,360 8,710 8,570 8,450 8,100 7,800 7,580 7,520 7,470 7,390 7,060 7,030 6,700 6,200 5,860 5,800 5,640 5,400 5,200 4,940 4,790 4,210 3,800

0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600 0.625 0.650 0.675 0.700 0.725 0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975

40.00 20.00 13.33 10.00 8.00 6.67 5.71 5.00 4.44 4.00 3.64 3.33 3.08 2.86 2.67 2.50 2.35 2.22 2.11 2.00 1.90 1.82 1.74 1.67 1.60 1.54 1.48 1.43 1.38 1.33 1.29 1.25 1.21 1.18 1.14 1.11 1.08 1.05 1.03

the exceedence probability (p) and the return period (T) for each discharge are calculated using Equations 12.43 and 12.44 and are tabulated in columns 3 and 4, respectively.

12.6.3

Data Plotting and Theoretical Distributions

As previously mentioned, a graphical representation of a hydrologic data series can be obtained by plotting the data points on specially designed probability paper. The type of probability paper used depends on the probability distribution appropriate to the data or one that is being tested. The data are plotted using the plotting positions discussed in the preceding section.

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A theoretical straight line representing the probability distribution may be plotted using the frequency factors discussed in Section 12.5.1. Although two points are adequate to draw a straight line, it is good practice to use at least three points to detect any computational errors. For a perfect fit, all the data points must fall on the straight line. We never see a perfect fit in actual applications; if the data points are close enough to the theoretical straight line, then the probability distribution being tested is acceptable. It is possible to quantify and test the suitability of the distribution being applied to the given data using a goodness-of-fit statistical test, as described in Section 12.5.2. In this case, the upper and lower limits of the class intervals corresponding to the selected probability limits can be determined directly from the graph using the theoretical (straight-line) probability distribution. As noted in Section 12.5.3, uncertainties are associated with the estimates made using statistical methods, and we usually present those estimates with a range called the confidence band. A confidence band is obtained by plotting the upper and lower confidence limits on probability paper. We obtain the lower boundary of the confidence band by drawing a line through the calculated lower confidence limits. The line defining the upper boundary passes through the calculated upper confidence limits. Obviously, the theoretical probability distribution must fall within this band. The width of the band will depend on the confidence level discussed in Section 12.5.3. A confidence level of 90% is common in hydrology. Example 12.10 Prepare a log-normal plot of the annual maximum discharge series of the Meherrin River. Draw the theoretical (straight-line) probability distribution and the 90% confidence band. Solution Figure 12.3 displays the Meherrin River data on log-normal probability paper. The observed data points were plotted using the plotting positions calculated in Example 12.9 and presented in Table 12.7. In Example 12.8 the discharges corresponding to various return periods were calculated for a lognormal distribution. These are theoretical values and are used to plot the theoretical probability distribution. Only two points are required to draw the straight line, but three points are used in this example. As to the confidence band, the lower and upper confidence limits were also calculated in Example 12.8 for various return periods. By plotting UT versus T on the probability paper, we obtain the upper boundary of the confidence band. Likewise, a plot of L T versus T will give the lower boundary, as shown in Figure 12.3.

12.6.4

Estimating Future Magnitudes

When a statistical distribution is fit to a data series and the confidence band is developed, the future expected magnitudes of the hydrologic variable being considered may be estimated quite easily. For instance, in Example 12.6 we have shown that the annual maximum discharge series of the Meherrin River fits a log-normal distribution with the corresponding theoretical straight line and the confidence band displayed in Figure 12.3. We can now use Figure 12.3 to estimate the discharge for virtually any return period even though the original data series only contains 39 years of data. Suppose we want to estimate the discharge that has a return period of 100 years. Using T = 100 years and the theoretical straight line in Figure 12.3, we could read Q100 directly off the log-normal graph or, as computed in Example 12.8, Q100 = 25,600 cfs. Likewise, from the 90% confidence band, U100 = 33,800 cfs and L 100 = 21,000 cfs, as computed in Example 12.8, shown in Table 12.6, and plotted in Figure 12.3. We can now interpret these results as follows: There is only a 5% chance that the actual 100-year discharge will be greater than 33,800 cfs.

Sec. 12.7

Rainfall Intensity-Duration-Frequency Relationships

485

Exceedence Probability (%) 99.99

99.9

99.8

0.5

0.2 0.1 0.05 0.01

10 9 8 7 6 5

50,000

Discharge (cfs)

20,000

10,000

1 9 8 7 6 5

5,000

Observed points Theoretical points Theoretical line Confidence limits

2,000

1,000

1.01

1.11

1.25

50 100 200

1000

Return Period (years) Figure 12.3

Log-normal probability plot of the Meherrin River data

Likewise, the probability that the actual value will be less than 21,000 cfs is 5%. The most likely value of the 100-year discharge is 25,600 cfs. The theoretical line of a probability graph can also be used to estimate the return period of a given magnitude of discharge at a given location. For example, the return period of 20,000 cfs for the Meherrin River can be read directly off Figure 12.3 as being 25 years. 12.7 Rainfall Intensity-Duration-Frequency Relationships Frequency analysis techniques can be used to develop relationships between the average rainfall intensity, duration, and return period. These relationships are often presented in chart form as intensity-duration-frequency (IDF) curves. IDF curves are used in engineering practice for designing a variety of urban hydraulic structures. To develop IDF curves for a given location, we first extract from rainfall records the annual maximum rainfall depths corresponding to selected rainfall durations. Then the data series for each duration are fit to a probability distribution. Next, the return periods are determined for various rainfall depths using this distribution. In the end, we divide these depths by the duration being considered to find the IDF relationships. Normally, the Gumbel distribution is used for rainfall frequency analysis. The following example problem will help clarify the procedure.

Pj (in.)

1.55 1.40 1.35 1.26 1.20 1.16 1.10 1.05 1.01 0.97 0.92 0.88 0.86 0.82 0.80 0.75 0.71 0.68 0.65 0.60 0.56 0.53 0.50 0.48 0.46 22.25 0.890

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Σ m s

2.20 2.00 1.85 1.72 1.60 1.53 1.47 1.40 1.34 1.28 1.24 1.20 1.14 1.09 1.04 1.00 0.95 0.90 0.86 0.82 0.78 0.74 0.71 0.68 0.65 30.19 1.208

Pj (in.)

0.430

0.985 0.628 0.413 0.263 0.154 0.104 0.069 0.037 0.018 0.005 0.001 0.000 0.005 0.014 0.028 0.043 0.066 0.095 0.121 0.150 0.183 0.219 0.248 0.278 0.311 4.436

(Pj - m)2 (in.2)

td = 30 Min

2.80 2.55 2.20 2.00 1.90 1.80 1.70 1.60 1.52 1.48 1.43 1.40 1.35 1.29 1.25 1.21 1.18 1.16 1.12 1.08 1.05 1.00 0.93 0.86 0.83 36.69 1.468

Pj (in.)

0.509

1.775 1.172 0.536 0.283 0.187 0.110 0.054 0.018 0.003 0.000 0.001 0.005 0.014 0.032 0.047 0.066 0.083 0.095 0.121 0.150 0.174 0.219 0.289 0.369 0.407 6.210

(Pj - m)2 (in.2)

td = 60 Min

Mean and Standard Deviation of Rainfall Depths for Example 12.11

3.20 2.80 2.60 2.47 2.40 2.29 2.18 2.07 2.00 1.90 1.81 1.71 1.64 1.60 1.53 1.46 1.40 1.35 1.29 1.22 1.16 1.11 1.09 1.07 1.06 44.41 1.776

Pj (in.)

0.598

2.027 1.048 0.678 0.481 0.389 0.264 0.163 0.086 0.050 0.015 0.001 0.004 0.019 0.031 0.061 0.100 0.142 0.182 0.237 0.310 0.380 0.444 0.471 0.499 0.513 8.594

(Pj - m)2 (in.2)

td = 120 Min

Statistical Methods in Hydrology

0.310

0.436 0.260 0.212 0.137 0.096 0.073 0.044 0.026 0.014 0.006 0.001 0.000 0.001 0.005 0.008 0.020 0.032 0.044 0.058 0.084 0.109 0.130 0.152 0.168 0.185 2.300

(Pj - m)2 (in.2)

td = 15 Min

Table 12.8

486 Chap. 12

Sec. 12.7

Rainfall Intensity-Duration-Frequency Relationships

487

Example 12.11 The annual maximum rainfall depth (P) series for a period of 25 years are given in columns 2, 4, 6, and 8 of Table 12.8 for storm durations (td) of 15, 30, 60, and 120 min., respectively. Develop IDF curves, assuming the data fit the Gumbel distribution. (A similar example was previously presented by Akan and Houghtalen.*) Solution The calculations can be performed in tabular form. The mean and standard deviation of rainfall depths (for each duration) are calculated in Table 12.8 using Equations 12.1 and 12.2. The frequency analyses of rainfall depths for the four storm durations are performed separately. Table 12.9 summarizes the calculations. The KT values shown in column 2 of Table 12.9 are obtained from Table 12.2 for Gumbel distribution. The corresponding precipitation depths are computed using Equation 12.24, as shown in columns 3, 5, 7, and 9 of Table 12.9. The average rainfall intensity, iavg, corresponding to the each precipitation depth is computed simply by dividing P by the duration, td, in hours. The results are plotted in Figure 12.4. Table 12.9 Rainfall Frequency Calculations for Example 12.11

td = 15 min. m = 0.890 in. s = 0.310 in. T 5 10 25 50 100

td = 30 min. m = 1.208 in. s = 0.430 in.

td = 60 min. m = 1.468 in. s = 0.509 in.

td = 120 min. m = 1.776 in. s = 0.598 in.

iavg

0.719 1.305 2.044 2.592 3.137

1.113 1.294 1.523 1.692 1.861

4.450 5.176 6.091 6.770 7.444

1.517 1.769 2.086 2.322 2.556

3.033 3.537 4.173 4.644 5.112

1.833 2.131 2.507 2.786 3.063

2.207 2.557 3.000 3.327 3.654

1.103 1.279 1.500 1.664 1.827

10 9 T (years) 100 50 25

Average Intensity (in./hr)

8 7 6

4 3 2 1 0

Figure 12.4

60 75 Duration (min.)

