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APPLIEDHYDRAULIC

TRANSIENTS

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APPLIEDHYDRAULIC

TRANSIENTS

by

M. Hanif Chaudhry, Ph.D.

British Columbia Hydro and Power AuthorityVancouver, British Columbia, Canada

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Van Nostrand Reinhold Cornpany Regional Offices:New York Cincinnati Atlanta Dalias San Francisco

Van Nostrand Reinhold Company International Offices:London Toronto Melbourne

Copyright © 1979 by Litton Educational Publishing, Inc,

Library of Congress Catalog Card Number: 78-4087ISBN: 0-442-21517-7

AHrights reserved. No part of this work covered by the copyright hereon may be repro-dueed or used in any form, either wholly or in part, or by any means=graphic, electronic,or rnechanical, ineluding photocopyíng, recording, taping, or information storage and re-trieval systerns=without permission of the publisher and the author.

Manufactured in the United Sta tes of America

Published by Van Nostrand Reinhold Company135 West 50th Street, New York, NY 10020

Published simultaneously in Canada by Van Nostrand Reinhold Ltd,

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Library of Congress CataJoging in Publication Data

Chaudhry, M. Hanif.Applied hydraulie transients,

Ineludes indexoL Hydraulic transients. 1. Title,

TCl63.C43 620.1'064 78-4087ISBN 0-442-21517-7

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PREFACE

In the last decade, the use of digital computers for analyzing hydraulic transientshas increased by leaps and bounds, and the graphical and arithmetical methodsfor such analyses have been replaced by sophisticated numerical techniques, Notonly has this change reduced the amount of laborious computations, but it hasresulted in more precise results and has made the analysis of cornplex systemspossible. Applied Hydraulic Transients provides a comprehensive and systematicdiscussion of hydraulic transients and presents various methods of analysessuitable for digital computer solution. The book ís suitable as a reference forpracticíng engineers and researchers and as a textbook for senior-Ievel under-graduate and graduate students. The field of application of the book is verybroad and diverse and covers areas such as hydroelectric projects, pumped-storage schemes, water-supply systems, nuclear power plants, oil pipelines, andindustrial piping systems.

Each chapter of the book is developed in a systematic manner from firstprincipies. A very strong emphasís is given to the practical applications, andadvanced mathematics and unnecessary theoretical details have been avoidedas much as possíble. However, wherever inclusión of such details was con-sidered necessary from the point of view of researchers, they are presented insuch a manner that a practicing engineer can skip them without losing continuityof the text. Several case studies, problems of applied nature, and design criteriaare included, which will be helpful to design engineers and will introduce stu-dents to the design of real-life projects. Solved examples are given for illustra-tion purposes, extensive lists of up-to-date references are included at the end ofeach chapter for further study , and sample computer programs and flowchartsare presented to familiarize the reader with digital computer applications.Approximate methods and design charts are appended to the text for quickcomputations during the preliminary design stages.

Because of the díverse nature of application, the various chapters have been

v

vi Preface

written so that they can be read individually. Sornetimes, however, other partsof the book had to be referred to in order to avoid duplication. This has be endone in such a manner that only the section referred to may be read and not thewhole chapter. This modeof presentation will allow practicing engineers to readonly those parts of the book that are of their immediate interest, and will allowteachers to select the chapters most relevant to their courses,

SI iSysteme Intemationale) units are used throughout the book. However,wherever empirical constants are involved or numerical constants are introducedin the derivations, their corresponding values in the English uníts are given in thefootnotes. With these footnotes and the conversion table of Appendix F, anyreader preferring to use the English units may do so without much difficulty.

The sequence of presentation is as follows: In Chapter 1, cornmonly usedterms are defined, a brief history of hydraulíc transients is presented, and funda-mental concepts are introduced. The dynamic and continuity equations for aone-dimensional flow in closed conduits are derived in Chapter 2, and varíousnumerical methods available for their solution are discussed. The details of themethod of characteristics are presented in Chapter 3. The next four chaptersare problern-oriented and discuss transients in pumping systems (Chapter 4), inhydroelectric power plants (Chapter S), in nuclear power plants (Chapter 6),and in oil pipelínes (Chapter 7). The analysis of transients in homogeneous,two-phase flows is also presented in Chapter 6. Resonance in pressurized pipingsystems is díscussed in Chapter 8, and the details of the transfer matrix methodare outlined. Transient cavitation and liquid colurnn separation are discussedin Chapter 9, and various methods for eliminating or alleviating undesirabletransients are presented in Chapter 10. The analysis of surge tanks using alurnped-system approach is presented in Chapter 11. In Chapter 12, transientsin open channels are discussed, and the details of explicit and implicit finíte-difference methods are outlined. A number of design charts and sample corn-puter programs are presented in Appendixes A through D.

The book presents in a systematic manner a collection of my own contríbu-tíons, sorne of which have not previously been published, as weLJ as materialdrawn from various sources. Every attempt has been made to identify theso urce of material; any oversight in this regard is strictly uníntentional.

M. H. Chaudhry, Ph.D.Vancouver, B.C.Canada

:.11. 1

ACKNOWLEDGMENTS

1 would like to express my gratitude to Professors V. L. Streeter, E. B. Wylie,C. S. Martín.and G. Evangelisti, and to J. Parmakian. Theír excellent publica-tíons were a source of inspiration duríng the preparation of this book. Myformer teacher and research supervisor, Dr. E. Ruus, Professor of Civil Engineer-ing, revíewed the entire manuscript and offered many useful suggestions for itsimprovement; in addition, 1 am thankful to him and to the Department of CivilEngíneering, University of Britísh Columbia (UBC), Vancouver, Canada, forgiving me the opportunity and privilege to participate in giving lectures, basedon parts of this book, to the graduate students.

Thanks are extended to my employer, British Columbia Hydro and PowerAuthority (BCHP A), for granting me permission to inelude the results of testsconducted on a number of their power plants and a number of design studiesfor their projects. These studies were undertaken under the technical and ad-ministrative supervision of F. J. Patterson, Manager, Hydroelectric Design Dí-VISIono J. W. Forster, Assistant Manager, Hydroelectric Design División, and1. C. Dirom, Manager, Development Departrnent, províded encouragementthroughout the preparation of the book, and each reviewed severa! chapters.Dr. P. W. W. Bell, Supervisor, Hydraulic Section, revíewed eight chapters anddíd the hydraulíc transient studies summarized in "Design" on page 324. Pro-fessor Emeritus E. S. Pretious and F. El-Fitiany, a Ph.D. candidate, both of UBeand D. E. Cass, Engineer, BCHPA, revíewed several chapters. Their commentsand suggestions for improving the presentation are appreciated. R. E. Johnson,Senior Engineer, was in charge of ínstrumentation for the prototype tests pre-sented in Sections 3.8, 5.8, and 10.7 and devísed the gauges for measuring waterlevels duríng the tests of Section 12.11; D. E. Cass assisted in the studies re-ported in Sections 4.9 and 4.10; R. M. Rockwell, Supervisor, System StandardsSection and 1. H. Gurney, Engineer, prepared Figs. 5.8 and 5.18; W. S. Me-illquham and G. Arczynski, Senior Engineers, prepared turbíne characteristics

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viii Acknowledgments

for the studies of Sections 5.5, 5.8, and 10.7; G. Vandenburg, Senior Engineer,prepared Fig. 10.5; Dr. E. A. Portfors, Head , Hydraulic Division, Klohn LeonoffConsultants Ltd., for several discussions during studies of Sections 3.8,5.8, and10.7; K. H. Kao, C. 1. Gibbs, and W. A. Johnson, Engineers, did the prograrn-ming for the case study of Section 12.15; F. E. Parkinson, Vice-President, andS. Alam, Senior Engineer, Lasalle Hydraulic Laboratory, conducted the hydrau-lic model studies of Section 12.15; and Engineers L. S. Howard, D. E. Cass, and1. H. Gurney assisted in the proofreading.

1 arn thankful to Westinghouse Electric Corporation for granting permission toinclude material from their reports in Section 6.8 and to J. M. Geets, Manager,BOP Systems Design, Westinghouse, who enthusiastically furnished a numberof relevant reports; to C. F. Whitehead, Vice-President, Engineering, EbascoServices Inc., New York, for providing photographs rOT Section 1.7 and to Ebascofor granting permission to incJude these photographs in the book; and to De. W.T. Hancox, Head, Thermalhydraulics Research Branch, Atomic Energy ofCanada; to Professor F. W. Wood, Queen's University, Kingston, Ontario, andGlenfield & Kennedy Ltd., Scotland, for granting perrnissíon to include materialfrom their publications. 1 am grateful to the Department of Water Resources,State of California, and to the folJowing members of the department: D. L.Smith, Senior Mechanical Engineer. M. L. Chauhan, Associate Mechanical Engi-neer, and L. O. Well, Assistant Mechanical Engineer, for providing the test dataof Section 4.9.

Thanks are also due to the American Society of Civil Engineers, AmericanSociety of Mechanical Engineers, International Association for Hydraulic Re-search, British Hydromechanic Research Association, the Water Power and DarnConstruction, and the L'Energia Elettrica for a1lowing incJusion of material fromtheir copyrighted publications.

1 would like to extend rny thanks to F. Ching and D. Reíd for typing themanuscript and to L. Myles and P. Wong for preparing the figures and illustra-tions. My wife, Shamim, helped in numerous ways in completing the manuscript,1 am thankful to her and to our son, Asif, for rnany hours that should have beenspent with thern but were required in the preparation of this book.

M. H. Chaudhry

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CONTENTS

PREFACE v

ACKNOWLEDGMENTS vü

INTRODUCTlON

1.1 Definitions l1.2 Historical Background 21.3 Pressure Changes Caused by an Instantaneous Velocity Change 7lA Wave Propagation and Reflections in a Single Pipeline II1.5 Classification of Hydraulic Transients 161.6 Causes of Transien ts 181.7 Systern Design and Operation 191.8 Sumrnary 21

Problems 22References 22

, ) 2 EQUA TlONS OF UNSTEADY FLOW THROUGHCLOSED CONDUITS 27

2.1 Assumptions 272.2 Dynamic Equation 272.3 Continuity Equation 30204 General Remarks on Dynamic and Continuity Equations 332.5 Methods for Solving Continuity and Dynamic Equations 342.6 Velocity ofWaterhammer Waves 342.7 Case Study 392.8 Summary 40

Problems 41References 41Additional References 43

ix

x Contents Contents xi

3 METHOD OF CHARACTERISTlCS 44Emergency 96

3.1 Introduction 44 Catastrophic 963.2 Characteristic Equations 44 4.9 Verification ofMathematical Model 963.3 Boundary Conditions 50 Plant Data 97

Constant-Head Reservoir at Upstream End 51 Tests and Instrumentation 97Constant-Head Reservoír at Downstream End 52 Mathematical Model 97Dead End at Downstream End 53 Comparison of Computed and Measured Results 98Valve at Downstream End 54 4.10 Case Study 99Orífice at Lower End 55 Water-Supply System 100Series Junction 55 Analysis 100Branching Junction 57 Results 102Centrifugal Pump at Upstream End 58 Discussion 103Francis Turbine at Downstream End 58 4.11 Summary 104

3.4 Stability and Convergence Conditions 58 Problems 1043.5 Selection of Time Incremen t for a Complex Piping System 59 References 1063.6 Combined Implicít-Characterístíc Method 61 Additional References 1073.7 Analysis of a Piping System 63

,13.8 Case Study 653.9 Summary 70 5 HYDRAULlC TRANSIENTS IN HYDROELECTRIC

Problems 70 POWER PLANTS 109References 71Additional References 73 5.1 lntroduction 109

5.2 Schematic of a Hydroelectric Power Plant 109

4 TRANSIENTS CAUSED BY CENTRIFUGAL PUMPS5.3 Upstream and Downstream Conduits 110

74 5.4 Simulation of Turbine 111

4.1 Introductíon 74 5.5 Hydraulic Turbine Governors 118

4.2 Transient Conditions Caused by Various Pump Operations 74 Actuator 122

4.3 Mathematical Representation of a Pump 75 Dashpot 122

4.4 Boundary Condítions for Pump Failure 77 Permanent Drop 122

Equations of Conditions Imposed by Pump 79 Distributing Valve 122

Differential Equation of Rotating Masses 81 Gate Servomotor 123

Characteristic Equation for Discharge Pipe 82 5.6 Computational Procedure 124

Continuity Equation 83 5.7 Causes of Transien ts 124

Solution ofGoverníng Equations 83 5.8 Verification of Mathematical Model 126

4.5 Boundary Conditions for Special Cases 86 Prototype Tests 126

Parallel Pumps 86 Prototype Data 127

Series Pumps 88 Comparison of Computed and Measured Results 129

4.6 Example 4.1 92 5.9 Design Criteria fOTPenstocks 130

4.7 Pump Start-Up 94 Normal 130

4.8 Design Criteria for Pipelines 95 Emergency 133

Normal 95 Catastrophic 133

I5.1 O Generator Inertia 133

xií Contents Contents xiii: t

5.11 Governing Stability 135 6.9 Surnrnary 186General Rernarks 135 Problems 186Differential Equations of the Systern 137 References 187Criteria for Stability 139 Additional References 189Example 5.1 140Transient Speed Curve 142

7 TRANSIENTS IN LONG OIL PIPELlNESOptimum Values of Governor Parameters 145 ' I

190'i~'.

Example 5.2 145 7.1 Introduction 1905.12 Case Study 147 7.2 Definitions 1915.13 Summary 153 7.3 Causes of Transien ts 194

Problems 153 7.4 Method of Analysis 195References 155 ; I

7.5 Design Considerations' ' 197Additional References 156 General Rernarks 197

6 HYDRAULlC TRANSIENTS IN NUCLEAR POWER PLANTS 158Control and Surge Protective Devices 198

7.6 Summary 199

6.1 Introduction 158 .. Problems 199

6.2 Terminology 158 References 199

General 158 Additional References 200

Types of Reactors 159Ernergency Core Cooling Systems 160 8 RESONANCE IN PRESSURIZED PIPING SYSTEMS 201

6.3 Causes of Transients 1616.4 Methods of Analysis 162 8.1 Introduction 201

General Remarks 162 8.2 Development of Resonating Conditions 201Formulation of Mathernatical Models 163 8.3 Forced and Self-Excited Oscillations 206

Numerical Solution 163 Forcing Functions 206

6.5 Boundary Conditions 165 Self-Excited Oscillations 206

Condenser 165 8.4 Methods of Analysis 208

Entrapped Air 168 Analysis in Time Domain-Method of Characteristics 209

Pipe Rupture and Failure of Rupture Discs 169 Analysis in Frequency Dornain 209

6.6 Loss-of-Coolant Accident 169 8.5 Terminology 210

6.7 Implicit Finite-Difference Method for Analyzing Two-Phase Steady-Oscillatory Flow 210

Transient Flows 170 Instantaneous and Mean Discharge and Pressure Head 211

General 170 Theoretical Period 211

Governing Equations 170 Resonant Frequency 212

Conversion of Governing Equations into Characteristic Form 172 State Vectors and Transfer Matrices 212

Formulation of Algebraic Equations 174 8.6 Block Diagrams 215

Computational Procedure 177 8.7 Derivation of Transfer Matrices 216

Verification 177 Field Matrices 216

6.8 Case Study 178 Point Matrices 226

Configuration of Feedwater Line 178 8.8 Frequency Response 235

Description of lncident 179 Fluctuating Pressure Head 237

Possible Causes of Shock 182 Fluctuating Discharge 238

xiv Contents Contents xv

Oscillating Valve 240 9.11 Surnrnary 298Procedure for Determining the Frequency Response 241 Problerns 298

8.9 Pressure and Discharge Variation along a Pipeline 246 References 2998.10 Location of Pressure Nodes and Antinodes 248 AdditionaI References 301

Series Systern 2498.11 Deterrnination of Resonant Frequencies 2508.12 Veriflcation of Transfer Matrix Method 256 10 METHODS FOR CONTROLLING TRANSIENTS 302

Experimental Results 257Method of Characteristics 257 10.1 Introduction 302Impedance Method 260 10.2 Available Devices and Methods for Controlling Transients 303Energy Concepts i64 10.3 Surge Tanks 303

8.13 Studies on Pipelines with Variable Characteristics 267 J, Description 3038.14 Surnrnary 269 Boundary Conditions 304

Problerns 269 lOA Air Chambers 306References 271 Description 306Additional References 272 Boundary Conditions 308

10.5 Valves 312

9 TRANSIENT CAVITATlON ANO COLUMN SEPARATION 274 Description 312Boundary Conditions 317

9.1 Introduction 274 10.6 Optimal Control ofTransient Flows 3239.2 General Rernarks 274 10.7 Case Study 3249.3 Causes of Reduction of Pressure to Vapor Pressure 276 Design 324904 Energy Dissipation in Cavitating Flows 278 Mathernatical Model 3259.5 Wave Velocity in a Gas-Liquid Mixture 279 Results 3269.6 Analysis ofCavitating Flows with Colurnn Separation 281 10.8 Surnmary 3289.7 Derívation of Equations 283 Problerns 328

Assurnptions 284 References 329Continuity Equation 284 AdditionaI References 330Mornenturn Equation 285Cavitating Flow 285Colurnn Separatíon 286 11 SURGE TANKS 332

9.8 NurnericaI Solution 287 11.1 Introduction 332Wave Equations 287 11.2 Types of Surge Tanks 333CoIurnn Separation 290 11.3 Derivation of Equations 334Gas ReIease 291 Dynarnic Equation 335Results 292 Continuity Equation 337

9.9 Design Considerations 2929.10 Case Study 293

11.4 AvailabIe Methods for Solving Dynarnic and ContinuityEquations 337

Project Details 293 11.5 Period and Arnplitude of Oscillations of a Frictionless Systern 338Field Tests 294 11.6 Stability 340Mathernatical ModeI 295

~ 11.7 Normalization of Equations 342Comparison of Cornpu ted and Measured Results 296

í 11.8 Phase-Plane Method 342

i

I.,xvi Contents E Contents xvii,H

11.9 Analysis of Different Cases of Flow Demand 344 12.6 Methods of Solution 394Constant-Flow 344 12.7 Method of Characteristics 395Constant-Gate Opening 348 12.8 Explicit Fínite-Difference Method 397Constant Power 352 12.9 Diffusive Scheme 398Constant Power Combined with Constant-Gate Opening 356 Formulation of Algebraic Equations 398Conclusions 360 Boundary Conditions 399

11.10 Orifice Tank 360 Stability Conditions 404Description 360 Computational Procedure 404Derivation of Dynamic Equation 361 12.10 Initial Conditions 405

1l.11 Differential Surge Tank 362 12.11 Verification of Explicit Finite-Difference Method-

Descríption 362 Diffusive Scheme 408Derivation of Equations 363 Mathematical Model 408

11.12 Multiple Surge Tanks 364 Prototype Tests 40811.13 Design Considerations 365 Comparison of Computed and Measured Results 411

Necessity of a Tank 365 12.12 lmplicit Fínite-Difference Method 414Location 365 .' Description 414Size 365

.Available Implicit Schemes 415~

11.14 Case Study 368 ~ Strelkoff's Implicit Scheme 415Project Details 368 !l Systems Having Branch and Parallel Channels 420Preliminary Investigations 369 ,j Stability Conditions 422Selectíon of Method of Analysis 370 12.13 Comparison of Explicit and Implicit Finite-Dífference Methods 422Program of Investigations 370 12.14 Special Topics 423Selection of Range of Various Variables 371 Dam-break 423Derivation of Equations 372 Tidal Oscillations 424Analog Simulation 372 Secondary Oscillations or Favre's Waves 425Results 373 Free-Surface-Pressurized Flows 426Selection of Tank Size 375 Landslide-Generated Waves 431

1l.l5 Sumrnary 376 12.15 Case Study 433Problems 376 i. : Project Details 433References 379 Mathematical Model 433

Results 440

12 TRANSIENT FLOWS IN OPEN CHANNELS 382 Operating Guidelines 44012.16 Summary 442

12.1 Introduction 382 Problems 44212.2 Definitions 382 References 44412.3 Causes of Transients 38412.4 Wave Height and Celerity 384 'ti APPENDICES

Rectangular Channel 387

f12.5 Derivation of Equations 389 Appendix A: Formulas and Design Charts for Preliminary Analysis 449

Continuity Equation 390 Appendix B: Computer Program for Analyzing Transients Caused byDynamic Equation 392 Opening or Closing a Val ve 469

i,lij'1

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xviii Contents

Appendix C: Computer Program for Analyzíng Transients Caused byPower Failure to Centrifugal Pumps 474

Appendix D: Computer Program for Determining FrequencyResponse of a Series Systern 481

Appendix E: Pump Characteristic Data 484Appendix F: SI and English Units and Conversion Factors 487

AUTHOR INDEX 489

SUBJECT INDEX 493

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APPLIEDHYDRAULIC

TRANSIENTS

CHAPTER 1

INTRODUCTION

In this chapter, a number of commonJy used terms are defined, and a briefhistory of the development of the knowledge of hydraulic transients is presented.The basic waterhammer equations for the change in pressure caused by an in-stantaneous change in flow velocity are then derived. A description of thepropagation and reflection of waves produced by cJosing a valve at the down-stream end of a single pipeline is presented. This is followed by a discussíon ofthe cJassificatíon and causes of hydraulic transients.

1.1 DEFINITlONS

Terms commonJy used are defined in this section; less common terms are de-fined in the text wherever they appear for the first time.

Steady and Unsteady Flow. If the flow conditions, such as pressure, velocíty,and discharge, at a point do not change with time, then the flow is said to besteady. If the conditions change with time, the flow is termed unsteady .Strictly speaking, turbuJent flows are aJways unsteady since the conditions at apoint are changing continuously. However, by considering temporal meanvalues over a short period, these flows are considered as steady if the temporalmean conditions do not change with time. When referring to the steady or un-steady turbulent flows herein, we will use the temporal mean conditions.

Transient-State or Transient Flow. The intermediate-stage flow, when the flowconditions are changed from one steady-state condition to another steady state,is called transient-state flow or transient flow.

Uniform and Nonuniform Flow. If the velocity is constant with respect todistance at any given time, the flow is called uniform flow, whereas if the veJoc-ity varies with distance, the flow is called nonuniform .

Steady-Oscillatory or Periodic Flow. If the flow conditions are varying withtime and if they repeat after a fixed time interval, the flow is called steady-

2 Applied Hydraulic Transients

osci/latory flow and the time interval at which conditions are repeating isreferred to as the periodo If T is the period in seconds, then the frequency ofoscillations, f, in cycles/s and in rad/s is lIT and 21TIT, respectively. Frequencyexpressed in rad/s is called circular frequency and is usually designated by W.

Column Separation. If the pressure in a closed conduít drops below the vaporpressure of a liquid, then cavities are forrned in the liquid and the liquid columnmay separate.

Waterhammer. In the past, terms such as waterhammer, oilhammer, and steam-hammer referred to the pressure fluctuatíons caused by a flow change dependingupon the fluid ínvolved, Nowadays, however, the term hydraulic transient isused more frequently.

The following discussion wíll be helpful in clarífyíng the preceding definitions.Let us assume that the downstream val ve of the pipeline (see Fig. 1.1a) isfully open, the water is flowing with velocity Va' and at time, t = to, the valve issuddenly closed. As a result of the valve closure, the flow through the valve is in-stantly reduced to zero, and because of the conversion of the kinetic energy intoelastic energy, pressure rises at the valve, and a pressure wave travels in the up-stream direction. This wave is reflected from the reservoir and travels back andforth betwecn the valve and the reservoí., Due to friction losses, this wave isdissipated as it travels in the pipeline, and finally-Iet us say, at time ti -thepressure in the entire pipeline becomes equal to the reservoir head, and flow iscompletely stopped.

Based upon the definitions given previously , the flow is steady when the con-ditions are constant with respect to time (i.e., for t < to and t > ti); and theintermediate flow (i.e., to ~ t ~ ti) when the conditions are changing from theinitial steady state to the final steady state is transient flow.

Now let us consider another situation. Let the valve be opened and closedperiodically at a frequency, wf. After a number of cycles, the flow conditionsin the pipeline wíll become periodic too, having frequency wf. This flow iscalled steady-oscillatory flow .

1.2 HISTORICAL BACKGROUND*

__. .The study of hydraulic transients began with the investigation of the propaga-tion of sound waves in air, the propagation of waves in shallow water, and theflow of blood in arteries. However, none of these problems could be solvedrigorously untiJ the development of the theories of elasticity and calculus, and

*Most o( the material presented in Section 1.2 ís based on Ref, 1; readers interested in thehistory of hydraulics should see Ref, 2.

Introduction 3

the solution of partial differential equations. Newton presented, in hisPrincipiai the results of his investigations on the propagation of sound waves inair and on the propagation of water waves in canals, Both Newton and Lagrangeobtained theoretically the velocity of sound in air as 298,4 mIs as compared totheir experimental value of 348 mIs. Lagrange erroneously attributed this dif-ference to experimental error, whereas Newton explained that the theoreticalvelocity was incorrect and that this discrepancy was due to spacing of the solídparticles of air and the presence of vapors in air. By comparing the oscillationsof a liquid in a U-tube to that of a pendulum, Newton derived an incorrect ex-pression for the celerity of water waves in a canal as 1T.JL7i, where L = thewavelength and g = acceleration due to gravity.

Euler" developed a detailed theory of the propagation of elastic waves andderived the following partial differential equation for wave propagation:

a2 y a2 y- = a2 - (l.Ia)a t2 ax2

in which a2 = gh; x = the equilibrium position of a particle; y = the particle dis-placement; and h = height of the air column. He also developed a general solu-tion of this equation as

y :::F(x + at) + f(x - at) (1. lb)

in which F and t= the travelling waves. Euler also tried, but failed, to obtain asolution for the flow of blood through arteries. s

Lagrange analyzed" the flow of compressible and íncompressíble fluids. Forthis purpose, he developed the concept of velocity potential. He also derived acorrect expression for the celerity of waves in a canal as e = .Jgd, in whichd = canal depth. In 1789, Monge developed a graphical method for integratingthe partial differential equations 7 and in troduced the term method of charac-teristics. About 1808, Laplace" pointed out the reasons for the difference be-tween the theoretical and measured values of the velocity of sound in airo Heexplaíned that the relationships derived by Newton and Lagrange were based onBoyle's law and that this law was not valid under varying pressures since the airtemperature did not remain constant. He reasoned that the theoretical velocitywould increase by about 20 percent if the adiabatic conditions were used insteadof the isothermal conditions.

Young? investigated the flow of bloodstrearns, friction losses, bend losses, andthe propagation of pressure waves in pipes. Helmholtz appears to be the first topoint out that the velocity of pressure waves in water contained in a pipe wasless than that in unconfined water. He correctly attributed this difference to theelasticity of pipe walls. In 1869, Rtemanrr'" developed and applied a three-dimensional equation of motion and its simplified one-dimensional forrn to such


fields as vibrating rods and sound waves. Weber!' studied the flow of an incorn-pressible fluid in an elastic pipe and conducted experiments to determine thevelocity of pressure waves. He also developed the dynamic and contínuityequations that are the bases of our studies. Marey 12 conducted extensive seriesof tests to determine the velocity of pressure waves in water and in rnercury andconcluded that the wave velocity was:

l. independent of the amplitude of the pressure waves2. three times greater in mercury than in water3. proportional to the elasticity of the tube.

Resal'? developed the continuity and dynamic equations and a second-orderwave equation. He used Marey's experimental results to verify his analyticalstudies. In 1877, Lord Rayleigh published his book 011 the theory of sound.!"which summarized the earlier studies and his OW11 research.

Korteweg! s was the first to determine the wave velocity considering the elas-ticity of both the pipe wall and the fluid; earlier investigators had consideredonly one of the two at a time.

Although Wood 1 lists Michaud 16 as the first to deal with the problern of water-hammer, recent investigations by Anderson 17 have shown that actuallyMenabrea'" was the first to study this problem. Michaud " studied the problemof waterhammer, and the design and use of air chambers and safety valves.Gromeka included the friction losses!" in the analysis of waterhammer for thefirst time. He assumed, however, that the liquid was incompressible and that thefriction losses were directly proportional to the flow velocity.

Westorr" and Carpenter;" both American engineers, conducted a number ofexperiments lo develop a theoretical relationship between the velocity reductionin a pipe and the corresponding pressure rise. However, neither one succeededbecause their pipelines were short. FrizelI22 presented an analysis of water-hammer based on studies undertaken while acting as a consulting engineer forthe Ogden hydroelectric development in Utah. This power plant had a 9449-m-long penstock. Frizell developed expressions for the velocity of waterhammerwaves and for the pressure rise due to instantaneous reduction of the flow. Hestated that the wave velocity would be the same as that of sound in unconfinedwater if the modulus of elasticity of the pipe walls was infinite. He also dis-eussed the effects of branch lines, wave reflections, and successive waves onspeed regulation. Unfortunately, Frizell's work has not been appreciated asmuch as that ofhis contemperarles, Joukowski and Allievi. .

In 1897, Joukowski conducted extensive experiments in Moscow on pipes wíththe following dimensions (expressed in length and diameter, respectively):7620 m, 50 mm; 305 m, 101.5 mm; and 305 m, 152.5 mm. Based on hisexperimental and theoretical studies, he published hís elassic reporr'? on the

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._ .

Introduetion ,5

basic theory of waterhammer. He developed a formula for the wave velocity,taking into consideration the elasticity of both the water and the pipe walls. Healso developed the relationship between the velocity reduction and the resultingpressure rise by using two methods: the conservation of energy and the con-tinuity condition. He discussed the propagation of a pressure wave along thepipe and the reflection of the pressure waves from the open end of a branch. Hestudied the effects of air chambers, surge tanks, and spring safety valves onwaterhammer pressures. He also investigated the effects of the variation ofclosing rates of a valve and found that the pressure rise was a maximum forclosing times, T.s;;,Zl.]«, in which L = length of the pipeline and a = wavevelocity.

AlIievi developed the general theory of waterhammer from first principies andpublished it in 1902.24 The dynamíc equation that he derived was more accu-rate than that of Korleweg. He showed that the term V(a v/ax) in the dynamicequation was not important as compared to the other terrns and could bedropped. He introduced two dimensionless parameters,

avo}p = 2gHa

e = aTe2/.

( 1.2)

in which a = waterhammer wave velocity; Va = steady-state vclocity; l. =Iengthof the pipeline; Te = valve-closure time; p = one-half of the ratio of the kineticenergy of the fluid to the potential energy stored in the fluid and the pipe wallsat pressure head Ha; and () = the valve-closure characteristics. For the valve-closure time, Te, AlIievi obtained an expression for the pressure rise at the valveand presented charts for the pressure rise and drop caused by a uniformlyelosing or opening valve. Braun2S,26 presented equations similar to those pre-sented by Allievi in his second publícatíon.?? In a later publication, Braurr"elaimed priority over AlIievi, and it appears that the so-called AJlievi's constanto, was actually introduced by Braun. However, Allievi is still considered to bethe originator of the basic waterhammcr theory. Allievf'? also studied therhythmic movemen t of a valve and proved that the pressure cannot exceed twicethe sta tic head. *

Joukowski's and Allievi's theories were mainly used in the first two decades ofthe 20th century. Camichel et al. 30 demonstrated that doubling of pressurehead is no! possible unless Ha >aVo/g. Constantinescu " described a mecha-nism lo transmit mechanical energy by using the waterhammer waves, In World

*For details of Allievi's work, interested readers should consult Refs, 24,27, and 29.


War 1, British fighter planes were equipped with the Constantinescu gear forfiring the machine guns. Based on Joukowski's theory, Gíbsorr'? presented apaper that included, for the first time, nonlinear friction losses in the analysis.He also invented an apparatus" to measure the turbine discharge using thepressure-time history following a load rejection.

Strowger and Kerr presented'" a step-by-step computational procedure to de-termine the speed changes of a hydraulic turbine caused by load changes.Waterhammer pressures, changes in the turbine efficiency at various gate open-ings, and the uniform and nonuniform gate movements were consídered in the

analysis.In his discussion of Strowger and Kerr's analysis, WOOd35 introduced the

graphical method for waterhammer analysis. L6wy36 independently developedand presented an identical graphical method in 1928. He also studíed resonancecaused by periodic valve movements and pressure drop due to gradual opening ofvalves and gates. He considered the friction losses in his analysis by including thefriction terms in the basic partial differential equations. Schnyder '? includedcomplete pump characteristics in his analysis of waterhammer in pipelines con-nected to centrifugal pumps. Bergerorr" extended the graphicalmethod to de-termine the conditions at the intermediate sections of a pipeline, and Schnyder "was the first to include the friction losses in the graphical analysis. At a syrn-posium"" sponsored jointly by the American Society of Civil Engineers and theAmerican Society of Mechanical Engincers in 1933 in Chicago, several paperswere presented on the analysis of waterhammer in penstocks and in dischargepipelines.

Angus'" outlincd basic theory and sorne applications of the graphical methodinc1uding "lumped" friction losses, and Bergeron'f presented a paper describingthe theory of plane elastic waves in various media. Another symposíum'? onwaterhammer was held in 1937 at the annual meeting of the American Socíetyof Mechanical Engineers. At this symposium, papers were presented on theanalysis of air chambers and valves, on the inclusion of complete pump charac-teristics, and on the comparison of the computed and measured results. Bylinearizing the friction term, WOOd44 used Heaviside 's Operational Calculus, andlater Rich4S used Laplace transforms for the analysis of waterhammer in pipe-lines. Angus'" presented in 1938 the analysis of compound and branching pipe-Iines and water-column separation. Other papers on water-column separationwere published by Lupton,"? Richard.t'' and DUC.49

From 1940 to 1960, in addition to books by Rich,50 Jaeger," and Par-makian,S2 numerous papers were published on the analysis of waterhammer.Because of their large number, they are not listed herein. Instead, importantcontributions are discussed and Iisted in the following chapters.

RUUS53a,53b was the flrst to present procedures for determining a valve-c1osure

Introduction 7

,,1

sequence, called optimum va/ve closure, so that the maximum pressure rernainedwithin the prescribed Iimits. Later on, Cabelka and Franc,54 and Streeter"independentIy developed the concept and the latter extended and computerizedit for complex piping systems.

GrayS6 introduced the method of characteristics for a computer-orientedwaterhammer analysis. Lais7 used it in his doctoral dissertation, and his jointpaper with Streeter'" was the pioneer publication that made this method andthe use of computers for the analysis of transients popular. Later on, Streeterpublished nurnerous papers on the method of characteristics as well as a texr"on hydraulic transients. These and important contributions of others are listedin Chapter 3.

On the theory of surge tanks, early European contributions were made byLéauté,"? Rateau.v' Prásil,62 and Vogt.63 Calame and Gaden,64 and Frank andSchüller'" summarized the earlier investigations and their own research. Thorna'"was tite first to show that the surge tank of a governed hydraulic turbine wouldbe stable only if the cross-sectional area of the surge tank were more than a cer-tain minimum value, now commonly known as tite Thoma area. Johnson"? in-vented the differential surge tank to develop accelerating or decelerating headsrapidly. Other coutributors to the theory of surge tanks are Escande,68Jaeger,5l,70 Gardel.t" Binníe,"! Evangelisti.I? Paynter,73,74 and Marris."!

1.3 PRESSURE CHANGES CAUSED BY AN INSTANTANEOUSVELOClTY CHANGE

Let us consider the piping systern of Fig. 1.1, in which a fluid is flowing withvelocity Vo' and the initial pressure upstream of the valve is Po' If the valvesetting is changed instantaneously at time t = 0, the velocity changes to Vo + .6V,the pressure at the valve becomes Po + Sp, the fluid density Po is changed toPo + .6p, and a pressure wave of magnitude Ap travels in the upstream direction.Let us designate the velocity of propagation of the pressure wave (commonlycalled waterhammer wave velocity) by a, and, to simplify the derivation, let usassume that the pipe is rigid, i.e., its diameter does not change due to pressurechanges.

The unsteady flow situation of Fig. 1.1a ís converted into a steady conditionby superimposing, on the control volume, velocity a in the downstream direction.This is equiva1ent to assuming that an observer is traveling in the upstreamdirection wíth velocity a. To this observer, the moving wave front appearsstationary (Fig. 1.1b), and the inflow and outflow velocities from the controlvolume are (Vo + a) and (Vo + AV + a), respectively.

Let us consider distance, x, and velocíty, V, as positive in the downstream


MOlling WOlltl'ronlIniliol sltlody stat«hydroulic 9rodtl lin.

~~---¡----'-----I JAH-

!?tlstlrlloir VollltlConlrol IIOIUmtl)