105

120

IDF Curves for Example 12.11

* Akan, A. O., and Houghtalen, R. J., Urban Hydrology, Hydraulics and Stormwater Quality. Hoboken, NJ: John Wiley and Sons, 2003.

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12.8 Applicability of Statistical Methods The statistical methods presented in this chapter, in particular the frequency analysis procedures, are applicable to a wide range of hydrologic problems. Much of this chapter examined the application of frequency analysis to discharges on a river. However, these techniques can be used to predict flood stage elevations, flood storage volumes, precipitation depths, pollutant loadings, and many other hydrologically related phenomena. For the techniques in this chapter to be used, an annual maximum or annual exceedence series must be given. We cannot, for example, take 27 discharge measurements from 13 years of streamflow data and use these 27 values as if we had 27 years of data. Furthermore, the data to be analyzed must represent hydrologically independent events. In other words, the magnitude of one event cannot be dependent or related to the magnitude of another event, nor can it be part of another event. For example, high flood stage elevations associated with a large rainfall event that occurs at the end of a year should only be used to represent the peak elevation in one of the two years. As long as the frequency distribution selected is appropriate for the type of analysis being performed, and the above conditions are satisfied, the statistical results should be valid. The log-Pearson type III and Gumbel distributions were developed to predict streamflows associated with rainstorms. They are not particularly adept at predicting drought conditions. In any case, we can and should test our statistical results using a goodness-of-fit test such as the chi-square test. Problems (secTion 12.2) 12.2.1. The annual precipitation measurements (Pi in inches) for the city of Atlanta, Georgia over a twenty year period are listed below. Determine the mean, standard deviation, and the skew coefficient for this series. Year

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

44.6

51.7

46.2

38.9

35.6

38.4

47.8

52.9

53.6

56.4

Year

2006

2007

2008

2009

2010

2011

2012

2013

2014

2015*

48.5

31.9

41.5

69.4

48.2

39.2

37.0

66.0

47.6

38.6

*Estimated based on gauge records for a portion of the year.

12.2.2. Determine the mean, standard deviation, and the skew coefficient for the log values of annual rainfall for the city of Atlanta, Georgia given in Problem 12.2.1. Also determine the mean precipitation (in inches) of the log transformed data. 12.2.3. A forensic engineer is studying the past history of flooding (prior to the construction of a flood control dam) on a river in southern Asia over a 20-year period. The annual maximum flood discharges (Qi in m3/s) from 1986 to 2005 are given below. Determine the mean, standard deviation, and the skew coefficient for this flood series. Year

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

4903

3751

4798

4290

4651

5050

6960

4366

3380

7826

Year

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

3320

6599

3700

4175

2988

2709

3873

4593

6761

1971

Problems

489

12.2.4. Determine the mean, standard deviation, and the skew coefficient for the log values of annual maximum flood discharges given in Problem 12.2.3. Also determine the mean annual flood (in m3/s) of the log transformed data set.

(secTion 12.3) 12.3.1. The annual precipitation mean and standard deviation in Atlanta, Georgia (for the 20-year period provided in Problem 12.2.1) are 47.1 in. and 9.60 in. respectively. Write out the probability density function assuming a Gumbel distribution for this precipitation series. 12.3.2. The annual precipitation mean and standard deviation in Atlanta, Georgia (for the 20-year period provided in Problem 12.2.1) are 47.1 in. and 9.60 in. respectively. The log mean and log standard deviation of the same data set (Problem 12.2.2) are 1.67 and 0.0863. Write out the appropriate probability density function assuming a normal distribution and a log normal distribution for this annual precipitation series. 12.3.3. Using the results of Examples 12.2 and 12.3, write out the probability density function assuming a log-Pearson type III distribution for the annual peak discharges, Qi, of the Meherrin River at Emporia, Virginia.

(secTion 12.4) 12.4.1. In Example 12.4, there is roughly a 40% chance of the culvert's capacity being exceeded in the 25 year service life. This has been determined to be too high a risk based on potential flooding concerns of nearby businesses. Therefore, determine the hydrologic risk of failure during the service life if the culvert was designed to accommodate the 100 year flood? Finally, determine the return period required to reduce the risk of exceedence during the service life to 10% and comment on its practicality. 12.4.2. A chemical industry needs to make repairs to its wastewater treatment plant which is located near the river. A levee is built to protect the construction work from the 10-year flood. What is the probability that the construction site will be flooded in the two year construction period? What is the probability that it will be flooded in the first and the second year? 12.4.3. A city depends upon a well field for its water supply. If the wells' yield during the dry season drops below a base level of 8,600 cfs, various emergency actions begin. After two consecutive days of sub-base flow, a small reservoir is brought on line, and after ten consecutive days of sub-base flow, water must be piped in from a nearby city. Well field yield records have been assessed to determine the exceedence probability of the two-day drought (40%) and the ten-day drought (10%). Determine (a) the probability of having to pipe in water at least once in the next two years; (b) the probability of not having to pipe in water in the next two years; (c) the probability of having to pipe in water during each of the next two years; and (d) the probability of having to pipe in water exactly once in the next two years. 12.4.4. A city depends upon the flow of a nearby river for its water supply. If the discharge drops below a base level of 25 m3/s, various emergency actions begin. After two consecutive days of sub-base flow, a small reservoir is brought on line, and after ten consecutive days of sub-base flow, water must be piped in from a nearby town. River discharge records have been assessed to determine the exceedence probability of the two-day drought (30%) and the ten-day drought (15%). Determine (a) the probability of having to utilize the reservoir at least once in the next two years; (b) the probability of not having to rely on the reservoir in the next two years; (c) the probability of having to rely on the reservoir during each of the next two years; and (d) the probability of having to rely on the reservoir exactly once in the next two years.

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12.4.5. A riparian power plant is scheduled for construction. A temporary cofferdam is being built to protect the construction site from the 10-year flood. What is the probability of the construction site being flooded during the first year of the three-year construction period? What is the probability of being flooded all three years? What is the risk of being flooded at least once during the three-year construction period? What is the probability of not being flooded during the construction period? If the owners of the power plant want to reduce the risk of flooding during the construction period to 20%, how quickly should the construction be completed in years?

(secTion 12.5) 12.5.1. Annual precipitation measurements (Pi in inches) for Atlanta, Georgia were given in Problem 12.2.1. The statistical parameters for this annual series were computed to be: m = 47.1 in., s = 9.60 in., and G = 0.736. Determine the magnitude of the 5-year precipitation depth if the data fits the (a) Normal distribution and (b) Gumbel distribution. How many times was the P5 (Normal) exceeded in the 20-year annual precipitation record given in Problem 12.2.1? 12.5.2. Annual precipitation measurements (Pi in inches) for Atlanta, Georgia were given in Problem 12.2.1. In Problem 12.2.2, the statistical parameters for this annual series were computed to be: ml = 1.67, sl = 0.0863, and Gl = 0.221. Determine the magnitude of the 5-year precipitation depth if the data fits the (a) Log-Normal distribution and (b) Log-Pearson Type III distribution. How many times was the P 5 (Log-Normal) exceeded in the 20-year annual precipitation record given in Problem 12.2.1? 12.5.3. Annual maximum flood discharges (Qi in m3/s) for a river in southern Asia from 1986 to 2005 were given in Problem 12.2.3. The statistical parameters for this annual series were computed to be: m = 4,533 m3/s, s = 1,513 m3/s, and G = 0.655. Determine the return interval of the 1995 flood (7,826 m3/s) if the data fits the (a) Normal distribution and (b) Gumbel distribution. 12.5.4. Annual maximum flood discharges (Qi in m3/s) for a river in southern Asia from 1986 to 2005 were given in Problem 12.2.3. In Problem 12.2.4, the statistical parameters for this annual series were computed to be: ml = 3.63, sl = 0.147, and Gl = -0.229. Determine the return interval of the 1995 flood (7,826 m3/s) if the data fits the (a) Log-Normal distribution and (b) Log-Pearson Type III distribution. 12.5.5. In Example 12.6 we determined that the log-transformed Meherrin River data fit the log-normal distribution at a significance level of α = 0.10. How many additional flows would need to be in the first class interval (lowest flows) transferred from the fifth class interval (highest flows) for the test to fail at the same significance level? How many additional flows would need to be in the fifth class interval (highest flows) transferred from the fourth class interval (second highest) for the test to fail at the same significance level? 12.5.6. In Example 12.6 we determined that the log-transformed Meherrin River data fit the log-normal distribution at a significance level of α = 0.10. Would our conclusion be different for α = 0.05 or α = 0.50. Which significance level is closer to failure? Explain. 12.5.7. Consider the annual maximum discharges on the Meherrin River given in Example 12.1 with sample statistics m = 9,820 cfs and s = 4,660 cfs. Test the goodness of fit of this data to the Gumbel distribution at a significance level of α = 0.05. Use five class intervals (k = 5). 12.5.8. Consider the annual maximum discharges on the Meherrin River given in Example 12.1 with sample statistics m = 9,820 cfs and s = 4,660 cfs. Test the goodness of fit of this data to the Normal distribution at a significance level of α = 0.50. Use five class intervals (k = 5). 12.5.9. Annual precipitation measurements (Pi in inches) for Atlanta, Georgia (a 20-year record) were given in Problem 12.2.1. The statistical parameters for this annual series were computed to be: m = 47.1 in., s = 9.60 in., and G = 0.736. Determine the 90% confidence limits for the

Problems

491

5-year precipitation depth assuming the data fits (a) the Normal distribution and (b) the Gumbel distribution. 12.5.10. Annual precipitation measurements (Pi in inches) for Atlanta, Georgia (a 20-year record) were given in Problem 12.2.1. In Problem 12.2.2, the statistical parameters for this annual series were computed to be: ml = 1.67, sl = 0.0863, and Gl = 0.221. Determine the 90% confidence limits for the 5-year precipitation depth assuming the data fits (a) the Log-Normal distribution and (b) the Log-Pearson Type III distribution. 12.5.11. Annual maximum discharges on the Meherrin River in Example 12.1 yielded these sample statistics: m = 9,820 cfs and s = 4,660 cfs. It has been determined that the data fits the normal distribution at the 10% significance level. Determine the 1.25-, 2-, 10-, 25-, 50-, 100-, and 200-year peak discharges and the 90% confidence limits for the Meherrin River. 12.5.12. Annual maximum discharges on the Meherrin River in Example 12.1 yielded these sample statistics: m = 9,820 cfs and s = 4,660 cfs. It has been determined that the data fits the Gumbel distribution at the 10% significance level. Determine the 1.25-, 2-, 10-, 25-, 50-, 100-, and 200-year peak discharges and the 90% confidence limits for the Meherrin River.