Vo ---

Vtllocily VoOtlnsily PoPrtlssurtl Po

Vo + AVPo + ApPo + Ap

(a) Unsteady flow

~""'WI'~I

~~~~------~-

!?tlstlrlloirCon/rol IIOIUmtl""\

Vo+a J-Vtllocily Vo+aOtlnsily PoPressurtl Po

Vo+ AV+aPo+APPo+AP

( b) Unsteady flow converted to steady flowby superimposino velocity a

Figure 1.1. Pressure rise in a pipeline due to instantaneous reduction of velocity (Ap =og a H}.

."

""i

\

Introduction 9

direction. Then the rate of change of momentum in the positive x-direction

7Po(Vo+a)A [(Vo+~V+a)-(V +a)]o

= Po(Vo + a)A ~ V (1.3)

Neglecting friction, the resultant force, F, acting on the fluid in the controlvolume in the positive x-direction, isPoA - (Po + Áp)A, i.e.,

F= -ÁpA (1.4)

According to Newton 's second law of motion, the time rate of change of mo-mentum is equal to the net force. Hence, it follows from Eqs. 1.3 and 1.4 that

( 1.5)

We will see in Chapter 2 that in most of the transíent conditions in metal or con-crete pipes or in the rock tunnels, a (approximately 1000 m/s) is much greaterthan Vo «lO mIs). Hence, Vo in Eq. 1.5 may be neglected. Also, since

p = pgH, (1.6)

in which H is the piezometric head, Eq. 1.5 may be written as

Áp=-poaÁV[ (1.7)

or

w=-~b.V7g ¡

The negative sign on the right hand side of Eq. 1.8 indicates that the pressureincreases (i.e., ~H is positive) for a reduction in velocity (i.e., for negative ~V)and vice versa. AIso note that Eq. 1.8 was derived for the case of velocítychanges occurring at the downstream end of a pipe and for the wave frontmoving in the upstream direction. Proceeding sirnilarly, it can be proved that, ifthe velocity was changed at the upstrearn end and the wave was moving in thedownstream direction, then

(I.8)

~H= E.~Vg

(1.9)

Note that there is no negative sign on the right-hand side of Eq. 1.9. This showsthat in this case, the pressure increases for an increase in velocity and the pres-sure decreases with a decrease in velocity.

It was assumed previously that the fluid density changes to Po + Áp as a resultof the change in pressure. For the control volume of Fig. 1.1 b,

Rate of mass inflow = PoA (Vo + a) (1.10)


Rate of mass outflow = (Po +~p) A (VO + ~V + a) (1.11 )

The in crease in the mass of control volume due to density change is small andmay be neglected. Therefore, the rate of mass inflow is equal to the rate ofmass outflow. Hence,

PoA(Vo + a) = (Po + ~p)A(Vo + ~V +a) (1.12)

which upon simplification becornes

~P~V= - - (Vo + .ó.V+a).

Po(1.13)

Since (Vo + ~V) «a, Eq. 1.13 may be written as

.ó.p.ó.V=--a

Po

The bulk modulus of elasticity , K, of a fluid is defined 76 as

(1.l4)

K=~~p/Po

lt follows from Eqs. 1.14 and 1.15 that

( 1.15)

~Va=-K-

~p'( 1.16)

On the basis of Eq. 1.7, Eq. 1.16 becomes

Ka=-apo

( 1.17)

which may be written as

~a= - /KL V PoNote that the expression of Eq. 1.18 is the velocity of waterhammer waves in acompressible fluid confined in a rigid pipe. In the next chapter, we will discusshow this expression is modified if the pipe walls are elastic.

(1.18)

Example l.l

Compute the velocity of pressure waves in a O.S-m-diameter pipe conveying oilfrom a reservoir to a valve. Determine the pressure rise if a steady flow of 0.4m3/s is instantaneously stopped at the downstream end by closing the valve.

~ , I

lntroduction 11

Assume that the pipe is rigid; the density of the oíl, P = 900 kg/rn ": and the bulkmodulus of elasticity of the oil, K = 1.5 GPa.

Solution:

1TA = 4" (0.5)2 = 0.196 m2

Vo = Qo/A = 0.4/0.196 = 2.04 m/s

a=v1

_/1.5 X 109

= V 900 = 1291 mis

As the flow is completely stopped, .ó.V=O - 2.04 = -2.04 m/s; therefore,

(Eq.1.I8)

a~H= - - ~V

g

1291= - 9.81 (-2.04) = 268.5 m

Since the sign of .ó.His positive, it is a pressure rise.

1.4 WAVE PROPAGATION AND REFLECTIONSIN A SINGLE PIPELINE

Let us consider the piping system shown in Fig. 1.2, in which flow conditions~re steady and at time t = O, the valve is instantaneously c1osed. If the systemIS assumed fri~tio~les~, then the initial steady-state pressure head along thelength of the pipeline IS H o. Let the distance x and the velocity V be positive inthe downstream direction.

The sequence of events following the valve closure can be divided into fourparts (Fig. 1.2) as follows:1. 0<t~L/a(Fig.l.2aandb)

As soon as the valve is closed, the flow velocity at the valve is reduced tozero, .which ca~ses a pressure rise of ~H = +(a/g) Vo. Because of this pres-~ure nse, the pipe expands (in Fig. 1.2, the initial steady-state pipe diameterIS sh.own by dotted lines), the fluid is compressed thus increasing the fluiddensity, and a positive pressure wave propagates toward the reservoir. Be-hind this ~ave, the .flow velocity is zero, and all the kinetic energy has beenconverted mto elas tic energy. Ir a is the velocity of the waterhammer waves

',-_,.


RlIsllr"oir

Hydroulic grodll linl

v-vo

l

(a) Conditions at t + E

~HY raulc gra I mI

--------------

RISITvairval""~

V=o --.. -

1 ..L

( b) Conditions at t = ~

H d r~' Y raulc gra I

r -----------

RlIsllrvoirVal", \

V=-Vo V=o- - --- --=1 ..

L

(e) Conditions at t = !:. + Ea

L>.H

.1

..1

Figure 1.2. Propagation of pressure waves caused by instantaneous elosure oC valve.

1ntroduction 13

rr: grad, lin,

--~

v= -Vo---~=========~--.1

R,sITvoir

l. l

(d) Conditions at t e 2La

~HYdraUhC gradl hn,

C=l~_1L>.H--r

v=-vo V=O

íHydrauliC grad, linll

---=;:~?=I- - -r - - - - - - -- - l.l--'---'L>.H-"T

RlISllr'lloir

______ -'F

1 ..

--l. .1L

(e) Conditions at t = 2L + ..a

v=O ---L

( f) Conditions at t = 3LaFigure 1.2. (Continued ¡


R6Stlrvoir

Hydrounc grodtl hn6

( g) Conditions at

Rtlstlrvoir

--

/ Hydroulic grade I~

(h) Conditions at t. ~L

Figure 1.2. (Colltinued)

and L is the length of the pipeline, then at time t = L/a, along the entirelength of the pipeline, the pipe is expanded, the flow velocity is zero, and thepressure head ís Ho + t:.H.

2. Lla < t'" 2L/a (Fig. 1.2c and d) .As the reservoir level is constant, the conditions are unstable at t~e reservoir

end when the wave reaches there because the pressure on. a s~ctJOn o~ t~ereservoir side is Ho while the pressure on an adjacent ~ect!On JO the pipe IS

H¿ + t:.H. Because of this pressure differential, the fluid starts to flow .fro~the pipeline into the reservoir with velocity - Vo' Thus, the velocity IS

changed from O to - Vo, which causes the pressure to drop from H¿ + t:.H to

'S' ",.,',.'_'....'"

Introduction 15.__ '1

'\

Ho' In other words, a negatíve wave travels toward the valve such that thepressure behind the wave (i.e., on the upstream side) is H¿ and the fluid ve-locity is - Va. At t = 2L/a, the pressure head in the entire pipeline is Ha,and the fluid velocity is - Va.

3. 2L/a < r« 3L/a (Fig. 1.2e and f)Since the valve is completely c1osed, a negative velocity cannot be main-

tained at the valve. Therefore, the velocity is instantaneously changed from- Vo to O. Because of this, the pressure is reduced to H o - t:.H, and a nega-tive wave propagates in the upstream direction. Behind this wave, the pres-sure is H¿ - t:.H, and the fluid velocity is zero, At time t = st.t«, the pressurehead in the entire pipeline is H¿ - t:.H, and the fluid velocity is zero.

4. s u« < t '" 4L/a (Fig. 1.2g and h)As soon as this negative wave reaches the reservoir, an unbalanced condition

is created again at the upstream end. Now the pressure is higher on thereservoir side than in the pipeline. Therefore, the fluid starts to flow towardsthe valve with velocíty Vo, and the pressure head is restored to Ha. At timet = 4L/a, the pressure head in the entire pipeline ís Hn, and the flow ve-locity is Va. Thus, the conditions in the pipeline are the same as during theinitial steady-state conditions,

As the valve is completely closed, the preceding sequence of events startsagain at t = 4L/a. Figure l.2 illustrates the sequence of events along thepipeline, while Fig. 1.3 shows the pressure variation at the valve end withtime. As we assumed the system ís frictionless, this process continúes andthe conditions are repeated at an interval of 4L/a. This interval after whichconditions are repeated is termed the theoretical period of the pipeline. Inreal physical systems, however, pressure waves are dissipated due to friction

Reservoir 111"111

o rim6

Figure 1.3. Pressure variation at valve; friction losses neglected.


Rtlstlrvoir Itlvtll

o !ha

Figure 1.4. Pressure variation at valve; friction losses considered.

losses as the waves propagate in the pipeline, and the fluid becomes stationaryafter a short time.

If the friction losses are taken into consideration, then the pressure varia-tion at the valve with time will be as shown in Fig. 1.4.

1.5 CLASSIFICATION OF HYDRAULIC TRANSIENTS

Dependíng upon the conduit in which the transient conditions are occurring,transients may be classified into three categories:

l. transients in closed conduíts2. transients in open channels3. combined free-surface-pressurized transient flow.

The analysis of transients in closed conduits may be further subdivided into twotypes: distributed systems and lumped systems. In the former case, the fluid isconsidered compressible, and the transient phenomenon occurs in the form oftraveling waves. Examples in which such transients occur are water-supply pipes,power plant conduits, and gas-transmission lines. In the analysis of lumped sys-tems, any change in the flow conditions is assumed to take place instantaneouslythroughout the fluid, i.e., the fluid is considered as a solid body. Example ofsuch a system is the slow oscillations of water level in a surge tank following aload change on the turbine ,

Mathematically, the transients in the distributed systems are represented bypartial differential equations, whereas the transients in the lumped systems aredescribed by ordinary differential equations. If wL/a is much less than 1, then

lntroductíon 17

Figure 1.5. View of burst penstock of Oigawa Power Station, Japan, due to excessívetransíent pressures caused by operating errors and malfunctioning of equipment, (AfterBoninso Courtesy of Ebasco Services Inc., New York.)

the system may be analyzed as a lumped systern 77; otherwise , the system mustbe analyzed as a distributed system. In the preceding expression, w == frequency,L == length of the pipeline, and a = wave velocíty.

Transients in open channels may be divided into two types depending uponthe rate at whích they occur: (1) gradualIy varied flow, such as flood waves inrivers, and (2) rapidly varied flow, such as surges in power canals, If the wavefront in the rapídly varied flow is steep, it is referred to as a bore,

Sometimes a free-flow becomes pressurized due ro priming of the conduitsduring the transient-state conditions. Such flows are called combined free-surface-pressurized flows. Examples of such flows are flow in sewers following arainstorm, and flow in the tailrace tunne! of a hydroelectric power p!ant follow.ing rapid acceptance of load on turbines.


Figure 1.5. (Continued)

1.6 CAUSES OF TRANSIENTS

As defined previously, the intermediate-stage flow, when the conditions arechanged from one steady state to another, is termed transient-state flow, Inother words, the transient conditions are initiated whenever the steady-stateconditions are disturbed. Such a disturbance may be caused by changes,planned or accidental, in the settings of the control equipment of a rnan-madesystem and by changes in the inflow or outflow of a natural system.

Common examples of the causes of transients in engineering systems are:

l. Opening, closing, or "chattering" of vaIves in a pipeline2. Starting or stopping the pumps in a pumping system3. Startíng-up a hydraulic turbine, accepting or rejecting load4. Vibrations of the vanes of a runner or an impeller, or of the blades of a fan

Introduction 19

Figure 1.6. View of collapsed section of penstock of Oigawa Power Station, Japan, causedby vacuum upstream of the bui:st section. (After Bonin80 Courtesy of Ebasco Services Inc.,New York.)

5. Sudden changes in the inflow or outflow of a canal by opening or cJosingthe control gate

6. Failure or collapse of a dam7. Sudden increases in the inflow to a river or a sewer due to flash storm

runoff.

1.7 SYSTEM DESIGN AND OPERATION

To design a system, the system layout and parameters are first selected, and thesystem is analyzed for transients caused by various possible operating conditions.If the system response is not acceptable, such as the maximum and minimumpressures are not within the prescribed limits, then either the system layout or


Figure 1.6. iContinued )

the parameters are changed, or various control devices are provided and the sys-tem is analyzed again. This procedure is repeated until a desired response isobtained. For a particular system, a number of control devices may be suitable,or it may be economical either to modify the operating conditions, if possible,or to change the acceptable response. However, the final aim is always to havean overall economical system that yields acceptable response.

The system must be designed for various normal operating conditions ex-pected to occur during its lífe. And, similarly, it is mandatory that the system beoperated strictly according to the operating guidelines. Failure to do so hascaused spectacular accidents 78-83 and has resulted in extensive property damageand many times loss of life. Figures 1.5 and 1.6 show the burst and the collapsedsections of the penstock of the Oigawa Power Station''? caused by operatingerrors and malfunctioning equipment.

Introduction 21

Figure 1.6. (Continued i

If the data for a system are not precisely known, then the system should beanalyzed for the expected range of various variables.

During the commissioning of a newly built system or after major modifications,the system should be tested for various possible operating conditions. ro avoidcatastrophes, it is usually advisable to conduct the tests in a progressive manner.For example, if there are four parallel purnping-sets on a pipeline, the tests forpower failure should begin with one purnping-set and progressively increase toa11four.

1.8 SUMMARY

In this chapter, the most commonly used terms were first defined. A briefhistorical background of the development of the knowledge of hydrauJic


transients was presented, and expressions were derived for the pressure rise ordrop due to an instantaneous increase or decrease in the Ilow velocity. Thechapter was concluded by a discussion of the c1assification and causes of hy-

draulic transien ts.

PROBLEMS

1.1 Derive Eq, 1.9 from first principies.

1.2 Derive Eq. 1.8 assuming that the pipe is inclined to the .horizontal at an

angle (J.

1.3 Compute the wave velocity in a 2-m-diameter pipe conveying seawater.Assume the pipe is rigid.What would be the pressure rise if an initial steady discharge of lO m

3/s

was instantaneously stopped at the downstream end of the pipeline ofProblem 1.3?A valve is suddenly opened at the downstream end of a l-m-diameter pipe-line such that the flow velocity is increased from 2 to 4 mis. Computethe pressure drop due to the opening of the valve. Assume the liquid is water.

Prove that if the fluid is incompressible and the pipe walls are assumedr igid, then the pressure rise

1.4

1.5

1.6

1.7

L dVtlH= - - -

g dt

in which L = length of the pipe and d V [dt = the rate of change of velocitywith respect to time. tHint: Apply Newton's second law of motion to thefluid volume.)Plot the pressure variation with time at the mid-length of the pipelineshown in Fig. 1.2 following instantaneous closure of the valve. Assume thesystem is frictionless.

ANSWERS

1.3 1488 mis

1.4 482.81 m

I.S 301.86 m

REFERENCES

l. Wood, F. M., "History of Waterhammer," Report No. 65, Department of Civil Engineer-ing ; Queeri's University at Kingston, Ontario, Cariada, April 1970.

19.

20.

21.

22.

23.

',1;:-~

! 24.

~25..;¡26..:)

i

Introduction 23

2. Rouse, H. and Ince, S., History of Hydraulics, Dover Publications, New York, 1963.3. Newton, l., The Principia, Royal Soc., London, Book 2, Proposition 44-46, 1687.4. Euler, L., "De la Propagation du Son," Mémoires de l'Acad. d. Wiss., Berlin, 1759.5. Euler, L., "Principia pro Motu Sanguinis per Arterias Determinando," Opera Postume

Tomus Alter, XXXIII, 1775.6. Lagrange,1. L.,Mecanique Analytique, Paris, Bertrand's ed., 1788, p. 192.7. Monge, G., "Graphical Integration," Ann. des Ing. Sortis des Ecoles de Gand, 1789.8. Laplace, P. S., Celestial Mechanics, 4 volumes, Bowditch translation.9. Young, T., "Hydraulic Investigations," Phil. Trans., Royal Society , London, 1808,

pp. 164-186.10. Riemann, B., Partielle Differentia/gleichungen, Braunschweig, 1869.11. Weber, W., "Theorie der durch Wasser oder andere incompressible Flüssigkeiten in

elastischen Rohren fortgepllanzten Wellen," Berichte über die Veriiandlungen derKonígtichen Sachsischen Gesselschaft der wissenschaften zu Leipzig, Leipzig, Gerrnany,Mathematisch-Physische Klasse, 1866, pp. 353-357.

12. Marey, M., "Mouvement des Ondes Liquides pour Servir a la Théorie du Pouls,"Travaux du Laboratoire de M. Marey, 1875.

13. Resal, H., "Note sur les petits mouvements d'un fluide incompressible dans un tuyauélastique," Journal de Mathematiques Pures et Appliquées, Paris, France, 3rd Series,vol. 2, 1876, pp. 342-344.

14. Rayleigh,J. W. S., Theory of Sound , 1877.15. Korteweg, D. J., "Ueber die Fortpflanzungsgeschwindingkeit des Schalles in elastischen

Rohren," Annalen der Physik und Chemie, Wiedemann, ed., New Series, vol. 5, no. 12,1878, pp. 525-542.

16. Michaud, J., "Coups de bélier dans les conduites. Étude des moyens employés pour enatteneur les effets," Bulletin de la Société Vaudoise des lngénieurs et des Architectes,Lausanne, Switzerland, 4e année, nos. 3 and 4, Sept. and Dec. 1878, pp. 56-64,65-77.

17. Anderson, A., "Menabrea's Note on Waterhammer: 1858," Jour., Hyd. Div., Amer.Soco Civ, Engrs., vol. 102, Jan. 1976, pp. 29-39.

18. Menabrea, L. F., "Note sur les effects de choc de I'eau dans les conduites," CompresRendus Hebdomadaires des Séances de L 'Academie des Sciences, Paris, France, vol. 47,July-Dec. 1858, pp, 221-224.Gromeka, 1. S., "Concerning the Propagation Velocity of Waterhamrner Waves inElastic Pipes," Scientiflc Socoof Univ. of Kazan, Kazan, U.S.S.R., May 1883.Weston, E. B., "Description of Sorne Experiments Made on the Providence, R. l., WaterWorks to Ascertain the Force of Water Ram in Pipes," Trans. Amer. Soc. of CivilEngrs., vol. 14,1885, p. 238.Carpenter, R. C., "Experiments on Waterhammer," Trans. Amer. Soco of Mech. Engrs.,1893-1894.Frizell, J. P., "Pressures Resulting from Changes of Velocity of Water in Pipes," Trans.Amer. Soco Civil Engrs., vol. 39, June 1898, pp. 1-18.Joukowski, N. E., Mem. Imperial Academy Soco of Sto Petersburg , vol. 9, no. 5, 1898,1900 (in Russian, translated by O. Simín, Proc. Amer. Water Works Assoc., vol. 24,1904, pp. 341-424).Allievi, L., "Teoría generale del moto perturbato dell'acqu anei tubi in pressíone," Ann.Soco Ing, Arch. italiana, 1903 (French translation by Allievi, Revue de Mécanique,1904).Braun, E., Druckschwankungcn in Rohrleitungen , Wittwer, Stuttgart, 1909.Braun, E., Die Turbine, Organ der turbinentcchnischen Gesellschaft , Berlin, 1910.

'c •


27. Allievi, L., "Teoría del colpo d'ariete," A tti Collcgio lng, Arch., 1913 (English trans1a-tion by E. E. Halmos, "The Theory of Waterhammer," Trans. Amer. Soco Mech. Engrs.,1929).

28. Braun, E., "Bermerkungen zur Theorie der Druckschwankungen in Rohrleitungen,"Wasserkraft und wasserwirtsch aft, vol. 29, no. 16, Munich, Gerrnany, 1939, pp.181-]83.

29. A llievi , L., "Air Chambers Ior Discharge Pipes," Trans. Amer. Soco Mech. Engrs., vol.59, Nov. 1937, pp. 651-659.

30. Camichel, C., Eydoux , D., and Garie1, M., "Étude Théorique et Expérimentale desCoups de Bélier ," Dunod, Paris, France , 1919.

31. Constantinescu, G., Ann. des Mines de Roumanie, Dec. 1919, Jan. 1920_32. Gibson, N. R., "Pressures in Penstocks Caused by Gradual Closing of Turbine Gates,"

Trans. Amer. Soc. Civil Engrs., vol. 83,1919-1920, pp. 707-775.33. Gibson, N. R., "The Gibson Method and Apparatus for Measuring the Flow of Water in

Closed Conduits," Trans. Amer. Soc. Mech. Engrs., vol. 45, 1923, pp. 343-392.34. Strowger, E. B. and Kerr, S. L., "Speed Changes of Hydraulic Turbines [or Sudden

Changes of Load," Trans. Amer. Soco Mech. Engrs., vol. 48, 1926, pp. 209-262.35. Wood, F. M., discussion of Ref. 34, Trans. Amer_ Soco Mech. Engrs., vol. 48, 1926.36. Lowy , R., Druckschwankungen in Druckrohrleitungen , Springer, 1928.37. Schnyder, O., "Druckstosse in Pumpensteigleitungen," Schweizerische Bauzeitung,

vol. 94, Nov.s-Dec. 1929, pp. 271-273, 283-286.38. Bergeron, L., "Varialions de regime dans les conduites d'eau," Comptes Rendus des

Travaux de la Soco Hvdrotech. de France , 1931.39. Schnyder, O., "Ueb~r Druckstosse in Rohrleitungen," wasserkraft u. Wasserwirlschofl,

vol, 27, Heft 5, 1932, pp. 49-54, 64-70.40_ "Symposium on Waterhamrner," Amer. Soco of Mech. Engrs. and Amer. Soco of Civil

Engrs., Chicago, lIIinois, June 1933.41. Angus, R. W., "Simple Graphical Solutions for Pressure Rise in Pipe and Pump Dis-

charge Unes," Jour. Engineering Institute of Callado, Feb. 1935, pp. 72-81.42. Bergeron, L., "Méthod graphique generale de calcul des propagations d'ondes pianes,"

Soco des l.'lg. Civils de France, vol. 90, 1937, pp. 407-497.43. Symposium on Waterhammer, Annual Meeting, Amer. Soco of Mecl!. Engrs., Dec. 1937.44. Wood, F. M., "The Application of Heavisides Operational Calculus lo Ihe Solution of

Problems in Walerhammer," Trans. Ami'r. Soco of Mech. t"ngrs., vol. 59, Nov. 1937,pp. 707-713.

45. Rich, G., "Waterhammer Analysis by the Laplace-Mellin Transformations," Trolls.Amer. Soco Mech. Ellgrs., 1944-1945_

46. Angus, R. W., "Waterhammer Pressure in Compound and Branched Pipes," Proc. Amer.Soco Civ. ElIgrs., Jan. 1938, pp. 340-401.

47. Lupton, H. R., "Graphical Analysis of Pressure Surges in Pumping Systems," Jour.Insl. Water Engrs., 1953.

48. Richard, R. T., "Water Column Separation in Pump Discharge Unes," Trans. Amer.Soco Mech. Engrs., 1956.

49. Duc, F., "Water Column Separation," Sulzer Tech. Rel'iew, vol. 41, 1959.50. Rich, G. R., Hydraulic Transients, 1st Ed., McGraw-HiII Book Co., Inc., New York,

] 951 (Dover Reprint).51. Jaeger, e., Engincering f1uid Mechonics, translaled from German by Wolf, P_ 0_,

Blackie & Son Lid., London, 1956.

II

Introduction 25

52. Parmakian, J., wat erhammer Anaiysis, Prentice-Hall, Inc., Englewood Cliffs N J 1955(Do ver Reprint, 1963). ' .. ,

533. RlIUS, E.. "Bestimmung von Schlicssfunctionen welche den kleinsten Wert des maxi-malen Drut"kstos~es e[g~ben," thesis presenred fa the Technica! Univ. of Karlsruhe,Gerrnany, in partial fulfillment al' the requirements Ior the dcgree 01" Doctor of En i-neering, 1957. g

53b. RlIUS, E., "Optimurn Rate of Closure 01" Hydraulic Turbine Gates," presented at Amer.Soco Mech. Engrs.i-Engíneeríng Inst. of Canada Conference, Denver Colorado April1966. ' ,

54. Cabelka, ,!., and f'm~c, L, "Closure Characteristics of a Valve with Respecl to Water-hammer, Proc., Eighth Congress, International Assoc. for Hydraulic RcsearcliMontreal, Canadu, Aug. 1959, pp, 6-A-1 to 6-A-23. '

55_ Streeter, V. L., "Valve Stroking to Control Waterhammer," Jour. Hvd, Div. AmeSoco Civil Engrs., vol. 89, March 1963, pp. 39-66. '-' r.

56. Gruy, C. A. M., "The Analysis of the Dissipation of Energy in Waterhammer " ProcAma. Soco Cirl. Engrs., Proc. paper 274, vol. 119, 1953, pp. 1176-1194_ ' .

57. Lai, C., "A Study of Waterhammer lncluding Effect of Hydraulic Losses," thesis pre-sented to the University of Michigan, Ann Arbor, Michigan, in partial fulfillment of therequirements for tite degree of Doctor of Philosophy, 1962.

58. Streeter, V. l., and Lai, c., "Waterhammer Analysís lncluding Fluid Friction." Trans.Amer. Soco ctv. Engrs., vol. 128,1963, pp. 1491-1524 (see also discussions by Gray,C. A. M., pp, 1530-1533, and by Paynter, H. M., pp. 1533-1538).

59. Streeter, V. L, and Wylie, E. B., Hydraulic Transients, McGraw-HiII Book Co., NewYork,1967.

60. Léauté, "Mémoires sur les oscillations a longues périodes dan s les machines actionnéespar des rnoteurs hydrauliques," J. Ecole Poly techn., Cahier XLVII, Paris, 1880.

61. Rateau, A., Traité des Turbomachines, Dunod, Paris, 1900.62. Prásil, F_, "Wasserschlossprobleme," Schweiz, Bouztg., 52,1908_63. Vogt, F., Berechnullg und KOlIslruktiotl des Wosserschlosses, Enke, Stuttgart, 1923.64. Clame, J., and Gaden, D., Théorie des Chambres d'équilibre, Paris, 1926.65. Frank, J., and SchüUer, J., Schwíngungen il1 den ZuleilllngS-llnd Ableitungskal1iilen 1'011

WasserkraJtanlagen, Springer, Berlin, 1938.66. Thoma, D., Zur Theoril' des Wosserschlosses bei selbsttiitíg gerege(ten Turbínenan!agen,

Oldenbourg, Munieh, 1910.

67. Johnson, R. D., "The Differential Surge Tank," Tra/ls. Amer. Soco Cív. J::ngrs., vol. 78,1915,pp.760-805.

68. Escande, L., "Recherches théoriques el expérimentales sur les oscillations de J'eau dansles chambres d'équilibre," Paris, 1943.

69. Gardel, A., Chambres d'Equilibre, Rouge et Cie, Lausanne, SWitzerland, 1956.70. Jaeger, C., "Present Trends in Surge Tank Design," Proc. Inst. of Mech_ Engrs.,

England, vol. 168, 1954, pp. 91-103.

71. Binnie, A_ M., "Oscillations in Closed Surge Tanks," Trans. Amer. Soco Mech. Engrs.,vol. 65, 1943.

72. Evangelisti, G., "Pozzi Piezometrici e Stabilita di Regolazione," Energia E/ettrica, voL27, nos. 5 and 6,1950, vol. 28, no. 6,1951, voL 30, no. 3,1953.

73. Paynte,~' H .. M., "Transient Analysis of Certain Nonlinear Systems in Hydroe1ectricPlan.ts, thesls presented to Massachusetts Institute o[ Technology, Massachusetts, inpartlal fulfllJment of the requirements of degree of doctor of philosophy, 1951.


74. Paynter , H. M., "Surge and Watcr-Hammer Problerns," Electrical Analogies and Elec-tronic Computer Symposium, Trans. Amer. Soco Civil Engrs., vol. 118, 1953, pp.

962-1009. "75. Marris, A. W., "Large Water Level Displacernents in the Simple Surge Tank, Jour,

Basic Engineering, Trans. Amer. Soco ofMech. Engrs., voL 81, 1959, pp. 446-454.76. Streeter, V. L., Fluid Mechanics, 4th Ed., McGraw-HilI Book Co., New York, 1966,

77. ~~!~dhry, M. H., "Resonance in Pressurized Pipíng Systems," thesis presented to theUniversíty of Brítish Columbia, Vancouver, Canada, in partial fulfillment of the re-quirements for the degree of doctor of philosophy , 1970, p. 24.. ."

78. Rocard, Y., "Les Phenornenes d'Auto-Oscillation dans les Installations Hydrauliques,

Hermann, Paris, 1937.79. Jaeger, c., "Water Harnmer Effects in Power Conduits," Civil Engineering Pub; Works

Rev: London, vol. 23, nos. 500-503, Feb.-May, 1948.80. Boni'n, C. C., "Water-Hammer Damage to Oigawa Power Station," Jour. Engineering [or

Power, Amer. Soco Mech. Engrs., April 1960, pp. 111-119. .81. Jaeger, C., "The Theory of Resonance in Hydropower Systems, Discussion of lnci-

dents Occurring in Pressure Systerns," Jour. Baste l:.'ngineering, Amer. Soco Mech,Engrs., vol. 85, Dec, 1963, pp. 631-640.

82. Kerensky, G., Discussion of "The Velocity of Water Harnmer Waves," by Pearsall, 1. S.,Symposium 011 Surges in Pipelines , Inst. of Mech. Engrs., London, vol. 180, Pt 3 E,1965-1966, p. 25.

83. Pulling, W. T., "Literature Survey of Water Hammer Incidents in Operatin~ Nuc~earPower Plants," Report No. WCAP-8799, westinghouse Electric Corporation, PIUS-burgh, Pennsylvania, Nov. 1976.

CHAPTER 2

EQUATIONS OF UNSTEADY FLOWTHROUGH CLOSED CONDUITS

Unsteady flow through closed conduits is described by the dynamic and continu-ityequations. In this chapter, the derivation of these equations is presented, andmethods available for their solution are discussed.

• 2.1 ASSUMPTIONS

The following assumptions are made in the derivation of the equations:

1. Flow in the conduit is one-dimensíonal,"? and the velocity distribution isuniform over the cross section of the conduit,

2. The conduit walls and the fluid are linearIy elastic, i.e.i'stress is proportionalto strain," This is true for most conduits such as metal, concrete and woodenpipes, and lined or unlined rock tunnels.

3. Formulas for computing the steady-state friction losses in conduits are validduring the transient state. The validity of this assumptíon has not as yet beenverified. For computing frequency-dependent friction, Zíelke! has developeda procedure for laminar flows, and Hirose'' has proposed an empirical proce-dure for turbulent flows. However, these procedures are too complex andcumbersome for general use, and we will not discuss them further.

• 2.2 DYNAMIC EQUATlON

We will use the following notation: distance, x, discharge, Q, and flow velocity,V, are consídered positive in the downstream direction (see Fig. 2.1), and H isthe piezometric head at the centerline of the conduit above the specified datum,

Let us consider a horizontal element of fluid having cross-sectional area A andlength ÓX, wíthín a conduit as shown in Fig. 2.1. If the piezometric head andthe velocity at distance x are H and V, then their corresponding values at x + ÓX

are H + (aH;ax) Bx and V + (a v;ax) óx, respectively. In the x-direction, three

27


Instantaneous hydraulicorade line

forces F F and S are acting on the element. F, and F2 are forces due to, 1, 2, ,

pressure while S is the shear force due to friction. If 'Y :: specific weight ofthe fluid, A = cross-sectional area of conduit, and z = height of conduit aboyedatum, then

11 11--- Positive x, v, 01I I I'-+~±-+-

oc110

xl"1O(l)

+x

z 8x"11'"

lt I I rDatum~ I

I1

x x + 8x

Q Q + ~~ 8x

H H+ ~~ 8x

(a)

s-

(b) Free body diaoram

Figure 2.1. Notation for dynamic equation.

F, = 'YA (H - z)

F2 = 'Y ~ - z + ~~ ox) A

(2.1)

(2.2)

1r'

f,

"I

. !·1·1

1

Equations of Unsteady Flow Through Closed Conduits 29

If the Darcy-Weisbach formula ¡ is used for compu ting the friction losses,then the shear force

-y [V2S=- --TTDóx

g 8 (2.3)

in which g = acceleration due to gravity, [= friction factor, and D = diameterof the conduit. The resultant force, F, acting on the element is given by theequation

F=F¡ - F2 - S (2.4)

Substitution of the expressions for F¡, F2, and S from Eqs. 2.1 through 2.3into Eq. 2.4 yields

sn -y ¡V2F= --yA -óx - - --TTDoxax g 8

According to Newton's second law of motion,

Force = Mass X Acceleration.

For the fluid element under consideration,

Mass of the element = z A ox}

Acceleration of the element = :Vdt

(2.5)

(2.6)

(2.7)

Substitution of Eqs. 2.5 and 2.7 into Eq. 2.6 and division by -yA ox yield

dV aH ¡V2

-= -g -- - (2.8)dt ax 2D

We know from elementary calculus that the total derivative

dV av aVdx-=-+--dt at ax dt (2.9a)

or

dV av av-=-+ V-dt at ax

Substituting Eq. 2.9b into Eq. 2.8 and rearranging,

av av aH ¡V2-+V-+g-+-=Oat ax ax 2D

(2.9b)

(2.10)


In most of the transient problerns,? the term vea v¡ax) is significantly smaIlerthan the term a v/ato Therefore, the former may be neglected. To account forthereverse flow, the expression V2 in Eq. 2.l0 may be written as VIVI, in whichIVI is the absolute value of V. By writing Eq. 2.1 O in terms of discharge, Q, andrearranging, we obtain

aQ sn fa¡+ gA ax + 2DA QIQI = O

In Eqs. 2.3, 2.5, 2.8, 2.10, and 2.11, the Darcy-Weisbach formula has beenused for calculating the friction losses. If a general exponential formula hadbeen used for these losses, then the last term of Eq. 2.11 could be written askQ IQ 1m¡Db, with the values of k, m, and b depending upon the formula em-ployed. For exarnple, for the Hazen-Williarns formula, m = 1.85 and b = 2.87,while, as derived aboye, for the Darcy-Weisbach formula, m = l and b = 3. Ifcorrect values of m and b are used," the resuIts are independent of the formulaemployed, i.e., the Darcy-Weisbach and the Hazen-Williams formulas would givecomparable results.

(2.11 )

23 CONTINUITY EQUATION

Let us consider the control volume shown in Fig. 2.2. The volume of fluidinflow, Vin, and outflow, t"out' during time interval Sr are

Vin = V 1T,2 ót

t"out = (v + ~: óx ) 1Tr2 o t

in which r = radius of the conduit. The increase in the fluid volume, oVin, dur-

(2.l2)

(2.13)

ing time ót is

(2.14)

The pressure change, Sp , during time interval ot is (ap¡at) St, This pressurechange causes the conduit waIls to expand or contract radiaIly and causes thelength of the fluid element to decrease or increase due to fluid compressibility(see Fig. 2.2).

Let us first consider the volume change, ot"n due to the radial expansion orcontractíon" of the conduit. The radial or hoop stress, a, in a conduit due to

*To sirnplify the derivation, we are neglecting the elongation or shortening of the fluid ele-ment due to Poisson ratio effects. Any reader interested in the derivation of the continuityequation including these effects should see Ref. 7.

ii-1

;~I-oo,.a

1;.,;

o

,

-

r----------,Equations of Unsteady F10w Through Closed Conduits 31

I

I IInflow I I

I I OutflowI II I.r

IL-_-ó~-----...J I

.... -Figure 2.2. Notation COI continuity equation.

the pressure p is given by the equation"

p,a=-

e

in which e = the conduit wall thickness. Hence, the change in hoop stress, So;caused by op may be written as

r ap rSo=Bp -= -ót-

e at eSince the radius r has increased to r + O" the change in strain

orOE=-

rIf the conduit waIls are assumed linearIy elastic, then

oaE=-OE

(2.15)

(2.16)

(2.17)

(2.18)

in which E = Young's modulus of elastícity. Substitution of expressions for oaand OE from Eqs. 2.16 and 2.17 into Eq. 2.18 yields

E = (ap/at) 01 (r/e)ór/r

or

ap ,2or=--otal eE

o_o

(2.19)

(2.20)

~e change in the volume of the element due to the radial expansion or contrae-tíon of the conduit is

of"y = 21Tr OX o, (2.21)


Substituting for Sr from Eq. 2.20 yields

ap r3

8-V = 2n - -8t 8xr at eE

Let us now derive an expression for the change in volume, 8-Vc, due to corn-pressibility of the fluid. The initial volume of the fluid element

(2.22)

(2.23)-v = nr2 8x

The bulk modulus ofelasticity of a fluid, K, is defined ' as

-8pK = 8V"cW

By substituting for -v from Eq. 2.23 and noting that 8p = (aplar) 8t, Eq. 2.24

(2.24)

becomes

-ap 8r 28V" =--nr Sx

e ar K

If we assume that the fluid density remains constant, then it follows from the

law of conservation of mass that

(2.25)

(2.26)

Substitution of expressions for 8fin, 8-V" and 8-¡;<cfrom Eqs. 2.14, 2.22, and2.25 into the aboye equation and division by nr2 lix lit yield

av 1 ap 2r ap-----=-- (2.27)

ax K at eE aror

av + ap(2r +.!..)=oax al eE K(2.28)

Let us define

Ka2 =------

p[l + (KDleE)]

in which p = mass density of the fluid. Noting that p = pgH, rearranging theterms, and substituting Q = VA, Eq. 2.28 becomes

a2 aQ aH--+-=0 (2.30)gA ax at

(2.29)

It will be shown in the next chapter that a is the velocity of waterhammer

.~,

Equations of Unsteady Flow Through Closed Conduíts 33

waves. Expressions for a for various conduits and support conditions are pre-sented in Section 2.6.

2.4 GENERAL REMARKS ON DYNAMIC ANDCONTINUITY EQUATlONS

The dynamic equation, Eq. 2.11, and the continuity equation, Eq. 2.30, are aset of first-order partial differential equations. In these equations, there are twoindependent variables, x and t, and two dependent variables, Q and H. Othervariables, A and D, are characteristics of the conduit system and are time-invariant but may be functions of x. Although the wave velocity, a, dependsupon the characteristics of the system, laboratory tests have shown that it is sig-nificantly reduced? by reduction of pressure even when it remains aboye thevaporpressure. The friction factor [varies with the Reynolds number. However,[is considered constant herein because the effects of such a variation on thetransient-state conditions are negligible .

Discussion about the type of Eqs. 2.11 and 2.30 now follows: any reader notinterested in the mathematical details may proceed to Section 2.5.

Since the nonlinear terms in Eqs. 2.11 and 2.30 involve only the first powerof the derivatives, the equations are called quasi-linear. These equations may befurther c1assified as elliptic, parabolic, or hyperbolic as follows:

Equations 2.11 and 2.30 may be written in matrix form as

in which

(2.31 )

and

G{':;}The eigenvalues, A, of matrix 8 determine the type of the set of equations.

The characteristic equatíon!? ofmatrix 8 ís

(2.32)

A2 - a2 = O (2.33)


Hence,

'A=±a (2.34)

Since a is real, both eigenvalues are real and dístinct, and hence Eqs. 2.11 and2.33 form a set of hyperbolic partial differential equations.

2.5 METHOOS FOR SOLVING CONTINUITY ANOOYNAMIC EQUATIONS

As demonstrated previously, the dynamic and contínuity equations are quasi-linear, hyperbolic, partial differential equations. A c)ose.d-for~~ solution of ~heseequations is impossible. However, by neglecting or hneanzmg the nonlmearterms, various graphical7,1 1,12 and analytical'3-'s methods have be en developed.These methods are approximate and cannot be used to analyze large systems orsystems having eomplex boundary conditions. Although sorne of these methodshave been programmed for analysis on a digital computer, 16,17 they are not pre-sented herein because their programming is difficult. We will discuss techniquesthatare more suítable for computer analysis, such as the implicit finite-differencemethod 18 and the method of characteristics. 8,18-24

In the implicit finite-difference method, the partial derivatíves are replaced byfinite differences, and the resulting algebraíc equations for the whole system arethen solved simultaneously. Depending upon the size of the system, this involvesa simultaneous solution of a large number of nonlinear equations. The analysisby this method becomes even more complicated in systems having complexboundary conditions, which must be solved by an iterative technique. Thernethod has the advantage that it is unconditionally stable. Therefore, largertime steps can be used, which results in economizing computer time. However,the time step cannot be increased arbitrarily beeause it results in smoothing thepressure peaks. Details of this method are presented in Section ?7. .

In the method of characteristics, the partial differential equatíons are first con-verted into ordinary differential equatíons, which are then solved by an explicitfinite-difference technique. Because each boundary condition and each eonduitsection are analyzed separately during a time step, this method is particularlysuitable for systems wíth complex boundary condítíons. The disadvantage ofthis method is that small time steps must be used to satisfy the Courant condi-tionl8 for stability. To overcome this, a combination of the implicit finitedifference and the method of characteristics23 may be used. This is discussed

in detail in Chapter 3.

2.6 VELOCITY OF WATERHAMMER WAVES

An expression for the velocity of waterhammer waves in a rigid conduit wasderived in Section 1.2. However, in addition to the bulk modulus of elasticity,

-¡


K, of the fluid, the velocity of waterhammer waves depends upon the elasticproperties of the conduit, as well as on the external constraints. Elastic proper-ties inelude the conduit size, wall thickness, and wall material; the externalconstraints include the type of supports and the freedom of conduit movementin the longitudinal directíon. The bulk modulus of elasticity of a fluid dependsupon its ternperature, pressure, and the quantity of undissolved gases. Pearsall'"has shown that the wave velocity changes by about 1 percent per 5°C. The fluidcompressibility is íncreased by the presence of free gases, and it has been found2S

that 1 part of air in 10,000 parts of water by volume reduces the wave velocityby about 50 percent. *

Solids in liquids have similar but less drastic influence, unless they are corn-pressible. Laboratory? and prototype tests2S have shown that the díssolvedgases tend to come out of solution when the pressure is redueed, even when itremains aboye the vapor pressure. This causes a significant reduction in the wavevelocity. Therefore, the wave velocity for a positive wave may be higher thanthat of a negative wave. Further prototype tests are needed to quantify thereduction in the wave velocity due to reduction of pressures.

Halliwell/" presented the following general expression for the wave velocíty:

a =1p[1 + (~/E) 1/IJ (2.35)

in which 1/1 is a nondimensional parameter that depends upon the elastíc proper-ties of the conduit; E = Young's modulus of elasticity of the conduit walls; andK and pare the bulk modulus of elasticity and density of the fluid, respectively.The moduli of elasticity of commonly used materials for conduit walls and thebulk moduli of elasticity and mass densities of various liquids are Usted in Tables2.1 and 2.2.

Expressions for ¡J¡ for various conditions are as follows:

l. Rigid Conduits

1/1=0 (2.36)

2. Thick-Walled Elastic Conduitsa. Conduit anchored against longitudinal movement throughout its length

R2 + R~ 2vR/f1/1 = 2(1 + v) o I

R~ - Rt R~ - R;(2.37)

in which v = the Poisson's ratio and Ro and R¡ = the external and internalradii of the conduit.

*For a derivation of expressions Ior the wave velocity in gas-liquid mixtures, see Section 9.5.


Table 2.1. Young's modulus of elasticity and Poisson's ratio for variouspipe materials.*

Modulus of Elasticity, E**Material (GPa) Poisson's Ratio

Alurninurn alloys 68-73 0.33Asbestos cernent, transite 24Brass 78-110 0.36Cast iron 80-170 0.25Concrete 14-30 0.1-0.15Copper 107-131 0.34Glass 46-73 0.24Lead 4.8-17 0.44Mild steel 200-212 0.27Plastics

ABS 1.7 0.33Nylon 1.4-2.75Perspex 6.0 0.33Polyethylene 0.8 0.46Polystyrene 5.0 0.4PVC rigid 2.4-2.75

RocksGranite 50 0.28Lirnestone 55 0.21Quartzite 24.0-44.8Sandstone 2.75-4.8 0.28Schist 6.5-18.6

.Compiled from Refs. 12,25, and 31 .• oTo convert E into lb/in.', mult iply the values given in this column by 145.038 X 10'.

b. Conduit anchored against longitudinal movement at the upper end

= [R~ + I.SR; + v(R~ - 3Rt)]!J¡ 2 R2 _ R~ R2 - R~o 1 o 1

(2.38)

c. Conduit with frequent expansion joints

!J¡ = 2 (R~ + R~+ v)Ro - R¡

(2.39)

3. Thin-Walled Elastic Conduitsa. Conduit anchored against longitudinal movement throughout its length

(2.40)

in which D = conduit diameter and e = wall thickness,


Table 2.2. Bulk modulus of elasticity and density of commonliquids at atmospheric pressure. *..

Bulk Modulus ofTernperature Density, pt Elasticity, K:I:

Liquid (OC) (kg/m3) (GPa)

Benzene 15 880 1.05Ethyl alcohol O 790 1.32Glycerin 15 1,260 4.43Kerosine 20 804 1.32Mercury 20 13,570 26.2OH 15 900 1.5Water, fresh 20 999 2.19Water, sea 15 1,025 2.27

'Compiled fro'm Refs. 12,25,32, and 33.tre det.ermine t~e speciflc weight of the liquid , in Ibrlft', multiply thevalues grven in thís column by 62.427 X 10-'.:l:To convert K ínto lb/in.', multiply the values given in this column by145.038 X 10'.

b. Conduit anchored against longitudinal movement at the upper end

D1/1 = - (1.2S - v)

e

c. Conduit with frequent expansion joints

!J¡ =12e

(2.41)

(2.42)

4. Tunnels Through Salid RackHalliwell'" has derived long expressions for 1/1 for lined and unlined rock

tunnels. Usually, the rock characteristics cannot be precisely estimated be-cause of nonhomogeneous rock conditions and because of the presence of fis-sures. Therefore, in our opinion, using Halliwell's expressions for practicalapplications is unwarranted. Instead, the following expressions based onParmakian 's equations 7 may be used,a. Unlined tunnel

I/I=l}E=G

in which G = modulus of rigidity of the rock.b. Steel-lined tunnel

(2.43)

1/1= DEGD+Ee (2.44)


in which e = thickness of the steelliner and E = modulus of elasticity ofsteel.

5. Reinforced Concrete PipesThe reinforced concrete pipe is replaced by an equivalent steel pipe havíng

equivalent thick.ness 7

(2.45)

in which ec::: thickness of the concrete pipe.ví , and ls are the cross-sectionalarea and the spacing of steel bars, respectively; and E; = ratio of the modulusof elasticity of concrete to that of steel. UsualIy the value of E; varíes from0.06 to 0.1. However, to alJow for any cracks in the concrete pipe, a value of0.05 is suggested.? Having computed ee, the wave velocity may be deter-mined from Eq. 2.35 using the modulus elasticity of steel.

6. Wood-Stave PipesThe thickness of a uniform steel pipe equivalent to the wood-stave pipe is

determined 7 from Eq. 2.45 using E; = ~,ec = thick.ness of wood staves, andAs and Is are the cross-sectional area and the spacing of the steel bands,respectively. The wave velocity is then computed from Eq. 2.35.

7. Polyvinyl Chloride (PVC) and Rein{orced Plastic PipeInvestigations reported in Ref. 27 show that Eq. 2.35 can be used for com-

puting wave velocity in the polyvinyl chloride (PVC) and reinforced plasticpipes, provided a proper value of the modulus of elasticity for the wall ma-te rial is use d .

8. Noncircular ConduitsThe following expression for 1/; is obtained from the equation for the wave

velocity in the thin-walled rectangular conduits derived by Jenkner''" byusíng the steady-state bending theory and by aUowing the corners of theconduit to rota te:

(2.46)

in which (3 = 0.5(6 - 50:) + 0.5 (dlb)3[6 - 5 (bld)2], o: = {1 + (dlb)3] 1[1 +(d/b)], b = width of the conduit (longer side), and d = depth of the conduit(shorter side).

Thorley and Cuymer " have included the influence of the shear force onthe bending deflection of the thick-walled (l/e < 20) rectangular conduitswhile deriving the equations for the wave velocity. From these equations, the


following expression is obtained f hi kor a t IC -walled conduit having a squarecross section:

1 (/)3 I ( E )1/;=- - +- 1 +-15 e e 2e (2.47)

in _which e = wall thickness, (1- e) = inside dimension of the conduit ande _shear modulus of the wall material. '

Based on the equations presented by Thorley and Twyrnan 30 the '"011 .. . b . , I.' owmgexpression IS o tained for 1/; for a thin-walled hexagonal conduit:

1/; = 0.0385 (;Y (2.48)

in which 1= mean width of one of the flat sides of the hexagonal section.

2.7 CASE STUDY

The data for the steel penstock of the Kootenay Canal hydroelectric power~Iant,. owned and operated by British Columbia Hydro and Power Authority arelísted m the following table: '

WallLength Diameter Thickness,

Pipe No. (m) (m) (mm) Remarks

1 244. 6.71 19 Expansion coupling at one end2 36.5 5.55 22 Encased in concrete

~or conducting a transient analysis, the waterhammer wave velocity in each sec-tíon of the penstock was determined as follows. The values of E for steel e forconcrete, and K and p for water were taken as 207 GPa, 20.7 GPa, 2.19 GPa,and 999 kg/m3, respectively.

Pipe No.l

D 6.71-;; = 0.019 = 353

40 Applied Hy draulic Transients

As the pipe is anehored at one end,

Dl/J == - (1.25 - v)e

(Eg.2.41)

== 353(1.25 - 0.30)

== 335.4

K (Eg.2.35)a=p[1 + (K/E)l/Jl

~ == 2.19 = 0.0106E 207

2.19 X 109

a=999(1 + 0.0106 X 335.4)

== 694 mis.

Pipe No. 2

Equations for a steel-lined tunnel may be used to compute the wave veloeity in

pipe No. 2.

DE (Eg.2.44)l/J==CD+Ee

5.55 X 207 X 109

= 20.7 X 109 X 5.55 + 207 X 109 X .022

= 9.62

2.19 X 109

a== 999(1 + 0.0106 X 9.62)

= 1410 m/s.

2.8 SUMMARY

In this ehapter, the derivation of the dynamie ~nd .eontínuity .equati~nsItweret d d the assumptions used in these derivatíons were dlseusse.. was

presen e , an . li h b li partial dífferen-d t ated that these equatíons are quasi- mear, yper o le, ./~o~~:tions and various methods available for their solution were dl~eussed.Expressíons f~r the velocity of waterhammer waves in the closed conduíts were

presented.


PROBLEMS

2.1. Derive the dynarnic equation eonsidering the eonduit is incJined at angle 8to the horizontal.

2.2. Compute the velocity of waterhammer waves in a 3.05-m-diameter steelpenstock having a wall thickness of 25 mm if it:

l. is embedded in a concrete dam2. is anchored at the upstream end3. has expansion joints throughout its length.

2.3. Determine the velocity of waterhammer waves in a reinforeed concretepipe having 1.25-m diameter, 0.15-m wall thickness, and carrying water.The 20-mm reinforcing bars have a spacing of 0.5 m, and the pipe has ex-pansion joints throughout its length.

2.4. A 0.2-m-diameter copper pipe having a wall thickness of 25 mm is convey-ing kerosene oi! at 20°C from a container to a valve. If the valve is closedinstantly, at what velocity would the pressure waves propagare in the pipe?Assume the pipe is anchored at the upper end.

2.5. Figure 5.10 shows the power conduits of an underground hydroeleetricpower station. Compute the wave velocity in each section of the conduit.Assume modulus of rigidity of rock = 5.24 GPa.

ANSWERS

2.2.1. 1413 m/s2. 992 m/s3. 978 mis

2.3. 9B mis2.4. 1232 mis

REFERENCES

l. Streeter, V. L., Fluid Mechanics, Third Edition, McGraw-HiJI Book Co., New York,1966.

2. Streeter, V. L. (ed.), Handbook of Fluid Dynamics, McG~aw-Hill Book Co., New York,1961.

3. Rouse , H. (ed.), Advanced Mechanics of Fluids, John Wiley & Son s, Inc., New York,1959.

4. Timoshenko, S., Strength of Materials, Second Edition, Purt 2, Van Nostrand Co., Inc.,New York, 1941.

5. Zielke, W., "Frequency-Dependent Friction in Transient Pipe Flow," Jour. Basic Engi-neering ;Amer. Soco Mech. Engrs., Man:h 1968.


6. Hirose , M., "Frequency-Dependent Wall Shear in Transient Fluid Flow Simulation ofUnsteady Turbulent Flow," Mast er's Thesis, Massachusetts Institute of Technology,Mass., 1971.

7. Parmakian, J., Waterhammer Analysis, Dover Publications, Inc., New York, 1963.8. Evangelisti, G., "Waterhammer Analysis by the Method of Characteristlcs," L 'Energia

Elettrica, Nos. 10-12,1969, pp. 673-692, 759-770, 839-858.9. Streeter, V. L., "Numerical Methods for Calculation of Transient Flow," Proc., First

International Conference on Pressure Surges, Canterbury, England, published byBritish Hydromechanic Research Association, Cranfield, England, Sept. 1972, pp.AH-AH!.

10. Wylie, C. R., Advanced Engineering Matbemattcs, Third Edition, McGraw-HiII Book Co.,New York, 1967.

11. Bergeron , L., Waterhammer in Hydraulics and Wave Surges in Electricity , John Wiley &Sons, Inc.,New York, 1961.

12. Pickford, J., Analysis of Surge, Macmillan and Co', Ltd., London, 1969.13. Allievi, L., "Theory of Waterhammer," translated by E. E. Halmos, Ricardo Garoni,

Rome, 1925.14. Rich , G. R., Hydraulic Transients, Dover Publications,lnc., New York, 1963.15. Wood, F. M., "The Application of Heavisides Operational Calculus lo the Solution of

Problerns in Waterhammer," Trans. Amer. Soco Mech. Engrs., vol. 59, Nov. 1937,pp. 703-713.

16. Harding, D. A., "A Method of Programming Graphical Surges in Pipelines," Proc. Insti-lurion of Mech, Engrs., London, Nov. 2-3, 1965, vol. 180, Par! 3E, pp. 83-97.

17. Enever, K. J., "The Use of the Computerized Graphical Method of' Surge Analysis withparticular Reference to a Water Supply Problem," Symp, on Pressure Transients , TheCity University, London, Nov. 25, 1970.

18. Perkins, F. E., Tedrow, A. C., Eagleson, P. S., and Ippen, A. T., "Hydro-power PlantTransíents," Part 11 and III, Dept, of Civil Engineering, Hydrodynamics Lab., ReportNo. 71, Massachusetts Institutc of Technology, Sept. 1964.

19. Lister, M., "The Numerical Solution of Hyperbolic Partial Differential Equations bythe Method of Characteristics," in Ralson, A., and Wilf, H. S. (eds.), Mathemafica/Method for Digital Computers, John Wiley & Sons, Inc., New York, 1960, pp. 165-179.

20. Abbott, M. B., An lnfroduction fo the Method of Characteristies, American Elsevier,New York, 1966.

21. Streeter, V. L. and Lai, C., "Waterhammer Analysis Including Fluid Friction," Jour.Hydraulics Dil'., Amer. Soco Civil Engrs., vol. 88, No. HY3, May 1962, pp. 79-112.

22. Streeter, V. L. and Wylie, E. B., Hydraulic Transients, McGraw-HilI Book Co., NewYork, 1967,pp. 22-52.

23. Streeter, V. L., "Waterhammer Analysis," Jour., Hydraulics Div., Amer. Sal'. CivilEngrs., vol. 95, No. HY6, Nov. 1969, pp. 1959-1971.

24. Evangelisti, G., Boari, M., Guerrini, P., and Rossi, R., "Sorne Applications of Water-hummer Analysis by the Method 01' Characteristics," L 'Energia Elerrrica, Nos. 1 and 6,1973,pp. 1-12,309-324.

25. Pearsall, 1. S., "The Velocity of Waterhammer Waves," Proc. Symposium on Surgesin Pipelines, Inst. of Mech. Engrs., England, vol. 180, Part 3E, Nov. 1965, pp. 12-27.

26. Halliwcll, A. R., "Velocity of a Waterhammer Wave in an E1astic Pipe," Jour., Hydrau-ties Dil'., Amer. Soco Civil Engrs., vo). 89, No. HY4, July 1963, pp. 1-21.

27. Watters, G. Z., Jeppson, R. W., and rJammer, G. H., "Water Hammer in PVC and Rein-forced Plastic Pipe," Jour., Hyd. Div., Amer. Soco Civil Engrs., vo). 102, July 1976,pp. 831-843. (See also Discussion by Goldberg, D. E., and Stoner, M. A., Junc 1977.)


28. J~nkncr, W. R., "~ber die Druc~stos~~eschwindigkeit in Rohrleitungen mit quadrati-schen und rech teckigcn Ouerschnitten, Schweizerische Bauzeitullg vol, 89 Feb 1971pp. 99-103. ' ,. ,

Th~rley, A. R. D. and Guvmer, G., "Pressure Surge Propagation in Thick-Wallcd Con-duits of Rectangular Cross Section," Jour. Fluid Engineering, Amer. Sal'. Mech, Engrsvol. 98, Sept. 1976, pp, 455-460. .,

Thorl:y, A. R. D. and Twyman, J. W. R., "Propagation of Transient Pressure Waves in~ SodlUm-C??led Fast Reactor," Proc. Second Con]. on Pressure Surges, London, pub-Ilshed by British Hydromechanic Research Assoc., 1977.Roark , R. J., Formulas for Stress and Strain, 4th ed. McGraw-HiIl Book Ne Y k1965. " w or,

29.

30.

31.

32. Hydraulic Mode/s, Manual of Engineering Practice No. 25, Cornrnittee ofHyd. Div, onHyd. Research, Amer. Soco Civil Engineers, July 1942.

33. B~umeister, T. (ed.), Standard Handbook [or Mechanical Engineers, 7th ed., McGraw-HIII Book Co., New York, 1967.

ADDlTlONAL REFERENCES

Joukowsky, N., ''Waterhammer,'' Translatcd by O. Simin, Proc. Amer. Water Works Assocvol. 24, 1904,pp.341-424. .,

Kennison, H. F., "Surge-Wave Velocítr -Concrete Pressure Pipe," Trans. Amer. Soco MechEngrs., Aug. 1956, pp. 1323-1328. .

Tison, G., "Permanen~c c~ niet-permanente stromingen door leidingen in Kunststof," Thesispresented to the University of Ghent, Ghent, Belgiurn, in part ial fulfillment of the require-rnents for the degree of doctor of cngineering, 1960.

Mo~k, ~. J., "Modellbeha~dlung von Druckstossprobleme mit Hilfe von Kunststoffrohren,"MItt.edung No. 56, Institut fur Wasserbau and Wasserwirtschaft, Technical UniversityBerlín 1962. '

R~hm, S. L. and LindvalJ, G. K. E., "A Laboratory Investigation of Transient Pressure WavesIn Pre-Stressed Concrete Pipes," Proc. 10th International Assoc. for Hydraulic ResearchLondon,1963,pp.47-53. '

Swaminathan, K. V., "Waterhammer Wave V I 't . C Te OCI y In oncrete unnels," Water Power,March 1965,117-121.

Thorley, A. R. D., "Prcssure Transients in Hydraulics Pipelines," Amer. Soco Mech. Engrs.,Paper No. 68-W AlFE-2, Dec. 1968,8 pp.

Sw~ffield, J. A., "The Intluence of Bends on Fluid Transients Propagated in IncompressiblePIpe Flow," Proc. Institution of Mech. Engrs., vol. 183 Part 1 No. 29 1968-69 pp603-614. ' , , ,.

Wood, D. J. and Funk, J. E., "A Boundary-Laycr Theory for Transient Viscous Losses inTurbulent Flow," Jour. Basic Engineeríng, Tralls. Amer. Soco Mech. Engrs., Series D,1'01. 92, No. 4, Dec. 1970, pp. 865-873.

We_Yler, M. E., Streeter, V. L., and Larsen, P. S., "An Invcstigation of the Effect ofCavita-tIon Bubble on the Momentum Loss in Transient Pipe Flow"'o D' E' .

, JI uro naSIC ngmeerzngAmer. SOCoMech. Engrs., MaTch 1971, pp. 1-10. '

Safwat, H. H., "Measurements of Transient Flow Velocities for Waterhammer Applications "Amer. Soco of Mech. Engrs., Paper No. 71-FE-29, May 1971,8 pp. '

Trikha,. A. K:, ':An Effi~,ient Method for Simulating Frequency-Oependent Friction inTranslent Llquld Flow, Jour., Fluid Engineering, Amer. Soco Mech. Engrs. vol. 97March 1975,pp. 97-105. ' ,

CHAPTER 3

METHOD OF CHARACTERISTICS

3.1 INTRODUCTION

II

In the last chapter, it was demonstrated that the equations describing thetransient-state flow in closed conduits are hyperbolic, partial differential equa-tions, and a number of methods available for their solution were discussed. Thedetails of the method of characteristics are presented in this chapter. The equa-tions for simulating a conduit are derived, and the boundary conditions for anumber of simple eríd conditions are developed. The stabiJity and convergencecriteria for the stability of the finite-dífference scheme are then presented, and aprocedure for the analysis of piping systems is outlined. The chapter concludeswith the presentation of a case study ,

We will endeavor to keep the derivation of the equations free of advancedmathematics. Readers having an elernentary knowledge of partial differentialequations should be able to follow the development of these equations; thoseinterested in a rigorous treatment should refer to Refs, 1 through 9. In derivingthese equations, we will follow the general approach proposed by Líster!' andlater adopted by Streeter and Wylie.lO A number of innovations presented byEvangellsti? will also be outlined.II

3.2 CHARACTERISTIC EQUATIONSIJi

To facílitate discussion, let us rewríte the dynamíc and continuity equations(Eqs. 2.11 and 2.30) deríved in the last chapter as111:

(3.1)

(3.2)

Let us consider a linear combination of Eqs. 3.1 and 3.2, i.e.,

L :::L¡ + XL2

44

Method of Characteristics 45

or

(aQ 2 aQ) (aH I aH) f-a + Aa -a + AgA -+ -- +--QIQI =0t x at A ax 2DA (3.3)

If ~ :::.H(x, t) and Q :::Q(x, t) are solutions of Eqs. 3.1 and 3.2, then the totalderivatives may be written as

dQ _ aQ aQ dx---+--dt at ax dt (3.4)

.1 and

dH :::aH + aH dxdt at ax dt

By defining the unknown multiplier A as

1.::: dx :::Aa'A dt (3.6)

(3.5)

or

1A:::±-a (3.7)

and by using Eqs. 3.4 and 3.5, Eq. 3.3 can be written as

dQ + gA dH + _f__ _dt a dt 2DA QIQI- O (3.8)

if

dx--:::adt (3.9)

and

dQ _ gA dH fdt -;;- dt + 2DA QIQI:::O (3.10)

if

dx-:::-adt (3.11)

Note .that ~q: 3.8 is valid if Eq. 3.9 is satisfied and that Eq. 3.10 is valid ifEq.3.11 IS satisfíed. In other words, by imposing the relations given by Eqs. 3.9~nd 3.1 ~, we h.ave con.verted the partíal differential equations (Eqs. 3.1 and 3.2)into ordinary differentíal equations in the independent variable t.


In the x-t plane , Eqs. 3.9 and 3.11 represent two straight lines having slopes±l/a. These are calIed characteristic lines. MathematicalIy, these lines divide thex-t plane into two regions, which may be dominated by two different kinds ofsolution, i.e., the soJution may be discontinuous aJong these línes.? Physically,they represent the path traversed by a disturbance. For example, a disturbanceat point A (Fig. 3.1) at time ta would reach point P after time !:::.r.

Prior to presenting a procedure for solving Eqs. 3.8 and 3.10, let us first dis-cuss the physícal significance of characteristic lines in the x-t plane. To facilita tediscussion, let us consider a single pipeline shown in Fig. 3.2. The compatibi/ityequations (Eqs. 3.8 and 3.10) are valid along the pipe length (í.e., for O<x <L)and special boundary conditions are required at the ends (i.e., at x = O and atx = L) (Fig. 3.3). In the example under consideration, there ís a constant-headreservoir at the upper end (at x = O) and a valve at the downstream end (at x = L),and the transient conditions are produced by closing the valve. Let us assumethat there is steady flow in the pipe at time t = O when the valve is instanta-neously closed. Thís reduces the flow through the valve to zero and results in apressure rise at the valve. Because of this pressure rise, a pressure wave travels inthe upstream direction. If the path of this wave is plotted on the x-t plane, itwill be represented by line BC as shown in Fig. 3.4. It is clear from this figurethat the conditions in Región 1 depend only upon the ínitial conditions becausethe upstream boundary conditions did not change, whereas in Región 11 theydepend upon the conditions imposed by the downstream boundary, Thus, thecharacteristic line BC separates the two types of soJutions. If excitations are irn-posed simultaneously at points A and B, then the region influenced by the initialconditions is as shown in Fig. 3.5; the characteristic line AC separates the regions

.c.l( = a.c.t

xFigure 3. t. Characteristic lines in x-t plane.


Reservior

F/ol!'

influenced by the upstream boundary and the initial conditions, and the Jine BCseparates the regions inñuenced by the downstream boundary and the initia)conditions. In other words, the characteristic lines on the x-t plane representthe traveling paths of perturbations initiated at various )ocations in the system.

Va/ve

L

Figure 3.2. Single pipeline.

Region of Validity

of Compoll/Jllily Equotlons

( Eqs. 3.8 ond 3.10)

/nlllo/ Condilions

Bx = L

t = o '--------'---------------'A

x=o

Figure 3.3. Regions of validity for a single pipeline.x


I = .0.1

B

Region I

1=0A

xFigure 3.4. Excitation at downstream end.

To solve Eqs. 3.8 through 3.11, a number of finite-dífference schemes havebeen proposed: Streeter and WylielO use a first-order finite-difference technique;Evangelístt? suggests a predictor-corrector method; and Lister " ernploys bothfirst- and second-order finite-difference schemes. Because the time intervals usedin solving these equations for practical problems are usually small, a first-ordertechnique suggested by Streeter and Wylie is sufficiently accurate and is dis-cussed here. However, if the friction Iosses are Iarge, then a first-order approxi-mation may yíeld unstable results. For such cases, a predictor-corrector methodor a second-order approximation (see Section 7.4) should be used to avoidinstability of the finite-dífference scheme.

Referring to Fig. 3.1 , let the conditions at time t = to be known. These areeither initially known (í.e., at t = O, these are initial steady-state conditions) orhave been calculated for the previous time step. We want to compute theunknown conditions at to + flt. Referring to Fig. 3.1, we can write along thepositive characteristic line AP,

(3.12)

1= Al e

t = o

xFigure 3.5. Excitation at upstream and downstream ends.


dH=Hp-HA

Similarly, we can write along the negative characteristic line BP.

(3.13)

dQ=Qp- QB

dH=Hp-HB

(3.14)

(3.15)

The subscripts in Eqs. 3.12 through 3.15 refer to the locations on the x-tplane. Substituting Eqs. 3.12 and 3.13 into Eq. 3.8 and Eqs. 3.14 and 3.15 intoEq. 3.10, computing the friction terrn at the points A and B, and multiplyingthroughout by flt, we obtain

(3.16)

and

(3.17)

Equation 3.16 can be written as

(3.18)

and Eq. 3.17 as

(3.19)

in which

_ gA ffltCp - QA + -;;- HA - 2DA QAIQAI

gA ffltCn = QB - -;-HB - 2DA QBIQBI

(3.20)

(3.21)

and

(3.22)

Note that Eq. 3.18 is valid along the positive characteristic linc AP and Eq. 3.19along the negative characteristic Une BP. The values of the constants Cp and Cnare known for each time step, and the constant Ca 'depends upon the conduitproperties, We will refer to Eq. 3.18 as the positive characteristic equation andEq. 3.19 as the negative characteristic equation. In Eqs. 3.18 and 3.19, we havetwo unknowns, nal11ely,Hp and Qp. The values ofthese unknowns can be deter-mined by simultaneously solving these equations, i.e.,

(3.23)


t

x

oA

Inlerior sectton«Downslreom boundor)'Upslreom boundor)'•Figure 3.6. Characteristic grid.

Now the value of Hp can be determined either from Eq. 3.18 or Eq. 3.19. Thus,by using Eqs. 3.18 and 3.23, conditions at all interior points (se e Fig. 3.6) at theend of the time step can be determined. However, at the boundaries, either Eq.3.18 or 3.19 is available. Therefore, as discussed aboye, we need special bound-ary conditions to determine the condition at the boundaries at time to + D.t.

To ilIustrate how to use the aboye equations, we will again consider the singlepipeline of Fíg. 3.2. The pipeline is divided into n equal reaches (Fig. 3.6), andthe steady-state conditions at the grid points at t = to are first obtained. Then,to determine the conditions at t = to + D.f, Eqs. 3.18 and 3.23 are used for theinterior points, and special boundary conditions are used for the end condi-tions. A close look at Fig. 3.6 shows that the conditions at the boundaríes att = to + M must be known for calculating the conditions at t = fo + 2D.t at theinterior points adjacent to the boundaries, Now conditions at t = to + D.t areknown at all the grid points, and the conditions at t = lo + 2D.t are determinedby following the procedure just outlined. In this manner, the computations pro-ceed step-by-step until transient conditions for the requíred time are determined.

3.3 BOUNDARY CONDITIONS

In the last section we díscussed that special boundary conditions are required todetermine the conditions at the boundaries. These are developed by solvingEq. 3.18, 3.19, or both, and the conditions imposed by the boundary. Equation


3.18 is used for the downstream boundaries and Eq. 3.19 for the upstreamboundaries.

A number of simple boundary conditions are developed in this section whilecornplex boundary conditions such as for pumps and turbines are derived inChapter 4 and 5 and for waterhammer control devices, in Chapter 10.

Constant-Head Reservoír at Upstream End (Fig. 3.7)

If the entrance losses as well as the velocity head are neglígíble, then

n; =Hres (3.24)

in which Hres = height of the reservoír water surface aboye the datum. Equation3.19 for the upper end thus becomes

(3.25)

However, if the velocity head or the entrance losses are not small, then thesemay be considered in the analysis as follows:

Let the entrance losses be given by the equation

_ kQ'j,he - 2gA2 (3.26)

i, ';,~'á.'~I~ cEnergy Grade Une

\:r-l-¡-_I_~ __'--Hydraulic Grade Une

FlawFlow- -Oalum Une Datum Une

(o) (b)

Figure 3.7. Constant-Ievel upstream reservoir.

S2 Applied Hydraulic Transients

in which k is the coefficient of entrance loss. Referring lo Fig. 3.7,

Q2Hp =Hres - (1 + k) ~ (3.27)

2gA

Solving Eq, 3.27 and the negative characteristic equation (Eq. 3.19)simultaneously,

Q -1 + VI + 4k¡ (Cn + CaHres) (3.28)p = 2k¡

in whích

(3.29)

Now Hp can be determined from Eq. 3.27.For the reverse flow, k is assigned a negative value in Egs. 3.27 and 3.29.

Constant-Head Reservoir at Downstream End (Fig. 3.8)

If the head losses al the entrance lo the reservoir are

kQ2h =-_p_

e 2gA2 (3.30)

'" 1'"8"~"'mN «Enerqy Grade Line ~ ~ ~I} N~~

,_.___t_ l' NEnergy Grade une" "'tt

Hydra-;;;'C Grade untl)FlowFlow

..~

::r::---Datum Line Oofum Line

(a) ( b)

Figure 3.8. Constant-level downstream reservoir.

en.,X


then referring lo Fig. 3.8a

Q2H.p=H - (1- k)-P_

res 2gA 2

In Eq. 3.30, k is assigned a negative value for the reverse flow ,Elimination of Hp from Eqs. 3.31 and 3.18 yields

k2 Q~ - Qp + k , = O

(3.31)

(3.32)

in which

Ca{l - k) }k -2 - 2gA2

and(3.33)

k3 = Cp - CaHres

Solving Eq, 3.32 for Qp,

(3.34)

Now Hp may be determined from Eq. 3.18. If the exit loss and the velocityhead are negligible, then

. (3.35)

and it follows from Eq. 3.18 that

Qp = Cp - CaHres (3.36)

Dead End at Downstream End (Fig. 3.9)

At the dead end , Qp = O. Hence, from the positive characteristic equation (Eq.3.18), it follows that

(3.37)

_______ O_e_a_d__E._nd_~

Figure 3.9. Dead end,


Valve at Downstream End (Fig. 3.10)

Steady-state flow through a valve can be written as

Qo = (CdAv)o../2gHo (3.38)

in which subscript o indicates steady-state conditions, Cd = coefficient of dis-charge, Ho = head upstream of the valve, and A v = area of the valve opening.

An equation similar to Eq. 3.38 may be written for the Iransient state as

(3.39)

Dividing Eq. 3.39 by Eq. 3.38, taking square of both sides and defining the rela-tive valve opening 7 = (CdAv)f(CdAv)o, we obtain

Q~ = (º;7)2 Hp (3.40)o

Substitution for Hp from the positive characteristic equation (Eq. 3.18) into Eq.3.40 yields

(3.41 )

Hydraulie Grade Une

(o)

I

(b) Openinq (e) Closinq

Figure 3_10. Valve at downstream end,

~'.~J __<__'

Method of Characteristics 5 S

in which Cv = (TQo)2/(CaHo)' Solving for Qp and neglecting the negative signwith the radical term

(3.42)

Now Hp may be determined from Eq. 3.18.To compute the transient-state conditions for an opening or a closing val ve , T

versus t curves (Fig. 3.1 Ob and e) may be specified either in a tabular form or byan algebraic expression. Note that T = 1 corresponds to a val ve opening at whichthe flow through the valve is Qo under a head of Ho'

Orífice at Lower End

For an orifice, the opening rernains constant. Therefore, the aboye equationsmay be used with T = 1.

Series Junction (Fig. 3.11)

In the preceding discussion, we considered only one conduit, and the boundarywas either at the upstream or at the downstream end. Therefore, no special ca rehad to be taken to designa te the variables al the boundary sin ce there was onlyone conduit section under consíderation. However, if the boundary is at thejunction of two or more conduits, then the variables at different sections ofvarious conduits have to be specified. For this purpose, we will use two sub-scripts. The first subscript will designate the conduit number, while the secondwill indicate the section number. For example, Qp .. indicates flow al the jlhsection of the ith conduit. For variables that have sa~e value at al! sections of aconduit , only one subscript will be used. For exarnple , Ca¡ refers to constant Ca(Eq. 3.22) for the ¡th conduit. Although Cp and Cn may have different values atdifferent sections of a conduit, only one subscript will be used with them to-indicate the conduit number. This simplifies presentation and at the sarne timedoes not result in any ambiguity sin ce each conduit can have only one end-sectionat a boundary. As discussed previously, the subscript P will indicate the un-known variables at the end of the time step.

Conduif i Conduif i+1- -I

i,n+l~'--i+l, 1

Figure 3.11. Series junction,


If the difference in the velocity heads at sections (i, n + 1) and (i + 1, 1) (Fig.3.11) and the head losses at the junction are neglected, then

Hp¡,n+l =Hp¡+I,1 (3.43)The positive and negatíve characteristic equations for sections (i, n + 1) and(i + 1, 1) are

(3.44)

(3.45)

Qp¡,n+l = c., - Ca¡Hp¡,n+l

Qpi+l,l = Cni+1 + Ca¡+lHPi+l,1

The continuity equation at the junction is

Qp¡,n+l = Qpi+l,1

It follows from Eqs. 3.43 through 3.46 that

Cp.- en' 1H = 1 1+ (347)Pi, 11+1 C + C .

ai ai+l

Now Hp. ,Qp. l' and Qp. 1 1 can be determined from Eqs. 3.43 through,+1,1 ',n+ z+ ,3.45.

(3.46)

However, if the difference in the velocity heads at sections (i, n + 1) and(i + 1, 1) or the head losses at the junction are not negligible, then Eq. 3.43 isnot valid. In such cases, the following equation for the total head may be usedinstead of Eq. 3.43:

Q~. Qk·H + III+l=H +(I+k) 1+~,1

Pi,n+l 2gA¡ Pi+l,l 2gAi+1

in which k = coefficient of head losses, h¡, at the junction

kQ2h - P¡+I,1

l - 22gAi+1

Simultaneous solution of Eqs. 3.44 through 3.46, and 3.48 yields

b + ..Jb2 - 4cdQpi,n+l = 2c

(3.48)

(3.49)

in which

(3.50)


Now Qp. 1 i ' Hp. l' and Hp. 1 1 can be deterrnined from Eqs. 3.44 throught+ ,f,n+ ,+ I

3.46.

Branching Junction (Fig. 3.12)

For the branching junction shown in Fig. 3.12, the following equations can bewritten:

l. Continuityequation

Qpi,n+l = Qpi+l,l + Qpi+2,l

2. Characteristic equations

(3.51)

(3.52)

(3.53)

(3.54)

Qpi+l,l = Cni+1 + Cai+IHPi+l,l

Qpi+2,1 = Cni+2 + Cai+2Hpi+2,1

3. Equation for total head

(3.55,3.56)

In Eqs. 3.55 and 3.56, the head losses at the junction are neglected, and it is as-sumed that the velocity heads in all conduits are equaJ.

Simultaneous solution of Eqs. 3.51 through 3.55 yíelds

(3.57)

Now Hp. 1 1 and Hp. 1 can be determined from Eqs. 3.55 and 3.56, andr+ , '+2,Qp. ,Qp. and Qp. 1 from Eqs. 3.52 through 3.54.

I,n+! 1+1,1 ,+2,

Figure 3.12. Branching junction.


Hp

Qp

Figure 3.13. Head-discharge curve for a centrifugal pump,

Centrifugal Pump at Upstream End

The head-discharge curve for a centrifugal pump running at constant speed isshown in Fig. 3.13. This curve can be approximated by the equation

Hp=C7- C8Q~

Solving this equation simultaneously with Eq. 3.19,

-1 + Vi + 4CaCs(Cn + CaC7)

Qp= 2CaCs

Now Hp can be determined from Eq. 3.58.

(3.58)

(3.59)

Francis Turbine at Downstream End

The head-discharge curve for a Francis turbine running at constant speed (i.e.,connected to a large system) and at constant gate opening can be approximatedas

(3.60)

Solving this equation simultaneously with the positive characteristic equation(Eq. 3.18) yields

-1 + Vi + 4CaCIO(Cp - CaC9)Qp=------------~--~--

2CaClO

Now Hp can be determined from Eq. 3.60.

(3.61 )

3.4 STABILlTY AND CONVERGENCE CONDITIONS

The finite-dífference scheme presented in Section 3.2 is termed convergent ifthe exact solution of the difference equations approaches that of the original


differential equations as tlt and tlx approach zero. If the round-off error due torepresentation of the irrational numbers by a finite number of significant digitsgrows as the solution progresses, the scheme is calIed unstable; if this error de-cays, the scheme is stable. It has been proved that convergence implies stabilityand that stability implies convergence.P'!"

Methods for determining the convergence or stability criteria for non linearequations are extremely difficuIt, if not impossible. Collatz'? suggests that theconvergence and stability may be studied by numericalIy solving the equationsfor a number of Sx] tlt ratios and then examining the resuIts. The convergenceand stability may, however, be st udied analyticalIy by linearizing the basic equa-tions. If the nonlinear terms are relatively small, it is reasonable to assume thatthe criteria applicable to the simplified equations are also valid for the originalnonlinear equations.

Using the procedure proposed by O'Brien et al. 14 and considering the linearizedequations, Perkins et a1.9 showed that for the finite-difference scherne of Section3.2 to be stable,

(3.62)

This condition irnplies that the characteristics through point P in Fig. 3.1 shouldnot fall outside the segment AB. For a neutral scheme,

tlt-=-tlx a

(3.63)

The criteria for convergen ce indicate that the most accurate solutions are ob-tained if Eq. 3.63 is satisfied. Thus, the convergence and/or stability criterionfor the finite-difference equations (Eqs. 3.16 and 3.17) is given by the expression

~~l_tlx "" a (3.64)

This is called Courant 's stability condition.

3.5 SELECTION OF TIME INCREMENT FOR A COMPLEX PIPINGSYSTEM

For a complex system of two or more conduits, it is necessary that the sarne timeincrement be used for all conduits so that boundary conditions at the junctionmay be used. This time increment should be selected such that Courant's stabil-ity condition (Eq. 3.64) is satisfied.

If the time interval, St , is su eh that the reach Jength for any conduit in thesystem is not equal to atlt, then tlx must be greater than atlt to satisfy Courant'sstability criteria. In other words, the characteristics through P pass through R


and S and not through the grid points A and B (Fig. 3.14). The conditions atevery time step are, however, computed at the grid points only while conditionsat R and S must be known to determine conditions at P.

Streeter and Lai in their pioneer paperlS and Streeter and Wylie ro proposed aninterpolation procedure for computing conditions at R and S from the knownconditions at A, B, and C. However, later investigations have shown that this pro-cedure srnooths the sharp transient peaks. ro avoid thís, Streeter 16 suggeststhat the original dífferential equations for short conduits may be written in animplicit form, whereas Kaplan et al.!? propose a pro ce dure called zooming inwhích Át for longer conduits may be íntegral multiples of Át for shorter con-duits of the system.

In the author's opinion, the implicit method combined with the characterlsticmethod should be used if a number of conduits in the system are very short rela-tive to others; otherwise, simple adjustment of wave velocities to satisfy the fol-lowing equation should give sufficiently accurate results.

L·Át = -' (i = 1 to N)

a¡l1¡(3.65)

in which ni must be an integer and is equal to the number of reaches into whichith conduit is divided , and N = number of pipes in the system. As the wavevelocity is not precisely known, minor adjustments in its value are acceptable.

Because of the Iimitations imposed on Át by the Courant's stability condition,a large amount of computer time is required for analyzing systems having veryslowly varyíng transients. For the analysís of such systerns, Yow'" has reporteda technique that allows larger time steps and at the sarne time satisfies theCourant's condition. In this technique, the inertial term of the dynamic equa-

8

ÁI(

xFigure 3.14. Notation for interpolation.


tion is multiplied by an arbitrary factor a2• The resulting equation and the con-

tinuity equation are then converted into the characteristic formo Because ofmultiplication by a 2, a time step equal to exÁt is perrnissible, in which Át is thetime step given by the Courant's condition. Different values of a may be usedfor different conduíts, and the value of a may be as large as 20. Yow's te eh-nique, however, is applicable only to those systems in which the inertial term issmall as compared to the friction term such as gas flow in pipes/s,21 flow inporous media,19 and floods in rivers. The validity of this technique is question-able20 because the original governing equations are arbitrarily altered; thus, ex-treme caution must be exercised while using this technique for the analysis ofthese systems.

3.6 COMBINED IMPLICIT-CHARACTERlSTlC METHOD

In the last section, it was pointed out that sometimes it is advantageous to usea combination of the implicit and characteristic methods while analyzing certainpiping systems. Details of this follow.

Let us consider a piping system in whích the ith reach of a conduit is to beanalyzed using the implicít method. In this method, the derivatives of the con-tinuity and dynamic equations (Eqs. 3.1 and 3.2) are replaced by the centered-implicit finite dífferences!" as follows (Fig. 3.15):

aH = (HP¡+1 + Hi+l) - (Hp¡ + H¡)

ax 2Áx(3.66)

aH (Hp'+1 + Hp.) - (H¡+I + H¡)-=' ,at 2Át

(3.67)

xl+l

Figure 3.1 S. Notation for implicit method.


aQ (QP¡+l + Q¡+I) - (Qp¡ + Q¡)-::::

ax 2Ax

aQ <QPi+l + Qp¡) - (Qi+l + Qi)-=at 2At

Q = 0.5(Q¡+1 + Qí)

(3.68)

(3.69)

(3.70)

To simplify presentation, only one subscript is used in this section to designatethe variables. Substitution of Eqs. 3.66 to 3.70 into Eqs. 3.1 and 3.2 and sirnpli-fication of the resulting equations yield

Q», + Qp. 1 - CIlHp. + CIIHp'+l + C12 ::::O1 t+ J I

(3.71)

(3.72)

in which

gAAtCn =--

Ax(3.73)

(3.75)

(3.76)

Note that there are four unknowns in Eqs. 3.71 and 3.72, namely , Qpi' QPi+l'Hp and Hp For a unique solution of these equations, there should be foureq~~tions. iih~se other two equations, in addition to Eqs. 3.71 and 3.72, areprovided by the end conditions of the reach. For example , íf there is a conduitat the upstream end and a constant-head reservoir at the downstrearn end, thenthe negative characteristic equation (Eq. 3.19) and Hpi+1 ::::Hres are the othertwo equations. For any other end conditions, similar equations are written.Thus, there are four equations in four unknowns, and their values are deter-mined by simultaneously solvíng these equations.

Note that the conditions imposed by the boundary are used for the additionalequations and not the boundary conditions developed in Section 3.3. Therefore ,while selecting the conduit or the conduit reach for which the irnplicit method isbeing used, care should be taken that its end conditions are simple.

Method of Characteristícs 63

3.7 ANALYSIS OF A PIPING SYSTEM

To compute transient-state condítions in a piping system, the shortest conduitin the systern is divided into a number of reaches so that a desired computa-tional time interval, At, is obtained. According to Evangelístí.! a time intervalof h to 2~ of the transit time, i.e., wave-travel time from one end of the systernto the other, should give sufficiently accurate results. In the author's opinion,however, this criterion should be used as a rough guide , and At should be in-creased or decreased depending upon the rate at which transients are produced,Having selected the value of At, the rernaining conduits in the system are dividedinto reaches having equallengths by using the procedure outlined in Section 3.5.lf necessary, the wave velocities are adjusted to satisfy Eq. 3.65 or the procedureoutlined in Section 3.6 is used so that characteristics pass through the grid points.The steady-state discharge and pressure head at all the sections are then corn-puted, and their values are printed. The time is now incrernented , and the tran-sient conditions are cornputed at all the interior points frorn Eqs. 3.23 and 3.18and at the boundaries from the appropriate boundary conditions. This process iscontínued until transient conditions for the required time are computed.

The flowchart of Fig. 3.16 shows the computational steps for deterrniningthe transient conditions in a series piping system. To ilIustrate this proce-dure, Iransient conditions in the piping systern shown in Fig. 3.17a were deter-mined. For this purpose, the computer program of Appendix B was developedin FORTRAN IV language. Transient conditions were caused by closing thevalve according to the r-t curve shown in Fig. 3.17 b.

As the valve-closure time is rather large as compared to the wave-transit timein the system, pipe No. 2 was divided into two reaches, thus giving At = 0.25 s.Pipe No. 1 was also divided into two reaches to satisfy Eq. 3.65, and the initialsteady-state conditions were computed. Time was incrernented by At, and theconditions at the interior sections were determined using Eqs. 3.23 and 3.18.The boundary conditions for the reservoír (Eqs. 3.24 and 3.25) were used todetermine the conditions at the upstream end, and Eqs. 3.43,3.44,3.46, and3.47 were used to determine conditions at the junction of pipes No. 1 and No. 2.Seven points on the r-t curve were stored in the computer, and the r values atthe intermediate times were parabolically interpolated. Equations 3.42 and 3.40were used to determine the conditions at the valve.

Conditions at t = Al at all sections of the systern were now known. Thesewere stored as conditions at the beginning of the next time step. This procedurewas repeated until transíents for the desired duration were computed. The con-ditions were printed every second time step by specífyíng IPRINT = 2.


Figure 3.16. Flowchart for a series piping system.

L = 550m L 450mD = 0.75m D = 0.6 mO = 1I00m/s O = 900m/sf = O.OIOm f = O.OI2m

(O) Piping system

1.0

o.S

0.6

Io.l 0.4

0.2

2 3 4 5 6Time (seconds)

( b) Volve closure curve

Figure 3.17. Series piping system.

Reservoir

3.8 CASE STUDY


Ini/iol steody sto/eHydraulic grade line

Valve

Pipe No. I PIpe No. 2Qo:: I m.J/sec

Figure 3.18 shows the schematic layout of the conduits of the Jordan RiverRedevelopment22,23 located in British Columbia, Canada, and owned by theBritish Columbia Hydro and Power Authority. It is a peaking power plant.


UJ;:'j~C!IZ

~...J:::>C!IUJ~a:EUJEQ: .!E

e:::> •a(/) o:

o: ~.~g:i

o0-z«UJZala::::>1-

."..e (/)Z

!! Qe 1-

Uu UJ'O (/).s::.(/) (/)

(/)

!=I Oa:

." u.....Jc:«

! u.. z... >-u 1-c:aU


The upstream conduit consists of a tunnel having a 5285-m-Iong, mainly D-shapedsection; 82-m-long, 3.96-m-diameter, and 451-m-Iong, 3.2-m-diarneter sections;and a 1400-m-Iong penstock reducing in diameter frorn 3.2 to 2.7 m. There isonly one Francis turbine rated at 154 MW and 265.s-m rated head. To reducethe maximum transient-state pressures, a pressure-regulatíng valve (PRV) is pro-vided. The rating curve for the PRV, at rated head (H,) of 265.5 m as deter-mined from the prototype tests, is shown in Fig. 3.19.

To determine the transient conditions caused by opening or closing ofPRV, acomputer program was developed by using the boundary conditions for the PRVderived in this section.* Analysis of transients caused by various turbine opera-tions is discussed in Chapter 5. Points on the PRV rating curve (Fig. 3.19) werestored in the computer at 20 percent intervals of the valve stroke, and the dis-charge at the intermediate valve openings was determined by linear interpolation.Assuming that the valve characteristics obtained under steady-state operationare valid during the transient state, the PRV discharge under net head H¿ is givenby the equation

(3.77)

in which Qu = PRV discharge under a net head of Hn, and Q, = discharge underrated net head H" both at valve opening T. Note that both H, and Hn are totalheads, i.e., Hn = H p + Q~/(2gA 2), in which A = cross-sectional area of the con-duit just upstream of the PRV.

To develop the boundary condition for the PRV, Eqs. 3.18 and 3.77 aresirnultaneously solved. Noting that Qp = Qu and eliminating Hn froru theseequations,

(3.78)

in which

(3.79)

Now Hp may be determined from Eq. 3.18.

*Boundary conditions for the simultaneous operation of the PRV and wicket gates are de-veloped in Section 10.7.


?O~----~----~----'-----~-----

'b<:

'"<> 50..........<:1.

~ 40~..1::s<> 30.!:;..'~t:) 20.c:::<>.~<::)

10

MOde'--L/l I}~rOIOlype

oVO 20 40 60 80 100

Gafe openinll (%)

Figure 3.19. Discl\atge cl\aracteristics oí pressute-tegulating valve.

In the computer analysis, the upstream conduit was represented by 11 pipeswhile the conduit downstream of the PRV was neglected because of its shortlength. Lined and unlined segments of the tunnel were combined into two linedand unlined reaches, and the D-shaped tunnel was replaced by a circular conduithaving the same eross-secticnal area. The waterhammer wave velocíty '" wascomputed by taking the modulus of rigidity of the rock as 5.24 GPa, and assurn-ing the penstock to be anchored at the lower end and free for longitudinal ex-pansion at the upper end. The friction factor for various conduits were corn-puted such that they included the friction and minor losses, such as expansion,contraction, and bend losses. Thus, although the minor losses are concentratedat varíous locations in the actual systern, these are assumed to be distributedalong the conduit length. In the author's opinion, this approximation should notintroduce large errors in the analysis. The head losses computed using thesevalues of friction factors and those measured on the prototype are in closeagreement ,

A number of transient-state tests were conducted on the prototype. Steady-state pressures were measured by a Budenberg deadweight gauge having a certi-fied accuracy of 0.35 m. Transient-state pressures were measured with a strain-

·T·· .•··'···

"1 Method of Characteristics 69

gauge-type pressure cell, which delivered linear output within 0.6 percent overits entire range. The natural frequency of the cell was greater than 1000 Hz andit was calibrated against the deadweight gauge. A multiturn potentiometermechanically connected to the PRV-stroke mechanisrn was used to measure thePRV openíng, and a Westinghouse leading-edge flowmeter" was used to measurethe transien t-sta te flows.

The computed and measured transient-state pressures and flows are shown inFig. 3.20. In the prototype test, the PRV was first opened from Oto 20 percent

-..~

F:EFfil~ O 8 1"1/6 24

Time (sec)

I32

1I40

'bt:)

~.~~~

o.: 330

320II

3/0I

2

Time (SIIC)

Figure 3.20. Comparison of computed and measured results,


at a very slow rate and was kept at this opening until steady flow was establishedin the upstream conduit , The PRV was then closed from 20 percent to O (Fig.3.20). The wicket gates were kept closed throughout the test. In the computeranalysís, however , the PRV was not completely closed but was held at 1 percentopening to simulate the leakage through the wicket gates.

As can be seen frorn Fig. 3.20, the cornputed and measured transient pressuresagree closely up to an elapsed time of about 18 s; afterward, there is goodagreement between the shapes of the pressure curves but the measured resultsshow that the pressure waves are dissipated more rapidly than that indicated bythe results of the mathematical model. In addition, the rneasured period of thepressure oscillations is less than the computed periodo These differences may bedue to using the steady-state friction formula for computing the transient-statefriction losses and the reduction of wave velocity at low pressures as discussed inSection 2.6. The computed and rneasured discharge agree c1osely.

3.9SUMMARY

In this chapter, the details of the method of characterisücs were presented, anda number of simple boundary condítions were developed. Stability and con-vergence conditions for the finite-difference scherne were díscussed , and a proce-dure was outlined for the selection of time interval for a cornplex system. Forillustration purposes, a computational procedure for analyzing transient condi-tions caused by closing a valve in a series system was presented. The chapter wasconcluded by comparing the computed and measured results for the transientconditions caused by the closure of a pressure-regulating valve in a hydroelectric

generating station.

PROBLEMS

3.1. Prove that the equations of the characteristic curves are dxldt = V ± a if theterm V(oV/ox) of the dynarnic equation (Eq. 2.10) is not neglected andthere is an additional term V(oH¡ox) in the continuity equation.

3.2. Develop the boundary conditions for a centrifugal pump running at ratedspeed, taking into consideration transients in the suction lineo

3.3. Write a computer program for the piping system shown in Fig. 3.17a. Runthe program for various values of At and plot a graph between the corn-puted pressure at the valve and At.

3.4. Develop the boundary conditions for an opening or elosing valve located atthe junction of two conduits (Fig. 3.21). (Hint: The following four equa-tions are avaílable: the positive characteristic equation for section i, n + 1;the negative characteristic equation for section i + 1, 1; the continuity equa-


Va/ve

Conduif i Conduit i+ I

I

i,n+I/~i+l, l

Figure 3.21. Valve at series junction.

tion, and the equation for flow through the valve. Solve these equationssimultaneously to obtain an expressíon for Qp. .)

I,n+l

Preve that if the valve in Fig. 3.21 is replaced by an orifice and the conduitsi and i + 1 have the same diameter, wall thickness, and wall material, then

Qp¡,n+1 = Qp¡+1 ,1 = -C + ..JC2 + C(Cp + Cn)

in which C = Q~/(Cal1Ho) and l1Ho is the orifice head loss for Q().

3.6. Is the equation for Qp¡ n+1 gíven in Problem 3.5 valid for the reverse flow?If not, derive a similar e'quation for the reverse flow.

3.5.

3.7. Develop the boundary conditions for the pressure-regu1ating valve and theFrancis turbine shown in Fig. 3.18. The transient conditions are causedby opening or elosing the valve. Assume that the turbine speed and thewicket-gate opening remain constant during the transient-state conditions.

3.8. Prepare a flowchart for programming the boundary conditions developed inProblem 3.7.

3.9. A procedure called zooming ís presented in Ref. 17 in which the time stepfor the long pipes may be an integral multiple of that for short pipes. How-ever, the procedure requires extrapolation at the junction of pipes havingdifferent time steps. Investigate the effect of extrapolation on the pressurepeaks for the piping system shown in Fig. 3.17a. Assurne that pipe No. 2 is90-m-long instead of 450 m as shown in the figure. (Hint: Solve the systemusing the zoorning procedure and then using the same l1t for the whole sys-tem as determined by the Courant's condition.)

REFERENCES

1. Ralston, A. and Wilf, H. S. (eds.), Mathematical Methads [or Digital Computers, JohnWiley & Sons, Inc., New York, 1960.

2. Evangelisti, G., "Waterharnrner Analysis by the Method of Characteristics," L 'EnergiaElettrica, Nos. 10-12, 1969, pp. 673-692, 759-770, 839-858.

3. Webster, A. G., Partial Differential Equations of Mathematical Pñysics, Dover Publica-tions, Inc., New York, 1950.

4. Stoker, J. J., Water Waves, lnterscience, New York, 1965.


5. Abbott, M. B., An Introductlon to the Method of Chnracteristics, American Elsevier ,

New York, 1966. .6. Mises, R., Mathematical Theory of Campressible Fluid Flow, Acadernic Press, New

York, 1958. '''1',7. Gray, C. A. M., "The Analysis of the Dissipation of Energy m Waterhammer, roe.

Amer. Soco Civil Engrs., vol. 119, 1953, pp. 1176-1194.8. Courant, R., Methods of Mathematícal Physics, Interscience, New York, 1962.9. Perkins, F. E., Tedrow, A. c., Eagleson, P. S., and Ippen,~. T., HbydRro-poweNrr:

Transients, Part 11, Dept. of Civil Engineering, Hydrodynarnics La. eport o. ,Massachussetts Institute of Technology, Sept. 1964.

10. Streeter, V. L. and Wylie, E. B., Hydraulic Transients, McGraw-HiII Book Co., New

York,1967. . . . . . b11. Lister, M., "The Numerical Solution of Hyperbolic Partial Differentíal Equations y

the Method of Characteristics," chapter in Ref 1, pp. 165-179. .12. Fox, P., "The Solution of Hyperbolic Partial Differential Equations by Dlfference

Methods," chapter in Re]. l. . .13. Collatz, L., The Numerical Treatment of Differential Equations, Third ed., Springer,

Berlin, 1960. . .14. O'Brien, G. G., Hyman, M. A., and Kaplan, S. "A Study of the Nurnerical Solution of

Partial Differential Equations," Jour. Moth. and Pñysics, No. 29, 1951, pp. 223-251.15. Streeter, V. L. and Lai, e., "Waterhammer Analysis Including Fluid Fríction," Jour.

Hyd. Div., Amer. Soco of Civ. Engrs., May, 1962, pp. 79-112. .16. Streeter, V. L., "Waterhammer Analysis," Jour. Hyd. Div., Amer. Soco of Civ. Engrs.,

Nov. 1969,pp. 1959-1971. .17. Kaplan, M., Belonogoff, G., and Wentworth, R. C., "Econornic Methods for Modelhng

Hydraulic Transient Simulation," Proc. First International COIl!erence on PressureSurges, Canterbury, England, published by British Hydrornechanic Research Assoc.,Sept. 1972, pp. A4-33-A4-38. .

18. Yow, W., "Nurnerical Error on Natural Gas Transient Calculations," Trans. Amer. Soc.of Mech. Engrs., vol. 94, Series D, no. 2, 1972, pp. 422-428. .

19. Wylie, E. B., "Transient Aquifer Flows by Characteristics Method," Jour. Hyd. Dtv.;Amer. Soco of Civil Engrs., vol. 102, March 1976, pp. 293-305. .

20. Rachford, H. H. and Todd, D., "A Fast, Highly Accurate Means of Modeling TransientFlow in Gas Pipeline Systems by Variational Methods," Jour. Soco of Petroleum Engrs.,April 1974, pp. 165-175. (See also Discussion by Stoner, M. A., and Authors' Reply,

pp. 175-178J '.21. Wylie, E. B., Streeter, V. L., and Stoner, M. A., "Unsteady-State Natural Gas Translent

Calculations in Complex Pipe Systems," Jour. Soc. of Petroleum Engrs., Feb. 1974, pp.

35-43. r22. Portfors, E. A. and Chaudhry, M. H., "Analysis and Prototype Verification of Hydrau IC

Transients in Jordan River Power Plant," Re! 17, pp. E4-57-E4-72.23. Chaudhry, M. H. and Portfors, E. A., "A Mathematical Model for Analyzing Hydraulic

Transients in a Hydroelectric Power Plant," }',roe. First Canadian Hydraulic Canference,published by the University of Alberta, Edmonton, Canada, May 1973, pp. 298-314.

24. Parmakian, J., Waterhammer Analysis, Dover Publications, Inc., New York, 1963.25. Fischer, S. G., "The Westinghouse Leading Edge Ultrasonic Flow Measurement Sys-

tem," presented at the spring meeting, Amer. Soco of Mech. Engrs., Boston, May 1973.


ADDITIONAL REFERENCES

Syrnposium, Waterhammer in Pumped Storage Projects, Chicago, Nov. 1965, published byAmer. Soco of Mech. Engrs.

Evangelisti, G., "On the Numerical Solution of the Equations of Propagation by the Methodof Characteristics," Meccanica, vol. 1, No. 1/2, 1966, pp. 29-36.

Contractor, D. N., "The Reflection of Waterhammer Pressure Waves from Minor Losses,"Trans. Amer. SOCoMech. Engrs., vol. 87, Series D, June 1965.

Miyashiro, H., "Waterharnmer Analysis of Purnp Systern," Bull. Japan Soco of Mech. Engrs.,vol. ID, No. 42,1967, pp. 952-958.

Combes, G. and Zaoui, J., "Analyse des erreurs introduites par l'utilisation pratique de lamethode des caracteristiques dans le calcul des coups de belier ," La Houille Blanche, vol.22,No. 2, 1967, pp. 195-202.

Fox, J. A., "The Use of the Digital Computer in the Solution of Waterhammer Problems,"Proc.Tnstitution of Civil Engrs., Paper 7020, vol. 39, Jan. 1968, pp. 127-131.

Brown, F. T., "A Quasi Method of Characteristics with Application to Fluid Lines withFrequency Dependent Wall Shear and Heat Transfer," Amer. Soco of Mech. Engrs., PaperNo. 68-WAI Aut.-7, Dec. 1968.

Fox, J. A. and Henson, D. A., "The Prediction of the Magnitudes of Pressure TransientsGenerated by a Train Entering a Single Tunnel," Proc., Institution of Civil Engrs., Paper7365, vol. 49, May 1971, pp, 53-69.

Streeter, V. L., "Unsteady Flow Calculations by Numerical Methods," Amer. Soco of Mech.Engrs., Paper No. 71-WA-FE-I3, Nov. 1971,9 pp.

Sheer, T. J., "Computer Analysis of Waterhammer in Power Station Cooling Water Sys-tems," Proc. First International Conference on Pressure Surges, published by British Hy-dromechanics Research Assoc., Sept. 1972, pp. DI-1-DI-16.

Martin, C. S., "Method of Characteristics Applied to Calculation of Surge Tank Oscilla-tions," Proc. First International Conference on Pressure Surges, published by BritishHydromechanics Research Assoc., Sept. 1972, pp. EI-I-EI-12.

Proceedings, First Intemational Conference on Pressure Surges, Canterbury England, Sept.1972, published by British Hydromechanics Research Assoc., England.

Lai, C., "Sorne Computational Aspects of One and Two Dimensional Unsteady Flow Sirnu-lation by the Method of Characteristics," Internat, Symposium an Unsteady Flow in OpenCharlnels, Ncwcastle-upon-Tyne, England, published by British Hydromechanics ResearchAssoc., April 1976.

Wylie, E. B. and Streeter, V. L., "One-Dimensional Soil Transients by Characteristics,"Proc., 2nd International Conference on Numerical Methods in Geomechallics, VirginiaPolytechnic Institut::, Virginia, June 1976.

CHAPTER 4

TRANSIENTS CAUSED BYCENTRIFUGAL PUMPS

4.1 INTRODUCTlON

The starting or stopping of pumps causes transients in pumping installations. Toanalyze these transients, the method of characteristics presented in Chapter 3may be used. Since the pumping head and flow depend upon the pump speed,transíent-state speed changes have to be taken into consideration in the analysis.For this purpose, special boundary conditions for the purnp end of a pipelinehave to be developed.

In this chapter, the analysis of transients caused by various purnp operations ispresented. A procedure for storing the pump characteristics in a digital computeris outlined, boundary conditions for a pump end are developed, and a typicalproblem ís solved. Design criteria for designing pipelines are then presented, andthe chapter concludes by a presentation of a case study.

4.2 TRANSIENT CONDITIONS CAUSED BY VARIOUS PUMPOPERATIONS

During a pump start-up, the discharge valve is usually kept closed to reduce theelectricalload on the pump motor; and as the pump specd reaches the rated speed ,the valve is gradually opened , Usually, in a normal purnp-stopping procedure,the discharge valve is first closed slowly, and then the power supply to the pumpmotor is switched off. Transients caused by both of these operations may beanalyzed by using the boundary conditions developed in Chapter 3 since thepump speed remains almost constant during the transients in the piping systern.However , if the purnps are not started or stopped as previously outlined, thenprocedures outlined in this chapter should be used for the transient analysis.

Transients caused by emergency pump operations (e.g., sudden power failure)are usually severe, and the pipeline should be designed to withstand positive and

74

Transients Caused by Centrifuga! Pumps 75

negative pressures caused by these operations. Following a power failure, thepump speed reduces since the purnp inertia ís usually small compared to that ofthe liquid in the discharge line. Because the flow and the pumping head at thepump are reduced, negative pressure waves propaga te downstream in the dis-charge line, and positive pressure waves propaga te upstream in the suction line.Flow in the discharge line reduces rapidly to zero and then reverses throughthe pump even though the latter may still be rotating in the normal direction. Inthis condition (i.e., when there is reverse flow through the pump while it is rotat-ing in the normal direction), the pump is said to be operating in the zone 01en-ergy dissipation. Because of the reverse flow, the purnp slows down rapidly,stops momentarily, and then reverses, i.e., the pump is now operating as a tur-bine. The purnp speed increases in the reverse direction until it reaches the ron-away speed. With the increase in the reverse speed, the reverse flow through thepump is reduced due to choking effect, and positive and negative pressure wavesare produced in the discharge and suctíon lines, respectively.

If the pipeline profile is such that the transient-state hydraulic grade line fansbelow the pipeline at any point, vacuum pressure may occur, and the water col-umn in the pipeline may separate at that point. Excessive pressure will be pro-duced when the two columns later rejoin. During the design stages, the possibilítyof water-column separation should be investigated, and, if necessary, remedialrneasures should be taken. This wíll be discussed in detail in Chapter 9.

4.3 MATHEMATICAL REPRESENTA TION OF A PUMP

As discussed in Chapter 3, the relationship between the discharge, Q, and thepressure head, H, at the boundary must be known in order to develop the bound-ary conditions. The discharge of a centrifugal pump depends upon the rotationalspeed, N, and the pumping head, H; and the transíent-state speed changes dependupon torque , T, and the combined moment of inertia of the pump, motor, andliquid entrained in the pump impeller. Thus, four variables-namely, Q, H, N,and T-have to be specified for the mathematical representation of a pump , Thecurves showing the relationships between these variables are called the pumpcharacteristics. Various authors have presented these curves in different graphi-cal forms suitable for graphicall-4 or computerS-8 analysis. Of all the rnethcdsproposed for storing pump characterístics in a digital computer, the method usedby Marchal et a1.6 appears to be the most suitable and is used herein.

Although pump-characteristics data in the pumping zone are usually available,HUle data, except that presented in Refs. 4 and 8, are avaílable for either the zoneof energy dissipation or the zone of turbine operation. If the complete character-istics data are not available, then the characteristics of a pump having aboutthe same specific speed rnay be used as an approximation.


Data for prototype pump characteristics are obtained from model test resultsby using homologous relatíonshíps." Two pumps (or turbines) are consideredhomologous if they are geometrically similar and the streamflow pattern throughthem is also similar. For homologous pumps, the following ratio s are valid

HN2D2 == Constant

(4.1)and

NQD3 == Constant

in which D = diameter of impeller. Since D is constant for a particular unit, itmay be included in the constants of Eq, 4.1, i.e.,

HN2 = Constant

and (4.2)

NQ = Constant

Equation 4.2 may be nondimensionalized by using the quantities for the ratedcondition as reference values. Let us define the folIowing dimensionless variables:

Qv=-QR

Hh==-

HR

Na=-

NR

T~=-TR

In this equation, T == torque and the subscript R designates the value of the vari-ables for the rated conditions. On the basis ofEq. 4.2, Eq.4.3 may be written as

(4.3)

~ = con",nt}- == Constantv

(4.4)

Transients Caused by Centrifugal Purnps 77

TABLE 4.1. Zones of pump operation.

Zone of Operation et Range of (Ju

+ ++

00.. (J" 900

900.. (J .. 1800

1800.. o .. 2700

PurnpEnergy dissipationTurbíneTurbine energy

dissipation + 2700.. e .. 3600

Since a becomes zero while analyzing transients for all four zones of operation,h/a2 becomes infinite. To avoid this, the parameter h/(a2 + v2) instead of h/a2

may be used. *The signs of v and a depend upon the zones of operation. In addition to the

need to define a different characteristic curve for each zone of operation, olv be-comes infinite for v == O. To avoid thís, a new variable (J may be defíned? as

(J == tan"! ~v

II

, I(4.5)

and then the characteristic curve may be plotted between e and h/(a2 + v2).

By definition, e is always finite, and its value varies between 00 and 3600 for thefour zones of operation (see Table 4.1).

Similar to the pressure-head curve, the torque characteristic curve may beplotted between f3/(cx2 + v2

) and O.Using the data presented by Thomas.P characteristic curves for pumps having

specific speed** of 25,t 147, and 261 SI units (1276,7600, and 13,500 gpmunits, respectively) are presented in Fig. 4.1 and in Appendix E.

4.4 BOUNDARY CONDITIONS FOR PUMP F AILURE

As discussed in Chapter 3, the characteristic equation (equations ¡[the boundaryhas pipes on both the upstream and downstream sides) and the conditions im-

*MarchaJ et al. suggest that sgn (h)J¡h¡/(et2 + u2) be used to increase accuracy for smallervalues of this parameter (sgn designates sign of h). However, h/(et2 + u2) is used hereinbecause it simplifies the derivation of the boundary conditions for the pump end (see Sec-tion 4.4).

ni} **Specific speed = NR.JQR/HR 3/4. In SI units, NR is in rprn , QR is in m3/s, and HR is inm; in gpm units, NR is in rpm, QR is in gprn, and HR is in ft. For a double-suction purnp,QR is divided by 2 while computing the specific speed.+Sorne authors erroneously use a specific speed of 35 SI units for this pump. As the purnphad a double suction, rated discharge should be divided by two to compute the specificspeed (see Closure of Ref. 8, p. A-124 and A-127).

\r:-"~

I ),


r;.lRt~(~~

t "...... ,~,~_......~._.,

".r 1..-\ :!.}4

Yi- [fl",j ~[L,,'.IJ-----l...l!S,íJt,.

],

Z,(J

~I~ JI

~1.

u..'"

-1.

-Z,

2.

ji

l.

,\r.

ea U;O ZOO Z4"'.. Z10

" '"\ ....

"\\\

IZO

(J. ton-I ': ,in deQrees

(a) Pressur«

'1..)

1...........I \.~{ ./ ;. \. . \ \I / y-N.=Z61.

I ___...,' \ \-N. = 147I ".,.' ."\

/, \ \/ . \ \./ .

ea 200120 _l60

(} • lon-I e:-

(bJ Torqutl

Figure 4.1. Characteristlcs of pumps of various speciñc speeds.

.~..i

Z70

Transients Caused by Centrifuga! Purnps 79

posed by the boundary are solved simultaneously to determine the boundaryconditions. For a pump end, the pump characteristics define the conditions irn-posed by the boundary, and a differential equation defines the variation of thepump speed with time following power failure. Thus, we have to simultaneouslysolve these equations to develop the boundary conditions for the pump end .

To facilita te understanding of the derivation, let us first consider a simple sys-tem having only one pump and a very short suction lineo We will develop theboundary conditions for more complex cases in the next section.

Equations of Conditions Imposed by Pump

As we outlined in Section 4.3, pump characteristics may be mathernatically rep-resented by curves between fJ and h/(a? + v2) and between fJ and ~/(a2 + v2), inwhich fJ = tan " «(X/v). To use these curves in a mathematical model, discretepoints on these curves at equal intervals of (J, between the range (J = O and (J =3600

, are stored in the computer. Each segment of these curves between thepoints stored in the computer may be approximated by straight lines (Fig. 4.2).If a sufflcient number of points (e.g., 73) are stored, then the error introducedby approximating the curves by segmental straight lines is negligible.

For any value of (X and v (except when both (X and vare simultaneously zero),the value of (J = tan " «(X/v) may be determined by using IBM function ATAN2.However, this function computes the value of fJ between Oand 7T and between O

(\1

>.s:. +

(\1

~

Approximalingslraíghl Iin6

Actual curv«

8 = tan-I~V

Figure 4.2. Approximation of pump characteristic curves by segmented straight lines.


and -1T, whereas our range of interest is between O and 21T. This limitation canbe circumvented by adding 21T to the computed value of e if e < O; e.g., if ogiven by this function is - 30°, then the value of o to be used for determining thepoint on the pump characteristic curve is 360 - 30 = 330°.

Let us assume that the calculation has progressed to the ;th time step; that thevariables a, v, h, and fj at the beginning of this time step are known; and that wewant to compute the values of these variables at the end of the time step. Let usdenote these unknown variables by ap, vp, hp, and fjp. To determine the valueof these variables, we have to first of all determine the equation of the segmentof pump characteristics corresponding to ap and vp. However, since the valuesof these variables are initially unknown, we may use, as a first estimate, theirvalues determined by extrapolation from the known values for the previous timesteps, i.e.,

(4.6)

in which ae and ve are the estimated values at the end of ith time step, a¡ andv¡ refer to known values at the beginning of the ith time step, and b.a¡_1 andb.V¡_1 are the variation of these variables during the (i - I)th time step. Sincethe pump speed and the pump discharge vary gradually, the preceding linearextrapolation should yield sufficientIy accurate estimates if the size of the comoputational time step, b.t, is small. Now, the grid points on either side ofO = tan "(ae/ve) are searched, and the ordinates h/(a2 + v2) and fj/(a2 + v2) for these gridpoints are determined from the stored values. From these, the constants* forthe equation of the segmental straight line are determined. Now, assuming thatthe points corresponding to ap, vp,hp, and (jp He on these straight lines, then

hp _1 ap(4.7)-2--2 = al + a2 tan

ap +vp vp

(jp -1 ap(4.8)-2 --2 = a3 + Q4 tan

ap + Vp vp

in which al and a-, and a3 and a¿ are constants for the straight lines representoing the head and torque characteristics, respectively.

Referring to Fig. 4.3, the following equation can be written for the total headat the pump:

Hp. = Hsuc + Hp - b.Hp1,1 u

(4.9)

*If y =o¡ +02X is the equation of a straight line passing through the points (x¡,y¡) and(X2' Y2), then 01 = (y ¡X2 - Y2X ¡)/(X2 - X ¡) and 02 = (Y2 - Y ¡)/(X2 - XI)'

Transients Caused by Centrifugal Pumps 81

..Q.

:J:<l

Q.:J:

u:o'":J: Oischar9~ YO/JltI

Pump

Oalum

Figure 4.3. Notation of boundary conditions for pump.

in which Hsuc = height of the liquid surface in the suction reservoir aboye datum,Hp = pumping head at the end of the time step, and b.Hpv = head loss in the dis-charge valve. Note that the velocity head in the discharge pipe, which is usuallysmall, is not taken into consideration in Eq. 4.9. The valve head loss is given bythe equation:

(4.10)

in which C¿ = coefficient of head losses in the valve. Note that in this equation,Q~. is written as Qp. 1 IQp. 1I to account for the reverse flow.

1,1 1, 1,Equations 4.7 to 4.10 represent the conditions imposed by the boundary.

,1 Differential Equation of Rotating Masses

The accelerating torque for a rotational system is equal to the product of theangular acceleration and the polar moment of inertia of the system. Since thereis no external torque acting on the pump following power failure, the decelerat-


ing torque is the pump torque. Hence,

2 dwT=-WR -dt

or

(4.11)

in which WR2 = combined polar moment of inertia of the pump, motor, shaft,and liquid entrained in the pump impeller, and w and N are rotational speed ofthe pump, in rad/s and in rpm, respectively. On the basis of Eq. 4.3, Eq. 4.11may be written as:

(3 = - WR2 2rrNR dCi60 TR dt

In this equation, TR = 60 rHRQR!(27TNR1'/R) in which r = specific weight ofliquid, and 1'/R = pump efficiency at rated conditions. By using an average valueof (3 during the time step, this equation may be written in a finite-differenceform as:

(4.12)

Cip - ci 60 TR--=-Át 27TWR2NR

(4.13)

which rnay be simplified to

(4.14)

in which

e - -15 TRÁt6 - rrWR2N

R(4.15)*

Characteristic Equation for Discharge Pipe

As the suction líne is short, it may be neglected in the analysis. Therefore, weneed only the characteristic equation for the discharge line, i.e.,for section U, 1),

(4.16)

*In English units, the right-hand sides, of Eqs. 4.11 and 4.12 have to be dívided by the ac-celeration of gravity, g. In SI units, WR2 is in kg m 2 and TR is in Nm; in the English units,WR2 is in Ib-ft2, and TR is in lb-ft. NR in both SI and English units is in rpm. In Englishunits, the right-hand side of Eq. 4.15 has to be multiplíed by g.

1 Transients Caused by Centrifuga} Pumps 83

Continuity Equation

Since there is no storage between the suction reservoir and section (i, 1)

Qpi,! = Qp

in which Qp = flow through the pump at the end of the time step.

(4.17)

Solution of Governing Equations

ro develop the boundary conditions, we have to solve Eqs. 4.7 through 4.10,4.14, 4.16, and 4.17 simultaneously. By eliminating Hp. , ÁHp , and Qp.!

l,l v 1,from Eqs. 4.9,4.10,4.16, and 4.17 and by using QR and HR as reference values,the resulting equation may be written as

QRvp = Cn + CaHsuc + CaHRhp - CaCvQh vplvpl (4.18)

Now we have four equations-i.e., Eqs. 4.7, 4.8, 4.14, and 4.18-in four un-knowns-cos, Vp, hp, (3p. ro simplify the solution, we will first eliminate hp and(3p from these equations as discussed in the following.

By substituting for hp from Eq. 4.7 into Eq. 4.18 and for (3p from Eq. 4.8 intoEq. 4.14 and simplifying, we obtain

(4.19)

(4.20)Equations 4.19 and 4.20 are nonlinear equations in two unknowns, Cip and Vp .

.These equations can be solved by using the Newton-Raphson method in which asolution of the equations is first guessed, which is then refined to a required de-gree of accuracy by successive iterations.

Let ci~) and v~) be the initially estimated values of solution, which may betaken equal to Cie and Ve as determined from Eq. 4.6.* Then, a better estima teof the solution of Eqs. 4.19 and 4.20 ís

ci~) = ci~) + Ocip

v~) = v~) + oVp

(4.21)**

(4.22)**

"Superseript (1) indicates estimated values and superscript (2) indieates values after firstiteration.**Thesc equations may be deduced from the general derivation presented on p. 91.


in whích

sr; _ F) oF2F2-

oap =oUp oUp

sr, oF2 aF) oF2---oap oUp oUp aap

(4.23)

oF¡ oF2F --FoUp =

20ap ) aap (4.24)aF¡ aF2_ sr, oF2-- -_oUp aap oap oUp

In Eqs. 4.23 and 4.24, functions F¡ and Fl and their derivatives wíth respect toap and up are evaluated at a~) and u~¡). Differentiation of Eqs. 4.19 and 4.20yields the following expressions for these derivatives:

ap)up

OlP)- -QRup

(4.25)

(4.26)

OlP)up

oF2 ( -r l OlP)-a =C6 -2a3up+a40lp-2a4uptan -Up up

If IOOlpl and loupl are less than a specífied tolerance (e.g., 0.00]), then 0l~2) andu~) are solutions of Eqs. 4.19 and 4.20. Otherwise , Ol~) and u~) are assumedequal to 0l~2) and U~2), and the aboye procedure is repeated until a solution is ob-tained. Having determined Olp and Up, it is verified whether the segment of thepump characteristic used in the computations corresponds to Olp and Up, If itdoes not, then Ole and Ve are assumed equal to Olpand Up, and the abovementionedprocedure is repeated.

However, if the correct segment was used, then hp and ~p are determined fromEqs. 4.7 and 4.8; Hp and Qp from Eq. 4.3; and Hp. and Qp. from Eqs. 4.91,1 1,1and 4.17. The values of a and u are initialized for the next time step (i.e., a = Olpand ~ = ~p), and the solution progresses to the next time step. To avoid an un-limíted number of iterations in the case of divergence of solution, a counter maybe used so that the computations are stopped if the number of iterations exceedsa specified value (e.g., 30). The flowchart of Fig. 4.4, illustrates this procedure.

(4.27)

(4.28)


NO

YES

Figure 4.4. FIowchart for boundary conditions for pump.


4.5 BOUNDARY CONDlTIONS FOR SPECIAL CASES

In Section 4.4, boundary conditions were developed for a systern having onlyone pump and a short suction lineo Because of the short length, the prop~gationof the waterhammer waves in the suction line was neglected. In this section, wewill develop boundary conditions for complex systems often found in practice.Boundary conditions for systems not covered herein may be developed by fol-lowing a similar procedure.

We will briefiy describe the system configuration, and then present the govern-ing equations and the expressions for F1, F2, aF,/aO/p• aF. /aup, "OF2/aO/p, andaF2/aup. Using these expressions, the solutions may be determined as outlinedin Section 4.4.

Parallel Pumps

Systems having paralle! pumps to which power fails simultaneously may be ana-Iyzed as follows: If the length of pipe between each pump and the dischargemanifold is long. then each pump may be handled as outlined in Section 4.4, andthe parallel piping system may be analyzed using the boundary conditions pre-sented in Chapter 3 (note that the díscharge manifold will be considered as ajunction of two or more pipes). However , if the pipe between e.ach pump a~dthe discharge manifold is short, then this pipe may be neglected In the analysis,and the combined discharge of alI pumps may be considered as the flow at theupstream side of the discharge manifold. Boundary conditions for the latter caseare developed in this section.

The continuity equation for this case is:

(4.29)

in which np = number of parallel pumps.Depending upon the length of the suction line, boundary conditions for parallel

pumps may be divided into the following two cases:

l. Short suction lineo If the suction line is short, then the waterharnmer wavesin this líne may be neglected. On the basis of Eq. 4.29. Eq. 4.18 becomes

npQRup = Cn + CaHsuc + CaHRhp - CaCuQk_ uplupl (4.30)

Equations 4.7,4.8, and 4.14 are valid for thís case as well. Proceeding similar1yas in Section 4.4, the following expressions are obtained:

FI = CaHRal(O/f, + uf,) + CaHRa2(0/f, + uf,) tan " O/p - npQRupUp

(4.31)


(4.32)

Expression for F2• aF1 /aflp, aF2 /aO/p, and aF2/aup are given by Eqs. 4.20, 4.25,4.27, and 4.28, respectively.

2. Long suction line (Fig. 4.5). If the suction line is not short compared tothe discharge line, then waterharnmer in the former has to be considered in theanalysis. Therefore, we have to inelude the characteristic equation for the suc-tion lineo Referring to Fig. 4.5,

H p = H p. I I - Hp. I1+ , ',n +

Qp¡.n+l = Cp - Ca¡Hp¡,n+1

Qp;+I,1 = Cn + Ca;+IHP;+I.l

Qp;,n+1 = Qp;+I.1 = npQp

(4.33)

(4.34)

(4.3 5)

(4.36)

In addition, Eqs. 4.7, 4.8, and 4.14 are valid for this case.By multiplying Eq. 4.34 by Ca. ,Eq. 4.35 by Ca" substituting for Qp. •

1+1 1 ',n+land Qp. from Eq. 4.36, and adding the resulting equations, we obtain

1+1,1

npQp(Ca; + Ca;+) = e.e; + CpCa;+1 + Ca/--"'a;+IHp

By using QR and HR as reference values, Eq. 4.37 may be written as

(4.3 7)

(4.3 8)

ÍIfnsfonfaneous hydrouficgrade fine

a,::t: _-

+ci:"

::t:Oischarge fine

PumpPipe i+1

I i+I,1i,n+1 Oalum

Figure 4.5. Pump with long suction line.


Elimination of hp from Eqs. 4.7 and 4.38 yields

(4.39)

in which

np(Ca· + Ca· 1 )QRC7

= 1 1+

Ca¡Ca¡+IHR

CnCa· + CpCa· 1C8

= 1 1+

Ca¡Cai+IHR

By differentiating Eq. 4.39 with respect to CXpand vp, we obtain

aF1 -1 CXp-él == 2a1CXP + 2alCXp tan - +a2Vp

CXp Vp

aF1 ~l CXp- == 2a1 vr + 2a2Vp tan - - a2cxp - C7a~ ~

Equations 4.20, 4.27, and 4.28 define the expressions for F2, aF2/acxp, andaF2/élvp.

(4.40)

(4.41)

(4.42)

(4.43)

Series Pumps (Fig. 4.6)

If the pipe length between the two pumping-sets is long, then each pumping-setmay be analyzed individually assuming the downstream pumps have a long suc-

~----,t:i Inslonloneous hydroutic-""""9<1:L._..L_ grade tine

...:

Suclion line

+ir

::t:

Moin IpumpOischorge valve

Pipe (i+ 1)

oischorge tine

Oolum

Figure 4.6. Notation for series pumps.


tion lineo However, if the pipe length between the pumps is short, then this pipemay be neglected in the analysis, and the combined boundary conditions forboth the pumping units may be developed as discussed in the following.

Referring to Fíg. 4.6, the following equations may be written for the system:

l. Pumping head

Hp¡+I.1 =Hp¡.n+1 +Hpb +HPm -IlHpv

2. Continuítyequations

(4.44)

Qp¡.n+1 = npQpb

Qpb = QPm

Qpi+1 •• = npQpm

3. Positive characteristic equation for suction line

(4.45)

(4.46)

(4.47)

Qp¡.n+1 = Cp - Ca¡Hp¡,n+,

4. Negative characteristic equation for discharge line

(4.48)

Qp¡+I.1 = Cn + Cai+IHPi+I,1

5. Equatioo for head loss ín the valve

(4.49)

IlH = e Q IQ IPv v P¡+I.I Pi+I,.

6. Equations for the pump characteristics

(4.50)

2 2 2 CXphp =a. (CI'.p +vp )+a2 (CI'.p +Vp2 )tao-· .sm.m mm m mm m VPm

(4.51)

(4.52)

Cl'.pPp = a3 (CI'.~ + v~ ) + a4 (CI'.~ + Vp2 ) tao -1 _.!!!. (4.53)m mm m mm m V

Pm

PPb = a3b (CI'.~b+ vh) + a4b (CI'.~b+ v~b) tan -1 ~ (4.54), vPb

7. Equation for the rotating masses (í.e., equations similar to Eq. 4.14)

CY.Pm- C6mPPm = Cl'.m + C6mPm

cxPb - C6bPPb = Cl'.b+ C6b{Jb

(4.55)

(4.56)


In the preceding equations, subscripts b, m, and v refer to the booster and mainpump and to the valve, respectively; np = number of pumping-sets connected inparallel; and Cu = coefficient ofhead loss in the valve.

To solve these equations, let us first reduce the number of unknowns from 13to three as follows:

El' . ti f H Q H Q Qp and Hp from Eqs. 4.44uruna Ion o P¡,n+l' P¡,n+l' P¡+I,I' P¡+I,I' b' uto 4.50 yields

UsingHRm, HRb, and QRm as reference values, Eq. 4.57 may be written as

npQR npQRmhpmHRm + hPbHRb = e m vPm + Vp

a¡+1 Caí m

By substituting expressions for hpm and hpb from Eqs. 4.51 and 4.52 into Eq.4.58 and simplifying the resulting equation, we obtain

(4.59)

Note that in Eq. 4.59, we have replaced VPb by vPm since both are equal asthe number of main and booster pumping-sets are equa!.

By eliminating (3Pm from Eqs. 4.53 and 4.55 and (3Pb from Eqs. 4.54 and 4.56,we obtain

F2 = ap - C6 [a3 (a~ + v~ ) + a4 (a~ + v~ ) tan -1 apm]

m m m m m m m m vPm

Transients Caused by Centrifugal Purnps 91

(4.61)

Now we have three nonlinear equations (Eqs. 4.59-4.61) in three unknowns,apm, vPm' and apb' ro solve these equations by the Newton-Raphson method,we have to obtain a solution of the following equations:

(aFI aFI aFI )(1)

-- oap + -- ovp + -- oap = -Ff!)aapm m aVPm m aapb b

(aF2 aF2 aF2 )(1) (1)-a-- oap + -a- ovp + -a- oapb = -F2aPm m vPm m apb

(aF3 aF3 aF3 )(1) (1)-- oap + -- ovp + -- oap = -F3aapm m aVPm m aapb b

(4.62)

(4.63)

(4.64)

In these equations, the functions FI, F2, and F3, and their derivatíves, are eval-uated for the estimated values of a~l) , V~l) , and a~l), and a better estimate ofm m bthe solution is determined from the following equations:

a(2) = a(1) + oaPm Pm Pm

V(2) = V(l) +OVPm Pm Pm

a(2) = a(l) + oaPb Pb Pb

As before, the superscript in the parentheses refers to the number of the itera-tion. The expressions for the derivatives obtained by differentiating Eqs, 4.59through 4.61 are

(4.65)

(4.66)

(4.67)

(4.68)

(4.69)

3F3 -1 flPb-3u = -2C6ba3bVPm - 2C6ba4bUPm tan -- + C6ba4bO/Pb

Pm VPm

If lócxp 1, lócxp 1, and lóup 1, obtained by simultaneously solving Eqs. 4.62m b m ) (2) (2)

through 4.64, are less than a specified tolerance (e.g., 0.001 , then cxPm' vPm'

and CXp(2)are solutions of Eqs. 4.59 through 4.61: otherwise, cx~) ,v~) ,and CX~lb)b m m

are assumed equal to cx~~, u~%, and cx~~, and the aboye procedure is repeateduntil a solution is obtained. Then, it is verified whether the segment of pumpcharacteristics used in the computations corresponded to CXp and Up. If it doesnot then CXe 'O/e , and Ve are assumed equal to CXp(2), cx~b)' and v~) , respec-

, m b m m mtively, and the aboye procedure is repeated; otherwise, the values of the rernainingvariables are obtained from Eqs. 4.44 through 4.56, and the solution progressesto the next time step.

To avoid an unlimited number of iterations in the case of divergence of itera-tions, a eounter should be used so that the eomputations are stopped if the nurn-ber of iterations exeeeds a specífíed value (e.g., 30).


aF2 C -} cx.Pm-- = 1 - 2C6 a3 cx.p - 2 6 04 cx.p tan3m m m m m m

cxPm

4.6 EXAMPLE


(4.70)Indiol sleody stateHydrolllic grade line

(4.71)

t;;1:) t. Two pllmps10 Reseryoir

"o::J:

(4.72)

(4.73) L .. 450mO = 0.75mO = 900 m/sec.f = 0.01

L ,. 550mO = 0.75mO = uoom/sec.f = 0.012

Qo = 0.5 m3/sec.

(4.74)

(4.75)

Pump 0010(4.76)

QR = 0.25 m3/sec.HR = 60 mNR = 1100 rpmWR2 = 16.85 kg - m2 per pump

Pump efficiency ot roted conditions 0.84

Figure 4.7. Piping system.

transient-state conditions are caused by simultaneous failure of power to bothpumps.

A computer program (Appendlx C) was developed using the boundary condi-tions derived in Section 4.5 for Parallel Pumps and the flowchart shown in Fig.4.4. The method of characteristics discussed in Chapter 3 and the boundaryconditions for the reservoir and series junction were used to analyze the transientconditions in the discharge lineo The waterhammer wave velocity for various sec-tions of the discharge line was determined using the equations presented in Sec-tion 2.6. The pump-characterístics data for N, = 25 SI units (1276 gpm units)shown in Fig. 4.1 were used in the analysis. At rated discharge and rated pumpspeed, the pressure head at the upstream end of the discharge line would be equalto the rated head. Starting with this flow and pressure head at the upstream

To illustrate the use of the aboye procedure, the piping system shown in Fig.4.7is anaIyzed. lnitially, both pumps are operating at rated eonditions, and the

94 Applied Hydraulic TransientsTransients Caused by Centrifuga! Purnps 95

end, the steady-state conditions in the discharge line were determined , Then,the power was assumed to faíl, and the resulting transient conditions were corn-puted. As the inertia of the liquid between the pump and the discharge manifoldwas small, the discharge of both pumps was lumped together and consídered asthe flow at the upstream end of the system.

Computed results are presented in Appendix C.

The pressure rise during a start-up may be reduced by having a slow start-up,This can be done by increasing the WR 2 of the pump motor, by reducing voltage,or by having a part-winding start. The overall economy of decreasing the maxi-murn pressure to reduce the pipe-wall thickness by these methods should be in-vestigated prior to their selection.

4.8 DESIGN CRITERIA FOR PIPELlNES4.7 PUMP START-UP

In some piping systerns, there is no control valve downstream of the pump; there-fore start-up procedures outlined in Section 4.2 cannot be used. Pump start-upin such installations may produce very high pressures, especially if the motor isof the induction type and is started across the line (i.e., without reducing thevoltage).

The transients caused by a pump start-up may be analyzed by selecting a start-up time, Ts' and by assuming that the pump speed increases linearly from zeroto the rated speed in time Ts. The motor manufacturer can supply the time takenby the motor to reach the rated speed. The time specified by the motor rnanu-facturer should be deereased by about 30 pereent 11 to obtaín a value for Ts.

Since the pump speed is known (it is assumed to inerease from zero to NR' intime Ts)' the data for the torque characteristics and moment of inertia, WR2, ofthe pump-rnotor are not required for the analysis. The pumping head and thepurnp discharge may be computed as outlined in the following.

Estimate the nondimensional purnp discharge, Ve' at the end of the time stepby extrapolation from the known values of V for the previous time steps. Fromthe pump characteristics, determine hp for the known value of Ci.p and for the esti-mated value of the pump discharge , Ve. Then, Hp = hpHR, in which the sub-

1,1

script (I, 1) refers to the first section on the discharge line just downstream ofthe purnp. Now, using thís value of Hp, l' compute the discharge at section(1, 1) from the negative characteristic equatíon (Eq. 3.19),

Once the layout and dimensions of a piping system have been selected, the rnaxi-mum and minimum pressures for various operating conditions can be determinedby using the procedures outlined in Sections 4.4, 4.5, and 4.7. In the safest de-sign , a11cornponents of the system would be designed for the possible máximumand minimum pressures with a liberal factor of safety. Such a design would,however, be uneconomícal. Therefore, a factor of safety is chosen dependingupon the risks and the probability of occurrence of a particular operating condi-tion during the Jife of the projeet, i.e.; the higher the probability of occurrence,the higher is the factor of safety .

Based upon the frequency of occurrence, various operating conditions may bec1assified as normal, emergency, or catastrophic. A diseussion of the operatingconditions included in each of these categories and the recommended factors ofsafety ' 2 follows.

Normal

Qp = e; -c;«;1,1 1,1

and then determine Vp = Qp /(npQR) in which np = number of parallel pumps.1,1

If Ivp - ve I~ e, In which e is a specified toleranee (e.g., 0.001), proceed to thenext time step. Otherwise, assume ve equal to the mean of the computed va!ueof vp and the estímated value, ve' during the previous iteratíon, and repeat theproeedure.

If the discharge line is under a statie head prior to the pump start-up, then therewilI be no flow into the discharge line until the pumping head exceeds this sta tichead. This condition can be included in the aboye analysis by assuming thatQp = 0.0 until Hp exceeds the static head.

1,1 1,1

(4.77)

Al! those operations that are likely to oceur several times during the !ife of thepumping system are terrned normal. Appurtenances or devices Ce.g., surge tanks,surge suppressors, and air valves) províded in the system to reduce severe tran-sients are assumed to be properly designed and to function as desígned duringthese operations,

The following are considered to be normal operatíng condítíons:

l. Autornatic or manual starting or tripping of pumps throughout the entirerange of pumping head. If there is more than one pump on the line , all aretripped simultaneously; however, only one rnay be started at a time.

2. If a check valve is present near the pump, it closes instantly upon flowreversal,

3. A surge tank does not drain and thus adrnit air into the pipeline, and it doesnot overflow unless an overflow spillway is provided.

4. If there is an air charnber, it is assumed to have a mínimum air volurne duringa power faílure.


As a result of any of the aboye operations, the water column does not separateat any point in the pipeline. However, if the water-column separation doesoccur, then appurtenances such as air charnbers, surge tanks, etc. should be pro-vided to avoid it. But, if it is impractical or too costly, then special devices willbe provided to minimize the transient prcssures when the columns subsequentlyrejoin.

A factor of safety of three* based on the ultimate bursting strength of themernber and a suitable factor of safety against collapse are recommended for thetransient pressures caused by normal operations.

Emergency

The emergency operating conditions in pumping systems are those in which oneof the pressure-control devices malfunctions during power failure. These condi-tions in elude :

l. One of the surge suppressors, surge tanks, or relief valves is inoperative.2. Closure of one of the check valves provided for shutting off return flow

through the pumps is delayed and occurs at the time of maximum reverse flow.3. Air-inlet valves, if present in the system, are inoperative.

Since the probability of occurrence of these conditions is rather srnall, a factorof safety of two based on the ultimate bursting or collapsing strength issuggested.

Catastrophic

Catastrophic conditions are those in which the protective equipment rnalfunc-tions in the most unfavorable manner, such as loss of all air in the air chamber,very rapid abnorrnal opening or elosing of a val ve or a gate, and pump-shaftfailure. Because the probability of occurrence of any of these conditions isextremely remote, a factor of safety of slightly more than one, based on theultimate bursting or collapsing strength, may be used.

4.9 VERIFICATION OF MATHEMATICAL MODEL

The transient-state prototype test data obtained on the Wind Gap Pumping Plantby the Mechanical Design Unit of the Departrnent of Water Resources, State ofCalifornia, Sacramento, was used to verify a mathematical model based on the

* A factor of safety of four is recommended in Ref. 12.

Transients Caused by Centrifuga] Purnps 97

boundary conditions developed in Sections 4.4 and 4.5. In this section, plantdata are first presented; the tests, instrumentation, and mathematical model arethen briefly described. This is followed by a comparison of the computed andmeasured results.

Plant Data

The Wind Gap Pumping Plant has five pumping units: three small units (Nos.1-3) and two Iarge units (Nos. 4, 5). Since test data for the large units only wereused for verification purposes, we will only list the parameters of these units andof theír discharge line. The large units (Nos. 4, 5) are manifolded together into a3.81-m-diameter pipe. Each unit has a combined pump-motor WR 2 of 99) 366kg m", and is rated at 17.84 m3/s at a total head of 159.7 m when operatingat 360 rpm. The specific speed of the pump is 33.8 (SI units). The pipeline is628 m long and varies in thickness from 11 to 27 mm. There is a discharge valveon the downstream side of each pumping unit. This valve closes in 22 ± 2 sfollowing power failure lo the unit. The minimum and maximum total pumpingheads are 159.06 and 160.03 m, respectívely, and the friction loss in the pipelinecorresponding to a flow of both units is 1.8 m. To preven! backflow from thedownstream canal, a siphon is provided near the downstrearn end of the pipeline,with the siphon having an air valve at its topo This valve opens as soon as powerfails to the purnp-rnotors.

Tests and Instrumentation

S¡ngle- and multiple-unit tests were conducted on both the small and large units.Runaway tests were conducted by subjecting the units to simulated powerfailure, with the discharge-valve elosure delayed until after the units had reachedthe steady-state runaway speed.

The straín-gauge-type pressure transducers were used to measure the transient-state pressures on the upstream and downstream sides of the discharge val ve . Avalve position transducer (displacement) and an rpm (analog) transducer wereinstalled on all units tested to record the díscharge-valve closure and the unitspeed.

Mathematical Model

A computer program was developed based on the boundary conditions for thepump end derived in Sections 4.4 and 4.5 and on the flowchart of Fig. 4.4. ThecJosure of the discharge valve following power failure may be incJuded in the


analysis, if desired. The pumping stations may have several parallel pumps, andthe purnps rnay have long or short suction lines. To compute the transient con-ditions in the pipeline, the rnethod of charaeteristies of Chapter 3 and the bound-ary eonditions for the downstream and upstream reservoirs and for the seriesjunetion, derived in Section 3.3, were used in the programo

Comparison of Computed and Measured Results

The computed and measured results were eompared for the tests simulatingsimultaneous power failure on Unit Nos. 4 and 5 with and without dosure of thediseharge valve. Results for the case when the diseharge valve remained open arepresented in Fig. 4.8a, and results for the case when the valve was closed areshown in Fig. 4.8b. As can be seen from the figure, the agreement between the

1.3 • 00

1\".:r-,

r\ =:/~·).-__"V -,'-- _-__,...)~ f~-_ "':::

"_ ...../

I\__1\

j ",.,.'N_/k--

\ V v

I 1\\ 1/

'\'1,..,..,.--~"",p 6P •• 4.\. II'H.I"'~~,v .V¡ \\

\ i\ I '\v :::;.,:;~~

~ ~f\.

<, ~ ?- ---.,

1.1 aoo

1.0 100

0.9

..L

o ':'o'"'"::;

-100 ...."L

o .•

0.7 -zoo

o .• -lOO

-'100

o.•o

'!IOOZ210 l. l.12 l.

TIME ArTER PQW[R rAILUAE - SEC

(a) Discharge valve remained apen

Figure 4.8. Wind Gap Pumping Plant. Comparison of computed and measured results,

Transients Caused by Centrifuga¡ Purnps 99

1.2

r.1.1 "-

~ ~/ ,-, .... ,. "<Id,

r ~¡::_- ...--_K\.0 r-'~w /' '-...,._.1(.) IIf#II"lJrwI

9

1/), 1

8- \ i\\ / '~

_PuIllP.lPNtI,"'.(I$Id~d

~~i._.7

Ctllftpv/,d

\V \\\

V

\~

-

'~ f..- .....-__ -_ -- __

,~ ..- -o 2 • • • 10

00

'00

lOO

aoo

::oL

"oo e

:'oL2:>

100 L

...; o.~.."

o.•200

o.•

12 14 ,.

TIM[ '''TER POWEA FAILUR[ .. SEC2Z

,. ao

(b) Discharge valve closed gradually in 22 seconds


computed and measured results is satisfaetory. However, the eomputed mini-mum pressure is lower than the measured minimum pressure. This difference is~ost probably due to the operation of the siphon valve, which was not simulatedin the mathematical mode!.

4.10 CASE STUDY

The hydraulie transient studies.P carried out by the Hydroeleetric Design Divi-sion of British Columbia Hydro and Power Authority for the water supply con-sultant* during the preliminary design of the makeup and cooling-water supplysystem for the Hat Creek Projeet of the Authority, are presented in this seetion.

*Sandwell and Company Limited (Sandwell), Vancouver. British Columbia, Canada.


Water-Supply System

As presently planned , the water-supply system (see Fig. 4.9) for pumping waterfrom the Thompson River to the plant reservoir would be comprised of an 800-rnm-diameter buried pipeline, approximately 23 km long; a pumping stationwith five pumping units at the river intake; two booster stations, each with fourpumping units and a free-surface suctlon tank; and a reservoir near the powerplant. Each booster station would have the free-surface tank on the suction side.The average and máximum discharges would be 0.725 and 1.60 m3/s, respec-tively, and the maximum total sta tic !ift from the river intake to the plantreservoir would be 1083 m. The river íntake would be located on the right bankof the Thornpson River, 2.4 km northeast of Ashcroft, British Columbia.

Both booster stations have three-stage pumps, each rated at 0.4 m3/s, 670m, and 3580 rpm. The specific speed of each pump is 39.2 (SI units}, and themoment of inertia of purnp, motor, shaft, and entraíned water in the irnpelleris equa! to 62 kg m", If required, total inertia for each unit can be increased to420 kg m2 without exceeding the limits set by the pump start-up time. Thepump manufacturer supplied the pump characteristics for the normal zone ofpump operation only. Since no data were available for the other zones and sincethese characteristics agreed closely with those of Fig. 4.1 for Ns = 25 (SI units),the characteristics of Fig. 4.1 were used for all zones of operation.

Analysis

Computer Program

A computer program fOTanalyzing the transient conditions in a pipeline causedby power failure and/or valve operation was developed. The boundary condi-tions and solution procedures presented in Chapters 3,4, and 10 were used tosolve the characteristic form of the dynarnic and continuity equations. ro avoiderrors introduced by ínterpolations, wave velocities were adjusted slightly, ífnecessary, so that the characteristics passed through the grid points. Becausethere is a free-surface tank 00 the suetion side of each booster station, transientsin the diseharge Une were analyzed neglecting the effects of transients in thesuetion line.

The program was verífied by cornparing the eomputed results with those mea-sured on a prototype (see Section 4.9) or those obtained by using other avail-able , símpler , problem-oriented eomputer prograrns.

Selection of Control De vices

The procedure outlined on p. 102 was used to select appropriate waterharnrnercontrol devices:

lf~.UJ /lDMUJOJ

e ·ON UO.'IDIS J~/S008

~I ~I §I 81 gl ~I ~I ~I °1S3!:!.l3W - NOI.l1t1\313


l. Column separation. The system was analyzed for the case of simultaneouspower failure to all purnps, assuming there were no control d.evices. Water-column separation occuredin the pipeline between Booster Stations No~ .. l and2 and in the pipeline downstream of Booster Station No. 2. The provision ofadditional inertia at the pumps and one-way surge tanks prevented columnseparation. The data for these devices are listed subsequently. . .

2. Maximum pressure. lt was assumed during the initial design of the pipelinethat with appropriate control devices, the maximum pressure rise* at the pumpend could be limited to 10 percent of the rated head. With check valves locateddownstream of the pumps, the pressure rise following power failure exceeded.lOpercent. However, it could be reduced to less than 5 pereent by slowly closingthe pump-discharge valves.

Results

The maximum and mínimum hydraulic grade lines following power failure areshown on Fig. 4.9 for the system containing suitable control equipment.

Column Separation

The following control devices would successfully prevent column separation inthe various segments of the pipeline: ..

l. Pipeline from Booster Station No. 1 to 2. Two alternatives are available :CA) Increase the WR 2 of each pump motor to 115 kg m2 , and provide a 4-m-diameter one-way surge tank at the top of Elephant Ridge with the steady-statewater level in tank at El. 627 (10m aboye the ground surfaee); and (B) Inereasethe WR 2 of each pump motor to 390 kg m2. With these controls, the minimumpressures in the pipeline remain aboye atmospheric pressure.

2. Pipeline Downstream of Booster Station No. 2. lncrease the WR 2 of eachpump motor to 370 kg m2

; provide a 4-m-diameter one-way surge tank alStation 1 I+175 with steady-state water level in the tank at El. 1252 (10m aboyeground level); and provide a 4-m-diameter one-way surge tank al Station 17+480with steady-state water level in the tank at El. 1345 (25 m aboye ground level).

With these measures, the minimum pressures along the pipeline were aboyeatmospheric pressures.

Maximum Pressures

The pressure rise following power failure could be reduced by slowly closing t~epump-discharge valves. With the closing times of about 100 s, the pressure nse

*Pressure rise " Maximum transient state pressure - Steady-state pressure,

Transients Caused by Centrifugal Purnps l03

at the pump following power failure was less than 5 percent of the rated head.A single rate closure was assumed in these computations. The rnaxímum reversopump speed following power failure for the cases when the discharge valveremained open and when the discharge valve was closed was less than the follow-ing máximum permissible limits specified by the purnp manufacturer: 130 per-cent of rated speed for Iess than 30 s and 120 percent of rated speed for longerperiods.

Emergency Conditions

As an emergency condition, the discharge valves were assumed to remain openfol1owing power failure. Because of the higher than normal inertia, the rnaxi-mum pressure at the pump in al1 cases remained less than the steady-statepressure.

Discussion

The aboye results were obtained using assumed friction factors and assumedpurnp characteristics. In addition, both the topographic information and thedata for the discharge valves were not precisely known. As the design operatingeonditions for each pumpíng station were different frorn the specified rated con-ditions for the pump, the water level in the suction reservoir had to be artificiallylowered to obtain the correct downslream head for the given pump speed ,Artificial lowering of the suction tank should have a negligible effect on thecornputed pressures in the discharge line. However, test runs using differentpump characteristics showed that variation in the pump characteristics and/orin the data for the discharge valve could substantially change the computedmáximum and minimum pressures. Similariy, significant changes in the groundlopography would change the hydraulic grade line relative to the pipeline, thuspossibly resulting in situations where column separation could oeeur. A differ-ence in the friction losses could a1so affect the rnaxímum and minimumpressures.

With the available data, the maximum pressures at the purnp could be keptbelow 5 percent of the rated head. However, as discussed, the pressure rise maybe higher due to significant variations in the data [or the system. If necessary,the pressure rise could be decreased by increasing the discharge-valve closingtime, which would result in an increase in the time period for which the pumpruns in the reverse direction. Although the máximum reverse pump speed waswithin the Iimits specifíed by the pump manufacturer, reverse flow through thepurnps for an extended period may partially draín the pipeline at high points.Therefore , it was recommended that, until better data were available and a sensi-


tivity analysis of the effects of changes in the variables affecting pressure rise wasmade, the maximum pressure rise at the pump end should be taken equal to 10percent of the rated head , and the elevation of the maximum hydraulic gradeline shown in Fig. 4.9 should be adjusted proportionately.

With the specified control measures, the minimum hydraulic grade line wasalways aboye the pipeline. At Elephant Ridge and at the summits downstreamof Booster Station 2, the minimum hydraulic grade line was less than 5 m aboyethe pipeline. During the final design, however, when better data should be avail-able, this should be investigated in detail; if necessary , the safety margin couldbe increased.

Air valves should be provided at high points along the pipeline. These wouldbe helpful during filling and draining of the line and would prevent collapse of along length of the pipeline should a break occur in the pipeline at a lower eleva-tion. In addi tion, valves could be provided along the line to isolate and drainsegments of the line for inspection , repair, etc. Transients caused by the opera-tion of these valves, if provided, would be studied during the final design.

The one-way surge tanks should have two pipes for water outflow. Thisshould considerably reduce the possibility of a tank becoming inoperative due tothe failure of a check valve to open.

Two alternatives are available to prevent column separation in the pipelinebetween Booster Stations Nos. I and 2. The alternative with increased inertiaonly is better from an operational point of view because the one-way surgetank is not as foolproof and in addition requires constant maintenance.

The inertia of the pump motors could be increased by adding flywheels orby a custom design of the electric motors. In order to provide operationalflexibility and ease in exchanging spare parts, etc., it was decided that all units atboth the booster stations should be identical and that each would have a WR 2

equal to 400 kg m2 •

4.11 SUMMARY

In this chapter, a procedure for storing the pump characteristics in a digitalcomputer was presented, an iterative procedure for analyzing transients in pipingsystems caused by various pump operations was outlined, and boundary con-ditions for a number of cases usually found in practice were developed. Criteriafor the design of pipelines were presented, and the chapter was concluded by apresentation of a case study.

PROBLEMS

4.1. Write a general-purpose computer program to determine the transient-statepressures in a discharge Jine caused by power failure. Using the pump char-

Transients Caused by Centrifugal Purnps lOS

4.3.

acteristic o~ Appendíx E, investigate the effect of increasing the value of WR 2on the maximum and minimum pressures.

Using the program of Problem 4.1, prove that the maximum pressure at thepump does not exceed 14 the pumpíng head if the friction losses are greaterthan 0.7a V ls in whích - t h .o , a - wa er ammer wave velocity V = steady-state flow velocity, and g = acceleration due to gravíty, ' o

De~elop the bo.undary conditions for a system having n parallel pumps, inWhICh power fails to nf pumps and no purnps keep operating.

Draw a flowchart for the boundary condition derivad in Problem 4.3, anddevelop a computer programo

t: reduce ~aximu~ pressures following a power faiIure, a pressure-regu-lating valve IS sometirnes provided just downstream of the pump. This valveopen~ .as the power fails and is closed slowly latero Develop the boundaryconditions Co~ such a system ; write a computer program and investigate theeffect of vanous rates of openíng and closure of the pressure-regulatingvalve.

A check valve is provided in a discharge Jine to prevent reverse flowst.hrough the pumps. When power fails to the purnp, water in the dischargehne decelera~es, and .t?e check .valve c1oses. A check valve having no dash-pot a~d having neglígible bearing friction losses closesl5 according to theequation

4.2.

4.4.

4.5.

4.6.

d20 _. (BV e dO)2 (G dO)2 (FV)2J --;¡2 - Ws' sin 0+ - + - - + - - + - = ot Kf Kd dt Kd dt Kf

in which O = angle between the center of gravity of disk and vertical' J =mome~t of inerti.3 of the disk; Ws = weight of disk in water; r = dist~cefro~ pívot to weI.ght-center o.f gravity of disk; V = mean pipeline velocity ;kf -=- flow coeffi.cl.ent for stationary disk in moving water (function of 0);kd - flow coefficient for moving disk in still water (function of O); andB, e, e, and F are constants, Expression for these constants are

_ (AR)0.25B- -p3

(A 2 P3) 0.25

e=J --R

F= (AR - J219t5


in which A = area of disk; R = distance from ~ivot to c;nte~ of_ di~k; P =distan ce from pivot to the point of concentrabon ~f f ~ dA,! - dlst~ncefrom pivot to point of concentration of moment of inertía of disk area, andr = moment arm measured from disk pivoto .

Develop the noundary conditions for the check valve, assummg that K¡and Kd are given in a tabular formo .

Write a computer program for the check valve, and run it for ~e fol~oWl~!data: J = 0.235 lb-ft-sec2; B = 0.548; C = 0.357;F = 0.11;G - ~.07, Wsr o

10.74 lb-ft ; (J = 16.10 + ex; initial steady~tate.(J and ex are 60.1 and 44 ,respectively. K¡ and K d are listed in the following:

4.7.

ex k¡ kd(degrees)

O O. 0.0

4 0.16 0.23

8 0.28 0.4012 0.40 0.4916 0.49 0.5520 0.56 0.58

24 0.62 0.5428 0.67 0.4932 0.71 0.44

36 0.71 0.3840 0.84 0.2744 0.95 0.09

Use the pipeline and pump data given in Ex. 4.1 (Section 4.6), except that

the diameter of the pipelines is 9 in.

REFERENCES. . if l P d Their Use in Predio-l. K napp R T "Complete Characterisucs of Centn uga umps an 9 1937

. ,. ". B havi "Trans Amer SOCo of Mech. Engrs .• vol. 5 • , pp.non of Transient e aviour, . .

683-689. .. S t " Trans Amer.Kitt d J P "Hydraulic Transients In Centrifuga! Pump ys erns, .2. 1 re ge , . .•Soco of Mech. Engrs .• vol. 78. 1956. ~p. 1307-132_2. . 78-81

3 P ki n J WaTerhammer AnalySlS. Dover Publícattons, 1963. pp. .

4· Darm~y a B ··..Complete Purnp Characteristics and the Effect of Specific Speeds on. ons • .• T A e Soc of Mech EngrsHydraulic Transients," Jour. Basic Engineering, rans. m r.· . .•

s. ~~:~:!~\,:PL.~8'~:~:rhammer Analysis of Pipelines." Jour. Hydraulics Div .. Proc.

Amer Soc ofCiv. Engrs .• July 1964. pp. 151-171. b. . FI h G d Suter P. "The Ca1culation ofWaterhammer Problems y

6 Marchal, M., esc, ., un ., . l S i 1m on. f h Disit l Computer" paper presented at Intemationa ympos tMeans o te' igr a ,


Waterhammer in Pumped Storage Projects, sponsored by Amer. Soc, of Mech. Engrs.,1965, pp. 168-188.

7. Streeter , V. L. and Wylie, E. B., Hydraulic Transient s, McGraw-HiII Book Co., NewYork, 1967,pp. 151-159.

8. Thomas, G., "Determination of Pump Characteristics for a Computerized TransientAnalysis," Proc. First International Conference on Pressure Surges, organized by BritishHydromechanic Research Assoc, al Canterbury, England , Sept. 1972. pp. A3-21-A3-32.

9. Streeter, V. L., Fluid Mechanics, Third Edition, McGraw Hill Book Co., New York, pp.466-468.

10. Miyashiro, H., "Waterhammer Analysis for Pumps Installed in Series." Re]. 6, pp. 123-133.

11. Joseph, (. and Hamill, F., "Start-Up pressures in Short Pump Discharge Lines," Jour.,Hyd. Div .• Amer. Soco of Civil Engrs .• vol. 98, July 1972. pp. 1117-1125.

12. Parrnakian, J., "Waterhammer Design Criteria," Jour., Power Div., Amer. Soco of CivilEngrs .• Apri11957, pp. 1216-1-1216-8.

13. Chaudhry, M. H., Cass, D. E., and Bell, W. W., "Hat Creek Project: Hydraulic TransientAnalysis of Make-up Cooling Water Supply Systern," Report No. DD 108, HydroelectricDesign Division, British Columbia Hydro and Power Authority , Vancouver, Canada,February 1978.

14. Stephenson, D.• Pipeline Design for Water Engineers, Elsevier Scientific Publishing Co.,Amstcrdam, The Nethcrlands, 1976, p. 58.

15. Parrnley, L. J., "The Behaviour of Check Valves during Closure," Research report, Glen-field and Kennedy Ltd., Kilmarnock, England, Oct. 1965.

ADDlTIONAL REFERENCES

Schnyder, O., "Comparison Between Calculated and Test Results on Waterhammer inPumping Plant," Trans. Amer. Soco ofMech. Engrs., vol. 59, Nov. 1937, pp. 695-700.

Linton, P., "Pressure Surges on Starting Pumps with Ernpty Delivery Pipes," British Hydro-mechanic, Research Assoc., TN402, 1950.

Rich, G. R., Hydraulic Transients , McGraw-HiIl Book Co., New York, 1951.Parrnakian, J., "Pressure Surges at Large Pump Installations," Trans. Amer. Soco uf Mech.

Engrs., vol. 75, Aug. 1953, pp. 995-1006.Parmakian, J., "Pressure Surge Control at Tracy Purnping Plant," Proc. Amer. Soco o] Civil

Engrs., vol. 79, Separa te No. 361, Dec, 1953.Swanson, W. M., "Complete Characteristic Circle Diagrams for Turbo Machinery," Trans.

Amer. Soco of Mech. Engrs., vol. 75,1953. pp. 819-826.Stepanoff, A. J., Centrifugal and Axial Flow Pumps, 2nd Edition, John Wiley & Son s, New

York, 1957.Jaeger, c., Engineering Fluid Mechanics, Blackie and Sons, London, 1959.Jaeger, C., "Waterhammer Caused by Pumps," WaTer Power, London, July 1959, pp. 259-

266.Miyashiro, H., ''Waterhammer Analysis of Pumps in Parallel Operation," Bull. Japan Soco of

Mech. Engrs., vol. 5, No. 19, 1962, pp. 479-484.Miyashiro, H., ''Water Leve1 Oscillations in a Surge Tank When Starting a Pump in a Pumped

Storage Power Station," Proc. International Assoc. for Hydraulic Research, London,1963, pp. 133-140.


Kinno, H. and Kennedy, 1. F., "Waterhammer Charts for Centrifugal Pump Systerns," Proc.Amer. Soc. al Civil Engrs., vol. 91, HY3, May 1965, pp. 247-270.

Duc, J., "Negative Pressure Phenomena in Pump Pipelines," Proc. Intemational Symp.Waterhammer in Pumped Storage Projects, Amer. Soco of Mech. Engrs., 1965, pp.154-157.

Miyashiro, H., ''Waterhammer Analysis of Pump Systern," Bull. Japan Saco of Mech. Engrs.,vol. 10,1967, pp. 952-958.

Kinno, H., "Waterhammer Control in Centrifugal Pump Systems," Proc. Amer. Saco a[ CivilEngrs., vol. 94, HY3, 1968, pp. 619-639.

Brown, R. J., ''Water-Column Separation at Two Pumping Plants," Jour. Basic Engineering,Amer. Saco of Mech. Engrs., Dec, 1968, pp. 521-531.

CHAPTER 5

HYDRAULIC TRANSIENTS INHYDROELECTRIC POWER PLANTS

5.1 INTRODUCTION

In Chapter 3, boundary conditions for a Francis turbine connected to a large sys-tem were derived. In this chapter, a mathematical model is developed for ana-Iyzing hydraulic transients caused by various turbine operations, su eh as start-up,load acceptance, or load rejection.

The schematic representation of a typical hydroelectric power plant is firstpresented. Details of the mathematical simulation of the conduit system,turbogenerator, and governor are then outlined. Various turbine operationsthat produce hydraulic transients in the water passages of a power plant arediscussed. Prototype test results used to verify the mathematical model arethen presented, followed by a discussion of the governing stability of hydro-turbines, and the selection of generator WR2 and optimum governor settings.The chapter concludes with the case study of the governing stability studiescarried out for a 500-MW hydroelectric generating station.

5.2 SCHEMA TIC OF A HYDROELECTRIC POWER PLANT

Figure 5.1 shows the schematic diagram of a typical hydropower plant. Asshown in the figure, the upstream conduits convey water from the upstreamsource, such as a reservoir, lake, or canal, to the turbine. Outflow from theturbine is carried downstream through the downstream conduit system. Anelectrical generator is mechanically coupled to the turbine, and the electricaloutput of the generator is carried by the transmission lines to the load centers.A governor is provided to correct any changes in the system frequency byopening or closing the wicket gates of the turbine.

109


R6S6rvoir

Transmission /in,

P6nslock

Tai/rae,Turbin.

Figure 5.1. Schematic diagram of a hydroelectric power plant.

Thus, the folIowing components have to be mathematicalIy simulated todevelop the mathematical model of a hydroelectric power plant:

1. upstream and downstream water conduits2. turbine and generator3. governor.

Details of the simulation of these components are presented in the followingsections.

5.3 UPSTREAM AND DOWNSTREAM CONDUITS

As shown in Fig. 5.1, water is carried from the upstream reservoir or canal tothe turbine scroll case through a tunnel and/or a penstock, and the outflowfrom the turbine is discharged through the draft tu be to the downstream waterpassages, which may consist of either a free-surface flow or a pressurized tunnel,a tailrace canal, a ríver, or a downstream reservoir. Depending upon the conduitlengths, surge tanks may be provided to improve the governíng characteristicsor to reduce the maximum waterhammer pressures.*

The method of characteristics and the boundary conditions presented inChapter 3 are used to simulate the upstream and downstream conduits. Ifthe downstream conduit system is comprised of a free-surface flow tunnel, anopen channel, or a short pressurized conduit, then the draft tube as well as thedownstream conduit system may be neglected to simplify the analysis.

*This will be discussed in detail in Chapters 10 and 11.

Hydraulic Transients in Hydroelectric Power Plants 1II

Boundary conditions to analyze the turbine end of the conduit system are de-veloped in the followíng section.

5.4 SIMULATION OF TURBINE

The relationship between the net head and discharge has to be specified tosimulate a turbine in a hydraulic transient model. Flow through a turbinedepends upon various parameters: for example, the flow through a Francisturbine depends upon the net head, rotational speed of the unit, and wicket-gate opening, while the flow through a Kaplan turbine depends upon thesevariables as well as the runner-blade angle. In an impulse turbine, however,the flow is a function of the head and the nozzle opening only. Curves rep-resenting the relationship between these parameters are called turbinecharacteristics.

To the author's knowledge, except for Krivehenko et al., I no data have beenreported in the literature for the turbine characteristics during transient-stateconditions. Therefore, steady-state model test results are used to plot the ex-pected prototype turbine characteristics and these are assumed to be valid duringthe transient state as well. As shown by Perkin et al.,? this is a valid assumption.

The data Ior the turbine flow and power output, obtained from the modeltests, are presented in a graphical form known as hill charts (Fig. 5.2). Theprototype efficiency is usual!y more than that determined from model tests be-cause of scale effects. Therefore, while plotting the prototype output, this factis taken into account by stepping up the model efficiency. Various empiricalformulas have been proposed for this purpose, of which the Moody formula?appears to be the most suitable.

Usually very little data are available for smal! wicket-gate openings, and, tocover this range, the characteristic curves are extrapolated. To do this, flowshould be known when the turbine rotational speed is zero, and the windageand friction losses should be known at wicket-gate openings below the speed-no-load gate (SNL). SNL gate is the lowest gate opening at which turbine rotatesat synchronous speed with zero output.

Typical characteristics for a Francis turbine are shown in Fig. 5.3. In thisfigure, the abcissa is the unit speed, ¡P, and the ordinates are the unit flow, q,and the unit power, p. Definitions of rjJ, p, and q are given in Table 5.1.

In the expressions of Table 5.1, D = diarneter of the runner; N = rotationalspeed; Hn = net head; Q = turbine discharge; and P = power output. In En-glish units, D is expressed in in., N in rpm, Hn in ft, Q in ft3/sec, and P in hp.In SI units,Hn and D are in 111, N in rpm,P in kW, and Q in m3 /s.

112 Applied Hydraulic Transients Hydraulic Transients in Hydroelectric Power Plants 113

./4

./3

./2

./1

.10

Cl. .09~'~ .08~....~ .07:::)

.06

.05

0.5

0.4CT

.;~ 0.3'"..,".'"'> 0.2....

" Runaway ;S0./

(a) Unit speed

0.0/

.0245 .50 .55 .60 .65 .70 .75 .80 o

Figure 5.2. Typical hill chart for a Francis turbine (in It-Ib-sec units),

Figure 5.3. Characleristics of a Francis turbine (in f't-lb-sec units),

At SNL gate, the turbine output is equal to the turbogenerator windage andfriction losses at the synchronous speed. Therefore, if the wicket gates aresteadily open al the SNL gate opening, the unit rota tes at the synchronousspeed, and the net turbine output is zero. The abcissa axis on the unit powercurves (Fig. 5.3b) represents the conditions at SNL gateo It is clear from thisfigure that the value of SNL gate varíes with the net head. To keep the unitrunning at the synchronous speed when the wicket gates are open less than theSNL gate openíng, power has to be supplied to the unit from an outside source

because the windage and friction losses are greater than the turbine output. Thisis called motoring of the unit.

During steady-state model tests, unit speed cannot exceed the runaway speedfor a particular net head and gate opening. Therefore, model data are ~otobtained for cp higher than cp at runaway conditions (CPrun)' However , dunngthe transient state, the prototype speed may exceed the runaway speed for ashort duration. To account for this, the curves are extended for cP values higher


Table 5.1. Definition of unit values.

English Units SI Units

DN DN

1838~ 84.45~

Q Qq

(D/12)2 ~ D2~

P PP

(D/12)2 H~/2 D2H~/2

than cfJrun assuming that they follow the same trend as that at cp values less than

cfJrun·A grid of points on the characteristic curves for various gate openings are

stored in the computer, and the unit discharge and unit power at intermediategate opening and cfJ values are determined by parabolic interpolation.

The boundary conditions for a Francis turbíne" are derived below. For aKaplan turbine, the variation of the turbine characteristics with the runner-blade angle would have to be considered. For a Pelton turbine, however, bound-ary conditions for a valve developed in Chapter 3 may be used.

Referring to Fig. 5.4,

Q;,Hp =H; + Htail - --2

2gA(5.1)

b:.~:~~,~_;jHydraulic Transients in Hydroelectric Power Plants lIS

Hllod 105$115inplln510ck

roitrac»

Dolum linll

Figure 5.4. Notation for boundary conditions Ior a Francis turbine.

extrapolation. To determine the range of cfJ for which turbine characteristicshave to be used during the time step,Hn is a1so extrapolated.

Let the values of Tp, N«, and Hn estimated by extrapolation be Te, Ni" andHile and the value of cfJ for the estimated values of N¿ and Hile be cfJe· The char-acteristics for Te for cfJ between cfJl and cfJ2 may be approximated by the straightline EF as shown in Fig. 5.5. The values at E are interpolated from the knownvalues at the grid points A and B, and the values at F are interpolated from theknown values at points e and D.

The equation of straight line EFmay be written as

in which Hp = instantaneous piezornetric head at the scroll-case entrance; Hn =instantaneous net head; Htail = tailwater level aboye datum; Qp = instantaneousflow al entrance to the scroIl case; and A = cross-sectional area of the pressureconduit at the turbine inlet. Hp, Qp, and HIl are the values of these variables atthe end of the time step under consíderatíon. Note that the velocity head atthe draft-tube exit has been neglected in Eq. 5.1 during computation of the nethead. This is a valid assumption since the exit-velocity head is usually negligible.However, if the velocity head is not small as compared to Hn' it should be in-cluded in the analysis. Let the wicket-gate opening at the end of the time stepbe Tp.

The values of four variables, namely, Qp, Hp , Tp, and Np are unknown at theend of the time step under consideratíon and may be determined by the follow-ing iterative procedure. Because the transient-state turbine speed and gate open-ing vary gradually, these can be estimated as a first approximation by parabolic

(5.2)

in which ao and al are determined from the known coordinates of E and F.Substituting for q and cfJ from Table 5.1 (SI units) into Eq. 5.2 and simplifying

(5.3)


in which

and

• Stored grid pointo Interpoloted poinl - (c C H Caa~)a6 - p - a tail - -;;r- (5.9)

Solu tion of Eq. 5.6 yields

(5.1 O)eat« opening, 1;'"

Turbine choroclenslics Note that the positive sign with the radical term is neglected. Now Hn is de-termined from Eq. 5.3 and HP from Eq. 5.1 .

Because of the instantaneous unbalanced torque , Tu = Ttur - Tgen, the speedof the turbogenerator-set changes according to the equation

T = WR2 dwu dt

e

(5.11)*

orel> 1 elle cJ>2

UNIr SPEED, .", T T - 2 2rr dNtur - gen - WR --

60 dt(5.12)*

Figure 5_5. Interpolation of turbine characteristics.in which Ttur = instantaneous turbine torque ; Tgen = instantaneous generatortorque; w = rotational speed of lhe turbogenerator , in rad/s; N = speed, in rpm;WR2 = total mornent of inertia of the turbine and generator, in kg m2• IflIg = generator efficiency, and assurníng that the load is only resistive, thenEq. 5.12 may be written as

P _ Pgen _ WR2 ( 2rr )2 dNtur --_ - N-

lIgen 60 dt

(5.4)*

(5.13)*

Combiníng Eqs. 3.18 and 5.1,

CaQ]'Qp = (Cp - CaHtail) + ? A2 - CaHn

-g

Squaring both sides of Eq. 5.3, eliminating Hn from the resulting equation andEq. 5.5, and simplifying,

in which

(5.5)

in which Pgen = generator load, and Ptur = power developed by the turbine, bothin kW.

Integrating both sides of Eq. 5.13,

ftP( p) j'NPPtur - ~ dt = 1.097 X 10-2 WR2 N dNti lIgen NI

(5.14)*

(5.6)Sirnplifying,

(p +P P +P )turl turP _ genI genP ~ t = 0.548 X 10-2 WR2 (Nj, - Nf)

2 211gen .Ca Ca

a4 = -- --2gA 2 a~

2a3Caas = --2 - -a2

(5.7)

(5.15)

(5.8)*If WR2 is in Ib-ft2, repIace WR2 of Eqs. 5.11 through 5.13 by WR2/g; if Ptur and Pgen arein horsepower, then divide the right-hand side of Eq, 5.13 by 550, and replace 1.097 X 10-2

of Eq. 5.14 and 0.548 X 10-2 of Eq. 5.15 by 0.619 X 10-6 and 0.3096 X 10-6, respectively.*ln English units, a2 = aoD2/144, and a3 = ND3al/264,672.

118 Applied Hy draulic Transients Hydraulic Transients in Hydroelectric Power Plants 119

in which subscripts I and P indicate the values of the variables al the beginningand at the end of time step. Solving for Np,

{6. t [ 0.5 1}o. s

Np:: N~ + 182.38 WR2 0.5 (Pturl + PturP) - 1)gen (Pgelll + Pgenp)J

(5.16)*

In Eqs. 5.15 and 5.16, it is assumed that the generator load is graduaJly varied.However, if the transients are caused by a step change in the generator load,then the aboye equation may be simplified as

{2 6. t [ ~ ] }o.sNp = NI + 182.38 --2 O.5(Pturl + PturP)-

WR 1)gen(5.17)*

The speed droop is a governor characteristic that requires a decrease in thespeed to produce an increase in the wicket-gate opening (Fig. 5.6). For hy-draulíc turbines, a large value of speed droop is required for stability. However,this ís not permissible from the point of víew of system operation. Therefore,two types of speed droop are provided: permanent and temporary. Permanentspeed droop is usually about 5 pereent, is fixed during a transient, and is utilizedfor sharing loads on parallel units. The value of ternporary droop may be large.It is made ternporary by means of a dashpot. .

Three types of governors are used for hydroelectric units: (1) dashpot, (2) ac-celerometric, and (3) pmportional-integral-derívattve (PID). The dashpot gove.r-nors have been more commonly used in North America, and the accelerornetricin Europe; the PlD has been recently íntroduced. In the dashpot governor, thecorrective action of the governor is proportional to the speed deviation, n; inthe accelerometric governor, it is proportíonal to dnldt ; in the PlD, it is pro-portional to n, dnldt , and time integral of n (see Problern 5.3). We will discussonly the dashpot governor (Fig. 5.7) herein.

Following a load increase (decrease), the sequence of events is as follows:The speed of the unit decreases (increases) because of the load change, and theflyballs move inward (outward). This displaces the piston of the pilot valve,and the oil is admitted into the hydraulic servo that opens (eloses) the wicketgates. As a result of the wicket-gate movernent , the dashpot spring is comopressed, whích changes the position of the pilot valve. After sorne time, thedashpot spring returns to its original position because of oil flow through thesmall orifice in the dashpot, even though the servo and the wicket gates arenow al a different position.

in which Pgenf = final generator load.A cornputatíon procedure for using Eqs. 5.10 and 5.17 ís presented in

Section 5.6.

5.5 HYDRAULlC TURBINE GOVERNORS

As discussed in Section 5.2, a governor is provided to keep the speed of theturbogenerator at the synchronous speed. The main components of a governorare a speed-sensing device and a servomechanism for opening or elosing thewicket gates.

Various mechanical and electrical speed-sensor devíces" have been used.Of the mechanical devices, a centrifugal ballhead in various configurations hasgained rnuch popularity because of its simplicity, sensitivity, and ruggedness.The electrical speed sensors include a dc generator with permanent magneticfield, a permanent rnagnet alternator, a perrnanent magnet alternator feedinginto a frequency-sensitive network , and a speed-signal generator. The output ofthese speed sensors is the deviation from the reference speed. This output isusually small and is amplified by means of a pilot valve before feeding it intothe servo-mechanisrn provided for opening or elosing the wicket gates. Sincea large force ís required lo move the wicket gates, a hydraulic servo is providedfor this purpose.

A governor, having a ballhead, pilot valve, and hydraulic servo, has onlyone equilibrium position in which the ports of the pilot valve are elosed so thatno oil is admitted into the servo. Such a governor is called an isochronousgovemor and is inherently unstable. Therefore, for stabílíty, speed droop isf·ovided.

Droop Une

5% Droop"O

:: 100Q.(/)

OL---------~-----------r----50 lOO

*I[ WR2 is in lb-ft2, and Ptur and Pgen are both in hp, then replace 182.38 in these equa-tions by 3.23 X 106. .

Wicket Gate Opening

Figure 5.6. Speed droop.


spring

Pi/o/ Oosllpo!

ro op,n go/,

Figure 5.7. Dashpot govemor (permanent speed droop not shown).

Figure 5.8 shows the block diagram for a dashpot governor. In this diagram,conventional notation of control systerns" is used with different blocks repre-senting various components of the governor. Input and output of various blocksis shown by means of arrows, and the transfer functions listed in the blocksshow the relationship between the input and output of various components.In the transfer functions, s is the Laplace variable. If initial condítions are zero ,then s is equivalent 7 to the time derivative dldt .

The differential equations for different components of the governor can bewritten using the transfer functions listed in Fig. 5.8. Readers not interested inthe derivation rnay proceed directly to Eqs. 5.31 through 5.34.

The following notation is used in the block diagram shown in Fig. 5.8:

T¿ = actuator time constantT; = dashpot time constanto = temporary speed droop0= permanent speed droop

Td = distributing valve time constantkd = distributing valve gainks = gate-servomotor gainn = normalized transient-state turbine speed.

The synchronous speed of the turbogenerator-set, NR , is used to normalize theturbine speed, i.e., n = N/NR. If 10 = initial steady-state wicket-gate opening,then Ilref = 1.0 + 010, The outputs of various governor components and theirsaturation limits are shown in Fig, 5.8. Typical values of these constants are:

Ta = 0.05 to O.ls0=0)3to0.5

0,0) Lo 1).05

Hydraulic Transients in Hydroelectric Power Plants 121

'OI;'!?...+

... -,a~ .~....

~a::. :~...~ ~...

'<t ....~... ~a.... ~ -I~c ~

'O1:) ... •~ .. CI..... ..

_'<t 'O• •...e -•

s:

..

122 Applied Hydraulic Transients Hydraulic Transienls in Hydroelectric Power Plants 123

Td = 0.05 lo 0.1 skd = 10 to 15ks = 0.2

etmax = 0.2 lo 0.5

For a given set of power planl pararneters, the optimum values 01' ó and T, maybe selected by using the procedures outlined in Section 5.11.

and

-1v :--dmin k Ts e

in which T¿ and Te are the effective wicket-gate opening and closíng times,which are defined as twice the time taken by the wicket gates to open or closebetween 25 and 75 percent openings.

(5.26)

Actuator

1v =-eaTas

Gate Servomotor(5.18)

(5.27)or

e = T dVaa dt (5.19)

or

Dashpot(5.28)

s r,,e¡ = 1 + Trs va

The followíng equations may be written because of two feedbacks(5.20)

(5.29)

or and

(5.21)Vi = va - T (5.30)

(5.22)

Note that the output of various components may saturate and that thesesaturation limits must be taken into consideration in the analysis of large loadchanges.

By eliminating e and ed from Eqs. 5.19,5.22, and 5.29, eliminating vi fromEqs. 5.24 and 5 JO, and rearranging Eqs. 5.24 and 5.28, we obtain

du¿ 1dt = T (nref - n - e, - oVa)a

(5.31)

Permanent Drop

Distributing Valve

(5.23) det = __!_ (óT dVa - e )dt Tr, r dt t

dVd 1dt = Td

[kd(Vd - T) - Vd]

dTdt = ksVd

(5.32)

(5.24) (5.33)

The gate-servomotor rate limits are often applied by restricting the maximumtravel of the distributing valve in the positive and negative directions. Therefore,in the aboye inequality, we have

(5.34)

1Vdmax = k T

s o(5.25)

The preceding four dífferential equations in four variables, namely , Va' et, Vd,

and T, may be integrated by any standard numerical technique; a closed-for:nsolution is not possible because of nonlinearities introduced by the saturation of

124 Applied Hydraulic Transients Hydraulíc Transients in Hydroelectric Power Plants 125

various variables. We have used the fourth-order Runge-Kutta method" in ouranalysis.

5.6 COMPUTATIONAL PROCEDURE

Boundary conditions for a Francis turbine and equatíons tor a dashpot governorwere derived in the preceding sections, A computational procedure for usingthese equations is presented in this section.

Let us assume that the transient-state conditions have been computed for (i -1) time steps. The values of Ne, 'Te' and Hne at the end of the ¡th time step maybe estimated from the known values for the previous three time steps byparabolic extrapolation from the equation

Yi = 3Yi-l - 3Yi-2 + Yi-3 (5.35)

in which y is the variable to be extrapolated? and the subscript indicates thetime step. Note that this equation is valid only if the time steps are equal. Sincethere are no previous time steps at t = O, the steady-state values may be used forprevious time steps for extrapolation purposes. For the estimated values of Hne,

Ne, and 'Te, the grid points A through D (Fig. 5.5) are searched from the storedcharacteristic data. The coefficients ao and al are computed, and the values ofcoefficients a2 to a¿ are determined from Eqs. 5.4 and 5.7 through 5.9. NowEq. 5.1O is used to determined Qp, and Eq. 5.3 to determine Hn' The value of¡Pe ls computed using the estimated value of N; and the computed value of Hn,

and the turbine output, PturP, is then determined from the turbine characteristicdata for ¡Pe and 'Te' The turbine output, PturP, and the generator load, Pgenp, nowbeing known, the value of Np ís determined frorn Eq. 5.16 or 5.17. If INp-

Nel>0.002 NR, then Ne is assurned equal to Np, and the aboye procedure isrepeated; otherwise, the governor equations (Eqs. 5.31-5.34) are solved for 'Tp

by the fourth-order Runge-Kutta method. If l'Tp - 'Te 1> 0.005, 'Te is assumedequal to 'Tp, and the aboye procedure is repeated; otherwise, time is incrernented,and the transient conditions are computed for the system. To avoid unlimitedrepetítion of iterations in the case of a divergence in solution, a counter is in-troduced in both the iterative loops.

The flowchart of Fig. 5.9 illustrates the preceding computational procedure.


The following turbine operations produce transient-state condítions in the waterconduits of a hydroelectric power plant:

Figure 5.9. Flowchart for boundary conditions fOI a Francis turbine.

126 Applied Hydraulic TransientsHydraulic Transients in Hydroelectric Power Plants 127

l. Unit synchronízed to a large systema. load acceptanceb. load reduction or total load rejection

2. Isolated unita. unit start-upb. load acceptancec. load reduction or total load rejection.

A unit connected to a large system runs at the synchronous speed during loadacceptance or rejection because of the large inertia of the system. However, thespeed of an isolated unit rises during load rejection and decreases during ~oadacceptance. For Kaplan and Francís turbines, the turbine speed has a consider-able influence on the transients. Speed changes should therefore be taken intoconsideration in the transient-state computations for these turbines.

The boundary conditions and the computation procedure described in Sec-tions 5.4 through 5.6 are for the isolated units only. These conditions can beused for .units eonnected to a large system by keeping the speed constant and bybypassing the loop for computing the speed changes.

While starting a unit, the wicket gates are opened to the breakaway gateopening to gíve the unit a "kick " to overcome statie friction. Gates are usualIykept at thís opening until the unit speed is about 60 percent of the rated speed ;then the gates are closed to speed-no-load gate , and the unit is allowed to run atthe synchronous speed for a short period of time. It is then synchronized to thesystem and is ready for load acceptance.

For load aeceptance, the wicket gates are opened at the prescribed rate to theopening at which turbine output will be equal to the final output , Similarly,the gates are closed from one opening to another for a load reduction. Wicket-gate closure following total load rejection, however, depends upon the type of

rejection.

ducted. The upstream reservoir level during the tests was at an elevation of671.0 m, and the downstream manifold level was at an elevation of 503.2 m.

The Westinghouse leading-edge flowrneter? was used to measure the steady aswell as the transient-state flows, Locations of the flow transducers are shown inFig. 5.10. The flowmeter display exhibited average flow every 2.1 s. As ob-serving and recording the flowmeter readings at this rate was difficult, theflowmeter digital display along with dock time were recorded on a videotape.After each test, the tape was replayed at a slower speed in order to note thereadings.

The speed of the turbogenerator was measured by a de tachometer generator.A rubber-faced drive wheel was fastened to the shaft of the tachometer. Thetachometer was mounted on a horizontal arm, which was free to turn about avertical pivot anchored to the upper bearing of the turbine. A tensioning deviceheld the drive wheel of the tachometer in contact with the turbogenerator shaft.The voltage output of the tachometer, which was proportional to the turbogen-erator speed, was recorded on a Sanborn recorder.

The transient-state pressures were measured by a strain-gauge pressure cellattached to the turbíne-inlet piezometer manifold. The output of the trans-ducer, when appropriately conditioned through its strain-gauge amplifier, wasrecorded on a chart recorder.

The wicket-gate opening was recorded as follows. The motion of one of theservomotors of the wicket-gate rnovíng mechanism activated a precision voltagedivider (potentiometer). The change in voltage was then recorded on an oscillo-graph, which was calibrated against O and 100 percent gate openings.

5.8 VERIFICATION OF MATHEMATICAL MODEL

Prototype Data

G. M. Shrum Generating Station is located on the Peace River in British Colum-bia, Canada. The power plant consists of 10 Francis units. Each unit has itsindividual power conduit (penstock and power intake). In the tailrace , fiveunits discharge into one manifold, and a free-flow tunnel conveys water fromeach manifold to the tailrace channel. A schematic of the upstream water pas-sages ís shown in Fig. 5.10, and of the downstream water passages in Fíg. 12.20.

Data for Unit No. 4, on which tests were conducted, are given in Table 5.2.The friction faetors were calculated such that other minor losses were includedin the friction losses. The conduit between the trashrack and the downstreamend of the transition was replaced by an equivalent 5.49-m-diameter pipe.* Thelength of the scrolJ case was taken as one-half of the actuallength to account for

Prototype Tests

To obtain data for verifying the aboye mathematical model,load-rejection testswere conducted on Unit No. 4 of the G. M. Shrum Generating Station, ownedand operated by British Columbia Hydro and Power Authority. The unit wasloaded to the amount to be rejected, and was kept at this load until pressure andflow at the turbine inlet became steady. Then , to simulate an isolated load rejec-tion , the speed-no-load solenoid was blocked, and the load was rejected. Fivetests involving load rejections of 50, 115, 118,202, and 250 MW were con-

*See Appendix A.


Table 5.2. Data for Unít 4, G. M. Shrum Generating Station.

"..1;~~

~ .... 1;.~...

~i: s., :::¡¡~ t:.~~ ....~

lo ......" ~all: ~~ El) E

E/i5.e

o <::>- ....• -:o '":§

~ "l

" .. '".. ti <::i.. .. ~.!; ..ti .~.... .... ....~ ~

Turbine and Generator

Rated turbine outputRated headSynchronous speedFlow at rated head for rated outputWR2 of turbine and generatorRunner diameter

231 MW152.4 m150 rpm164m3/s9.27 Gg m2 (i.e., 9.27 X 106 kg m2)

4.86 m

=o.::~cñ

'0.10..§~lUó,,·~ :;e41oe2

oCen:ic.:i...o

]41

"O"O;41

iEoQ::

o-lI'l~

~ .§'"'

¡;,.<l

Governor Settings

Dashpot time constant, T,Temporary droop, liPermanent droop, oDashpot saturation limit, etmaxSelf-regulation constant, a

8.050.40.050.250.15

Conduits

Wave VelocityPipe No. Diameter (m) Length (m) (mis) Friction Factor

1 5.49 207 1244 0.0162 5.49 78 1290 0.0103 4.9 36.5 1300 0.009

reduction of flow along its length. Waterhammer wave velocities were computedusing the equations presented in Section 2.6.

The draft tube was not inc1uded in the analysis because of its short length .This simplification should not introduce large errors. The downstream mani-fold, being a free-surface area, was assumed as a constant-level reservoir.


A computer program based on the preceding mathematical model was devel-oped. The flow in the conduit was analyzed using the method of characteristicsof Chapter 3, and the turbogenerator and governor were simulated úsing theequations derived in Sections 5.4 and 5.5.

Pipe No. 3 was divided into two reaches. To satisfy the Courant's conditionfor stability of the finite-difference scheme, i.e., !!.t ~ Sx]«, a time interval of0.014 s was used. The wave velocities in the upper pipes were slightly adjustedso that there was no interpolation error. Sta tic head and an estimated initial

130 Applied Hydraulic Transients Hydraulic Transients in Hydroelectríc Power Plants 131

steady-state gate opening were the input data to the programo Correspondingflow and turbine output were computed from the turbine characteristics. Ifthe power output was different from the actual value, the initial gate openingwas sJightly changed, and the aboye procedure was repeated. By using this trial-and-error procedure, the initial steady-sta te gate opening, which gave the re-quired turbine output, was determined.

Load was rejected at time, t = O. As the unit was assumed to be isolated fromthe system following load rejection, it was allowed to overspeed, and the wicketgates were closed under governor control. Computed turbine speed, flow, gateopening, and pressure at the downstream end of each pipe were printed afterevery 35 time intervals, i.e., after every 0.5 s of prototype time.

Computed and measured results are plotted in Figs. 5.11 and 5.12. As can beseen, the computed and measured pressures agree closely. The computed andmeasured maximum unit speed agree cJosely; however, the computed resultsshow a faster speed reduction than that shown by the measured results. Itshould be noted that this deviation starts when the wicket-gate opening is small.This difference may be due both to an error in the estimation of the windageand friction losses and due to lack of data for the turbine characteristies at smallwicket-gate openings. The computed pressures show sorne oscillations that werenot recorded during field measurements. The cause of this difference has notbeen explained.

60...!>"~ 40..

-- M~asured

r....... - - - - Compuled

~~,

~ ~ ......

o 2 4 6 8 10 12 14Time afler load rejecliOfl, seconas

21OI

'5 -- Measured _/" ~

---- Compuled/

/j \\~ti \ , -.y ~\

.¡""# \ -.

fl \ I \~'\.... \..1"- "\r '--'.J \7 ,-/

....O 2 4 6 8 10 12 14

20

Il: /95~-~ /90.,..~ 185

<t 180

175

/70

5.9 DESIGN CRITERIA FOR PENSTOCKS 165

As discussed in Section 4.7, the factor of safety to be used during design de-pends upon the risks involved and the probability of oeeurrence of a particularoperation during the life of the project. Based upon the frequency of occur-rence , various operating conditions may be classified as normal, emergency, orcatastrophic .10,11 A discussion of the operating conditions included in eaehof these categories and recomrnended factors of safety"? follows.

Time afler load rejeclion, seconas

190-- Measured---- Compuled- -=:::_---,--

~ ~ -_r ---~

/O 2 4 6 8 10 /2 14

Normal

... !> 160~.e

AIl operations that are Jikely to occur several times during the Jife of the pen-stock are termed normal. During these operations, appurtenances or devices-such as surge tanks, pressure-regulating valves, and cushioning stroke devices-provided for reducing excessive pressure-rise or pressure-drop function properlyfor whieh they are designed. The following are consídered to be normal opera-tions:

150

Time aller load rejeclkM, secoflds

Figure 5.11. 150·MW load rejection: comparison of computed and measured results,

l. Full-load rejection and cJosure of the wicket gates in effective gate-closingtime * with the sta tic head on the turbine up to its maximum value.

2. Openi~g of the wicket gates from the speed-no-load tu full opening ineffective gate-openíng time, with the static head on the turbine as low asits minimum value.

3. The surge tanks do not overflow, unless an overflow weir is provided, nordo they drain,

*Effective wicket-gate opening and closing times are defined as twice the time token bv thewicket gates to open 01' close between 25and 75 percent openings. .


~, * 100o:;

.!!: 80c::-:} 60

~ 40tio::.

--- Mtlosurtld _- - - - Campultld

f-,.

............... !'-............

~""~ ., 16 ¡,

... 20~~OO 1 '884 6 102 12

Timtl afltlr load rtljtlclion, stlconds

(o) Turbine oote openino

210

20~

200l:;

19~",,'ti 190-oC::

18~~::o 180....

IT~~tl

ITO

16~

- --- Meosurtld _7í/ [\\ ---- Compuled

l' \ ",¡ x., ......1"-......"\

// '\1/ \\

iJ \

v \ <,! \1/ .....- ...... t-/""'J \..r- f-_

16 ¡,160 O 8 10 12 14 '82 4 6Timtl afltlr load rejeclion, seconas

(b) Penstoek pressure

210

/' i= -_ -_h/ r---- t---

7 --_ ~

/V --- Measured

---- CampuledI I

16 ¡,

",,' 190--~- ITO.5:~~

2 4 6 8 10 12 14 '8Time afltlr load rejtlclion, secona«

(e) Turbine speed

Figure 5.12. 250·MW load rejection: comparison of computed and measurcd results.


The penstock and scroIl case are designed to withstand the rnaximurn andminimum pressures caused by the preceding operatíons wíth a minimum factorof safety of four, based on the ultimate bursting and coIlapsing strength.

(C('~¡ A5t'-1[jEmergency

The emergency conditions are those in which one of the pressure-control equip-ment malfunctions. These condítíons include:

l. The pressure-regulating valve is inoperative on one unit.2. The cushioning stroke de vice is inoperative on one unit.

A factor of safety of two, based on the ultimate collapsing or burstíngstrength, is recommended for pressures produced by emergency operations.

Catastrophic

Catastrophic conditions are those in which various control devices malfunctionin the most un favorable manner. For example, if a pressure-regulating valve isprovided, then the wicket-gate closing mechanism is designed such that thewicket gates wíll close at a very slow rate in case the pressure-regulating valve isinoperative during a load rejection. However, if the pressure regulating valvemalfunctions followíng a load rejection and the wicket gates do not close at aslow rate, then this operation is considered catastrophic_11

Because of very low probabílity of occurrence, a factor of safety of slightlymore than one, based upon the ultimate bursting or colIapsing strength, issuggested.

5.1 O GENERA TOR INERTlA

For stable governing of a hydroelectric power plant and to keep the speed riseof the unit within permissible limits foIlowing a load rejection, it is necessarythan an adequate amount of generator and turbine (unit) inertia be provided.Turbine inertia is smalI compared to the generator inertia; therefore, if neces-sary , only the latter is increased. Increasing the generator inertia increases thecost of the project. Although the increase in the generator cost due to increasingits inertia may not be large, other associated costs, su eh as increasing the cranecapacity or increasing the powerhouse dimensions, a~e usually high. Therefore ,the generator inertia is kept as smalI as possible while still maintaining acceptablegoverning characteristics.

The folIowing factors are considered in -selecting the generar or inertia:

l. A llowable frequency fluctuation. The alIowable frequency fluctuationdepends upon the type of load. For example, a frequency deviation of 0.1


percent is not perrnissible for paper milis, while a deviation as large as 5 per-cent may be al!owed for mining equipment.

2. Size al the system. A unit should be designed to be stable in isolated opera-tion if it is supplying 40 percent or more of the system load, or if there arepossibilities of the unit becoming isolated because of failure of the transmis-sion lineo The overal! stability of the system is increased if the majority ofthe units in the system are stable in isolated operation.

3. Type al load. Periodically changing loads, such as electric trams and miningshovels, contribute to system instability. Therefore , more inertia should beprovided if such loads are present in the system.

4. Water passages. One of the major factors in the selection of the inertia is thesize, length, and layout of the water passages of the power plant. By increas-ing the size of the water passages, the generator inertia may be decreased.However, the former is usually more costly. Therefore, the size ofthe waterpassages is first selected based on the cost-benefit ratio of reducing the headlosses, and the required generator inertia is then determined.

5. Governor times. By decreasing the governor opening and elosing times, thestability of the system can be improved. However, they cannot be arbitrarilydecreased since they are set so that the waterhammer pressure ís within thedesign limits (Section 5.9), and so that the water column does not separate athigh points of the penstock or in the draft tube.

No analytical method is available for determining the generaior inertia re-quired for a given set of plant parameters. Therefore , a number of empiricalformulas and experience curvesl2-14 have been proposed. The normal or stan-dard generator inertia depends upon the unit rating'? and is given by theequation,

Normal generator WR 2 = ) 5 ,970 (:;~) 1.25

in which N, = the synchronous speed, in rev/min; kua = the generator rated out-put; and WR 2 = moment of inertia, in kg m2 •

Depending upon the factors just outlined, the value of WR2 may be increasedor decreased. For good regulation, the United States Bureau of ReelamationrecomrnendsP that the ratio of the mechanical starting time, Tm' and the water-starting time, Tw, be greater than 8. Units with Tm /T w of 5 or less may beintegrated into a system but it may be necessary to compensate this deficiencyon the other units of the system. Tm is the time in seconds for the ra~ torqueto accelerate the rotating masses from zero to rated speed, and Tw is the time

(5.36)*

*In English units, replace 15,970 by 379,000.

Hydraulic Transients in Hydroelectríc Power Plants 135

for the rated head to accelerate the flow from zero lo rated velocity. Expres-sions for Tm and Tw are

WR2 X N2T = r

m 90.4 X 106 MW

Qr LTw =-~-gHr A

in which g = acceleration due to gravity; Qr and H; are turbine flow and net headat rated conditions; L and A are the length and cross-sectional area of the waterpassages; N, = synchronous speed; and ~ L/A is computed from the upstreamintake to the downstream end of the draft tube. The cross-sectional area of thescroll case at its upstrearn end is used for computing ~L/A, and its length istaken as one-half to account for the reduction of flow along its length.

Experience curves proposed by the Tennesee Valley Authority relate Tm andTw and show stability limits for various ratios of the unit size to that of thesystem. Gordon!" has taken into consideration the effect ofthe governor timeswhile plotting his curves (se e Fig. 5.13), which are based 011 experience with 40Kaplan, Francis, and propeller turbine installations. In the author's opinion,Gordon's stability curves should be used because they take into account most ofthe factors upon which the generator inertía depends, During the final designstages, however, the mathematical model developed in Sections 5.4 through 5.6should be used to confirm the results of the preliminary analysis. To use thesecurves, the wicket-gate opening time, Tg, is computed by adding the time of thecushioning stroke (about 1.5 s) to the effective gate-opening time, To.

(5.37)*

(5.38)

5.11 GOVERNING STABILITY

General Remarks

As discussed previously, a dashpot, a PID, or an accelerometric governor is pro-vided to control the speed oscillations of a turbogenerator of a hydroelectricpower plant. The speed oscillations are stable or unstable depending upon thevalues of the parameters of the hydro-unit, penstock, and governor.

Paynter!" presented a stability Jimit curve and suggested optimum values ofthe governor settings by solving the problem on an analog computer. Hoveyl6,17

derived a similar stability curve theoretically. However, both Paynter and Hoveyneglected the permanent speed droop of the governor and the self-regulatíon ofthe turbine and of the load. In most cases, permanent'S¡Jeeádroopa1is not zero. 2...:.)

*Ir WR2 is in lb ft2 and the output is in hp, then Tm = WR2 X N; /(1.6 X 106 hp).

136 Applied Hydraulíc Transients

...... .~ ".!i: ...~ ~ ·s ~l} " ...... ¡¡;~ .~~ -;;" & "'oC::ti ti'· !ii... " 1; b~ .~.. ~ ~~:It:i" ~ ..--.: .. ~ .. i~~.~ ~ i:j! ~!~:;: ~~t11

....... ...." ¡._lt ....~

s

e...oC.J

~=!§ze .."O,,¡:

1'-- ~~zioz"'-

.............. -II:~

~~

"OC

~'g~~t~ i'--.r-,::z.,;;0>-