(secTion 12.6) 12.6.1. The tenth largest discharge recorded on the Meherrin River at the gauge site of interest (Example 12.10) during the 39-year study period was 12,600 cfs. Based on a Log-Normal distribution, determine the return period for this discharge in two different ways noting that ml = 3.95 and sl = 0.197. 12.6.2. A bridge will be built over the Meherrin River near the site that the discharge measurements were taken in Example 12.10. The service life of the proposed bridge is 40 years. If a design discharge of 23,000 cfs is used for the bridge, based on the Log-Normal distribution: (a) what is the probability that the bridge will be flooded in any one year? (b) What is the probability that the bridge will be flooded at least once over its service life? 12.6.3. Annual precipitation measurements (Pi in inches) for a 20-year record at Mythical City are displayed below. Determine the return periods assigned to these data values for plotting purposes. Prepare a probability plot (Normal distribution) of the annual maximum precipitation series. Use Normal distribution (probability) paper found on the Internet or use Figure 12.2. In addition, draw the theoretical (straight line) probability distribution on the plot. The statistical parameters for this annual series were computed to be: m = 40.0 in., s = 3.50 in., and G = 0.296. Year

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

44.2

47.6

38.5

35.8

40.2

41.2

38.8

39.7

40.5

42.5

Year

2006

2007

2008

2009

2010

2011

2012

2013

2014

2015

39.2

38.3

46.1

33.1

35.0

39.3

42.0

41.7

37.7

38.6

12.6.4. Obtain annual precipitation depths from the Internet (e.g., the National Weather Service) for the full gauge record at a location of interest. Using this data, determine (a) the mean, standard deviation, and the skew coefficient for this data series; (b) the mean, standard deviation, and the skew coefficient for the log transformed data; (c) the goodness of fit of this data to the Normal distribution (α = 0.10, k = 5); (d) the goodness of fit of this data to the Log-Normal distribution (α = 0.10, k = 5); (e) the 2-, 10-, 25-, 50-, and 100-year precipitation depths and the 90% confidence limits for the Normal distribution and the Log-Normal distribution;

492

Statistical Methods in Hydrology

Chap. 12

(f) a Normal probability plot of the precipitation series with the theoretical (straight line) probability distribution and the 90% confidence band; and (g) a Log-Normal probability plot of the precipitation series with the theoretical (straight line) probability distribution and the 90% confidence band. Note: Special probability paper is available on the Internet at various locations. 12.6.5. Obtain annual peak discharge values from the Internet (e.g., the U.S. Geological Survey) for the full gauge record at a location of interest. Using this data, determine (a) the mean, standard deviation, and the skew coefficient for this data series; (b) the mean, standard deviation, and the skew coefficient for the log transformed data; (c) the goodness of fit of this data to the Gumbel distribution (α = 0.10, k = 5); (d) the goodness of fit of this data to the Log-Pearson Type III distribution (α = 0.10, k = 5); (e) the 2-, 10-, 25-, 50-, and 100-year peak discharge values and the 90% confidence limits for the Gumbel distribution and the Log-Pearson Type III distribution; (f) a Gumbel probability plot of the annual peak discharge series with the theoretical (straight line) probability distribution and the 90% confidence band; and (g) a Log-Pearson Type III probability plot of the annual peak discharge series with the theoretical (straight line) probability distribution and the 90% confidence band. Note: Special probability paper is available on the Internet at various locations.

Symbols

A BHP b C

Cd CHW CN °C D DRH d de E Eb Ec Ep

cross sectional area, watershed area brake horsepower channel width, confined aquifer depth Chezy's coefficient, celerity (surface wave speed), weir coefficient, runoff coefficient (rational method) discharge coefficient Hazen–Williams coefficient SCS curve number degree Celsius diameter, hydraulic depth direct runoff hydrograph water depth (occasionally), soil interstitial space dimension depth of rainfall excess energy per unit weight of water (energy head), water surface elevation, evaporation modulus of elasticity (bulk) composite modulus of elasticity modulus of elasticity (pipe material)

Es EGL e em, ep F Fs f fp G g H Ha Hp Hs H′s HSH Hv HGL

specific energy energy grade line pipe roughness height, efficiency, pipe wall thickness motor and pump efficiency force, channel freeboard, cumulative infiltration specific force friction factor, rate of infiltration infiltration capacity skew coefficient gravitational acceleration total head approaching head pump head static pressure head, elevation rise net positive suction head system head velocity head hydraulic grade line 493

494

elevation (position) head, depth of water in an aquifer bend loss hb contraction loss hc discharge (exit) loss hd entrance loss he enlargement loss hE friction loss hf head loss hL friction loss in clockwise direction Σhfc Σhfcc friction loss in counter-clockwise valve loss hv I moment of inertial, linear impulse, infiltration, rainfall intensity, reservoir inflow moment of inertia about the neutral Io (centroidal) axis i rainfall rate rate of rainfall excess, rate of effective ie rainfall K coefficient of permeability, hydraulic conductivity, coefficient of energy loss SCS unit hydrograph constant Kp KT frequency factor k number of class intervals kk number of sample statistics L liters L hydraulic length LT lower confidence limit M moment, momentum, total mass m meter m mass, model, mean, side slope, number of flow channels (flow net) M, N exponents of the gradually varied flow function N Newton N number of values in sample Nf Froude number NR Reynolds number, rotational speed Nr pump rotational speed NS specific speed of pump Nw Weber number NPSH net positive suction head n Manning's coefficient, number of equipotential drops (flow net) O reservoir outflow P wetted perimeter, pressure, precipitation, SCS cumulative rainfall depth Pi, Po power input and power output Pf characteristic suction head PT total rainfall depth p pressure, probability

Symbols

Q Qi, Qo q qp R Rh Si r ri, ro ro, rw S Sc Se Sf So Sw Sy sp. gr. s sl T

TRH Tc TL Tp Tt1 t td UT UH u,v,w ui, uo V VG Vi, Vo Vs Vol υi , υ o υti, υto W W(u) w x,y,z Y y y1, y2

volumetric flow rate (discharge) inflow and outflow discharge per unit width SCS unit hydrograph peak discharge radius, risk (hydrologic), SCS cumulative runoff depth hydraulic radius initial degree of saturation variable radius, homologous ratio, rank of numbers in a listing inside, outside radius radius of influence, well radius slope, shape number, storage, SCS maximum potential retention critical slope, aquifer storage constant slope of the energy grade line energy (friction) slope channel slope water surface slope unconfined storage coefficient specific gravity stream slope, aquifer drawdown, standard deviation standard deviation of logs temperature, channel top width, aquifer transmissibility, transpiration, torque, return period or recurrence interval total runoff hydrograph time of concentration hydrograph lag time time-to-peak sheet flow travel time time, storm duration time, storm duration upper confidence limit unit hydrograph velocity in x, y, z direction impeller speed (inside, outside) mean velocity mean square error of sample skew inlet, outlet velocity seepage velocity volume radial velocity (inside, outside) tangential velocity (inside, outside) work, weight well function weight coordinate axes average watershed slope water depth initial, sequent depth

Symbols

yc, yn yp Z z α β γ ∆D ∆P ∆t

critical, normal depth depth to center of pressure depth of the wetted zone elevation, slope (run per unit rise) porosity, energy coefficient, angle, statistical significance level vane angle, momentum coefficient, confidence level (as a fraction) specific weight effective storm duration incremental precipitation time increment

495

ε θ μ v ρ σ τ τo ϕ χ2 ω

Poisson's ratio angle absolute viscosity kinematic viscosity density surface tension, cavitation parameter shear stress wall shear stress effective porosity Chi square test statistic angular velocity

Answers to Selected Problems

Chapter One 1.2.1 1.2.3 1.2.5 1.3.1 1.3.3 1.3.5 1.3.7 1.3.9 1.4.1 1.4.3

1.4.5 1.4.7

496

Etotal = 3.26 * 108 calories P = 0.692 atm T = 40.8°F γ = ρ#g W = 1,110 lb (4,940 N); SG = 0.889 ρ = 1.72 slug/ft3 (887 kg/m3) d = 3.00 ft Vol2 = 100.3 m3 (0.3, change) 1 Pascal = 1.450 * 10-4 psi 1 lb # s/ft2 = 478.9 poise 1 ft2/s = 929.0 stokes 1.002 * 10-3 N # s/m2 = 2.092 * 10-5 lb # s/ft2 1.003 * 10-6 m2/s = 1.080 * 10-5 ft2/s F = 0.702 lb Fshear resistance = 188 N

1.4.9 1.5.3

1.5.5 1.6.1 1.6.3

Torque = 32.7 N # m for h = 1.5 in., D = 0.0301 in. for h = 1.0 in., D = 0.0452 in. for h = 0.5 in., D = 0.0904 in. h = 0.158 cm ∆ρ/ρ = - 0.0642, (density increase) ∆γ/γ = - 0.727, (specific wt. increase) γs = 9,810 N/m3; γb = 9,880 N/m3

Chapter Two 2.2.1 2.2.3 2.2.5 2.2.7 2.2.9 2.2.11 2.4.1

h = 3.00 ft; P = 187 lb/ft2 P = 89.2 kN/m2, Error = 4.71, F = 62,300 lb, F = 31,200 lb (sides) Ptank = 8.03 psi = 5.54 * 104 Pa Pair = 21.3 psi (gauge), Pabs = 36.0 psi Psystem = 180 kPa; Foutput = 4.50 kN h = 3.35 in.