~~~~ <, r-,!:~~2--- ~~~er--- ~::~=

r-r-- ....r------- r---zoIoeo.J

oZ-e..oIo..o.J

"... -O".. zz ...o"oC.. '"z- '"... '"oC ..."'c~ '"0-

.í.'" N -; o..

.;

.o

,......... l·i:: ~O"i! iOO..~...<:._,.;Q,I¡:;="~::=

:E!"'."'eO"i!OO...;

....~I.......-vi

~IJ.

..o

N,;

o


Table 5.3. Values of self-regulation coefficient.a

Ij'

r

(l/ (lturb (l = (l/ - (lturb

TurbineIn generalHigh specific speed

LoadGrid loading: Motors only (constant torque)Ohmic resistance only with voltage regulationOhrnic resistance without voltage regulation

about -1up to -0.6

o +1-1 0.0

1 to 4 2 to 5

8Taken from Ref. 18.

while theself-regulatíon coefficient, a;18 rnay or may not be zero dependingupon the type of load.* (For values of a, see Table 5.3.)

The stability criteria are forrnulated herein 19 by taking into consideration theperrnanent speed droop and self-regulation. It is found that, if a and a areallowed for, the stability is considerably increased, and the optimum values ofthe temporary speed droop, 5, and the dashpot time constant, Tr, are reduced .

Differential Equations of the System

In formulating the differential equations, the following assumptions are made:

l. The changes in the turbine speed, head, and gate opening are small; thus,nonlinear relationships can be assumed linear.

2. A single hydro-unit supplies power to an isolated load .3. The governor has no dead band, backlash, or hysteresis .4. The walls of the penstock, and the water in the penstock and scroll case are

rigid. Thus, waterhammer pressure caused by changes in the gate openingcan be computed by using the rigid water-colurnn theory.

By making these assumptions, the following differential equations! S-18 can bewritten for the hydroelectric power plant shown in Fig. 5.1.

l. Machine acceleration

dnT - = g + 1.5h - cm - ·f:.m

m dt (5.39)

"'The turbine self-regulation coefficient, atup is defined as the slope of the graph relatingthe per unit deviation of the turbine torque frorn the rated value, to the per unir deviationof the turbine speed frorn the rated value. The load self-regulation coefficient al, is definedas the slope of the graph relating the per unit deviation of the torque of the electricalloadfrom the rated value, to the per unit deviation o] the [requency of the electricalload [romthe rated value. The self-regulation coefficient is defined as the algebraic difference betweenthe load self-regulation coefficient and the turbine self-regulation coefficient.


2. Water acceleration

dh dg-0.5 T w dt = T w dt + h (5.40)

3. Governor response

dg dn(a + Ii) T - + ag = - T - - n

r dt r dt

in which n = relative speed deviation = (N - No )/N o; h = relative pressure-headrise = (H - Hn)/Ho; g = relative gate-opening change = CG - Go)/Go; Sm = rela-tive load-torque change = AM/Mo; 11M = step-load torque cha~ge (negat~ve forload rejection); M¿ = initial steady-state load torque; G = transíent-state Ins~an-taneous gate opening; andN = transient-state instantaneous speed of the turbine ,in rpm. The subscript o refers to the initial steady-state values.

Let s = dldt . Then Eq. 5.41 may be written as

[(a+Ii)Trs+a]g=-(Trs+ 1)11 (5.42)

(5.41 )

or

-(Trs+ I)ng = (a + Ii)Trs + a

(5.43)

It follows frorn Eq. 5.40 that

-(0.5 Tws+ I)h = Twsg (5.44)

or

Tws(Trs+l)nh = --_":':"""'.;....-':---'----:-[(a +sit;» + a] (0.5 r;, + 1)

Substitution of Eqs. 5.43 and 5.45 into Eq. 5.39 yíelds

-(Trs + l)n 1.5 Tws(Trs + l)nTmpn= (a+Ii)Trs+a + (a+Ii)Trs+a] (0.5 Tws+ 1)

By sirnplifying and replacing S3 by d3/dt3 ; S2 by d2/dt2 ; and s by dldt , Eq.5.46 takes the forrn

d3n05 T T T (a+Ii)-3 + [0.5aTwTm +(a+8)TmTr- TwTr. w m r dt

(5.45)

-cm - Sm (5.46)

d2n dn+ O 5"'T T (a+ Ii)] -+ [aTm +T; - Tw + 0.5ao:Tw +(a+ li)o:Tr] -d

• u. w r dt2 t

+ (1 + ao:)n = -aAm (5.47)


Criteria for Stability

According to the Routh-Hurwitz critería.i? the oscillations represented by thethird-order differential equation (Eq. 5.47) are stable if

(5.48)

(5.49)

(5.50)3. [aTm + T, - Tw + 0.5aaTw + (a+ o)aTrJ > O

4. (1 + o:a) > O (5.51 )

5. [aTm + T" - T w + 0.5 aaT w + (a + li)aTrJ [0.5 aT w Tm +(a+Ii)TmTr- TwTr+0.50:TwTrCa+Ii)] >[0.5 T w Tm Tr(a + Ii)] (1 + oo). (S .52)

The inequalities 5.48 and 5.51 are always satisfied. To plot the stability-limitcurves, we have to consider the expressions given by inequalities 5.49, 5.50, and5.52. There are six parameters in these expressions, namely, a, O, a, Tw' T,«,and Tr· To reduce the nurnber of parameters and to present the criteria in anondimensional form, the following nondimensional pararneters are introduced:

TwA =-I óTm

TwA =-2 T,

«r;A3 =--

Tm

aTmA4 =--Tw

(5.52)

By substituting the aboye parameters into inequalities 5.49, 5.50, and 5.52 andsimplifying the resulting expressions, we obtain the following equations for thelimit of stability:

(5.53)

(5.54)

140 Applie d Hy draulic Transients

AfP'4 + O.5A~A~ - O.sA~A4 - 1+1..2 - 21..21..4- 0.51..31..4

- 1..21..31..4 + 1..3 Al + A2A3Al + 0.5A~A~ + 0.25A2A~A~

+0.25AªA3A~ +A2A¡)+AI(I -1.51..2 +A2A4 +2A3A4

+ A2/..3/..4 - 0.5A3 - 0.SA2A3 + A~A4 + 0.25A2A~A4)

+ (1..3 + 0.5 Aj) = O (5.55)

Equations 5.53 through 5.55 represent the stability criteria. Based on theseequations, the stability Iimit curves for different values 0[;\1, A2, A3, and 1..4 areplotted in Fig. 5.14. Speed oscíllations corresponding to those values OfAI and1..2, which lie in the region enclosed by the stability Iimit curve and the positivecoordina te axes, are stable. For A) = O and 1..4 = O, Hovey's stability curve isobtained. This is shown as a dotted curve in Fig. 5.14.

The following exarnple illustrates that the results obtained by neglecting a anda are conservative.

Example 5.1

For the Kelsey hydroelectric plant, Tw = 1.24 s and Tm = 9.05 s. Hoveyreported+' that , according to his criteria, the speed oscillations caused by a stepload change are unstable Ior 8 = 0.28 and Tr = 2.25 s. The following analysisshows that the oscillations are stable for these values of 8 and T; if the perrna-nent speed droop and the self-regulation are taken into consideration.

The Kelsey plant supplies power to an isolated load consisting of furnaces,blowers, and cornpressors.!" Thus, a may be taken equal to 1 (se e Table 5.3).As reported by Hovey in another paper ,22 a = 0.035. Stability calculations canbe done as follows:

Tw 1.24Al = 8Tm :::0.28 X 9.05 = 0:49

r; 1.241..2 =-=-=0.55r, 2.25

«r; 1 X 1.241..3 =-= =0.137r; 9.05

«r; 0.035 X 9.051..4 = T

w= 1.24 =0.255.

From Fig. 5.14, it follows that, for Al :::0.49, A2 = 0.55 and

1. A3 = 0.0 and 1..4 = 0.0, oscillations are unstable (Hovey's criteria)2. A3 = 0.0 and A4 = 0.255, oscillations are stable

i\ .

/.0

L 08

-< ,,0.6

0..4

0.2

/.4,..---,--,.---,-----¡-----,


0.8).,-Figure 5.14. Stability Iimit curves,

/.21---+---I---+---t---1

1.0.1---i----f--+--I1---1

0.4

2.0.

t-<

o

3. A3 = 0.1 and A4 = 0.0, oscillations are stable4. A3 = 0.1 and A4 = 0.255, oscillations are stable

To check the validity of these results, Eqs. 5.39 through 5.41 were solved on adigital computer , The plotter output obtained from the computer ís presented


.12L 1 = .490L2 = .551L 3 = .000L4 = .000

.~ .08

~'"~ .04

~~ O

~ -.04

-.08'-f---r---r--....,O

.08

20.0 40.0 60.0Timtl (Stlc)

(a)

L 1 = .490L2 = .551L3=.137L4 = .000

'"~ .06.~'"~ .04

"..~ .02

o

-.02+-~,--,--...,O 20.0 40.0 60.0

Time (Stlr:)(e)

.ª .04....

.~~ .02

".... o~-.;11:: -.02

.06L 1 = .490L2=.551L3 = .000L4 = .225

-.04-1--,---;---,o .20.0 40.0 600

Time (Stlr:)( b)L 1 = .490L2 = .551L3 = .137L4 = .225

....--....34I

~lo,: .24'-

.ª..... /4

.~'"~""..~ -.06

.04

-.16+--,---;---,O 20.0 40.0 60.0

Tim« t sec )(d)

Figure 5.15. Unstable and stable speed oscillations.

in Fig. 5.15. In this figure, LI, L2, L3, and L4 denote Al, 1..2, 1..3, and ~,respectively.

Transient Speed Curve

The differential equation , Eq. 5047, may be solved as follows to determine anexpression for n:

lnitial Conditions

At time t = O,N =No• C = Co' and H = Ho' Therefore,

nlt=o=O; glt=o=O; hlt=o=O.O (5.56)


By substituting these values into Eq. 5.39, we obtain

dnlTm - = -11mdt t=O

or

(5.57)

Note that for load-off condition , 11m is negative. Differentiation of Eq, 5.39yields

d2n dg dh dnT -=-+ 1.5 -- 0:-

m dt2 dt dt dt(5.58)

By substituting Eq. 5.56 into Eg. 5 Al, we obtain

dgl dnl(a + o) T; - = - Tr-dt t=O dt t=o

(5.59)

which upon simplification becomes

dgl __ -_1 dnldt t=O (a + o) dt t=O

(5.60)

It follows from Eqs. SAO and 5.60 that

dhl __ 2_ dnldt t=O (a + o) dt t=O

Substítutíon of Eqs. 5.57,5.60, and 5.61 into Eq. 5.58 and sírnplífícatíon of theresulting equation yields

(5.61)

d2n I 2 - ao: - 00: (-11m)dt2 t=O = a + o T;",

(5.62)

Solution of the Dlfferential Equation, Eq. 5.47

The general solution of the third-order dífferential equatíon, Eq. 5047, is equalto the sum of the complementary function, ne, and the particular integral, np(i.e.,n=np+ne)·ForEg.5.47, .

- al1mn =---

p (1 + co)(S .63)

144 Applied Hydraulic Transients Hy draulic Transients in Hy droelectric Power Plants 145

To determine the complementary function, compute the roots of the char-acteristic equation ,

+ (o + a) T,.aJ J.1 + (l + oo) = O (5.64)

Solution of Eqs. 5.69 to 5.71 for A, B, and C yields

-213 2 - (a+o)a a 2 2r::: + (a + ó)T~ - (1 + aa) ('Y + 13 )A= (a'-f3)2+'Y2 (-.1m)

B = [- .1m _ (a' _ f3)A _ af3.1m] _!_r; 1 +aa 'Y

otsmC=---A.

1+ oa

(5.72)

0.5 TwT mTr(ó + a) p,3 + [0.5 TwT m a + (o + a) TrT m - TwTr

+ O.SQTwTr(ó + a)J J.12 + [Tma + T; - Tw + 0.5 Twaa

Let the roots of Eq. 5.64 be a', 13± i'Y. Then,

n¿ =Ae'h + De({3+i-y)t + Ee«(3-i-y)t (5.65)

where A, D, and E are arbitrary constants. Equation 5.65 can a1so be written as Optimum Values of the Governor Parameters

(5.66) For a specifíc value of A3 and a specific value of A4, Eqs. 5.39 through 5Al aresolved for various values of Al and 1..2, Those values of Al and 1..2 are consideredoptimum values, which gíve the shortest settling time, but slightly underdampedresponse. This procedure is repeated for A3 = 0.0 and 0.25, and 1..4 = 0.0 to 004.The optimum values of Al and A2 for different values of A3 and A4 are presentedin Fig. 5.16. The following example illustrates the procedure to select the opti-mum values from this figure.

where A, B, and C are arbitrary constants. Therefore, the complete solution is

• 1> oSmn =AeO! t + e"t(B sin 'Yt + C cos 'Yt) - ---

(l + aa)(5.67)

The values of the arbitrary coefficients are determined from the initial conditions

I oSmn t=O = A + C - =: O

(l + aa) (5.68)Example 5.2

Determine the optimum values of ó and T; for Kelsey hydroelectríc power plant.The following are the values of different pararneters:

Tw = 1.24 s (computed from the dimensions and geometry of the penstock)Tm == 9.05 s (computed from the known value of WR2 and ratings of the

turbine and generator)o= 1.0 (determined from Table 5.3 for the type of load)o= 0.035.

Hence,

A + C= a.1m(l + co) (5.69)

By differentiating Eq. 5.67, substituting t = O, and using the initial conditiongiven by Eq. 5.57, we obtain

, .1ma A +'YB+f3C==--.

Tm

l. The optimum values of ó and T; may be determined as follows:a. Compute 1..3 and 1..4 :

(5.70) «rw 1.0 X 1.241..3 = - = = 0.137r; 9.05

aTm 0.035 X 9.05 O1..4 = -= = .255.r; 1.24

b. Determine the optimum values OfAI and 1..2 from Fig. ).16:From this figure, Al = 0.430 and 1..2 = 0.27 for 1..3 = 0.137 and A4 =: 0.255.

By differentiating Eq. 5.67 twice, substituting t = O, and using the initial condi-tion gíven by Eq. 5.62, we obtain

a'2 A + 2{tyB + (132_ 2) C= 2 - aa- oa (-.1m).'Y a+5 T,~ (5.71)

146 Applied Hydraulic Transients Hydraulic Transients in Hydroe1ectric Power P1ants 147

038L-----~-----L----~----~

e- ::-- ::--I

II

1:) .40 LI =.4301:) .40 LI =.500

~ .40 L 1=.400~ i:( ~'- L2= .270 <; L2=.250 '- L2=.170

~ .32 L3= .137 ~ .32 L3=.137 ~ .32 L3=.137s ~ ~L4=.255 1..: L4 =.255

~24~ .24 ~24::..

~ ~ ~~ .16 e ./6 ~ ./6

~ ~ ~~08

s,lIj lIj .08

~ id idQ:: Q:: Q::

o 200 400 60.0 o 200 400 60.0 o 200 40.0 600

TIME (slIconds) TIME (SlIconds ) TIME Iseconas]

(o) (b) (e)

0.3

~ )..3=0.25

)..3=0-

Figure 5.17. Speed deviation for varíous governor settings.

3. Optimum values according to Paynter's relationships:

0.2 ó=~= 1.24 =0.342OATm OAX9.05

r; 1.24Tr = --- = --- = 7.3 S

0.17 0.17

0.1 O 0./ 0.2 0.3)..4-

0.4

For cases 1 through 3, n - t curves for tsrn = - 0.1 are presented in Fig. 5.17.It is clear that, if a and o: are taken into consideration, the optimum vaJuessuggested by the author give a better transíent response.

Figure 5.16. Optímurn governor settings.5.12 CASE STUDY

c. Compute ó and T; from the values 00.'1 and A2 determined in step b:

ó =~= 1.24 _A1Tm OA30X9.05-0.319

T = Tw = 1.24_r A2 0.27 - 4.6 s

2. Optimum values suggested by Hovey:

ó = 2T w = 2 X 1.24 _r; 9.05 - 0.274

T, = 4 Tw = 4 X 1.24 = 4.96 s

For illustration purposes , governing stability studies carried out for the KootenayCanal Development are presented in this section. Kootenay Canal Developmentis a 500·MW hydroelectric power plant owned by British Columbia Hydro andPower Authority. There are four units in the plant wíth each unit having its ownpower intake and penstock.

Data for the turbine , generator , and penstock follow:

1. TurbineType: FrancisSpecific speed:* SS (English units), 209 (SI units)

*Specific speed = n'¡¡;/H1•25• In the English units, Pis in hp, H in ft, and n in rprn; in SIunits, P is in kW, H in m, and 11 in rpm.

148 Applied Hydraulic TransientsHydraulic Transients in Hydroelectric Power P1ants 149

Rated turbine output: 127.5 MWRated head: 74.6 mSynchronous speed: 128.6 rpmFlow at rated conditions: 191 m3/sRunner-throat diameter: 4.95 m

2. GeneratorRated output: 125 MW

3. Penstock. The diameter of the penstock was determined from an economicanalysis so that the incremental benefits from the decreased head losses weremore than the increase in the penstock costs. The length, diameter, and waIlthickness for various sections of the penstock are listed in Section 2.7.

Computations were done as folIows:

WR2 X N2

Tm = 90.4 X 106 ~W

4.83 X 106 X (128.6)290AX 106 X 127.5

= 6.93 S

2. Water starting time, Tw

(kva)1.25

Normal generator WR2 = 15,970 N; .5 (Eq. 5.36)

LLength, L Cross-sectional Area, A 2:;-

AConduit (m) (m2) (m-1) Remarks

Intake 7.6 9.1 X 9.1 = 82.8 0.09112.8 4.9X 7.3=35.8 0.357

Penstock 244 ~(6.71)2 = 35.4 6.894

36.5 ~(5.55)2 = 24.2 1.514

Scroll case 14.5 ~(5.55)2 = 24.2 0.60 Tota11ength ofscroll4 case = 29 m.

Draft tube 15.2 ~(4.88)2 = 18.7 0.814

13.7 0.5(18.7 + 14.6 X 5.33)= 48.3 0.28

L¿ - = 10.54;

A

L:¡; - excluding draft tube = 9.45

A

1. Mechanical starting time, Tm

MW X 103kva=-----

Power factor

kva= 125,000/0.95 = 131,579

(131 579 )1.25= 15970 '

, (128.6)1.5

= 4.44 Gg m2

_ ( 127,500 )1.25-1446 (128.6)1.5

T =_g_J;!=_W gHr A

191 X 10.549.81 X 74.6

=2.75s

3. Experience curves. Since there is a strong possibiJity of this power plantbeing isolated from the system, the generator inertia was selected such thatthe units would be stable in isolated operation. For this purpose, the folIow-ing empirical relationships and curves were used.

(kW )1.25

Turbine WR2 = 1446 Nr1.5

= 0.39 Gg m2

Total WR2 = 4.44 + 0.39 = 4.83 Gg m2


a. U.S. Bureau of Reclamation criteria.t? For the normal WR2 of thegenerator and turbine and for the selected conduit sizes, the values of Tm

and Tw were computed aboye as 6.93 and 2.75 s, respectively. Hence,

Table 5.4. Speed rise.

Speed rige, pereent

Tm 6.93-=-=2.5.r; 2.75

Te(s) Tm = 6.93 s Tm = 10.27 s Tm = 11.48 s

As this ratio is less than 8, the unit would be unstable in isolatedoperation.

b. Tennessee Valley A uthority curves. 13 The values of Tm and Tw corn-puted in (a) were plotted on the TVA experience curves. It was foundthat the units would be unstable in isolated operation ,

c. Cardan s curves. 14 As both the USBR criteria and the TVA experiencecurves indicated that the units would be unstable in isolated operationwith the normal WRl , total WRl of 7.2 and 8 Gg ml , in addition to thenormal WRl of 4.83 Gg ml , were considered. The values of Tm for WR2

of 7.2 and 8.0 Gg m2 are 10.3 and 11.5 s, respectively. Water startingtime, T;", excluding that for the draft tube, is

68

10

56.463.891.8

42.34751.6

38.642.947.8

Therefore ,

295730 mis

, 191 X 9.45T w = 9.81 X 74.6 = 2.46 s

ae = 244 51 20-+--+--694 1410 1244

2L 295-=-=0.81 saeq 730

LAe=L

1:-A

2959.45

= 31.22 m2

Let us assume that the wicket-gate opening and closing times are equal.Then , allowing 1 sec for the cushioning stroke, Tg = Te +1.0, in whichTe = effective gate-closing time and Tg = total opening time. Now fordifferent assumed values of Te and fOTtotal WR2 equal to 4.83, 7.2, and8.0 Gg m2, points were plotted on the Gordon's curves. Of these threecurves, the curve for 4.83 Gg m2 did not intersect the curve dividing thestable regions for the isolated and system operation. The values of TefOT WR2 = 7.2 and 8.0 Gg m2

, which would result in stable isolatedoperation, were determined from the intersection of the other two curves.These values were 8.6 and 10.2 s, respectively.

4. Speed rise. Using the procedure outlined in Ref', 12 (p. 29), speed rise forfull-load rejection was computed for varíous values of Te and Tm' Thecomputed values are listed in Table 5 A.

5. Waterhammer pressure. Waterhammer wave velocities in the penstock werecomputed in Section 2.7. Wave velocity and cross-sectional area for an equiv-alent 295-m-Iong (including half the length of scroll case) pipe23 was corn-puted as follows:

191V = --= 6.12 mIs

o 31.22

aVoAllievi's parameter, p = 2gH

= 730X 6.12 =3.052 X 9.81 X 74.6

The waterhammer pressures for various values of Te were computed fromthe charts presented in Appendix A. The computed values are as follows:

L L-=1:-ae a

D.HTe Pressure rise, -

Hr

6 0.488 0.36

10 0.27


6. Selection of WR2 and governor times. From the preceding computations,the values of WR 2 and governor times were selected as follows:a. The maximum effective governor time was selected as the minimum of

i, The governor time required for isolated stable governing from Gor-don 's stability curves.

ii. The governor time so that the speed rise following total load rejectiondoes not exceed 60 percent.

b. The minimum value of the effective governor time is the maximum ofi. The gate-opening time such that negative pressures do not occur in the

penstock for the minimum forebay water level.ii. The waterhammer pressure rise following total load rejection does not

exceed 50 percent of static head.Based on these criteria, the following values were selected:

Total WR2 of generator and turbine = 7.2 Gg m2

Turbine WR2 = 0.2 Gg m2 (specified by the turbine manufacturer)Generator WR2 = 7.2 - 0.2 = 7.0 Gg m?Governor closing time = 8 s

7. Governor settings. For the selected conduit sizes and WR2 of the generator,Tw = 2.75 s and Tm = 10.27 s. Assuming the permanent speed droop, a,equal to 5 percent and the self-regulation constant, a, egual to 0.5,

TwDashpot time constant, r, =~

2.75=-= lOs

0.27

8. Final check. During the final design stages, the turbine characteristics wereavailable from the model tests conducted by the turbine manufacturero Themathematical model presented in Sections 5.3 through 5.6 was used to checkthe maximum and minimum transient-state pressures, maximum speed risefollowing total load rejection, and the speed deviation following large loadchanges. The maximum and minimum pressures and speed rise were found tobe within the design limits, and the unit was stable following large loadchanges.

5.13 SUMMARY

CJ.T w 0.5 X 2.751..3 =--= = 0.13r; 10.27

aT m 0.05 X 10.271..4 =-T :: =0.19

w 2.75

In this chapter, the details of the mathematical simulation of the conduit sys-tem, hydraulic turbine, and governor were outlined. Various turbine operationsthat produce the hydraulic transients were discussed. Prototype test results toverify the mathematical model were presented. Procedures for the selection ofthe generator inertia and for determining the optimum governor settings werethen described. The chapter was concluded by the presentation of a case study.

PROBLEMS

0.43 X 10.27 :: 0.622.75

5.1. Develop the boundary conditions for a Francis turbine having a long pres-surized downstream conduit.

5.2. How would the boundary conditions of Problem 5.1 be modified if therewas a downstream surge tank?

5.3. The block diagram for a proportional-integral-derivative (PID) govemor isshown in Fig. 5.18. Proceeding similarly as in Section 5.5, derive the differ-ential equations for this governor.

5.4. Figure 3.18 shows the layout for the Jordan River Power Plant in which apressure-regulating valve (PRV) is provided to reduce the transient pres-sures. In a load-rejection test on the prototype, the wicket gates closed,and the PRV opened as shown in Fig. 10.12. Develop a mathematicalrnodel to analyze the transients caused by a load rejection, and compare thecomputed results with those measured on the prototype (Fig. 10.13)Assume that the unit is isolated from the system, and that there is a delayof 0.4 s between the opening of the PRV and the closure of the wicketgates. Use the turbine characteristics shown in Fig. 5.3.

For these values of 1..3 and 1..4, optimum governor settings as deterrnined fromFig.5.16are

A) :: 0.43

and

1..2 :: 0.27

Hence,

Temporary speed droop, ó :: ~A)Tm


.... ~1> sc::.0.o:: ....... :¡ bo~ I§d:: ~ct


S.S. Develop the boundary conditions for a Kaplan turbine taking into con-sideration the variation of the blade angle.

5.6. Determine the unit WR2 required for stable governing of a hydroelectricpower plant in isolated operation. The data for the power plant follow:

Rated output = 39 MWSynchronous speed = 500 rpmRated head = 240 mTurbine discharge at rated conditions = 38 m3

/5

Length of the penstock = 640 mLength of the scroll case = 36 mCross-sectional areas of the penstock, and the scroll case at the upstream

end = 7.9 m2

Governor opening and closing time = 5 sNeglect the length of the draft tube and assume a power factor of 0.95while computing the kva of the unit.

5.7. What are the optimum governor settings for the unit of Problem 5.67

REFERENCES..1. Krivehenko, G. L, Zolotov, 1. A., and Klabukov, V. M., "Moment Characteristics of

Cascades under Non-stationary Flow Conditions," Proceeding, Annual Meeting, Inter-nOliona/Association [or Hydraulic Research, vol. 2, Paris, 1971, pp. B 11-1-B 11-9.

2. Perkins, F. E., et al., "Hydropower plant Transients," Part II and 1lJ, Dept. of CivilEngineering, Hydrodynarnics Lab., Report 71, Massachusetts Inst. of' Tech., Sept. 1964.

3. Streeter, V. L., Fluid Mechanics, 4th Edition, McGraw-HiIl Book Co., New York, N.Y.,1966.

4. Chaudhry, M. H. and Portfors, E. A., "A Mathematical Model for Analyzing HydraulicTransients in a Hydroelectric Powerplant," Proceedings, First Canadian HydraulicConference, published by the University of Albcrta, Edmonton, Canada, May 1973, pp.298-314.

5. "Speed Governor Fundamcntals," Bulletin 25031, Wordward Governor Cornpany,Rockford,lllinois.

6. D'Azzo, J. J. and Houpis, C. H., Feedback Control System Analysis and Synthesis,Second Edition, McGraw-Hill Book Co., New York, 1966.

7. Lathi, B. P., Signals, Systems and Communications, John Wiley & Sons, Inc., NewYork,1965 .

8. McCracken, D. D. and Dorn, W. S., Numerical Methods and Fortran Programming , JohnWiley & Sons, Inc., New York, 1964 .

9. Fischer, S. G., "The Westinghouse Leading Edge Ultrasonic Flow Measurement Sys-tem," presented at the spring meeting, Amer. Soco of Mech. Engineers, Boston, Massa-chusetts, May 15-16,1973.

10. Parmakian, J., "Water Hammer Design Criteria," Jour. Power Div., Amer. Suco ofCivi/Engrs., April 1957,pp. 1216-1-1216-8.

11. Portfors, E. A. and Chaudhry, M. H., "Analysis and Prctotype Verification of HydraulicTransients in Jordan River Powerplant," Proceedings, First Imernational Conference onPressure Surges, Canterbury, England, published by British Hydromechanic ResearchAssociation, Sept. 1972, pp. E4-57-E4-72.


12. Krueger, R. E., "Selecting Hydraulic Reaction Turbines," Engineering Monograph No.20, United States Bureau of Reclamation, 1966.

13. "Mechanical Design of Hydro Plants," Tennessee Valley Authority Projects, TechnicalReport No. 24, vol. 3, 1960.

14. Gordon, J. L., "Determination of Generator lnertia," presented to the Canadian Elec-trical Association, Halifax , January 1961.

15. Paynter, H. M. HA Palimpsest on the Electronic Analogue Art," A. Philbrick Re-searches, Inc., Boston, Mass., 1955.

16. Hovey , L. M., "Optimum Adjustment of Governors in Hydro Generating Stations," TheEngineering Joumal, Engineering Inst. of Canada, pp. 64-71, November 1960.

17. Hovey, L. M. "Speed Regulation Tests on a Hydrostation Supplying an Isolated Load,"Trans. Amer. Inst. o] Elect, Engrs., 1962, pp. 364-368.

18. Stein, T., "The lnlluence of Self-Regulation and of the Damping Period on the WR2

value of Hydroelectric Power Plant," The Engineers' Digest , May-June 1948. (Trans-lated frorn Schwcizerische Bauzeitung, Sept .-Oc!. 1947.)

19. Chaudhry, M. H., "Governing Stability of a Hydroelectric Power Plant," Water Power,London, April 1970,pp. 131-136.

20. Wylie, C. R., Advanced Engineering Mathematics, McGraw-HiIl Book Co., 3rd Edition,New York, p. 721.

21. Hovey , L. M., "The Use of an Electronic Analog Computer for Determining OptimumSettings of Speed Governors for Hydro Generating Units," Eng. Inst. of Canadá, AnnualGeneral Meeting Paper No. 7, 1961, pp. 1-40.

22. Hovey, L. M., "Optirnurn Adjustmenl of Hydro Governors on Manitoba Hydro Sys-tern," Trans. Amer. lnst . of Elect. Engrs., Dec. 1962, pp. 581-586.

23. Parrnakian, J., waterhammcr Analysis, Dover Publications, New York, 1963.

Lein, G. and Parzany, K., "Frequency Response Measurements al Vianden," Water Power,July 1967, pp. 283-286,Aug. 1967, pp. 323-328.

Newey, R. A., "Speed Regulation Study for Bay D'Espoir Hydroelectríc Generating Sta-tion," Paper No. 67-WAfFE-17, presented at Winter Annual Meeting, Amer. Soc. of Mech.Engrs., Dec. 1967.

Schleif, F. R. and Bates, C. G., "Governing Characteristics for 820,000 Horsepower Unitsfor Grand Coulee Third Powerplant," Trans., Inst, of Elect. and Electronics Engrs., PowerApparatus and Systems, Mar.-AprilI971, pp. 882-890.

Stein, T., "Frequency Control under lsolated Network Conditions," Water Power , Sept.1970.

Araki, M. and Kuwabara, T, "Water Column Effect on Speed Control of Hydraulic Turbinesand Governor Improvement," Hitachi Review, vol. 22, no. 2, pp. 50-55.

Chaudhry, M. H. and Ruus, E., "Analysis of Governing Stability of Hydroelectric PowerPlants," Trans., Engineering Inst. of Canada, vol. 13, June 1970, pp. I-V.

Wozniak, L. and Fett, G. H., "Conduit Representation in Closed Loop Simulation of Hydro-electric Systems," Jaur. Basic Engineering, Amer. Soco af Mech. Engrs., Sept. 1972, pp.599-604. (See also discussion by Chaudhry, M. R., pp. 604-605).

Goldwag, E., "On the Influence ofWaler Turbine Characteristic on Stability and Response,"Jour. Basic Engineering, Amer. Saco of Mech. Engrs., Dec, 1971, pp. 480-493.

Vaughan, D. R., "Speed Control in Hydroeleetric Power Systems," thesis submitted toMassachusetts Institute of Teehnology in partial fulfillmenl of the requirements for thedegree of doctor of philosophy, 1962.

Thorne, D. H. and HiU, E. F., "Field Testing and Simulation of Hydraulic Turbine Gover-nor Performance," Trans., Inst. of Elect. and Electranics Engrs. (U.SA), Power Appara-tus and Systems , July -Aug, 1974, pp. 1183-1191.

Brekke, H., "Stability Studies for a Governed Turbine Operating under lsolaled Load Con-ditions," Water Power, London, England, Sept. 1974, pp. 333-341.

Thorne, D. H. and Hill, E. F., "Extension of Slability Boundaries of a Hydraulic TurbineGenerating Unit," Trans., Inst. of Elect. and Electronics Engrs. (U.S.A), vol. PAS-94,JulyfAug. 1975, pp. 1401-1409.

ADDITlONAL REFERENCES

Almeras, P., "lnfluence of Water Inerlia on the Stability of Operation of a HydroclectricSystem," Engineer's Digest, vol. 4, Jan. 1947, pp. 9-12, Feb. 1947, pp. 55-61.

"IEEE Recommended Practice for Preparation of Equipment Specifications for SpeedGoverning of Hydraulic Turbines Intended to Drive Electric Generators," Amer. InSI. afElect. and Electronics Engrs., April 1977.

Dennis, N. G., "Water-Turbine Governors and the Stability of HYdroelectric Plant," WaterPower, Feb. 1953, pp. 65-76; Mar. 1953, pp. 104-109; Apr. 1953, pp. 151-154; May1953, pp. 191-196.

Concordia, C. and Kirchmayer, L. K., "Tie-Une Power and Frequency Conlrol of ElectricPower Syslems," Par! I and lI, Trans., Amer. Inst. of Elect. Engrs., June 1953, pp. 562-572, April1954, pp. 1-14. (See also discussion by H. M. Paynter.)

Sweiecicki, l., "ReguJalion of a Hydraulic Turbine Calculated by Step-by-Slep Method,"Jaur. Basic Engineering, Amer. Saco oi Mech. Engrs., Scpt. 1961, pp. 445-455.

Oldenburger, R., "Hydraulic Speed Governor with Major Governor Problems Solved,"Jaur. Basic Engineering, Amer. Saco uf Mech. Engrs., Paper No. 63-WA-15, 1964, pp. 1-8.

"Waterhammer in Pumped Storage Projects," Internatialla/ Sympasium, Amer. Saco Mech.Engrs., Nov. 1965.

lnternational Code of Testing of Spced Governing Systems for Hydraulic Turbines, Tech-nical Committee No. 4, Hydraulic Turbines, Internationol Electratechnical Cammission,Feb. 1965.

Hydraulic Transients in Nuclear Power Plants 159

HYDRAULIC TRANSIENTS INNUCLEAR POWER PLANTS

CHAPTER 6

CDDlonl pump

Figure 6.1. Schematic arrangemen t of a nuclear power plan t.

6.1 INTRODUCTlON

Types of Reactors

Reactors are classífied ' according to the type of fuel, reactor coolant, and thetype of rnedíurn, called moderator, used to slow down or moderate the high-energy neutrons produced by the fission process. Three rnain types of liquid-cooled reactors are:

l. Pressurized water reactor (PWR). In this reactor, ordinary water is used asa reactor coolant and as a moderator. Water coolant is circulated at high pres-sure (about 14 MPa) by the externa! pumps through the reactor vessel. Waterflows upward through the fuel clusters, then from the vessel into the heat ex-changers and back to the pumps. This loop is called the primary loop. Apressurizer? is provided in this loop to control the pressure of the cooJant. Thetemperature of the coolant rises as it flows through the reactor vessel. There-fore, the pipe on the inlet side of the reactor is called the cold leg, and the pipe00 the exit side is called the hot leg (Fig. 6.2). In the secondary loop, water is

In nuclear power plants, various piping systems are used for cooling and for heattransfer. It is necessary for the design and for the safe operation of these sys-tems that the transient-state conditions caused by various operating conditionsbe accurately known ,

In this chapter, terminology is introduced first; various operations which mayproduce transients are then outlined. Different methods for analysis are dis-cussed, and special boundary conditions for the method of characteristics arederived. The details of a numerical scheme, suitabJe for the analysis of two-phase transient flows, are then presented.

Since various types of reactors are avaílable and each reactor type differs indetails frorn one manufacturer to another, the analysis techniques are ernpha-sized instead of the specifics of design details.

6.2 TERMINOLOGY

R.DCIDTcootanr pump

General

Figure 6.1 shows the schematic arrangement of a nuclear power plan t. Thermalenergy, generated in the nuclear reactor by splitting the atoms, is transferredfrom the reactor by a fluid rnedium called the reactor coolant. The reactorcoolant may be a liquid (e.g., ordinary water, heavy water [D20), liquidsodium); it may be a liquid-vapor mixture, (e.g., boiling ordinary water); or itmay be a gas (e .g., helium, carbon díoxide [C02]). Figure 6.2. Pressurized water reactor.

158

160 Applied Hydrau1ic Transients Hydraulic Transients in Nuclear Power Plants 161

boiled in the heat exchangers forming saturated steam, which is used to drívethe steam turbine.

2. Boiling water reactor (BWR). This reactor is similar to the PWR exceptthat ordinary (líght) water coolant is permitted to boíl in the reactor coreo Thesteam thus produced is separated from the coolant by centrifugal separators(Fig. 6.3) located in the reactor vessel aboye the coreo This steam is then directlyfed into the turbine.

3. Liquid-metal fast breeder reactor (LMFBR). In this reactor, liquid sodium I

is used as the reactor coolant, which is cooled in the intermediate heat ex-changers (Fig. 6.4) and is returned to the reactor. The intermediate heat ex-changers are cooled by a second flow of liquid sodium, which in turn is cooledin a second set ofheat exchangers in which steam is produced for the turbine.

The steam exhausted from the turbine is condensed by means of a condenser,and the condensed water is pumped back into the heat exchangers. A largeamount of water has to be pumped from a lake, river, or ocean to the condenserloop (Fig. 6.1) for this purpose.

/nl.rm.dial~"'01 'Jtchan(lrr

R~aclar core

SI.am lo lurbin.

SI.am (I.n.ralor

F.. dwal.,

coo/anl pump circu/Olin(l pump

Figure 6.4. Liquid-metal fast-breeder reactor.

R.aclor ,,~ss.1

in the primary loop. This system is activated during the early stages of depres-surization.

2. Low-pressure coolant injection system. This system employs low-head,high-capacity pumps; it is intended for supplying coolant following large-sizebreaks in the primary loop and provides coolant during later stages of LOCA.

3. Accumulator System. This is a passive system and is intended to providelarge volumes for large-size breaks. A check valve, whích isolates the accumu-lator from the reactor system during normal operation, opens whenever thepressure in the reactor system drops below the pressure of the accumulator andprovides coolant to the prírnary loop.

Emergency Core Cooling Systems

The emergency core cooling (ECC) systerns? are used to provide coolant forpossible loss-of-coolant accident (LOCA). A number of subsystems are ern-ployed for this purpose:

l. High-pressure coolant injection system. This system employs high-head,low-capacity pumps and is intended to provide coolant during small-size breaks

SI.om- wol.rs.para/or ~ __ +-'"

¡.:::::::_-=:::::.h----,


F•• dwaf.r

Transients in the piping systems of a nuclear power plant are caused by thefollowing (only events initiated in the hydraulic system are consideredj.Y'"

l. Planned or accidental starting or stopping of pumps; power failure to purnps2. Planned or accidental opening or closing of the control valves3. Instability of pumps4. Release of entrapped air or collapse of vapor bubbles5. Wave action at the reservoir water surface (for the condenser piping system)6. Rupture of pipeline.

The starting or stopping of the pumps, the opening or closing of controlvalves under controlled conditions, and power failure to the pumps are con-Figure 6.3. Boiling water reactor.

162 Applied Hydraulic Transients Hydraulie Transients in Nuclear Power P1ants 163

sidered normal operations. Flow variations may be caused by pump instabiJitydue to abnonnalities in the pump characteristics. If the pumps are started intoan empty pipeline or into a partly drained discharge line, severe pressure osciJIa-tions may be produced.

Waterhammer pressure generated by air release or due to collapse of vapor orsteam bubbles has resulted in damage to the pipeline s of a number of nuclearpower plants.'?: 14

Rupture of a pipe in the primary 100p2 and the loss of reactor coolant, i.e.,loss-of-coolant accident, is one of the worst conditions to be considered in thedesigno Rupture of, or a leak in, the pipe in the secondary loop of a LMFBRwill result in sodium-water reaction; transients produced by such an incidenthave to be considered in the design stages.

If heat is added or removed from the system during a transient, then theenergy equation has to be considered in addition to the eontinuity and momen-turn eq uations.

Fonnulation of Mathematical Models

6.4 METHODS OF ANALYSIS

Extreme caution must be exercised in formulating a mathematical model since amodel based on questionabIe simplifying assumptions may yield totally incorreetresults no matter how sophisticated are the numerical techniques employed tosolve the goveming equations. Similarly, using too complex a model, where asimplified model would suffice, results in waste of manpower as welI as eom-puter time. A lumped-system one-dimensional model is one of the simplest ,whereas a multidimensional, distributed-system, two-phase model with heataddition or removal is the most eomplicated.

The following factors are considered whiJe selecting a model for the analysisof a particular system:

1. For a slow transient phenomenon, the compressibility effects may beneglected and the system may be analyzed by using a lumped-system approach.This approach may be used18 for systems in which wl/a« l (see p. 219). Inthis expression w = forcing frequency; 1 = length of the pipe, and a = wavevelocity. As a rough rule of thumb, wl/a = 0.05 may be considered as the upperlimit for the validity of the lurnped-system approach.

2. If the void fraction, a Ca = volume of the gas and vapors per total volumeof the gas-vapor-liquid mixture), is small, then Its effects can be totally neglectedor may be taken into consideration by reducing the waterhammer wave velocity.

3. Flows havíng a large void fraction, have to be analyzed as two-phase flows.If the phases are not separated, then a homogeneous flow model may be used inwhich the gas-vapor-liquid mixture is treated as a pseudofluid and the continuity,momentum, and energy equations for a single-component flow are used in theanalysis. However, if the phases are separated, then the continuity, momentum,and energy equation for each phase have to be used, and the transfer of mass,momentum, and energy between the phases has to be considered. Such a modelis called a separated flow model.

4. If there ís significant heat input or output from the system during the tran-sient conditions under consideration, then it is necessary to inelude the energyequation in addition to the continuity and momentum equations in the analysis.

General Remarks

The followíng two approaches are available for the analysis of hydraulic tran-sients in the piping systems of nuclear power plants:

l. Lumped-system approach2. Distributed-systern approach.

In a lumped-system approach, sometimes referred to as control-volume ormacroscopic approach,' 5 the system is divided into a number of con trol volumes,and an integrated form of the continuity and momentum equations (and theenergy equation , if requíred) is used. Thus, there is no continuous spatial varia-tion (í.e., with respect to distance x in a one-dimensional model) of variousvariables, and they vary with respect to time only. Therefore, a system ofordinary differential equations, instead of partial differential equations, describethe transíents.l" Since the initial steady-state conditions are specified, thetransient analysis using a lumped-system approach may be considered as aninitial value problem.

In a distributed-system approach, both the temporal and spatial variations ofvarious variables are taken into consideration. Thus, a system of partial differ-ential equations subject to appropriate boundary conditions describe the behav-ior of the systern. If there is simultaneous flow of two phases (e .g., water andvapor, air and water), then the flow is called a two-phase flow. 17 Two-phaseflows may be further elassified as homogeneous or separated flows. We willconsider only the homogenous two-phase flows in which the mixture of the twophases may be treated as a pseudofluid. The separated flows are beyond thescope of this book.

Numerical Solution

The numerical method to be used for the solution of the equations describingthe system behavior depends upon whether a lumped-system or a distributed-


system approach is being used. Various available methods are discussed in thefollowing paragraphs.

Lumped-System Approach

As discussed previously, a mathematical model based on a lumped-systern ap-proa eh is comprised of a system of ordinary differential equations. A number offinlte-difference methods 19-20 are available to solve these equations. The author'sexperience with the solution of these equations indicates that the fourth-orderRunge-Kutta method20•21 is quite versatile and accurate. In addition, mostcomputer installations have standard package s available for this method.

Distribu ted-System Approacñ

Distributed-system approaches may be further subdivided into two categories,depending upon whether the flow is single-phase or two-phase:

l. Single-phase flows. The method of characteristics presented in Chapter 3may be used for single-phase f1ows. To account for a small amount of gaseousphase in the liquid, a reduced value of the waterhammer wave velocity (seeSection 9.5) may be used in the analysis.

The transients caused by opening or closing of valves, by starting or stoppingof pumps, or by power failure to the pump-motors may be analyzed by usingthe method of characteristics. A number of eommonly used boundary condi-tions were derived in Chapters 3 and 4, and a few additional ones are developedin the next section and in Chapter 10.

2. Two-phase flows. The two-phase flows may be analyzed by consideringthern as homogeneous or separated f1ows. In the case of homogeneous flow, theliquid mixture is treated as a pseudofluid, and the averaged values of the variousvariables (such as pressure, flow velocity , and void fraction) over a cross sectionare used. The spatial variation ofvoid fraction may be included in the analysis.

In the analysis of separated flows, eaeh phase is treated separately , and thetransfer of mass, momentum, and energy between each phase is taken intoconsideration. Analyses of these flows are beyond the scope of this book.

The following numerical methods have been used for the analysis of homog-eneous two-phase flows:

l. Method of charaeteristics2. Lax-Wendroff's two-step finite-dlfference method3. Explicit finite-difference methods4. Implicit fíníte-dífference methods.

~"': ...., ..."-.

'~-~. 'HydraúlicTráñsléñtsIn Nuclear Power Plants 165

In the method of charaeteristics,22-25 the discontinuities in the derivativescan be handled, and the boundary conditions are properly posed. The method,however, fails because of the convergence of characteristics curves if the wavevelocity is highIy dependent upon pressure, and a shock is formed in the solu-tion. In addition, if an explicit finite-difference scheme is used to solve the totaldifferential equations obtained by this method, the Courant-Fredrich-Levyconditíon+f for the stability of the numerical scheme has to be satisfied. Thiscondition requires the use of small computational time steps, thus making themethod unsuitable for solving real-Jife large systems. The method may, however,be used to verify other numerical schemes by analyzing small, simple systems.

The Lax-Wendroff two-step finite-difference scheme" is the most suitablefor analyzing systems in which a shock forrns. However, the scheme producesoscillatory solution behind the wave front, and a smoothening parameter'" hasto be introduced to avoid this. This introduces numerical damping, which is notpresent in the actual system and which, íf not properly taken care of, maysmoothen the transient peaks. Because the Courant's stability condition has tobe satisfied, the size of time steps is restricted, which makes the scheme un-economical for general analyses.

Explicit finite-difference methods are very easy to program. However, as thestep size is limited by the Courant's stability condltion.i" a large amount ofcomputer time is required. Thus, the method is not suitable for analyzing largesystems.

In the implicit finite-difference methods, the size of the time step is governedby accuracy only and not by the stability considerations.29-32 These methodsare therefore useful for the analysis of large systems. Details of this method arepresented in Section 6.7.

6.5 BOUNDARY CONDITIONS

To analyze a piping systern by the method of characteristics, the boundaryconditions should be known. A number of boundary conditions commonlyfound in the piping systems of nuclear power plants are derived in this section.Note that these conditions are valid only for single-phase flows and are requiredif the method of characteristics of Chapter 3 is used for the analysis.

Condenser

A condenser is comprised of a large number of tubes with water boxes on theends (Fig. 6.5). To derive the boundary condition, the cluster of tubes may bereplaced by an equivalent pipe having a cross-sectional area , Ae, equal to the

166 Applied Hydraulic Transients Hydraulic Transients in Nuclear Power Plants 167

e nde e f be A in which Á V is the change in volume due to change in pressure , Áp. For thepressure changes usually encountered in practice, Á V is srnall, and therefore Vmay be assumed constant. The change in volume, ÁV, during a time step Át,may be determined frorn the con tinuity equa tion

1:ons r u , t

Water bO/f-

Volume

Pipe i.y.

C~.(1, n+l)

Á.JL:: !Át [(Qpi, 11+1 + Qi, n+l) - (Qpi+l, 1 + Qi+l, 1)] (6.2)

in which Q and Qp are discharges at lhe beginning and al the end of the limestep, and subscripts (i, n + 1) and (i + 1, 1) refer lo the section numbers (se e Fig.6.5). If ít is assurned that the pressure is sarne throughout the box, then

(6.3)

in which HP = píezornetric head aboye the datum al the end of lime step. Now,

Áp = 'YÁH = 'Y(Hp. 1 - H¡ n+l)1, n+ , (6.4)

E< iva! t 'Pein which 'Y = specific weight of water.

By substituting Eqs. 6.2 and 6.4 into Eq. 6.1 and simplifying the resultingequation, we obtaín

qu en pI ,Ae= nt At

Water bO/f- i--Woter bO/f

Volume ,.y.

Pipe i (i+ 1, 1) Pipe i+2,

C.

KÁtHpi,n+1 =Hi,I1+1 + 2'Y"~ [(Q;,n+1 - Qi+l.l)+ (Qpi, 11+1 - Qpi+I,I)] (6.5)

The positive and negative characteristic equations (Eqs. 3.18, 3.19) for sections(i,n + 1) and (i + 1, 1) are

(1, n+ 1)

Figure 6.5. Condenser.Qpi, n+1 = ep - eaiHpi, n+1

Qpi+I,1 = en + eai+! Hpi+1, 1

(6.6)

(6.7)

cornbíned area of all the tubes, i.e., Ae = ntA t, in which A t = cross-sectional areaof one tube and nt = number of tubes in the condenser (Fig. 6.5). The headloss in the equivalent pipe is, however, assumed equal to the head loss in anindividual tube. The water boxes may be considered as lumped capacitances,and the compressibility of water and the elasticity of the walls of lhe boxes maybe taken into consideration.

Equations for the upstream water box are derived below; equations for thedownstream box may be derived in a similar manner.

Let the volume of water in the box be .JL,and let the combined effective bulkmodulus ofwater inside the box and the vessel walls be K. Then , by definition ,

in which ep, en, and ea are as defined by Eqs. 3.20 through 3.22. Substitutionof Eqs. 6.3, 6.6, and 6.7 into Eq. 6.5 yields

KMHpi,n+1 =H¡,n+1 + 2'Y-Jl [(Q¡,n+l - Q¡+I,I)+ (ep - en)]

(6.8)

Hence,

(6.1)

2'Y.JL { KÁtH = H· + - . -.Pi,n+l 2 .JL+KÁt(C .+e.) ',n+1 2.JL[(Q"n+1 Q'+I,I)'Y a, a,+1 'Y

+ (ep - en)]} f6.9)


Now, Hpi+I,1 ' Qp¡, 11+1' and Qpi+l.l may be determined from Eqs. 6.3, 6.6,and 6.7, respectively.

Entrapped Air

Let us consider a volume of air en trapped in a pipe having liquíd on either sideas shown in Fig. 6.6. If the entrapped air follows the polytropic law for perfectgases, then

(6.1 O)

in which H;. and -J.Lp. are the absolute pressure head and volume of the en-air arrtrapped air, respectively, and m = exponent in the polytropic gas law. The valueof the constant, e, in Eq. 6.10 is determined from the initial steady-stateconditions.

From the continuity equation, the following equation may be written for thevolume of the air:

-J.LPair=-JLair+-t t.t{(Q¡+I, 1 +Qp¡+I,I)- (Q¡,n+l + Qp¡,n+l)} (6.11)

The positive and negative characteristic equations (Eqs, 3.18, and 3.19) forsections (i, n + 1) and (i + 1,1) are

Qp¡,n+l =ep - ea¡Hp¡,n+l

Qp¡+I,1 = en + ea¡+l Hp¡+1 ,1

(6.12)

(6.13)

in which ep, en, ea¡, and ea¡+1 are as defined by Eqs. 3.20 through 3.22. Ifthe pressure of air at any instant is assumed lo be same throughout its volume,then

(6.14)

In addition,

H*. =H +H . - ZPa.r b P"n+1 (6.15)

in which Hb = barometric head, and z = height of the pipeline aboye datum.

AirPip, Pip, i+1..•.•.•••.•.•.•..•.•.•....·;;.··.·;I·:t:¡••IIItl

(i, n+l) ( i+ 1, 1)

Figure 6.6. Entrapped airo

Hydraulíc Transients in Nuclear Power Plants 169

Now we have six equations (Eqs. 6.10-6.15) in six unknowns, namely,Hp* . ,air-J.Lp . , Qp. +1' Qp. 1 i ' Hp. 1 l' and Hp. +1' Elimination of the first fiveau 1, n 1+ • 1+ , 1, nunknowns from these equations yields

(6.16)

in which

(6.17)

Equation 6.16 rnay be solved for Hp¡ 11+1by an iterative technique, such as thebisection method.i? The values of th~ other unknowns may then be determinedfrom Eqs. 6.1 O through 6.15.

Pipe Rupture and Failure of Rupture Discs

Sometimes, rupture discs are installed, which fai! if the pressure inside the pipe-line exceeds a specified Iimit. Because of this controlled failure , extensive dam-age to the pipeline is avoided.

The ruptured discs or pipe-break may be analyzed as an orifice, and the bound-ary conditions derived in Chapter 3 may be used for this purpose. If there isback pressure from outside the pipeline, then t.H is the difference between thepressure inside and outside the pipeline.

Total pipe-break may be analyzed considering it as a fixed-opening valvelocated at the ends of pipe at the location of the pipe-break.

6.6 LOSS-OF-COOLANT ACCIDENT

A sud den rupture of a pipe in the primary loop and the resulting loss of reactorcoolant is referred to as loss-of-coolant accidento The size and location of thepipe-break in this hypothetical accident is selected, which results in the maxi-mum c1adding temperatures. In a PWR, a complete rupture of the pipe connect-ing the pump to the reactor (i.e., cold leg) is assumed.? In a BWR, a completeand instantaneous circumferential rupture is assurned? of the largest pipe in oneof the suction lines of the recirculation system.

The analysis of LOCA is done by using a number of mathematical models, andthe output of one model serves as input to the other model. For example, theaverage conditions in the core, such as flow, pressure, and temperature, aredetermined by a mathematical model of the primary loop. Using these com-puted conditions as input, the maximum cladding temperature is determinedusing another mode!. These two models are used in an iterative manner atevery time step or after a number of time steps.

In the mathematical model of the primary loop, the continuity , dynamic, and


energy equations are sol ved . The heat addition in the core to the fluid of theprimary loop and the heat loss in the heat exchangers from the primary loop tothe secondary loop are assumed to be distributed along the pipe lengths repre-senting the core and the heat exchangers. The secondary loop ís considered as aheat sink and is therefore decoupled from the primary loop. Similarly, thecontainrnent vessel is decoupled by specifying a back pressure at the hypotheti-cal pipe-break.

A numerical method is presented in the next section to solve the equationsdescribing the flow conditions in the primary loop following a pipe-break.

6.7 IMPLlCIT FINITE-D1FFERENCE METHOD FOR ANALYZINGTWO-PHASE TRANSIENT FLOWS

General

In this section, the details of an implicit finite-difference method, presented byHancox et a1.28, 30, 31 are outlined. In this method, the characteristic form ofthe equations governing the homogeneous two-phase flows are used. Therefore,the method is as similar as possible to the method of characteristics. However,as it is not necessary to satisfy the Courant's stability condition, a larger size oftime steps can be used, thus making the method suitable for the analysis of largesystems.

Governing Equations

The following equations describe28,30 the homogeneous, two-phase flow inpipes including the heat addition or loss:

l. Momentum equation

av av 1 ap-+V-+--=c¡at ax p ax (6.18)

2. Energy equation

ah av ah-+a2-+ V-=cal ax ax 2

(6.19)

3. Continuityequation

ap 2 av ap-+ pa -+ V-=cal ax ax 3(6.20)


in which

el = - F - g cos e (6.21)

[ ap I v dA] (6.22)C2 = a2 (q + VF) - - - -ap h A dx

[ ap I pV dA 1 (6.23)C3 = -a2 (q + VF) - + - -ah p A dx

a=Cpl +_!_~I rl

/

2(6.24)ap h p ah Ip

q = kwAw (Tw - T¡) (6.25)p

F = (4f T* + ~ ) VIV¡ (6.26)d 1 2

and in which a = wave velocity in the fluid; q = wall-to-flow heat transfer; p =density of the fluid; F = friction force per unit mass; kw = wall-heat transfercoeffícient; Tw = wall temperature; T¡ = temperature of the fluid; f = frictionfactor; k/l = distributed loss coefficient; A = cross-sectional area of the pipe;de = equivalent pipe diameter; g = acceleration due to gravity ; h = mixtureenthalpy; p = pressure; t = time; V = fluid velocity; x = distance along the pipeaxis; and e = angle between the pipe axis and horizontal.

In addition,

T* = 1 for x' < O and x' > 1 }

T* = appropriate two-phase multiplier for O < x' < 1(6.27)

f= 0.046RÑo.1 for RN > 2000 }

64t= - for RN < 2000RN

h =x'hg + (1 - x') h¡

p = o.Pg + (l - 0.) PI

(6.28)

(6.29)

(6.30)

In these equations, x' = thermodynamic quality; RN = Reynolds number; o. =void fraction = volume of vapor per total volume of the vapor-liquid mixture;and the subscripts g and 1 refer to the vapor and the liquid, respectívely.

) 72 Applied Hydraulic Transients

Conversion of Governing Equations into Characteristic Form

By following the procedure outlined in Section 2.4, it can be preved that Eqs.6.18 ~hrough 6.20 form a set of hyperbolic partíal differential equations. Theseequations can be con verted into ordinary differential equations by the methodof characteristics as follows:

Multiplying Eq. 6.18 by a linear multíplier, ~'(, adding the resulting equationlo Eq. 6.20, and rearranging the terrns, we obtain

AI[aa~ +(v+ P:12)~:]+[:~+(v+ ~1):~]=C3+AICI (6.31)

Let

pa2 dx AV+-=-=V+___.!.

Al dt p(6.32)

or

Al = ±pa

Substitution of X¡ = pa and Al = -pa into Eqs. 6.31 and 6.32 yields

pa[~~ +(V+a) ~:]+[:~+(V+a) :~]=c3+pacl

(6.33)

(6.34)

if

dx-= V+adt

(6.35)

and

[av av] [ap ap]-pa -a +(V-a)- + -+(V-a)- =c3-pact ax at ax I

(6.36)

if

dx-= V-adt

(6.37)

Note thal Eq. 6.34 is valid only if Eq. 6.35 is satisfied; i.e., Eq. 6.34 is validalo~g characteristic curve, dx/dt = V + a, in the x-t plane. Similarly, Eq. 6.36 isvalid along the characteristic curve dxldt = V-a.

Now, multiplying Eq. 6.20 by a linear multiplier (A2)' adding the resulting

Hydraulic Transients in Nuclear Power Plants ) 73

equation lo Eq. 6.19, and rearranging the terrns, we obtain

(ah ah) (ap ap) av-+V- +Az -+V- +a2(l+PA2)-=A2c3+C2at ax at ax axLet

or

Substitution of Eq. 6.40 into Eq. 6.38 yields

(ah + V ah) _ _l__( ap + V ap) = C2 - _l__ C3at ax p at ax p

and

dx-=Vdt

(6.38)

(6.39)

(6.40)

(6.41)

(6.42)

Equation 6.41 is valid if Eq. 6.42 is satisfied. Note that, unlike Eqs. 6.34 and6.36, which were valid along the characteristic curves dxldt = V ± a, Eq. 6.41 isvalid along the path of a particle in the x-t plane.

Equations 6.34,6.36, and 6.41 can be written as:

1. Along dxldt = V + a,

pa dV + dp = C3 + pac¡

2. Along dxldt = V - a,

-padV+dp =C2 - pac ,

3. Along dxldt = V,

pdh - dp = PC2 - C3

(6.43)

(6.44)

[6.45)

These ordinary differential equations (Eqs. 6.43-6.45) are called compatibilityequations. Thus, Equations 6.34, 6.36, and 6.41 are ·in the so-called character-istic form, These equations can be combined into matrix form as

au auB-+C-=Dat ax (6.46)


in which

(6.47)

(6.48)

[

pa(V+a)

c= O

-pa(V - a)

~ (v+t]O (V-a)

(6.49)

{e3 +~ael}

D = e2 - -e3p

C3 - pacI

(6.50)

Formulation of Algebraic Equations

Referring lo Fig. 6.7, Eq. 6.46 can be expressed in the flnite-dífference form as

U~+I- U~ U~+1- U~+1Bi.-'--' + ei.' ,-1 = D{

, Át ' ÁX¡_I(6.51 )

U/+1 - Ui. Ui.+1 - Ui.+1Bi.' , + Cl '+1 , = D{

, Át I ÁX¡

Note that the spatial derivatives are approximated by the left-hand finite differ-ences (Fig. 6.7) in Eq. 6.51 and by the right-hand finite differences in Eq. 6.52.Which of these two equations should be used , depends upon the flow directionand whether the flow is subsonic or supersoníc. For subsonic flows, the left-hand finite differences are used for the positive characteristic (V + a equation),and the right-hand finite differences for the negative characteristic (V - a equa-tion); in the Vequation, the right- or left-hand finite differences are used depend-ing upon the flow direction. This conventíon makes the finite-difference schemeas similar as possible to the method of characteristics.

(6.52)

Hydraulic Transients in Nuclear Power P1ants 175

j+1,~----l\.~--1 I

II II I"

i-I 1+1

(a) (Y+al Equation

j+IL..-----l"\-, --1 I

1 II I1 I-,

r:¡_---- .....I r :: :I II II I-,

j+1

i-I 1+1 i-I 1+1

(1) Positiva V(2) Nagativa V

(b) V Equotion

j+1(.¡,----~)I r= f.'I II II I-,

i-l 1+1

(e) (Y-a) Equation

Figure 6.7. Notation for implicit finite-difference scheme (subsonic f1ow).


By intro.duci~g a weighting rnatrix, W, and by using the precedíng conventionfor the finite-difference approximations Eqs 6 51 and 6 52 b bi, " . may e com ined as

M·· ui"l + M ui+1 M j+1I,I-! ¡-I t.iv¡ + ¡,;+1 u¡+1 =N¡ (6.53)

Usually, W = 1, but where the property derivatives are changing rapidly W maybe used as a weighting matrix. The components of the 3 X 3 matrices' M ..M¡,¡, and M¡,i+! and the column vector N¡ are 1,1-1,

M¡,¡_I =E[-WCi+(I- W)e~~t-l IBi-I- C¡_l)] (6.54)

M w[( Sx¡ Ax· ]¡,¡ = 1- E) ~ B¡ + E A~-I B¡ - (I - 2 E) C¡

+ (1- W) [ECi_l - (1 - E) C¡+l] (6.55)

M¡,¡+l = (1 - E)[ WCk + (1 - W)( ~~i IB¡+1 + Ci+l )] (6.56)

N¡ = W[(1 - E) áx¡ + E AXi-l) D¡ +~I - E) ~~i + E A~~ -1) B¡u{ ]

+ (1 - W)[ED¡_1 AXi-I + (I- E) D¡+l Ax. + AX¡_1 EB. Uf1 At 1-1 1-1

Ax· ]t A,' (I - E) B¡+I uf+l (6.56)

. The matrix E depends upon the type (subsonic or supersoníc) and the dírec-tion of flow as follows:

1. Supersonic flowa. From left to right

E = I (6.57)b. From right to left

E=O (6.58)2. Subsonic flow

a. From left to right

(6.59)


b. From right to left

(6.60)

Equation 6.53 is al so used at the boundaries except that those flnite-differenceequations having subscripts of either O or n + 2 are dropped frorn the set and arereplaced by the appropriate conditions imposed by the boundary.

Computational Procedure

The pipeline is divided into a number of reaches. Equation 6.53 is used at theinterior points. This equation is also used at the boundaries except that theequations having the terms with subscripts O and n + 2 are dropped and are re-placed by the conditions imposed by the boundary. (If these conditions arenon linear , they are linearized.) Since the coefficients of these equations are ex-pressed in terms of the known variables al the beginning of the time step, theresulting algebraic equations are linear. If the pipe is divided into n reaches, thenwriting Eq. 6.53 at the interior points and at the boundaries as described aboyeresults in 3(n + 1) linear equations in 3(n + 1) unknowns. These equations havea banded matrix structure as shown in Fig. 6.8. The structure of the bandmatrices, showing the flow direction dependence, is given in Fig. 6.9. Due tothe banded structure, these equations can be efficiently solved using a special-purpose algorithm based on the Gauss elimination method, which takes advan-tage of the zero elements.

Verification

The preceding finite-difference scheme has been verified by comparing theresults computed by using thls scherne fOI standard problems with those corn-puted by using the method of characteristics and with the experimental re-sults.28, 30, 31 Figure 6.10 shows the comparison with the experimental resultsreported by Edwards and O'Brien.33 The experimental set-up was the blow-down of a closed-end, cylindrical pipe filled with high enthalpy water, whichwas suddenly opened to the atmosphere. The tube was 4 m long and had aninside diameter of 32 mm. The initia! pressure was 7 MPa, and the initia! tern-perature was 243°C. It is clear from Fig. 6.1 O that the comparison between thecomputed and measured results is satisfactory.


r:

Mil MI2

M21 M22 M231

\\

M¡,HI

Mn.I,n Mn+1"+1'

• • • •t;> t;> • • O O

• • • •• • • •

t;> t:> • • O O

• • • •• • • •

t:> t:> • • O O

• • • •

Hydraulic Transients in Nuclear Power Plan ts 179

k-2 k-I k k+1 k+2

UI NIk-I

U2 Nz

k• • ,1• •• • k+1

Ni=

0 "0" rtlro tlltlmtl"f for V = posifi"tI o"d "tlgOfi"tI

• •• • G "0" rtlro tlltlmtl'" o"ly "htl" V = posifi"lI

• • ~ tlltlmtl'" o"'y whtl" V = nllgofi",non rero

Figure 6.9. Structure of diagonal elements. (After Banerjee et aI.28

)

layout of the motor-dríven feedwater systern is shown in Fig. 6.11: the turbine-driven auxiliary feedwater system is not shown. The pipe-break in feedline No.22 occurred inside the eontainment near the eontainment penetration (Fig. 6.12).

Un Nn

Un•1

Description of Incident

On Novernber 13, 1973, the Indian Point Unit No. 2 was operating at about8 pereent power, with one rnain boller feed pump in service and the main turbinerunning at 1750 rpm. At about 7:38 A.M., the main turbine and the operatingrnain boiler feed purnp were tripped due to high level in Steam Generator (SG)No. 23. Simultaneously, both motor-driven auxiliary purnps were started. At7:44:41 A.M., the reactor tripped due to low-low level in SG No. 21. Severa]minutes after the reactor trip, there was a severe shock in the feedwater line No.22 and water level in SG No. 22 could not be restored. At about 8:30 A.M.,

there were indications that the containment dew point and temperat ure wererising, and a second shock was observed when some adjustments were made toincrease the auxiliary feedwater flow to SG No. 22. A third shock was observedat about 8:40 A.M. when the stearn-driven auxiliary was started up to feed SGNo. 22. Loop 22 was isolated, and eooldown was ínítiated at 10: lOA .M.

Figure 6.8. Structure of system of algebraic equations.

6.8 CASE STUDY

Studies undert~ken by the Westinghouse Electric Corporation.P>" following the~upture of a pipe (0.46-m diameter, 25-mm wall thickness) due to waterhammer10 the feedwater line of Unit No. 2 ~Consolidated Edison Company of NewYork's nuclear power plant at Indian Point, are summarized in this section.

Configuration of Feedwater Une

The Indian Point Unit No. 2 is equipped with a four-loop Westinghouse PWRsteam su~ply system. There are two turbine-driven main-feed pumps, and twomotor-dnven and one turbine-driven auxiliary feedwater pumps. The sehematie


7

~I--- Measurtld

\ - - - - Compu'~d\

\\\

\\\\\

\.'-- -------- 1---

v> -r

LI

6

5

........~4

...."":_,...

11..

2

2 4Time (ms)

6 8

4 I--- "'easured

---- ---- t---_-... ---- Compuled

r",--~ -~'-:::::

~" 1----

""'" <,<,---

~2

".."....11..

/

50 /00 250/50 200rim« (ms)

300 250

Figure 6.10. Comparison of computed and measured results, (After Hancox, andBanerjee30)

TII


t

21

Mofor dri"fln nAuxiliary (:)Fflfldwafflr pump

Cond,nsaffl pumps

13Figure 6.11. Schematic layout of feedwater system. (After Aanstad )


S/(ltlfn 9/1n",alor __ ~

El 29.26 m

~ SuPPOrl or rrslroi", 1101111 (',pI,./..._______

-SI.p, "---1----4..---

EI/7.9/m

i->

Figure 6.12. Feedwater piping of steam generator No. 22. (After Aanstad 13)

Possible Causes of Shock

Three possible shock mechanisms were considered:

l. Waterhamrner due to quick closure of a feedwater regulating valve.2. Shock. due t.o.ba~kflow of hot water into the main feedline from se No. 22.3. Flow instability In the feedwater line when the feed ring was uncovered.

A brief description of the investigation for each of these causes follows.

Waterhammer

Since the line break occurred at about 8 percent power level, an instantaneouscIosure. of a feedwater control valve or a feedwater check valve would not causeappreclable waterhammer. Thus, it would have to be assumed that the fe dliwas .da~aged during a previous incident and that the pipe finally ruptured durínzthe mCI~ent on November 13. Actually, shocks had been noted during a previou;plant trip from 360 MWe (30 percent power) due to low-low level in one steamgenerator.


A computer program was used to determine the maximum pressure due to:

1. instantaneous closure of the control valve with the check valve acting2. control-valve closure in 50 ms with check valve acting3. instantaneous control-valve closure without check valve.

Analytical investigations indicated that the maximum stress for 8 percentpower could not cause pipe rupture; but for 30 percent power, the pipe stress atthe containment penetration exceeded the specified minimum-yield strength ofthe pipe.

Backflow

Field investigations indicated that backflow from se No. 22 could occur underconditions prevalent during the incident on November 13. The hot water couldflow into line No. 22 for a period of time, and when the reactor tripped, thesecondary side could have gradually depressurized, and the hot water could haveflashed into steam and produced a trapped steam bubble in feedline No. 22.This trapped stearn bubble could have collapsed abruptly when it came in con-tact with the cold auxiliary feedwater, and the resulting shock could have rup-tured the lineo

Steam-Header Flow Instabiiity

Following the faJl in the steam-generator liquid level below the feedring, theinlet sections of the feedline and the feedring could have been drained. If theauxiliary purnp does not keep these sections full of water, a free surface willforrn, which wíll be available for condensation of counterflow stearn moving upthrough the feedwater-ring injection holes. If the flow of stearn is sufficient,ripples will forrn on the free surface , which will be followed by flooding of thecross-section. This local flooding will cause isolation of a condensing stearnbubble, which will then collapse, causing a high-pressure pulse.

An analyticaI analysis of this phenomenon is impossible. Therefore, investi-gations were carried out on al: 12-scale rnodel in which the low pressure of thecondensing stearn was simulated by a vacuum lineo These experirnents revealedthat a periodíc slug of water would form within the horizontal section of thefeedwater lineo

Figure 6.13 shows the sequen ce of events. As the auxiliary feedwater isinitiated to cornpensate for the falling steam pressure, cold water pours into thefeedpipe, and a standing wave builds up at the juncture of the therrnal sleeveto the feedring (Fig. 6.13a). Depending upon the flow rate, the standing wave


Oulsidtl steam gtlntlrolor ' /nsidtl sltlom gtlntlrofor

Ttlt!, ¡Ftltldring

;Flltld/intl

\

ho/tls

,),---!:;.

---Oulsidtl sltlom gtlntlrolor /nsidtl sltlom gtlntlrofor

,------(d) ~ I --_ Woltlr s/ug

•••'.,,~.(e)

Figure 6.13. Schematic representation of a slugging mechanism. (After Pulling 14)

seals the feedring (Fig. 6.l3b) yielding an entrapped steam bubble. This en-trapped steam condenses due to rapid heat transfer, and a slug of water acceler-ates from the steam generator into the feedwater pipe (Fig. 6.l3c) until it hits anobstruction, usually the first elbow in the feedpipe. A great amount ofenergy


is dissipated at the sleeve, and the remainder is transmitted into the pipe. Theslug then returns to the steam generator, and the cycle is repeated with a dimin-ishing energy (Fig. 6.13d and e).

Plant Modifications

Based on the results of the investigations, the following modifications were madeto prevent recurrence of the November incident:

l. The feedwater control valves were equipped with snubbers to increase theclosing time.

2. The main feedwater isolation valves were rnodified so that they will closeby motor-operated actuators when the possibility of backflow exists.

3. The long horizontal run of feedwater line No. 22 was modified (Fig. 6.l4)to reduce the entrapped steam volume in case the water level drops belowthe feedwater ringo

Verification Tests

To verify the effectiveness of the plant modifications, a systematic series of testswere conducted. lt was found that low-intensity shocks still occurred in sornecases. The tests indicated that the shock mechanism was related to the drainage

507 m NDrilDnfal 'un

Figure 6.14. Modified feedwater piping of steam generator No. 22. (After Aanstad13)

186 Applied Hydraulic TransientsHydraulic Transients in Nuclear Power Plants 187

of the feedring and that the shocks occurred only al higher auxiliary feedwaterflow rates, In additlon, it was found that:

l. Prior to the shock, with the line partially drained, thermocouples located atthe top and at the boUom of the feedwater line showed steam and auxiliaryfeedwater temperatures, respectively.

2. Shortly after the shock, the thermocouples located on the outside surfaceof the pipe showed that the ternperature inside the pipe changed suddenly,which is indicative of a sudden condensation.

3. a junction of three pipes4. a constant-head reservoir.

Neglect the form losses for cases 1, 3, and 4, and Iinearize the equation for

case 2.

REFERENCES

6.1 Derive the boundary conditions for an air pocket entrapped in a pipeline.Assume that the air is released slowly through an air valve as the air pres-sure increases.

1. Lecture Notes, Nuclear Power Syrnposium, Atomic Energy of Canada Ltd., Sheridan

Park, Ontarío, Canada, 1972.2. Ybarrondo, L. l., Solbrig, C. W., and Isbin, H. S., "The Calculated Loss-of-Coolant

Accident: A Review," Motlograph Series No. 7, Amer. Inse. of Chem. Engrs., New

York, 1972, pp. 1-99. .3. Belonogoff, G., "Computer Simulation of Waterhammer Effects," Jour., Transportatton

Engineering o«; Amer. Soco of Civü Engrs., vol. 98, Aug, 1972, pp. 521-530.4. Okrent, D., Holland, L. K., Moffette, T. R., and Moore, J. S., "On Calculational Methods

in Power Reactor Safety ," Amer. Nud. Soc., Top. Meet., Univ. of Mich., Ann Arbo~,April 1971 (published by U.S. Atomic Energy Cornmission, CONF-730414l, April

1971, pp. DOI-25.S. Gwinn, J. M. and Wender, P. J.. "Start-up Harnrner in Service Water Systerns," Paper

No. 74-WA/PWR-8, presented at Winter Annual Meeting, Amer. Soco of Mech. Engrs.,

Nov. 1974,8 pp.6. Pool, E. B., Porwit, A. J., and Carlton, J. L., "Prediction of Surge Pressure Frorn Ch~ck

Valves for Nuclear Loops," Paper No. 62-WA-219, presented al Winter Annual Meeting,

Amer. Soco of Mech. Engrs., Nov. 1962,8 pp. .7. Harper, C. R., Hsíeh, J. S., and Luk, C. H., "An Analysis of Waterhammer Loads m a

Typical Nuclear Piping System," Paper No. 75-PVP-53, presented at Second NationalCongress on Pressure Vessels & Piping, San Francisco, Amer. Soco of Mech. Engrs., June

1975,8 pp,8. Liebermann, P. and Brown, E. A., "Pressure OsdUations in a Water-Cooled Nuclear

Reactor Induced by Water-Hammer Waves," Jour., Basic Engineering ; Amer. Soco ofMech. Engrs., vol. 82, Dec.1960, pp. 901-911.

9. Shin, Y. W. and Chen, W. L., "Numerical Fluid-Hammer Analysis by the Method ofCharacteristics in Cornplex Piping Networks," Nucl. Engineering and Design, vol. 33,

1975, pp. 357-369. . .10. Penzes, L. E., "Theory of Purnp-Induced Pulsating Coolant Pressure In Pressunzed

Water Reactors," Nucl. Engineering and Design , vol. 27, May 1974, pp. 17~~188.11. O'Leary , J. R. and Patel, Y. A., "Transient Analysis of a Class A Nuclear Piping Sys~e~:

A Comparative Study of Three Large Computer Codes,' Pressure Vessels & PipingConf., Amer. Soco o{ Mech. Engrs., Miami Beach, June 1974, pp. 51-58.

12. Bordelon, F. M., "CalcuIation of Flow Coastdown After Loss of Reactor CooIantPump," Report No. WCAP-7973, Westinghouse Electric Corporation, Pittsburgh,

Pennsylvania, Aug. 1970.13. Aanstad, O., Investigation of a Feedwater Line Incident at Consolidated Edison Corn-

pany of New York's Indian Point No. 2 Plant," Report No. WCAP·8433, WestmghouseElectric Corporation, Pittsburgh, Pennsylvania, Feb. 1975. .

14. Pulling, W. T., "Literature Survey of Water Hammer Incidents in Operatmg Nuclear

To reduce drainage of the feedwater ring following normal plant trips, theoriginal nozzles in the bottom of the feedwater ring were plugged and then re-placed with 3S larger holes in the top of the ringo A short J-bend pipe waswelded to each hole to direct the feedwater vertically downward. Extensivetests conducted after rnaking this modification showed that no shocks wereobserved.

6.9SUMMARY

In this chapter, the terminology was fírst introduced, and causes that may pro-duce transient conditions were then outlíned. Various numerical methods avail-able for the transient analysís were discussed. A nurnber of boundary conditionscommonly found in the pipíng systems of the nuclear power plants were derived.The loss-of-coolant accident was then briefly described. This was followed bythe presentatíon of the detaíls of an implicit flnite-dífference scherne that issuítable for analyzing transíents in homogeneous two-phase flows.

PROBLEMS

6.2 Develop the boundary conditions for a condenser in which the condensertubes al higher level are unsubmerged as the water level in the water boxesfalls. (Hint: Replace the condenser tubes with a number of paraIlel pipeslocated at different levels. Consider the flow through only those pipes thatare submerged.)

6.3 For the implicit finíte-difference method for analyzing two-phase flowspresented in Section 6.7, write the end equations for:

1. a series junction2. a val ve

188 Applied Hydraulic Transients Hydraulic Transients in Nuclear Power Plants 189

Power Plants," Report No. WCAP-8799, Westinghouse Electric Corporation, Pittsburgh,Pennsylvania, Nov. 1976.

15. Esposito, V. J., "Mathernatical Aspects of Reactor Blowdown," paper presented atComputational Methods in Nud. Engineering Meeting, Charleston, South Carolina,Amer. Nucl. Soc., April1975, 5 pp.

16. Wylie, C. R., Advanced Engineering Matñematics, Third Ed., McGraw-Hill Book Co.,New York, 1965, p. M5.

17. Wallis, G. B., One-Dimensional Two-Phase Flow, McGraw-HiII Book Co., New York,1969.

18. Chaudhry, M. H., "Resonance in Pressurized Píping Systerns," thesis presented to theUniversity of British Columbia, Vancouver, British Columbia, Canada, in 1970, inpartial fulfillment for the degree of Doctor of Philosophy.

19. Richtrnyer, R. D. and Morton, K. W., Difference Melhads [or lnitial Value Problems,Interscience Publications, New York, 1967.

20. McCracken, D. D. and Dorn, W. S.• Numerical Methods and FORTRAN Programming,John Wiley & Sons, Inc., New York, 1964.

21. Butcher, J. C., "On Runge-Kutta Procedures of High Order," Jour. Austr. Math. Soc.,vol. 4,1964.

22. Hancox, W. T., Mathers, W. G., and Kawa, D., "Analysis of Transient Flow-Boiling;Application of the Method of Characteristics," paper presented at 15th National HeatTransfer Conference, San Francisco, Amer. Inst. of Chem. Engrs., Aug, 1975.

23. Martin, C. S..and Padmanabhan, M., "The Effect of Free Gases on Pressure Transients,"L 'Energia Elettrica, no. 5, 1975, pp. 262-267.

24. Martin, C. S., Padmanabhan, M., and Wiggert, D. C., "Pressure Wave Propagation inTwo-Phase Bubbly Air-Water Mixtures," Proc., 2nd Conference on Pressure Surges,published by British Hydromechanics Research Association, England, Sept. 1976.

25. Abbott, M. B., An Introduction to the Method of Characteristics, American Elsevier,New York, 1966.

26. Lax, P. D. and Wendroff, B., "Systern of Conservation Laws," Camm. on Pure andApplied Math., vol. 13, 1960, pp. 217-227.

27. Kranenburg, C., "Gas Release During Transient Cavitation in Pipes," Jaur., Hyd. Div.,Amer. Soco of Civ. Engrs., vol. 100, Oct. 1974, pp. 1383-1398.

28. Banerjee, S. and Hancox, W. T., "On the Development of Methods for AnalysingTransient Flow-Boiling," paper submitted to Inter. Jour. Multiphase Flow.

29. Spencer , A. C. and Nakarnura, S., "Irnplicit Characteristie Method for One-DimensionalFluid Flows," Trans. Amer. Nuclear Society , vol. 17, Nov. 1973, pp. 247-249.

30. Hancox, W. T. and Banerjee, S., "Numerical Standards for Flow-Boiling Analysis,"Nuclear Science and Engineering, 1978.

31. Mathers, W. G., Zuzak, W. W., McDonald, B. H., and Hancox, W. T., "On Finite-Differ-ence SoIutions to the Transient Flow-Boiling Equations," paper presented lo Cornrnit-tee on the Safety of Nuclear Installations, Specialists Meeting on Transient Two-PhaseFlow, Toronto, Ontario, Aug. 1976 (proceedings 10 be published by Pergamon Press).

32. Harlow, F. H. and Amsden, A. A., "Numerical Calculation of Almost IncornpressibleFlow," Jour. Comp. Physics, vol. 3, 1968, pp. 80-93.

33. Edwards, A. R. and O'Brien, T. P., "Studies of Phenomena Connected with the Depres-surization of Water Reactors," Jour. British Nuclear Energy Society, vol. 9, 1970, pp.125-135.


Arker, A. J. and Lewis, D. G., "Rapid Flow Transients in Closed Loops." Reactor HeatTransfer Conference, U.S. Atomic Energy Comrnission, 1956.

Morí, Y., Hijikata, K., and Komine, A., "Propagation of Pressure Waves in Two-PhaseFlow," Internacional Jour. Multiphase Flow, vol. 2, 1957, pp. 139-152.

Standart, G., "The Mass, Momentum, and Energy Equations for Heterogeneous Flow Svs-tems," Chem. Eng. Sci., vol. 19, 1964, pp. 227-236.

Fabie, S., "BLODWN-2: Westinghouse APD Computar Program for Calculation of FluidPressure, Flow, and Density Transients during a Loss-of-Coolant Accident," Trans. Amer.Nucl. Soc., vol. 12, 1969, p. 358.

Henry, R. K, Grolmes, M. A., and Fauske, H. K., "Propagation Velocity of Prcssure Waves inGas-Liquid Mixtures," Callcurrcnt Gas-Liquid Flow, editcd by Rhodes, E. and Seott, D. S.,Plenum Press, Ncw York, 1969, pp. 1-18.

Kalinin, A. V., "Derivation of Fluid-Mechanics Equations for a Two-Phase Medium withPhase Changes," Heat Transfer, Soviet Research , vol. 2, May 1970, pp. 83-96.

Boure, J., "Two-Phase Flow Transients and the Stability of Once-Through Steam Genera-tors," International Heat Exchange Conf., Paris, France, no. 22, June 1971.

Albertson, M. L. and Andrews, J. S., "Transients Caused by Air Release," in Control ofFlow in Closed Conduits, edited by Tullís, J. P., Colorado State University, Fort Collins,Colorado, 1971, pp. 315-340.

Rose, R. P. and Cooper, K. F., "Analysis of a PWR System for a Complete Loss of Feed-water Accident Withoul Reactor Trip," Top Meet OH Water Reactor Safety; Salt LakeCity, published by AEC (Conf. 730304), March 1973.

Gonzalez, S., "Exact Solutions fOI Flow Transients in Two-Phase Systerns by the Method ofCharacteristics," Jour. Heat Transfer, Amer. Soc. of Mech. Engrs., vol. 95, Nov. 1973, pp.470-476.

Porsching, T. A., "Fluid Network Analysis," Trans. Amer., Nucl. Soc., vol. 17, 1973, p. 242.Hold, A., "Analytical One-Dimensional Frequency Response and Stability ModeJ for PWR

Nuclear Power Plant," Nucl. Engineering and Design, vol. 33, 1975, p. 336.Martin, C. S., "Entrapped Air in Pipelines," Proc., Second Conf on Pressure Surge, London,

published by British Hydrornech. Research Assoc., Cranfield, England, Sept. 1976.Mikielewicz, J., Chan, T. C., Wilson, D. G., and Goldfish, A. L., "A Method for Correlating

the Characteristics of Centrifugal Pumps in Two-Phase Flow," Paper No. 76-WA/FE-29,Amer. Socoof Mech. Engrs., 1976, 11 pp.

Transients in Long Oil Pipelines 191

CHAPTER 7

TRANSIENTS IN LONG OIL PIPELINE S

Pumpi"g heod Hydroulic grade ¡¡"e

Figure 7.1. Schematic diagram of a long oil pipeline.

7.1 INTRODUCTION

pumping units. In addition to I, the values of the bulk modulus of elas~icity,K, and the density, p, of the oil should be known to compute the velocity ofthe pressure waves. The friction factors can usually be estimated closely duringthe design stages; the values of K and p, however, are unknown and may varyfrom one oil batch to another. Therefore, a range of the expected values ofK and p should be used in the analysis during design, and the values that yieldworst conditions should be selected. As soon as the pipeline is commissioned,the value of these variables should be determined by conducting prototypetests. Based on the results of these tests, operating guidelines should be pre-pared for a safe operation of the pipeline.

In this chapter, a number of terms commonly used in the oil industry.arefirst defined. Different operations of vaíous appurtenances and control devicesthat may produce transient conditions in the pipeline are discusse~. ~ com-putational procedure to analyze the transient conditions in long pipelines bythe method of characteristics is then presented.

Cross-country pipelines transporting crude oil or refined products are usualIyseveral hundred kilometers long and have several pumping stations locatedalong their length (Fig. 7.1), with each having a nurnber of pumps in series.The pumping head of these stations is mainly used to overcome the frictionlosses in the pipeline; in a mountainous terrain, however, sorne gravity lift maybe required in addition to the friction losses.

The analysis of transients in oil pipelines, sometimes called oil-hammer anal-ysis or surge analysis, is rather complex because the pipeline friction losses arelarge compared to the instantaneous pressure changes caused by a sudden vari-ation of the flow velocity. Prior to the availability of high-speed digital com-puters, these analyses were approximate, and a large factor of safety had tobe used to allow for uncertainties in the computed results. Nowadays, pressuresfor various operating conditions can be accurately predicted, thus allowing areduction in the factor of safety.

During the design of a pipeline, the maximum and minimum pressures arecomputed in order to select the pipe-wall thickness necessary to withstandthese pressures, A small reduction in the pipe-wall thickness can result in sig-nificant savings in the initial cost of the project; therefore, a detailed analysis ofthe transients caused by various possible operating conditions is necessary for aneconomic designo A detailed anaJysis of an existing pipeline may indicate thepossibility of increasing its throughput by increasing the normal working pres-sures, which in the original design might have been set too low to allow for un-certainties in the prediction of the maximum and minimum pressures.

A very important parameter in the design of an oil pipeline is the frictionfactor, [, which should be precisely known to determine the initial steady-state pressures along the pipeline and hence the required pumping heads of the

7.2 DEFINITlONS

The following terms'<" are commonly used in the oil industry:

Potentlal Surge. The instantaneous pressure rise caused by instantaneous~ystopping the flow (i.e., reducing the flow velocity to zero) is defined as potentialsurge. The amplitude of the potential surge, Z, may be computed as f~llo,:s:If Va is the initial steady-state velocíty, then f1V = O - Va = - Va. Substitutingthis value of f1 V into Eq. 1.8, we obtain

a aZ = f1H = - - f1V = - Va

g g(7.1)

in which f1H = instantaneous pressure rise due to reduction of flow velocity Vato zero; a = velocity of pressure waves; and g'" acceleration due to gravity. IfVa and a are in m/s,g is in m/s2, then Z is in m.

190

192 Applied Hydraulic TransientsTransients in Long OH Pipelines 193

Line Packing. The increase in the storage capacity of a pipeline due to anincrease in pressure is called fine packing, The following analogy will be helpfulin the understanding of this phenomenon.

Let us assurne that the flow velocity in the canal shown in Fig. 7.2 is suddenlyreduced from Va to zero at the downstream end by closing a sluice gateo Thissudden reduction in the flow velocity produces a surge, which travels in theupstream direction. For simplification purposes, let us assume that the surgeheight does not change as it propagates in the upstrearn direction. If the watersurface behind the wave front does not change, then it will be parallel to theinitial steady-state water surface (line cd in Fig. 7.2), except that the watersurface will be at a higher leve!. Because of slope in the water surface, waterwill keep on flowing toward the sluice gate even though the surge has passed aparticular point in the canal. As the velocity at the sluice gate ís zero, thewater flowing behind the surge will be stored between the surge loeation andthe sluice gateo Due to this storage, the water surface downstrearn of the wavewill be almost horizontal. With further upstream movernent of the surge frombe to fh, more water will flow behind the surge, and the water level at the sluicegate will rise from e to j. Thus, the shaded area in Fig. 7.2 is due to storage ofwater flowing downstream of the surge. At the downstream end, water levelrose from a to d due to initial or potential surge, whereas it rose first from dto e and then from e to i due to storage of water between the surge locationand the sluice gateo

Conditions are analogous in a long pipeline. Due to sudden closure of a down-stream valve (Fig. 7.3), pressure rises instantaneously at the downstream end,and a wave having amplitude equal to the potcntial surge travels in the up-stream direction. J ust like the flow behind the surge in the canal, oil flowsdownstream of the wave front, pressure rises gradually at the downstream end,the hydraulic grade line becomes almost horizontal, and more oi! is storedbetween the wavefront and the valve. This increase in storage is called fine

Wavefronl

Reservoirhydrau/ic grade /ine

Va/veF/ow

Figure 7.3. Potential surge and line packing.

Surge Sluice gale

packing . Referring to Fig. 7.3, the pressure rise at the valve ís made up oftwoparts: (1) potential surge, Z, produced due to instantaneous closure of thevalve, and (2) D.p, due to líne packing.

Depending upon the length of the pipeline, the pressure rise due to line pack-ing may be severa] times greater than the potential surge.

Attenuation. As discussed previously, the oil flows toward the valve eventhough the wave front has passed a particular location in the pipeline. In otherwords, the velocity differential (.6 V) across the wavefront is reduced as thewave propagates in the upstream direction. Hence, it follows from Eq. 7.1 thatthe amplitude of the surge is reduced as it propagates along the pipeline dueto reduction in the velocity differential across the wave front. This reductionin the surge amplitude is referred to as attenuation.

The amplitud e of the surge is also reduced due to friction losses. However,thís reduction is usuaJly small as compared to that due to decrease in the ve-locity differential across the wave front.

Pyramidal Effect. The superposition of one transíent state pressure uponanother is referred to as pyramidal effect . For example, if a line were packeddue to closure of a downstream valve or due to power fai!ure to the pumpsof a downstream pumping station, and the pumps of an upstream station werestarted, then the pressure due to pump start-up will be superimposed on thepressure due to line packing.

Put-and-Take Operation. In a put-and-take station operation, each stationpumps oi! from a tank located on its suction side into a tank located on thesuction side of the next station.

Reservoír

Figure 7.2. Propagation of surge in a canal.

Float- Tank Operatíon. Float- tan k operation is similar to the put-and- takestation operation except that a float tank is open to the suction line of each


station. The size of the float tank is usually small compared to those in theput-and-take operation.

Tighr-Line Operation. In tíght-line station operation, each station pumpsdirectly into the suction manifold of the downstream station, and no tanksare provided on the suction side. Such a system is called a tight-Iine systemor a closed system.

Station Regulation. Maíntenance of the pressures within the safe limits ofthe pipe and equipment by means of pressure controllers at each pumpingstation is called station regulation.

Line Regulatton. Maintenance of an identical pumping rate at each pumpstation of a tight-line or closed system is referred to as Une regulation,

Rarefaction Control Planned reduction of flow at an upstream station toreduce pressure rise in the pipeline following a sudden accidental flow reduc-tion at the downstream station is called rarefaction control. Flow at the pump-ing stations may be reduced by shutting down a pump or by closing a valve.By reducing flow at the upstream station, a negative wave is produced, whichtravels toward the downstream station. This wave nullifies part of the pressurerise caused by the upset at the downstream station.


The followíng operation of various appurtenances or control devices!" 3, 7-9 pro-duces transient conditions in oil pipelínes:

1. Opening or closing the control valves2. Starting or stopping the pumps3. Power failure to the electric motors of pumping units4. Change in the pumping rate and discharge pressure of pumping stations5. Operation of the reciprocating pumps6. Pipeline rupture,

Starting a pump or opening a valve at an intermediate station produces apressure rise on the downstream side and a pressure drop on the upstreamside, whereas a pump shutdown or closure of a valve produces a pressure riseon the upstream side and a pressure drop on the downstream side.

The flow and pressure on the suction and discharge sides of a reciprocatingpump are periodic. If the period of the flow or pressure oscillations matchesthe natural period of the piping system, resonance will deveJop (se e Chapter 8)resulting in high-amplitude pressure fluctuations, wlúch may damage thepipeline.


Air entrapped during filling or following major repairs may produce surgesof high magnítude.!? In addition, if air is present in the pipeline, there isalways a danger of an explosion, which may rupture the pipeline.

7.4 METHOD OF ANALYSIS

The dynamic and continuity equations (Eqs, 2.l1, 2.30) derived in Chapter 2describe the transient-state flows in oil pipelines. Note that these equationsare not valid for a simultaneous flow of gas and oil in a pipeline for wlúchequations for a two-phase flow will have to be used.

To determine the transient conditions in a pipeline, the dynamic and con-tinuity equations are solved subject to appropriate boundary conditions. Asdiscussed in Chapter 2, these equations can be integrated only by numericalmethods since a closed-form solution is not possible because of the presenceof nonlinear terms. The method of characteristics presented in Chapter 3 maybe used for the numerical integration of these equations. However, as thefriction losses in long oil pipelines are large compared to the potential surge,a first-order approximation* of the friction term, fQIQI/(2DA), of Eqs. 3.8and 3.10 may yíeld an lncorrect and unstable solution. An examination ofthecomputed results will reveal any instability, while the validity of the resultsmay be checked by comparing the results obtained by using larger and smallertime steps. Since oil pipelines are usualIy very long, small time steps wouldrequire excessive amount of computer time. To avoid this, the friction termmay be considered in the analysís by using either of the following procedures,wlúch allow use of larger time steps:

l. Second-order approxímation=": II

2. Predictor-corrector scheme.12,13

In a second-order approximation, an average value of the friction term com-puted at points P and A (Fig. 3.1) is used for Eq. 3.8, and an average value ofthe friction term computed at points P and Bis used for Eq. 3.10. This resultsin two nonlinear algebraíc equations in Qp and Hp. These equations may besolved by the Newton-Raphson method. In the predictor-corrector scheme,presented by Evangelisti, 12,13 a first-order approximation is used to determinethe discharge at the end of the time step. This predicted value of the dischargeis then used in the corrector part to compute the friction termo

....

*f XI ¡(x) dx '" ¡(xo) (xI - xo) is called a first-order approximation, andXo

f XI ¡(x) dx '" t flexo) + ¡(XI)] (xl - xo) is called a second-order approximation,Xo


Because the predictor-corrector scherne is easy to program, requires lesscomputer time than the second-order approximation, and yields sufficientlyaccurate results, details of this scherne are presented below.

Referring to Fig. 3.1, let us assume that the conditions (e.g., pressure andflow) are known at time lo and that we have to determine the unknown con-ditions at point P. (If to = O, these are initial steady-state conditions, and iflo > O, then these are computed values for the previous time step.)

For the predictor part, integration of Eqs. 3.8 and 3.10 by using a first-orderapproximation yields

Qp- QA +ea(Hp-HA)+RQAIQAI=O

Qp - QB - ea (H p - H B) + R QB IQB I = O

(7.2)

(7.3)

in which R = ftit/(2DA). In these equations, the notation of Section 3.2 is usedexcept that an asterisk is used to designate the predicted values of variousvariables.Equations 7.2 and 7.3 can be written as

Q* - e* *p - p - ea Hp (7.4)and

Q; = C! + Ca H; (7.5)

in which e; and e! are equal to the right-hand sides of Eqs. 3.20 and 3.21,respectively. Elimination of H; from Eqs. 7.4 and 7.5 yields

Q; = 0.5 (e; + e!) (7.6)

Now this value of Q; may be used in the corrector part to calculate the frictiontermo Integration of Eqs. 3.8 and 3.10, by using a second-order approximationand by using Qp for computing the friction term, yields

Qp - QA + Ca(Hp - HA) + 0.5 R(QA IQA 1+ QplQpl) = O (7.7)

and

Qp - QB - ea(Hp - Ha) + 0.5 R(Qa IQB I + QplQpl) = O (7.8)

Equations 7.7 and 7.8 rnay be written as

(7.9)and

(7.10)in which

(7.11 )

Transients in Long Oil Pipeline s 197

(7.12)

By eliminatingHp from Eqs. 7.9 and 7.10, we obtain

Qp = 0.5 (ep + en) (7.13)

Now Hp may be determined from either Eq. 7.9 or 7.10.To determine conditions at the boundaries, the boundary conditions derived

in Chapters 3 and 10 are first used to compute Q; for the predictor part. Then,this value of Q; is used to compute ep and Cn from Eqs. 7.l1 and 7.12, andthe same boundary conditions are used again to determine Qp and Hp in thepredictor part befare proceeding to the next time step. The other computa-tional procedure is the same as described in Section 3.2.

7.5 DESIGN CONSIDERATIONS

General Remarks

With a proper design and provision of autornatic-control and protective devices,such as pressure controlIers, pump-shutdown switches, and relief valves, a pipe-line can be safely operated to its maxímum capacity. The main functions ofthese devices are to detect asevere upset in the system and to take appropriatecorrective action so that the pipeline pressures remain within the design limits.The time lag between the detection of an upset and the corrective action shouldbe as smaIl as possible. In addition, the action of the protective facilities shouldbe rapid.

The rnaximum pressure in a blocked line having a centrifugal pump is equalto the shutoff head. However, there is no such upper limit on the máximumpressure in a blocked line having a reciprocating pump; the pressure in thiscase keeps on rising until either the pipeline ruptures or the pump fails.

Power failure to the electric motors of a pumping station will shut down aIlpumping units simultaneously. Since this condition may occur a number oftimes during the lífe of the project, it should be considered as a normal opera-tion during designo In an engine-powered pumping station, however, the proba-bility of simultaneous shutdown of alJ pumping units is rather small under nor-mal conditions although it is possible if sorne control equipment malfunctions.

While analyzing the controlling action of a valve, a proper flow versus valve-opening curve should be used since the flow in certain types of valves does notchange until the valve has been opened or closed by approximately 20 to 30percent. If it is assumed that the flow begins to change as soon as the valveopening is changed, it may yield incorrect resuIts.

Since normal transient operating conditions are likely to occur several timesduring the lífe of the project, a factar of safety should be used that is largerthan that for the emergency conditions, whose probability of occurrence is


rather small. Malfunctioning of the control equipment in the most unfavor-able manner may be considered as an emergency condition.

Control and Surge Protective Devices

A discharge-pressure controller and a pump-shutdown switch are commonlyprovided in a pumping station having centrifugal pumps. A pressure controllercontrols the discharge pressure by reducing the pump speed or by elosing a con-trol valve. lf the pressure exceeds the limits set on the controller, the purnp-shutdown switch shuts down the entire station.

In pumping stations having reciprocating pumps, a pressure controller is alsoused 10 operate a bypass val ve or 10 change the pump stroke or the pumpspeed. As an added protection, a relief valve with a capacity equal to the pumpdischarge should be installed. The resonance conditions may be avoided asfollows: In a pumping station with two or three pumps operating at 20 cycles/sor less and having a discharge pressure of less than 6 MPa, tying the di s-charge lines a short distance from the pump headers or providing an air chamberhas been reported ' to be successful in reducing the pressure fluctuation byabout 90 percent. On high-speed multiplex pumps operating at more than20 cycles/s or having discharge pressures more than 6 MPa, the provisionof air chambers does not adequa1ely suppress the pressure fluctuations. Insuch cases, pulsation dampeners of special design have been found to besatísfactory.!?

The pressure caused by line packing in a closed system may be several timesgreater than the potential surge, and it may stress the entire pipeline to the fuIldischarge pressure of the upstream station. These pressures can be kept low byan advance action or rarefaction control, as discussed in the following. Pressuremonitors, provided on the suction side of a pump station, detect any excessivepressure rise and transmit a sígnal to the supervisory control, which reducesthe discharge pressure and/or outflow of the upstream station. Due to reduc-tíon of the díscharge pressure or flow of the upstream station, a negative wavetravels in the downstream direction and reduces pressure built up in thepipeline. Another method for reducing the pressure rise is to provide a reliefvalve at the downstream station.

Upon power failure to the pumping units of an in1ermediate pumping station,pressure rises on the suction side and decreases on the discharge side. Excessivepressure rise or drop may be prevented by providing a check val ve in the pumpbypass, As soon as the rising pressure on the suction síde exceeds the fallingpressure on the discharge side, oil begins to flow from the suction side to thedischarge side of the pump. This flow through the check valve helps in pre-venting further increase and decrease in the pressure on the suction and dis-charge sides, respectively.


In a mountainous terrain, a surge tank may be provided at the peaks to avoidcolumn separation following pump failure at the upstream station.