Answers to Selected Problems

2.4.3 2.4.5 2.4.7 2.4.9 2.4.11 2.5.1 2.5.3 2.5.5 2.5.7 2.5.9 2.5.11 2.5.13 2.6.1 2.6.3 2.6.5 2.6.7

2.6.9 2.6.11

2.6.13 2.8.1 2.8.3 2.8.5 2.8.7 2.8.9

2.8.11 2.8.13

h = 3.87 cm PA = 65.1 lb/ft2 = 0.452 psi P = -9.83 kPa; h = - 1.00 m of H2O PA - PB = -0.302 psi EA = 30 m F = 332 lb; yp = 0.853 ft 7 y = 0.848 ft F = 27.2 kN, yp = 5.01 m F = 841 lb; yp - y = 0.500 ft F = 6,450 lb; yp - y = 0.0652 ft h = 1.40 m F = 154 kN, yp = 2.75 m T = 21,000 lb (lifting force) d = 8.77 ft; depths greater makes the gate open; less makes it close. FH = 0; FV = 1080 lb FH = 783 lb d ; F = 825 lb; θ = 18.4° FH = 68.4 kN d ; FV = 77.2 kN c ; (larger force) FV = 4.07 * 106 N c ; which is also the total tension force in the bolts holding the top on. F = 4,090 lb; θ = 31.5° passing thru the center of the gate radius. FH = 6,030 lb S yp = 9.44 ft FVTriangle = 1,100 lb c 1.47 ft from wall FVquadrant = 947 lb c 1.87 ft from wall γcone = 19,100 N/m3; S.G. = 1.95 B = 242 N 6 W = 248 N, it will sink The lake level will fall. Why? γA = 0.5γ; γB = γ R = 1.02 m M = 1.11 * 106 [email protected] (heel angle = 5°) M = 2.21 * 106 [email protected] (heel angle = 10°) M = 3.29 * 106 [email protected] (heel angle = 15°) It is not stable. Why? d = 0.171 m

Chapter Three 3.3.1 3.3.3 3.3.5 3.3.7 3.5.1

Choose the flat shield; but why? V = 32.6 m/s P = 270 kPa F = 453 kN; θ = 44.3° f = 0.049; turbulent – rough pipe

497

3.5.3 3.5.5 3.5.7 3.5.9 3.5.11 3.5.15 3.7.1 3.7.3 3.7.5 3.7.7 3.7.9 3.11.1 3.11.3 3.11.5 3.11.7 3.11.9 3.11.11 3.12.1 3.12.3 3.12.5 3.12.7

∆P = 14.9 kPa h = 7.85 m Q = 95.6 m3/s L = 871 ft D = 1.28 ft (use an 18 in. pipe) QLeak = 8 L/s Hazen: hf = 11.4 m; Manning: hf = 15.2 m; Darcy: hf = 12.0 m Hazen: Q = 178 m3/s; Manning: Q = 163 m3/s; Darcy: Q = 156 m3/s Q = 98.6 m3/s (H); Q = 76.5 m3/s (M) CHW = 110 Q30 = 0.156 m3/s; Q20s = 0.108 m3/s hc = 0.275 m; hE = 0.353 m Kv = 2.66 PA = 510 kPa Q = 567 ft3/s Pressure Head Gain = 26.1 m P1 = 55.8 kPa [(DE5.33)/(nE2L E)]1/2 = Σ[(Di5.33)/(ni2L i)]1/2 hfAF = 40.6 ft; Q1 = 87.0 cfs; Q2 = 33.0 cfs L E = 4,040 m; hfAF = 275 m L E = 2,340 m

Chapter Four 4.1.3 4.1.5 4.1.7 4.1.9 4.1.11 4.1.13 4.1.15

4.2.1 4.2.3 4.2.5 4.2.7 4.2.9 4.2.11

V = 1 m/s and D = 2 m; V = 2 m/s and D = 1 m; etc. hA - hB = 35.7 ft Q = 5.88 m3/s and Q = 5.92 m3/s Q8 = 3.75 cfs; Q16 = 16.1 cfs ( +429,) D = 0.44 m; hA - hB = 121 m D = 11.5 mm QAB = QCD = 18.0 cfs, HB = 209.7 ft; HC = 117.1 ft; QBC1 = 6.84 cfs; and QBC2 = 19.2 cfs Q = 96.4 L/s, PS /γ = -49.0 kPa PS = -14.7 lb/in.2; cavitation will occur PS /γ = -5.61 m ( 7 - 7.0 m; no cavitation) hS = 9.26 ft L = 51.6 m P2 = 38.2 psi; PS /γ = -21.6 ft ( 7 -23.0 ft; no cavitation concerns)

498

Answers to Selected Problems

Q1 = 7.822 m3/s, Q2 = 4.220 m3/s Q3 = 12.041 m3/s, HJ (total head at junction) = 1567.9 m; (P/γ)J = 17.9 m 4.3.5 Q1 = 75.1 ft3/s, Q2 = 172.7 ft3/s Q3 = 247.4 ft3/s, HJ (total junction head) = 5163.7 ft; Elev(J) = 5133.7 ft. 4.3.7 Q = 2.19 L/s 4.4.1 (a) Q1 = 0.321 m3/s; Q2 = 0.615 m3/s hL = 2.51 m; (b) Q1 = 0.321 m3/s, Q2 = 0.615 m3/s, hf = 2.50 m 4.4.3 (a) PF = 167.4 kPa (b) Node F 4.4.5 QAB = 464 L/s, QAC = 536 L/s QBD = 505 L/s, QCE = 495 L/s QCB = 41 L/s, QED = 45 L/s 4.4.7 QAB = 12.4 cfs, QAC = 15.6 cfs QBD = 8.25 cfs, QCE = 8.04 cfs QBF = 4.14 cfs, QCF = 1.56 cfs QFG = 5.70 cfs, QGD = 1.75 cfs QGE = 3.96 cfs, PD = 28.7 psi ( 630) 4.4.11 QAB = 130 L/s, QFA = 30 L/s QBC = 25 L/s, QBD = 105 L/s QDE = 70 L/s, QEC = 6 L/s QFG = 270 L/s, QGD = 215 L/s QGH = 55 L/s, QEH = 65 L/s 4.4.13 Q1 = 8.367 cfs, Q2 = 0.515 cfs Q3 = 1.852 cfs, Q4 = 0.877 cfs Q5 = 1.779 cfs, Q6 = 1.983 cfs Q7 = 6.387 cfs, Q8 = 8.859 cfs 4.5.3 t = 2.00 s 4.5.5 ∆P = 8.79 MPa 4.5.7 ∆P = 3,030 psi; Pmax = 3,090 psi 4.5.9 e = 32 cm (this is unreasonable) 4.6.3 Ds = 33.9 ft 4.6.5 Ds = 8.58 m 4.3.3

Chapter Five 5.1.3 5.1.5 5.1.7 5.1.9 5.5.1 5.5.3 5.5.5

Drawdown = 0.0302 m (3.02 cm) Pm = 787 kW T = 9,270 ft # lb Q = 5.89 m3/s; Pi = 28,000 kW Q ≈ 17.5 cfs and Hp ≈ 242 ft Q = 0.75 m3/s and V = 3.82 m/s Q = 400 L/s and Hp = 18.0 m

5.5.7 5.6.1

5.6.3 5.6.5 5.7.1 5.7.3

5.9.3 5.9.5 5.9.7 5.10.1 5.10.3 5.10.5 5.11.1 5.11.3 5.11.5

5.11.7

Q = 13.5 cfs and Hp = 52.2 ft Q1 = 12.5 cfs and Q2 = 1.0 cfs (d) two pumps in series (e) four pumps, two parallel pipes with two pumps in series in each pipe Q = 0.30 m3/s and Hp ≈ 50 m. Po = γQHp = 73.4 kW Parallel: Q ≈ 0.60 m3/s; Hp ≈ 85 m Series: Q ≈ 0.75 m3/s; Hp ≈ 105 m Qsys = 4.2 m3/s; Hp = 21 m Q1 = 2.7 m3/s; Q2 = 1.5 m3/s QAB = 1.00 m3/s; HpA = 13 m. QBD = 0.50 m3/s; HpB = 19 m EC = 56.6 m hp … 9.86 ft L = 10.3 m (Pi /γ + V 2i /2g) = -1.06 m Po = 37.8 hp e = 0.741; Pi = 63.5 kW Q = 9.56 m3/s, e = 0.891 Hmax = 46 m when Q = 0; the match point is Q = 75 L/s; Hp = 43 m Pump I, ω = 4350 rpm, Q ≈ 22 L/s, Hp ≈ 51 m, e ≈ 43, Pump I S ω = 4350 rpm, e ≈ 38,, Q ≈ 40 L/s, Hp ≈ 35 m, & Pump II S ω = 3850 rpm, e ≈ 54,, Q ≈ 42 L/s, Hp ≈ 38 m Best: Use two pumps (IV) in parallel: ω = 3850 rpm, e ≈ 61,, Q ≈ 170 L/s (each pump), and Hp ≈ 34 m

Chapter Six 6.1.1 6.2.1 6.2.3 6.2.5 6.2.7 6.2.9 6.3.1 6.3.3 6.3.5 6.3.7 6.4.3

(a) steady, uniform; (b) steady, varied S = 0.000742 y = 4.90 m, b = 9.80 m yn = 1.33 ft; from Fig 6.4a. yn = 1.26 ft m = 2.19 m/m do = 2.00 m; do = 1.54 m b = 2y (half square) do = 2.00 m (flowing half full) y = 9.31 ft; b = 10.7 ft m = 1; θ = 45° S = 0.00567; yc = 1.37 m; Nf = 0.720 (Nf 6 1, therefore flow is subcritical)

Answers to Selected Problems

6.4.5 6.4.7 6.4.9 6.5.3 6.5.5

499

yc = 1.90 m; S = 0.00264 m/m Nf = 0.233 ( 61; subcritical); E = 8.46 ft; yc = 3.65 ft ∆z = 0.39 m ∆E = 1.62 ft, NF1 = 3.56 (supercritical) Here are some of the plotting points: Depth (ft)

Area (ft)2

0.5 1.0 1.5 etc.