7.6 SUMMARY

In thís chapter, a number of terms commonly used in the oil industry weredefined, and different causes of transient conditions in oil pipelines were out-lined. A computational procedure was presented to analyze transients in longpipelines by the method of characteristics. The chapter was concluded bydiscussing the use of various protective and control devices to keep the pressureswithin the design limits.

PROBLEMS

7.1 Write a computer program to analyze a long pipeline having a reservo ir atthe upstream end and a valve at the downstream end. Inelude the frictionterm of Eqs. 3.8 and 3.10 by using (1) a first-order finite-difference ap-proximation outlined in Chapter 3 and (2) a predietor-eorreetor sehemeof Section 7.4.

7.2 Use the computer program of Problem 7.1 to compute the potential surge,and the maximum transient-state pressures at the valve and at midlengthof a 32,OOO-m-Iong, 2-m-diameter pipeline carrying 3.14 m3/s of erudeoil. Assume that the downstream valve is instantaneously closed.

7.3 Compare the maximum pressures of Problem 7.2, obtained by includingthe friction terrn, by using: (1) a first-order, finite-difference approxi-mation and (2) a predictor-corrector scheme.

7.4 Develop the boundary conditions for a purnping station having a bypassline fitted with a check valve. The check valve opens as soon as the pres-sure on the suction side exceeds the pressure on the diseharge side.

REFERENCES

1. Ludwig, M. and Johnson, S. P., "Prediction of Surge Pressures in Long Oil TransmissionLine," Proc. Amer. Petroleum Inst., Annual Meeting, New York, Nov. 1950.

2. Kaplan, M., Streeter, V. L., and Wylie, E. B., "Computation of Oil Pipeline Transients,"Jour., Pipeline Div.. Amer. Soco o/ Civil Engrs., Nov. 1967, pp. 59-72.

3. Streeter, V. L. and Wylíe, E. B., Hydraulic Transients, MeGraw-HiII Book Co., NewYork,1967.

4. Streeter, V. L., "Transíents in Plpelines Carrying Liquids or Gases," Jour., Transpor-tation Engineering Div., Amer. Socoof' Civil Engrs., Feb. 1971, pp. 15-29.

5. Burnett, R. R., "Predicting and Controlling Transient Pressures in Long Pipelines,"Oil and Gas Jour., U.S.A., vol. 58, no. 10, May 1960, pp. 153-160.

6. Oil Pipeline Transportation Practices, issued by Univ. of Texas in cooperation withAmer. Petroleum Inst., Division ofTransportation, vol. 1 and 2,1975.

200 Applied HydrauJic Transients

7. Lundberg, G. A., "Control of Surges in Liquid Pipelines," Pipeline Engineer, March1966, pp. 84-88.

8. B,agwell, M. U. and Phillips, R. D., "Pipeline Surge: How It is Controlled," Oil andGasJour., U.S.A., May 5,1969, pp. ¡15-120.

9. Bagwell, ~. U. ~nd PhiJIips, ~. D., "Liquid Petroleum Pipe Line Surge Problems (Realand Irnaginary), Transportation Div., Arner. Petrol. Inst., April 1969.

10. Ludwig, M., "Design of Pulsation Dampeners for High Speed Reciprocating Purnps,"Conference of the Transportatlon Div., Amer. Petroleum Inst. Houston Texas May1956. ' , , ,

¡1. Lister, M., "The Numerical Solution of Hyperbolic Partial Differential Equations bythe Method -: ~haracteristics," in Ralston, A. and Wilf, H. S. (eds.), Mathematical7;;~ods for Digital Computers, John Wiley & Sons, Inc, New York, 1960, pp. 165-

12. Evangelisti, G., "Waterhammer Analysis by the Method of Characteristics " L 'EnergiaElettrica, nos. ID, n, and 12, 1969, pp. 673-692, 759-771, and 839-858.'

13. Evangelisti, G., .Boari, M., Guerrini, P., and Rossi, R., "Sorne Applications of Water-hammer Analysis by the Method of Characteristics, " L 'Energia Elettrica, nos. 1 and6, pp. 1-12, 309-324.


Green: J. E., "Pressure Surge Tests on OH Pipclines," Proc. Amer. Petroleum Inst., vol. 29,Section 5,1945.

Kerr, S. L., "Surges in Pipelínes, OH and Water," Trans. Amer. Soco of Mech Engrs 1950p.667. . ., ,

Binnie,. A. M., ''The. Effect of Friction on Surges in Long Pipelines," Jour. Mechanics andApplied Mathematics , vol. IV, Part 3, 1951.

Ker.ste~, R. ~. and Waller, E. J., "Prediction of Surge Pressures in Oil Pipelines," Jour.,Pipeline DIV., Amer. Soco of Civil Engrs., vol. 83, no. PL 1 March 1957 pp 1195-1-1195-22. ' , .

Wostl, W. J. and Dresser, T., "Velocimeter Measures Bulk Moduli " Oil and Gas JourDec. 7,1970, pp. 57-62. ,.,

Wylie, E. B., Streeter, V. L., and Bagwell, M. U., "Flying Switching on Long Oil Pipelines "Amer. lns~: ~hel.n. Engrs. Sy'!"'posium Series, vol. 135, no. 69, 1973, pp. 193-194. '

Techo, R., Pipeline Hydraulic Surges are Shown in Computer Simulations," Oil and GasJour., Novernber 22, 1976, pp. 103-111.

Kaplan, M., "Analyzing Pipeline Transients by Method of Characteristics," Oil and GasJournal, Jan. 15, 1968, pp. 105-108.

CHAPTER 8

RESONANCE IN PRESSURIZEDPIPING SYSTEMS

8.1 INTRODUCTION

In Chapters 4 to 6, we considered transient-state flows that represented the inter-mediate-flow conditions when the flow is changed from one steady state toanother. However, sometimes when a disturbance is introduced into a pipingsystem, it is amplified with time instead of decaying and results in severe pres-sure and flow oscillations. This condition, which depends upon the character-istics of the piping system and of the excitation, is termed resonance.

In this chapter, the development of resonating conditions and the availablernethods for their analysis are discussed. The details of the transfer matrixmethod, derivation of the field and point matrices, and procedures for de-terrnining the natural frequencies and frequency response of piping systems arethen presented. To verify the transfer matrix method, its results are comparedwith those of the characteristics and impedance methods and with those mea-suredin the laboratory and on the prototype installations.

8.2 DEVELOPMENT OF RESONATING CONDITIONS

We know from fundamental mechanics! that the natural frequency, Wn, of the1

spring-mass system shown in Fig. 8.1 is equal to -..jk[iñ, in which wn = natural27T

frequency of the systern in rad/s, m = mass, and k = 'spring constant. If a sinus-oidal force having frequency w¡ (Fig. 8.1b) is applied to the mass, initially a beatdevelops (transient state) and then the system starts to oscillate (Fig. 8.lc) atthe forcing frequency w¡ and with a constant amplitude. These oscillations,having a constant amplitude, are called steady vibrations, The amplitude of the

201


(a) SprinV - masa syatem

T=~Fn

(b) Periodle force

Tronsillnl slolll SllIody .-i/JlroliDns.-ibrttHDns

(e) Vibration of ma..

Figure 8.1. Vibrations of a spring rnass system.

vibrations depends upon the ratio ca; = wf/wn. If the forcing frequency wf isequal to the natural frequency wn and the system is frictionless, then the ampli-tu de of steady vibrations becomes infinite. The reason for this is that the totalenergy of the system keeps on increasing with each cycle because no energy is

Resonance in Pressurized Piping Systems 203

dissipated in the system. Hence, the oscillations are amplified without any upperbound. However, ir the system is not frietionless, the amplitude of the osctlla-tions grows until the energy input and energy dissipation during a cycle areequal. Since at that time there is no additional energy input per cyc1e, the systernoscillates with a finite amplitude.

Now let us eonsider a pipeline having a reservoir at the upstrearn end and avalve at the downstream end (Fig. 8.2a). Let us assume that the valve is initiallyin a closed position but that we open and close it sinusoidally at frequency wfstarting at time t = O (Fig. 8.2b). Similar to our spring-rnass system, a beat de-velops first (transient state), and then the flow and pressure oscillate at a con-stant amplitude but with frequency Wf (Fig. 8.2c). Such a períodic flow istermed steady-oscillatory flow.

Let us compare the characteristics of the steady-oscillatory flow in our simplehydraulic system with the steady vibrations of the spring-mass system. The dís-placement of the spring at the fixed end in our spring-rnass system is zero.Similarly, the water leve! in the upstream reservoir of the hydraulie systern isconstant. Therefore , the amplitude of pressure oscillations at the reservoír iszero , or , in other words, there is a pressure node at the reservoir. In the spring-mass system, there is only one rnass and one spring; therefore , there is only onemode of vibrations or one degree of freedom, and hence the system has only onenatural frequency (or natural period). If the eompressibility of the fluid is takeninto consideration , the fluid in the pipeline of our hydraulic system is comprisedof an infinite number of rnasses and spríngs. Therefore, our hydraulíc systern hasinfinite modes of oscillations or degrees of freedom* and henee has infinitenatural periods: the first is termed as fundamental, and the others are caJledhigher harmonics,

Figure 8.3 shows the variation of the amplitude of the pressure oscillations atvarious harmonics along the piping systern of Fig. 8.2. Since the reservoir levelis constant, a pressure node always exists at the reservoir end. At the valve, how-ever, there is a pressure node during even harmonies and an antinode during oddharmonics. The location of the nodes and antinodes along a pipeline dependsupon the harmonic at which the system is oscillating.

Let us now consider another significant difference between our spríng-mass,and hydraulie systerns. In the former, the source of energy is the external peri-odie force acting on the mass. In the hydraulic system, although the valve is theforcing function, it is not the source of energy. The val ve is just eontrolling theefflux of energy from the system, whereas the upstream reservoir is the source of

*Distributed systemsl representad by partial differential equations have infinite degrees offreedorn, whereas lumped systems represented by ordinary differential equations have finitedegrees of freedom.


Reservoir

(a) Single pipeline

(b) Periodic volve operotion

H

Trunsient state -+- Steady osc/ttarory --f/ow f/ow

(e) Pressure oscillotions ot volve

Figure 8.2. Development of steady-oscillatory flow in a single pipeline.

energy. Since the volume of the fluid in the pipeline is constant, the outflow percycle at the valve must be equal to the inflow per cycle at the reservoir. Thereservoir level being constant, the energy input into the system is at a constanthead. However, there is no such restriction on the energy efflux at the valve. lf


the valve operation is such that there is outflow when the pressure at the valvc islow and there is little or no outflow when the pressure is high, net influx ofenergy occurs during each cycle. This causes the pressure oscillations to grow.When the steady-oscillatory flow is fully developed, a discharge node exísts atthe valve during oscillations at odd harmonics if the head losses in the system areneglected. Once a discharge node is formed at the valve, opening or closing of

Third hormonic

H

x

L"3

(a) Odd hormonics

L"4

x

L"4

L"4

L"4

(b) Even harmonics

Figure 8.3. Pressure oscillations along pipeline at various harmonics.


the valve has no effect on the energy efllux, and thus the amplitude of the pres-sure oseillations does not inerease further even though it is assumed that there isno energy dissipation in the system,

Allíevi' was the first to prove that the maximum possible amplitude of thepressure oseillations at thevalve was equal to the statie head. However, la ter on,Be rgeron" proved graphieally that for large values of Allievi's parameter ,p = (a Va)/(2gHa), it is possible to have amplitude greater than the statie head.Also, Camichel s demonstrated that doubling of the pressure head is not possibleunless Ha> (a Va)/g. In the preeeding expressions, a = waterhammer wave ve-locity; g = acceleration due to gravity; Ha = static head; and Va = steady -stateflow velocity.

8.3 FORCED AND SELF-EXCITED OSCILLATIONS

Steady-oscíllatory flows in piping systems may be eaused by a boundary thataets as a periodic forcing function or by a self-excíted excitation. The systemoscillates at the frequency of the forcing funetion during forced oscillations andat one of the natural frequencies of the system during self-excited oseillations.

Forcing Functions

There are three common types of forcing funetions in hydraulic systerns: peri-odie variation of the pressure, flow, and the relationship between the pressureand the flow.

A typieal example of the periodie pressure variation is a standing wave on thereservoir water surface at a pipe intake. If the period of the surface waves corre-sponds to one of the natural periods of the piping system, steady-oscillatoryeonditions are developed in the system in just a few cycles,

A reciprocating pump has periodic inflow and outflow. If the períod of a pre-dominant harmonic of either the inflow or of the outflow corresponds to anatural period of the suction or diseharge lines, severe flow oseillations willdevelop.

A periodicaIly opening and elosing valve is an example of a periodie variationof the relationship between the pressure and the flow. The development ofsteady-oseillatory flows by a periodic valve operation was discussed in Section 8.2.

Self-Excited Oscillations

In the self-excited or auto- oscillations, a component of the system aets as anexciter, whieh causes the system energy to inerease following a smalI disturbaneein the system. Resonanee develops when the net energy influx to the system pereyc1e is more than the energy díssipated per cyele.


A typical example of an exciter is a leaking valve or a leaking seal." Let usexamine how the oscillations develop. Figure 8.4 shows the valve eharaeteristiesfor a normal and for a leaking valve (Fig. 8.4a and b). For a normal valve, theflow inereases as the pressure inereases; for a Ieaking valve, the flow decreases asthe pressure íncreases. In Chapter 1, we derived the following equation for theehange in pressure eaused by a flow variation, i.e.,

at:.H = - ~ t:. V

g(l.8)

Va/ve cñaracteristics

(Q) Normal valve

Va/ve cñaracteristics

I .

9

H

(b) Leaking valve

Figure 8.4. Self-excited oscillations.

208 Applied Hydraulic Transients Resonance in Pressurized Piping Systerns 209

On the H - V diagram, this equation plots as straight lines having slopes ±a/g fora decrease or increase in velocity, respectively. Let us assume that the initialsteady-state velocity is Vo and that, due to a disturbance, it is decreased to VI.The decay or amplification of this disturbance using the graphical waterhammeranalysis is shown in Fig. 8.4.7 lt can be seen that the disturbance decays in afew cycJes for a normal valve whereas it amplifies for a leaking valve. As theflow through a leaking valve cannot be less than zero and it cannot increaseinfinitely, the amplification of the pressure oscillations is therefore finite.

Den Hartog" reported self-excited penstock vibrations caused by the dís-turbances produced by the runner blades passing the guide vanes of a Francisturbine. Based on a simplified theoretical analysis, which was confirmed by ob-servations on eight hydroelectric installations, he concluded that such vibrationsare to be expected if the number of the runner blades is one less than the num-ber of the guide vanes. Self-excited vibrations of the guide vanes of a centrifugalpurnp is considered to be the cause of the accident at the Lac Blanc-Lac Noirpumped storage plant.? in which several testing personnel died.

The possibility of the self-excited vibrations in a liquid rocket-engine-propellant feed system were determined by a computer analysis;'? experimentaldetermination of such a possibility in the piping system of a control system isdescribed by Saito.i '

Improper settings of a hydraulic turbine governor can result in self-excitedoscillations called governor hunting.

Abbot et al." measured self-excited vibrations in Bersimiss II power plant. Aslight leak in a 3.7-m-diameter penstock valve due to a reduction of seal pressureresulted in the vibrations of the valve. The valve vibrations and pressure oscilla-tions were sinusoidal and were eliminated by opening a bypass valve.

McCaig and Gibson 12 reported vibrations in a pum p-discharge line caused by aleak under the static head in a 0.25-m-diameter spring-cushioned check valve.Measurements showed that the pressure osciJIations were approximately sinu-soidal with sharp impulses of large magnitude every third cycJe. The vibrationswere prevented by instaIJing a weaker cushioning spring in the check valve andremoving the air valves from·the pipeline.

Analysis in Time Domain-Method of Characteristics

As discussed in Chapter 3, in this method, the partial differential equationsdescribing the unsteady flow are converted into ordinary differential equations,which are then solved by a finíte-difference technique , Nonlinear friction lossesand the nonlinear boundary conditions may be included in the analysis.

To analyze the steady-oscillatory flows by this method ," the initial steady-state discharge and pressure head in the piping system are assumed equal to theirmean values or equal to zero-flow conditions. The specifled forcing function isthen imposed as a boundary condition, and the system is analyzed by consideringone frequency at a time. When the initial transients are vanished and a steady-oscillatory regime is established, the amplitudes of the pressure and dischargefluctuation are determined. The process of convergence to the steady-oscillatory conditions is slow (it takes about 150 cycles) and requires a con-siderable arnount of computer time, thus making the method uneconomical forgeneral studies. The main advantage is that the nonlinear relationships can beincJuded in the analyses.

Analysis in Frequency Domain

8.4 METHODS OF ANAL YSIS

By assuming a sinusoidal variation of the pressure head and the flow, thedynamic and continuity equations describing the unsteady flow in the timedomain are converted into the frequency domain. The friction term and thenonlinear boundary conditions are linearized for solution by these methods. Ifthe amplitude of oscillations is smaIJ, the error introduced by linearizationís negligible.

Any periodic forcing function can be handled by these methods. The forcingfunction is decomposed into varíous harmonics by Fourier analysis," and eachharmonic is analyzed separately. Since a1l the equations and relationships arelinear, the system response is determined by superposition 16 of individualresponses.

As the frequency response is determined directly, the cornputer time requiredfor the analysis is smalJ. Therefore , these methods are suitable for generalstudies. The folJowing two methods of analysis in the frequency domain areavailable:

The steady-oscillatory flows in a hydraulic system may be analyzed either inthe time domain or in the frequency dornain'" by using the folJowing methods:

1. Time domain: method of characteristics2. Frequency domain: (a) lmpedance method and (b) Transfer matrix method.

A discussion of the advantages and disadvantages of each method follows.

Impedance Method

The concept of impedance was introduced by Rocard? and was used later byPaynter'" and WalJer.18 Wylie19,20 extended and systemized this method forthe analysis of complex systems.


In this method, the terminal impedance, Zs' which is the ratio of the oscilla-tory pressure head and the discharge, is computed by using the known boundaryconditions. An impedance diagram between wf and IZs I is plotted. The fre-quencies at which IZs I is rnaximum are the resonant frequencies of the system.

Because of the lengthy algebraic equations involved, the method is suitable fOIdigital computer analysis only. For a parallel piping system, a procedure is sug-gested that requíres solution of a large number of simultaneous equations. Thisbecomes cumbersome if the system has many parallelloops. For example, eightsimultaneous equations have to be solved for systems having only two parallel100pS.19

Transfer Matrix Method

The transfer matrix method has been used fOI analyzing structural and me-chanical vibrations21,22 and for analyzing the electrical systems.F" This methodwas introduced24-27 by the author for the analysis of steady-oscillatory flowsand for determining the frequency response of hydraulic systems.

Similar to the impedance method, the transfer ma trix m ethod is based on thelinearized equations and on sinusoidal flow and pressure f1uctuatíons. However,in the author's opinion, transfer matrix method is simpler and more systematicthan the impedance method. Other advantages of the transfer matrix method are(1) the analysis of the parallel systems does not require any special treatment;(2) the method is suitable for both hand and digital computations; (3) thestability of a system can be checked by the root locus technique r" and (4) sys-tems, in which oscillations of more than two variables (e.g., pressure, flow,density, temperature) have to be considered, can be analyzed.

Details of the transfer matrix method are presented herein. Any reader havingan elernentary knowledge of the matrix algebra should be able to follow thederivation of matrices and their application. Block diagrams are used to achievean orderly and concise formulation and analysis of problerns involving corn-plex systems.

8.5 TERMINOLOGY

The terminology established by Camichel et a1.5 and later used by Jaeger30,31

and Wyliel9,20 is followed herein.

Steady-OsciUatory Flow

A flow in which a permanent regime is established such that the conditions at apoint (e.g., pressure, discharge) are periodic functions of time is called steady-

··'1


oscillatory flow. In the theory of vibrations, the steady-state oscillations refer tothe oscillations that have constant amplitude. However, the term steady-oscillatory is used herein to avoid confusion with the steady flow in which con-ditions at a point are constant with respect to time.

Instantaneous and Mean Discharge and Pressure Head

In a steady-oscillatory flow, the instantaneous discharge, Q, and the instanta-neous pressure head, H, can be divided into two parts:

o=üs+a" (8.1)

H=Ho+h* (8.2)

in which Qo = average, or mean, discharge; q* = discharge deviation from themean (see Fig. 8.5); Ho = average, OI mean, pressure head; and h* = pressurehead deviation from the mean. Both h* and q* are functions of time, t, anddistance, x. It is assumed that h* and q* are sinusoidal in time, which, in prac-tice, is often true or a satisfactory approximation.6,12, 30,31 Hence , by usingcomplex algebra, we can write

q* = Re (q(x)eiwt)

h* = Re (h(x)eiwt)

(8.3)

(8.4)

in which W = frequency in rad/s; j = y::I; h and q are complex variables andare functions of x only; and "Re" stands fOI real part of the complex variable.

Theoretical Period

For a series piping system,

n !ir.. = 4¿;=1 a;

(8.5)

Q

t

Figure 8.5. Instantaneous, mean, and oscillatory discharge.

212 Applied Hydraulic Transients Resonance in Pressurized Piping Systems 213

and

21T-r :

th(8.6)

in which the input variables are combined into a single column vector, x, andthe output variables are combined into a column vector, y. In other words, itcan be said that the system converts, or transfers , the input variables x into theoutput variables y, and the transfer takes place in accordance with Eq. 8.8. Thematrix U in Eq, 8.8 is called the transfer matrix, and x and y are called thestate vectors.

In the previous example, the system has only one component. The physicalsystems, however, are usually made up of several subsystems or components. Insuch cases, each component is represented by a transfer matrix, and the overalltransfer rnatrix for the system is obtained by mul tiplying the individual transfermatrices in a proper sequence (see pp. 214-215).

The general system of Fig. 8.6 has n input and output variables. In hydraulicsystems, however, the quantities of interest at the section i of a pipeline areusually h and q, which can be combined in the matrix notation as

in which Tth = theoretical period; Wth = theoretical frequency; n = number ofpipes; and a = velocity of the waterhammer waves .. The subscript ¡denotesquantities for the rth pipe. In a branch system, L]«¡ is calculated along themain pipeline.

Resonant Frequency

The frequency corresponding to the fundamental or one of the higher har-monics of the system is called the resonant [requency .

State Vectors and Transfer Matrices(8.9)

YI = UIIXI + UI2X2 + +u1nxn

Y2 = U2¡X¡ + U22X2 + + u1nxn

The colurnn vector Z¡ is called the state vector at section i. The statevectors just to the left and to the righ t of a section are designated by the super-scripts L and R, respectively. For example, zf refers to the state vector just tothe left of the ¡th section (Fig. 8.7).

To combine the matrix terms in sorne cases (se e p. 229) the state vector isdefined as

Let us consider a general system (Fig. 8.6) whose input variables XI,X2," 'xnand output variables YI ,Yl, ... 'Yn are related by the following n simultaneousequations:

(8.7)

(8.10)

In the matrix notation, these equations can be written as

y=Ux (8.8)Because of the additional element with unit value, the column vector z; is calledthe extended state vector. A prime is used herein to designa te an extendedsta te vector.

XIX2

ux-oi+x~I~ Pipe iI !

L.'R1I i

i + IXn-Ixn

Yn-I

Yn ~------"--------~Figure 8.6. Block diagram for one-component system, Figure 8.7. Single pipeline.

214 Applied Hy draulic Transients Resonance in Pressurízed Piping Systerns 215

A matrix relating two state vectors is called a transfer matrix . The upper-caseletters F, P, and U are used to designate the transfer matrices; the correspondinglower-case letters with the double subscripts refer to the elements of the matrix:the first subscript represents the row, and the second subscript represents thecolumn of the element. For example, the element in the second row and thefirst column of the matrix U is represented by U21.

Transfer matrices are of three types:

The intermedia te field and point matrices are:

zf = FIZ~

z~ = P2zf

Zt=F2Z~

l. Field transfer matrix, or field matrix, F. A field-transfer relates thestate vectors at two adjacent sections of a pipe. For example, in Fíg. 8.7,

zf = F¡-¡z1-1

z1 = p¡zf(8.11 )

in which F¡ = field matrix for the ith pipe.2. Point transfer matrix, or point matrix, P. The state vectors just to the

left and to the right of a discontinuity, such as at a series junction (Fig. 8.8) orat a valve, are related by a point-transfer matrix. The type of the discontinuityis designated by specifying a subscript with the letter P. For example, in Fig. 8.8,

z~ = Pnz~

Z~+1 = Fn z!:'

(8.13b)

(8.12)Elimination of z~ ,z~, ... ,z~, and z~ from Eq. 8 .l3b yields

(8.13c)Z~+1 = (FiíPn ... F¡P¡ ... F3P3F2P2F;r)z~~

Hence , it follows from Eqs. 8.13a and 8.l3c thatin which Psc = point matrix for a series junction.

3. Overall transfer matrix, U. The overall transfer matrix relates the statevector at one end of a system, or a si de branch, to that at the other end. Forexample if n + 1 is the last section, then

A block diagram is a schematic representation of a system in which each corn-ponent, or a combination of components, of the system is represented by a"black box." The box representing a pipeline of constant cross-sectional area,wall thickness, and wall material is charactcrized by a field matrix, while thatrepresenting a discontinuity in the system geornetry is represented by a pointmatrix. The block diagram for a system can be simplified by representing ablock of individual boxes by a single box. This procedure is iIlustrated in thefollowing sections by a number of typical examples.

A section on a block diagram is shown by a small circle on the líne joining thetwo boxes. The number of the section is written below the circle and the left-and right-hand sides of the section are designated by writing the letters L and R

(8.13a)8.6 BLOCK DIAGRAMS

in which U = overall transfer matrix. This is obtained by an ordered multiplica-tion of all the intermediate field and point matrices as follows:==-

Pipe I L R:1:l.,

Pipe i+1

- : ,1'1, 11

i+1Figure 8.8. Seriesjunction.

(8.14d)

216 Applied Hydraulic Transients Resonance in Pressurized Piping Systems 217

~

RF· p.

i 1 i+1 1+1 ¡+I

2. Dynamic equation

Figure 8.9. Blockdiagram.

aH 1 aQ fQn-+--+---=0ax gA at 2gDAn

in which A = cross-sectional area of the pipeline; g = acceleration due to gravity;D = inside diameter of the pipeline;!= Darcy-Weisbach friction factor; n = expo-nent of velocity in the friction loss term; x = distance along the pipeline, mea-sured positive in the downstream direction (see Fig. 8.7); and t = time.

As the mean flow and pressure head are time-invariant and as the mean flow isaQo aQo su;

constant along a pipeline, --, -a-' and -a- are all equal to zero. Hence, itax t t

(8.15)

aboye the circle. For example, in Fig. 8.9, i and i+ l denote the number of thesections, and L and R denote the left- and right-hand sides of the section. In thecase of a branch pipe, the number of the section is written to the right of thecircle, and the left- and right-hand sides of the sections are identified by writingthe letters BL and BR to the left of the circle (see Fig. 8.13b).

The block diagrams are of great help for {l) the concise and orderly formula-tion and analysis of problems involving complex systems, (2) an easy understand-ing of the interaction of different parts of the system, and (3) determining thesequence of multiplication of transfer matrices while doing the calculations byhand or while writing a computer programo

follows from Eqs. 8.1 and 8.2 that

aQ aq*ax =¡¡:;-;aH ah*-=_.at at'

(8.16)

8.7 DERIVATION OF TRANSFER MATRICES

ro analyze the steady-oscillatory flows and to determine the resonating charac-teristics of a piping system by the method presented herein, it is necessary thatthe transfer matrices of the elements of the system be known. In this section,field matrices for a simple pipeline and for a system of parallel Ioops are derived.A numerical procedure is presented to determine the field matrix for a pipehaving variable characteristics along its length. The point matrices for a seriesjunction, for valves and orifices, and for the junction of a branch (branch withvarious end conditions) and the main are developed.

aHHowever since we are considering the friction losses, __ o is not equal to zero., ax

For turbulent flow,

aHo fQ~--=----ax 2gDAn (8.17)

and for laminar flow,

aQ gA aH-+--=0ax a2 at

(8.14)

su; 32vQoax = - gAD2

in which v = kinematic viscosity of the fluid. Ir q*«Qo, then

Qn = (Qo + q*t ~Q~ + n Q~-I q*

in which higher-order terms are neglected.It follows from Eqs. 8.14 through 8.19 that

aq* gA ah*-+--=0ax a2 at

ah* 1 aq*-+--+Rq*=Oax gA at

in which R = (nfQ~-I)/(2gDAn) for turbulent flow and R = (32v)/(gAD2) for

(8.18)Fíeld Matrices

Single Conduit

The field rnatríx for a conduit having a constant cross-sectional area, constantwalI thickness, and the same wall material is derived in this section. In the deri-vatíon, the system is considered to be distributed, and the friction-loss term islinearized.

The continuity and dynamic equations describing the flow through closed con-duits were derived in Chapter 2. For an easy reference, let us rewrite these equa-tions as1. Continuity equation

(8.19)

(8.20)

(8.21)

laminar flow.

218 Applíed Hy draulic Transients Resonance in Pressurized Piping Systems 219

The field matrix for a pipe can be derived by using the separation-of-variabletechnique? or by using the Cayley-Hamilton theorem.é ' The former ís usedherein because of its sirnplicity.Í

Elimination of h" from Eqs. 8.20 and 8.¿ 1 yields

a2q* 1 a2q* gAR aq*--=---+----ax2 a2 at2 a2 at

In addition, at the (i + 1)th section (at x = I¡), h = hT+ 1 and q = qT+ t : The sub-stitution of these values of h and q, and el and e2 from Eq. 8.28 into Eqs. 8.26and 8.27 yields

(8.22)qf.¡ = (cosh /.1.¡I¡)qf - -}- (sinh /.1.¡/¡)hf

ehf+ 1 = - Z¿ (sinh /.1.¡I¡) qf + (eosh /.1.¡/¡) hf

in whieh characteristic impedance 14 for the pipe [Zc = (·-I!._-¡a-.l-_~-L(_-j_-~-~-~)JEquations 8.29 and 8.30 can be expressed in the matrix notation as

(8.29)

(8.30)

Now, if it is assumed that the variation of q* is sinusoidal with respect to t,then on the basis of Eq. 8.3, Eq. 8.22 takes the forro

(8.23)(8.31 )

oror

(8.24) L - F R1¡+1 - ¡I¡. (8.32)

2 w2 jgAwR/.l. =--+--

a2 a2 (8.25)

Henee, field matrix for the ith pipe is

[

eOSh /.l.·I·F·= "

, - Zc sinh /.l.¡l¡

- _1 sinh /.1..1-]Z 'le

cosh /.l.¡l¡

(8.33)

in which

The soIution of Eq. 8.24 is

q = el sinh /.l. x + e2 cosh /.l. x (8.26) If frietion is neglected, i.e., R¡ = O, then F¡ becomes

in which el and c2 are arbitrary eonstants.If h* is also assumed sinusoidal in t, then by substituting Eqs. 8.26 and 8.4

into Eq. 8.20 and solving for h, we obtain

a2/.l.h = - -. - (el cosh /.l. x + e2 sinh /.l. x)¡gAw (8.27)

- !_ sin b'W]e¡ '.eos b¡w

in whíchlé, = ¡;;-;;; and !C¡-;_a;/(gA¡)/ Note that b¡ and el are eonstants for a pipeand are o.ot funCilons ciCw: and that e¡ is the eharacteristie ímpedance!" for theith pipe if friction is neglected.

If w/¡Ja¡« 1, then the system may be analyzed as a lumped system. In thiscase, for a frietionless system, field matrix F¡becomes

_ [ _gA:iWi

]F¡- .

l,wJ--'- 1

gA¡

reos b¡wF·:

, -je¡ sin b¡w(8.34)

The field matrix relating the state veetors at the ith and at the (i + 1)th seetionof the ith pipe (see Fig. 8.7) of length l¡ is to be derived. It is known that at theith seetion (at x = O), h = hf and q = qf. Henee, it follows from Eqs. 8.26 and8.27 that

and

_ jgA¡WhR}el -- -2- ¡a¡ /.1.¡

e2 = qf

(8.35)

(8.28)

tFor the derivation using the Cayley-Hamilton theorern, interested readers should see Ref.24 or 26.

which follows from Eq. 8.34 sinee cos (wl¡/a¡) "" 1 and sin (wl¡/a¡) "" wl¡/a¡ fOIsmall values of wl;/a¡.

While doing the analysis, one first ealeulates the elements of the field matrixfor each pipe. A eomparison of the field matrices of Eqs. 8.34 and 8.35 shows

220 Applied Hydraulic Transients Resonance in Pressurized Piping Systerns 22)

that this idealization of a distributed system as a lumped systern does not greatlysimplify the computations.

aC=-

gA

Example 8.1

For w = 2.0 rad/s, compute the elements of the field matrix for a pipe that is400-m-long and has a diameter of 0.5 m. The velocity of waterharmner waves inthe pipe is 1000 m/s. Assume

1. The liquid inside the pipe as a lumped mass.2. The liquid inside the pipe as distributed, and consider the system is

frictionless.

1000 _ -2

= 9.81 X 0.196 - 520.08 s m

bw = 0.4 X 2 = 0.8

Substituting these values into Eq. 8.34,

Solution

ti I = t22 = cos bw

= cos 0.8 = 0.697

t = - Lsin bw12 C

Data:

1= 400 mD= 0.5 ma = 1000 mIss= 9.81 m/s2w = 2 rad/s

::- -j-sin O 8 = -O 0014j520.08 . .

t21 :: -jC sin bw

= -j X 520.08 X sin 0.8 = 373.08 j

1. Lumped System Conduit Having Variable Characteristics Along Its Length

A:: ¡(0.5)l = 0.196 m2 A conduit is said to have variable characteristics if, A, a, waIl thickness, or wallmaterial vary along its length. Equations 8.14 and 8.15 describe the transient flowthrough sueh conduits; the only difference ís that A and/or a are functions of xinstead of being constants, i.e.,

From Eq. 8.35,

gAlwjtl2 = ---2-a aQ gA(x) aH

-+----=0ax a2(x) at (8.36)_ 9.81 X 0.196 X 400 X 2j- - (1000)2

= -0.00154 j

lwt21 = - gA j

400 X 2

en 1 aQ-+---=0ax gA(x) at (8.37)

= - 9.81 X 0.196j

= -416.07 j

in which A(x) and a2(x) denote that A and a2 are functions of X. In these equa-tions, nonlinear terms of higher order and friction are neglected. By substitutingEqs. 8.1 through 8.4 into the preceding equations and simplifying, we obtain

aq jgA(x)w-+ h=Oax a2(x)(8.38)

2. Distributed System

b = lla = 400/1000 = 0.4 sah jw-+--q=Oax gA(x)

(8.39)


These equations can be expressed in the matrix notation as

dz-=Bzdx

(8.40)

in which z is the column vector as defined in Eq. 8.9 and

[

OB=

jwgA(x)

_jgA(X)W]a2(x)

O

(8.41 )

Since the elements of the matrix B are functions of x, the procedure that hasbeen outlined to determine the field matrix for the conduit fails. To analyzesuch cases, recourse is made to either of the following procedures'" :

l. The actual pipeline is replaced by a substitute pipeline having piecewiseconstant elements (see Fig. 8.10), and the system is analyzed by using thefield matrices given by Eq. 8.34. This gives satisfactory results at lowfrequencíes.!"

2. A numerical procedure ís adopted to determine the elernents of the fieldmatrix.

The determination of the field matrix for a pipeline having variable character-istics is equivalent to integrating the differential equation, Eq. 8.40. This maybe done by the Runge-Kutta method'" as follows.

The pipeline is divided in to n reaches as shown in Fig. 8.1O. First, the fieldmatrix for each reach is computed, and then the field matrix for the pipe is

Substitute pipe

Actual pipe

2 i+1 n n+1

Figure 8.10. Actual and substitute pipe for a pipe having variable eharacteristics along itslength.

Resonance in Pressurized Piping Systerns 223

determined by multiplying these matrices in a proper sequence. If the lengthof the reach between the sections i and i + l is s, then the fourth-order Runge-Kutta method gives29

(8.42)

in which

k¿ = sB(x¡)z¡ }k¡ = sB(x¡ + s/2) (z, + ko/2)

k2 :: sB(x¡ + s/2) (z, + k¡ /2)

k, :: sB(x¡+ ¡)(z¡ + k2)

in which B(x¡), B(x¡+ 1)' and B(x¡ + s/2) are, respectively, the values of thematrix B(x) at section i,at i + 1, and at the middle of sections i and i + 1.

By substituting Eqs. 8.43 into Eq. 8.42, we obtain

(8.43)

(8.44)

in which the field matrix for a pipe having variable characteristics along its lengthis

. sFuc=I+"6 [B(x¡)+4B(x¡+s/2)+B(x¡+¡)]

S2+ - [Btx, + s/2)B(x¡) + B(x¡+ ¡)B(x¡ + s/2) + B2(x¡ + s/2)]

6

(8.45)

in which 1= identity or unit matrix.

ParaJ/elSystem

Let there be n loops in parallel (Fig. 8.11) whose overall transfer matrices are

u(m) = F(n~)P~~)... P(;')F~~) m = 1,2, ... .nm m m m

(8.46)

The superscript in the parentheses refers to the number of the loop. The matrixU(m) relates the state vectors at the l ~ st and at the (n~ + l )th section of themth loop (see Fig. 8.1 lb), í.e.,

Z(m)L - U(m)z(m)RnÍn+¡ - I'm (8.47)

224 Applied Hydraulic Transients Resonance in Pressurized Piping Systerns 22S

I I IU(m)I=1 m = 1,2, ... ,n (8.50a)

~

I

=~ ~t r-G:H í.e.,= i Fp i+1

:u(m) u(m) _ u(m) u(m) _ 1¡ ¡ 11 22 12 21 _ m = 1,2, ... .n. (8.50b)

"

Pipe "n A prime on the subscript denotes a section on the parallelloop.The elements of the field matrix, Fp» for the parallelloops relating the state

vectors zf+ 1 and zf (Fig. 8.11 b) can be determined from the followingequatíons.P

,2' 3'

Longitudinal seetion of n1h loop

I_~

11 _-

11

~t112 = - - 11

11

1121 =-

11

t122 =-11

(8.48)

,1+1

in which

(8.49)

(a) Piping systemn 1

11 = L (m)m=l U21

n u(m)" 22t = L... (m)

m=1 U21

In deriving the aboye expressions, use has been made of the relation that

Note that Eq. 8.48 is valid only if the elements of the overall transfer matrixfor each parallel loop satisfy Eq, 8.50b. It is known from the theory of matricesthat, for square matrices, the determinant of the product of matrices is equalto the product of the determinants of matrices. Hence, if IP~m)1= 1, k =2,3, ... ,nm, and IF~m)I=1,k=1,2, ... ,nm, 'for m=I,2, ... ,n, thenIU(m)1= l. It is cJear from Eqs, 8.33 and 8.34 that IFI = 1. Furthermore, thedeterminants of the point matrices for series junctions and for valves and oríficesare a1so unity (see the point matrices derived in the following paragraphs). Thus,if there is a discontinuity other than a series junction, valve, or orifice in any of

lb) BIDck dlagram

Figure 8.11. Parallel system.


the parallel loops, the determinant of the point matrix for the discontinuityshould be checked to ensure that it has unit value before using Eq. 8.48.

Point Matrices

When there is a discontinuity in the geometry of the system at a section (e.g.,series junction, orifice, valve, branch junction), we have to derive a point matrixrelating the state vector to the left of the discontinuity with that to the right.This point matrix is required in the calculation of the overall transfer matrixfor the system, which is then used to determine the resonant frequencies and/orfrequency response of the system.

Point matrices for various boundary conditions usualIy found in hydropowerand in water-supply schemes are derived in the following sections.

Series Junction

A junction of two pipes having different diameters (see Fig. 8.8), wall thick-nesses, wall materials, or any combination of these variables is calIed a seriesjunction.

It follows from the continuity equation that

qf =qf (8.51 )

In addition,

hf == hf (8.52)

if the losses at the junction are neglected. These two equations can be expressedin the matrix notation as

ZR -P Li - scz i (8.53)

in which the point matrix for the series junction is

(8.54)

Since Psc is a unit matrix, it can be incorporated into the field matrix whiledoing the calculations.

Va/ves and Orifices

The point matrix for a valve or an orifice can be derived by linearizing the gateequation. This linearization does not introduce large errors ifthe pressure rise at


the valve is small as compared to the static head. For an oscillating valve, a sinu-soidal valve motion is assumed. It is possible, however, to analyze non sinusoidalperiodic valve motions by thís method. The periodic motion is decomposed intoa set of harmonics by Fourier analysis," and the system response is determinedfor each harmonic. The individual responses are then superimposed to determinethe total response for the given valve motion (since aIl the equations are linear,the principie of superposition 2,16 can be applied).

Oscillating Va/ve Discharging into Atmosphere. The instantaneous and meandischarge through a valve (Fig. 8.12a) are given by the equations

QL - C A ( L )1/2n + 1 - d v 2gH n + 1

Qo == (CdAv)0(2gHo)I/2

(8.55)

(8.56)

in which Cd = coefficient of discharge, and Av = area of the valve opening. Divi-sion of Eq. 8.55 by Eq. 8.56 yields

Q~+!= ":"'(H~+I)I/2Qo ro u; (8.57)

in which the instantaneous relative gate openíng T = (CdAv)!(CdAv)s, and themean relative gate opening ro = (CdAv)o!(CdAv)s' The subscript s denotessteady-state reference, or index, values.

The relative gate opening may be considered to be rnade up of two parts, í.e.,

r = ro + r* (8.58)

in which r* = deviation of the relative gate opening from the mean (Fig. 8.12b).Substitution of Eqs. 8.1,8.2, and 8.58 into Eq. 8.57 yields

(*L ~ ( r* ) ( h*L J 1/2

l+q'Q:lr l+

ro1+ ;:1) (8.59)

If the valve motion is assumed sinusoidal, then

r* == Re (keiwt) (8.60)

in which k = amplitude of valve motion. The phase angle between any otherforcing [unction in the system and the oscillating valve can be taken into consid-eration by making k a complex number; otherwise, k is real.

By expanding Eq. 8.59, neglecting terms of higher order (this is valid only ifIh!~1I «Ho), and substituting Eqs. 8.3, 8.4, and 8.60 into the resulting equa-tion, we obtain

(8.61 )


Va/ve

L R

n+1

(a) Valva at downstream end of pipeline

t

(b) Sinusoidal valve motlon

PiplI I

OrificlI

PiplI 1+1

1+1

(e) Orlfle. at interm.diate .. ctlon

Figure8.12. Valves.


Since h~+1 = O, on the basis ofEq. 8.61, we can write

R _ L 2Hok 2Ho Lhn+1 -hn+1 +--- -Q qn+1

To o(8.62)

In addition, from the continuity equation it follows that

q~+1 = q~+1 (8.63)

Equations 8.62 and 8.63 may be expressed in the matrix notation as

{q}R [1

= 2Hh o

n +1 Qo

(8.64)

The two matrix terms on the right-hand side may be combined as follows:

{:L = [2~0 O 2:0k]{:t11+1 ~o O T; 11+1

Note that the expansion of Eq. 8.65 yields Eqs. 8.62 and 8.63, and 1 = l. Thus,the additional element 1 in the column vector aids in writing the right-hand sideof Eq. 8.64 in a compact formo As defined in Section 8.5 (Eq. 8.10), the columnvector with l as an additional element is called an extended-state vector, z'. Theextended-state vectors and extended-transfer matrices are denoted by a prime.

On the basis of Eq. 8.1 O, Eq. 8.65 may be written as

(8.65)

(8.66)

in which p~v = the extended point matrix for an oscillating valve and is given by

p' = [21

HOov Qo

O

O

(8.67)

o

Val!!.(;J1qJ!_W.g_CQllllant Gg~__Qp_enil1g_j2j_$_chq[gJngj_nto A tmospher_~, In this case,k = O. Hence, Eq. 8.64 takes the form

{q}R ~ 1= 2H

h n+1 - Qoo ~]{:}:., (8.68)


or

ZRn+1 - p zL- u n+l (8.69)

in which Pu = the point matrix for a valve or orifice discharging into atmosphere,and is given by .

(8.70)

Note that Pu is not an extended-point matrix.If a valve of constant gate opening, ar an orifice, is at an intermediate section

(Fig. 8.l2c) then Eq. 8.70 becomes

p.,= ~ 2~o :] (8.71)

in which t::.Ho = the mean head loss across the valve corresponding lo the meandischarge, Qo'

Branch Pipelines

In the piping systems shown in Fig. 8.l3a, pipeline abe ís the main branch, andbd is the side branch. The transfer matrix for the pipeline ab can be computed

P¡p~ i

e

Main pipe

(a) Piping system

Figure 8.13. Branch system,

f Resonance in Pressurized Piping Systerns 231

-1II

B¿+I~

r--G--GJ------11

Li+1

11I

I-r

(b) Block diagram


by using the field and point matrices derived previously. To calculate the overalltransfer matrix for abe, the point matrix at the junction b, relating the state vec-tors to the left and to the right of the junction, must be known. This matrix canbe obtained if the boundary conditíons at point d are specified.

Point matrices for the junction of the main and the branch having variousboundary condítions are derived in this section.


Le! U be the overall transfer matrix for the branch (refer to Fig. 8.13b), i.e.,

-L - U--Rzn +1 - ZI (8.72)

or

{cit' [U"hJn+1 = U21

(8.73)

in which U=FnPn ... P3F2P2F,. The quantities relating to the branch aredesignated by a tilde (l.

Expansion of Eq. 8.73 yields

-L - -R - h-Rqn+1 =Ullq, +U12 ,

h;;+1 =u2,cif +unhf(8.74)

(8.75)

If the flow direction is assumed positive as shown in Fig. 8.13a and the losses atthe junction are neglected, then the following equations can be written

qf = qf ;. cirhf = hf = iíf

(8.76)

(8.77)

By substituting appropriate boundary conditions into Eqs. 8.74 and 8.75 andmaking use of Éqs. 8.76 and 8.77, the point matrices at the junction of themain and the branch can be derived. The following examples illustrate theprocedure.

l)e{ld~f:,·Ylc/._fl!lm(:J¡_. In this case, íj;;+, = O. Hence , it follows from Eqs. 8.74 and8.77 that

(8.78)

Substitution of this equation into Eq, 8.76 yíelds

R L ul2 hLq¡ = q¡ +:;;- ¡.ulI

(8.79)

Equation 8.77 can be written as

hf = O qf + hfEquations 8.79 and 8.80 can be expressed in the matrix notation as

(8.80)

tf t ~;:']{:r (8.81)


or

(8.82)

in which Pbde = the point matrix for the side branch with dead end and is givenby

(8.83)

Branch wjt_I!__CQI)'§_!f!!!:t-..&_arLf!...!!,§!!!y"q_!r, In this case, lí*+, = O. Hence, it fol-)ows from Eqs. 8.74 through 8.77 that

q~ = qI- + U22 M·1 1 ...... 1

U21(8.84)

Equations 8.84 and 8.80 can be expressed in the matrix notation as

{q}R = [1 ~::l{qLLh¡ O IJhJ (8.85)

or

(8.86)

in which Pbres = the point matrix for Ihe branch with constant-head reservoirand is given by

(8.87)

Branch_wiJh_Q.s_d1lalif1g__y_{ll!,g_gL(hg_P_QJY.n~tre_a_rtJ_t;nq_. If the frequency of theoscillating valve on the branch is the same as that of the forcing function on themain (these rnay not be in phase), the system can be analyzed by using the pointmatrix derived in this section.' However , if the frequencies of the forcing func-tions are not the same, then the system is analyzed considering each forcingfunction at a time and the results are then superimposed to determine the totalresponse.

Since an extended overall transfer matrix relating the state vector at the firstand at the las! section on the branch is required, the extended transfer matriceswill also have to be used for the main. Let the extended overall transfer matrix

234 Applied Hy draulíc Transients

for the branch be

o ~] (8.88)

In writing U' in Eq. 8.88, it is assumed that there is no other furcing function onthe branch. If this is not the case, then at least one of the elements ¡¡ 13 , ¡¡23 ,

¡¡31, and Un is not equal to zero, and the point matrix derived in this sectionshould be modified accordingly.

For the branch pipeline,

(8.89)

and

(8.90)

By substituting Pov from Eq, 8.67 and Z;+I from Eq. 8.90 into Eq. 8.89 andexpandíng the resulting equation, we obtain

(8.91 )

(8.92)

AH the notations defined in the previous sections apply except that the tilde (-)refers to the branch. For example, ro = the mean relative gate opening of thevalve on the branch. Any phase shift between the valve on the branch and theforcing function on the main can be taken into 'consideration by making k acomplex number; otherwise, k is real.

Since Fí~+1 = O, and Fí~ = hf, it follows from Eq. 8.92 that

(8.93)""in which

(8.94)

and

PI3 = (8.95)

rJ)


By substituting ¡¡f from Eq. 8.93 into Eq. 8.76, we obtain

qf =qf +P12hf +P13 (8.96)

Moreover, we can write

1 = O qf + O hf + 1 (8.97)

Equations 8.80, 8.96, and 8.97 can be expressed in the matrix notation as

PI2l

O

(8.98)

or

(8.99)

I¡I~:

in which P~ov = the transfer matrix at the junction of the side branch having anosciJIating valve and is given by

P12 p~J (8.100)

O

., If there is an orifice, or a valve having constant-gate opening at the downstreamend of the branch, then k = O. Hence , P13 = O, and the point matrix for thebranch can be written as

P'o,' ~[~ p;,]Note that this is not an extended-point matrix.

(8.101)

8.8 FREQUENCY RESPONSE

The transfer matrix method can be. used to determine the frequency responseof a system having one or more periodic forcing functions. The equationsderived in this sectíon can be directIy used if the forcing functions are sinusoidal.Nonharmonic periodic functions are decomposed into different harmonics byFourier analysís.? By considering each harmonic at its particular frequency, the

-system response is determined, and ..the results are then superimposed to de-termine the total response.

Systems having more than one exciter may be analyzed as follows: If a1lexciters have the same frequency, then the system oscillates at this particularfrequency. To analyze such a system, the extended transfer matrices are used,

[

r


which allows all the exciters to be considered símultaneously. The concept ofextending the matrices facilitates the development of a point matrix, which in-eludes the effect of the forcing function. If', however, the forcing functions havedifferent frequencíes, then the system response for each forcing function has tobe determined separately, and the concept of extended transfer matrices be-comes invalido The following explanation will further clarify this point. Systemshaving more than one exciter can be classified into three categories as shown inFig.8.14. The systern of Fíg. 8.14a can be analyzed considering all the excite essimultaneously by using the extended transfer matrices with frequency w l' Forthe system of Fig. 8 .14b, the exciters are divided into groups, with each groupcomprised of exciters that have the same frequency. Considering one group ofexciters at a time, the system is analyzed by using the extended-transfer matriceswith the frequency of the group. The results are then superimposed to deter-mine the total response of the system. For the systern of Fig. 8.14c, the systernresponse for each exciter is determined separately, and the total response is then

(a) Al! exclters have same frequency

(b) Exciters divided lnto two groups,according to frequency r

(e) Each exciter has Q dlfferent frequency

Figure 8_14. Systems having more than one exciter.


calculated by superposition. As in this case only one forcing function is con-sidered at a time, and ordinary transfer matrices (i.e.; 2 X 2) are used.

Expressions to determine the frequency response of typical piping systems f?rthe following exciters are derived in this section: f1uctuating pressure head, oscil-lating valve , and fluctuating discharge By proceeding in a similar manner, ex-

pressíon for other exciters can be derived.

Fluctuating Pressure Head

Consider the system shown in Fig. 8.15, which has a dead end at the right end.A wave on the surface of the reservoir produces pressure oscillations in the sys-temo Due to the wave, the pressure head at section 1 fluctuares sinusoidallyabout the mean-pressure head. Let this pressure·head variation be given by

hrR =Re(hfeiwt)=Kcoswt=Re(Keiwt) (8.t02)

and let U be the transfer matrix relating the state vectors at the 1st and

(n + 1)th section , i.e.,(8.103)

lt is assumed that there is no other forcíng function in the systern; otherwise, anextended.transfer matríx , U', wil\ have to be used. Expansion of Eq, 8.103

yields

L R + hRqn+l = Ull ql Ul2 1

L R + hRhn+1 = U21 ql Un 1

Since q~+1 = O (dead end), it follows from Eq. 8.104 that

(8.104)

(8.105)

R Ul2 hRq¡ =-- 1Ull

(8.106)

Restlrvoir

Figure 8.15. Series system with dead end.


which on the basis of Eq. 8.102 becomes

R U12Kq1 = ---Un

(8.107)

Substitution of Eq, 8.107 into Eq. 8.105 and simplification of the resultingequation gives

(8.108)

Hence , the amplitude of the pressure-head fluctuation at the dead end is

ha = Ih*+1 1= I(U22 - U~I~21) KIThe amplitude of the pressure head at the dead end rnay be nondimensional-

ized by dividing the amplitude of the pressure fluctuations at the reservoir end,

(8.109)

i.e.,

-1 ha 1-1 Ul2

U21 Ih-- -u ----

r K 22 U11(8.1 lO)

Fluctuating Discharge

Flows are periodic on the suction and on the discharge side of a recíprocatíngpump. These fluctuations can be decomposed into a set of harmonics. Severepressure oscillations may develop if any of these harmonícs has a period equal toone of the natural periods of either the suction or discharge pipeline.

Expressions are derived below to determine, by the transfer rnatrix method,the frequency response of systerns having a reciprocating pump, The suctionand the discharge pipeline rnay have stepwise changes in diameter and/or wallthickness and may have branches with reservoírs, dead ends, or orífices.

Suction Line

Let the transfer matrix relating the state vectors at the lst and (n + I)th sectionof the suction line (Fig. 8.16) be D, i.e., ,

Z*+1 = Uzf

By expanding Eq. 8.111 and noting that hf = O, we obtain

q~+1 =uuqf

(8.111)

(8J 12)

and

(8.113)


Pipo n-I Pipo n

Figure 8.16. Suction and discharge pipelines-

Hence ,(8.114)L _~ Lhn+1 - qn+1

Ull

The inflow-versus-time curve for one period can be decomposed int? a set ofharrnonics by Fourier analysis.' Let the discharge for the mth harrnonic be

q*L =A' sin(mwt+¡J!m) (8.115)n+1 m

or (8.116)q*L :: Re (A e; m w t)n+] m

in whích Am = A;" exp [j(l/Im - } 7T)]; A;" and l/Im are the amplitude and theh se angle respectively , Ior the mth harmonic; and w = frequ~ncy .of the

p a, d 8 116 th t l. - A 10 which Afundamental. u follows from Eqs. 8.3 an ". ~ ~n+1 - m . m

Substl'tution of this relatlOnshlp IOta Eq. 8.114 yieldsis a complex constant.

L U21hn+1 = -AmulI

Hence , the amplitude of pressure-head fluctuation at the suction ñange is

»; = Ih~+llm =! u~~m I

(8.117)

(8.118)

and the phase angle for the pressure head is

-1 [Im(h~+l )m] (8.119)rpm = tan L )Re(hn+1 m

The head-versus-time curve rnay be obtained by vecto~iaJly adding the head-versus-time curve for each harmonic. For the mth harmomc,

h*L = Re [h ei(mwt+l/>m)1 (8J20)n+1 m


or

h*L - h (n+! - m COS mwt + </Jm) (8.121)Hence, the pressure-head-versus_time curve may be t d f

compu e rom the equatíon*L M

hn+! = m"I;! hm cos(mwt+</Jm) (8.122)

thewhiCh ~ =dnumber of harmonics into which the inflow-versus-time curve forpump IS ecomposed.

Discharge Une

By proceeding in a similar rnanner and noting that hL - O tI r 1I .ti . b . d f n+! - , le JO owing egua-

thlon

IS O taín e or the pressure-head-versus_time curve at the discharge side ofe pump:

Mh*!R = " h'L.. m COS (mwt + </J~I)

m=! (8.123)

in which

h'm = I(hf)ml = IU21Am Ilu I (8.124)!2U2! - UIIU22

</J:n ::: tan-I [lm(h~)]Re(h~) (8.125)

and Am :: complex amplitude of mth h .the pump, armomc of the discharge-time curve of

Oscillating Yalve

In oscillating val ves , the area of the .periodically. Since the gate eguation ~:em;g of the .gate or valve is variedand the gate area is non linear this case n .55) r~latmg the head, discharge ,preceding ones However as' di d IS more diffículj to analyze than theIinearized if h ~.<H In t'h d I.SCU~se in Section 8.7, thís eguation can be

o' e envatíon of expressi . thi .matrix of Eq, 8.67 is used Thi " . ons In IS secnon, the point

• IS matríx IS denved by . hment as sinusoidal and by linea" h assunung t e valve move-síon derived herein for the no I1hzmg t ~ gate ~gu.ation. Thus, to use the exprés,

. n armomc períodíe val vemotíon is decomposed into a s t f harmoní movements, the valve

e o armomcs by Fourier lvsí hresponse is determined considering h harmoní ana ySIS, t e system

l eac arrnoníc at its fre d htata systern response is calcuJated b '. quency , an t en the

Let U' be the extended overall t y s~peflmp?smg the individual responses.rans er matrix reJating the state vectors at the

':t.;.":


lst and the (n + l)th section of the system, i.e.,

(8.126)

In addition,

(8.127)

Hence,

'R _ p' U' 'RZn+1 - QV ZI (8.128)

By substituting p~v from Eq. 8.67; multiplying the matrices P~v and U'; andexpanding and noting that hf = O, h~+1 = O, and q~+l = q~+l ; we obtain

2Ho 2HokU23 - -- U13 + -- Un

R Qo To (ql = - 2H 2H k 8.129)

U21 - __ 0 ull

+ __ 0_ Ul3

Qo To

q*+1 = ull qf + un (8.130)

in which u 11 ,u 12 , ••• ,U33 are the elernents of the matrix, U'. By expandingEg. 8.1 26 and noting that hf :::O, we obtain

h~+I=U21qf+U23 (8.131)

To determine the frequency response, the extended field and point matricesare first computed. Then, the extended overall transfer matrix is determined bymultiplying the field and point matrices starting at the downstream end, i.e.,

(8.132)

The value of qf is determined from Eq. 8.129, and q~+1 and h~+! are com-puted from Eqs. 8.130 and 8.131. The absolute values of h~+1 and q~+l are theamplitudes of pressure head and discharge fluctuations at the valve, and theirarguments are, respectively, the phase angles between head and T* and betweendischarge and T* •

If there is no other forcing function except the oscillating valve at the down-stream end of the system, ordinary fleld and point matrices may be used insteadof the extended ones. In this case, Ul3 = un ::::U31 = O and U33 ::::I in Eqs.8.129 through 8.131.

Procedure for Determining the Frequency Response

The frequency response of piping systems may be determined as follows:

1. Draw the block diagram and then the simplified block diagram for thesystem. In the case of simple systems, this step may be omitted.


2. lf the forcing funetion is nonharmonic, deeompose it into a set of har-montes by Fourier analysis, and eonsider one harrnonic at a time. For thespecified frequency, compute the point and field matrices. To write anextended transfer rnatrix , simply add the following elements to the ordinarytransfer matrix derived in Section 8.7: u13 :: U23 :: U31 :: un ::O and U33 :: l.Note that extended transfer matrices are used only if there is more than oneforeing function in the system and each has the same frequency.

3. Calculate the overall transfer matrix by an ordered multíplication of thepoint and field matrices, starting at the downstream end. For this ealculation,the block diagram of step 1 is very helpful, For multiplication of matrices, thescherne outlined in Example 8.2 may be followed if the calculations are beingdone by hand , slide rule, or desk calculator. This scherne reduces the amountof computations.

4. Use the expressions developed in this section to determine the frequencyresponse.

S. If a frequency-response diagram is to be plotted , repcat steps 3 and 4 bytaking different frequencies.

The following example illustrates the preceding procedure for determíning thefrequeney response at the valve end of a branch systern having an oscillatingvalve at the downstream end.

Example 8.2

Plot a frequency-response diagram for the valve end of the branch systemshown in Fig. 8.17a and for the following data:

Qo :: 0.314 m3/sR ::0.0ro :: 1.0

Tth :: 3.0 sk:: 0.2

H¿ :: 100 m

Do the calculations by using a slide rule or a desk caIculator.Computations for wy:: 2.0 are sumrnarízed in the following. By taking dif-

ferent values of ca; and proceeding similarly as for Wy :: 2.0, the frequency-response díagrarn shown in Fig. 8.l7c can be plotted.

Components 01 Transfer Matrices

Wth :: 211/3 :: 2.094 rad/sca= WyWth :: 2 X 2.094 = 4.189 rad/s


------=---

Pip~ I

ReservO;' Ho

> 11 = 500 mal = 1000 mIsO) = 1.81 m

Oeod ~nd

12= 250m02= 1000 mI.02= 1.05m

lal PipinO system

BR

11

11

lbl Block diooram

Frequency response oí a branch piping system. Branch with dead end.Figure 8.11.


hr• 2lh~ItHo

Qr·2/QS/IQo

o ¡fftl,IIodof characlluislics

o"8Ifir---I----i, __-+-=~'-=-~r.::_ra~n~s(,~lIr~m~a:!-lr~iJ(~m~fI~~»a~d~_J

°O~"5¡-~~--~200----3~"O~---~4f.o~~~~~-~~6~"O~~--J7.0wr=""/Wth -

(e) Frequency response diaoram


Pipe 1:

b, = 1] [a¡ = 500/1000 = 0.5 sA] = rrDf¡4 = 7T(1.8 1)2 /4 = 2.578 m2

e] = a] /(gA]) = 1000/(9.81 X 2.578) = 39.542 s m-2

Substitution of these values into Eq. 8.34 yields

fu = f22 = cos (0.5 X 4.189):::: -0.5f2] ::::-39.542 sin (0.5 X 4.189)j= -34.244jf Il:::: - j sin (0.5 X 4.189)/39.542 ::::- 0.022j

Pr~ceeding in a similar manner, the following field matrix F for pipe 2 isobtained: ' 2,

[

0.500F2 =

-102.732j

-0.007/J0.500

Branch pipe:

Since the branch pipe is made up of a single pipe, iJ = F.

""1

I


Proceeding similarly, the following values of the elernents of the fíeld matrixfor the branch are obtained:

Uu ::::0.5U]2 = -0.0154j

Substitution of these values into Eq. 8.83 yields the following point matrix forthe junction of the branch and the main:

_ [1.0 -0.o308/JPbde -

0.0 1.0

lt is clear from the block diagram shown in Fig. 8.17b that

U::::F2PbdeF¡

These matrices may be multiplíed in a schematic manner as shown in Table 8.1 .Since hf ::::O (constant-head reservoír), the second column in the matrices F ¡,

PbdeF¡, and F2PbdeF¡ is multiplied by zero. Thus, the elernents in the secondcolumn of these matrices are unnecessary and therefore may be drapped. Theunnecessary elements in Table 8.1 are indicated by a horizontal dash.

Note that ordínary transfer matrices have been used because there ís only oneforcing function. Hence, U¡3' U23' and U3¡ are zero, and U33 is unity in Eqs.8.129 through 8.131. Substitution of these values and those for Uu and u2]

calculated in Table 8.1 into Eq. 8.129 yields

q~ ::::-0.584 + 0.0127 j

Hence,it follows frorn Eqs. 8.130 and 8.131 that

qt =0.0600- 0.0131j

Table 8.1. Scheme for multiplication of transfer matrices.

[0.500 :]{:r-,f-34.244 i

[1.000 -0.031 i] L1.555 :J {:r ',fPbde0.000 1.000 -34.244 t

[ 0.500 -0.0073] [-1.0273 :J{:}>,¡F2

-102.732i 0.500 142.595 j


and

hr = -1.S134- S.3215j

Hence ,

h, = 2 Ihr I/Ho = 0.170qr = 2 Iqr I/Qo = 0.390

The phase angle between the pressure head and the relative gate opening

-1 [-S.3215]= tan-1.S134

=-102.29°

The phase angle between the discharge and the relative gate opening

=t -1 [-0.0131]an 0.0600

= -12.29°

8.9 PRESSURE AND DISCHARGE V ARIA TION ALONG A PIPELINE

The previous sections dealt with the determination of the pressure and dischargeoscillations at the end sections of a system. However, sometimes it is necessaryto determine the amplitudes of the discharge and pressure fluctuations along thelength of the pipeline. In this section, a procedure to determine the amplitudesof the discharge and pressure fluctuations along the length of the pipeline isoutlined.

To analyze a píping system, two of the four quantities-discharge and pressureor their relationship at either end of the system-must be known. The other twoquantities can then be calculated by using the equations derived in the last section.The amplitudes of the díscharge and pressure fluctuations at the upstream endbeing known, their amplitudes along the pipeline may then be determined.The procedure ís illustrated by discussing a system that has a reservoír at theupstream end and an oscillating valve at the downstream end. Similarly, equa-tions for other systems having different boundary conditions can be developed,

Suppose that the amplitudes of the discharge and pressure oscillations at thekth section on the ¡th pipe (see Fig. 8.1Sa) are to be determined. Let the transfermatrix relating the state vectors at the first section of the first pipe and the firstsection of the i th pipe be designated by W, i.e.,

(z1)¡ = W(z1)1 (8.133)


r-_p~iP_e_1 ----....:.P.:.t.iP::.::e:....2=-¡tt Pipe i- I

r----- K -----ii p'. I

ti 'pe 1 :

I¡ k¡

(o) Pipino system

r&-G-------11

_Q _n_______ _~

\ I11

~ki

lb) Block dioorom

Figure 8.18. Designation of k th section on ¡th pipe.

and the field matrix relating the state vectors at the first and the kth section ofthe ith pipe by Fx, i.e;

(S .134)

In these equations, the subscript within the parentheses refers to the pipe nurn-ber. The matrix W is cornputed by multiplying the point and field matrices forthe first (i - 1) pipes (see the block diagram of Fig. 8 .18b), i.e.,

W=P¡F¡_IPi-l ... F1 (8.135)

and the matrix Fx is calculated by replacing 1 with x in Eq. 8.33. Note that theelements of the matrix W for a specifled frequency are constants, while those ofthe matrix Fx depend upon the value of x as well.

It follows from Eqs. 8.133 and 8.134 that

(zf)¡ = S(Zf)1 (S .136)

in which

(8.137)


111evalue of (q 1)1 is calculated from Eq. 8.129. Furthermore, it is known that(h1)1 == O. Substitution of these values into the expanded form of Eq. 8.136yields

(8.138)

and

(8.139)

The amplitudes of the discharge and pressure fluctuations at any other sectíoncan be determined by proceeding in a similar manner.

8.10 LOCATION OF PRESSURE NODES AND ANTINODES

The location of pressure nodes and antinodes is an important aspect of theanalysis of resonance in pipelines at higher harmonics. The amplitude of thepressure fluctuation is a mínimum at a node and a maximum at an antinode.For a frictíonless system, the amplitude of the pressure fluctuation at a no deis zero.

The pipeline s may be subjected to severe pressure fluctuations at the antinodes.Thus, the pipe may burst due to pressure in excess of the design pressure or maycollapse due to subatmospheric pressure. A surge tank becomes inoperative inpreventing the transmission of pressure waves upstream of the tank if a node isformed at its base. Jaeger explained the development of fissures in the Kander-grund tunnel30,31 due to the establishment of a pressure node at the tank, whíchmade the tank inoperative although it was overdesigned.

The locations of the nodes and antinodes may be determined as follows:Equation 8.139 gives the amplitude of the pressure fluctuation at a point. By

making use ofthe fact that for a frictionless system, the amplitude of the pressurefluctuations at a node is zero and q1 f= O for nontrivial solutions, we obtain

Su (x) == O (8.140)

The solution of this equation for x gíves the location of the nodes on the ithpipe.

The amplitude of the pressure fluctuation is a maximum at the antinodes.The location of these points may be determined by differentiating Eq. 8.140with respect to x, equating the result to zero, and then solving for x, Le.; theroots of the equation

d- S21(x) == Odx

(8.141)

give the location ofthe antinodes.


By making use of Eqs. 8.140 and 8.141, expressions for the location of nodesand antinodes in simple systems can be derived. The procedure is illustrated inthe following by deriving expressions for a single pipe and for two pipes inseries. Expressions for other systems may be derived in a similar manner. How-ever, for complex systems, it is better to solve Eq. 8.140 and 8.141 numericallyrather than to derive the expressions and then solve them.

On the basis of Eq. 8.34, for a frictionless single pipeline having constantcross-sectional area, Eq. 8.140 becomes

- ¡CI sin (wx/a¡) == O (8.142)

or

sin (wx/a¡) == O (8.143)

whose solution gives

x=mra¡/w (n=0,1,2, .... ) (8.144)

The values of x> l¡ represent the locations of the imaginary nodes, which arediscarded. It follows from Eqs. 8.141 and 8.143 that

cos (wx/a¡) == O (8.145)

The solution of this equation gives the locations of the antinodes, i.e.,

x=(n+!)1I"a¡/w (n=0,1,2, .... ) (8.146)

Agaín, the values of x> l¡ are the locations of the imaginary nodes and arediscarded.

Equations 8.143 and 8.145 show that a standing wave is formed along thelength of the pipeline.

Series System

In a series system having two pipes (Fig. 8.7), the locations of nodes and anti-nodes in the pipe leading from the reservo ir are given by Eqs. 8.143 and 8.145.However, their locatíon in the second pipe can be. determined by using Eqs.8.140 and 8.141.

By substituting the expressions for Fx' F 1, and P2 into Eq. 8.135, multiplyingthe matrices, and using Eq. 8.140, we obtain

- C2sin (:)COS(:~I) - el cos (:) sin (~~I)=0 (8.147)



wx alA2 wlltan - = - -- tan --

. a2 a2AI al

Note that Eqs. 8.147 and 8.148 are valid for a frictionless system only. Solutionof Eq. 8.148 for x gives the location of the nodes.

(8.148)

8.11 DETERMINATION OF RESONANT FREQUENClES

To reduce the possibility of resonance in piping systems, it is important to knowtheir resonant frequencies so that possible forcing functions or exciters havingsimilar frequencies can be avoided. However, if lt is not possible to avoid thesefrequencies, then remedial rneasures can be taken. As the forcing function isusually unknown in the case of self-excited systems, the frequency response ofthe system cannot be determined. For the investigation of such systems, reso-nant frequencies can be calculated using the procedure presented by Zielke andRosl32 or by using the transfer matrix method " as follows.

The transfer matrices for the system components are written for the free-damped oscillations. This is done by replacing jw by the complex frequencys = o + jw in the transfer matrices presented in Section 8.7. By multiplying thesematrices, the overall-transfer matrix is obtained. Then, applying the free-endconditions (e.g., constant-head reservoirs, dead ends, orifices), two homogeneousequations in two unknowns are obtained. For a nontrivial solution, the deter-minant of the coefficients of these equations should be zero. A trial-and-errortechnique is used to solve the determinant equation to determine the resonantfrequencies of the system.

The following example of a dead-end system having a constant-head reservoirat the upper end is presented for illustration purposes:

Let U be the overall transfer matrix for the system, íe.,

Z~+I =Uz~

By substituting the free-end conditions, i.e., h~becomes

(8.149)

O and q~+1 = O, Eq. 8.149

UII q~ = O (8.l50)

Note that Ull and q~ are complex variables. By using the superscripts r and ito represent the real and imaginary parts of a complex variable, i.e., u 11 = u~ 1 +ju{¡ and simplifying, Eq. 8.150 takes the form

( r rR i iR) + '( i rR + r iR) - OUII ql - uuql / ullq¡ ullq¡ - (8.151)

For a complex nurnber to be zero, the real and imaginary parts must be zero.


Hence,

u{¡ qiR - u{¡ q(R = O

ufl q~R + uf¡q~R = O

(8.152)

(8.153)

For a non trivial solution of Eqs. 8.152 and 8.153,

(8.154)=0


(Uil)2 + (U{I)2 =0

Equation 8.155 can be satisfied if and only if

(8.155)

(8.156)

and

U{I = O (8.157)

The Newton-Rapson rnethod'" may be used to determine the values of o andw that satisfy Eqs. 8.156 and 8.157. If o¿ and Wk are the values after kthiteration, then a better estima te of the solution, 0k+1 and wk+¡ of Eqs. 8.156and 8.157 is

(8.158)

aufl i aui¡uf¡ ac;- - Ull a;

Wk+1 = Wk - . .aufl au~¡ _ aui¡ au~¡aw au au aw

In Eqs. 8.158 and 8.159, ufl and ui¡ and their partial derivatives are computedfor 0k and wk' If Iuk+ ¡ - 0k I and IWk+ 1 - wk I are less than a specified toler-ance, then 0k+ ¡ and wk+ 1 are solutions of Eqs. 8.156 and 8.157; otherwise, Uk

and Wk are assumed equal to Uk+1 and Wk+1 and this process is continued untilthe difference between the two successive values of u and W is less than thespecifíed tolerance.

This procedure is general and is not limited to simple and frictionless systems.

(8.159)


Also, note that the intermediate-state vectors have been eliminated by multiply-ing the transfer matrices, and only a second-order determinant has to be solvedas cornpared to an n X n determinant (value of n depends upon the number ofpipes and appurtenances in the systern) in Zielke and Rosl's method.

The aboye procedure is simplified considerably for a fríctionless system sinceu is a function of w only. In a frictionless system having a constant-head reser-voir at the upstream end and an oscillating valve at the downstream end, the arn-plitude of the discharge f1uctuation at the valve is zero during resonating condi-tions at the fundamental or one of the higher odd harmonics. This was observedby Camichel et al.s and reported to be true by Jaeger.30,31 The frequency-response diagrams obtained by theoretical analysis of a number of series, parallel ,and branch systems (a branch system with a side branch having an orífice or anoscillating valve being an exceptíon) done by Wylie19,20 and by the author "confirm this resulto Expressions for the resonant frequencies of the simple frie-tionless systems and their numerical values for the simple or complex systemscan be determined by using this result as follows:

Let U be the overall transfer matrix for a systern having a constant-head reser-voir at the upstream end (section 1) and an oscillating valve at the downstreamend (section n + 1), Le.,

(8.160)

By expanding Eq. 8.160 and noting that h~ = O (constant-head reservoir), andq ~+ 1 = O (discharge f1uctuation node) at a resonant frequency, we obtain

(8.161)

Recall that u 11 is the element in the first row and the first column of the matrixU. For a nontrivial solution, q ~ =1= O; therefore,

UII = O (8.162)

To determine the resonant frequencies, we have to solve Eq. 8.162. To do this,UII is computed for different trial values of w, and the Un -versus-w curve isplotted. If the chosen value of W is equal to one of the resonant frequencies,then u 11 = O. This requirement is not normally met by the first guess for w,and the resulting numerical value of u 11 is referred to as the residual. The pointsof intersection of the UII -versus-w curve and the w-axis are the resonantfrequencies.

Example 8.3

Derive an expression for the frequencies of the fundamental and odd harmonicsofthe system shown in Fig. 8.19. Assume that the system is frictionless.


'\1 1-

R~s~rvoir Ha

Pip~ I

1I

i3

OSClllofillfl volv~

Pipe 2

I2

Figure 8.19. Series piping system.

For the series system shown in Fig. 8.l9 ,

U = F2P2F¡ (8.163)

By substituting F2 and F¡ from Eq. 8.34 and P2 from Eq. 8.54, multiplying the

matrices, and using Eq. 8.162, we obtain

cos b, W. cos b2 W - :: (~;Ysin b¡w sin b2w = O (8.164)

in which b 1 and b2 are constants ror pipe No. 1 and No. 2, respectively, as de-

fined in Eq. 8.34.

Example 8.4

Determine the frequencies of the fundamental and odd harrnonics of the Toulouse

pipeline shown in Fig. 8.20.

Pipe 1 Pipe 2

l' 201.63 mD' 80 mma • 1300 mI.

I • 105.85 mD' 40 mma • 1356 mIs

• 1a I loulou.e pipeline

Pipe 1 Pipe 2

1 • 227.8 mD' 0.6 mQ' 1075 mIl

i • 234.9 mO • 0.5 ma = 1256 mI.

1b I Fully pipeline

Figure 8.20. Longitudinal proflle, (After Camichel et al. s)


:;'"i:!.. .~

o ª•eO• "!eL 00

Q.. eu l

1: ~1: 1:

.5; 1: .5;

~ .5; "~ ~~ "" ~.I:! ~t: ...., ~


1:iti

1: 'Ci

'" ......: ")

" "

,., :;'"Q.. ::s

• 'St:e ª•e,

Q.. OMoci

"t:I '"....~¡;¡"

1: ~1: ".5; 1: .5;.. .!: ....:.. .... -c:: >.

'" ~ ~....I:! '" ~t: ......,


.6

1\ ~

\ I \BB ... K / \.AA/\- CCI1 \ 1 I\,\1 \ I \ ~1'" ..- -_

1\ /o 5 /0 "---d' 20 25 30

-.2

-.4

/5

.003~

0008.85 8.88 w

-.003

EnlarQlmlnt AA

Enlarglmlnt B8

.0/6~.008

..000 31.5 3/.8 w

008

.0/6

Figure 8.21. P10t of residual versus (&1 for Toulouse system,

EnlorQlmlnt ce

The overall transfer matrix, U, for the system of Fig. 8.20 is given by Eq.8.163. For an assumed value of w, the elements of the matrices F 2 , P2 , and F I

are computed from Eqs. 8.34 and 8.54 using the system dimensions shown inFig. 8.20, and these matrices are multiplied to compute the elements of Umatrix. The value of U¡¡ is the residual. By assuming different values of w, thevalues of the residual are determined, and the residual-versus-ce curve is plottedas shown in Fig. 8.21. The intersection of this curve with the ce-axis yields thefollowing frequencies in rad/s:

1. Fundamental: 8.8632. Third harmonic: 20.143. Fifth harmonic: 31.6

8.12 VERIFICATION OF TRANSFER-MATRIX METHOD

To demonstrate the validity of the method presented herein, the results ob-tained by the transfer matrix method are compared with the experimentalvalues and with those determined by the method of characteristics, by theimpedance theory, and by energy concepts.


Experimental Results

Except for the laboratory and field tests reported by Carnichel et al} fewexperimental results on the resonating characteristics of pipes are available inthe Iiterature. In the tests reported by Camichel et al., the resonating conditionsin series pipes were established by a rotating cock located at the downstream endof the pipeline. Each system had a constant-head reservoir at the upstream end.The data for these systems are given in Fig. 8.20.

The values of the periods of the fundamental and higher harmonics determinedexperimentally and by the procedure outlined in Section 8.11 are Iisted in Table8.2. As can be seen, close agreement is found between the experimental valuesand those determined by the transfer matrix rnethod,

Method of Characteristics

A number of systems-series, parallel, and branch systems with the side branchhaving various boundary conditions-were analyzed using the transfer-matrixmethod and the rnethod of characteristics. The data for four of these systemsand the frequency-response diagrams are presented in Figs. 8.22 through 8.25.

The frequency-response diagrarns are presented in a nondimensional formoThe frequency ratio, Wy, is defined as W/Wth; the pressure head ratio, h, as2Ih~+11/Ho; and the discharge ratio, qr, as 2Iq~+¡j/Qo. The values of h; andqr determined by the method of characteristics represent the amplitud e of theswing from the minimum to the maximum value. The frequency of the forcingfunction is designated by w.

The oscillating valves are the forcing functions in all the systems except thedead-end series system of Fig. 8.22 in which the f1uctuating pressure head atthe upstream end is the forcing function. The valve movement is taken assinusoidal with To = 1.0 and k = 0.2. The f1uctuating pressure head in Fig.8.22 is also sinusoidal with K = 1.0. In the branch systems of Fig. 8.25,ro = 1.0 and k = 0.2.

If the friction losses are taken into consideration, the analysis of varioussystems by the method of characteristics showed that the amplitudes of thepositive swing of pressure head and negative swing of the discharge are largerthan the corresponding negative and positive swings. This is caused by thefriction loss term of the governing differential equatíons. In the transfermatrix method, however, the amplitudes of the positive and negative oscillationsare equal because a sinusoidal solution is assumed.

To check the values of the phase angles between different quantities of interest,the oscillatory discharge and pressure head at the valve are computed by usingthe method of characteristics. The qr* - t, h: - t, and T* - t curves are plotted


(j ¡Qo

·c N

O ., I r r "<T

E :::!:1 -c:i..::I: 00

.s J r '";; I r '"- u c:i

(j ¡Qe--'-C'2 ., I r r '-C

o :::!:1

~c:i

::r: Jo

.s I."

;; r r 'Da-. U c:i

¡Qo

u ."'2 ., I r r -o ;:!!; N

~c:i

,-. ::I: Jr-.-~ .s ;; '" r r -

'8 r- U I Nc:i

'C¡f U ¡Q O'2 o- r ro ., .... <'1

5 ;:!!; c:i c:i

::r: J 00 '-C

.s O'> O'>a r r Non c:i c:i(.j

~ on'§ r r O., '" ."

~;:!!; c:i c:i

::r: J N

""..... O... ;; '" r I ."

r<l U c:i c:i..;

"Q¡Q

Nooo ;; 0'\000-..0.¡:: .... ., ~I.(")OOM

11.1 ¡:::::!:1 OM""';""':

CI.~"Qa -e 000'\1'-\1")

§ Ü 0,-000

'" ;; r-r---oo~

"" U. U OM""';""';u .....e

"Q ¡;; .!:! "d N OO"<Te .g ,-. <'1\00\0

"Q O\O\OvU o ~ OuiN_;.... .,¡f -"" ~'3CJca ..~ e ~ NNV'aO'.

'"Z¡¡:; :or'! "=-=(lO e Q) '"'" >J! ::s.!l .E e-, '",.Q ~ o"" ::S::l

f-< ti.) o ::t q- fI"'I Z¡""U.uu. ..


Resllrvoi'r Otlod IMd

I K 304.8 m0.1.22 m0= 1219 mIs

1 = 609.6 m0= 1.1 m0.914 mIs

1.487.7 m0= 0.91 ma· 609.6m/s

l· 655.3 m0= 0.76 ma • 1310.6 mIs

(a 1 Pipinc¡ system

40

lJ\_ l/\L) \~-

30

10

OO 2 3 4

w (Radians/sJ6

{b 1 Frequency response d iac¡ram

Figure 8.22. Frequency response of a series piping system with dead end.

in Fig. 8.26. In this diagram, h~ = h*/Ho and q~ = q*/Qo' The phase anglesdetermined by the transfer matrix method and by the method of characteristicsare presented in Table 8.3. Close agreement is found between the results ob-tained by the two methods.


Impedance Method

The data and the solution, by the irnpedance method, of examples used for com-parison purposes in this study have been taken from the Iiterature .17,19 The firstexample is that of a 490.7-m-long, 0.61-m-diameter, simple pipeline connected

HlIsllrvoir

Pipe 2

Oscilloling VO/VII

Pipe /

,2

1I = 609.6 m01 = 1219 mIsDI = 0.61 m

12= 228.6 mO2= 914.4 mIsO2= 0.3 m

(o) Piping syslem

-,-- Tronsltlr matri« mIJlhodo Melhod 01 choroclerislics

.1:

000~.5r-~~~~--;-------~3~------~~--~~------~6~:_ ~7

""r = w/wth--

(b) Frequency response diogrom

Figure 8.23. Frequency response of a series piping system.


"7----

1 = 670.6 mRlIsllrvoir 0= 1295 mIs Oscll/oling VO/VII ~

0= 0.3 mI I, I I

I2/

1 = 335.3 m0= 1295 mIs0= 0.61 m

4

1 = 335.3 m0= 1295 mIs0= 0.61 m

1 = 762 m0= 1097 mIs0= 0.61 m

( o) Piping system

I I

h, = 21h¡11Haq, = 2lq¡I/Qa

¡--h,

rV"..... , '"'1 , <,

I~I '\11\

V :i\ V¡l\ l/¡\11

1\..,1\" ./ "• U./1 • r-,

.1:-06

0.4

0.2

0.00.5 4 6 7

wr=w/Wth-2

( b) Frequency response diogrom

Figure 8.24. Frequency response of a parallel piping system.

to a reservoir. There is an oscillating valve at the downstream end with To = 0.5and k = 0.5. The steady-state mean discharge is 0.89 m3/s; the wave speed, a =981.5 mis; and the static head, 30.5 m. Using the Paynter's díagram.!" thefollowing values are obtained: h; = 1.92, and the phase angle between the pres-


R~S~'VDir Osciflaling va/ve No. 2

,\'j..

va/ve No. /

.J12 = 226.5 ma2 = 914 misD2= 0.3 m

system

1, = 609.5 mal = 1219 mIsDie 0.61 m

(a) Pipino

-,-- rransfer matri« metno«o Methad 01 charaClers

~~o.6r--it-------~-------i~--------~------4i--------+-------~

7'wr=w/Wth-

( b) Frequency response diaoram (val ves No. I and No. 2 are in phase}

Figure 8.25. Frequency response of a branch piping system. Branch with oscillating valve.


/.2 0.2¡ I*~

<T

*¡.. 1.0

/

0.8 -0.2

1.4 0.4

11

*~<T

*1- 1.2

1.0II

//

//

//./_---

as

0.6(iil ",' 3.0

(e) Branch .y.Iom 01 Fig. 5.Ba (Val •• No. I and 2 in phas.)

Figure 8.26. Time history oCh~, q~, and T*.

Table 8.3. Phase angles.

Phase Angles, tP (in degrees)

Between h and T* Between q and r*

Transfer TransferFrequency Matrix Method of Matrix Method of

System Ratio, Wr Method Characteristics Method Characteristics

Series (Fig. 8.23a) 2.5 -110.99 -110.50 -20.99 -20.53.0 -180.ül -180.00 -270.01

Branch (Fig. 8.1 7a) 2.5 -117.90 -119.00 -27.90 -29.50(Side branch with 3.0 -180.ül -180.00 -270.üldead end)

Branch (Fig. 8.25a) 2.5 -117.13 -118.00 -18.17 -18.00(Side branch with 3.0 -180.01 -180.00 -270.01oscillatíng valve)


sure head and the relative-gate opening, Q>h = 164°. The analysis by the transfermatrix method gives h, = 1.929 and rf>h= 164.5°, which are in close agreernentwith those obtained from the Paynter's diagram. The piping systems presentedby Wylie in Ref. 19 are solved by the transfer-matrix method. The impedance,Z, at the valve is determined by computing the ratio h~+ 1/q;;+ t : Approximatelythe same values of the impedance are obtained as those given by Wylie. Two ofthese piping systems and the impedance diagrams are presented in Figs. 8.27 and8.28. The normalized impedance, Zr, was computed by dividing IZI by the char-acteristic impedance, Ze, at the valve, i.e., z; = IZ I/Zc'

Energy Concepts

In steady-oscillatory flows in piping systems, the energy input during a cycle isequal to the energy output plus the losses in the system. If the losses in the sys-tem are neglected, then the energy input is equal to the energy output duringone periodo This result may be used as follows to verify the numerical values ofthe amplitudes of the pressure head and discharge oscillations, and of the phaseangles obtained by the transfer-matrix method.

The energy entering the system during time interval t:. t is

Ein = rQHt:.t (8.165)

in which r = specific weight of the fluid and the subscript in refers to the inputquantities. Substitution of Eqs. 8.1 and 8.2 into Eq. 8.165 and expansion of theresulting equation yield

Ein = r(QoHo + qtnHo + h~nQo + q~nh~n)t:.t

Let q~n and htn be sinusoidal, i.e.,

i: = hin cos wt

qtn = qln cos (wt - rf>in)

(8.166)

(8.167)

(8.168)

in which rf>in = phase angle between q~n and h~n and hin and qin are the ampli-tudes of the pressure and discharge fluctuations. Note that both hin and qin arereal quantities. The energy input during one cycle may be calculated by substi-tuting Eqs. 8.167 and 8.168 into Eq. 8.166 and by integrating the resultingequation over period, T. This process gives

s.; = rQoHo T+ rqínhín jT cos wt cos (wt - rf>in)dto

(8.169)

~ j_

i~""'Resonance in P~~s'suri~edPiping Systerns 265

R~$"Yoi, Oscillaling valv~

,2 3

1, .. 579 ma, = 1158 misD, = 0.61 m

. 12" 1097 m02= 1097 mIsD2" 0.3 m

tal Pipino system

u,.......-;;;4,..-N

I \ I 1\

/ \ I \ / 1\/ f¡J 'vv \

3

2

oO 2 3 4 5 6

"r=.,I"'th-

t b1 Impedonce diogram

Figure 8.27. Impedance diagram Corseries piping system,

J I

j I \ I \/ \ 1/ \ I / \

v V ~ 1/ \V.5 1.0 20 3.0

266 Applied Hy draulíc Transients

Reservolr

2. 312= 304.8 m02 = 1219.2 mIsD2= 0.61 m

1I " 304.8 m01 = 1219.2 mIsDI = 0.91 m

(o) Piping system

uN<,

N'4•~N

2.

oO 4.0 5.0

Oeod end

Oscll/a/lng va/vII

6.0 T.O"'r=",I"'th-

(b) Impedonce diogrom

Figure 8.28. Impedance diagram for branch piping system. Branch has dead end.

".':~.r~.,~

\~ Resonance in Pressurized Piping Systems 267

If there is a constant-level reservoir at the upstream end, then h¡n = O. Hence ,

Eq. 8.169 becomes

Ein = 'Y QoHo T

By proceeding in a similar manner,

(8.170)

Eout = 'YQoHoT+ 'Yh~utq~ut jT cos wt cos (wt - cf>out)dt (8.171)o

The subscript out designates output quantities.If the losses in the system are neglected , then Ein = Eout' Hence , it follows

from Eqs. 8.170 and 8.171 that

iTcos wt cos (wt - cf>out) dt = O (8.171)

o

which yields

cf>out = 90° (8.172)

For a11 the systerns analyzed in this study, cf>out was 90°. The only exceptíonswere the branch systems in which the side branch had an orífice or an oscillatingvalve. Equations 8.171 and 8.172 do not hold in these cases because there isenergy output at more than one point (see Problem 8.6).

8.13. STUDIES ON PIPELINE WITH VARIABLE CHARACTERISTICS

The resonating characteristics of a pipeline with linearly variable characteristics-A and a-along its length, a constant-head reservoir at the upstream end , and anoscillating valve at the downstream end (Fig. 8.29a) are studied by using thetransfer matrix method. Frequency response is determined by using the transfermatrix given in Eq. 8.45. Then the actual pipe is replaced by a substitutepipe having stepwise changes in characteristics, as shown in Fig. 8.29a.The expressions presented in Section 8.8 are used to determine the frequencyresponse. ro compute w" the theoretical period is calculated from the equation

(8.173)

in which am = velocity of waterhammer waves at the midpoint of the pipeline,and 1= length of the pipeline. The results for both these cases are presented in

Fig.8.29b.


Rest1rvoirx

Ooto for actual pipe .' Dlx) = 1.22 - 0.004 xalx) = 975.4 + 1.6x

OscÜloti"9volvt1

[

- Suósfltutt1 pipe

~ ~. -- . -1-- ----+-- . -- -+-- -- --+--. -- ~ m E

__ ~~~F __ ~~~--~~~C~W~I-dActual pipe

152.5 m

1 = 30.5 mD= 1.16 m0= 999.7 mIs

30.5 m1.04 m1048.5 mIs

30.5 m0.91 m1097.3m/s

30.5 m0.79 m1146.1 mIs

30.5 m0.67 m1194.8 mIs

(o) Piping system

Actual pipeo- - -o Subsltlvle pipe

(b) Frequency response diagram

Figure 8.29. Frequency response of pipeline having variable characteristics.

-····1.······:···'···· ...········••••• Resonance in Pressurized Píping Systems 269

Table 8.4. Resonant frequencies of piping system of Fig. 8.29a.

Resonant Frequencies (rad/s)

Transfer Matrix Method

Mode Favre's Expression Actual Pipe Substitute Pipe

Fundamental 15.127 15.075 14.905Third 35.683 35.702 35.001Fifth 57.647 57.856 56.375Seventh 79.963 74.130 77.749

Resonant frequencies for the system of Fig. 8.29a were determined by con-sidering the pipe per se and then replacing it with a substitute pipe (shown bydotted lines in Fig. 8.29a), and by using the following expression for the resonantfrequencies of a pipeline having linearly variable characteristics derived byFavre" :

wl wltan-=---

am ama(8.174)

in which a = (1 + 1/1/2) [J.I(1+ 1/1!2) + 1/1]; 1/1 = (ao - am)/am; and J.I= (DA - Do)!Do. The subscripts o, m, and A refer to the values at the valve, at the midpoint,and at the reservoir end of the pipe, respectively. The results are tabulated inTable 8.4. Close agreement is found between the results obtained in these casesup to the fifth harmonic. The higher harmonics can be predicted to a reasonabledegree of accuracy by increasing the number of reaches into which the pipelineis divided.

j¡

., 8.14 SUMMARY

In this chapter, the development of the resonating conditions in piping systemwas discussed; available methods for determining the frequency response andthe resonant frequencies were presented; and the details of the transfer matrixmethod were outlined. The transfer matrix method was verified by comparingits results with those of the characteristics and impedance methods and withthose measured in the laboratory and on the prototype installations.

PROBLEMS

8.1 Prove that if (wlla)« 1, then the system may be analyzed as a lumpedsystem. Assume the systern is frictionless. (Hint: Compute and compare


the elements of the field matrix for a lumped systern [Eq. 8.35] and fora distributed system [Eq. 8.34] for cal]a :::::.0.01.)

8.2 Derive the point matrix for an orifice located at the junction of ith and(i + l)th pipe (Fig. 8.l2c). The mean head loss, dHo, across the orificecorresponds to the mean discbarge, Qo'

8.3 Compute the elements of the field matrices for the pipes of the systemshown in Fig. 8.22 for wr = 2.0, and compute the overall transfer matrix.

8.4 Derive expressions for tbe location of the nodes and antinodes for a systemhavíng three pipes in series, a constant-level reservoir at the upstream end,and an oscillating valve at the downstream end. Assume the system to befrictionless. (Hint: Proceed as in Section 8.10.)

8.5 Derive an expression for the natural frequencies correspondíng tothe oddharmonics of a frictionless system having tbree pipes in series, a constant-level reservoir at the upstream end, and an oscillating valve at the downstreamend.

8.6 Prove that for a branch system having an oscillating valve or an orifice onthe branch

, , h.....' -, ....hout q out cos rpout + out quut COS rpout = O

in which h~ut and q~ut are the amplitudes of the pressure and discharge os-cillations, and </Jout is the phase angle between the pressure head and dis-charge. A tilde (~) on various variables refers to the branch; other variablesare for thejnaín. i Hint: Ein = Eout + Eout' Substitute expressions Ior Fj., ,Eout, and EOU! in terms of the mean and oscillatory parts, integrate over theperiod T, and simplify the resulting equation.)

8.7 Derive a point matrix for a simple surge tank and for an air chamber.

8.8 Figure 8.30 shows a Helmholtz resonator. Derive a point matrix for theresonator.

8.9 A short dead-end pipe, called tuner , is sometimes connected to a pipelineto change its frequency response at a particular frequency. Determine

A

Volume. "f

l. L.. 1

Figure 8.30. Helmholtz resonator.

n, I~'~i~.~'

¡I,

,

~


the length, diameter, and wave velocity of a tuner to be connected at thejunction of pipes No. 1 and No. 2 of the series system of Fig. 8.23 so thatthe resonating conditions do not occur at ta, = 3.0. (Hint: Select arbitrar-ily the length, diameter, and wave velocity of a tuner, and analyze the sys-tem as a branch system with the branch having a dead end. If the resonatingconditions still occur at wr == 3.0, change the characteristics of the tuner,and repeat the aboye procedure until a suitable tuner is obtained.)

REFERENCES

1. Thomson, W. T., Vibration Theory and Appllcations, Prentice-Hall, Inc., EnglewoodCliffs, New Jersey, 1965, p. 5.

2. Wylie, C. R., Advanced Engineering Mathematics, Third Ed., McGraw·HiII Book Co.,New York, 1965,p.145.

3. Allievi, L., Theory of Water Hammer (translated by E. E. Halmos), Riccardo Garoni,Rome, Italy, 1925.

4. Bergeron, L., "Etude des variations de regirne dans les conduits d'eau: Solutiongraphique generale," Revue hydraulique , Vol. 1,.1935, París, pp. 12-25.

5. Camichel, C., Eydoux, D., and Gariel, M., "Elude Théorique el Expérimcntale desCoups de Bélier," Dunod, Paris, Frunce, 1919.

6. Abbot, H. F., Gibson, W. L., and McCaig, l. W., "Measurements of Auto-Oscillationsin a Hydroelectric Supply Tunnel and Penstock Systern," Trans. Amer. Soco of Mech,Engrs., vol. S5, Dec. 1963, pp. 625-630.

7. Parrnakiun, 1., Water Hammer Analysis, Dover Pubticatíons, lnc., New York, 1963.8. Den Hartog, J. P., "Mechanical Vibrations in Penstocks of Hydraulic Turbine In-

stallations," Trans., Amer. Soco of Mech. Engrs., vol. 51, 1929, pp. 101-110.9. Rocard, Y., Les Phénornéns d'Aulo-Oscillation dans les lnstallations Hydrauliques,

Paris, 1937.10. Fashbaugh, R. H. and Streeter, V. L., "Resonance in Liquid Rocket Engine System,"

Jour., Basic Engineering, Amer. Soco of Mech. Engrs., vol. 87, Dec. 1965, p. 1011.11. Saito, T., "Self-excited Vibrations of Hydraulic Control Valve Pipelines," Bull., Japan

Soco of Mech. Engrs., vol. 5, No. 19. 1962, pp. 437-443.12. McCaig, 1. W. and Gibson, W. L., "Sorne Measurements of Auto-Oscillat.ions Initiat~d

by Valve Characteristics," Proceedings, 10th General Assembly , International Assocla-tion [or Hydraulic Research, London, 1963, pp. 17-24.

13. Hovanessian, S. A. and Pipes, L. A., Digital Computer Methods in Engineeríng, McGraw-Hill Book Co., New York, Chapter 4,1969.

14. Streeter, V. L. and Wylie, E. B., Hydraulic Transients, McGraw Hill Book Co., Inc., NewYork,1967.

15. Blackwall, W. A., Mathematical Modelling of Physical Networks, Macmillan Co., NewYork, 1965, Chapter 14.

16. Lathi, B. P., Signals, Systems and Communication, John Wiley & Sons, 1965, pp. 2,13.17. Paynter, H. M., "Surge and Water Hammer Problems," Trans. Amer. Soco of Civ, Engrs.,

vol. liS, 1953,pp. 962-1009 .lS. Waller, E. J., "Prediction of Pressure Surges in Pipelines by Theoretical and Experimen-

tal Methods," Publication No. 101, Oklahoma State University, Stillwater, Oldahoma,June 1958.


19. Wylie, E. B., "Resonance in Pressurizcd Piping Systerns," thesis presented lo the Univer-sity of Michigan, Ann Arbor, Michigan, in 1964, in partial fulfillment of the require-ments f'or the degree of doctor of philosophy.

20. Wylie, E. B., "Resonance in Pressurized Piping Systerns," Jour., Basic Engineering,Trans., Amer. Soco of Mech. Engrs., vol. 87, No. 4, Dec, 1965, pp. 960-966.

21. Pestel, E. C. and Lackie, F. A., Matrix Methods in Elastomechanics , M¡;Graw-Hill BookCo., New York, 1963.

22. Molloy, C. T., "Use of Four-Pole Parameters in Vibra tion Calculations," Jour. Acousti-cal Society of America, vol. 29, No. 7, July 1957, pp. 842-853.

23. Reed, M. B., Electrical Network Synthesis, Prentice-Hall, Inc., New York, 1955, Chap-ter 2.

24. Chaudhry , M. H., "Resonance in Pressurized Piping Systerns," thesis presented lo theUniversity of British Columbia, Vancouver, British Columbia, Canada, in 1970, inpartial fulfillment of the requirements for the degree of doctor of philosophy.

25. Chaudhry , M. H., "Resonance in Pipe Systerns," Water Power, London, July/August1970,pp.241-245.

26. Chaudhry , M. H., "Resonance in Pressurized Piping Systems," Jour., Hydraulics Div.,Amer. Soco of Civil Engrs., vol. 96, Sept. 1970, pp. 1819-1839.

27. Chaudhry, M. H., "Resonance in Pipes Having Variable Characteristics," Jour., HydraulicsDiv., Amer. Soco of Civil Engrs., vol. 98, Feb, 1972, pp. 325-333.

28. Thorley, A. R. D., Unpublished lecture notes, course on Hydraulic Transients arrangedby British Hydromechanics Research Assoc., England, Sept. 1972.

29. McCrak~n, D. D. and Dorn, W. S., Numerical Methods and FORTRAN Programming ;John Wlley & Sons, New York, 1964, p. 156.

30. Jaeger, C., "Water Hammer Effects in Power Conduits," Civil Engineering and PublicWorks Review, vol. 23, No. 500-503, London, England, Feb.-May 1948.

31. Jaeger, c., "The Theory of Resonance in Hydropower Systems, Discussion of Incidentsand Accidents Occuring in Pressure Systerns," Jour. of Basic Engineering, Trans., Amer.Soco ofMech. Engrs., vol. 85, Dec. 1963, pp. 631-640.

32. Zielke, W. and Rosl, G., Discussion of Ref 26, Jour., Hydraulics Div., Amer. Soco ofCiv. Engrs., July 1971, pp. 1141-1146.

33. Chaudhry , M. H., Closure oC Ref 26, Jour., Hydraulics Div., Amer. Soco ofCiv. Engrs.,April 1972, pp, 704-707.

34. Favre, H., "La Résonance des Conduites a charactéristiques Linéairement Variables,"Bulletin Technique de la Suisse Romande, vol. 68, No. 5, Mar. 1942, pp. 49-54.(Translated into English by Sinclair, D. A., "Resonance of Pipes with Linearly VariableCharacteristics," Tech. Translation J 511, National Research Council of Cariada, Ottawa,Canada, 1972.)


Jaeger, C., "Theory of Resonance in Pressure Conduits," Trans, Amer. Soco of Mech. Engrs.,vol. 61, Feb. 1939,pp. 109-115.

Evangelisti, G., "Determinazione Operatoria Delle Frequenze di Risonanza Nei SistemiIdraulici in Pressione," Letta Al/a R. Accademia Delia Scienze Dell'Istituto di Bologne,Feb.1940.

Deriaz, P., "Contributions to the Understanding oCFlowin Draft Tubes in Francís Turbines,"Symposium, International Assoc. for Hydraulic Research, Paper N-l, Sept. 1960.


Katto, Y., "Sorne Fundamental Nature of Resonant Surge," Japan Soco of Mech, Engrs.,1960, pp. 484-495.

Blade, R. J. and Goodykootz, J., "Study of Sinusoidally Perturbed Flow in a Line Includinga 90° Elbow with Flexible Supports," NASA Report No. TN-D-1216, 1962.

Oldenburger, R. and Donelson, J., "Dynamic Response of Hydroelectric Plant," Trans.Amer. Inst, of Elect. Engrs., Power App. and Systems, vol. 81, Oct. 1962, pp. 403-418.

Roberts, W. J., "Experimental Dynamic Response of Fluid Lines," M.S. Thesis, PurdueUniversity, 1963.

Lewis, W. and Blade, R. J., "Study of the Effect of a Closed-End Side Branch on SinusoidallyPerturbed Flow of a Liquid in a Line," NASA Report TN-D-1876, 1963.

D'Souza, A. F. and Oldenburger, R., "Dynamic Response of Fluid Lines," Trans. Amer.Soco of Mech. Engrs., Series O, Sept. 1964, pp. 589.

Vibrations in Hydraulic Pumps and Turbines, Symposium, Inst, of Mech. Engrs., Pr oc. V, pt.3A,1966-67.

Weng, C., "Transmíssion of Fluid Power by Pulsating Flow Concept in Hydraulic Systerns,"Jour., Basic Engineering, Amer, Soco of Mech. Engrs., June 1966.

Tadaya, L et al., "Study of Self-Sustained Oscillations of a Piston-Type Valving Systern,"Japan Soco of Mech. Engrs., 1967, pp. 793-807.

Florio, P. J. and Mueller, W. K., "Development of a Periodic Flow in a Rigid Tube ," Jour,Basic Engineering, Sept. 1968, p. 395.

Holley , E. R., "Surging in a Laboratory Pipeline with Steady Inflow," Jour. Hyd. Div.,Amer. Soco of Civ. Engrs., May 1969, pp. 961-980.

Chaudhry , M. H., Discussion of preceding paper by Holley, E. R., Jan. 1970, pp. 294-296.Moshkov, L. V., "Natural Frequencies of Water Pulsations in a Pipe of Uniforrn Cross

Section in the Case of Aerated Flow," Trans. Vedeneev AII Union Scientific Res. ¡IISt. ofHyd. Engineering , vol. 88, 1969, pp. 46-54 (translated frorn Russian Israel Program Sci.Trans., Jerusalem, 1971).

CHAPTER 9

TRANSIENT CAVITATION ANDCOLUMN SEPARATION

9.1 INIRODUCTlON

In the previous chapters, we assumed that the transient-state pressures through-out the system remained aboye the vapor pressure of the Iiquid. However,this is not always the case. In many low-head systems or systerns in whichtransients are produced rapidly, the pressure may be reduced to the vaporpressure of the Iiquid. This may produce vapor cavities in the flow or maycause the liquid column to separate. Rejoining of the separated columns orcollapse of the cavities results in a large pressure rise, which may damage thepiping system ,

The term transient cavitation is used herein to refer to the phenomenon of theformation and growth of cavities within a liquid due to reduction of transient-state pressures to the vapor pressure of the liquido Depending upon the pipelinegeometry and the velocity gradient , the cavity may become so large as to fillthe entire cross section of the pipe. This is called column separation . Theliquid is divided into two columns at the location of column separation (seeFig. 9.1). Sorne authors also refer to the formation of a large cavity at the topof a pipe as column separation,

In this chapter, column separation and transíent cavitation are briefly de-scribed. Various causes that may reduce the liquid pressures to the vaporpressure are then discussed. Expressions for the dissipation and for the velocityof pressure waves in a gas-liquid mixture are presented. Various methods avail-able for the analysis of cavitating flows or flows in which column separation mayoccur are listed, and the details of two of these methods are presen ted. Thechapter concludes with a case study.

274

Transient Cavitation and Column Separation 275

catumn No. 2Liquid column No. I

Figure 9.1. Column separation.

9.2 GENERAL REMARKS

Almost all industrial liquids and especially natural water contain a smaIl gaseousphase either in the form of free bubbles or as nucleí adhering .t~ or hid~en ,inthe físsures of solids. The solids rnay form a boundary contaírung the liquid ,or they rnay be present as contaminants in the liquido The nuclei grow in sízewhen the liquid pressure is reduced to the vapor pressure and may becorne bub-bles of sufficient size to act as nuclei for cavitation. The growth of a bubbledepends upon the force acting on the bubble due to surface tension, the arnbientliquid pressure, the vapor pressure of the liquid, the gas pressure inside the bubble ,and the tirne-pressure history to which the bubble has been exposed. Further-more, the molecules of free gases may enter the bubble, and two or more bub-bIes may coalesce to form a large cavity. The síze of this cavity increases untilthe dífference between its internal pressure and the decreasing external pressureis sufficient to offset the surface tension , Once this critical size is reached,the vapor-filled cavity becornes unstable and expands explosively , This hypoth-esized sequence of events, i.e., frorn the pressure reduction to the onset of ex-plosive cavitation, occurs in a very short time períod , probably in a few milli-seconds.'

Depending upon the system geometry and the velocity g~adient, the c~~tymay become so large as to fill the entire cross section of the pl~e and ~hus dlv~dethe liquid into two columns. This usually occurs in vertical p~pes, PlP~S ha~tngsteep slopes, or pipes having "knees" in their profile , Expenmental mves.hga-tions2-6 have shown that bubbles are dispersed in the pipeline over a consider-able distance on either side of the Iocation of the column separation.

In horizontal pipes or pipes having mild slopes, a thin cavity confining to thetop of the pipe and extending over a long distance may be forrned. In addition,in thís case, cavitation bubbles are produced over a considerable length of thepipe. Such a flow is referred to as cavitating flow.


Time

Figure 9.2. Time history of pressure following colurnn separation.

The low pressures that may lead to column separation or cavitating flows areproduced by negative or rarefaction waves. These waves are reflected as positivewaves from various boundaries (e.g., a reservo ir) in the system, and compressthe bubbles in the cavitation-flow region and progressively reduce the size ofthe cavity where column separation had occurred. When the cavities collapse orwhen the separated columns rejoin, very high pressures are produced. Thesepressures may burst the pipe if they are not allowed for in the designo

The pressure inside a cavity is equal to the sum of the partial pressures of theliquid vapors and the released gases. If the ternperature of the liquid is assumedconstant, the partial pressure of the Iiquid vapors is constant. The partial pres-sure of the gases can, however, increase or decrease if their mole fraction in thecavity increases or decreases. If a cavity forms and collapses several times duringa transient, Weyler's experimental measurernents ' show that the pressure withinthe cavity increases with successive cavity formations.

If water-column separation occurs at more than one location, Tanahashi andKasahara's experimental resulta? show that the second pressure peak may behigher than the first pressure peak, i.e., Hmax2 >Hmaxl (Fig. 92), althoughgenerally the first pressure peak is the highest.

9.3 CAUSES OF REDUCfION OF PRESSURE TO VAPOR PRESSURE

The transíent-state pressure in a pipeline may be reduced to vapor pressure bypower failure to a pump or by rapid closure of a valve. The sequen ce of eventsfor these cases follows.

Upon power failure, negative-pressure waves generated at the pump travel inthe downstream direction. If the pumping head is small and the pump has asmall moment of inertia, the pressure in the pipeline may be reduced to vaporpressure. For high-head pumping systems, the pressure at a high point of thepipeline may be reduced to vapor pressure (Fig. 9.3a) .


Sfeady srat« hydrau/icgrade fine (HGL)

IMinimum transientsrat« HGL

Reservair

Pump

(o) Pumping system

Sfeady sra fe HGL~

Reservoir--

I ~Va/ve _ ---V __--Reservo/'"

----__,/r<Minimum transtent srate HGL

(b) Upstreom volve

<,<,

<, <, fMinimum rronsient"<, stat« HGL

I

........ Va/ve<,

Reservoir

(e) Downstreom valve

Figure 9.3. Reduction of transient-state pressure to vapor pressure.


If a valve is closed rapidly at the upstream end of a pipeline (Fig. 9.3b),the pressure downstream of the valve may be reduced to vapor pressure. Simi-larly, the pressure upstream of a rapidly closing valve located at the downstreamend of a pipeline may be reduced to vapor pressure 2Lla s after the valve closure(a = velocíty of waterhamrner waves, and L = length of the pipeline). The se-quence of events is as follows: A positive pressure wave produced by the closureof the valve travels in the upstream direction. It is reflected as a negative wavefrom the reservoir. This negative wave is again reflected as a negative wave at thevalve. If the initial steady-state pressure were low or if the magnitude of thepressure waves were large, the pressures at the valve may be reduced to vaporpressure.

A rapid opening of a valve at the downstream end of a pipeline may reduce thepipeline pressures to vapor pressure.

9.4 ENERGY DISSIPATION IN CAVITATING FLOWS

Because of the presence of the bubbles, the liquid in a cavitating flow is amixture of the released gases and the liquid. Experimental investigations ha veshown that there is more dissipation of the pressure waves in a gas-liquid mixturethan in apure liquido This additional dissipation is due to the heat transfer tothe liquid when the bubbles are expanded and compressed. Bernardis el a1.7

showed how mechanical work is transferred in the form of heat energy into theliquid during each compression-and-expansíon cycle of a single spherícal bubblecontaining a perfect gas, confined in an unbound incompressible liquid, and sub-jected to a sudden pressure impulse of short duration ,

Weyler ' developed the following equatíon for the shear stress due to non-adiabatic behavior of a spherical bubble

(9.1)

in which ao = void fraction at reservoir pressure; p = mass density of the liquid;g = acceleration due to gravity; D = inside diameter of the pipe; ¿lx = fixedlength of the pipe; ¿lH = change in the piezometric head; V = flow velocity;and e = an unknown constant. The void fraction, 0:, for a gas-liquid mixtureis defined as

o:=~Vg + VI

in which Vg and V, are the volumes of the gas and the liquid in the mixture.Weyler determined the value of e by trial and error by comparing his computed

(9.2)


results with Baltzer's experimental data.' It was found that CO:o varied slightlyfor a wide range of 0:

0, thus allowing the use of a mean value of CO:o = 225.

To compute energy dissipation in the cavitating flows, the total shear stress, T,

is determined by adding Tb to the wall shear stress, To' i.e., T = Tb + To·

9.5 WAVE VELOCITY IN A GAS-LIQUID MIXTURE

The wave velocity in a liquid having a small quantity of undissolved gases is con-síderably lessS-13 than in the pure liquido Pearsall, based on measurements takenon two sewage plants,!' reported that this reduction in the wave velocity'? canbe as much as 75 percent depending upon the gas content.

1 . . d . 12 14 • fBy making the fol owmg assumptions , we can erive . an expression or

the wave velocity in a gas-liquid mixture:

1. The gas-liquid mixture is homogeneous, i.e., the gas bubbles are uniforrnly

distributed in the liquido2. The gas bubbles follow an isothermallaw .3. The pressure within the bubbles is independent of the surface tension and

the vapor pressure.

Let us consider a volume of gas-liquid mixture at pressure Po confined in anelastic conduit, and assurne that the pressure ís instantaneously in.creased by

dp. Then,

(9.3)

in which the subscripts m, g, 1, and e, respectively , refer to the quantities for thegas-liquid mixture, gas, liquid, and conduit. The syrnbol V denotes volume, andthe letter d in front of V indicates the change in the volume due to increase inpressure, dp. For example, dVg is the change in the volume of gas, Vg. Now,

(9.4)

Hence , Eq. 9.2 becornes

(9.5)Vo: = __JL

Vm

Since Yg ís a function of pressure p ; o: is also a function of p. If the bubbleexpansion follows an ísotherrnal law, then

«p = O:oPo (9.6)

in which subscript o indícates initial conditions, and variables without any sub-script refer to conditions at pressure p.


If M and P refer to the mass and the mass density, then

Mm =M¡ +Mg

It follows from Eq. 9.4 and 9.5 that

Y¡=(l-a)Ym

Dividing Eq. 9.7 by Ym and making use of Eqs. 9.5 and 9.8, we can write

Pm = o¡ (l - a) +apg

On the basis of Eq. 9.6 and the fact that pYg = PoYgo' Eq. 9.9 becomes

(9.7)

(9.8)

(9.9)

(9.10)

Let us now write expressions for dYg, dY¡, and dYc' If the conduit wallsare thin, then

du o.v;• c = - ---dp (9.11)

Ecein which E¿ = modulus of elasticity of the conduit walls, and D¿ = diameter ofthe conduit. If the void fraction is small and K¡ = bulk modulus of elasticity ofthe liquíd , then •

dpdi/¡ = -y -m K¡

Since the gas bubbles are assumed to follow the isothermal law, pY = p. V .Diff ti ti hi . g o go1 reren la mg t IS equatíon, we obtain

(9.I 2)

dpdY :::-y -g g p

Making use of the fact that Yg = PoVg)p and Ygo = 0:0Y m, Eq. 9.13 becornes

(9.13)

dV = - O:oPoy dpg p2 m

The bulk modulus of the gas-liquid mixture, Km , may be defined as

(9.14)

dpKm::: -_!..__

_dYmYm

Substitutíng expressions for dYc, dY¡, and dYg from Eqs. 9.1 1,9.12, and 9.14into 9.3, substítutíng the resulting expression for dYm into Eq. 9.15, and

(9.15)


simplifying, we obtain

Km = ------aopo 1 o,--+-+-p2 K¡ Ece

In Chapter 1, we derived the following expression for the wave velocity (sub-script m is added to denote values for the gas-líquíd mixture)

(9.16)

a = - (if;;m V p: (9.17)

Substituting expression for Pm from Eq. 9.1 O and for Km from Eq. 9.16 intoEq. 9.17, we obtain

(9.18)

Ir the compressibility of the liquid, the elasticity of the conduit walls, and theterrns of smaller magnitud e are neglected, then the precedíng expression , on thebasis of Eq. 9.6, may be written as

~

pa -m - p¡(l- 0:)0:

(9.19)

In this derivation, we assumed that the pressure inside the bubble did notdepend upon the surface tension as well as on the vapor pressure , Raiteri andSiccardi 14 derived an expression for the wave velocity without making thisassumption. Kalkwijk and Kranenburg" presented a similar, but simplifiedexpression.

9.6 ANALYSIS OF CAVITATING FLOWS AND COLUMN SEPARATION

From the preceding discussion, it is clear that a system in which cavitating flowand column separation occur can be divided into three regions or phases: (l)waterhammer, (2) cavitation, and (3) column separation.

In the waterhammer regíon, the void fraction is so small that it can be ne-glected. Hence, the velocity of the pressure waves does not depend upon thepressure. In the cavitation region, gas bubbles are dispersed throughout theliquido Thus, the liquid behaves like a gas-liquid mixture. In such flows, there isadditional damping due to therrnodynamic effects (se e Section 9.4). In addi-


tion, the velocity of waterhammer din turn depends upon the pressure. waves epends upon the void fraction, which

The three regions may be present simultaneousl in a . .case, cavitation occurs in one part of th y pipmg system. In such acritical locatíon, and the remaining t e system , colu~n separates at Sorneharnmer reoí sys em can be consídered as the water-

mer region. These phases can occur in a sequence as well Fsystern can be considered initially as waterh . he or ex~mple, theincreases due to the reduction of pressure a:~~:r :::I~n, t ben the voíd fr.act~onflow. With further reduction of pressure the colu ow ecomes a cavltatlOgpoint. Then, as the transient-state pressur~ increase:u:t

a: s:parate at ~ critical

separated columns rejoin, the cavities collapse and the who7e ;yes:efle~hons, thewaterhammer region again. ' em ecomes a

In the analysis of systerns in which liquid I .occurs there are m co umn separa tes or cavitating flow

~~ ~~fee;!:,e:n": s~e~onn:e~hB~as~e~d~aO~n~:~~~~~~~a::r~:t:~~t~~ :~: :~~~:~a~it:~;aatli~~~. vanous sImpl'fY' .

methods for the analysis of column se arati 1 mg ass.um~tlOns, a number ofreported in the literature We will b .pfl disc and cavuatmg flows have beenpresen t details of two of them, ne y ISCUSSsome of these methods and

In the graphical methods 16-19 th '.. . ' e cavltahon region is I t dIiquíd column is assumed to separate over the whole i neg e~ e , and thethe pressure at that location drops below th li .d p pe cross section as soon asof the cavity is calculated from the . ~ iqui vapor pressure. The volurnethe Iiquid colurnns on either side of t~n~~:~Ity equation usin.g t.he velocities ofassumed to be equal to the li .d ty. The pressure inside the cavity isanalyzed as for a normal water

1hqall

m1

mVeaporIPr~ssure, and the remaining system israna yS1S.

Streeter and Wylie20 used a similar rocedure í •

teristics was used for the waterham PI' m which the method of charac-and collapse of a cavity was consi;eer ;na ySIS~nd the development, growth,Pressures following rejoining of th re ed an mternal boundary condition.approach were found to be high:r s~~arateh columns computed by using thisSwaffield P h d I an tose measured on a prototype.U

s owe t iat a better agree t btaiputed and measured results fo I men was o tamed, between the com-a short pipeline when th el co urnn separation behind a closing valve in

, e gas re ease was conside d tI hwas assumed. In the latter case ti d re ian w en no gas release. . , re erre to as a vapor- 1 hinside the cavity was assumed equal to the v on y case, t e pressurewas consídered, the pressure inside th . apor pressure. When gas releasepressure plus the partial pressure of t~e ca:lty ~as. assumed equal to the vaporKranenburg 23 B 19 g s. Similar results were obtained by

. rown reported that the maví .of two pumping systerns computed b max~mum pressures m the pipelines

y assumlOg that colurnn separation oc-


curred at only one critical location were less than those measured on the pro-totype. However, he obtained better agreement between the computed andmeasured results when discrete air cavitíes were assumed along the pipeline.

For horizontal pipes, Baltzer, 1 Dijkman and Vreugdenhíl," and Síernons "assumed that a thin cavity was formed at the top of the pipe, and analyzed theflow below the cavity, considering it as an open-channel flow. Vreugdenhilet a1.26 presented two mathematical models for the analysis of cavitation in longhorizontal pipelines: In the first model, a homogeneous gas-liquid mixture wasassumed in the cavitation region, while the regular continuíty and dynamicequations of waterhammer were used for the waterhammer region. In thesecond model, cal!ed separated flow model, a thin cavity was assumed at the topof the pipe in the cavítation region. Gas release into the cavíties was neglected inboth the models. Based on experimental observations, Weyler et al? developeda semiempirical formula for predicting additional momentum loss in the cavita-tion region.

Kranenburg'" presented a mathematical model in whích al! three regions-namely., column separation, cavitation, and waterharnmer-were considered.Equations were derived such that they are valid both in the cavitation regionand in the waterhammer región. A finíte-difference scherne was outlined that issuitable for the analysis of shocks. Comparison of the computed and measuredresults showed good agreement.

Of all the methods listed previously, Kranenburg's mathematical model "appears to be the best because of the inclusion of gas release, consideration ofall three possible regions, and its suitability for the analysis of shocks withoutisolating them. (A shock is a steep wave front.) We will present the details ofthis model in Section 9.8. In addition, the procedure outlined by Brown 19 isvery simple, and the comparison of the computed results with those measuredon the pipeline of two pumping stations has shown good agreement. Detailsof this method will be presen ted in Section 9.1 O.

9.7 DERIVA TION OF EQUATIONS

In this section, we will present the continuity and momentum equations thatdescribe the flow in the cavitation region, and the equations of state for the gasrelease. These equations were derived by Kranenburg in Ref. 27.

The continuity and momentum equations presented in this section become thecontinuity and momentum equations for the waterhammer region if the voidfraction, 0:, approaches zero, Hence, these equations can be used for both thecavitation and the waterhammer regions.


Assumptíons

The following assumptions are made in the derivation of the equations:

l. The flow is one-dirnensional.2. The gas-vapor-liquid mixture is homogeneous.3. The vapor pressure and the gas temperature in the cavities are constant.

This is a valid assumption since the heat-transfer processes related tothe cavities are fast processes. compared to the time span of the pressurechanges.

4. The momentum of the gas and vapor phases is small compared to that ofthe liquid phase and may be neglected.

S. The gravity term for the density gradíent of the fluid along the pipeline issmall and may be neglected.

6. The bubbles are spherical.

Continuity Equation

Applying the law of conservation of mass to a control volume comprising of asegment of the pipe yields

~ J (1 - ex)o¡ dA + ~ J (1 - ex)p, V dA = O (9.20)at ít ax A

in which ex= local void fraction; x = distance along the pipeline; t = time; V=liquid velocity; and A = cross-sectional area of the pipe. Note that exís a func-tion of x , t, and the position in the cross section.

The mean void ratio, a, for a cross section may be written as

a=..!.. f exdAA ít

The equations of state for the liquid phase and for a circular pipe are

dp, = p,dp K,

(9.21)

and

dA 1/1A-=--dp eE/D

in which K, = modulus of compressibility of the liquid; E = modulus of elasticityof pípe-wall material; e = wall thickness of the pipe; and 1/1 = a coefficient ac-counting for the anchorage system of the pipeline. By substituting Eqs. 9.21and 9.22 into Eq. 9.20, neglecting higher-order terms, and simplifying, we

(9.22)


obtain

~ [CI - a) (1 +~)] +~ r(1- a) (1 +-;.) v] = Oat p,a, ax ~ p,a,

(9.23)

in which a, = wave velocity in the liquid (assuming no gas release) and is given bythe expression

_ /; K¡/p,a, -: I/IDK1+--'

eE

(9.24)

and p = absolute pressure in the top of the cross section of the pipe.

Momentum Equation

Applying the law of conservation of momenturn in the positive x-direction

a f a f t:- (1 - ex)o¡ V dA + - (l - ex)P, v2 dA + -a dAat A ax A A X

=- [0-ex)(p¡gSine+2~ VIVI)dA (9.25)

in which e = angle of inclination of the pipeline, and A= friction pararneter.By substituting Eqs. 9.21 and 9.22 into Eq. 9.2S, neglecting higher-order terms,and simplifying, we obtain

~ [(1 - a) (1 + ~) v]+ ~ [o - a) (1 + ---;.) V2 +.E.]at p,a¡ ax p,a, o¡

= - (1 - a) (g sin e + 2~ v IVI) (9.26)

Note that Eqs. 9.23 and 9.26 are in the conservatíon formo

Cavitating Flow

As discussed previously, small bubbles or cavities are present in the region ofcavitating flow. Assuming the gas inside the cavity follows the perfect gas lawand neglecting surface tension, we can write for the dynamic equilibrium of a


spherícal bubble,

(9.27)

in which Pv = vapor pressure of the liquid; R = radius of the bubble; k = universalgas constant; T = absolute temperature; and Nb = quantity of gas in a bubble,

Let us assume that the bubbles are spherical, uniformly distributed, and ofthe same size. Then, the average void fraction

Q=nb~1fR3A 3

(9.28)

in which nb = n umber of bubbles per unit length of the pipeline.Because of release or re-solutíon, the quantity of gas, Nb, in a bubble is a

function oftime, i.e.,

dM r dR(T) ]d: = "r(ps - p)v'rfF lR(T), ---;¡¡-, U(T) (9.29)

in which 'Y = proportionality constant in Henry's law relating the gas pressureand equilibrium concentration of the dissolved gas; Ps = saturation pressure ofthe liquid; (3 = diffusion coefficíent ; T = dummy variable with respect to time;and U = bubble velocity with respect to the surrounding liquid.

If IdRldtl« U and UR/(J» 1, then the function F may be approxímated'"as

F z- 4 R (t) y21fU(t)R (t) (9.30)

The value of the bubble velocíty, U, may be selected?? as follows: If thebubble is not influenced by the wall of the pipe, then U may be assumed equalto the rise velocity of the bubble, If the bubble is attached to the wall, U isapproximately equal to the liquid velocity V; but, íf the bubble is in the topmostpart of the pipe without being attached, then U is considerably less. Experi-mental measurements show that Uz-O.OI mis.

Column Separation

As previously díscussed, liquid column separates only at the critical sections,such as sharp vertical bends, of a pipeline. Thus, it is a local phenomenon andis governed by the continuity equation only, i.e.,

(9.31 )

Transient Cavitation and Colurnn Separation 287

in which Ve = volume of the cavity at the location of colurnn separation; Ve 1

and Ve'2 are the velocities of the liquid column on the upstream and downstreamsides of the cavity, respectively. According to the perfect gas law,

(9.32)

in which Pe = pressure at column separation and N¿ = quantity of the releasedgas at the column separation. Gas release or re-solution is represented by

dNe dNb-=n -dt e dt

in which flc = nurnber of bubbles that together form the column separation anddNb/dt is determined from Eq. 9.29, assuming P = Pe.

(9.33)

9.8 NUMERICAL SOLUTION

Wave Equations

In the last section, we developed equations that describe the transient conditionsin the cavitating flows. If the transient-state pressure is significantly higher thanthe vapor pressure , then the bubble size , and hence the void fraction, become sosmall that the free-gas content may be neglected. In this case, Eqs. 9.23 and9.26 reduce to equations describing the transient-state condítions in closed con-duits without cavitation. Hence , Eqs. 9.23 and 9.26 may be used in both thewaterhammer and cavitation regions.

Equations 9.23, and 9.26 through 9.29 form a nonlinear, second-order , hyper-bolic systern. The following numerical methods are available for the solution ofEqs. 9.23 and 9.26:

l. Method of characteristics.2. Finite-difference methods.

As the velocíty of waterhammer waves in the cavitation region depends uponthe pressure, a fixed grid cannot be used in the method of characteristics, sin ce afixed grid requires interpolations at each time step which smoothen the sharppeaks. Interpolations can be eliminated by using a flexible grid. However, if ashock is formed, the method fails because of the convergence of the characterís-tic curves.i? This difficulty can be círcumvented by isolating the shock in thecornputations. Although, isolation of the shock may be feasible for the analysisof a simple system," the procedure is too cumbersome to be practical.

Only those Iinite-difference methods are suitable for the numerical integrationof Eqs. 9.23 and 9.26, in which isolation of the shock is not necessary and which


add some numerical dissipatíon terms30 so that the non linear instability due tothe pressure dependence of the waterhammer wave velocity is damped. Twosuch schemes are available: Lax's diffusive scheme and Lax-Wendroff's two-stepscheme. Lax's scherne " is of first-order accuracy and introduces additional dif-fusion terms, which are not present in the original differential equations. Thismay result in reducing the maximum pressures. Lax-Wendroff's scheme'P is ofsecond-order accuracy. It can be shown that the resulting difference equations,based on this scheme, are consistent with the original differential equations. De-tails of this scheme are presented herein.

To apply the Lax-Wendroff's scheme, it is necessary that the governing equa-tions be in the so-called conservation or divergence formo The continuity andmomentum equations, Eqs. 9.23 and 9.26, are in conservation form and may bewritten as

aYI; ah;"""""3t + --ax- = y 3;

in which Yji (j = 1, 2, 3 and i = 1 , 2) is a function of the pressure, velocity, andvoid fraction. By replacing the partial derivatives of Eq. 9.34 by finite differ-ences (Fig. 9.4a) as given by the Lax scheme, we obtain

(9.34)

y¡¡(x + Ax, t + At) = 0.5 [y¡¡(x + 2AX, t) + y¡¡(x, t)]

{-At

+ 2Ax [Y2i(x + 2Ax, t) - h¡(x, t)]

At }+ "2 [Y3;(X + 2Ax, t) + Y3¡(X, t)] (9.35)

in which Ax = length of a reach into which a pipeline is divided and At = com-putational time step. By adding the following second step to Eq. 9.35, a second-order accuracy can be obtained:

AtY Ii(X, t + 2At) = Yli(X, t) - AX [Y2¡(X + AX, t + At) - Y2i(x - AX, t + At)]

+ M[Y3¡(X + Ax, t + At) + Y3¡(X - Ax, t + At)] (9.36)

Equations 9.35 and 9.36 are known as Lax- Wendroff's two-step scheme. Thenumerical damping introduced by this scherne is acceptablej ' provided Ax issmall.

The aboye finite-difference scheme is stable if

Ax ;;;.At(a¡ + IVI) (9.37)

t +2.d


o Lox scñeme.,0 Lax-Wendroff scbem«

I II II I

---- --t-- --.-- ---I I

I

¡I

I

x x

(a)

x

Figure 9.4. Notation for Lax-Wendroffs two-step scheme. (After Kranenburg27)

( b)


and

Dt::.t< XlVI (9.38)

If the transient-state pressures drop to the vapor pressure of the liquid, then thesolution obtained by using the Lax-Wendroff's scherne show sorne oscillations,which are caused by the nonlinearities of the governing equations. These oscilla-tions can be suppressed by applying a srnoothing operator at those grid points atwhich a pararneter, °¡, characterizing the oscillations in the variable YIi' exceedsa prescribed value, 0r' The pararneter O¡is defined as

0.5 [y¡¡(x + 2t::.x, t) + y¡¡(x - 2t::.x, t)] - y¡¡(x, t)O· (x t) = -~~ __ ~__.::....!..!....:: __ _.:_..:..=._......:.....;:.;_;_--,-

1 , y,¡ (9.39)

in which y,¡ = reference interval of the variable y¡¡(x, t). (See Fig. 9.4b.) IfIO¡1 > 0n the smoothened value of YIi is deterrnined frorn the equation

YIi(X, t) = YIi(X, t) + O.5YriO¡(X, t) (9.40)

However, if IOil < 0n the value of YI¡(X, t) is not srnoothened. Kranenburg used10,1 = 0.01 in his cornputations.

Colurnn Separation

As discussed previously , liquid colurnn separates only at the criticallocations ofa pipeline. This can, therefore, be explicitly taken into account at the pointswhere it is expected to occur. The differential equations describing the colurnnseparation, Eqs. 9.31 and 9.32, can be integrated by using an explicit finite-difference scherne. However, the time steps required for such a scherne forsmall arnounts of free gas, Ne, is considerably less than that given by Eq.9.37.This difficulty can be overcorne by elirninating Vel and y';2 by using the corn-patibility equations27.33.34 for the characteristic directions following from Eqs.9.23 and 9.26.

Vel + _l-(Pe - Pu) = BI = [V + _1_ (p - Pv) + t::.t (2-DX VIVI- g sin 4>PIaR p¡a

anbkT dNb)] (9.41)+ A(p - Pv) dt R

Ve2 - _1_ (Pe - Pu) =B2 = [V - _1_ (p - Pv) + t1t (2-DX VIVI- g sin 4>p¡aS p¡a

anbkT dNb)] (9.42)A(p - Pv) dt S


in which the subscripts R and S denote conditions at points R and S, which areassumed equal to those at the grid points A and B on either side of columnseparation (Fig. 9.5). By eliminating Vel, Ve2, and Pe from Eqs. 9.31, 9.32,9.41, and 9.42 and integrating over a time step, 2t::.t, assuming Ne to be constant,we obtain

in which

e = (_!_ + _!_)N kTw al a2 e (9.44)

Equation 9.43 allows larger time steps without leading to instability.

Gas Release

A simple explicit finite-difference scheme rnay be used to integrate Eqs. 9.29and 9.33 describing the gas release and re-solution.

o Grid poinls

t /COlumn separation

"

A

x

Figure 9.5. Notation for liquid-column separation.


Results

Kranenburg/" eompared the results computed by using the mathematieal modelof Sections 9.7 and 9.8 with Baltzer's experimental results! and with those mea-sured on a laboratory pipeline .26 The agreernent between the computed andmeasured results was satisfaetory.

Based on the results of his studies, Kranenburg concluded that:l. Including the gas release has ínsignifícant influence in cases where only

cavitating flow occurs; however, the influence is considerable in cases in whichboth column separation and cavitating flow occur.

2. Gas release in the cavitating flow adjacent to the column separation reducesthe duratíon of the subsequent column separations and hence the maximumpressures following column separation,

3. Gas release at the separation eavity inereases the duration of column separa-tion and slightly in creases the pressure.

4. For a quantitative prediction of the gas release, further investigations,probably on prototypes, are required of various parameters, such as number ofbubbles, nb and nc, and relative bubble velocity.

9.9 DESIGN CONSIDERA TIONS

If the analysis of transient eonditions shows that the liquid column wíll separateor transient cavitation will occur in a pipeline, then it has to be decided whetherthe pressures generated when the separated columns rejoin or when the cavitiescollapse are acceptable. Of course, it is possible to design a pipeline to withstandany pressure. Such a design will, however, be uneconornical. Therefore, pro-vision of various control devices or appurtenances should be investigated toobtain an overall eeonomic design.

The following are some of the common appurtenances usually employed topreven! column separation or to reduce the pressure rise when the separatedcolumns rejoin:

l. Air chambers2. Surge tanks3. One-way surge tanks4. Flywheel5. Air-inlet valves6. Pressure-relief or pressure-regulating valve.

Providing an air chamber or a surge tank is usually costly. Increasing the WR2

of the pump-motor by means of a flywheel inereases the space requirementsand may require a separate starter for the motor, thus increasing the initial


costs. Caution must be exercised if air-inlet valves are used, because onceair is admitted into a pipeline, it has to be removed from the line prior to refillingsince entrapped air can result in very high pressures. By providing a pressure-relíef valve or a pressure-regulating valve, the pressure rise can be reduced byletting the colurnns rejoin under controlled conditions.

In addition to the initial costs, the cost and ease of maintenance should betaken into consideration while selecting any of the preceding appurtenances fora particular installa tion.

9.10 CASE STUDY

Brown 19 reported analytical studies and prototype test results on the water-column separation in the discharge lines of two pumping plants designed by theUnited States Bureau of Reclamation. Details of the mathematical model andcomparison of the computed and measured results for one of the pumpingplants are presented in this section.

Project Details

The pipeline profile for the 7.2 Míle Pumping Plant is shown in Fig. 9.6. Otherdata for the pumping plant are:

Type of pump:Rated head:Flow at rated head:Rated speed:Peak efficiency:Specific speed of equivalent

single suction pump:Length of pipeline:Diameter of pipeline:Thiekness of steel pipe:Output of motors:WR2 of one motor:WR2 of one pump:No. pumping units:

Single stage, double suetion72.24 m0.237 m3/s1770 rpm86 pereent

1270 (gpm units); 25 (SI units)1078 ID

0.61 m4.8 mm224kW10.11 kg m2

1.59 kg m2

2

The pump characteristics were obtained from a double suction-type pump,3S,36and therefore its equivalent, single-suction, specífic speed is 1270 (gpm units)instead of the listed value of 1800.


430.---------------------

425

420415

fi 410

.~405~400~395

390385

380375

370365~~--------------------------------------~

t. 0.61 m Dio pipe

Profile

Pressure Ironsducer,----------I El. 36776I

El 366.9<,

Canal W. S.El 364.54 WS. El 427.02

361.87I

Inlet Outlet

Figure 9.6. Profile of 7.2·mile pumping plant. (After Brown 19)

Field Tests

The locations of the test stations are marked in Fig. 9.6. The purnping-plantstation is located essentially at the pumping units, and station 3 and 4 bracketthe "knee" in the pipeline where the water column was expected to separate.The resistance-type pressure cells were used to measure the pressure, and aphotoelectric revolution counter was used to measure the pump speed. High-speed oscillographs were used to record the data.


Mathematical Model

A mathematical model was developed based on the method of charaeteristics.The upstream boundary condition was a centrifugal pump. The pump charac-teristies for all four zones of operation were sto red in the eomputer. The down-stream boundary condition was a constant-head reservoir. The effeet of theentrained air and column separation was taken into eonsideration as outlinedbelow. In order to be compatible with the text, the notation used herein is dif-ferent from that of Ref. 19.

The total volume of the entrained air in the pipeline is assumed to be coneen-trated at diserete air eavities. Let us eonsider the air eavity located at the ithjunetion (see Fig. 9.7). The volume ofthis air eavity is

-Vi = exA¡L¡ (9.45)

in whieh ex = void fraction and L¡ and A¡ are the length and eross-seetional areaof the ith pipe, respectively.

The expansion and eontraetion of the air poeket is assumed to follow the

Oe/oil Z

.s: üarum

Detail Z

Figure 9.7. Notation for air cavity. (After Brown19)


polytropic equation for a perfect gas

(JiPi, n+1 - hu) V'PI = C (9.46)

in which Vp. = volume of the air pocket at the end of time step; Hp. = piezo-I r,n+l

metric head aboye the datum at sectíon ti, n + 1) at the end of time step; C = aconstant determined from the initial steady-state conditions for the air pocket;and hu = pressure head between the datum and the lowest possible absoluto-pressure head level that a gauge at the top of the pipe at junction i could measure(Fig. 9.7). The vaJue of exponen t m is equaJ to 1.0 for a slow isothermal process,and it is equal to lA for a fast adiabatic process. An average value of m = 1.2may be used.

The continuity equation at the cavity may be written as

(9.4 7)

in which I1t = size of the time step; Vi and VPi are the volumes of the air cavityat the beginning and at the end of the time step; Qi n+1 and Qp. are the flow

I l,n+lrates at the upstream end of the air cavity at the beginning and at the end of thetime step; and Qi+1 ¡ and Qp. ¡ ¡ are the flow rates at the downstream end of, 1+ •

the air cavity at the beginning and at the end of the time step. Note that thevalues of the variables at the beginning of the time step are known.

lf the method of characteristics (see Chapter 3) is used for the analysis oftransient conditions, then the positive and negative characteristic equations (Eqs.3.18 and 3.19) for the (i, n + 1) and (i + 1, 1) .section are

(9.48)

and

Qp'+1 I = Cn + Ca· ¡Hp. I 1I t 1+ 1+,

If the head losses at the junction are neglected, then

(9.49)

(9.50)

Therearefiveunknowns,namely,-Yp.,Qp. ¡,Qp. J!», andHp. I l'I I,n+ 1+1,1 l,n+l 1+ ,

in Eqs. 9.46 through 9.50, which can be solved by an iterative technique.


In the computations done by Brown, Q at an absolute pressure head of 10.4 mwas assumed equal to l. x 10-4

, and the value of exponent, m, was taken equalto 1.2. The pipeline was divided into 30 pipes, and an air cavity was assumed atthe end of each pipe. The effects of the entrained air were considered only inthe effective head range, hui - h,¡ in which hui is defined as the upper Iimit of


07 eo

1 Meosured ---r-1 ~. i Computed - - --

:\~, JJI I \~Curlle No. .J 11

: Jl ~ Tesl No. 91 1 !\~ lh:!?;r-f-'WJ_ L --

Curve No, 2--1 .[ti I I

Test No. 1/ r-\ ~rf-I 11.:~::r _Ll

~ V ~ ! 111 j j

11 12 13 '4 I~ " 11 " " 20 Zr U 23.4 Z!JI~OOO I 2 J <1 " I , roTime I seconas

(a) Pumping plant station

1700

Meosured ----r-

1 Compuled----

Curve No. 3~},' E Tesl NO.~

JitL Jli- Tes! No, .J

:~~Iy': i i;\ ~ I=>h-' ,

~-Ijl '_\ ' . 1.

1-:.1'"

_' , 1'-'l"urve No_2\~- '.

r- l±I 2- Z!J

'(000

1200 o I 1 1 <1 !J (o r • 'J 10 11 12 IJ 14 1I 16 U 1I "lO Z 12 II

Time, seconds

(b) Line station :3 - one pump operation

Meosured-r-

r\ Compuled - - - -

~r-r--r-

o ~1\1*" Curve No. 2 , r- r-¡- r--r-

, Tesl No. 9':;t'~

v-Test No. 11riJ -+11; 1- _-- -r-t-t- r-r-

1"-' __ ,

Curve No. 3-n~V'1:

¡-- _r---t- -. , . " 1211 .... sr '1 " lO 21 Zl l) 24 a

2000

1000

")0000 I Z l 7 • ,

Time, seconds

(e) Pump speed~-

19Figure 9.8. Comparison of computed and measured results. (After Brown )

298 Applíed Hydraulic Transients

the effective head range. If the hydraulic grade line was aboye hui, the effects ofthe air at that location were neglected. An effective head range , hui - hu, oflOA m was used in the computations.

The measured and computed results are shown in Fig. 9.8. In one case, shownas Curve No. 2, a very small amount of air was considered only at the criticalpoint, i.e., at the knee in the pipeline pro file . This was done to determine theeffects of the air entrainment. For Curve No. 3, o: = 0.0001 was assumed. Itis clear from these figures that a better agreement is obtained between thecomputed and measured results if the effects of entrained air are taken intoconsiderat ion.

9.11SUMMARY

In this chapter, flows in which liquid-column separation and cavitating flowsmay occur were considered, and the causes of the reduction of pressure that mayproduce these flows were outlined. The continuity and momentum equationsdescribing the cavitating flows were derived, and the methods available for theirsolution were listed. Details of one of these methods were presented. Thechapter concluded with a case study.

PROBLEMS

9.1 Derive Eq. 9.19 from first principies.9.2 At low pressures and ternperatures, the expansion of the entrained gases in a

gas-liquid mixture is isothermal, and thus the bulk modulus of the gas, Kg, isequal to the absolute pressure of the gas, Pg.16 From first principies, provethat, for a low gas content (i.e., void fraction, o: < 0.001),

u _ LEL_uo = Vo:-~+;~

in which Uo = wave velocity in a gas-free liquido Neglect the effect of thepipeline anchorage system. (Hint: For small values of 0:, O:Pg« (I - 0:) PIand (I - 0:) PI ~ PI')

9.3 Assuming different values of 0:, compute the wave velocity, u, in an air-watermixture at atmospheric pressure. Plot a graph between u and 0:.

9.4 Write a cornputer program for the analysis of the piping system shown inFig. 9.3a. Transient conditions are produced by power Iailure to the pump-motors. Assurne that the water-column separation occurs as soon as thepressure at the summit of the pipeline is reduced to the vapor pressure ofthe liquido


9.5 By using the program of Problem 9.4, investigate the effect of increasing theWR2 of the pump-motors on the duration of column separation and on themaximum pressures in the pipeline.

REFERENCES

1. Baltzer, R. A., "Column Separation Aecompanying Liquid Transients in Pipes," Jour.of Basic Engineering ; Amer. Socoof Mech. Engrs., Dec. 1967, pp. 837-846.

2. Tanahashi, T. and Kasahara, E., "Comparison between Experimental and TheoreticalResults of the Waterhammer with Water Column Separations," Bull. Japan Soco ofMech. Engrs., vol. 13, no. 61, July 1970, pp. 914-925.

3. Weyler, M. E., Streeter, V. L., and Larsen, P. S., "An Investigation of the Effect ofCavitation Bubbles on the Momentum Loss in Transient Pipe Flow," Jour. Basic Engi-neering, Amer. Socoof Mech. Engrs., March 1971, pp. 1-7.

4. Sharp, B. B., Discussion of Ref', 3, Jour. Basic Engineering ; Amer. SOCoof Mech. Engrs.,March 1971, pp. 7-10.

5. Safwat, H. H., "Photographic Study of Water Column Separation," Jour., Hyd. DiI,.,Amer. Soco of Civ. Engrs., vol. 98, April 1972, pp. 739-746.

6. Safwat, H. H., "Water-Column Separation and Cavítation in Short Pipelines," Paper No.75-FE-33, presented at the Joint Fluid Engineering and Lubrication Conference,Minneapolis, Minn., organized by Amer. Soco of Mech. Engrs., May 1975.

7. Bernardinis, B. D., Federici, G., Siccardi, F., "Transient with Liquid Column Separa-tion: Numerical Evaluation and Comparison with Experimental Results," L'EnergiaElenrica, no. 9, 1975, pp. 471-477.

8. Wood, A. B., A Textbook of Sound, Bell & Sons, London, 1955.9. Ripken, J. F. and Olsen, R. M., "A Study of the Gas Nuclei on Cavitation Scale Effects

in Water Tunnel Tests," Sto Anthony Falls Hyd. Lab, Proj., Rcport No. 58, Minneapolis,Univ. of Minnesota, 1958.

10. Silberman, E., "Sorne Velocity Attenuation in Bubbly Mixtures Measured in StandingWave Tubes," Jour. Accaust. Soco af Amer., vol. 29, 1957, p. 925.

11. Whiteman, K. J. and Pearsall, 1. S., "Reflex-Vaíve and Surge Tests at a Station," HuidHandling, vol. XIII, Sept. and Oct. 1962, pp. 248-250 and 282-286.

12. Pearsall,l. S., "The Velocity of Water Hammer Waves," Symposium on Surges in Pipe-Unes, Institution of Mech. Engrs., vol. 180, part 3E, Nov. 1965, pp. 12-20 (see alsodiscussions, pp. 21-27 and author's reply, pp. 110-111).

13. Kobori, T., Yokoyarna, S., and Miyashiro, H., "Propagation Velocity ofPressure Wavesin Pipe Line," Hitachi Hyoron, vol. 37, no.l0,Oct.1955.

14. Raiteri, E. and Siccardi, F., "Transients in Conduits Conveying a Two-phase BubblyFlow: Experimental Measurements of Celerity," L 'Energia Elettrica, no. 5, 1975, pp.256-261.

15. Kalkwijk, J. P. Th. and Kranenburg, C., "Cavitation in Horizontal Pipelines due toWater Harnmer," Jour., Hyd. Div., Amer. Soco Civ. Engrs., vol. 97, Oct. 1971, pp.1585-1605.

16. Bergeron, L., Waterhammer in Hydraulics and Wave Surges in Electricity, John Wiley& Sons, New York, 1961 (translated from original French text published by Dunod,Paris, 1950).

1'i


17. Parmakian, J., "One-Way Surge Tanks for Pumping Plants," Trans. Amer. Soco of Mech.Engrs., vol. 80, 1958, pp. 1563-1573.

18. Sharp, B. B., "Rupture of the Water Column," Proc. Second Australian Conf. of Hy-draulics, Fluid Mech., Auckland, New Zealand, 1965, pp. A 169-176.

19. Brown, R. J., "Water-Column Separation at Two Pumpíng Plants," Jour. Basic Engi-neering ; Amer. Soco of Mech. Engrs., Dec. 1968, pp. 521-531.

20. Streeter, V. L. and Wylie, E. B., Hydraulic Transients, McGraw-Hill Book Co., NewYork,1967.

21. Joseph, L, "Design of Protective Facilities for Handling Column Separation in a PumpDischarge Line," in Control of Flow in Closed Conduits, edited by Tullís, J. P., FortCollins, Colorado, 1971, pp. 295-313.

22. Swaffield, J. A., "Column Separation in an Aircraft Fuel Systern," Proc. First Interna-tional Conference on Pressure Surges, British Hydrornechanics Research Assoc., En-gland, 1972, pp. (C2) 13-28.

23. Kranenburg, C., "The Effects of Free Gas on Cavitation in Pipeline s," Proc. FirstInternational Conference on Pressure Surges, British Hydrornechanics Research Assoc.,England, 1972, pp. (C4) 41-52.

24. Dijkman, H. K. M. and Vreugdenhil, C. B., "The Effeet of Dissolved Gas on Cavitationin Horizontal Pipe-Lines," Jour. Hyd. Research, International Assoc, of Hyd. Research,vol. 7,no. 3, 1969, pp. 301-314.

25. Siemons, J., "The Phenomenon of Cavitation in a Horizontal Pipe-line due to SuddenPurnp-Failure," Jour. Hyd. Research, International Assoc. for Hyd. Research, vol. 5, no.2,1967,pp.135-152.

26. Vreugdenhil, C. B., de Vries, A. H., Kalkwijk, J. P. Th., and Kranenburg, C., "Investiga-tion into Cavitation in Long Horizontal Pipeline Caused by Water-Hammer," 6th Sym-posium,Internationa/ Assoc. for Hyd. Research , Rorne, ltaly, Sept. 1972, 13 pp.

27. Kranenburg, C., "Gas Release During Transient Cavitation in Pipes," Jour., Hyd. Div.,Amer. Soco of Civ. Engrs., vol. lOO, Oct. 1974, pp. 1383-1398.

28. Boussinesq, J., "Calcul du Pouvoir Refroidissant des Courants Fluídes," Journal deMathématiques Pures et Appliquées, París, France, vol. 6, no. 1,1905, pp. 285-332.

29. Martin, C. S. and Padmanabhan, M., "The Effect of Free Gases on Pressure Transients,"L 'Energia Elettrica, no. 5, 1975, pp. 262-267.

30. Richtmeyer, R. D. and Morton, K. W., Difference Methods [or Initial- Va/ue Problems,Interscience Publications, New York, 1967.

31. Lax, P. D., "Weak Solutions of Nonlinear Hyperbolic Equations and Their NumericalComputation," Communications on Pure and Applied Mathematics, vol. 7, 1954, pp.159-163.

32. Lax, P. D. and Wendroff, B., "Systern of Conservation Laws," Communications on Pureand Applied Mathematics, vol. 13,1960, pp. 217-237.

33. Kranenburg, C., "Transíent Cavitation in Pipelines," Report No. 73-2, Laboratory ofFluid Mechanics, Dept. of Civil Engineering, Delft University of Technology, Delft, TheNetherlands, 1973.

34. Streeter, V. L., "Transients in Pipeline s Carrying Liquids or Gases," Jour., Transp. Div.,Amer. Soco of Civ. Engrs., vol. 97, Feb. 1971, pp. 15-29.

35. Stepanoff, A. M., Centrifuga/ and Axial Flow Pumps, John Wiley & Sons, New York,1948, pp. 271-295.

36. Swanson, W. M., "Complete Characteristic Circle Diagrams for Turbomachinery,"Trans. Amer. Soco of Mech. Engrs., vol. 75, 1953, pp. 819-826.

Transient Cavitation and Column Separation 30 l


de Haller, P. and Bedue, A:;-uThe Break-away of Water Columns as a Result of NegativePressure Shocks," Sulzer Tech. Review, vol. 4, pp. 18-25,1951.

Duc, J., "Water Column Separation," Sulzer Tech, Review, vol. 41, 1959.Lupton, H. R., "Graphical Analysis of Pressure Surges in Pumping Systerns," Jour Inst. of

Water Engrs., vol. 7, 1953, p. 87.Richards, R. T. et al., "Hydraulic Design of the Sandow Pumping Plant," Jour., Power Div.,

Amer. Soco of Civ. Engrs., April1956.Richards, R. T., "Water Column Separation in Pump Discharge Lines," Trans. Amer. Soco of

Mech. Engrs., vol. 78, 1956, pp. 1297-1306.Kephart, J. T. and Davis, K., "Pressure Surge F ollowing Water-Column Sepuratíon," Jour.,

Basic Engineering, Amer. Soco of Mech. Engrs., vol. 83, Sept. 1961, pp. 456-460.Li, W. H., "Mechanics of Pipe Flow Following Column Separation," Trans. Amer. Soco of

Civ. Engrs., vol. 128, part 2, 1963, pp. 1233-1254_Li, W. H. and Walsh, J. P., "Pressure Generated in a Pipe," Jour. Engineering Mech. Div;

Amer. Soco of Civ, Engrs., vol. 90, no. EM6, 1964, pp. 113-133.Carstens, H. R. and Hagler, T. W., "Water Hammer Resulting from Cavitating Pumps," Jour.,

Hyd. Div., Amer. Soco ofCiv. Engrs., vol. 90, 1964.Duc, J., "Negative Pressure Phenomena in Pump Pipelines," International Symposium on

Waterhammer in Pumped Storage Projects, Amer. Soco of Mech. Engrs., Nov. 1965.Walsh, J. P. and Li, W. H., "Water Hammer Following Column Separation," Jour., Applied

Mech. Div., Amer. Soco ofMech. Engrs., vol. 89, 1967, pp. 234-236.Tanahashi, T. and Kasahara, E., "Analysis of Water Hammer with Water Column Separa-

tion," Bull. Japan Soco of Mech. Engrs., vol. 12, no. 50,1969, pp. 206-214.Knapp, R. T., Daily, J. W., and Hammitt, F. G., Cavitation, McGraw-HiIl Book Co., New

York,1970.Proc. Meeting on Water Column Separation, Organized by ENEL, Milan, Italy, Oct. 1~J71.

CHAPTER 10

METHODS FOR CONTROLLINGTRANSIENTS'

10.1 INTRODUCTlON

A piping system can be designed with a liberal factor of safety to withstand themaximum and minimum pressures caused by any possible operating conditionexpected to occur during the !ife of the system. Such a design in most caseswill, however, be very uneconomical. Therefore, various devíces and/or controlprocedures are used to reduce or eliminate undesirable transients, e.g., excessivepressure rise or drop, column separation, pump or turbine overspeed. Suchdevices are usually costly, and there is no single device that is suitable for allsystems or for all operating conditions. There fore , while designing a pipingsystem, a number of alternatives should be consídered. The alternative thatgives an acceptable system response and an overall economical system shouldbe selected. An acceptable systern response may be defined by specifyinglimits on the maximum and minimum pressures, maximum turbine speed fol-lowing full-load rejection, or maximum reverse pump speed following powerfailure.

In Chapter 1, we derived the following equation for pressure change, S.H, as aresult of an instantaneous change in the flow velocity, tlV,

atlH= - - tlV

g(10.1)

in which a = waterhammer wave velocity and g = acceleration due to gravity.This equatíon indicates that the main function of a device used for the reductionof the magnitude of pressure rise or pressure drop would be to reduce tlVand/or a. In addition, the flow velocity, V, may be varied in such a mannerthat the pressures are kept within the prescribed limits. Such a controlledvariation of the flow conditions, which results in a required system response,

302

Methods for Controlling Transients 303

is referred to as optimal control al transient flows. A discussion of the optimalcontrol of transient flows wilI be presented in Section 10.6.

In this chapter, we will first present various devices available to reduce or toelirninate undesirable transients. Boundary conditions for these devices will bedeveloped, which are required for the analysis of a system by the method ofcharacteristics presented in Chapter 3. The chapter concludes with a case study,

10.2 AVAILABLE DEVICES AND METHODS FOR CONTROLLINGTRANSIENTS

The following devices are commonly used to reduce or to eliminate the unde-sirable transients, such as excessive pressures, column separation, and pump orturbine overspeed following a power failure or a load rejection: (1) surge tanks,(2) air chambers, and (3) valves.

In addition, the severíty of undesirable transients rnay be reduced by changingthe pipeline profile, by increasing the diameter of the pipeline, or by reducingthe waterhammer wave velocity.

A brief description of and the boundary conditions for the three devices arepresented in the following sections.

In the 'derivation of the boundary conditions, the notation of Chapter 3 isused. Two subscripts are used with the variables to designate their values at asection of a pipe: the first subscript refers to the pipe, while the second refersto the section. The subscript P is used to designate an unknown variable at theend of the time step under consideration, i.e., at time to + tlt, while a variablewithout the subscript P refers to its known value at the beginning of the timestep, i.e., at time, to (se e Fig. 3.1).

10.3 SURGE TANKS

Description

A surge tank is an open chamber or a tank connected to the pipeline. Thistank reflects the pressure waves, and supplies or stores excess Iiquid. Varioustypes of surge tanks are presented in Section 11.2.

The transients in a piping systern having a surge tank may be analyzed byusing the method of characteristics lf the transients are rapid and are of shortduration-e.g., waterhammer. However, for slow transients-e.g., oscillationsof the water level in a surge tank following a load rejection on a turbine-thismethod requires an excessíve amount of computer time and is not suitable.In such cases, the system may be analyzed as a lumped system, details ofwhichare given in Chapter 11. In this sectíon, we will derive the boundary conditions


for a surge tank. These boundary conditions are required for the analysis ofa system having a surge tank by the method of characteristics.

Boundary Conditions

Let us consider a surge tank having a standpipe, as shown in Fig. 10.1. Thelength of the standpipe is usually short as compared to the lcngths of the pipesof the system. Therefore, the Iiquid inside the standpipe may be consideredas a lumped mass.

The following eguations may be written for the junction of the stand pipewith the pipeline (see Fig. 10.1a):

1. Positive characteristic equation (Eq. 3.18) for section (í, n + 1)

Qpi,n+l = Cp - CaiHpi,n+l2. Negative characterístic equatíon (Eq. 3.19) for section (i + 1, 1)

Qpi+l, I = Cn + Cai+1 Hpi+1, 1

3. Continuityeguation

(10.2)

(10.3)

Qpi,n+l = Qpi+I,1 + Qpsp (1004)in which Qp = flow in the stand pipe (flow in the upward direction is con-

spsidered positive) at the end of the time step; Qp = discharge at the end of timestep; Hp =: piezometric head aboye datum; Cp, Cn and Ca are constants as de-fined by Eqs. 3.20 through 3.23. The subscripts i and i + 1 refer to the pipenumbers, and the subscripts 1 and n + 1 refer to the section numbers.

4. If the losses at the junction are neglected, then

Hpi,n+l =:Hpi+I,I' (10.5)5. Referring to the freebody diagram of Fig. 10.1b, the following differential

equation may be written for the acceleration of the liquid in the stand pipe atthe end of the time step:

A Lsp dQsp -_ A [H ( ]'V 'V P - zp - Lp) - W - ~fI sp gAsp dt I sp i,n+1 • (10.6)

in which 'Y = specific weight of the Iiquid; W = weight of the líquid in the stand-pipe =: 'YAspLsp; F¡ = force due to friction = 'YAspfLspQp IQp~ 1/(2gDspA~p);and Dsp =: diameter of the standpipe. Substituting express~ns ftr W and Ff intoEq. 10.6 and dividing by 'YAsp, we obtain

Lsp dQsp. ¡Lsp-- =: Hp - Zp - Q IQp IgAsp dt i,n+l 2gA;pDsp Psp sp (10.7)

fr t-~~~r~ •, ,'-"t;fethods for CoiitiólIirig Transients 305

HYdr~

a..:J:

r--

------Surg~ As

lank- ~- .__ r--'-

Sfondpip~--

Q.a,

- .. NN

i'2 Asp...J

Pip~ i Pip~ i+1--- I I --- -(i, n+l) (i+I,1) >-

OOfum~

e

e::J:

+e

(o) Nototion

'YAsp (Hp. I-Y)l. n+

(b) Freebody diogrom

Figure 10.1. Notation for surge tank with standpipe.


6. Let Z and Zp be the heights of the liquid surface in the tank aboye thedatum at the beginning and at the end of the time step. Then,

0.5 Mzp=z + -- (Qp + Qsp)

As sp

in which As = horizontal cross-sectional area of the tank; Qsp = flow in thestand pipe at the beginning of the time step; and /1t is the size of the compu-tational time step.

If the size of the time step is small, then dQsp/dt ~ (Qp - Qsp)/M. Inaddition, since the variation of the flow in the standpipe iS:usually gradual,the friction term may be approximated as f Lsp Qsp IQsp 1/(2gDspA~). Sub-stitution of these relationships into Eq. 10.7 yields

(10.8)

gMAspQ = (H C )+QPsp L P¡,n+l - zp - sp sp

sp(10.9)

in which Csp = fLsp Qsp IQsp 1/(2gA.i:,Dsp)'

Now we have six linear equations, í.e., Eqs. 10.2 to 10.5, 10.8, and 10.9, insix unknowl~s-namely, Qp¡,n+l' ~P¡+l.l' Hp¡,n+l' Qpsp, Hpi+1•1, and zp.These equatíons may be solved símultaneously by any standard numericalmethod.

Note that if we had not written the friction term in terms of the knowndischarge at the beginning of the time step, we would have a nonlinear systemof equations. In that case, the Newton-Raphson method may be used to solvethe system of equations.

10.4 AIR CHAMBERS

Description

An air chamber (Fig. 10.2) is a vessel having compressed air at its top andhaving liquid in íts lower part. To restrict the inflow into or outflow fromthe chamber, an orifice is usualIy provided between the charnber and the pipe-line. An orifice, which is shaped such that it produces more head loss forinflow into the chamber than for a corresponding outflow from the chamber, ísreferred to as differential orifice (Fig. 10.2). To prevent very low minimumpressures in the pipeline and hence column separation, the outflow from thechamber should be as free as possible, while the inflow may be restricted toreduce the size of the chamber. A ratio of 2.5: 1 between the orifice headlosses for the same inflow and outflow is commonly used.' As the air volumemay be reduced due to leakage or due to solution in the liquid, an air com-pressor is used to keep the volume of the air within the prescribed limits.


___Air

CompressorCompressor

Pump

Check va/ve

Figure 10.2. Air chamber.

It is a common practice to provide a check valve between the pump and the airchamber (see Fig. 10.2). Upon power failure, the pressure in the pipeline drops,and the liquid is supplied from the chamber into the pipeline. When the flow inthe pipeline reverses, the check valve closes instantaneously, and the liquid flowsinto the chamber. Because of the inflow or outflow from the chamber, the airin the chamber contracts or expands, and the magnitude of the pressure rise anddrop are reduced due to gradual variation of the flow velocity in the pipeline.

An air chamber has the following advantages over a surge tank:

1. The volume of an air chamber required for keeping the maximum andminimum pressures within the prescribed limits is srnaller than that of an equiv-alent surge tank.

2. An air chamber can be installed with its axis parallel to the ground slope.This reduces the foundation costs and provides better resistan ce to both wind andearthquake loads.

3. An air chamber can be provided near the pump, which may not be practicalin the case of a surge tank because of excessive height. This reduces the pressurerise and the pressure drop in the pipeline.

4. To prevent freezing in cold climates, it is cheaper to heat the liquid in anair chamber than in a surge tank because of smaller síze and because of prox-imity to the pumphouse.

The main disadvantage of an air chamber is the necessity to provide air com-pressors and auxiliary equipment, which require constant maintenance.

Engler.é Allíeví," and Angus" discussed the use of air chambers in pumpingsystems to control transients generated by power failure to the pumps. Charts(see Appendix A) to determine the size of an air chamber for a pipeline to keepthe maximum and minimum pressures within design limits are reported in the


literature.1,s-s These charts rnay be used to determine the approximate sizeof a chamber for a pipeline. During the final design stages, however, a detailedtransient analysis should be carried out. The method of characteristics incor-porating the following boundary conditions rnay be used for this analysis.

Boundary Conditions

Referring to Fig. 10.3, the following equations are available at the junction ofthe chamber with the pipeline:

1. Positive characteristic equation (Eq. 3.18) for section (i, n + 1):

Qp. +1 = Cp - Ca·Hp. 11, ni', n+

(10.10)

2. Ncgative characteristic equation (Eq. 3.19) for section (i + 1, 1):

Qpi+1, 1 = Cn + Cai+1 Hpi+1, 1

3. If the losses at the junction are negIected, then

(10.11)

(10.12)

4. Continuityequation

Qpi,n+1 = Qpi+I,1 + QPorf (10.13)

in which QPorf = tlow through the orifice (considered positive into the charnber).Assuming that the air enclosed at the top of the chamber folIows the poly-

tropic relation for a perfect gas, i.e.,

(10.14)

Air

D..N N Pipe i+1-

(i, n+l) (i+l, 1)

üarom

Figure 10.3. Notation for air chamber.


in which H;. and -JLp. are the absoIute pressure head and the volume of thearr 81f

enclosed air at the end of the time step; m is the exponent in the poIytropicgas equation; C is a constant whose value is deterrnined from the initiaI con-ditions, i.e., C = -JLo

nl. H;. ; and the subscript o refers to the initial steady-air arr

state conditions. The values of m are equal to 1.0 and 1.4 for an isothermaland for an adiabatic expansion or contraction of the airo The expansion orcontraction process is almost adiabatic for small-size chambers and rapid tran-sients, and it is almost isothermal for slow transients and large air volumes.For design caIcuIations, an average vaIue of m = 1.2 may be used because thetransients are usualIy rapid at the beginning and they are slow at the end.

For modeling the behavior of the air volume, Graze9-11 recommends usinga differentiaI equation based on rationaI heat transfer rather than using thepolytropic equation, Eq. 10.14. The main difficulty in using the equationrecommended by Graze is that the rate of heat transfer is not precisely knownand has to be assumed.

The orifice losses may be expressed as

(10.15)

in which Corf = coefficient of orifice losses, and hporf = head Ioss in the orificefor a tlow of QPorf' Note that if the orifice is of differentiaI type, then Corfhas different vaIues for the intlow into and outtlow from the chamber.

The following equations may be written for the enclosed air volume:

H* = Hp + H - Z - hpPair i,Il+1 b P orf (10.16-a)

(lO.l6-b)

t!.tZp = Z + 0.5 (Qorf + QPorf) A

e(10.16-c)

in which H b = barometric pressure head; Ac = horizontal cross-sectíonal area ofthe chamber; z and z pare the heights of the liquid surface in the chamber aboyethe datum at the beginning and at the end of the time step (measured positiveupwards); Qorf = orifice tlow at the beginning of time step; and -JLair = volume ofair at beginning of time step.

We ha ve nine equations, Eq. 10.10 to 10.16 in nine unknowns-namely,

Qpi,ll+t' Qpi+t,l' QPorf' Hpi,n+1' Hpi+t,t' hporf' -JLpair, H;air' and Zp. Anumber of these unknowns may be eliminated as foIlows: Substituting Eqs.10.10 through 10.12 into Eq. 10.13, we obtain

(10.17)


It follows from Eqs. 10.14 through 10.16b that

(Hp¡,n+l + Hb - zp - CorfQPorfIQporf1) Wair - Ac (zp - z)]m = C

(10.18)

In Eqs, 1O.16c through 10.18, we have three unknowns, namely, Qpoñ'Hp. , and Zp. The value of these three unknowns may be determined bysol~i~g\hese equations by an iterative technique, such as the Newton-Raphsonmethod. The known values of these variables at the beginning of the time stepmay be used as a first estimate for starting the iterations.

Sometimes, the air chamber is connected to the pipeline by a standpipe asshown in Fig. 10.4. In such a case, the length of the stand pipe should be asshort as possible, and it should be sízed so that there is little delay betweenthe pressure change in the pipeline and the start of inflow or outflow from thechamber. To account for the liquid in the standpipe, the preceding equationsfor an air chamber having a stand pipe are modified as discussed in the folIowing.