2.5 5.0 7.5

V (ft/s) 16.0 8.0 5.3

7.4.9

E (ft)

Fs (lb/ft)

7.5.3 7.5.5 7.5.7 7.8.1

4.5 2.0 1.9

256 155 153

7.8.3

6.8.1

(a) reservoir flowing into a steep channel (S-2); etc. for (b) through (d) 6.8.3 Classification is M-1; explain why? 6.8.5 M-2, 5.18 ft 6.8.7 1.74 m, 1.66 m (normal depth) 6.8.9 746 m and 1,616 m 6.8.11 M-1, 5.38 m, 4.74 m, 3.31 m, 1.18 m 6.8.13 M-1, 3.45 ft, 3.35 ft, 3.22 ft 6.8.15 M-2, 6.19 ft, 27.87 ft 6.9.1 b = 8.87 m and y = 0.45 m 6.9.3 y = 2.59 m and b = 5.18 m

Chapter Seven α = 0.315 (a) t = 12.3 min; (b) V (actual) 7 Vs 7.1.7 K = 5.79 * 10-3 ft/s 7.1.9 Q = 1.79 cfs (ft3/s) 7.2.3 K = 7.82 * 10-4 m/s. 7.2.5 K = 1.30 * 10-4 m/s; T = 1.30 * 10-3 m2/s; S60 = 8.89 m 7.2.7 K = 4.01 * 10-4 ft/s; sw = 100 ft 7.2.9 s1 = 7.51 ft; s1 + s2 = 11.8 ft 7.2.11 T = 1050 m2/day 7.3.1 s = 14.6 ft 7.3.3 s (t50) = 5.39 m, s (t250) = 6.92 m 7.3.5 t = 2.2 days, s Total = 1.00 m s1 = 0.33 m, and s2 = 0.67 m 7.3.7 s = 4.44 ft 7.4.3 K = 2.42 * 10-4 m/s s = 5.32 m, ro = 88.7 m 7.1.3 7.1.5

7.4.5 7.4.7

7.8.5 7.9.1 7.9.3 7.9.5

T = 447 ft2/hr, s = 1.67 ft T = 1.90 m2/hr; S = 1.08 * 10-4 s = 1.97 m T = 1.53 m2/hr, S = 1.01 * 10-4 s = 5.16 m sw = 13.2 m, sb = 0.0 m T = 0.0262 ft2/s s = 5.54 m H1 = 70.0 ft, VS1 = 1.00 ft/day H2 = 65 ft, VS2 = 2.86 ft/day H4 = 7.5 ft, VS4 = 0.837 ft/day q = 1.08 gpm per foot VS = 3.74 * 10-4 ft/s q = 1.78 * 10-6 m3/s per meter 20% seepage reduction Q = 13.2 gpm Q = 21.7 ft3/day VS = 1.34 * 10-7 m/s q = 1.08 m3/day

Chapter Eight 8.3.1 8.3.3 8.3.5 8.3.7 8.3.9

8.5.3 8.5.5 8.5.7 8.6.1 8.6.3

8.6.5 8.7.1 8.7.3 8.7.5 8.8.1 8.8.3

ΣF = 0; ΣM = 0 H = 12.3 m FRslide = 5.61 (safe) FRover = 5.70 (safe) PT = 3,360 lb/ft2 PH = 2,210 lb/ft2 @25 ft, R = 37,200 lb/ft @50 ft, R = 47,400 lb/ft @75 ft, R = 30,800 lb/ft yc = 0.380 m; Q = 5.14 m3/s x = 2.75 ft Q = 2.06 m3/s; H = 1.45 m; Actual H = 1.44 m , Error = 1.36 , The design flow can't be met under these conditions. A little longer spillway is needed or a more efficient spillway (C). Q = 8100 cfs; xP.T. = 12.7 ft; yP.T. = -10.4 ft y = 7.62 ft. y5 = 6.61 ft; y10 = 7.21 ft y = 10.7 m, 11.5 m, and 11.8 m Pc /γ = - 6.71 m ( 7 - 7.0 m, OK) W = 6.52 m

500

Answers to Selected Problems

Pc/γ = -37.5 ft; yes cavitation danger A = 1.65 m2 (a) hL = 3.10 m 7 2.35 m (won't work) (b) hL = 1.98 m 6 2.07 m (will work) 8.9.3 Q = 645 cfs 8.9.5 Q = 129 cfs (category (b) flow) 8.9.7 D = 3.25 ft (or next larger standard size operating under inlet control) 8.10.1 d2 = 3.5 ft; L = 8.1 ft; ∆E = 3.9 ft Efficiency = 0.54 or 54, 8.10.3 Type II; d2 = 5.2 m; L = 22.4 m; ∆E = 23.3 m; Efficiency = 18, 8.8.5 8.8.7 8.9.1

Chapter Nine 9.1.1 9.1.3 9.2.1 9.2.3 9.2.5 9.3.1 9.3.3 9.3.5 9.3.7 9.4.1 9.4.3 9.4.5 9.4.7 9.4.9 9.4.13

SG Oil = 0.811 P = 1.47 psi = 10.1 kN/m2 h = 320 cm; V = 1.38 m/s (not avg.) ∆h = 0.734 ft; Q = 2.13 cfs V = 1.77 m/s ∆h = 2.81 ft Q = 0.0818 m3/s ∆h = 7.93 in. ( ≈ 8 in.) ∆h = 0.0527 m = 5.27 cm Q = 29.6 cfs ∆h = 1.77 ft p = 2.08 m L = 0.476 m Q = 2.28 m3/s Q = 3.09LH 3/2 (BG unit system)

Chapter Ten 10.2.1 10.2.3 10.2.5 10.2.7 10.2.9 10.3.3 10.3.5 10.4.3

Am = 0.125 m ; Vm = 0.20 m/s Qm = 0.025 m3/s Tp = 3.05 hr L r = 241, but use L r = 250 Qm = 1.70 * 10-3 cfs; Vm = 0.11 ft/s Fp /L p = 510 lbs/ft; Qr = 9,000 Tr = 2,500; Fr = 1; Pr = 0.02 Er = MrV 2r = 50 Vp = 1.1 m/s; Tp = 200 N # m Qm = 2.63 cfs Vm = 150 ft/s; Vm = 0.424 ft/s

10.4.5 Qp = 1,200 m3/s; yc = 2.17 m; Cp = 2.19 10.5.1 σp = 4.07 * 103 dyn/cm; Fr = 156 10.5.3 Qr = 0.192; Fr = 0.333 10.7.1 Vm = 0.5 ft/s; NRm = 1.77 * 105 10.7.3 μm = 8.06 * 10-5 N # s/m2 10.8.1 Xr = 124, Vm = 0.475 m/s 10.8.3 (a) nm = 0.012; (b) nm = 0.029 10.8.5 Xr = 206 10.9.1 No dimensionless variables–all are repeating and Π = [P/(ω # T)] 10.9.3 FD = (B2V 2ρ) ∅′(BVρ/μ) 10.9.5 Π1 = [V/(dg)1/2]; Π2 = [d/ε] Π3 = [(d 3g)1/2ρ/μ]

Chapter Eleven 11.1.1 Clouds–water (vapor) holding element, Precipitation–liquid transport, Interception/Depression storage/Snow pack– water (or ice) holding, etc. 11.2.1 Return Period–the average number of years between occurrences of a hydrologic event with a certain magnitude or greater. 11.2.3 P = 1/5 = 0.20 (20,) P (3 yrs in a row) = 0.008 (0.8,) P (not exceeded in 3 yrs) = 0.512 11.2.5 (a) 60% of R/F occurs when t/td = 0.5 (b) 80% of R/F occurs when t/td = 0.5 11.2.7 These are the ratios requested: t/td

0.25

0.5

0.75

1.0

P/PT

0.0

0.19

0.64

0.88

1.0

11.2.9 R/F intensities (in./hr) in 10-minute increments from t = 0 to t = 60 min.: I = 1.8, 3.2, 7.0, 2.4, 1.2, 0.6 11.2.11 R/F intensities (cm/hr) in 10-minute increments from t = 0 to t = 60 min.: I = 3.08, 7.68, 15.3, 4.61, 2.20, 1.65 Peak I would increase for smaller ∆t. 11.2.13 Peak intensity = 5.436 in./hr 11.2.15 P(total) = 3.81 cm; Imax = 9.20 cm/hr; Vol = 3,810 m3 11.3.3 Infiltration losses (in./hr) in 0.25 hr increments from t = 0 to t = 1.5 hr:

Answers to Selected Problems

501

f = 1.20, 2.05, 0.93, 0.78, 0.70, 0.60 Total R/F = 3.5 in. Total f = 1.57 in. 11.3.5 R = 6.57 in.; R/O Vol. = 54.8 [email protected] 11.3.7 R = 67.1 mm; R/O Vol. = 67,100 m3 11.3.9 H a lf h o u r R / O d e p t h s ( i n . ) ; predevelopment: ∆t ∆R

11.4.1 11.4.3 11.4.5 11.4.7 11.4.9 11.4.11 11.4.13 11.5.1 11.5.3 11.5.5

11.5.7 11.5.9 11.6.1 11.6.3 11.6.7 11.6.7

11.6.9 11.6.11 11.6.13

1st

2nd

3rd

4th

5th

6th

0.00 0.03 0.42 0.52 0.34 0.14

R/O depth increases to 1.88 in. from 1.45 in.; ∆Vol = 5.38 [email protected] Tc = 0.083 hr (5.0 min.); 25 m of overland flow (%) S 52, of Tc Tc decrease from 0.92 hr to 0.59 hr Vol = 20.6 [email protected]; R/O depth = 0.99 in. ≈ 1.0 in. (i.e., unit hydrograph). S = 19.1 mm; Tc = 52.4 min Tp = 35.2 min; qp = 5.75 cms/cm ∆R/O depth = -36,; ∆Qp = - 24, TRH peak Q = 5,350 cfs at t = 5 hr UH1 peak Q = 30 cms at t = 1.25 hr Speak = 4.49 [email protected]; Elevpeak = 884.3 ft. At depth = 2 ft, (2S/∆t) + O = 8.45 cfs At t = 100 min.; Outflow = 0.74 cms; At t = 120 min.; Outflow = 0.86 cms; S = 4,950 m3; and H = 0.6 m Peak outflow = 0.80 cms (t = 32 min.) and h = 0.59 m above the spillway At t = 4 min, O = 31.5 cfs, h = 4.75 ft Q25 = 58.0 cfs Q10 = 0.163 m3/s Q2 = 12.7 cfs (a) For a 5-ft-spread, the inlet would be located further to the east (less flow in the gutter before an inlet is required). The second inlet is placed about 193 feet down the street from the first inlet. Dr = 0.311 m (use a 40 cm pipe dia.) y = 20.0 cm; t = 21 s (0.35 min) Pipes sizes (inches): [email protected] = 18, [email protected] = 30, [email protected] = 15; [email protected] = 30

11.6.15 Pipe sizes (inches): AB = 12, CB = 15, BD = 24, and DR = 30.

Bonus Problems (Water Budgeting) B1 B3

Leak = 411 gal/30 days = 13.7 gal/day P = 8,800 [email protected]; ∆S = +2,800 [email protected]

Chapter Twelve 12.2.1 m = 47.1 in.; s = 9.60 in.; G = 0.736 12.2.3 m = 4,533 m3/s; s = 1,513 m3/s; G = 0.655 12.3.1 f X(x) = (0.134) # exp[z - exp[z]], where z = - 0.134 (x - 42.8) 12.4.1 R = 22.2,; T = 238 years (probably unrealistically expensive) 12.4.3 (a) R = 19.0,; (b) p = 81.0,; (c) p = 1.0,; (d) p = 18.0,; 12.4.5 (a) p = 10,; (b) p = 0.1,; (c) R = 27.1, (d) p = 72.9,; (e) n = 2.12 years 12.5.1 (a) P5 = 55.2 in. (b) P5 = 54.0 in. (c) Exceeded 3 times in 20-yr record 12.5.3 (a) T ≈ 75 years; (b) T ≈ 30 years The answers are quite different; using the appropriate distribution is critical. 12.5.5 (a) Four additional flows, and (b) three additional flows 12.5.7 Since χ2 6 χ2α (i.e., since 1.385 6 5.99), the Gumbel distribution does adequately fit the annual maximum discharge data at the 5% significance level. 12.5.9 U5 = 60.2 in.; L 5 = 51.5 in. (Normal) U5 = 58.7 in.; L 5 = 50.4 in. (Gumbel) 12.5.11 Q10 = 15,800 cfs; U10 = 17,700 cfs; L 10 = 14,300 cfs Q100 = 20,700 cfs U10 = 23,500 cfs; L 10 = 18,600 cfs 12.6.1 T = 4.2 years (Figure 12.3) T ≈ 4.5 years (Equation 12.24)

Index Absolute pressures, 14–17 Absolute viscosity, 6, 55 Acceleration, 375 angular, 158, 181, 375 gravitational, 4 Acoustic Doppler velocity, 352 Acre-feet, 431 Adverse channel, 222–224 Alternate depths, 212 Angular acceleration, 375 Angular deformation, 5 Angular momentum conservation, 156 Angular velocity, 158, 181 Annual exceedence series, 464 Annual maximum series, 464 Antiseep collar, 316 Apparent velocity, 255 Aquifer, 253, 398 boundaries, 279–284 characteristics, 270–279 coastal, 286–291 confined, 253, 254 drawdown, 258 homogeneous, 258 hydraulic gradients in, 253 isotropic, 258 recovery, 269 storage coefficient, 263 transmissivity, 260 unconfined, 253, 254 Arch dam, 310. See also Gravity dam Archimedes' principle, 31 Area of influence (well), 259 Artesian spring, 253 Artesian well, 253 Artificial recharge, 289 Atmosphere of earth, 2 Atmospheric pressure, 2, 14, 15 Average rainfall intensity, 400, 436 Axial flow pump, 161–164 Backwater curve, 226, 229 Base flow (BF), 415–416, 419, 425, 426, 446 Bend meter, 356–357 Bernoulli's principle, 62, 348 Best hydraulic section, 207 Body forces, 10 502

Boiling point, 3 Booster pump, 164–165 Bourdon tube gauges, 348 Brake horsepower, 165 Branching pipes, 103–110, 173–176 Broad-crested weir, 360–361 Bubble collapse, 3, 99, 178, 329–330 Buckingham Pi theorem, 363, 389, 391 Bulk modulus of elasticity, 9 Buoyancy force, 31–32 Buttress dams, 309, 310 Calibration (meter), 352, 355 Calorie, 2 Capillary action, 7–8, 263 Catchment, 399 Cavitation, 3, 99, 330 parameter, 178 in water pumps, 177–180 Celerity, 126 Center of buoyancy, 33 Center of gravity, 33, 35 Center of pressure, 24–27 Centrifugal force, 156, 210, 356 Centrifugal pumps, 155–161, 165–166 Centroid of plane areas, 24 Channel classification, 222 Channel liners, 235 Channel roughness, 203, 418 Characteristic curves (pump), 165–166 Chezy's formula, 202 Chi-square test, 476–478 Cipolletti weir, 359 Closure error, 112 Coastal aquifer, 286–291 Coefficient of permeability, 255, 257 Coefficients (discharge), 375 bend meter, 356 nozzle meters, 354 orifice, 333 orifice meters, 354 Venturi meters, 352, 356 weirs, 357–361 Coefficients (energy loss) bend (pipe), 80 confusor, 76

contraction, 76 diffusor, 78 entrance loss, 76 exit, 79 valves, 81–84 Colebrook–White equation, 67 Compressibility of water, 8–9, 126 Condensation, 3, 397 Cone of depression, 259 Confidence band, 484 Confidence interval, 478 Confidence limits, 478–481 Confined aquifer, 253, 254 drawdown curve in, 281 equilibrium test in, 271–272 steady radial flow in, 259–261 unsteady radial flow in, 263–267 Confusor, 76 Conservation angular momentum, 156 energy, 95–98, 101, 103, 110 mass, 57–58, 103–104, 110, 114, 427 momentum, 58, 156, 161, 217, 327, 360 Continuity equation, 57–60 Contracted horizontal weir, 358 Control section, 222, 224, 225 Control volume, 57–58, 161–162, 201, 217 Critical channel, 224 depth, 212–214, 221–222, 317, 318, 361 flow, 212 Critical flow flume, 361 Critical zone (Reynolds number), 56 Culvert, 331–336 Cumulative density function, 472, 473 Current meters, 350–352 Curve number (CN), 413–415 Cutoff wall, 293, 312 Dams, 308 arch, 310, 314–315 buttress, 309 classifications, 308–310 earth, 309, 316–317 earthquake forces on, 312

Index functions, 308–310 gravity, 309–314 low head, 320, 321 Darcy's law, 255, 261, 292, 410 Darcy–Weisbach equation, 64, 65, 98, 110 Darcy–Weisbach friction formula, 173–176 Datum, reference, 54, 210 Delineation of watershed, 399–400 Density, 1, 3–5 Depression storage, 397 Depths, 198 alternate, 212 critical, 212–214, 221–222, 318, 361 flow, 198 hydraulic, 199, 213 initial, 216 normal, 204, 205, 220 sequent, 216 uniform, 204, 205, 220 Design event (storm), 400–408 design-storm selection, 402–404 HEC-HMS Model, 444–445 hyetograph, 401–402, 409 intensity–duration–return period relationships, 402 runoff hydrographs, 415–426 SCS, 405–408 synthetic block design-storm hyetograph, 404–405 Design flow rate, 398 Diffusor, 78 Dimensional analysis, 371, 388 Dimensional homogeneity, 372–373 Direct runoff hydrograph (DRH), 419–420, 425 Direct step method, 227–234 Discharge head loss, 79 Discharge measurements in open channels, 357–366 in pipes, 352–357 Discharge of spillway, 323 Distorted model, 387 Dominant forces, 376 Drag, 384 force, 384–386, 389, 390 form, 384 Drainage area, 421, 424, 435, 438–440 Drainage blanket, 295 Drawdown of water table, 258 Drop inlet, 316 Dynamic similarity, 376–377