Let us consider the flow in the stand pipe in the upward direction as positive(Fig. lOA) and assume the liquid inside the standpipe as a lumped mass. Then,the dynarnic equation for the stand pipe may be written as

¡LspAsp dQsp = A [H - (H;31'r - Hb) - (zp - Lsp)] - Ff - WgAsp dt ¡ sp P¡,n+l

(10.19)

-: -ber_ ;.,.__ -Air

--_--_

'---., ,---'

Sfondpipe---...

o. o,

'"z¡ Asp

...J NN

Pipe i Pipe i+1- I I --(i,n+l) (i+I,1)

OOlum~

Air chom

Figure lOA. Notation for air chamber with standpipe.


in which W = weight of the liquid in the standpipe; Lsp and Asp are the lengthand cross-sectional area of standpipe, respectively; zp = height of the liquid sur-face in the charnber aboye the datum; and Ff = frictional force in the stand pipeincluding the entrance or exit losses from the chamber into the stand pipe andis equal to ¡Asp hp[" In this expression,

h - (fLsp + C ) Q IQ Ip f - 2 D A 2 orf Psp P spg sp sp(l0.20)

in which Dsp = diameter of the standpipe and Corf = coefficient of the orificehead losses. If the size of the time step, !!.t, is smalI, then hpf may be approxi-mated as

(10.21)

in which k = (fLsp)/(2gDspA;p) + Corf' Substituting W = ¡LspAsp and theexpression for hpf from Eq. 10.21 into Eq. 10.19, approximatíng dQsp/dt =(Qp - Qsp)/M, and simplifying the resuIting equation, we obtain

sp

(10.22)

The continuity equation at the junction of the stand pipe and the pipeline,and the equation for the liquid level in the chamber, may be written as

Qp¡,n+l = Qpsp + Qp¡+l,l

!!.tZp = Z + 0.5 (Qp + Qsp) -

sp Ac

Elimination of Qp¡,n+l' Qp¡+l,l' and Hp¡+l,1 from Eqs. 10.10 through 10.12and 10.23 yields

(10.23)

(10.24)

e C Qpspa¡ + a¡+l

(10.25)

It follows from Eqs. 10.14 and 10.16b that

H;. W"air - (zp - z)Ac]m = e (10.26)air

Now we have four equations (i.e., Eqs. 10.22, 10.24, 10.25, and 10.26) in fourunknowns, namely, QpsP" H;air' zp, and Hp¡,n+l' Because Eq. 10.26 is non-linear, the Newton-Raphson method may be used to solve these equations.The value of Hp'+1 l' Qp. l' Qp'+1 l' and -Yp. may then be determined from" l,n + 1, arrEqs.1O.12, 10.10, 1O.11,and 10.16b,respectively.


10.5 VALVES

Description

Depending upon the type, a valve is used to control the transients by either ofthe following operations:

1. The valve opens or closes to reduce the rate of net change in the flowvelocity in the pipeline.

2. lt allows rapid outflow of the liquid from the pipeline if the pressure ex-ceeds a set limito This outflow causes a pressure drop, thus reducing the maxi-mum pressure.

3. The valve opens to admit air into the pipeline, thus preventing the pressurefrom dropping to the liquid vapor pressure.

A number of valves commonly used to control transients are:

1. Safety valves2. Pressure-relief valves3. Pressure-regulating valves4. Air-inlet valvesS. Check valves.

A safety va/ve or an overpressure pop-off va/ve (Fig. IO.Sa) is a spring orweight-Ioaded valve, which opens as soon as the pressure inside the pipeline ex-ceeds the pressure head set on the valve. The valve closes abruptly when thepressure drops below the limit set on the valve (Fig, IO.6a). A safety valve iseither fully open or fully closed.

The operation of a pressure-relief va/ve or surge suppressor (Fig. 1O.Sb) issimilar to that of a safety valve except that its opening is proportional to theamount by which the pressure in the pipeline just upstream of the valve exceedsthe prescribed Iimits. The valve closes when the pipeline pressure drops andis fully closed when the pressure is below the limit set on the valve. There isusually some hysteresis in the openíng and the closing of the valve, as shownin Fig. IO.6b.12

For a pumping system having more than one pump discharging into a commonheader, a battery of smaller-size reliefvalves or surge suppressors may be usedl3

instead of one large surge suppressor. A suppressor may be installed on eachpump or the entire battery of the suppressors may be mounted on the maindischarge lineo In the latter arrangement, the overpressure setting of each valveshould be set such that the valves open in sequence one after the other ratherthan simultaneously.

Pipeline

Pipeline


(a) Safety valve

(b) Pressure relief valve

Figure 10.5. Schematic diagramsof safety, relief, and pressure-regulating valves,

(e) Pressure regulating valve


1-J~ 1.0 ------

.!::~:}

oL_------~-L------~Pipeline aressure

(a) Safety valve

Pipeline pres sure

(b) Pressure relief valve

1-J~ 1.0 ----

.!::~:}

Time, I

(e) Pressure regulating valve

Figure 10.6. Discharge characteristics of safety, relief, and pressure-regulating valves,(After Evangelistil2)

/ Ii

Methods TórCóntrolling Transients 315

A pressure-regulating valve (PRV) is a pilot-controlled throttling valve, whichis opened or closed by a servomotor, and the opening and closing times of thisvalve can be individually set. It is installed just downstream of a purnp in apumping systern and upstream of a turbine in a hydropower scheme. Followingpower failure to the purnp-motor, this valve rapidly opens and then graduallycloses (Fig. 10.6c) to reduce the pressure rise. The operation of this valve ina hydropower scheme is as follows: If the power plant is isolated from thegrid system, the PRV is kept partly open to provide for the rnaximurn antici-pated rapid load increase. When accepting rapíd load changes, the PRV isclosed in approximate synchronism with the turbine wicket gates to maintainan essentially constant flow velocity in the penstock. Following a load rejec-tion, the PRV is opened as the wicket gates are closed, and then the PRV isclosed at a slow rate. In sueh an operation, sorne water is wasted. However,as isolated operation is an emergency condition, the amount of water wastedis insignificant.

Upon full-load rejection, either in the normal or isolated operation of theturbine, all the turbine flow is switched from the turbine to the PRV, whichis then closed slowly,

Figure 10.7 illustrates the synchronous operation of a PRV and a turbine.Figs. 10.7a and b are for a turbine isolated from the gríd system. Figure 10.7cis for a turbine connected to or isolated from the grid system.

Ideally, the net change in the penstock flow may be reduced to zero bymatching the discharge characteristics of the PRV with that of the turbine.However, this is usualIy not possible because of the nonlinear flow charac-teristics of the turbínes and valves and because of the dead or delay time be-tween the opening (closing) of the pressure regulator and the closing (opening)of the wicket gates. This dead time should be as small as possible to rninimizepressure rise or drop in the penstock.

Air-inlet valves are installed to admit air into the pipeline whenever the pres-sure inside the pipeline drops below the atmospheric pressure. Therefore, thepressure differentiaI between the outside atmospheric pressure and the pressureinside the pipeline is reduced, thus preventing the collapse of the pipeline. Air-inlet valves are also used to reduce the generation of high pressures when theliquid columns rejoin following column separatíon by providing an air cushionin the pipe.

Once air has been admitted into the pipeline, extreme care must be exercisedwhile refilling the lineo The air pockets should be eliminated gradually frorn theline because the entrapped air can resuIt in very high pressures.14-17

Check va/ves are used to prevent reverse flow through a pump and to preventinflow into a one-way surge tank from the pipeline. They are installed imrne-


Turbine

, PRV',~__L _OL---------------~D~m~e----

(a) Partial load acceptanee in isolated operation

....!;c::..:}::..ctQ.. /.0...()

r':": PRV"x--C----/

/---_../

.........to

~ OL-----------------;T'"'"¡m-e-:---

(b) Partia I load rejection in isolated operation

i"ii

(e) Total load rejeetion in normal or isolated operation

Figure 10.7. Synchronous operation of turbine and pressure-regulating valve.


diately downstream of a pump or at the bottom of a one-way surge tank (seeSection 11.2). A check valve in its simplest form is a flap valve , although some-times dashpots and springs are provided to prevent slamming of the valve.

Boundary Conditions

For the analysis of a system in which a valve is used to control transients, theboundary conditions and the time history of the valve opening, r, should beknown.

Boundary conditions for a valve (developed in Section 3.3) may be used forsafety and relief valves, and for a PRV if the pump is isolated from the pipelinefollowing power failure by means of a check valve. The boundary condítionsfor a PRV become slightly more complex if simultaneous flow is permittedthrough the valve as well as through the pump or the turbine. Boundary con-ditions for a pump and a PRV are derived in this section; and for a Franeisturbine and a PRV in Section 10.6.

A check valve may be considered as a dead end for negative flows, while ítspresence may be ignored for positive flows. For a more elaborate analysis,however, the differential equation of Problem 4.6 for the closure of a checkvalve when the flow in the discharge line decelerates may be used.

To use the boundary conditions, the variation of valve opening, r, with timeshould be known. For a pressure relief valve or surge suppressor, t is a functionof the pressure in the pipeline just upstream of the valve. Therefore, the value oft is determined as the calculations progress. The r-versus-time curve for a PRV isspecified. Discrete values on this r-t curve may be stored in the computer, andthe r values at intermediate times may be determined by parabolíc interpolation.

PR Vand a Pump

Let us consider the PRV and pump arrangement shown in Fig. 10.8 in whichthe pipes between the pump and section (i, 1) and between the valve and section(i, 1) are very short and therefore may be neglected. Assuming the downstreamflow direction as positive, the continuity equation at section (i, 1) may bewritten as

Qp¡,l = npQpp - Qpv (10.27)

in which Qp = discharge at the end of the time step; the subscripts p, v, and(i, 1) refer to the pump, PRV, and section (i, 1); and np = number of parallelpumps.

Referring to Fig. 10.8,

Hp. 1 = Hp + Hsuc - 6.Hpd1, P(10.28)


Pr,ssur,r,guloling "oh"

Dolum

Ru,r"oir

e,::r Pip,

u:>..

::I:

Figure 10.8. Notation for a pressure-regulatlng valve- centrifugal pump system.

in which Hsuc = height of the suction reservoir aboye the datum;Hp. 1 = piezo-l.

metric head at section (i, I) aboye datum; I:::.Hpd= head loss at the dischargevalve; and Hp = pumping head.

The flow th~ough the PRV, Qp , is given by the equationu

Qp = CpuTPVHp. 1 - Zo (10.29)u "

in which Cpu = Qou/";~H""'o---zo-; Qou = flow through the fully open valve undera head of (H¿ - zo); Tp = effective valve opening at the end of time step =(CdAv)/(CdAv)o; Cd = coefficient of discharge; Av = area of valve opening;Zo = height aboye the datum of the reservoir into which PRV is discharging;and the subscript o refers to steady-state conditions. If the PRV is discharginginto the suction reservoir, then Zo = Hsuc, and if the PRV is discharging intoatmosphere, then Zo = height of PRV aboye datum.

The following equations are available for the pump-PRV end:

1. negative characteristic equation for section (i, 1), Eq. 4.l62. equations for the head and torque characteristics, Eqs. 4.7 and 4.83. equation for the rotating masses, Eq. 4.14.

Now, we have seven equations in seven unknowns, i.e., Qp , Qp. 1 ' Qp ,H r¡ 1 'P " v ICip, hp, and (jp. To simplify their solution, let us eliminate all other variab es

except two so that we have two equations in two unknowns. Then, we can usethe Newton-Raphson method to solve these equations. In this section, we will


only derive the expressions for FI and F2 and for their derivatives; the pro.cedure for using these expressions in the Newton-Raphson method was out-lined in Section 4.4.

By eliminating Qp. and Qpu from Eqs. 10.27, 10.29, and 4.16, and writina1,1 bI:::.Hpd = CuQpplQppl, we obtain

npQpp = CpvTPVHpp + Hsuc - CvQpplQppl- Zo + Cn

+ Ca¡(Hpp + Hsuc - CuQpplQppl) (10.30)

By using the relationships, Vp = Qp/QR and hp = Hp/HR, Eq. 10.30 may bewritten as

+ Ca¡(HRhp + Hsuc - CuQk vplvpl) (10.31)

By eliminating hp from Eqs. 4.7 and 10.31, and simplifying the resulting equa-tion, we obtain

e H (2 2) C H (2 2' 1 Cip+ a· ROl Olp+ Vp + a' R02 Olp+ Vp) tan- -, 1 Vp

- Ca¡CuQkvPlvpl- npQRvp + Cn + CaHsuc = oDifferentiation of Eq. 10.32a with respect to Olp and Vp yields

(10.32a)


(10.32c)

Expressions for F1, éJF1/éJcxp, and éJF1/éJup are given by Eqs. 4.20, 4.27, and4.28, respectively.

The value of Tp is determined from the specified valve-openíng-versus-timecurve, and then Eqs. 10.32a-c, 4.20, 4.27, and 4.28 are used to determine thevalues of CXp and Up by following the procedure outlined in Section 4.4 (see Fig.10.9).

Air-In/et Va/ve

Let the air-inlet valve be located at the junction of the ith and (i + 1)th pipe asshown in Fig. 10.1 O. Then the positive and negative characteristic equations forsections (i, n + 1) and (i + 1,1) are

(10.33)

Qp¡+l.l =Cn+Ca¡+lHp¡+l.l (10.34)

In addition, if the head losses in the pipeline at the valve are neglected, then

Hp¡.n+l =Hp¡+l.l (10.35)

In the subsequent diseussion, we will use Hp. 1 as the piezometric head at thel,n+

valve loeation.When Hp¡.n+l drops below a predetermined value,y, set on thevalve.P the

valve opens, and the air flows into the pipeline. Later, when Hp¡ n+1 >y, thevalve closes and entraps air inside the pipeline. Thus, dependíng upon thehistory of the pressure at the valve location, the valve may open or close severaltimes during a transient. The mass of the entrapped air will inerease with eachopening of the valve.

We will make the following assumptions in the derivation of the boundaryconditions:

l. The airflow into the pipeline is isentropic.2. The entrapped air remains at the valve loeation and is not carried away by

the flowing liquid.3. The expansion or contraction of the entrapped air is isothermal.

i:


from Eq.4.6

NObF1 bFIFDT Qe alld y. ~ compul. F1, F2• -~ - • ...-

bF l>F "ClIp "Vp_2 antl 2 fram Cqs. 1032, 4.20, 4.21' antl 4.28bCllp

YES

NO

NO

6V·".-VA a • ere - a

a • ere

Figure 10.9. Flowchart for boundary conditions for a pressuce-regulating valve- centrifugal

322 Applied Hydraulic Transients Methods for Controlling Transients 323

Ai,. in/el va/ve"'O ir

Elimination of mPa from Eqs. 10.36 and 10.41 yie!ds

(ma + d;a Llt) RT=p [Cair + 0.5 Llt(Cai + Cai+I)( ~+ Z - Hb)] (10.42)

In this equation, al1 variables are known except P and dm.fdt . If the absolutepressure, p, inside the pipeline is !ess than 0.53 Pa (Pa = barometric pressure),then the airflow through the valve is at sonie velocity, and, if P is greater than0.53 Pa but less than Pa, then the air ve!ocity through the valve is subsonie. Theexpressions for dma/dt are: 19

1. Subsonie air velocity through the valve t», > P > 0.53 Pa)

N

Oalum

Figure 10.10. Notation fOl air-inlet valve.

dl11amp =m + --Llta a dt (10.36)

dma _ v' (p) 1.43 [ ( P )0.286]d -CdAv 7paPa - 1--t Pa Pa

2. Sonie air velocíty through the valve (p ~ 0.53 Pa)

=«. PaT- 0.686 CdAv ~ f15"'f't ver,

If Llt is small and ma is the mass of the air entrapped in the pipeline at thebeginning of the time step, then the mass of air, mp at the end of time step isa (10.43)

in which dma/dt is the time rate of mass inflow of air through the valve into thepipeline.

The entrapped air volume should satisfy the continuity equation, i.e.,

-fLPair= -fLair + 0.5 .1t [(Qpi+I,1 + QH, 1) - (Qpi, n +1 + Qi, n +1)] (10.37)

By substituting Eqs. 10.33 through 10.35 into Eq. 10.37, we obtain

-fLPair= Cair + 0.5.1t (Cai + Cai+1 )Hpi, n+1 (10.38)

in which Cair = -fLair + 0.5 Llt (Cn + Qi+1. 1 - Cp - Qi, n +1)'If the expansion and contraction of the air volume inside the pipeline is iso-

therma!, then

(10.44)

(10.39)

in which Cd = coefficient of discharge of the va!ve; Av = are a of the valve open-ing at its throat; Pa = mass density of air at absolute atmospheric pressure , Pa,and absolute temperature, Ta, outside the pipeline. Equations 10.43 and 10.44are obtained by substituting k = 1.4 into the equations presented in Ref. 19 inwhich k is the ratio of the specific heats for airo

Substitution of Eq, 10.43 or 10.44 into Eq. 10.42 yields a nonlinear equationin p, which may be solved by an iterative technique such as the Newton-Raphsonmethod.ThevaluesofHp. ,-fLp.,mp,Hp. Qp. andQ mayh . l,n+1 air a 1+11' In+l' Pi+11

t en be determined from Eqs. 10.40, 10.38, 10.39, 10:35, 10.33, and'IO.34,respectively.

Prior to the time the air-inlet val ve opens for the first time, ma = O. After-ward, however, the value of ma increases with subsequent valve openings.

Note that in the development of aboye boundary conditions, it was assumedthat there is no air outflow through the valve. However, if the air is a1lowed toescape through the air-inlet valve when the pressure inside the pipeline exceedsthe outside pressure, then this should be taken into consideration in the deriva.tion ofthe boundary conditions (see Problem 10.6).

in which R = universal gas constant; and P and Tare the absolute pressure andtemperature of the air volume inside the pipeline. The absolute pressure, p, isrelated to Hp. 1 through the equationl,n+

P = 'Y(Hp. 1 - z + Hb) (10.40)l,n+

in which z = heíght of the valve throat aboye the datum; 'Y = specific weight ofthe liquid inside the pipeline; and Hb = barometric pressure head.

Substituting Hp. 1 from Eq. 10.40 into Eq. 10.38, eliminating -fLp. from'1 n+ arrthe resulting equatíon and Eq. 10.39, we obtain

mPaRT = P [Cair + 0.5 Llt (Cai + Cai+1) (~+ z - Hb)] (10.41)

10.6 OPTIMAL CONTROL OF TRANSIENT FLOWS

The mode of operation of various appurtenances and control devices such thata desired system response is obtained is called optimal flow control. A desiredsystem response may be keeping the maximum and minimum transient-state

324 Applied Hydraulic TransientsMethods for Controlling Transients 32S

pressures within specified limits, changing the flow conditions from one steadystate to another steady state in a minimum of time, changing the flow conditionsfrom one steady state to another without flow oscillations, and so on. Forexample , a valve at the downstream end of a pipeline may be closed such thatthe pressure does not exceed a specified limit and the transients in the pipelineare vanished as soon as the valve movement ceases. Such a valve operation has

1 1 20 d 1 t k' 21-25been referred to as optimum va ve e asure an va ve s ro tng.Optimal flow control is a design or synthesís approach in whích the variations

of boundary conditions are determíned to obtain a desired system response.This approach is dífferent from the usual analysis approach in which the varia-tions of the boundary conditions are specified and the system response iscomputed.

Following are sorne of the practical applications of optimal control of tran-sient flows:

1. Changing the outflow at different locations in water supply or oil pipeline swithout affecting the outflow to other clients.

2. Establishing flow in the conduits of a pumped-storage scheme in minimumtime while switching from generation to pumping mode or vice versa.

3. Opening or elosing the control valves in a piping system without violatingthe upper or lower pressure limits.

4. Closing the wicket gates of hydraulic turbines to minimize pressure andspeed rise following load rejection.

5. Accepting or rejecting load on hydraulic turbines in a minimum of timewithout exceeding the specified pressure drop or pressure rise.

6. Utilizing the storage capacity of sewer systems to obtain constant outflowfor treatment by properly operating the control devices.I"

To conserve space , we are not presen ting details of computational proceduresfor determining optimal flow control; interested readers should see Refs. 15 and20 through 25 for problem-oriented procedures, and Refs. 25 and 26 for appli-cation of varíous operations-research techniques.

In the optimization studies, a waterhammer pressure rise equal to 30 percentof the static head was assumed. However, subsequent studies indicated thatsubstantial savings would result from a reduction in the pressure rise withoutgreatly affecting the plant regulation. Therefore, a 20 percent pressure rise wasadopted for the final designo

To keep the maxírnurn transient pressures within the design limits, the follow-ing two altematives were considered:

1. Provision of an upstream surge tank.2. Provision of a PRV.

The topography at Jordan River does not favor locating a surge tank near thepowerhouse, as the cost of a tower-type surge tank would be very high. Inaddition, computer studies indicated that, with a surge tank more than 1585 maway from the powerhouse , the plant would not be stable under isolated opera-tion up to its full generating capacity.

In comparison to a surge tank, it was found that a PRV operating in syn-chronous operation with the turbine would provide good governing characteris-tics. The estimated cost of the PRV was about the same as that of a 90-m-highsurge tank. However, the PRV was seJected because it alone would meet theoperational requiremen ts.

MathematicaJ Model

A mathematical model was developed é": 28 to analyze the transient conditions inthe power conduit of the power plant. The method of characteristics was usedfor the analysis. Special boundary conditions, which are presented hereafter,were developed for the synchronous operation of the turbine and the PRV.(Boundary condítíons for the PRV acting alone are presented in Section 3.8.)

Let us designate the last sectíon on the penstock adjacent to the turbine as(i, n + 1). Then, referring to Fig. 10.11, the continuity equation may be writtenas

10.7 CASE STUDY(10.45)

Studies carried out for the analysis of transients in the power conduit of theJordan River Redevelopment, reported earlíer by the author ,27,28 are presentedin this section .

Design

During the design of this power plant, a number of plant layouts were con-sidered.?" The project layout as shown in Fig. 3.18 was selected because it wasthe most econorníc.'

( i, n + I )Pipe i

Pressure. regulo!ing lIollle

Turbine

Qp. +11, n

Figure 10.11. Notation for pressure·regulating valve and Francís turbine.

326 Applied Hydraulíc Transíents

in which Qp and Qp are the turbine and PRV discharge at the end of timetu r v for secti (' 1) .step. The posítíve characteristic equation or sectíon 1, n + IS

(10.46)Qp¡,n+l = Cp - CaHP¡,n+l'

Equation 10.46 rnay be written in terrns of the net head, Hn, as

- C C (H _ Q~¡ n+1 )Qp¡, n+l - p - a n 2gAl

in whlch A¡ = cross-sectional area of the penstock at turbine inlet.If the turbine characteristics are used in the analysís as outlíned in Section 5.4,

then

(l0.47)

QPtur = a3 + a2 $nin which a2 and a3 are constants as defined by Eq. 5.4

The flow through the PRV is

(10.48)

(10.49)

in which Qr and Qp are the PRV discharges, both for valve opening, Tp, under"net heads of H; and Hn » respectively ; and the subscript r refers to the rated

conditions.Substituting Eqs. 10,48 and 10.49 into Eq. 10.45, we obtain

(l0.50)Qp. 1::: a3 + a4 ..JH;;1, n+

in which a4 :::a2 + Qr/"¡¡¡;'Elimination of Hn from Eqs. 10,47 and 10.50 yields

asQ~¡,n+l +a6Qp¡,n+l +a7 ==0 (l051)

in which as ==Ca 11}(2gA;) - l/a~); a¿ :::2a) Ca/a~ - 1; and a- :::Cp - CaaVai.Solution of Eq. 10.5 1 gives

(10.52)

Now the values of Hn, Qp , Qp , and Hp. are determined from Eqs. 10.50,tur v I,n+l10.48,and 10,49,and 10,46,respectively.

The iterative procedure of Section 5.6 was used to refine the solution and to

determine the turbine speed.

Results

The transíent-state pressures computed by using the preceding mathematicalmodel were compared with those measured on the prototype following lS0-MW


~¡-----¡-----¡------¡-----,-----,------,-----~--~

~60r-----~~--~------+------+------+---~~~--_j~--__J~......lIlo~~40¡------i------~----~~--~~F------+------~------¡_----_j~~"t1Ilo~r------r----~~----+------+~~~+-----~----~~--~

2

Figure 10.12. Time history of wicket gates and pressure-regulating valve opening follow-ing 150-MW load rejection.

34'TIME" (sflconds)

76 8

load rejection. The PRV and the turbine wicket-gate-opening-versus-tímecurves (Fig. 10.12) recorded during the prototype tests were used in the analysis.The turbine and generator WR2 was taken equal to 1.81 X 106 kg m2 to com-pute the transien t-state turbine speed.

The computed and measured transient-state pressures at the turbine inlet areshown in Fig. 10.13. There is close agreement between the computed and mea-

375

~'- 350~~.~...!¡

~ 325

ti

""..<: '00

1~......ol:

275

tE

¡/~,1\\ _--,

/Compu/~d .,-_, -if/ \ 1/ V~

,,'" , ,";' J._ I

//- ,~ //\

AMeosu"d .\~

250o 128 16 20 28

Ti",e (SIC)

Figure 10.13. Comparison of computed and measured pressure head at turbine inlet.


sured maximum pressures; however, there ís sorne phase shitt , and the measuredresults show more rapid dissipation of transíent pressure than that indicated bythe computed results. Probably, the phase shift is caused by incorrect vaJues ofthe wave velocity , and the difference in the dissipation is due to computing thetransient-state head losses by using the steady-state friction formula.

10.7 SUMMARY

In this chapter, a number of de vices and methods were presented to control thetransient conditions in pipelines. A brief description of the operation of thesedevices was given,and the boundary conditions for these devices were developed,The chapter concluded with a case study.

PROBLEMS

10.1 Derive the boundary condition for an orifice surge tank. In this tank,an orifice is provided between the pipeline and the surge tank.

10.2 How would the boundary conditions for the simple surge tank of Fig.10.1 be modified if there were no standpipe?

10.3 Write a computer program for the analysis of a piping system having anair chamber as shown in Fig. 10.2. Assurne that the check valve closesas soon as power fails. The pipeline has a const an t-head reservoir at thedownstream erid.

10.4 Prove that an air chamber behaves Iike a virtual surge tankl2 having across-sectional area

1

1 .y H*m0air o

A s = m ---"':::':""_--m-:-:"+7"1

[Hp. 1 + Ho]--;nt .n+

in which the variables are as defined in Section 10.4.

10.5 Develop the boundary conditions for the centrifugal-pump-PRV systemshown in Fig. 10.8. Assume that the PRV is discharging into the suctionline of the pump. tHint: Equation 10.29 will be modified to Qpv =TpQo..¡r¡¡;;: - Hp )/Ho in which Hp is the piezometric head on the

1,1 s ssuction side of the purnp, and Ho = steady-state pumping head.)

) 0.6 Develop the boundary conditions for an air-inlet val ve that allows outflowof air, but not of liquid, frorn the pipeline when the pressure inside thepipeline exceeds the outside atmospheric pressure. (Hint: Use ex pressionssimilar to Eqs. 10.43 and 10.44 for the time rate of mass outflow of airthrough the valve. As the aiT inflow was assumed positive in Eqs. 10.36and 10.4 2, dma/dt fOT aiT outflow should be consideTed negative in theseequations.)


REFERENCES

1. Evans, W. E. and Crawford, C. C., "Design Charts for Air Charnbers on Pump Unes,"Trans. Amer. Soco of Civil Engrs., vol. 119,1954, pp. 1025-1045.

2. Engler , M. L., "Relíef Val ves and Air Charnbers," Symposium on Waterhammer AmerSoco of Mech. Engrs. and Amer. Soco of Ctvil Engrs., 1933, pp. 97-115. ,.

3. Allievi, L., "Air Charnber for Discharge Unes," Trans. Amer. Soco of Mech. Engrs., vol.59, Nov. 1937, pp. 651-659.

4. Angus, R. W., "Air Charnbers and Valves in Relation to Waterhammer," Trans. Amer.Soco of Mech. Engrs., vol. 59, Nov. 1937, pp. 661-668.

5. Combes, G. and Borot, G., "New Chart for the Calculation of Air Vessels Allowing forFriction Losses," Laliouille Blanche , 1952, pp. 723-729.

6. Tucker, D. M. and Young, G. A. 1., "Estirnation of the Size of Air Vessels," BritishHydromechanics Researeh Assac., Report SP 670,1962.

7. Graze, H. R. and Forrest, J. A., "New Design Charts for Air Charnbers," Fifth Austra·lasian Conference on Hydraulics and Fluid Mechanics, Christchurch New Zealand1974,pp.34-41. ' ,

8. Ruus, E., "Charts for Waterhammer in Pipeline s with Air Charnber," Canadian Jour.af Civil Engineering, vol. 4, no. 3, Sept. 1977, pp. 293-313.

9. Graze, H. R., "A Rational Therrnodynamic Equation for Air Charnber Design," Proc.Third Austra/asian Conference on Hydraulics and Fluid Mechanics, Sydney , Australia,1968, pp. 57-61.

10. Graze, H. R., Discussion of "Pressure Surge Attenuation Utilizing an Air Charnber,"by Wood, D. 1., Jour., Hyd. Div ; Amer. Soco of Civil Engrs., vol. 97, March 1971, pp.455-459.

11. Graze, H. R., "The Importance of Ternperature in Air Charnber Operations," Proc.First Intemational Conference on Pressure Surges, British Hydromechanic ResearchAssac., England, 1972, pp. F2-13-F2-21.

12. Evangelisti, G., "Waterhammer Analysis by the Method of Characteristics," L 'EnergiaElettrica, no. 12, 1969, pp. 839-858.

13. Lescovich, J. E., "The Control of Water Hamrner by Automatic Valves," Jour. Amer.Water Works Assac., May 1967, pp. 832-844.

14. Gadenberger, W., "Grundlagen der graphíschen Ermittlung der Druckschwankungenin" Wasserversorgungsleitungen," R. Oldenbourg Verley , 1950.

15. Streeter, V. L. and Wylie, E. B., Hydraulic Transients ; McGraw-HiII Book Co., NewYork, 1967.

16. Albertson, M. L. and Andrews, J. S., "Transíents Caused by Air Release," in ControlofFlow in Closed Conduits, edited by Tullís, J. P., Colorado State University , 1971.

17. Martin, C. S., "Entrapped Air in Pipelines," Proc. Second Conference on PressureSurges, British Hydromechanic Research Assoc., England, 1976.

18. Papadakis, C. N. and Hsu, S. T., "Transient Analysis of Air Vessels and Air Inlet Valves,"Jour., Fluid Engineering ;Amer. Soc. of Mech. Engrs., 1977.

19. Streeter, V. L., Fluid Mechanics, McGraw-HiIl Book ce., 4th edition, 1966, p. 304.20. Ruus, E., "Optimum Rate of Closure of Hydraulic Turbine Gates," presented at Arrier.

Soco of Mech. Engrs.-Engineering Inst. of Canada Conference, Denver, Colorado,Apri11966.

21. Streeter, V. L., "Valve Stroking to Control Water Hammer," Jour., Hyd. Div., Amer.Soco of Civil Engineers, vol. 89, March 1963, pp. 39-66.

22. Streeter, V. L., "Valve Stroking for Complex Piping Systems," Jour., Hyd. Div., Amer.Soc. ofCiv. Engrs., vol. 93, May 1967, pp. 81-98.


23. Driel, M., "Valve Stroking in Separated Pipe Flow," Jour., Hyd. Div., Amer. Soco ofCiv. Engrs., vol. lOO, Nov. 1974, pp. 1549-1563.

24. lkeo, S. and Kobori, T., "Waterhammer Caused by Valve Stroking in Pipeline With TwoValves," Bull. Japan Soc. of Mech. Engrs., vol. 18, October 1975, pp. 1151-1157.

25. Driels, M., "Predicting Optimum Two-Stage Valve Closure ," Paper No. 75-WA/FE-2,Amer. Soco of Mech, Engrs., 1975,7 pp.

26. Bell, W. W., Johnson, G., and Winn, C. B., "Control Logie for Real Time Control ofFlow in Combined Sewers," Proc., 9th Canadian Symp., Water PolI. Res., Canadá,1974,pp.217-234.

27. Portfors, E. A. and Chaudhry, M. H., "Analysis and Prototype Verification of HydraulicTransients in Jordán River Power Plant," Proc. First Conference on Pressure Surges,published by British Hydrornechanics Research Assoc., England, Sept. 1972, pp. E4-57-E4-72.

28. Chaudhry, M. H. and Portfors, E. A., "A Mathematical Model for Analyzing HydraulieTransients in a Hydro Power Plant," Proc. First Canadian Hydraulic Conference , pub-líshed by the University of Alberta, Edmonlon, Albecta, Canada, May 1973, pp. 298-314.

29. Forster, J. W., Kadak, A., and Salmen, G. M., "Planning of Ihe Jordan River Redevelop-rnent," Engineering Jour., Engineering Inst. 01 Callada, Oct. 1970, pp. 34 -43.


Strowger, E. B., "Relatlon of Relief Valve and Turbine Characteristics in the Determinationof Walerhammer," Trans. Amer. Soco of Mech. Engrs., vol. 59, Nov. 1937, pp. 701-705.

Parmakian, 1., "Air Inlet Valves for Steel Pipe Lines," Trans. Amer. Soco of Civ, Engrs.,vol. 115, 1950, pp. 438-443.

Jacobson, R. S., "Charts for Analysis of Surge Tanks in Turbine or Pump Installations,"Special Report 104, Bureau of Reclamation, Denver, Colorado, Feb. 1952.

Parmakian, J., "Pressure Surge Control at Traey Pumping Plant," Proc. Amer. Soco of CivilEngrs., vol. 79, Separa te No. 361, Dec. 1953.

Parmakian, J., "Pressure Surge at Large Installations," Trans. Amer. Soco of Mech. Engrs.,1953, p. 995.

Lindros, E., "Grand Coulee Model-Pump Investigation of Transient Pressures and MethodsIor Their Reduction," Trans. Amer. Soco of Mech, Engrs., 1954, p. 775.

Parmakian, J., "One-Way Surge Tanks for Pumping Plants," Trans. Amer. Soco of Mech.Engrs., 1958, pp. 1563-1573.

Kerr, S. L., "Effect of Valve Operation on Waterhammer," Jour. Amer. Water Works Assoc.,vol. 52, Jan. 1960.

Lundgren, C. W., "Charts for Determining Size of Surge Suppressors for Purnp-DischargeLines, " Jour. Engineering for Power, Amer. Soco of Mech. Engrs., Jan. 1961, pp. 43-46.

Whiteman, K. J., and Pearsall, 1. S., "Reflex Valve and Surge Tests at a Pumping Station,"Fluid Handling, vol. 152, Sept. and Oet. 1962, pp. 248-250,282-286.

Widmann, R., "The Interaction Between Waterhammer and Surge Tank Oscillations,"International Symposium on Waterhammer in Pumped Storage Projects, Amer. Soco ofMech. Engrs., Chieago Nov., 1965, pp. 1-7.

Bechteler, W., "Surge Tank and Water Hammer Calculations on Digital and Analog Com-puten," Water Power, vol. 21, no. lO, Oct. 1969, pp. 386-390.

Ruus, E., and Chaudhry, M. H., "Boundary Conditions for Air Chambers and Surge Tanks,"Trans., Engineering Inst. of Canada, Engineering Jour., Nov. 1969, pp. I-VI.


Meeks, D. R. and Bradley , M. J., "The Effect of Differential Throttling on Air VesselPerformance," Symposium on Pressure Transients , The City University , London, Nov.1970.

Wood, D. J., "Pressure Surge Attenuation Utilizing an Air Charnber ," Jour., Hyd. Div;Amer. Soco of Civil Engrs., vol. 96, May 1970, pp. 1143":1156.

Kinno, H., "Waterhammer Control in Centrifugal Pump Systerns," Jour., Hyd. Div.,Amer.Soco of Civ. Engrs., May 1968, pp. 619-639.

Chaudhry, M. H., "Boundary Conditions for Waterhammer Analysis," thesis submitted tothe Univ. of British Columbia, Vancouver, Canadá, in partial fulfilIment of the require-ments of M.A.Se., April 1968.

CHAPTER 11 Surge Tanks 333

Tunnel

SURGETANKSReservoir

11.1 INTRODUCTION

A surge tank is an open standpipe or a shaft connected to the conduits of ahydroelectric power plant or to the pipeline of a piping system. This is alsoreferred to as a surge shaft or surge chamber.

The main functions of a surge tank are:l. It reduces the amplitude of pressure fluctuations by reflecting the incoming

pressure waves. For example , the waterhammer waves produced in a penstockby load changes on a turbine (Fig. 11.1) are reflected back at the surge tank.Thus, the conduit length to be used in the waterhammer analysis is between theturbine and the surge tank rather than between the turbine and the upstreamreservoir. Due to this reduction in the conduit length, the pressure rise or dropis less than if the surge tank were not provided. In addition, if a surge tank werenot present at the junction of the penstock and the tunnel, then the tunnelwould ha ve to be designed to withstand the waterhammer pressures.

2. A surge tank improves the regulating characteristics of a hydraulic turbine.Because of the surge tank , the length of the power conduit to be used for de·termining the water-starting time (se e Section 5.10) is up to the surge tankrather than up to the upstream reservoir. The water-starting time of a hydro-power scheme is therefore reduced , thus improving the regulating characteristicsof the power plant.

3. A surge tank acts as a storage for excess water during load reduction ina hydropower plant and during start-up of the pumps in a pumping system.Similarly, it provides water during load acceptance in a hydropower plantand during power failure in a pumping system. Therefore , the water is ac-celerated or decelerated in the pipeline slowly, and the arnplítude of the pressurefluctuations in the system is reduced.

In this chapter, various types of surge tanks are reviewed. Differential equa-

Figure 11.1. Schematic diagram of a hydroelectric power plant.

tions describing the oscillations of the water level in a simple surge tank are de-rived, and methods available for their solution are presented. The phase planemethod is then used to study the stability of a simple surge tank. This is fol-lowed by the analysis of orifice, differential, and a system of surge tanks. Thechapter is concluded by surnmarizing the studies carried out for the design of theChute-des-Passes surge-tank system.

11.2 TYPES OF SURGE T ANKS

332

Depending upon its configuration, a surge tank may be classified as simple,orifice , differential, one-way , or e/osed. A brief description of each follows.

A simple surge tank is just a shaft or standpipe connected to the pipeline.If the entrance to the surge tank is restricted by means of an orífice, it is calledan orifice tank. An orifice tank having a riser is termed differential. In a one-way surge tank, the liquid flows from the tank into the pipeline only when thepressure in the pipeline drops below the liquid level in the surge tank. Follow-ing the transíent-state conditions, the tank is filled from the pipeline. If the topof the tank is closed or if there is a valve or orifice in the vent stack connectingthe tank to the outer atmosphere, the tank is called a closed surge tank. De-pendíng upon the requirement that the tank must fulfill, a simple tank mayhave upper or lower galleries. Figure 11.2 shows a number of typical surgetanks.

Ir necessary, a combination of different types of surge tanks may be providedin an installation.


Orifice

(a) Simple tank (b) Orif ice tank

ffi'iser

Orifice

(e) Differential tank (d) One-way tank

(e) Closed tank ( f) Tank wlth galleríes

Figure 11.2. Types of surge tanks.

11.3 DERIVATlON OF EQUATIONS

Figure 11.3a shows a typical surge-tank system in which a control valve is usedto represent a pump or a hydraulic turbíne. A change in the valve openingcauses the flow variation, which results in the oscillations of the liquid level inthe surge tank.

To simplify the derivation of the dynamic and continuity equations, let LIS

make the following assumptions:

1. The conduit walls are rigid, and the liquid is incompressible. This meansthat a flow change is felt throughout the system instantaneously and that

Surge Tanks 335

(a)

(b) Freebody díagram

Figure 11.3. Simple surge tank: notation for dynamic and continuity equations.

the liquid in the conduit is moving like a solid slug.2. The inertia of the líquid in the surge tank is small cornpared to that of the

liquid in the tunnel and can therefore be neglected.3. The head losses in the system during the transient state can be computed

by using the steady-state formulas for the corresponding tlow velocíties,

Dynamic Equation

A freebody diagram of a horizontal conduit havíng constant cross-sectional areais shown in Fig. II.3b. Forces acting on the líquíd are:

F, = rAtCHo - hu - h¡)

F2 = rAtCHo + z)

F3 = "tAth¡

(11.1)

(11.2)

(11.3)

336 Applied Hydraulic Transients Surge Tanks 337

In these equations, At = cross-sectional area of the tunnel; H¿ = static head;-y = specific weight of liquid ; hu = velocity head at the intake; ñ¡ = intake headlosses; hr = friction and form losses in the tunnel between the intake and thesurge tank ; and z = water level in the surge tank aboye the reservoir level (posí-tive upward). Considering the downstream flow direction as positive, the netforce acting on the Iiquid elernent in the positive direction is í:,F = FI - F2 - F3'Hence, from Eqs. 11 J to 11.3,

Continuity Equation

Referring to Fig, 11.3, the continuity equation for the junction of the tunneland the surge tan k may be written as

Qt = Qs + Qtur (11.7)

(1104)

in which Qs = flow into the surge tank (positive into the tank), and Qtor =turbine flow. Note that Eq. 11 .7 is equally valid for the pumping system if the flowof the pump is designated as Qtur' Writing Qs as A s(dz/dt), Eq. 11.7 beco mes

dz 1dt = As (Qt - Qtur) (11.8)The mass of the Iiquid element is -yAtL/g, in which L = length of the tunneI and

g = acceleration due to gravity. If Qt is the tunnel flow and t = time, then therate of change of momentum of the element may be written as Note that Eqs. 11.6 and 11.8 are for a surge tank located at the upstream side

of a turbine. These equations are valid for a tailrace surge tank (i.e., a tank 10-cated on the downstream of the turbine), provided the velocity head, hu, is prop-erly taken into consideration.

or

-yL dQt

g dt

11.4 AVAILABLE METHODS FOR SOLVING DYNAMIC ANDCONTINUITY EQUA TIONS

~;t= g~t (-z - cQtlQtl) (11.6)

Due to the assumption that the tunnel and the liquid inside are rigid, we do nothave spatial derivatives (i.e., variation with respect to x) in the dynamic andcontinuity equations, and the flow and the liquid level in the tank vary withrespect to time only. Therefore , Eqs. 11.6 and 11.8 are a set of ordinary differ-ential equations. Because of the presence of the term cQtlQtl, Eq. 11.6 is non-linear. Also, note that turbine flow may be a function of time.

A c1osed-form solution of Eqs. 11.6 and 11.8 is available only for a few specialcases.' Therefore, graphical+' 3 and arithmetical methods suitable for hand com-putations4,s have been used in the past to integra te these equations. However,with the invention of the analog and digital computers, graphical and arithrneti-cal methods of integration have be en superseded by analog simulation oc digitalcomputation. When a large nurnber of parameters have to be optimized, analogcomputers are very handy6,7; with these, once the required circuitry has beenwired, it is just a matter of twisting various knobs to change various parameters.As compared to the analog, digital computers are more readily accessible. There-fore, we will discuss only the methods suitable for digital computer analysis ..

A number of finite-difference techniques are avaílable+" to solve the dynamicand continuity equations on a digital computer. Although more sophisticatedand accurate methods 10,11 are reported in the Iiterature, in the author's opinion,

According to Newton's second law of motion, the rate of change of momentumis equal to the net applied force. Therefore,

-yL dQt- - = -yArC-z - hu - h¡ - hr).g dt(i 1.5)

By defining h = hu + h¡ + hr= cQt IQt 1, in which e is a coefficient, Eq. 10.5 maybe written as

Note that h is expressed as cQtlQtl to account for the reverse flow,In the preceding derivation, the tunnel was assumed to be horizontal and

to have constant cross-sectional area throughout its length. Equation 11.6 isalso valid for a sloping tunnel (see Problem 11.3). However, for a tunnel havingdifferent cross-sectional areas for various lengths, the term At/L of Eq. 11.6 isreplaced by í:,(At/L)¡ (see Problem 11.5).


the fourth-order Runge-Kutta rnethod'? may be used, whích yields sufficientlyaccurate results that are of the same arder of accuracy as that of the input data.In addition to being simpler to program than other methods, subroutines areavailable as a standard package for this method at most of the computerinstallations.

11.5 PERIOO ANO AMPLITUOE OF OSCILLA TlONS OF AFRICTlONLESS SYSTEM

The head losses were taken into consideration while deriving the dynamic equa-tions. If the system is considered frictionless, i.e., e = O, then Eq. 11.6 becomes

dQt gAt-=--zdt L

(11.9)

Let us assume that an initial steady-state turbine flow, Qo, is instantaneouslyreduced to zero at t = O, i.e.,

at t < O, Qtur = Qo (11.1 O)

and

at t ~ O, Qtur = O

On the basis of Eq. 11.11, Eq. 11.8 can be written as

dz 1dt = 7Qt

s

(11.11)

(11.12)

Differentiating Eq. 11.12 with respect to t and eliminating dQtldt from theresulting equation and Eq. 11.9, we obtain

d2z gAt+-z=Odt? LAs

(11.13)

As the coefficient of z in Eq. 11.13 is a positive real constant , we know fromthe theory of ordinary differential equatíons''? that a general solution ofEq. 11.13 ís

z = el COS' fiI¡ t + e2 sin' íff!¡ tV LA:: V LAs

in which el and e2 are arbitrary constants and are determined from the initialconditions. For thís case, at t = O,

(11.14)

z=O (11.15)

and

Surge Tanks 339

dz Qo-=-dt As

Substituting these conditions into Eq. 11.14,

el =0 }and

e2=QO'~Vg~Hence, it follows from Eq. 11.14 and 11.17 that

(11.16)

(11.17)

(11.18)

Equation 11.18 describes the oscillations of the water surface in the surge tank.The period, T, and the amplitude, Z, of these oscillations (Fig. 11.4) are

T= 27Tt

and

Z=Q o

The amplitude,Z, is referred to as free surge for load rejection.

zPeriod, T

Figure 11.4. Period and amplitude of oscillations of a frictionless system.

(11.19)

(11.20)

l-c,


11.6 STABILlTY

If the hydraulic system of Fig. ¡1.3 is disturbed (e.g., by changing the turbineflow), the water level in the surge tank begins to oscillate. These oscillationsare stable or unstable depending upon the parameters of the power plant andthe type and magnitude of the disturbance. Oscillations are saíd to be stableir they dampen to the final steady sta te in a reasonable time and unstable iftheir magnitude increases with time (see Fig. 11.5).

In addition to oscillatory instability , a condition called tank drainage has to beavoided. In this case, following a large load increase, the water in the tunnel does

z

(a) Stable oscillations

z

(b) Unstable oscillations

Figure 11.5. Stable and unstable oscillations.

Surge Tanks 341

not accelerate fast enough to meet the turbine demando Therefore, the surgetank supplies the water, and its water level keeps on falling until the tank drains.This condition usually occurs if the tunnel Iosses are large.

The following four cases of change in the turbine flow are of interest:l. Constant Flow . The turbine flow is changed from one steady-state value

Q 1 to another steady-state value, Q2' Since the turbine flow varies with thechange in the surge-tank level, constant flow is possible only on very high headinstalla tions where oscillations in the surge tank are small compared to the sta tichead.

2. Constant-Gate Opening . Constant-gate opening occurs when the plant isunder manual control after a load change, or when the governor is inoperativedue to malfunctioning of sorne of its parts, or when the turbine gates are openedto the maximum position following a load increase while the governor is tryingto maintain constan! power.

3. Constant Power. In this case, it is assumed that an ideal governor main-taíns constant power input to the turbine or maintains constan! turbine outputif the turbine efficiency is considered constant. Following a load increase, thegovernor opens the wicket gates to increase the flow. As a result, the waterlevel in the surge tank is lowered, and the net head on the turbine is reduced.Therefore, the governor has to open the gates further to keep the power con-stant. No restriction on Ihe turbine-gate opening is assumed, which impliesthat the turbine discharge can be increased to any required arnount to main-tain constant power. Thus, it can be seen that the action of the governor isdestabilizing.

4. Constant Power Combined with Full Gate Opening. In case 3, we assumedthat the turbine gates can be opened to any value to maintain constant power.On actual installations, however, gates cannot be opened lo more than their fullyopen position. Therefore , the governor maintains constant power ir the net headon the turbine is more than or equal to the rated head. When the net head isless than the rated head, the case of constant-gate opening applies. Note thatfor the net heads less than the rated head, the turbine power output decreaseswith the lowering of the surge-tank level, and, as a result, the systern frequencydecreases if the plant is isolated. Usually the load is tripped if the system fre-quency drops below a prescribed value. In our analysis, however, we are assum-ing that the load is not tripped, and the turbine keeps on generating power irre-spective of the systern frequency.

By integrating Eqs. 11.6 and 11.8 graphically , Frank and Schüller ' demon-strated that the oscillations are always stable in case 2 and stable in case 1 if thetunnel friction losses are taken into consideration. Case 3 has been studied by anumber of investigators: Thoma 14 linearized the governing differential equa-tions and demonstrated that the oscillations are unstable if the tank area is less


than a minimum. This minimum area is now called the Thoma area, Ath·PaynterlS,16 solved the equations analytically and on an analog computer andpresented a stability diagram. Using the phase-plane method,I,,18 Marrisl9,20

and Sideriades21,22 demonstrated that Thoma's stability criteria do not holdfor large oscillations. RUUS23 analyzed case 4 on a digital computer and showedthat small, rather than large, oscillations are critical for the stability of a tank.The author and RUUS24 used the phase-plane method to investigate the stabilityfor all four cases listed. These studies are presented herein. A number of quan-titative results are obtained from the analysis of the singularities, and a numberof phase portraits are presented to show the effect of the changes in differentparameters on the qualitative behavíor of the system.

11.7 NORMALIZATION OF EQUATIONS

In order to reduce the number of parameters, we will normalize Eqs. 11.6 and11.8 as follows:

Let

y =z/Zx = Qr/Qo

q = Qtur/Qo

r = 21Tt/T

By substituting these variables into Eqs. 11.8 and 11.6, and simplifying theresulting equations, we obtain

(11.21)

dy-=x -qdr

(11.22)

dx I 2dr = -y - 'lRx

in which R = 2h¡0/Z = 2eQ~/Z; and h¡o = head 1055 corresponding to flow Qo,in the tunnel.

(11.23)

11.8 PHASE-PLANE METHOD*

To facilitate discussion in the following section, a summary of the necessaryequations follows; for a detailed description of the method, see Ref. 17.

*Readers who are not in terested in the mathematical details may proceed directly to theconclusions summarized at the end of Section ti .!r, p. 360.

Surge Tanks 343

Let the dífferential equations describing a system be

dxdr =P(x,y) (11.24)

and

dydr = Q(x ,y) (1l.25)

in which the functions p(x, y) and/or Q(x, y) may be nonlinear. By combiningEqs. 11.24 and 11.25, we obtain

dy Q(x,y)-=---dx P(x,y)

(11.26)

The points (xs, Ys) for which : = % are called singular points. The location

of these points is obtained by simultantously solving the equations P(x, y) = O,and Q(x, y) = O. The type of singularity may be determined by substitutingx = Xs + u andy = Ys + v into Eq. 11.26, which after simplification yields

do Q(xs,ys) + e'u + d'v + e"u2 + d"v2

du P(xs,Ys) + a'u + b'v + a"u2 + b"v2(11.27)

in which a', a", b', s", e', e", d'; and d" are real constants. If both the linearterms and the higher-power terms in u and vare present in the denominatorand in the numerator, the singularity is called a simple singularity. For sucha singularity, the hígher-power terms can be neglected because their effect onthe solution in the neighborhood of the singularity is small compared to that ofthe linear terms. However, if the linear terms are missing, the singularity isnonsimple, and the higher-power terms cannot be neglected.

To study the propertíes of the solution in the neighborhood of a simple singu-larity , Eq. 11.27 may be written as

dv e'u + d'vdu a'u + b'o

(11.28)

The characteristic roots, Al and A2, of the two equations equivalent to the aboyeequation, i.e.,

dv , ,-=eu+dvdt

(11.29)


and

du , ,- =au+bvdt

(11.30)

are

Al, A2 = i [(a' + d') ±.Jea' + d')2 + 4(b'c' - a'd')1

The roots, Al and A2 ,determine the type of a singularity as follows:

1. Nade, if both roots are real and have the same sign.2. Saddle, if both roots are real and have the opposite signs.3. Vortex , if both roots are irnaginary.4. Focus,if the roots are cornplex conjugates.

If the real part of the roots is negative , the singularity of node and focus istermed stable; if positive, it is called unstable. Note that Eqs. 11.28 through11.31 are valid only for simple singularities.

(11.31)

11.9 ANAL YSIS OF DlFFERENT CASES OF FLOW DEMAND

For the cases discussed in Section 11.6, the normalized flow demand characteris-tics are shown in Fig. 11.6. Stability of these cases may be studied as discussedin the following.

Constant Flow

Let the initial steady-state flow be Q 1, and the final steady-state flow, Qz. Thenin normalized form, q = x*, in which x* = o.io: By substituting this into Eq.11.22, and dividing the resulting equation by Eq. 11.23, we obtain

dy x - x*-= (11.32)dx -y - !Rx2

The coordinates of the singularity are obtained by solving the equations

x - x* = O {l1.33)

and

(11.34)

simultaneously. This yields Xs = x* and y s == - t Rx *2.Substitution of x :::x* + u and y = - tRx *2 + v into Eq. 11.32 and following

Surge Tanks 345

y "~..,<:~ ~

Constant power <: '"'" ..~ .~

" '".. ....

~constont ~ .!:;diseh orge ~ ~

r Q

I/ // I

S /

I// ....

'",\ §'" '"'"" .~..

<:::/ Constant gate openinfl .._ '" '".. ....

/ ~ !:; ....../ '" '" ~Q.. !;:

Figure 11.6. Flow demand characteristics,

the procedure outlined in Section 11.8 yield

dv-:::---- (11.35)u

du -Rxvu - vBy comparing this equation with Eq. 11.28, a' = -Rx*, b'::: -1, e'::: 1, and d' = Oare obtained. Therefore , it follows from Eq. 11.31 that

I\), A2 = t [-Rx* ± v'(Rx*)Z - 4] (11.36)

Both roots are real and negative if Rx* > 2. The singularity is then a stablenade (Fig. 11.7a). Ir the final steady-state flow , Qz, is used as the referenceflow, (Qz =1= O), then Rx* > 2 impiles that hfo >Z. Ir Rx* < 2, then the rootsare complex conjuga tes with negative real parts. Therefore , the singularity is astable foeus (Fig. 11.7b). lf the friction Iosses are neglected, i.e., R = O, thenthe roots are imagínary, and the singularity is a vortex .

1:


y

0.5

- 2. O

(a) k = 0.35. R = 2.8

Figure 11.7. Phase portrait COI constan t discharge,

Thís analysís confirms the results obtained by Frank and Schüller ' whosolved the differential equations by graphical integration.

In the equation corresponding to Eq. 11.27 for the case of total rejection (í.e.,x * = O), there is a u2 term in the denominator, but there is no linear term in

Surge Tanks 347

(b) k ·0.025, R 110.2


u. Thus, the equation represents a nonsímple singularity. To solve this case,isoclines may be plotted as described subsequently, and thereafter the solutiontrajectories may be drawn. From the shape of the trajectories, the type of thesingularity can be ascertaíned.

The phase portraits may be plotted by the method of isoclines.!" Anisocline is the locus of the points at which the solution trajectories have thesame slope. Let m be the slope of the solutíon trajectories for an isocline.


Then, it follows from Eq. 11.32 that

dy x - x*= =m

dx -y - tRx2

point on the stability of the system depends upon its distance from the stablesingularities.

(11.37)Singularity (l, - iR)

By substituting x = 1 + u and y = - iR + v into Eq. 1IAO, and following theproeedure outlined previously for determining the type of a singular point, thefollowing equation is obtained:

or

1 x - x*y = - - RX2 - ---

2 m(11.38)

is the equation of the isocline. To obtain a graphieal solution, the isoc1ines arefirst plotted for different values of m. Onee this has been done, the solutiontrajeetories for any initial conditions can be drawn. This procedure is iIlustratedby the phase portraits of Figs. 11.7 through 11.1 O. The data for plotting thephase portraits ha ve been selected to ilIustrate the different types of singularities.

dv u - bsv

du -Ru - v(11.43)

Cornparison of Eqs. 11,43 and 11.28 yields a' = - R; b' = -1 ; e' = 1; and d' = - bs.Thus,

)'1, A2 = i [-(R + bs) ±v'(R + bS)2 - 4(1 +Rbs)] (11.44)Constant-Gate Opening

Since R, b, and s are all positive constants, both roots are real and negative if(R + bS)2 > 4(1 + Rbs)-i.e., (R - bs) > ± 2. The roots are complex conjugates

.,with negative real part if (R - bs) < ± 2. In the former case, the singular point isa stable nade (Fig. 11.8a); in the latter, it is a stable focus (Fig. 11.8b).

Note that the singular point is a stable nade for bs > ± 2, and a stable focus(Fig. 11.8e) for bs > ± 2, even if flow is eonsidered frictionless, i.e., R = O. Thisis beeause of the damping effeet of the turbine gates beíng he Id in a fixedposition.

The head-discharge relationship for a reaction turbine running at constant speedcannot be represented by a simple mathematieal function. As can be seen fromthe turbine charaeteristies given in Ref. 25, the net head aeting on and the dis-charge through the turbine are approximately linearly related for a constant-gate opening. To simplify the analysis, the head-discharge relationship is as-sumed as indicated in Fig. 11.6.

The equation for the flow through a reaction turbine may be written as

q = b(1 + sy)

in whieh b = 1/(1 - k);s =Z/Ho;k = h¡o/Ho;andHa = static head.Frorn Eqs. 11.22, 11.23, and 11.39, it follows that

dy x - b (1 + sy)dx = -y - tRx2

The eoordinates of the singular points are determined by solving simultaneously

(11.39)

(11.40)

Singularity [-l/k, -(l/ks)]

Substitution of x = -( l/k) + u, and y = -I/(ks) + v into Eq. 11.40 results in

dv u - bso-=---- (1l,45)

x - b(] + sy) = O (11.41)

du (R/k)u - v

Comparison of Eqs. 11,45 and 11.28 yields a' = R/ k; b' = -1; e' = 1; and d' =-bs. Henee,

(11.42)(11.46)

and

The solution of these two equations give the eoordinates of two singular points:(1, - iR) and (-l/k, -lf(ks)]. The second singular point is virtual becauseEqs. 11.22 and 11.23 are valid only for x > O. A singular point is virtual if itdoes not lie in the región to which it belongs. The effeet of a virtual singular


(11.47)

3 SO Applied Hydraulic Transients

y

(a) k = 0.35, R = 2.8

Figure 11.8. Phase portrait for constant gate opening.

Surge Tanks 3 S1

y

1.0

0.5

-0.5

- /.0

- 1.5

(b) k -0.15, R· 1.2

(e) k = O, R = O



Since Zb > 1, both the roots are real with opposite signs. Hence, the singularpoint is a saddle point. It is a virtual singularity because Eqs. 11.22 and 11.23are not valíd for x < O. The effect of this singular point on the stability of os-cillations depends upon its location. For small friction losses, l/k and I/ks arelarge quantities, and thus the point lies at a substantial distance from the stablesingular point (l, - tR). Hence, its destabilizing effect is negligible. For largefriction losses, however, this virtual singuJarity affects the stability of the sys-tem because of íts proximity to the stable singularity (1, - tR). For a friction-less flow, the singular point [-l/k, - I/(ks)] Hes at an infinite distance fromthe origin and thus has no destabilizing effect on the system.

Constant Power

In this case, it is assumed that an "ideal governor" ensures constant powerinput to the turbine. From Fig. 11.6, it can be seen that, as the water level inthe tank is lowered, the governor has to open the gates to increase the dischargefor maintaining constant hydraulic power. No restriction on turbine-gate open-ing is assumed, which implies that the turbine discharge can be increased to anyrequired amount lo maintain constant hydraulic power.

The variables of interest are x and y. Therefore , instead of using the y -dyldr plane, as done by Marris,l9,20 the x - y plane is used herein. In additionlo giving the variables of interest, the use of the x - y plane has the advantagethat one can clearly see the region in which Eqs. 11.22 and 11.23 are valido

If the efficiency of the turbine is assumed constant and the penstock frictionlosses are neglected, then, for constant hydraulic power,

Qtur(Ho + z) = Qo(Ho - h/o)

From Eq. 11.48, it follows that

(11.48)

Qtur Ho - h¡oq=-=

Qo H¿ + z


(11.49)

1- kq=--

1+ sy( 11.50)

in which k and s have the same meaning as defined previously. By substitutingEq. 11.50 into Eq. 11.22, dividing the resulting eguation by Eq. 11.23, andsimplifying,

dy = x + sxy - I + kdx -sy2 - (I + kx2)y - ~R.~2

(11.51)

Surge Tanks 353

ro determine the coordinates of the singular points, the following two equations,

x + sxy - 1 + k = O

sy2 + (1 + kx2)y + tRx2 = O

(! 1.52)

(11.53)

are solved simultaneously. The solution of these equations gives the coordinatesof the following three singular points:

1. (1,- !R)2. [-~+e¡,-~R(e2-el)]3. [- !-e¡,- ~R(e2 +e¡)]

in which e¡ = ..J(l/k) - ~, and e2 = (l/k) - t.By substituting x = Xs + u and y = Ys + v into Eq. 11.51, and neglecting the

terms in u and v of power higher than one, the equation

(11.54)

is obtained. Comparison of Eqs. 1 I .28 and 11.54 yields

a' = -(R + 2kys)xs

b' = -(kx; + 2sys + 1)

e' = (1 + sYs)(11.55)

Singularity (1, - !R)

Substituting Xs = I and Ys = - iR into Eq. 11.55, noting that k = !Rs, andsimplifying, we obtain: a' = R(k - 1); b' = k - 1; e' = 1 - k; and d' = S. Hence,

A¡,A2 = -! {R(k- 1)+s±..J[R(k- 1)+s]2 +4[-(k-l)2 -sR(k- I)]}

(11.56)

or

A¡,A2 =! [R(k- 1)+s±~] (11.57)

in which D¡ = [R(k - 1) + sF + 4[2k(l - k) - (k - 1)2]. lf 2k(l - k) -(k - 1)2 > O (i.e., k> i), then the singularity is a saddle (Fig. 11.9a). Fork < j, the singularity is a node if DI > O, and a spiral if DI < O. The nodeor spiral is stable if R (k - 1) + s < O. For small friction, this inequality takes theform s <R, or 2h¡oHo >Z2. The followíng expression for the Thorna area,


y y

2.0 \ 2.0

1.01.0

-5.0

(a) k=O.35, R=2.8

Figure 11.9. Phase portrait for constant power.

(b) k = 0.025, R = 0.2


A th , may be obtained from s = R :

LAh=--

t 2egAt

If2hfoHo <Z2, the singularity is unstable (Fig. 11.9b).

(11.58)

Singularity [el - !,- !R(e2 - e¡)]

Substitution of the coordinates of the singularity into Eq. 11.55 and simplifica-tíon of the resulting expressions give: a'=-R(l-k); b'=-k(t+el); e':


k(t+e,);andd'==s(e, - !). Hence ,

AI,A2 = t [-R(l-k)+s(c¡ - t)±yn;-] (11.59)

in which D, = [-R(I - k) + s(e¡ - t)]2 .¡. 4[-k2(c¡ + t)2 + 2k(1 - k) (e¡ - t)J.The singularity is a saddle ir 2k(1 - k) (e¡ - t) > k2(c¡ + t)2, which reduces tok < 1 (Fig. 11.9b). Note that for k == j, this singularity shifts to the pre-vious point, i.e., to (1, - iR). For k> {-, the singular point is a nade ir D2 >O, and a focus if D2 < O. The node or focus is stable if R( I - k) > s(c¡ - !),andunstableifR(1-k)<s(c¡ - t).

It is apparent from Fig. 11.9a that all trajectories starting inside the separatrixreach the stable node. For initial conditions such that the corresponding pointon the phase portrait Hes ou tside the separatrix, the tank will drain.

Singularity [(-el - t),- tR(e2 +e¡)]

Because Eqs. 11.22 and 11.23 are not valid for x < O, the singularity is virtual.By substi tu ting the coordinates of the singularity in to Eq. 11.55, we obtaina' =R(k - I);b' = -k(t - c¡);c' = k(t - el); and d' = -s(c¡ + t). Hence,

A¡, A2 :: ![-R(J - k) - s(! + c¡) ±.JD;""] (11.60)

in which D3 = [R(J - k) + s(! + CI)] 2 - 4[k2(! - C¡)2 + 2k(1 - k) <! + CI)]'Since O ~ k < 1, s >O, and R > O, both roots are real and negative if D3 > O,and cornplex conjugares with negative real part ir D3 < O. In the forrner case,the singularity ís a stable nade; in the latter, a stable [acuso

Constant Power Combined With Constant-Gate Opening

In the last section, it was assurned that the turbine gates can be opened to anyvalue to rnaintain constant power. On an actual installation, however, the gatescannot be opened beyond their fully open position, and therefore the dischargecannot be increased indefinitely to maintain constant power as the leve! in thesurge tank falls.

Referring to Fig. 11.6, for heads greater than the rated head -í.e., for y> -Yo,in which Yo == final steady-state water level in the tank-the governor operatesthe gates in such a manner that turbine discharge corresponding to constantpower is obtained. For heads less than the rated head, i.e., for y < -Yo, thegovernor keeps the gate open at the maximum value , and the discharge throughthe gate is determined by the maximum gate characteristics (Fig. 11.6). Theflow in this case is less than that required for constant power. Thus, poweroutput cannot be rnaintained constant for y < -Yo, and the oscillations for thiscondition should be analyzed considering the gate opening as constant.

Surge Tanks 357

For this cornblned governing case, the phase plane is divided into two regions:(1) constant power region Ior y > -Yo, and (2) constan t-gate-opening regionfor y<-yo. There are five singular points: two in the constant gate, two inthe constan! power region, and the singular point at (1, -iR), which is corn-mon to both the constant-gate and constant-power regions. The latter is calleda compound singularity, All the singular points have been analyzed above forconstant-gate opening and constant power, and the results are sumrnarized inTable 11.1.

For k <: j, there is only one real singular point at (I, -1R), hereafter calIedthe ~rst singularity, This is a compound singular point: for y < -Yo (region ofmaximum gate opening), it is always stable; for y > -Yo (region of constantpower), it may be stable or unstable depending upon whether the Thoma cri-terion is satisfied or not (see Table Il.l). Thus, if the Thoma criterion is satis-fied, the oscillations are stable whether they are large or small. In the case ofthe sub- Thoma area, the oscillations may be stable, unstable, or of constant

Table 11.1. Characteristics of singular points.

StableCoordina tes of or

Singularity Type Unstable Required Conditions Miscellaneous

Constant gate opening (y < - tR):(L, -!R) Node Stable (R - bs) > 2 Real

Focus Stable (R - bs) < 2 Real(-l/k, -l/ks) Saddle Always Virtual

Constant power (y> -tRi:(l, -tRi Saddle k>! Real

Nade k < t and D¡ > O RealStable R(k - 1) + s < OUnstable R(k - 1) + s > O

Focus k < t and D¡ < O RealStable R(k - 1) + s < OUnstable R(k - 1) + s > O

(el - L Saddle k<! Virtual-!R(e2 - el») Nade k> l. andD2 > O Real

Stable RO - k) > s(el - t)Focus 1 .

Realk > 3, and D2 < OStable RO - k) > s(e, - t)Unstable R(l-k)<s(e¡-!)

(-el - t. Node Stable D3 > O Virtual-fR(e2 +el») Focus Stable DJ <O Virtual

1.0

... .....l:

," ~... <:>

'¿~.

~ 0.0.....

... '"t:: 2.0 X '"(l .......'"<:>'->


magnitude (called the limit cycle in phase-plane terminology) depending uponthe stabilizing action of the gate and the point from which the trajectory em~-nates. The trajectories emanating inside the limit cycle are unstable, and theíramplitude increases until it is equal to that of the limit cycle. The oscillations

y

y

- 1.0

-2.0

- 3.0

- 4.

(b) k· o. 35 • R = 2. B


Surge Tanks 359

outside the limit cycle are stable and their amplitude decreases until it is equalto that of the limit cycle.

For k> !' the second singularity becomes real and is either a stable or un-stable nade or focus , while the first singularity is a saddle (see Table 11.1).Since such a high value of friction loss is not economical, this case is usuallyof little practical importance.

Phase portraits for k = 0.35 and R = 2.8, and k = 0.025 and R = 0.2 are pre-

(al k -0.025. R· 0.2

Figure 11.10. Phase portrait for constant power combined with constant gate opening.


sented in Fig. lUO. Oscillations in the latter case are unstable (Fig. 11.9b)as given by Paynter's stability diagram," if it is assumed that the constant poweris a1ways maintained. If, however, it is considered that the governor can openthe gates up to a maximum limit, and then the gates remain fully open as longasy < -Yo, then the osciI1ations are stable as shown in Fig. 11.1 Da.

that of an equivalent simple tank, and the development of the accelerating ordecelerating head on the tunnel is more rapid than in the case of a simple tank.If the orifice area is the same as that of the tunnel, then the orífice losses arenegligible and the tank acts like a simple surge tank. However, if the orificesize is very small, then there is very little inflow or outflow, and the systembehaves as if there were no surge tank.

Because of the restrictive effects of the orifice, the volume of inflow or out-flow from the tank is small compared to a simple tank, and therefore the tanksize can be reduced. This reduction in the tank area, however, depends uponthe orífice size. Disadvantages of an orifice tank are: (1) the waterhammerwaves are not completely reflected back at the tank and are partly transmittedinto the tunnel (this must be taken into consideration while designing thetunnel), and (2) the governing of turbines with an orifice tank is not as goodas wíth a simple surge tank because of the more rapid development of accelerat-ing and decelerating heads following a change in the turbine flow.

Conclusions

From the preceding analysis of the oscillations in a simple surge tank by thephase-plane method, the foIlowing conclusions can be drawn:

1. Oscillations are always stable in the cases of constant-discharge and constant-gate opening. This conclusion has been drawn by earlier investigators by usíngother methods.

2. For the case of an ideal governor , which ensures constant power but canopen the gates only to their specified maxirnum limit, the phase plane is dividedinto two regions: (1) The region in which power can be maintained constant, i.e.,y > -Yo, and (2) the region of máximum gate opening in which power cannotbe maintained constant, i.e., Y < -Yo. The solution trajectories in the formerregion correspond to the stable oseillations if As >A th » and to the unstableoscilIations if As <A ti.. while in the latter region they are always stable. Hence,the oscillations, large or small, are stable if As >Arh. For As <Arlo the solu-tion trajectories in the phase plane correspond to stable oscilIations for Y < -Yoand to unstable oscillations for y> -Yo. Due to these stabilizing and destabiliz-ing effects, a solution trajectory corresponding to perpetual oscillations is ob-tained, whieh in the phase-plane terminology is called a limit cycle. The region en-elosed by the limit cyele depends upon the stabilizing effect of the constant-gate opening and upon the destabilizing effect of the governor. The oscillationsinside the limit cycle are unstable, and their amplitude inereases until it is equalto that of the limit eyele. The oscillations ou tside the limit cycle are stable, andtheir amplitude decreases until it is equal to that of the limit cyele.

3. The danger of drainage of the tank for the case of constant power com-bined with constant-gate opening is considerably less than that indicated by thestability analysis assuming constant power only.

Derivation of Dynamic Eq uation

Let us consider the orifice tank shown in Fig. 11.lla in which the orifice frie-tion loss, horf = corf QsIQsl. The following forees are aeting on the liquid in theconduit (see the freebody diagrarn of Fig, 11.1 lb):

FI = 'YAt(Ho - h¡ - hu)

F2 = 'YAr(Ho + z + horf)

F3 = 'YAthf

(11.61 )

(I1.62)

(11.63)

In the aboye equations, the notation of Seetion 11.3 is used. Considering thedownstream direction as positive, the net force acting on the liquid elementin the positive direction is "í:.F= FI - F2 - F3. As shown in Section 11.3, therate of change of momentum of the liquid in the tunnel is ('YL/g) (dQt/dt).Applying Newton's second law of motion and substituting expressions for F, ,F2 , and F) from Eqs. 1 1.61 to 11 .63, we obtain

11.1 o. ORIFICE T ANK'YL vo,--='VA(-z-h -h.-h¡-h r)g dt t- tUI or (Il.64)

Description

In an orifice tank, there is an orifice between the tunnel and the tank (Fig.II.2b). As this orifice restricts the inflow into the tank or outflow from it,the amplitude of the oscillations of the liquid level in the tank is less than

By defining h = hu + h¡ + hf = cQrlQrl in which e is a coefficient, substitutingexpression for horf, and simplifying, Eq. 11.64 becomes

(11.65)

362 Applied Hydraulic Transients+l

Ho

hy r-t""=r-L_,,:,:,:

hf~t- __ ~~~~~~==~

Reservoir

\¡....,___ At L-----,...-{1(o)

_F3= YAth¡

F¡ = YAt( Ho-h¡ -hv) -II..- +- __.I- YAt( Ho+ l thorf )tw(b) Freebody diogrom

Figure 11.11. Orifice tank: notalion for dynamic equalion.

The continuity equation for the simple tank (Eq. 11.8) is also valid for theorifice tank.

The fourth-order Runge-Kutta method may be used to solve the dynamicand continuity equations for an orifice tank.

11.11. DIFFERENTIAL SURGE TANK

Description

In a differential tank (Fig. 11.2c), the riser acts like a simple tank while the maintank acts like an orifice tank. Thus, a differential tank is a compromise betweena simple tank and an orifice tank. In this tank, followíng a change in' the pen-stock flow, the accele rating or decelerating head on the tunnel develops slowerthan in an orifice tank but faster than in a simple tank. Because of this, the area

Surge Tanks 363

of the outer tank can be reduced as compared to that of an equivalent simpletank, and the regulation capabilities of the turbine are not as adversely affectedas in an orifice tank.

As shown in Fig. 11.2c, an orifice is provided between the tunnel and theouter tank. During initial steady state, the water surface in the riser and thatin the tan k are at the same leve!. If the turbine gates are opened to acceptload on the turbogenerator set, the water is initially provided by the riser. Be-cause of small cross-sectional area, the water level in the riser falls rapidly,thus creating an accelerating head on the tunnel in a short periodo The waterlevel in the tank falls slowly to supply additional water. If the turbine gatesare closed to reject load, then the water level in the riser rises rapidly to storewater, thus creating in a short period a decelerating head on the tunnel and adifferential head on the orifice of the outer tank. The water rejected by theturbine is then forced through the orifice into the tank.

Depending upon the terrain, the riser and the main tank may not be as closeto each other as shown in Fig. 11.2c.

Derivation of Equations

Referring to Fig. 11.12 and proceeding similarly as for the simple and orificetanks, the dynarnic equation may be written as

L dQ,--= -z - cQ IQ,IgA, dt r t

in which z; = water level in the riser above the reservoir level (positive up-wards) and e is a coefficient as defined in Section 11.3. The continuity equa-tion at the junction of the tunnel and the tank is

(11.66)

(11.67)

Reservo,,,

Energy grade fine

Hydraufio gradefine

.1hy

----------~~--------~~QturFigure 11.12. Differential tank: notation for dynamic equation.


in which Qr::: flow into or outflow from the riser, and Qs::: flow into or out-flow from the outer tank. The value of Qs depends upon the difference ofwater levels in the riser and in the tank, and upon the size and characteristicsof the orifice at the bottom of the tank, and may be computed from the fol-lowing equation:

(11.68)

in which Cd :::coefficient of discharge of the orifice, and Aorf::: cross-sectionalarea of the orifice. lf Zr > z , the flow is in to the tank and Qs is posi tive; if Z r <z, then Qs is negative , The coefficient of discharge may have different valuesfor flow into or out of the tank. In addition, note that in the Eq. 11.68, it isassumed that there is no spill from the riser into the tank.

To compute the variation of the level in the riser and in the tank, the fol-lowing equations are available:

(11.69)

and

(11.70)

11.12. MULTlPLE SURGE TANKS

The hydraulic system of a turbine or pump may have more than one surge tank.Such a system is referred to as a multiple surge tank system or a system vi surgetanks. Multiple surge tanks are used in the following situations:

l. To provide additional water into the tunnel upstream of the main surgetank through adits.

2. To increase the cross-sectional area of the tank in order to increase the out-put of the turbogenerator. This increase may be accomplished more easilyby adding another tank than by increasing the area of the existing tank.

3. To provide a tank on the tailrace tunnel of an underground hydroelectricpower plant or a pumped storage project to reduce waterhammer pressuresand/or to improve the governing characteristics of the turbogenerators.

4. To split the main surge tank into two or more shafts for the economy ofconstruction or to suit the rock conditions.

To conserve space and due to a large number of possible configurations of themultiple-surge-tank systems, the equations describing the tank oscillations arenot derived in this section. Two typicaI systems usualIy found in practice are

Surge Tanks 365

presented in the problems at the end of the chapter Equations ñ, thb . . . . L' r ese systems

may e easily denved m a rnanner similar to that used for a simple o ifitank. r an on Ice

1U3. DESIGN CONSIDERATIONS

Necessity of a Tank

The first q~estion that arises is whether a surge tank is required. For this u osethe followmg criteria25 may be used: p rp ,

1. Surge tanks should be provided where the resulting reduction in the water-~ammer. pressure provides a more economical penstock-surge-tankínstallatíon.

2. ~ surge tank should be províded if the maximum speed rise following rejec-tíon of the maximum turbine output cannot be reduced to less than 45 per-cent of th~ rat~d speed by othe~ practical mcthods, such as increasing thegenerator. lner~la or penstock diameter or decreasing the effective wicket-gate C~OSlng tl~e. Th~ speed rise will be computed assuming one unitoperatíng alone if there IS more than one unit on the penstock.

3. As ~ roug.h rule of thumb, the provision of a surge tank should be in-vestigated 1f

~L.v.zxs:» >3 t05

Hn(SI units)*

in. ~hich ~LiVi ís computed from the intake to the turbine, and Hn is therrurnrnum net head. In general, a surge tank should be preferred to a pres-sure regula~or although the latter may be preferred for high-head plantsfor econorruc reasons.

Location

A surge tank should be located as near to the turbine as the local topographypermits.

Size

The cross-sectiona] area of a tank is determined to safisfy the followíng criteria:

l. The tank is stable.

2. The tank does not drain (i.e., the water leve] does not fall to the tunnel

:u·v·*In the English units, --,f--!- > lOto 20.n

I!


crown) following maximum possible load acceptance with the reservoir atits mínimum level,

3. The tank does not overflow followíng load rejection unless an overflowweir is provided.

The minimum cross-sectional area required for stability has been a rnatter ofgreat discussion. Jaeger26,27 proposed a safety factor n such that the atea ofthe surge tank should be n times the Thoma area, A th, with n > l. As a roughguide for prelírninary design, n may be taken as 1.5 for a simple tank, and 1.25for an orifice and a differential tank. Duríng the final design when variousplant parameters have been selected, detailed investigations should be con-ducted to check the stability of the tank by using digital or analog computersor by arithmetical or graphical integration. In these ínvestigations, the varia-tion of the turbíne efficiency with head and gate opening and the fact that theturbine gates cannot be opened more than their maximum opening should betaken into consíderation. If these calculations show that the tank is unstableor that the díssipation of the tank oscillations is very slow, then the tank aresmay be increased and the aboye procedure repeated. If, however, the tank isstable , and the oscillations are dissipated at a faster than an acceptable rate,then the possibility of decreasing the tank area should be investigated.

Figure 11.13 may be used to determine the type of instability to be expected,i.e., oscillatory or drainage. In this figure, compiled by Forster28 using re-sults of various investigators, the abcissa, h, is hfolHo, and the ordinate,y, ish¡oIZ; the curves represent the condition of critical stability where oscilla-tions, once begun , continue with constant amplitude. This figure should beused for the most critical operating conditions, i.e., rninírnum reservoir level andmáximum possible turbine output at that level. In general, if h and y for a sys-tem plot aboye the upper envelope of curves, the system will be free of bothoscillatory instability and tank drainage for all conditions includíng full-loadacceptance from zero load. A system plotting below the lower envelope will besubject to instability or tank drainage, or both.

To determine the maximum tan k level, the turbine should be assumed to re-ject maximum possible load. If there is more than one unit on the pen-stock, then all units should be assurned to reject load sírnultaneously , Suchunloading conditions are possible íf there is a fault on the transmission line. Ex-perience with the operation of large grid systerns shows that such severe unload-íng conditions occur a number of times during the life of the project.

The selection of the critical loading conditions is more complex and difficultthan the unloading conditions. Sorne authors suggest that a load acceptance of50 to 100 percent of the rated load be used for determining the maximum down-surge. In the author's opinion, however, no set critería can be recornmended be-

Surge Tanks 367

1.4 ,-------,-------.-------"-T--Ir-:-----,hfo 1/

h :: Ho /11.2+--- hfo I

y = T 11 IPoynfer LO/seof /9J--¡ I

I /~ /.O+---------+------~I~---/~-+~~~~I-----~... Sfoble A / / / :

~ Morris L 19J"v/7' ~/ I~ 08+--------+--------.-~L-·~~-+--~1---~~ L? / I- _~~,/~- Poynter D6] I~ o..6+-------+-----~-~~~------+--_r---~~ Osctitatory /.f~~" ronge /7,.~~ h~~ l _J:=ji==~~~~~an~k~d~ro~m~a~g~e~r~a~ng~e~=t==:;~~~~

11) 0..4 ./ -,-..7V~ Ungovernable range--¡----

~ Morris L/9J Io. 2 +----,,.,.;:;Y.:;¿~~T-Thomo 04]

I g A I Unsfoble Io.~:::<"-P~ LO/seof 19J 1

O 0..1 0..2 0..3 0..4

Relofive heod loss, h

Figure 11.13. TYI!esof instability for various valúes of h andy. (After Forster28)

cause they depend upon the size of the grid system, the amount and the rate ofmaximum load that the plant may be required to accept because of isolation frornthe grid system, and the maximum load that can be added to the system at agiven rate , Therefore, the maximum load and the rate of acceptance for whichthe tank should be designed should be decided in consultation with the engi-neers responsible for operating the grid systern.

If the tunnel head Iosses are not precisely known, then the minimum probablevalue of friction factor should be used for cornputing the maximum upsurge, andthe maximum probable value of friction factor for computing the maximumdownsurge.

If the tank area has to be increased to keep the maximum upsurge or theminimurn downsurge within acceptable limits, then the provision of upperor lower gallery may be economical instead of increasing the area of the tank.

The size of the orifice of an orífice tank or a differential tank should beselected with careo For the orifice tank, the orífice is usually designed" sothat the initial retarding head for full-Ioad rejection is approximately equal tothe maximum upsurge. Johnson's charts30,31 may be used to determine theapproximate dimensions of the tank, riser, and the ports.


11.14. CASE STUDY

Design studies, carried out for the Chute-des-Passes surge-tank system and re-ported in Ref. 28 by Forster, are presented herein for illustration purposes.

Project Details

Figures 11.14 and 11.15 show the plant layout and the details of the hydraulicelements. The upstream tunnel is 9.82-km long, is concrete-líned, and has adiameter of 10.46 m. The 2.73-km-long downstream tunnel is also pres-surized , is unlined, and has a diameter of 14.63 m. The maximum and mini-mum reservoir elevations are 378.2 m and 347.7 m, respectively. There arefive uníts rated at 149.2 MW each at a net head of 164.6 m.

The plant supplies power to electric smelters that have a capacity of 746 MWor more and that are used for the production of aluminum. The nature of thesmelters is such that power shutdown of a few hours could irnpose great op-erating difficulties in addition to large monetary losses, The plant cou1d beísolated from the systern due to major systern disturbances, although the pos-sibility of such an event occurring was considered remote.

At the upstream end of the tailrace tunnel, the 144.9-by-14.64-m tailracemanifold running paralleI to the 144.9-m-long powerhouse served as the down-strearn surge tank.

.....~'-.~

~~

el ....1 Q; <o ~I , ..

~ .. e~l l! ~ <o ..

." '" ... ..ti ti l! ti.:; ~ '" ~

Figure 11.14_ Projecllayout of Chute-Des-Passes development (After Forster28)

Surge Tanks 369

Ptlnslock--..... ,

(a) Seellon Ihrouoh Powerhouseand Draft Tube Manifold

EZ /69.3

W~ir eresl- fE!:385827

........:I

(e) Upslream suroe lank

$upply lunn.1

(b) Manifold and Pensloeks(Plan view)

-( I

-~II

Tol/roes Supply

(d) Tunnel seellona

Figure 11.15. Details of hydraulic elements. (After Forster28)

Preliminary Investigations

1. As the downstream surge tan k is about seven 'times the Thoma area, thevariation of water level of the tank was considered to be ofsecondary importan ceand could be neglected in the preliminary analysis. Similarly, the losses in theorifice of the upstream tank could be neglected, and the tank could be con-sidered a simple tank.