503 Earth dams, 309, 316–317 Earthquake force (dams), 312 Efficiency (pump), 159, 165 of open-channel sections, 207–209 Elasticity, modulus of composite water-pipe system, 126 pipe materials, 126 water, 8–9, 126 Electrical resistivity method, 285 Elevation head, 61 Embankment, 316–317 Emergency spillway, 316–317, 323 Energy coefficient, 210 dissipators, 336 equation, 62 grade line (EGL), 62, 71, 98, 101–102, 197 kinetic, 60 latent, 2 open-channel flow, 210–216 in pipe flow, 60–63 potential, 60 pressure, 60 specific, 211 EPANET model, 137–140 EPA-SWMM model, 447–450 Equal pressure, surfaces of, 17–19, 349 Equilibrium test (aquifers), 271–275 Equipotential line, 291, 292 Evaporation, 2–3 Exceedence probability, 472, 478, 482–483 Expansion joint, 126 Extrados, 314 Extreme value type I distribution. See Gumbel distribution Eye of pump, 156, 178 Federal Highway Administration (FHWA), 334 Fixed-bed model, 386 Flexible liners, 235 Flotation stability, 33–37 Flow net, 291, 293 FLT system. See Force–length–time (FLT) system Flumes Parshall, 361–366 Venturi, 361–366

Force body, 10 buoyancy, 31–32 centrifugal, 156, 210, 356 earthquake, 312 gravity, 381–382, 384, 385 hydrostatic, 23–28, 310, 311 line, 10 pressure, 23 resistance, 5, 201 sedimentation, 311 shear, 5 specific, 218 surface, 10 uplift, 311, 312 viscous, 64 weight (dam), 311, 312 Force ratio against overturning, 313 Force ratio against sliding, 312 Force–length–time (FLT) system, 389 Form drag, 384 Freeboard, 208, 235 Free-body diagram, 28, 201 Frequency analysis, 473–481 using probability graphs, 481–485 Frequency factor, 473–476 Friction factor for laminar flow, 64–65 for turbulent flow, 65–71 Frictionless weir, 318, 319 Froude number, 213, 336, 381–382, 384, 389 Gauge pressure, 14–17, 99 Gauging station, 465 Gaussian distribution. See Normal distribution Geometric similarity, 373–374 Ghyben–Herzberg relation, 288 Goodness of fit, 476–478, 484 Gradually varied flow, 199, 203, 219–221 classifications, 221–224 Gravitational acceleration, 4 Gravity dam, 309. See also Arch dam cross-section of, 311 stability of, 311–314 top-view representation of, 310 Gravity forces, 381–382, 384, 385 Green and Ampt model, 410–412, 445 Gumbel distribution, 469, 473, 474, 480

504 Hagen–Poiseuille law, 65 Hardy–Cross method, 111–121 Hazen–Williams equation, 71, 72 Head elevation, 61 energy, 169 net positive suction, 178 pressure, 16, 61 pump, 158, 165 suction, 178 system, 167 velocity, 61 Head loss, 61–62 bends, 79–80 contractions, 75–77 entrance, 76 exit (discharge), 79 expansions, 78–79 friction, 63, 71–75 major, 63 minor, 63 valves, 81–84 Heat (energy), 2 fusion, 2 latent, 2 specific, 2 vaporization, 2 HEC-HMS model, 444–447 HEC-RAS model, 240–245 Heel of dam, 312 Horizontal channel, 222 Hydraulic, 1, 397 conductivity, 255 depth, 199, 213 efficiency, 207–209 jack, 17 jumps, 216–219 length, 418 radius, 71, 199, 202 similitude, 371–378 structures classification, 307–308 Hydraulic grade line (HGL), 62, 98, 101–102, 197, 328 Hydraulically rough pipe, 66 Hydraulically smooth pipe flow, 66 Hydrograph parameters of SCS, 420 streamflow, 416 unit, 419–424 Hydrologic cycle, 397, 398 Hydrologic methods, 398 EPA-SWMM model, 447–450 HEC-HMS model, 444–447

Index Hydrologic modeling, 443–450 Hydrologic risk, 473 Hydrologic simulation, 446–447 Hydrology, 397 estimating future magnitudes, 484–485 statistical methods (See Statistical methods, hydrology) Hydrostatic force, 310, 311 on curved surfaces, 28–31 on flat surfaces, 23–28 Hydrostatic pressure, 14, 15, 23 Hyetograph, 401 design storm, 400–409 SCS, 405–408 storm, 401–402 synthetic block design-storm, 404–405 IDF curve, 402, 436, 485, 487 Impeller, 156 Impulse-momentum, 58, 161 Inclined manometer, 348 Infiltration, 397 Infiltration capacity, 408, 410 Inflow path calculations (Hardy– Cross method), 117 Intensity–duration–frequency (IDF) curve, 402, 436, 485, 487 Interception, 397 Jacob formulation, 265 Jacob solution, 275–276 Jet pump, 164–165 Joint (expansion), 126 Junction (branching pipes), 103 Junction equation, 110 Keyway, 313, 316 Kinematic similarity, 374–376 Kinematic viscosity, 6 Kinetic energy, 60, 210 Lag time, 420 Laminar flow friction factor for, 64–65 in pipe, 55 Laminar sublayer, 66 Latent energy, 2 Line forces, 10 Linear impulse, 161 Linear momentum, 161 Linearity, 423, 424

Logarithmic method (surge tanks), 135 Log-normal distribution, 469, 473, 474, 480, 485 Log-Pearson type III distribution, 470–472, 474, 475, 480 Loop equation, 110 Losses from rainfall, 408–415 Low head dam, 320, 321 Low-level outlet, 316, 317 Major head loss, 63 Manning equation, 73, 203, 386–388, 417, 439 Manning's coefficient, roughness, 73, 203, 418 Manning's kinematic solution, 416 Manning's "n" values, 73, 203, 418 Manometer, 19–22, 347–349 differential, 19, 20, 22 inclined, 348 open, 19, 347 single-reading, 21, 22 Mass (density), 3–5 Mass–balance relationship, storage routing, 427–428 Mass–length–time (MLT) system, 389, 390 Match point, 167 Maximum permissible velocity method, 236 Mean, 464, 466–467, 486 Mean sea level (MSL), 54 Mean velocity, 54 Metacenter, 33 Metacentric height, 33 Method of images, 279 Mild channel, 222 Minor head loss, 63 Mixed-flow pump, 164–165 MLT system. See Mass–length– time (MLT) system Model fixed-bed, 386 movable bed, 387 open-channel, 386–388 scaled, 371 undistorted, 386 Modified Puls routing method, 428–430 Modulus of elasticity, 9, 126 Moment of inertia, 24–26, 35

Index Moment, righting, 35 Momentum, 57–60, 156, 161 Moody diagram, 67, 100–101, 105 Movable channel bed models, 387 Multistage propeller pump, 163 Nappe, 320 Negative gauge pressure, 99 Negative pressure, 98–103, 129, 320, 323, 328–329 Net positive suction head (NPSH), 178 Newton, 4 Newton method (pipe networks), 122–125 Newtonian fluids, 6 Newton's law of viscosity, 6, 379 Newton's second law of motion, 125, 129, 134, 379, 389, 390 Nonequilibrium test (aquifers), 275–279 Non-exceedence probability, 472 Non-Newtonian fluids, 6 Normal depth, 204, 205, 220 Normal distribution, 468–469, 473–474, 480 Nozzle meters, 354–355 Open-channel flow, 197, 417 classifications, 199–201 control section, 222 cross-sectional relationships for, 200 discharge measurements in, 357–366 energy principles in, 210–216 hydraulic design, 234–239 hydraulic efficiency, 207–209 modeling, 239–240 models, 386–388 uniform flow in, 201–207 Orifice equation, 333 Orifice meters, 354–355 Overflow spillway, 323–326 Overturning (dams), 311–313 Parshall flumes, 361–366 definition, 361 dimensions, 362–364 discharge equations, 365 flow-rate correction, 365 Pascal, 2

505 Pascal's law, 17 Pavement encroachment (spread), 437–439 Pearson type III distribution, 470–472 Performance curves (pump), 165 Peripheral velocity, 159, 378 Permeability coefficient, 255 Phases of water, 2–3 Phreatic line and surface, 294 Pi theorem, 388–392 Piezometer, 103, 347 Piezometric surface, 253, 254, 258, 263, 264, 287 Pipe bends, 79–80 contractions, 75–77 discharge measurements in, 352–357 elasticity, 126 equivalent, 84–87 expansions, 78–79 friction, 56, 63–71 pressure in, 54 systems (branching), 103–110 valves, 81–84 wall roughness, 56 Pipe flow continuity and momentum equations, 57–60 description of, 54 energy in, 60–63 Reynolds number, 55–57 Pipe network, 94 Hardy–Cross method, 111–121 junction equation, 110 loop equation, 110 modeling, 136–140 Newton method, 122–125 pumps and, 176–177 Pipeline analysis, 166–169 connecting reservoirs, 94–98 using energy equation, 100–102 water hammer phenomenon in, 125–133 Pipeline anchoring, 126 Pipes in parallel, 85–87 Pipes in series, 84–85 Piping (through earth dams), 294 Pitot tube, 350–351 Poise, 6 Poisson's ratio, 128

Polar vector diagram, 159 Population, 464 Porosity, 254, 255 Porous media, 255, 256, 292 Positive-displacement pumps, 155 Potential energy, 60, 210, 211 Power (pump), 156, 158–159 Prandtl Pitot tube, 351 Precipitation, 397 Pressure, 14 absolute, 14–17 atmospheric, 2, 14, 15 cell, 349 center of, 24–27 energy, 60, 210 force, 23, 27 gauge, 14–17, 99 head, 16, 61 hydrostatic, 14, 15, 23 measurement, 347–349 negative, 98–103, 320, 323, 328–329 openings, 348 in pipe, 54 pipe flow, 54 stagnation, 350 surface of equal, 17–19 vapor, 2, 99 water hammer, 125–133 wave, 125–133 zero reference, 198 Pressure ridge, 290–291 Prismatic channels, 199, 227 Probability concepts, 464 data plotting, 483–484 exceedence, 472, 478, 482–483 graphs, frequency analysis using, 481–485 non-exceedence, 472 plotting position, 481–483 theoretical distributions, 483–484 Probability density function, 469, 472 Gumbel distribution, 469 log-normal distribution, 469 log-Pearson type III distribution, 470 Probability distribution, 464, 468–472 Propeller pump, 161–164 Propeller-type current meter, 351, 352

506 Prototype, 371 Pump, 155 axial flow, 161–164 booster, 164 brake horsepower, 165 and branching pipes, 173–176 cavitation, 177–180 centrifugal, 155–161 characteristic curves, 165–166 efficiency, 159, 165 head, 158, 165 housing, 156 impeller, 156 jet, 164–165 multistage propeller, 163 output power, 159 overall efficiency, 159–161 in parallel/series, 169–173 performance curves, 165 peripheral velocity, 159 and pipe networks, 176–177 positive-displacement, 155 power, 156, 158, 159 propeller, 161–164 radial flow, 155–161 rated capacity, 165 selection, 183–187 similarity, 181–183 suction side, 101 tangential velocity, 159 turbo-hydraulic, 155 Pumping trough, 290 Radial flow equation, 258 in confined aquifers, 259–261, 263–267 in unconfined aquifers, 261–263, 267–270 Radial flow pumps, 155–161 Radius of influence, 259 Rainfall excess, 408–415 Rainfall intensity, 400, 401 Random variable, 464 Rapid closure (valves), 129, 133 Rapid flow, 212 Rapidly varied flow, 199, 220 Rate of effective rainfall, 408 Rate of rainfall excess, 408 Rated capacity (pump), 165 Rational method, 435–443 Recharge well, 280 Recurrence interval. See Return period

Index Reference datum, 54 Relative roughness, 65 Reservoir routing, 428, 434 Resistance force, 5, 201 Resultant force, 24 Return period, 400, 402, 472–474, 482 Reynolds number, 55–57, 100, 378–381, 384, 389, 390 Righting moment, 35 Rigid boundary channels, 238–239 Rigid liners, 235 Riser and barrel assembly, 316 Risk (hydrologic), 473 Roughness coefficient (Manning's n), 73, 203, 418 Roughness height (pipe), 65–66 Runoff coefficient, 435 Runoff hydrographs design, 415–426 direct runoff hydrograph, 419–420, 425 HEC-HMS Model, 446 time of concentration, 416–419 total runoff hydrograph, 415, 424–426 unit hydrograph, 419–424 Sample, 464 Scaled model, 371 Seawater intrusion, 286–291 Secondary current, 80 Sedimentation, 312, 372, 387 Seepage definition, 291 through dam foundations, 291–294 through earth dams, 294–295 Seepage velocity, 255 Seismic wave propagation methods, 285–286 Sequent depth, 216 Service spillway, 316 Shallow concentrated flow, 417 Shallow water wave, 384 Shape number, 181 Sharp-crested weir, 320, 357–360 Shear stress, 5 Sheet flow, 416, 417 Shutoff head, 165 Side-channel spillways, 326–329 Significance level, 476 Single pump, 166–169

Siphon spillway, 328–331 Skew coefficient, 465–468, 470 Skewness, 464 Sliding (dams), 312, 313 Sloughing, 294 Slow closure (valves), 130, 133 Slug, 5 Soil Conservation Service (SCS), 405–408, 413–415 hydrograph parameters, 420 synthetic unit hydrograph procedure, 420 unit hydrograph, 419–424, 446 Soil piping, 294 Specific energy, 211, 227 Specific force, 218 Specific gravity, 4 alcohol, 19 concrete, 340 mercury, 19 Meriam Red Oil, 19 seawater, 36 water, 4 Specific heat, 2 Specific speed, 181–183 Specific weight, 3–5 Spillway discharge of, 323 emergency, 316–317, 323 overflow, 323–326 service, 316 side-channel, 326–329 siphon, 328–331 Spiral flow, 80 Stagnation pressure, 350 Standard deviation, 464–468, 486 Standard step method, 225–227 Station skew, 465, 471 Statistical methods, hydrology, 463 applicability of, 486 concepts of probability, 464 frequency analysis, 473–485 IDF curves, 485–487 probability distributions, 468–472 return period, 472–473 risks, 472–473 Statistical parameters, 464–468 Steady flow, 199 Steady incompressible flow, 58 Steep channel, 222 Stilling basin, 336–340 Stokes, 6

Index Storage coefficient (aquifer), 263 Storage routing, 426–434 data requirements, 428–429 elevation–area-storage relationship, 431 elevation–outflow relationship, 431 graphical representation, 428 hole-in-the-barrel experiment, 427 mass–balance relationship, 427–428 storage–outflow relationship, 432 Storativity, 263 Storm drainage inlets, 437 Storm Water Management Model (SWMM). See also EPASWMM model definition, 444 hydrologic modeling options, 448–449 Stormwater pipe design, 439–443 Stormwater-collection systems, 437–439 Streamflow hydrograph, 415, 416, 424–426 Streamline, 75, 291 Subcritical flow, 212, 213 Submerged weir, 320 Subsurface barriers, 291 Suction head, 178 Supercritical flow, 212, 213, 318 Superposition, 260, 279, 423, 424 Surface force, 10 pitting, 178, 323 of saturation, 294 tension, 383–384 wave, 213, 372 Surface tension, 7–8 Surfaces of equal pressure, 17–19 Surge tanks, 134–136 Swamee–Jain equation, 67 Synthetic design storm, 404–405 Synthetic unit hydrograph, 420–423 System head curve, 167 Tangential velocity, 159 Theis equation (well flow), 265 Three-reservoir problem, 103 Tilt angle, 33–35 Time of concentration, 416 Time to peak, 420–421 Toe of dam, 311

507 Top width, 198 Total depth of rainfall, 400 Total runoff hydrograph (TRH), 415, 424–426 Transitional zone (pipe flow), 67 Transmissivity, 260 Transpiration, 397 Trapezoidal weir, 359 Turbo-hydraulic pumps, 155 Turbulent flows, 55–56, 63 in circular pipes, 56 friction factor for, 65–71 open-channel models, 388 Unconfined aquifer, 253, 254 equilibrium test in, 273–275 permeability coefficient in, 273 steady radial flow in, 261–263 unsteady radial flow in, 267–270 Uncontracted horizontal weir, 358 Undistorted model, 386 Uniform depth, 204, 205, 220 Uniform flow, 199 in open channels, 201–207 Unit hydrograph, 419–424 Unlined channels, 236–238 Unsteady flow, 199 Uplifting force, 311, 312 U.S. Army Corps of Engineers (ACE), 444 U.S. Bureau of Reclamation (USBR), 336–340 U.S. Waterways Experimental Station, 323 Vapor pressure, 2, 3, 99 Variance, 465 Varied flow, 199 Varied flow equation, 221 Varied unsteady flows, 201 Velocity apparent, 255 distribution (pipes), 54, 56 head, 61, 197, 210 mean, 54 measurements, 349–352 in pipe, 54 seepage, 255 vector diagram (pumps), 158 Vena contracta, 75, 354 Venturi flumes, 361–366 Venturi meters, 352–353, 356 Vessel (pressure), 14, 18, 19

Virtual mass, 385 Viscosity, 5 absolute, 6, 55 kinematic, 6, 55 of water, 5–7 Viscous dissipation, 56, 63 Viscous force, 64, 378–381, 384 V-notch weir, 359 Volume modulus of elasticity, 9 von Kármán equation, 66 Wall friction, 56, 63 Wall shear stress, 202 Water compressibility, 8 density, 1, 3–5 depth model, 386–387 elasticity, 8–9, 125–129 free surface of, 14 hammer phenomenon, 125–133 phases of, 2–3 physical properties of, 1 specific gravity, 4 specific weight, 3–5 surfaces, 2 table, 253–254 temperature, 1 vapor pressure of, 3 viscosity, 5–7 Water Resources Council, 470, 471, 474, 482 Water surface profile, 220, 224–234 Water-bearing formation, 253 Watershed definition, 444 delineation, 399–400 Waterways Experiment Station, 386 Wave disturbance, 213, 224 flood, 201, 372 forces, 384 front, 126–130, 133 gravity, 213, 384 negative pressure, 129 pressure, 126–130, 133 seismic, 285–286 shallow water, 384 shock, 285 speed, 126, 224, 285 surface, 213, 372, 383–384 Weber number, 383–384, 389 Weight, specific, 3–5

508 Weighted skew coefficient, 470–472, 474 Weir, 318–323 acceleration of flow over, 318 broad-crested, 360–361 Cipolletti, 359 contracted, 358 frictionless, 318, 319 horizontal, 358 sharp-crested, 320, 357–360

Index submerged, 320 trapezoidal, 359 uncontracted, 358 use of, 318 V-notch, 359 Well, 253 area of influence, 259 cone of depression, 259 discharge, 259, 262 drawdown, 258

function, 265, 267–268 radius of influence, 259 recharge, 280 steady radial flow to, 258–263 unsteady radial flow to, 263–270 Wetted perimeter, 71, 199 Wide rectangular channel, 220 Zero pressure reference, 198

Fundamentals of Hydraulic Engineering Systems Solutions Manual

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Brough Blithical1978

Fundamentals of Hydraulic Engineering Systems Solutions Manual

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