For full plant output and at the minimum reservoir level, the value of h was


computed as 0.065 for the minimum expected head losses and 0.091 for themaximum expected losses. On Fig. 11.13, these values lie within the rangewhere the oscillatory and the drainage types of instability overlap, thus indicat-ing that both types of instability were possible.

2. Analysis using the procedure outlíned by Jaeger '? for a double surge-tanksystern showed that íf the system was to be stable, the area of either tank couldnot be reduced significantly below its corresponding Thoma area even if the areaof the other tank were enlarged.

3. Preliminary analysis showed that, for a stable system and for a 144.9-by-14.64-m downstream surge tank, the diameter of the upstream tank wouldrange from 33.5 to 52 m, and detailed studies were necessary for determiningthe most economical tank diameter.

Selection of Method of Analysis

The following methods were available for a detailed investigation:

1. Arithmetical or graphical integration methods2. Hydraulic model studies3. Digital computer simulation4. Analog computer simulation.

Because of a large number of cases to be studied, the arithrnetical or graphicalintegration methods would have been time-consumíng and were therefore re-jected. In the hydraulic model studies, regulating the turbine flow for main-taining constan! power is difficult. Such studies are desirable for those systemsin which the velocity head and the form losses are comparable in magnitude tothe tunnel friction losses. Since this was not the case for the Chute-des-Passessystem, hydraulic model studies were not considered suitable. Digital computersimulations were rather Tare at that time, and difficulties had been reported byBarbarossa " for the analysis of tank oscillations following smallload changes.However, the use of analog computers for such studies had been demonstratedby Paynter.6,16 Since a large number of cases involving variation of manyvariables had to be analyzed, this method was considered to be the most eco-nomic and was selected for the detailed investigations.

Program of Investigations

For the assurned values of various variables, the following information had tobe obtained:

l. Maximum and minimum water levels in the upstream and in the down-stream surge tank

Surge Tanks 371

2. Stability of the system and the rate of surge damping3. Discharge and volume of overflow over the spillway of the upstream

tank4. Permíssíble load acceptance for different tank sizes over full operating

range of the reservoir.

Selection of Range of Various Variables

Size of Upstream and Downstream Tanks

For the downstream surge tank comprising the tailrace manifold, the area wasassumed to be fixed at 144.9 by 14.64 m.

Preliminary analysis had indicated that upstream tank diameters shouldrange ~rom 33.5 to 52 m. Based on preliminary computer investigations, threetank diameters, 33.5, 39.7 m, and 45.75 m, were selected for a detailed ana1ysis.

Tunnel Resistance

From the data publíshed" on the Niagara Falls Development,35 Appalachiatunnel,36 and Swedísh unlined tunnels.!" the following maximum and mini-mum values for Manning's n were selected:

Tunnel

Concrete-lined upstream tunnelUnlined downstream tunnel

0.0110.035

Maximum Minimum0.0130.038

Orífice Size

The orifice size was selected such that the waterhamrner pressure head in thetunnel did not exceed 0.5 of the rock cover.

The orifice loss coefficíent for the flow into the tank was computed fromthe expansíon losses for the flow from the orifice into the standpipe and frorn thestandpipe into the tank. The loss coefficient for the outflow from the tank wasdetermined from the contraction losses for the flow from the tank into the stand-pipe, from the standpipe into the orifice, and a 45° cone-diffuser expansiono Dou-bling these estimated loss coefficients or assuming them equal to zero had anegligible effect on the stability of the system.

Reservoir Levels

Four reservoir levels-EI. 347.7, 356.8, 366 and 378.2 m-were selected fordetermining the permissible amounts of load acceptance.

*For more up-to-date data on head Iosses in tunnels, see Ref. 34.


Table 11.2. Assumed conditions.

Assumption

Critical ConditionTunnel LossCoefficien t

ReservoirLevel

Stability

Possibility of drainageof upstream tan k followingload acceptance

Maximum upstrearn tanklevel following fu 11loadrcjection

Minimum Minimum

Maximum Minimum

Minimum Maximum

Table 11.2 summarizes the assumptions made regarding the reservoir leveland the tunnellosses for determining different critical conditions.

Derivation of Equations

The following equations were derived for the double-surge-tank system:

1. The dynamic equation for the upstream tunnel2. The dynamic equation for the tailrace tunnel3. Continuity equation at the upstream tank4. Continuity equation at the downstream tank5. Equation for turbine flow to maintain constant power at various net heads,

taking into consideration the variation of the turbine efficiency and the re-action time of the wicket gates

6. Equation for the net head on the turbine.

Analog Simulation

A modelADO Reeves "REAC" electronic analog computer was used. Changesin the value of the tank area, reservoir elevation, and Iosses in the tunnels andin the tan k orifice could be rnade easily and quickly. The variation of sixvariables-namely, ZI, QI ,Z2, Q2, and Qtur *-and net head on the turbine withtime were recorded on a Sanborn recorder. In addition, plots of ZI versus QI

and Z2 versus Q2 were recorded by an oscillographic recorder. These weresimilar to the phase portrait presented in Fig. 11.10a and were especially use fuiin showing the stability of the system.

*For notation, see Fig. 11.18a.

Results

Permissible Load Acceptance

Surge Tanks 373

Figure 11.16 shows curves between the permissible arnount of load, t::,.Pmax'

that may be accepted and the initial load, P. Figure 11.16a is for mínimumreservoir level of El. 347.7 m and a range of tank diameters, and Fig. 11.16b isfor 39.65-m tank diameter and a range of reservoir levels. These curves were de-termined by trial and error so that the rnaximum permissible load, t::,.p rna x s forthe given initial load, P, yielded critical stable oscillations, i.e., surges of con-stant amplitude, or tank drainage, whichever occurred first.

From these curves, the following observations can be made:

l. At low reservoir level, an undersized tank reduces the effective plant ca-pacity. For example, for 33.55-m tank diameter, the maximum per-missible base load (i.e., load increment, t::,.p m a x s approaches zero) is582 MW, whereas with a 45.75-m diameter tank, the plant is stable to fulloutput.

2. The tank drainage condition becomes critical at low base load and withlarge load incrernents, whereas instability becomes critical at large baseload and with smallload increments.

o300 375 450 525 600 675 300 375 450 525 600 675 750

450

;¡,::e.1;

375~<f.....' 300.,"..ti.1;'o 225"~~~.,.~ 150~..Q.

~·11 75

"~

- - -- ~/milOll~n,Ii'SlO~ililY- - LimilOliof1,lonk drainoge

-,,"\.....'-, .,:¡,'?:>..,

~ ...."">~.\"'.$)-s-.... 0<!o..),.'VJ?'~-"::-,( Untl corrtlsponding

~'. fo fuI! gole output'

"\~

~ \,"\ ~,

"-. ~"- ' 1\

Q. "'o

o Minimum heod 10$$6$6. Moximum heod 10$$e$

Lin6 corresponding/o sto/iongenera/o,copocily

Bas" lood on planl, P, in MW(o) (b)

Figure 11.16. Variation of plant operating flexibility. (After Forster28)


3. The advantages of larger tank diameter decreases as the reservoir levelincreases.

4. As the reservoir level increases, the limitation on the load increment gradu-ally changes from instability to tank drainage.

Effeer of Tank Area on Damping

Data listed in Table 11.3 show the effect of the tank diameter on surge dampingfor the critical case of minimum reservoir level and mínimum tunnellosses. Foreach tank diameter, the number of surge cycles, the period of each cycle, andthe time for thc surge to be damped to one tenth of the amplitude of the firstcyc1e are listed. It is clear frorn this table that, for fulI-Ioad rejectíon, the timefor surge damping is independent of tank diameter since the turbine gates arenot reacting to the changes in the tank leve!. For normal load changes, how-ever, increasing the tank size reduces the time required for damping and henceincreases stability.

Effect of Tank Area 011 Overflow

Table 11.4 shows the maximum overflow rate and the volume of overflow forvarious tank diameters. It is clear that increasing the tan k size does not signifi-cantly reduce the overflow volume.

Effeet of Tank Area 011 Excavation

Because the range between the maximum and mínimum surge levels decreaseswith increase in the horizontal tank area, the volume of excavation is not pro-portional to the tank area. Due to overflow, the maximum water level was un-

Table 11.3. Variation of surge damping with tank size.

149-MW Load lncrease Full-Load Rejection

Tank BaseDiarneler Load No. Period Tirnea No. Period Timea

(m) (MW) Cycles (s) (s) Cycles (s) (s)

33.55 407 7.6 630 4860 13.7 580 840039.65 425 4.0 770 3060 11.8 700 840045.75 440 2.8 930 2580 10.1 800 8400

BTime required for surge to be damped down lo one-Ienth of the amplitude off'irst cycle.

Surge Tanks 375

Table 11.4. Volume of overflow. a

Tank Diarneter(m)

Maximum WaterLevel in Tank

(m)Volume of Overflow

(x 103 m3)

Maxímum OverflowRate

(m3/s)

33.5539.6545.75

388.94388.78388.57

358328300

53.550.747.0

a Following full:l~ad rejection (746 MW) at maximum reservolr level (El. 378.2 m)and assurmng nurumurn tunnellosses.

affected by variation in the tan k area. The minimurn water level, howevervaried with increase in the tank diarneter , Table 11.5 lists the required volumeof excavation for various tank diameters.

Selection of Tank Size

Upstream Tank

As discussed previously, increasing the tan k diameter from 33.55 to 39.65 m in-creases the firm capacity of the isolated plant at low reservoir levels by about37.3 MW. This advantage decreases rapidly as the tank diameter is increasedaboye 39.65 m and disappears entirely at higher reservoir levels. Consideringsuch facto~s as permissible amount of load acceptance, degree of surge damping,and effective plant capacity, a tank diameter of 39.65 m was selected. Thebottom elevation of the tan k was set at El. 321.5 m. This level allowed suddenacceptance of one unit or a small amount of load acceptance following full-load

Table 11.5. Effect of tank area on required excavation volumes.

149-MW LoadF ull Load Rejection a Acceptance b

Tank Minimum Tank Minímum TankDiameter Water Level Volume Water Level Volume

(m) (m) (m3) (m) (m3)

33.55 320.34 63,200 325.07 59,00039.65 325.0 82,500 326.81 80,20045.75 329.4 102,700 328.45 104,300

~At mínimum reservoir level and assuming minimum tunnellosses.Fr(}~ maxi"_lum permissíble base load at minimum reservo ir level and

assurrung maxrrnurn tunnellosses.


rejection. A surge-tank-level indicator was instal!ed in the control room so thatthe operators could avoid accepting the load during that part of the surge cyclethat would cause excessive downsurge.

Overflow from the tank would have been carried by a stream course throughthe permanent town site for the project. Because of the potential danger associ-ated with sudden rushes of water through an inhabited area, it was later decided "to excavate a basin in rock at the upper level to reta in the overflow until itcould discharge back into the surge tank.

Downstream Tank

As outlined aboye, the tailrace manifold was selected to act as the tailrace surgetank. For normal operation , the maximum and minimum water levels in thistank were computed to be at El. 192.15 m and El. 181.2 m, respectively. Thefloor was set at El. 170.5 m. To avoid letting the water level fal! so low as tounwater the draft tubes following total-load rejection, a weir was constructed inthe tailrace tunnel downstream of the manifold.

11.15 SUMMARY

In this chapter, the description and analysis of various types of surge tanks werepresented. The phase-plane method was used to investigate the stability of asimple surge tank. Design criteria for determining the necessity of a surge tankand for selecting the tank size were presented, and the details of the studiescarried out for the design of the Chute-des-Passes surge-tank system wereoutlined.

PROBLEMS

11.1. Compute the free surge and the period of oscillations of a simple surgetank following sudden total rejection if the initial steady flow is 1200m3/s. The length of the tunnel is 1760 m, and the cross-sectional areaof the tunnel and of the tank are 200 m2 and 600 m 2, respectively ,

11.2. Prove that dynamic equation (Eq. 11.6) is valid for an inclined surge tank(Fig. 11.17) if As = horizontal area of the tank.

11.3. Derive the dynamic equations for simple, orífice, and differential tanksassuming that the tunnel is inclined at an angle (J to the horizontal. (Hint:Draw a freebody diagram of the tunnel, and apply Newton's second lawof motion. Because of the cancellation of the component of the weight ofwater in the tunnel by the difference in the datum head on the ends oftunnel, Eqs. 11.6, 11.65, and 11.66 are valid.)

Surge Tanks 377

Tunne/Reservo/r

, Penslock

Figure 11.17. Inclined surge tank.

11.4. Prove that the oscillations of the tunnel flow and the water level in asimple tank following load rejection are 90° out of phase. Is the flowleading the water level or vice versa? Assume that the system isfrictionless.

11.5. If the cross-sectional area of a tunnel varies in steps along its length, provethat the actual tunnel may be replaced by an equivalent tunnel havinglength, Le, and area, Ae, such that

in which L¡ and A¡ are the 1ength and the cross-sectional area of the ithsection of the tunnel (i = 1 to n).

11.6. If the inertia of the water in the tank is taken into consideration, provethat the expression for the free surge (Eq. 11.20) for a simple tank isvalid; however, the expression for the period, T, becomes

T=21T - ~V g At

in which A = L + HaA dAs, and Ha = height of the tank.

11.7. Figure 11.18 shows two multíple-surge-tank systems usually found inpractice. Derive the dynamic and continuity equations for these systems.

11.8. Preve that the critica] area, A en for perpetua] oscillations in a closed surgetank (Fig. 11.19) is


Reservoir

Tunnel

Upsfreamsurge tonk

Downstreamsurge tank

(a)

Reservoir

Tank Na 1,

Tunnel No. 1

~-LI

(b)

Figure 11.18. Multiple surge tanks.

in which Ase is critical are a for an open surge tank and is given by theexpression

Ase

= 2g(hf~ +_2_) (Ho-ht, + V~)+2 V~Vo 2g o 2g 2g

Po = steady-state air pressure, and zao = distan ce between the roof of thetank and the initial steady-state water surface in the tank. Assume that

Surge Tanks 379

ReservoirSleady statewater /ever-; Ho

Figure 11.19. Closed surge tank.

the expansion and contraction of the air follow the law, P U~ir = constant;the governor maintains constant power; and the efficiency of turbine isconstant. iHint: Write the dynamic equation for the tunnel, the con-tinuity equation, and the governor equation in terms of small deviations,.lz, .lQt, and .lQtur> from the steady-state values; neglect terms of secondand higher order, and combine the resulting equations by eliminating .lQtand .l Qtur' For perpetual oscillations, the coefficient of the term d[.lz ]/dtof this equation should be equal to zero.)

11.9. Write a computer program for determining the water-level oscillations in asimple tank following a load acceptance or rejection. Using this program,compute the minimum downsurge for a surge-tank systern in which flow issuddenly increased from 56 to 112 m3/s. The length of the tunnel is1964 m, the cross-sectional area of the tunnel and the surge tank are 23.25and 148.8 m2

, and the initial steady-state tunnellosses are equal to 1.22 m.

Answers

11.1. Free surge, Z = 46.42 m; period T = 145.84 s.11.9. 16.05 m below the reservoirlevel.

REFERENCES

1. Jaeger, C., Engineering Fluid Mechanics, translated from German by Wolf, P. O.,Blackie and Sons Ltd., London, 1961.

2. Frank, J. and Schüller, J., Schwingungen in den Zuleitungs=und AbleitungskanalenVDn Wasserkrafranlagen, Springer, Berlin, 1938.

3. Mosonyi, E., Water Power Development, Vol. 1 and ll, Publishing House of HungarianAcademy of Sciences, Budapest, Hungary, 1957, 1960.

4. Rich, G. R., Hydraulic Transients , Dover Publications, New York, 1963.5. Pickford, J., Analysis of Surge, MacMilIan and Co. Ltd., London, England, 1969.


6. Paynter, H. M., "Transient Analysis of Cer tain Nonlinear Systerns in HydroelectricPlants," thesis presentcd to Massachusetts lnstitute of Technology, Cambridge, Mass, inpartial fulfillment of the requirernents of degree of doctor of philosophy, 1951.

7. Paynter, H. M., "A Palimpsest on the Electronic Analogue Art," A. Philbrick, Re-searches, lnc., Boston.

8. Gear, C. W., Numerical tnitial Value Problems in Ordinary Differential Equations ,Prentice-Hall, lnc., Englewood Cliffs, New Jersey, 1966.

9. McCalla, T. R., Introduction to Numerical Methods and FORTRAN Programming;John Wiley & Sons, New York, 1967.

10. Butcher , 1. C., "On Runge-Kutta Procedures of High Order ," Jour. Austr. Math. Soc.,Vol. 4,1964.

11. Fehlberg, E., "Classical 5th, 6th, 7th and 8th Order Runge-Kutta Formula with StepSize Control," N.A.S.A. Tech, Repare R-287, Oct. 1968.

12. McCracken, D. D. and Dorn, W. S., Numerical Methods and FORTRAN Programming ,John Wiley & Sons, Inc., New York, 1964.

13. Wylie , C. R., Advanced Engineering Mathematics , 3rd Edition, McGraw-Hill Book Co.,New York, 1967.

14. Thoma, D., Zur Theorie des wasserschlosses bei Selbsttaetig Geregelten Turbinenanlagen ,Oldenburg, Munchen, Gerrnany, 1910.

15. Paynter, H. M., "The Stability of Surge Tanks," thesis presented to Massachusetts In-stitute of Technology , Cambridge, Mass., in partial fulfillment of the requirements ofdegree of master of science, 1949.

16. Paynter, H. M., "Surge and Water Hammer Problerns," Electrical Analogies and Elec-tronie Computers Syrnposiurn, Trans. Amer. Soco Ov. Engrs., vol. 118, 1953, pp.962-1009.

17. Cunningham, W. J., Introduction to Nonlinear Analysis , McGraw-Hill Book Cornpany,lnc., New York, 1958.

18. Li, Wen-Hsiung, Differential Equations of Hydraulic Transients, Dispersion, andGroundwater Flow, Prentice-Hall, lnc., Englewood Cliffs, New Jersey, pp. 22-36.

19. Marris, A. W., "Large Water Level Displacements in Ihe Simple Surge Tank," Jour. BasicEngineering, Arner. Soco of Mech, Engrs., vol. 81, 1959.

20. Marris, A. W., "The Phase-Plane Topology of the Simple Surge Tank Equation," Jour.Basic Engineering; Arner. Soco of Mech. Engrs., 1961, pp. 700-708.

21. Sideriades, L., "Qualitative Topology Methods: Their Applications lo Surge Tank De-sign," La Houille Blanche, Sept. 1962, pp. 569-80.

22. Sideriades, L,; Discussion of "A Review of Surge Tank Stability Criteria," Jour. BasicEngineering, Arner. Soco of' Mech. Engrs., Dec.1960, pp. 778-781.

23. Ruus, E., "Stability of Oscillation in Simple Surge Tank," Jour., Hyd. Div., Amer. Soc.of Civ. Engrs., Sept. 1969, pp. 1577-1587.

24. Chaudhry, M. H., and Ruus, E., "Surge Tank Stability by Phase Plane Method," Jour.,Hyd. Div., A mero Soco of OV. Engrs., April 1971, pp. 489-503.

25. Krueger, R. E., "Selecting Hydraulic Reaction Turbines," Engineering Monograph No.20, Bureau of Rec1amation, Denver, Colorado, 1966.

26. Jaeger, C., "Present Trends in Surge Tank Design," Proc., Inst, of Mech. Engrs.,England, vol. 168, 1954, pp. 91-103.

27. Jaeger, C., "A Review of Surge Tank Stability Criteria," Jour. Basic Engineering, Amer.Soco of Mech. Engrs., Dec. 1960, pp. 765-775.

28. Forster, 1. W., "Design Studies fOT Chute des-Passes Surge-Tank System," Jour., PowerDiv., Amer. Soco ofCiv. Engrs., vol. 88, May 1962, pp. 121-152.

Surge Tanks 381

29. Ruus, E., "TIte Surge Tank ," mimeographed lecture notes, University of BritishColumbia, Vancouver, Canada.

30. Johnson, R. D., "The Surge Tank in Water Power Plants," Trans. Amer. Soco of Mech .Engrs., vol. 30, 1908, pp. 443-501.

31. Johnson, R. D., "The Differential Surge Tank," Trans. Amer, Soc. of Civ. Engrs., vol.78, 1915,pp.760-805.

32. Jaeger, C., "Double Surge Tank Systern," Water Power, July-August, 1957.33. Barbarossa, N. l., "Hydraulíc Analysis of Surge Tanks by Digital Computer," Jour.,

Hyd. Div.,Amer. Soco of Civ. Engrs., vol. 85, April1959, pp. 39-78.34. "Factors Influencing J710w in Large Conduits," Report of the Task Force on Flow in

Large Conduits of the Committee on Hydraulic Structures, Jour., Hyd. Div., Amer.Soc. of Civ. Engrs., vol. 91, Nov. 1965, pp. 123-152.

35. Bryce, J. B. and Walker, R. A., "Head LOS5 Coefficients for Niagara Waler SupplyTunncls," Engineering Journal , Engineering Inst. of Canada, July 1959.

36. Elder, R. A., "Friction Measurements in Appalachia Tunnel," Jour., Hyd, Div., Amer.Soco of Civ. Engrs., vol. 82, June 1956.

37. Rahrn, l., "Flow Problems with Respect to Intake and Tunnels of Swedish Hydro-electric Power Plants," Bulletin No. 36, lnst. of Hydraulics, Royal Inst. of Tech.,Stockholrn, Sweden, 1953.

38. Matthias, F. T., Travers, F. J., and Duncan, J. W. l., "Planning and Construction ofthe Chute-des-Passes Hydroelectric Power Project," Engineering Jour., Engineeringlnst. of Canada, vol. 43, Jan, 1960.

CHAPTER 12

TRANSIENT FLOWS INOPEN CHANNELS

12.1 INTRODUCTlON

In the previous chapters, we considered transient-state flo~s in the closedconduits, In this chapter, we will discuss the transient flows m open channels.A flow having a free surface is considered open-channel flow even though thechannel may be closed at the top, e.g., a tunnel flowing partially full.

In this chapter, a nurnber of commonly used terms are first defined, and thecauses of transient flows are discussed. The dynamic and continuity equationsdescribing such flows are then derived, and various methods available for theirsolutíon are discussed. Details of two of these-the explicit fi/1ite-differencemethod and the implicit fi/1ite-differe/1ce method-are then presented. This isfollowed by a discussion of a number of special topícs on open-channel tran-

sients. The chapter concludes with a case study .

12.2 DEFlNITlONS

If the depth and/or velocity varies at a point with time, the flow is terrnedunsteady [low . Examples of unsteady flow are: floods in rívers, tides in oceansand in estuaries, surges in power canals, and storm runoff in sewers.

Depending upon the rate of variation of the flow and the depth, the unsteadyflows may be cJassified as rapidly varied or gradual/y varied flow. 1 In the caseof rapidly varied flows, the water-surface variation is rapid; and usually, the watersurface has a discontinuity called bore or shock. Examples of such flows aresurges in power canals that are caused by load changes on turbines or tidalbores in estuaries. In the gradually varied flows, the variation of the free surfaceis gradual-e.g., river floods, tides without bore formation.

Transient flows in the open channels are usually associated with the propaga-tion of waves. A wave is defined as a temporal (í.e., with respect to time) orspatial (i.e., with respect to distance) variation of ñow or water surface. The

382

Transient Flows in Open Channels 383

A=: Wovelenglh

Ys

Figure 12.1. Wavelength and amplitude.

wavelength, A, is the distance from one crest to the next, and the amplitude ofa wave is the difference between the maximum leve! and the stiJI water leve! (se eFig.12.1).

The wave speed relative to the medium in which it is traveling is called wavece/erity, c. Note that it is different from the flow velocity, V, with which theparticles of the fluid move as a result of the wave propagatíon. The absolutewave velocity ; Vw' is equal to the vectorial sum of the wave celerity and the flowvelocíty, i.e.,

Vw =V + C (12.1)

in which boldface type indica tes that the variables are vectors. In a one-dimensional flow, there is only one flow dírection. Therefore, the wave celerítyis either in the direction of the flow (downstrearn), or it is opposite to the flow(upstream). Considering the downstream direction as positive, Eq. 12.1 may bewritten as

(12.2)

In this equation, the positive sign is used for a wave traveling in the downstreamdirection, and the negative sign ís used for a wave traveling upstream.

By using different characteristics as the criterion of classification the wavesmay be classified as described in the following. '

A wave having a wavelength more than twice the flow depth is termed ashallow-water wave, and a wave having a wavelength less than twice the flowdepth is called a deep-water wave. Note that it is the ratio of the wavelength,A, to the depth, Ys' and not the flow depth alone, which defines the type ofwave.? For example, depending upon the ratio of the wavelength to the flowdepth, a short wave, such as a ripple, can be a deep-water wave in otherwiseshallow water; a long wave, such as a tide, in the deepest part of an ocean canbe a shallow-water wave.


In a shallow-water wave, the fluid particles at a cross-sectíon have the sameflow velocity; the wave celerity depends upon the flow depth, and the verticalacceleration of the fluid particles is usually negligible compared to the horizontalacceleration. The wave celerity of a deep-water wave depends upon the wave-length; the particle motion is negligible at depths equal to the wavelength fromthe surface; and the horizontal and vertical accelerations are comparable inmagnitude and decrease rapidly with distance from the surface.

If the wave surface is higher than the initial steady-state surface, the wave iscalled a positive wave, while if the wave surface is lower than the steady-statesurface, it is ca11eda negative wave.

If the fluid particles translate spatially with the wave, the wave is calledtranslatory (e.g., surges, tides, floods), while, if their is no such translation, thewave is called a stationary wave (e.g., a sea wave).

A wave having just one rising or falling limb is called a monoc/inal wave. Asolitary wave has gradually rising and falling (or recession) Iimbs. A number ofwaves traveling in succession are called a wave train.


Transient-state conditions are produced in open channels whenever either theflow or the depth of flow or both are changed at a section. These changes maybe planned or accidental; they may be natural or produced by human action.Following are sorne of the most common examples and causes of open-channeltransients:

1. Floods in the rivers, streams, and lakes caused by snow-melt, rainstorrn, oropening or closing of control gates

2. Surges in the channels caused by loading or unloading the turbines, startingor stoppíng the pumps, opening oc closing the control gates

3. Surges in the navigation canals caused by the operation oflocks4. Waves in a river or a reservoir created by a dam-break5. Lake and reservoir circulation caused by wind or density currents6. Storrn-runoff in sewers7. Tides in estuaries oc inlets.

Depending upon the rate at which the flow oc the depth changes, a bore orshock rnay be formed during the transíent-state conditions.

12.4 SURGE HEIGHT AND CELERITY

In the last section, celerity was defined as the wave speed reJative to the fluidin which it is traveling. Let us now derive expressions for the celerity and heightof a surge.


Referring to Fig. 12.2a, let the flow in the channel be steady at time t = Owhen a sluice gate located at the upstream end of the channel is suddenly openedand the flow is suddenly increased from Q 1 to Q2' Thís increase in flow pro-duces a wave of height, z, which travels in the downstream direction.

rotume

2 I

(a) Unsteady flow

Sluice gale

(b) Equivalent steady flow

Fa .y..y;rTl¡ [¡YO _ _~_ §-i'F1-rA1YI

Weighl, W

(e) Freebody diagram of control volume

Figure 12.2. Wave height and celerity.


Let us designate the flow depth and the flow velocity to the right of the wavefront (i.e., undisturbed conditions) by y I and VI, and the corresponding vari-ables to the left of the wave front by h and V2 (Fig. 12.2a). If Vw is theabsolute wave velacity , then the unsteady flow (Fig. 12.2a) can be con vertedinto a steady flow by superimposing velocity Vw on the control volum~ in ~heupstream direction (Fig. 12.2b). The velocity in the downstream flow dírection

is considered positive.Referring to Fig. 12.2b, the continuity equation may be written as

AI(VI - Vw)=A2(V2 - Vw) (12.3)

Assuming that the pressure distribution at section 1 and 2 is hydrostatic andthat the channel is horizontal, and neglecting friction, the forces acting on thecontrol volume (Fig. 12.2c) are:

Force in the upstream direction, FI = -yJ\A I

Force in the downstream dírectíon.F¡ =-yJi2A2

in which Jil and Jiz are the depths of the cen troids of areas A I and A 2·The rate of change of mornentum of the water in the control volume

(I2.4)

(I2.5)

=.lA I (VI - Vw) [(VI - Vw) - (V2 - Vw)]g

= .lA,(V, - Vw) (VI - V2)g

The resultant force, F, acting on the water in the control volume in the down-

(12.6)

stream direction, is

F = F2 - FI = -y(A2Ji2 - A IJiI)

Applying Newton's second law of rnotion ,

(12.7)

(12.8)

Eliminating V2 from Eqs. 12.3 and 12.8 and rearranging the resulting equation,

we obtain(12.9)

Since the wave is moving in the downstream direction, its velocity must begreater than the initial flow velocity VI. Hence, it follows from Eq. 12.9 that

(12.10)


If there is no initial flow in the channel (i.e., VI = O), then the absolute wavevelocíty Vw is equal to the radical term of Eq. 12.10. Transposing VI to theleft-hand side,

(12.11)

We previously defined the celerity , e, of a wave as its velocity relative to themedium in which it is traveling. Since Vw - VI is the velocity of the wave rela-tive to the initial flow velocity VI, the following general expression for e is ob-tained from Eq. 12.11 :

gA2 CA - A - )AI(A2

- A¡) 2Y2 - IYI

A positive sign is used if the wave ís traveling in the downstream direction, and anegative sign is used if it is traveling upstream.

The relationship between the velocíties and the depths of flow at sections 1and 2 is obtained by eliminating Vw frorn Eqs. 12.3 and 12.8, i.e.,

c=± (12.12)

- - _ AIA2 2A2Y2 -AIYI - g(A2 -Al) (VI - V2)

The height of the wave, z, is equal to Y2 - Y l. If h >Y I , then the wave is apositive wave, and if h <y I , then it is a negative wave.

There are five variables=namely, YI, VI' h, V2, and Vw-in Eqs. 12.3 and12.13. The value of V2 or Y2 may be determined by trial and error from theseequations if the values of other three independent variables are known.

Note that Eqs. 12.12 and 12.13 are general and may be used for channelshaving any cross-sectional shape, Let us see how these equations are simplifiedfor a rectangular channel.

(12.13)

Rectangular Channel

For a rectangular channel having width B, Jil = t YI ; Y2 = t Y2; A I = BYI ; andA2 = BY2. Substituting these expressions into Eq. 12.12 and simplifying theresulting equation,

gh ( )-2- YI +hYI

Ifthewave height is small as compared to the flow depth,y, then v¡ ::::Y2 ::::y.Hence, it follows from Eq. 12.14a that

c= (12.14-a)

c=ygy (I 2.14-b)


For a rectangular channel, the continuity equation (Eq. 12.3) may be written

as

By¡(V¡ - Vw)=BY2(V2 - Vw)

from which it follows that

(12.15)

y¡ V¡ - Y2V2V = ::.....:..__:..--=--=-~w y¡ - Y2

Noting that for a wave traveling in the downstream direction, Vw = V¡ + C 1

substituting expression for e from Eq. 12.14a and eliminating Vw from theresulting equation and Eq. 12.16, we obtain

. '---'---"- .. -.... _- .._ .._--'-;: (V - V)2 = g(y¡ - Y2) (y¡ - yD í (12.17)i ¡ 2 2y¡Y2 J

.. - .. --""-'----- --~-~'" ''-_ ,-_' ."-""~""._""~. ,., __ ._., o., .... _.,,_,_:. '''''''',

(I2 .16)

This equationv" was derived by Johnson and may be solved by trial and error

to determine the surge height.Figure 12.3 shows a positive wave. Because the depth at the leading edge of

2

(a)

2

~~~--------------

( b)

Figure 12.3. Variation of the wave front of a positive wave.

. ,


the wave front (point 1) is smaller than at the trailing edge (point 2), it followsfrorn Eq. 12.l4b that the wave celerity is higher at point 2 than that at point l.Thus, as the wave travels, the trailing edge of the wave front tends to overtakethe front edge. Therefore, the wave front gradually becomes steeper until abore forms. Using a similar argument, it is c1ear that a negative wave frontflattens as it travels in a channel.

We know that for the subcritical flows, the Froude number, F < 1, i.e.,

V--<1v'iY (12.18)

or

V<v'iYOn the basis ofEq. 12.l4b, Eq. 12.19 can be written as

V<c

(12.l9)

(12.20)

Hence, it follows from Eq. 12.2 and 12.20 that Vw is negative if the wave istraveling in the upstream direction. In other words, a disturbance travels bothin the upstream and in the downstream direction. For supercritical flows(F> 1), however, a wave can travel only in the downstream direction, sincethe flow velocity is more than the wave celerity and Vw is always positive.

12.5 DERIVATION OF EQUATIONS

The dynamic and continuity equations* describing the one-dimensional transientflows are derived in this section.

The following assumptions are made in deriving these equations:

1. The slope, (J, of the channel bottom is sma11 so that sin (J ~ tan (J ~ (J andcos (J ~ l.

2. The pressure distribution at a section is hydrostatic. This is true if thevertical acceleration is small, Le., if the water-surface variation is gradual.

3. The transient-state friction losses may be computed using formulas forthe steady-state friction losses.

4. The velocity distribution at a channel cross section is uniformo5. The channel is straight and prismatic.

Let us consider the control volume shown in Fig. 12.4. The x-axis Hes alongthe bottom of the channel and is positive in the downstream direction. Thedepth of flow, y, is measured vertically from the channel bottom. Thus, the

"These equations are usually referred to as Saint- Venant equattons.i For a general deriva-tion 01 these equations, see Refs. 6 through 8.


r---'"I ~1~-------'-r--=__I - I

I IInflowl 1 O~flowy-l :

I II 1

l. .11---+x

Oisfonce .. )( )( +l\.)(

Flow oreo,' A A+ !~l\.)(

Velocify" V V + 8V l\.)(8V

(a)

8yy+ 8il\.)(

~-----.8yl--;=------r---H----~8)( l\.X

FI = rAy

fF5

s-Xw(b) Freebody diagram

Figure 12.4. Notation for dynamic and continuity equations.

x and y axes are not orthogonal. However, as the channel is assumed to have asmall bottom slope, this discrepancy does not introduce significant errors.

Continuity Equation

If'Y is the specific weight of the water, then referring to Fig. 12.4:

The rate of mass inflow into the control volume

(12.21)

TransientFlows in Open Channels 391

The rate of mass outflow from the control volume

'Y ( oA ) ( av )=- A+-~x V+-~xg ox ox (12.22)

Hence, the net rate of mass inflow

Neglecting second-order terrns,

. 'Y oA -y avNet rate ofmass mflow = - - v- ~x - -A -a ~xg ax g x

The rate of increase of the mass of the control volume

(12.23)

'Y aA=--~xg al (12.24)

As the time rate of increase of the mass of the control volume must equal thenet rate of mass inflow into the control volume, it follows from Eqs. 12.23 and12.24 that

'Y aA 'Y aA 'Y a v- - ~x = - - v - ~x - -A - ~xg al g ax g ax

Dividing both sides by (-y/g)t.x and rearranging, Eq. 12.25 becomes

(12.25)

aA aA av-+ V-+A -=0al ax ox (12.26)

Since the channel is assurned prismatic, the flow area, A, is a known function ofdepth,y. Therefore, the derivatives of A may be expressed in terms ofy as follows:

aA = dA ~=8(y) ay}ax dy ax Bx

aA = dA ay =8(y) ayal dy al al

(12.27)

For a channel having continuous side slopes, dA/dy is equal to the channel width8 at depth y. If the values of A(y) and 8(y) are obtained by independent mea-surements, the measurement error may cause 8(y) to be different from the valuesof channel width obtained by differentiating area A(y) with respect to depth, y.For numerícal stabílíty,? it is important that A(y) and 8(y) should be com-patible-i.e., if either A(y) or 8(y) is obtained by measurement, then the othershould be determined by caJculus.


Substituting Eq. 12.27 into Eq. 12.26, we obtain

ay A av ay-+--+v-=oat B ñx ax

Since discharge Q = VA,we can write

(12.28)

aQ = V aA +A avax ax ax

On the basis of Eq. 12.27, Eq. 12.29 becomes

aQ =BV ay +A avax ax ax

Hence , it follows from Eqs. 12.28 and 12.30 that

(12.29)

(12.30)

aQ ay-+B-=Oax al (12.31)

Dynamic Equation

The following forces are acting on the water in the control volume shown inFig.12.4b:

F( = F2 = 'YAji

ayF3 = 'YA- t.x

ax

F4 = -yASft.x

(12.32)

(12.33)

(12.34)

Note that the pressure force acting on the downstream face is divided into twocomponents, F2 and F 3, and the terms of higher order are not included in theexpression for F3' In Fig. 12.4b, F(, F2, and F3 are forces due to pressure;F4 = force due to friction; Fs = x-component of the weight of the water in thecontrol volume; (}= angle between the channel bottom and horizontal axis (posi-tive downward); and Sf = slope of the energy grade lineo

The value of Sr may be computed using any standard formula for the steady-state losses. such as Manning's or Chezy's formula. Since (} is assumed small,sin {}::::=_{} = So in which So = bottom slope. Hence,

(I2.35)

Referring to Fig. 12.4b, the resultant force acting on the water in the controlvolume in the positive x-direction is F = ¿ F = F( - F2 - F3 - F4 + Fs. Sub-


stituting expressions for F¡ io F« fromEqs.12.32to 12.35,weobtain

avF = -'YA a~ t.x + 'YASot.x- 'YASft.x (I2.36)

Momentum entering the control volume = 2.A V2

g(12.37)

Momentum leaving the control volume = 2. [A V2 + ~ (AV2) t.x ]g ax

(12.38)

Therefore, net influx of momentun into the control volurne

= - 2.~(AV2)t.Xg ax

The time rate of increase of momentum

(12.39)

=~(2.AVt.X)al g

(12.40)

According to the law of conservation of momentum, the time rate of increaseof momentum is equal to the net rate of momentum influx plus the sum of theforces acting on the water in the control volume. Hence, it follows from Eqs.12.36,12.39, and 12.40 that

a ('Y ) -y a 2 ay-a -A Vt.x = - - -(A V )t.x - 'YA-t.x + 'YASot.x- -yASft.x19 gax ax

(12.41)

Dividing throughout by ('Y/g)t.xand simplifying, Eq. 12.41 becomes

a a ay-a (AV) + -(AV2) + gA - =gA(So- S,)t ax ax

(12.42)

Expansion of two terms on the left-hand side, dívision by A, and rearrangementof the terms yields

ay av av V(aA aA av)gax +V ax +a¡+A" ar+Va;+Aa;- =g(So-Sf) (12.43)

On the basis of the continuity equation, Eq. 12.26, the sum of the terms withinthe brackets on the left-hand side of Eq. 12.43 is equal to zero. Hence, Eq. 12.43becomes

ay av avg-+-+ V-=g(S -S)

ax al ax o, (12.44)


By expanding the terms and rnaking use of the faet that Q = VA, Eq. 12.42may be expressed6,10 in the following forrn

aQ +2VaQ +(1- F2)gA ay =gA(So -Sr)at a.x axin which F1 = V1J(gAJB).

Equations 12.28 and 12.44 are referred to as Sto Venant equations,Note that Equations 12.31 and 12.44 are derived assuming that the channel

is prisma tic and that there is no lateral inflow or outflow, Proceeding as before,the dynamic and continuity equations can be derived for non~rismatic channelshavíng lateral inflow or outflow (Problern 12.8).

(12.45)

12.6 METHODS OF SOLUTlON

The following numerical methods suitable for a computer analysís are availableto salve the contínuíty and dynamie equations describing the unsteady flow in

open channels:

l. Method of characteristicsll-12

2. Fínite-difference methods9,12-14,20,22-33

3. Finite-elernent method.34-36

In the method of characteristics, the equations are first converted into charac-teristie form , whieh are then solved by a finite-difference scheme. In the finite-difference rnethods, the partial derivatives are replaced by finite-differeneequotients, and the resulting algebraic equations are then solved to ~ete.r~inethe transient conditions. In the finite-element method, the systern IS dividedinto a number uf elements, and the partial differential equations are integratedat the nodal points of the elements.

The flnite-elernent method ís in the infancy stage for application to open-channel transients and will not be discussed further herein. The method ofcharacterlstics faíls due to convergence of the characterístic curves once a boreforms. Therefore, the bore has to be isolated and treated separately in thecomputational procedures. Using the advances rnade in gas dynamic and thefact that the equations of gas dynarnícs and St. Venant equations are analogous,a number of flníte- dífference schemes have been proposed in which it is notnecessary to isolate the bore.

The isolation of a bore in automatic computations is very cumbersome andcomplex, especially if there are several geometrical changes in the system sincethe transmission and reflection of the bore at each change has to be considered.Therefore, it is highly desirable to have a computational procedure in which nospecial treatment is required if a bore forms during the transient conditions.Such a scherne is presented in Section 12.9.


12.7 METHOD OF CHARACTERISTlCS

As described previously, in the method of characteristics, the St. Venant equa-tions are converted into characteristic equations, which are then solved by afinite-difference scheme. This method is not suitable for systems havíng numer-ous geometrical changes, and it fails because of the convergence of the charac-teristic curves whenever a bore or a shock forms. Although this method wasquite popular in the 19605, it is being replaced by the finíte-difference methods.However, in the latter methods and especially in the explicit finite-differencemethods, the characteristic equatíons are required to develop the boundary con-ditions. Herein, we will not present the details of the method but will developonly those equations that will be used in the next section. Readers interestedin the details of the method should see Refs, 11 through 15; and for its appli-cation to the open-channel transients, they should see Refs. 16 through 19.

Multiplying Eq. 12.28 by ±cB!A, adding it to Eq, 12.44, and rearranging theterms, we obtain the following so-called characteristic equations

[av av] g[ay ay]- + (V + e) - + - - + (V + c) - == g(S - Sr)ot ax e al ax o(12.46)

and

[av oV ] s [ay ay ]-+(V- c)- - - ·_+(V- c) - ==g(S - S)al ax e al ax o r (12.47)

in which e =...jgA/B. Equation 12.46 can be converted into an ordinary differ-ential equation by defining dxldt = V + e; similarly, Eq. 12.47 can be convertedinto an ordinary differential equation by defining dxldt = V-c. Since dxldt=V + c is the equation of a positive characteristic curve in the x-t plane , Eq. 12.46is called the [orward or positive characteristic equation . Similarly, dx[dt = V -c is the equation of a negative characteristic curve in the x-t plane, and Eq.12.47 is called the negative characteristic equation, Referring to Fig. 12.5, theseequations may be expressed I in the finite-difference form as

Vp - VM ( ) VM - VI,-=---=-+ VM + CMAt Ax+-.g [YP-YM +(V +c )YM-YLL (S -S ) (12.48)

cM At M M Ax] g o rM

and

Vp - VM + (V: _ e ) VR - VMAt M M Ax

_..L[YP-YM+(V - )YR-YM]= (S -S·) (12.49)cM A t M cM Sx g o JM


.o. x

x

(o) Upsfreom boundory

.o. x

x

(b l Downsfreom boundory

Figure 12.5. Notation for positive and negative characteristic equations.

In these equations, the subscripts L,M,P, and R refer to the variables at variouspoints in the x-t plane (Fig. 12.5).

The terms of Eqs. 12.49 and 12.48 can be rearranged to yield the followingequations:

l. Negative characteristic equation (Fig. 12.5a):

(12.50)


in which

Ca = L and CM = - ~CM lf BM

2. Positive characteristic equation (Fig. 12.5b)

Vp = Cp - CaYp

(12.52)

(12.53)

in which

e, = VM - ~t (VM + CM) (VM - Vd + L [YM - ~(VM + CM)(YM - Yd]...x CM Ll.x

+ g(So - SrM)Ll.t (12.54)

In Eq. 12.50 through 12.54, conditions at M are those at the boundary at thebeginning of the time step. Since the values of al! variables are known at L, M,and R, the value of constants Cn and Cp can therefore be computed. In Eq.12.50 or Eq. 12.53, there are two unknowns (i.e.,ypand Vp). These equationsand the conditions imposed by a boundary will be solved simultaneously inSection 12.9 and 12.12 to develop boundary conditions for the finite-differenceschemes.

12.8 EXPLICIT FINITE-DIFFERENCE METHOD

In the explicit finite-difference method , the partial derivatives of the St. Venantequations are replaced by finite differences such that the unknown conditions ata point at the end of a time step are expressed in terms of the known conditionsat the beginning of the time step. The following explicit finite-differenceschemes are available to solve the SI. Venant equations:

1. diffusive scherne2. two-step Lax-Wendroff scherne3. Dronker's scheme.

Of these, the diffusive scherne is the simplest and easiest to program, and it givessatisfactory results.7,20,29,30,31,33 In addition, a bore does not have to be isolatedin the computations. Details of this scherne are presented here. Readers in ter-ested in the other schemes should see Refs, 7, 23, 31, and 37 for the Lax-Wendroff scheme, and Refs. 7 and 38 for Dronker's scherne.


12.9. DlFFUSIVE SCHEME

Fonnulation of Algebraic Equations

In this scheme, the partial derivatíves of the Sto Venant equations are replacedby the following finite differences (Fig. 12.6):

ay YP-YM av Vp-VM (12.55)al = llt ; ¡¡= llt

ay YR - YL av VR - VL-= ;-=ax 2& ax 2&

(12.56)

and Sf ís replaced by Sfm' The conditions at point M are calculated from

VM = ~(VL + VR) }

YM = !(YL + YR)

SfM = !CSfL + SIR)

Substituting Eqs. 12.55 and 12.56 into Eqs. 12.31 and 12.44 and solving forVp andyp, we obtain:

(J2.57)

1 lltVP=VM+2&[VM(VL- VR)+g(YL-YR)]

+gllt[So- t(Sh + SIR)]

I llt 1YP=YM+"2 &B

M(QL-QR)

(12.58)

(12.59)

p

MIL M R

.o. x .o. x

x

Figure 12.6. Notation for diffusive scheme.


Note that the unknown conditions at point Pare expressed in terms of theknown conditions at points L and R. These two equations, Eq. 12.58 and12.59, are used to determine Vp andyp at the interior sections.

It is c1ear from Fig. 12.6 that the coordinates of points P, M, L, and R are(xo, to + M), (xo, to), (xo - Ax, to), and (xo + llx, to)' By expanding the termsof Eqs. 12.58 and 12.59 into a Taylor series -e.g.i j-, = y(xo, to) + llt(ayjat) +{[(At? /2!] (a2y/at2)}-and comparing with Eqs. 12.31 and 12.44, it can beproved that this difference scheme introduces additional diffusionlike terrns,t [(&)2 j At] (a2 yjax2) and t [(&)2 j M] (a2 yjax2). Therefore, this schemeis called a diffusive scheme.

Boundary Conditions

As discussed aboye, Eqs. 12.58 and 12.59 are used to determine the conditionsat the interior sections. At the boundaries, however, special boundary condi-tions are developed by solving the posítive or negative characteristic equations,or both, simultaneously with the conditions imposed by the boundary. Thepositive characteristic equation, Eq. 12.53, is used for a downstream boundary,and the negative characteristic equation, Eq. 12.50, is used for an upstreamboundary.

Two subscripts are used in this section to designate variables at various sec-tions. The first subscript refers to the channel, and the second refers to thesection number. For example, subscripts (i, 1) refer to the first section on the ithchannel. To designate the last section on the ith channel, which is assumed tobe divided into n reaches, subscripts (i, n + 1) are used. Note that subscriptP isused for the unknown quantities at time fo + llt(Fig. 12.6).

Four common boundary conditions are derived in this section; other boundaryconditions may be developed similarly.

Constant-Head Reservoir at Upstream End

If the entrance loss at the reservoir is k V~. j(2g), then, referring to Fig. 12.7a,1,1

vfi·+ (1 + k) __!Z!_.Yres =YP¡,I 2g (12.60)

Substituting for Yp. from Eq. 12.50 into Eq. 12.60 and solving for Vp. ,we1,1 11

obtain '

= -1 + VI + 4Cr(Cn + CaYres)Vp'l1, 2C, (12.61)


Entranc~ 1055= kV2pi, I

~En~rl1YI1rad~ lin~

o..>.

1 Actual wat~r surfactl

L "\__~:::mat~d wattlr surrac«

Flow-Chann~1 i

77777/777777777

(a) Upstreom reservair

Energy grod~ I¡n~

Actual wat~r 5urfacII

Assum~d wa/~r sur~: 7-cyv2

Pi n+l.

(b) Dawnstream reservair

Energy grade linll

Water surrac« ~

2V Pi+I, I

t----~

r-¡--------:-::r.........--yPi+1

(e) Junction of two channels

Figure 12.7. Notation for boundary conditions.

Transient Flows in Open Channels 40 I

in which

C, = CAl + k)/(2g) (12.62)

Now Yp. can be determined from Eq. 12.50.1,1If the head losses and the velocity head at the entrance are negligible, then

Yp¡ 1 = Yres; Vp¡,1 may be computed from Eg. 12.50.Note that Eqs. 12.60 and 12.61 are valid for the positive flows onIy; similar

eguations may be written fOI the negative flows.

Constant-Head Reservoir at Downstream End

Ifthe head loss is CvV],. /2g, then, referring to Fig. 12.7b,I,n+l

(I - el) V],¡ n+l

YP¡,n+l = Yres + 2g (I2.63)

in which Yres = water depth in the reservoir, and CI) = coefficient of head loss.Simultaneous solution of Eqs. 12.53 and 12.63 yields

V = 1 + VI - 4er(ep - Cres)P¡,n+l 2e. (12.64)

in which er = (I - el) ea/2g.If the total velocity head is lost, then CI) = 1, and Eq, 12.64 cannot be

used since it will involve division by zero , In such a case, the folJowing equationshould be used:

YP¡,n+l = Yres

Vp¡,n+1 is then determined from Eq, 12.53.

(12.65)

Discharge Change at Upstream or Downstream End

Discharge changes may result from load acceptance or rejection by hydraulicturbines, startíng or stopping of pumps, or opening or closing of control gates.

Discharge is specified as a function of time, i.e., the function Q = Q(t) isknown. Hence,

Qp .. = A (yp .. ) Vp.. = Q(tp)1,' 1,1 i.¡ (12.66)

Note that A ts» .. ) denotes area at depth yp .. and Q(tp) denotes discharge atJtl r,

time (p. To determine Yp .. and Vp .. , Eqs. 12.53 and 12.66 are solved by an'II I,J

iterative procedure for the downstream end and Eqs. 12.50 and 12.66 for theupstream end.


Junction of Two Channels

At the junction of the channels (Fig. 12.7c), the energy equation can be writtenas

V2 V¡..+ P¡,n+l=z.+ +(1+k)~

YP¡,rI+1 2g ,YPi+I,1 2g

in which k is the coefficient of head losses at the junction, and z¡ = rise or dropin the channel bottom = [invert elevatíon of the (i + 1)th channel] - (invert ele-vatio n of the ith channel),

The following other equations are available:

(12.67)

Vp¡+I,1 = e, + Ca¡+1 YP¡+I ,1

Vp¡,n+1 = Cp - Ca¡YP¡,n+1

A (yp" n+l) Vp¡ n+1 = A (yP¡+1 ,1) Vp¡+I,1, ,

(l2.68)

(12.69)

(12.70)

There are four unknowns-namely, Yp¡ n+1 ' YP¡+I l' Vp¡ n +1' and Vp¡+I,1 -inEqs. 12.67 to 12.70. These unknowns may be determíned by solving theseequations by the Newton-Raphson method39,40 as follows:

To simplify the notation, let us designate

YP¡,n+1 =XI

(12.71)Vp¡,n+l =X3

Vp¡+I,1 = X4 _,

Equations 12.67 to 12.70 may then be written as

x2 x2F =x -x +.2.-(I+k)__i.-z.=o

1 1 2 2g 2g 1

F = - Ca. X2 + X4 - Cn = O2 ,+1

F3 =Ca¡XI +X3 - Cp =0

F4 =A¡(XI)X3 -A¡+I(X2)X4 =0

(12.72)

( 12.73)

(l2.74)

(12.75)

in which FI, F1, F3, and F4 are functions of XI' x2, X3, and X4, and A¡(xl)

and A¡+I (x2) denote that A¡ and A¡+I are functions of x 1 and X2 , respectively.Neglecting the second- and higher-order terms, F may be expanded in the

Taylor series as

{

aF aF aF aF }(O)F= F(o) + - D..xl + - ~X2 + - D..x3 + - D..x4 = O (12.76)ax 1 aX2 aX3 aX4

or


in which the derivatives aF/axl , aF/ax2, aF¡ax3, aF/aX4, and the functionF(o) are evaluated for the estimated values of x 1 to X4.

On the basís of Eq. 12.77, Eqs. 12.72 through 12.75 may be written as

x(o) (1 + k)x(o)~XI - D..x2 + -¿- ~X3 - g 4 D..x4 = -Ffo)

- Ca¡+I.1X2 + ~X4 = -FJo)

Ca¡~XI + ~X3 = -F~o)

B¡x~o)~xl - B¡+IX!O)~X2 +A¡~X3 - A¡+I ~X4 = -F~o)

(12.77)

(12.78)

(12.79)

(12.80)

(12.81)

In Eq.12.81, areas zí , and A¡+l are computed for x~o) and x~o), and it is assumedthat

The coefficients of Eq. 12.81 are much larger than the coefficients of Eq.12.78 through 12.80. To reduce their magnitude, both the left- and the right-hand sides of Eq. 12.81 may be divided by B¡. Thus, Eq. 12.81 becomes

Equations 12.78 to 12.80 and 12.82 are a set of four linear equations in fourunknowns-namely, ~x I to ~X4' These equations may be solved by anystandard numerica! technique, such as the Gauss elimination scheme.!? Then,

x~l) = x1°) + ~Xl

X~l) = x~o) + ~X2

X~l) = x~o) + ~X3

X~I) = x~o) + ~X4

x(o) ~x - B¡+l x(o).1x + A¡.1x _ A¡+I ~x = _ ]_ F(O)3 1 B¡ 4 2 B¡ 3 D¡ 4 B¡ 4 (12.82)

(12.83)

in which X~I) to X~I) are better approximations of the solution of the non-linear equations, Eqs. 12.67 through 12.70, than the initial estimated values ofx~o) to x~o). If I ~XI 1, r~x21 , IAx31, and I~X4 I are smaller than a specifiedtolerance, then xP) to X~l) are solutions of Eqs. 12.67 through 12.70; otherwise,

404 Applied Hydraulic Transíents

assume

X(o) = X(I)1 1

X(o) =X(I)2 2

X(O)-X(I)3 - 3

X(O)-X(I)4 - 4

(I2 .84)

and repeat the foregoing procedure until a solution is obtained. To avoid an un-limited number of iterations in the case of divergence, introduce a counter in theiterative loop so that the computations are stopped if the number of iterationsexceed a specified value, e .g., 30. To start the iterations, the first estimatedvalues of x ~O) to x ~o) may be taken equal to the known values al the beginningof the time step.

Stability Conditions

The finite-difference scherne presented aboye is said to be stable+? if smallnumerical errors due to truncation and round-off introduced at time (o are notamplified during successive applications of the difference equations, and theerror at subsequent time tare not grown so large as to obscure the valid part ofthe solution.

Using the technique presented by Courant et al. ,4 1 it has been shown 7,20,30

that the diffusive scherne is stable if

AX.ó.r,;;;;---

I VI±c(12.85)

This is called the Courant-Friedrichs-Lewy condition or simply the Courantcondition.

Computational Procedure

To determine the transient conditions, the channel is divided into n equalreaches such that, if the first section is called 1, then the last section will ben + 1 (see Fig. 12.8). Initial steady-state conditions (I.e., V,y, and Q) are com-puted at these sections. The time step, Al, is selected so that Eq. 12.85 is satis-fied. Equations 12.58 and 12.59 are used to determine yp and Vp at sectionnumbers 2 to n, and the special boundary conditions are used to compute Vp

and yp at the upstream and at the downstream ends, i.e., at section 1 and n + l.Thus, yp and Vp, al time t = O + Al, are known at all the sections, and the valueof Qp ís computed by multiplying Vp by the flow area corresponding to yp.


Sluice Reach lengfh, ll.X

Water sorrace

Section numbers : 1 2 :3 l+t n n+l

Figure 12.8. Division of channel into n reaches.

Now, assuming these computed values of Vp, YP, and Qp as V,y, and Q, thevalues of yp and Vp at time 2At are computed. This procedure is continueduntil the transient conditions for the required time are computed.

If there are two or more channels in the system, then the time step Al isselected for the shortest channel, and each remaining channel is divided intoequal-length reaches such that Eq. 12.85 is satisfied.

It is necessary that the Courant stability condition (Eq. 12.85) is satisfied ateach time step. If it is not satísfíed, then the time step is reduced (e.g., 0.75 ofthe previous value), and the conditions at the end of the time step are recorn-puted before incrementing the time. To avoid making Al too small in thisprocess, its current value is compared at each time step to the value required forstability, and, if perrnissible, Al is increased (e.g., by 15 percent) fOI the nexttime step.

12.10 INITlAL CONDITlONS

To compute the transient-state conditions, it is necessary that the initialsteady-state flow depth and velocity are known at aIl the sections of the systern.If these conditions are not compatible with the Sto Venant equations, thensrnall waves will be generated at each section as the transient conditions arecomputed. These hypothetical disturbances may rnask the actual solution of thesystem. To avoid this, either of the following procedures may be used:

1. Any initial conditions, such as zero velocity and a constant depth, through-out the system are assumed, and the boundary conditions are set egua! to theinitial steady-state conditions. Using the St. Venant equations, the system con-ditions are cornputed for a sufficient length of time until the variation of flowconditions is negligible and the conditions have converged to the steady valuescorresponding to the initial boundary conditions. The boundary conditions arethen set equal to the values for which transient conditions are to be computed.


2. The initial conditions are determined by solving the ordinary differentialequation describing the gradually varied flow in open channels.

The former procedure requires a large amount of computer time. In addition,in sorne finite-difference schernes, the solution may converge to incorrect initialconditions. Therefore, in the author's opinion, the second procedure should beused for determining the initial conditions. Details of this procedure follow.

The steady-state graduaIly varied flow in open channels is described by thefollowing equation: 1

dy So - SIdx Q2B1- -_

gA3

Equation 12.86 is a first-order differential equation. The flow depth along thechannel may be computed by integrating this equation. For this purpose, thefourth-order Runge-Kutta rnethod may be used as follows:

Let the depth of flow, y¡ at x = x., be known (Fíg, 12.9a); then the depth offlow at x e x.i , is

(12.86)

(12.87)

in which

al = Axf(x¡,y¡)

a2 =Sx ftx¡« tD.x,y¡+ tal)

a3 = Axf(x¡+ !D.x,y¡+ !a2)

a4 = D.xf(x¡ + D.x,y¡ + a3)

dyf(x,y) = dx

(12.88)

By starting from the known depth at a control section, and by repeated appli-cation of Eq. 12.87, the flow profile along the whole channel is computed insteps of length, D.x.

The aboye procedure is used to determine the water profile in a prismaticreach of a channel. However, at the junction of two channels, the depth in oneof the channels is initially known, and the depth in the other channel may becaIculated from the energy equation as foIlows:

Let us assume that, at the junction of channels i and i + 1, Y¡+I ,1 is known andY¡,n+1 has to be determined (Fig. 12.9b). Let the head losses at the junction begiven by k Vl+I ,1 /2g. In this expression, the fírst subscript represents the num-ber of the channel, while the second subscrípt represents the number of the


o er sur ace

:3 --- 1---_ ----- ->. Flolr +"-- ~

/ f/ / / / / /T 77x· '1

i+!

( o) Along chonnel

Energy grode line.....____2

kV i+I, I2g

Actual water 2Vi+I,l

--2g"-Flolr

z·1

"+c:

~water surrace

Cllannel i+1

Cllannel

(bl Junction of two chonnels

Figure 12.9. Notation for computing steady-state conditions,

section on the channel. Unlike the actual water-surface pro file , the computedwater-surface profile will have a sud den rise or drop at the junction because thelosses and the velocity head are assumed to change abruptly at the junction.

Energy equation at the junction may be written as

v~ V2l,n+1 ¡+I,IY¡,n+1 + ~=z¡ + Yi+I,1 + (1 + k) ~ (12.89)


in which z¡ = drop in channel bottom at junction i. In this equation, a rise isconsidered positive, and a drop as negative. Equation 12.89 can be written as

V7 V7=z·+· +(I+k)~- ~Y¡,n+1 I Y,+I,I 2g 2g (12.90)

and then solved by an íterative technique to determineY¡,n+l.

12.11 VERIFICA TION OF EXPLlCIT FINITE-DIFFERENCEMETHOD-D1FFUSIVE SCHEME

To verify the diffusíve scheme of Seetion 12.9, a mathematical model wasdeveloped using this scheme, and its results were compared with the prototypetest results. A brief description of the mathematieal model, prototype tests, andcornparison of the computed and measured results reported earlíer " by theauthor are presen ted in this section.

Mathernatical Model

The mathernatical model was developed using the equations derived in Section12.9. The following boundary eonditions were included in the model:

l. Flow or stage changes at the upstream or at the downstream end2. Constant-head reservoir at the upstream or at the downstream end3. Junction of two channels having different eross seetions, friction faetors,

and/or bottom slopes.

The model was designed to analyze transient conditions in a system having up to20 prismatie channels in series. As outlined in Computational Proeedure (Sec-tion 12.9), the value of t:.t was checked at each time step and its value was in-creased by 15 percent or deereased by 25 percent, so that the Courant's stabilitycondition was always satisfied and at the same time t:.t did not become too small.

Prototype Tests

Prototype tests were eonducted in 1971 on the Seton Canal owned andoperated by British Columbia Hydro and Power Authority, Vancouver, Canada.A brief description of the projeet and the tests is given below:

Project Data

The Seton Canal, concrete-lined throughout its 3820-m length and designed fora flow of 113 m3/s, conveys water from the Seton Lake to the Seton Generat-

0··'5

0·6 .. 8 "DI$

8·;:t-il·D/S6lZ1 'O/s

O"fJI$


'" :'"' s.. .¡¡

"'''', ... .E:J:,E ..,,,'" .o.!! ~:;:~u- ~"IDen ...

.2 e

'"o

or¿,..__ , .~ ~ED ..",'

• 6~¡;:: ~ :: "'" " D...

--./ -C¡>e .."'':: r; ~ "') O,,-0"0 ~ ...:~; "011.... .0'"'.!! ..

C>.", =• B

ai 011N "'-l

w ..011!!! .....,

, o ;S~ w .... ;;;

~

..... D;I " ".!! O;;¡ .2O; ~~ 011'--...¡ r~,--j . Uo iEen ....en ... O... ~

.... .. '" C>.

~ :¡ ~ :¡ "':l.. ;;¡ ;;¡ ;;¡ ¡;~ ¡;i¡j O

N ¡¡:U )0; OJ, O~U ...." " o N.2 D-O; :::N.. U", ~-

~~C/)

~~ ~ :: ,;

~ ;;;... .. 'i ...!! :¡ '" :¡"O ¡¡j i¡j ;;¡e•.cUen


ing Station. The alignment and typical cross sections of the canal are shown in

Fig.12.10. . .The Seton Generating Station has one 44-MW, vertical reactíon turbine rated. at

44.8-m net head with the centerline of the unit at El. 187.45 m. The effectivewicket-gate opening and closing times are 15 and 13 s, respectively.

While starting the turbine from rest, wicket gates are opened to 15 percent(breakaway gate). At this opening, the turbine begins to rotate, and the gatesare then closed to speed-no-load gate of 9 percent. The wicket gates are kept atthis opening until the unit is synchronized to the system.

Tests

Transient-state conditions in the canal were produced by accepting or rejectingload on the turbine. The following tests were conducted:

l. Acceptance of 44 MW2, Rejection of 44 MW .3. Acceptance of 44 MW followed by rejection of 44 MW after 42 mm and

then acceptance of 44 MW after 37 mino

After total-load rejection , turbíne gates were kept open at speed-no-load gate

(approximately 9 percent).

Instrumentation

To record water-level changes, a special gauge was developed. This gauge con-sisted of a 12.5-mm-diameter vertical aluminum tube, sealed and weighted at thelower end. The tube was suspended inside a 100-mm-diameter tube from acantilever. The outer tube was open at both ends and was fastened to the canalbank in a vertical posítion. Strain gauges were installed on the cantilever to de-termine its deflection. Water-level fluctuations changed the buoyant force onthe bottom of the inner aluminum tube, thus resulting in a change in the deflec-tion of the cantilever, which was picked up by the strain gauges and recorded ona brush-recorder. The gauge was calibrated prior to the tests.

In addition to continuous recording, the transient-state water levels were alsoread at intervals of 30 s to 1 min from the gauge plates attached to the verticalor inclined síde walls. Countdown for the start of the test was given over aVHF ¡UHF radio, In case there was any disagreement between the recorded andobserved water levels, the latter were assumed to be more reliable.

To determine the water levels, five stations were established along the lengthof the canal. The location of these stations is shown in Fig. 12.10.

Transient Flows in Open Channels 41 1


The observed and the computed water levels at various statíons are shown inFigs. 12.11 and 12.12_ In the computations, the canal was represented by sixchannels, each having' a constant cross section along its length. The Manningformula was used to calculate friction losses, and the load acceptance or rejec-tion on the turbine was simulated by assuming a linear discharge variation at thedownstream end of the canal. Seton Lake was represented by a constant-headreservoir at the upstream end.

The secondary fluctuations of the water surface (Favre waves) could not becornputed by the program because of the inherent limitations of the governingequations (Eqs. 12.31 and 12.44). Therefore, lo determine the maximum waterlevels, the amplitude of the secondary fluctuations were computed using datapresented by Benet and Cunge'" and were superimposed on the maximum-waterlevels computed by the programo The rnaximurn level of the computed surgefluctuations is marked in Fig. 12.l2a-c.

It is clear from these figures that the computed and the measured results agreeclosely for water levels following the inítial surge and for maximum water levelof the secondary fluctuations at the upper end of the canal. The computedresults for the maximum water level of the fluctuations at Station 3 are, how-

236.0

-....E;'- 235.5

<::~~-!!.. 235.0~~...:;,..... 234.5......~

234.0

- ---- ~--~.---- f-~--

'1-----

-Measured

\\\

\= ......fcomputed

~ <,-- ......:::,

~~

~

o 6 12 159 18 21 24 27 30 33

Time afler load acceptance (min)

Figure 12.11. Comparison of computed and measured transient-state water levels at StationS Collowing44-MW load acceptance.

412 Applied Hydrau1ic Transients

"' 237.0

~<::

~~ 236.5,...!!..

I rTop of secondaryI fluetuations (eompuled)

----t--- I=-'"="--1 . ___j_ __-+_--II--_~

~ hJlA& ~~I~~ 236.0 +-----,f----+---.I------t--,IJIH'-'I-!-ILX!~ ..__....::;;:'"-;);~=F:::::~,--.....¡",::-¡¡¡d

~~ ....-----

235.5

27o 6 9 2412 15 18 21

Time afler load rejee/ian (mili)

(a) Al st otion I

ITop of seeondaryflue/uafions (eompuled)

30

Figure 12.12. Comparison of computed and measured transient-state water levels foUowing44-MWload rejection.

~ 237.0+------+------+---,~'-

<::

.~~ 236.5+-----+----~I\_---__Ir------+--_1..~~~ 236.0 +-----+-----Hl-,t/-----l--=~:::..:..-l--_!

M,asured

235.5+-----+-

o 3Time arter load rejee/ion (min)

15 18 21 24 27126 9

(b) Al stalion 2

36


237.0

-...1;'- 236.5

<::.~>..!!.... 236.0

"~...::r...... 235.5..~

2350

O 3 6

236.5

-...1;'-

236.0<::.!?b'"-!!.... 235.5

"~...:::o...... 235.0~~

234.5

4-----4--4----

---~---- t----+----+--I

I \Time af/er lood rejeelion (mili)

(e) Al slalion 3

--- ---_._--_ l---- '"'~

~V ~~

Measure~~\

~./-~.,_

K~ ~

·1\7 "--- Compu/ed

1,,7IL_

,11, li,11

!!1I!¡

o 3 6 9 12 15 18 21 24 27 30 33 36Time off" lood rejeclian (min) ! !¡

(d) Al station 4:1Figure 12.12. (Continued)


2370~--- --- .---- -~-r_~~~,--~-____,

<::~ 236.0~~..~ 235.5

~..~¡235.0+t-~-

.._--+---+-----j,

!I

~

234.5

3 6 9 ~ m m a H v ~ ~ Brime alter load re/eelion (min)

(e) Al slalion 5


o

ever, too high, because the fluctuations were developed in the prototype as theinitia! wave propagated upstream.

I

1

12.12 IMPLlCIT FINITE·DIFFERENCE METHODS

Description

In the explicit finite-difference method , we replaced the x-derivatives of theSt. Venant equations by finite-difference quotients evaluated at time lo (Fig.12.6). The other coefficients, B and SI' were also evaluated at time to' Thus,we had two linear algebraic equations relatíng the two unknowns, V and y, atpoint P to the known conditions at pointsL,M, and R. Because of this explicitrelationship, the method was called explicit finite-difference methad.

In the finite-difference method presented in this section, we wilI replace thex-derivatives in terms of finite differences eva!uated at time to + t:.t; thus, theunknowns will appear implicitly (hence, the name, implicit finite-differencemethad) in the resulting algebraic equations, which are usually non linear . Solu-

,1

",,

1

1I

1\

!)

1

415

tion of these algebraic equations is more complex than that in the explicitmethod. However, the implicit method has the advantage that it is uncondi-tionaIly stable. This allows the use of larger values of the computational timestep, t:.t, thus economizing computer time. We will compare the advantages anddisadvantages of the explicit and implicit methods in the next section.

Available Implicit Schemes

Various implicit schernes have be en reported in the literature. Of these, the fol.lowing have been used for studying the open-channel transients: Priessmann'sscheme,'.24 ,26 Amein's scheme,9.32 Vasiliev's scheme,43.44 and Strelkoff'sscheme ."

While writing the finite-difference quotients for the x-partial derivative, aweighting factor, Q, is introduced in the Priessmann's and in the Amein'sschemes, The presence of Q in the difference schemes introduces artificial damp-ing in addition to the damping due to friction and other losses. To make thescheme stable, the value of Q must be greater than 0.5 and less than or egual lo 1.Since the value of Q is arbitrarily selected, care must be exercisedso that an ex-cessive amount of damping is not unintentionally introduced into the systern.Vasiliev's scheme has two steps and thus is more complicated and requires morecomputer time. Compared to the preceding schernes, in the author's opinion,Strelkoff's scheme is the simplest and is easy to programo In addition, the com-parison of the results computed by using this scherne with those measured onthe prototype and on a hydraulíc model has shown good agreement (see Section12.15). Details of this scheme are presented herein; readers interested in theother schernes should refer to Refs. 7,9,24,26, 30, 32, 43, and 44.

Strelkoff's Implicit Scheme

In Strelkoff's scheme."? the partial derivatives of the St. Venant equations arereplaced by finite-difference quotients as follows (see Fig, 12.13a):

aQ Qp¡+l - Qp¡-l-=ax 2t:.x

ay Yp¡ - Y¡

al t:.t

Qp¡- Q¡(12.91)aQ

at t:.t

ay YP¡+l - YP¡-l

ax 2t:.x


• Kno

~

.:.x

* Unknown conditionswn conditions

i -1 1+1 x

(a) Interior seelions

2 x

(b) Upslream boundary

~.__

óX

x n n+1

(e) Downslream boundary

Figure) 2.13. Notation for implicit finite-difference scheme.


lo this equation, the subscript P refers to the variables at time to + 6.t. Allcoefficients of the equations are evaluated at time to except the friction slope,Sr, which is evaluated at time to + D.t. However, if we use the value of SI at time(o, then the implicit scherne is not unconditionally stable.P? and the condition,6.( < Kol(Ag$o), has to be satisfied for stability requirements. lo this expres-síon, K¿ = conveyance factor at normal depth. If the Manníng formula is usedlo compute SIp' and R = hydraulic radius, then

S - QplQpl{p - Kj,

The resulting algebraic equatíons will be nonlinear if the expression for Slpgiven by Eq. 12.92 is used. We can linearize Eq. 12.92 as follows:

as/¡ aS/1 aKISlp; ~ SI; + aQ ¡ (Qp¡ - Qi) + aK ¡ ay ; (YPi - y;)

in which the cooveyance factor, Kp, in the SI uoits* may be written as

x; =]_ AR2/3

1n y=yp

(12.92)

(12.93)

(12.94)

Note that aSlloQ and aSllaK in Eq. 12.94 are evaluated at time {o, and the ex-pressions for these derivatives may be obtained by differentiating Eq. 12.92, i.e.,

aSI 2S{-=-aQ QaSIaK = -2SIIK

aK = dK =!5. (5E _ 3.. R dP)ay dy A 3 3 dy

lo Eq. 12.97, dP/dy is the rate of change of the wetted perimeter, and its valuedepends upon the side slopes of the channel. As the derivatives are evaluated attime lo, subscript P is dropped in Eqs. 12.95 through 12.97.

Replacing the partial derívatíves of Eqs. 12.31 and 12.45 by the finíte differ-ences listed in Eq. 12.91, replacing SI of Eq. 12.45 by the expression given inEq. 12.94 and substituting expressions for aSllaQ and aSflaK from Eqs. 12.95and 12.96, and simplifying the resulting equation, we obtain

"In the English units, Kp = (1.486/n) AR2/3Iy=y p'

(12.95)

(12.96)

(12.97)

(12.98)


e¡Qp¡+¡ +!¡Qp¡-e¡Qp¡_¡ +k¡yp¡+¡ +I¡yp¡-k¡yp¡_l =D¡ (12.99)

In these equations, the variables with subscript Pare unknowns, and the coeffí-cients a, b, C· e. r. k, ·1· and D,' are evaluated in terms of the known quantí-

l' l' l' "J p " l'ties at time too The expressions for these coefficients are

1 (12.100)a¡ = 2Áx

B· (12.101)b, =.....!..1 Át

By· (12.102)C. =_'_'I Át

V¡ (12.103)e·=-1 ÁX

1"

1 2gA¡SJ¡(l2.lO4)

1:

[; = M+ Q¡

gA- - B·Vf(12.105)k. = 1 1 1

1 2Áx

1.=- 2gSJ¡A¡(dk) (12.106)1 K¡ dy ¡

{ [~Y' (dK)]} Q. (12.107)D¡ = gA¡ So + SJ¡ 1 -~: dy ¡ + t:.~

For the upstream boundary, the negative characteristic equation (Eq, 12.47 wrít-ten in terms of Q instead of V)* may be written as

(12.108)

in which

1 AISJI VI -e¡a =-+2g---¡ t:.t Q I t:.x

(12.109)

VI - e¡b 1 = --'---'-

Áx(12.110)

*íaQ + (V _ e) aQ1_ (V+ c)B [ay + (V - e) ~Y1=gA(So - SJ)[al axj al uxJ


el = -(VI +c¡)B¡ + -,-(V--,-r_-_e...:cr..:..,)B.....;:_¡_ 2g[AIS{, (dK)]Át Áx K¡ dy ¡

f - -(n - cDB¡¡ - t:.x

D¡ =gA¡ {So +S [1 - 2YI (dK) J} .e.. (V¡ +c¡)B¡y¡JI K 1 dy ¡ Át Át

(12.112)

(12.111 )

(12.113)

In the preceding equations, the variables with subscript P refer to the unknownconditions at time to + Si, and the coefficients a¡, b¡, e¡,!¡, and D¡ are ex-pressed in terms of the known conditions at time too In addition, the conditionsimposed by the upstream boundary may be written in a generalized form as

(12.114)

If the conditions imposed by the boundary are nonlinear, these are linearizedand expressed in the aboye formo Usually, either El is zero and F¡ is unity, orF¡ is zero and El is unity.

Proceedíng similarly for the downstream boundary at section (n + 1), the posi-tive characteristic equation (Eq. 12.46 written in terms of Q instead of V)*may be written as

an+IQpn+bn+IQPn+¡ +en+¡YPn+!n+¡YPn+1 =Dn+¡ (12.115)

in which

_ (Vn+¡ + cn+¡)an + ¡ --

t:.x (12.116)

(12.117)

(12.118)

r = - (Vn+¡ - cn+¡ )Bn+¡Jn+l Át

(V~+¡ - e~+¡)Bn+1Áx

_ 2g[A SJn+1 (dK) ]n+l Kn+l dy n+l .

(12.119)

* [:~ + (V+ e) :~J-(V - C)B[:~ + (V+ e) :~j=gA(So - Sr)

420 Applíed Hydraulic Transients

(12.120)

The conditions imposed by the downstream boundary may be written as

(12.121)

If the channel is divided into n reaches, then there are (n + 1) sections and sincethere are two unknowns (namely,yp and Qp) for each section, the total numberof unknowns is 2(n + 1). For a unique solution, there must be 2(n + 1) equa-tions. As discussed above, two equations (Eqs. 12.108 and 12.1 14) are providedby the upstream boundary, two equations (Eqs. 12.115 and 12.121) are pro-vided by the downstream boundary, and there are two equations (Eqs. 12.98and 12.99) for each interior section i, i = 2, n. Since there are (n - 1) interiorsections, we have 2(n - 1) + 2 + 2 = 2(n + 1) equations. Hence, a unique solu-tion can be obtained for each time step. These equations for a system may beexpressed in the matrix notation as

Ax =b (12.122)

in which A is a coefficient matrix, x is a column vector comprised of the un-known quantities Qp and YP, and bis a column vector comprised of the eon-stants on the right-hand side of Eqs. 12.98,12.99,12.108,12.114,12.115, and12.121. Equation 12.122 may be solved by any standard numerical technique ,such as the Gauss elimination technique.P"

A close examination of matrix A shows that the nonzero coefficients lie nearthe diagonal, and thus A is a banded matrix. Special standard computer pro-grams are available for the solution of such a system of linear equations. Suchprograrns not only save computer time and storage requirements, but they alsogive more aecurate results. We used a subroutine called GELB developed by IBMfor the case study presented in Section 12.15.

Systems Having Branch and ParalIel Channels

The coefficients matrix, A, is banded only for a system of channels conneetedin series. For a branching system (Fig. 12.14), a number of coefficients do notHe on the diagonal, and the proeedure for solving a banded matrix cannot beused. In sueh cases, however, if the channel seetions are numbered as shown inFig. 12.14, the resulting matrix is a banded matrix.45 Note that the numberingof the channel sections on the branch channel increases in the upstream direc-

Transíent Flows in Open Channels 421

• Series ¡vne/ion* Bronehing [unctian

~ Section nvmber

i+4 i+6 i+IO i+12

i+8 i+14

Main channel

Figure 12.14. Designation of channel sections on a branch system.

tion, and the difference between the consecutive section numbers of a channelis 2. These faetors have to be properly taken into consideration in thecomputations.

Símilarly, a matrix for systems having parallel channels (Fig. 12.15) is notbanded. For such systems, a procedure46 may be used in which the matrix A isfirst reduced to the upper triangular forrn, and then back substitution is done tosolve the system of equations; or the sections may be numbered as shown inFig, 12.15, which results in a banded matríx."!

• Channel junction

i+4Section number

i+7 i+9

1t

i+17. 18' Main cl10nnel1+ 1+19 i+20

i+16"et-....__-+----+r i+ 14

i+IO i+12

Figure 12.15. Designation ot channel sectíons on a parallel system,

Moin chonnel


Stability Conditions

Unlike the explicit scherne of Section 12.9, the implicit scherne presented in thissection is unconditionally stable, i.e., any arbitrary value of !::.t may be used inthe computations without making the solution unstable. This fact can be

utilized as follows:l. The channel need not be divided into reaches of equal length. (Note that

Eq. 12.91 and hence the coefficients of Eqs. 12.98 and 12.99 will be modified ifthe value of ~x is not the same for all reaches of a channel.)

2. The size of the time step may be varied during the computations, i.e., whenthe conditions are varying rapidly, a smaller value of ~t may be used to increasethe accuracy of the results, and, when the conditions are varying slowly, the sizeof the time step may be íncreased.

Note that the computational time step, ~t, cannot be arbitrarily increased eventhough the difference scherne is unconditionally stable. Actually, the size of thestep is selected taking into consideration both the accuracy of the results and thestability of the scheme. If too large a value of time step is used, then the finitedifferences no longer approximate the partial derivatives of the original equa-tions. Particularly, if there are sharp peaks, then these are truncated if a largevalue of ~t is used. Therefore, while investigating a particular case, the accuracyof the results should be checked by reducing the size of ~t and determining thedifference between the computed results obtained by usíng larger and reducedvalue of ~t. If this dífference is negligible, then a larger value of !::.t rnay be

used.

12.13 COMPARISON OF EXPLlCIT AND lMPUCITFlNITE-DlFFERENCE METHODS

The advantages and disadvantages of the difference methods presented previouslyare compared in this section.

l. Stability. Courant's stability criterion (Eq. 12.85), i.e., ~t ,;;;;~x/(I VI ± e),must be satisfied in the explicit method. There is no such restriction on ~t inthe implicit method.

2. Ease of Programming. The explicit method is easier to program than theimplicit method. Therefore, when the time available for developing a program islimited, the explicit method should be used.

3. Eeonomy. Because the size of ~t for an implicit scheme is not restrictedby any stability criterion, a larger value of ~t is permissible, thus requiring lesscomputer time as compared to the explicit scheme in which ~t is restricted bythe Courant's stability condition.


. 4 .. ~omputer-M_emary Requirements. The computer storage required in an~mphclt method IS usually more than that required in an explicit method. Thus,if the stor.age ca~acity of the computer is limited, one may have to use magnetictapes ~r dises to mcrease the effective storage for the implicit method. This mayresult m more computer time than that required for an explicit scherne becauseof reading and writing on the tapes during each time step even though a largervalue of ~t may be permissible in the implicit method.

5. Simulation of Special Cases. Explicit methods are not suitable for con-duits having closed tops in which the transient-state water surface either primeso~ reaches the top of the conduit. In such a situation, the free-water-surfacewidth becomes zero or very small, and the size of ~t has to be reduced to a verysIl_1allvalue. Examples of such cases are sewers and tailrace tunnels in hydroelec-tnc power plants. The implicit method should be used for the anlaysis of thesecases.

6. Simulation af Sharp Peaks. Because of the usually smaller size of M theexplicit methods are. more suitable for the analysis of transients in which sharppeaks of short duration occur. In the implicit methods, such peaks are usualiysmoothed out. If, however, time steps of the same size as that in the explicitmethod are used, the peaks would be reproduced, but the computer time re-quired in the implicit method would be greater than that in the explicit method.

7. Formation of Bores and Shocks. The explicit method is more suitablethan the implicit method for the analysis of transients in which a bore forms.Lax-Wendroff's two-step explicit scherne 7.31 presented in Chapter 9 may beused to obtain more accurate results.

12.14 SPECIAL TOPICS

A number of special topies are discussed in this section.

Dam-break

The investigation of the dam-break problem is similar to the analysis of open-channel transients in which a bore forms. Stoker"? studied it for those cases inwhich there is water in the channel below the dam. He assumed an instant boreformatíon and a rectangular, horizontal, and frictionless channel. Dronkers'"used ~he method of characteristics to determine the gradually varied flow onboth sides of the bore and used the conservation equations at the bore. Amein"?studie~ this problem both analytically and experimentally, and Shen so verifiedStoker s theory by conducting experíments on bores created by piston move-ment or by suddenly opening a gateo Terzidis and Strelkoff''" showed that the


diffusive scherne presented in Section 12.9 and a two-step Lax-Wendroff-Richtmyer explicit scheme can be used for the analysis of flows in which a boreforms without isolation of the bore. In these schernes, the positive and negativecharacteristic equations (Eqs, 12.50 and 12.53) are used at the boundaries if theflow there is subcritical, and both the depth and discharge are specified if theflow is supercritical. (lf a shock forms, only one of these is specífied; ~he jumpconditions yield the other.) Comparison of the computed results usmg theseschemes with the experimental results showed good agreement. It was alsoshown that the solution of St. Venant equations in their usual form, withoutinclusion of special dissipation terms, gave erroneous results when applied toflow with bores of height greater than one-half the flow depth. For a bore ofsmaller height, however, the S1. Venant equation yielded satisfactory results.Martin and Zovne " showed that the propagation and reflection of bores from asolid wall can be analyzed using the diffusive explicit scherne, and that the iso-lation of the bore and its treatment as an internal boundary is more of an aca-demic interest than required in real-lífe practical applications.

In the darn-break investigations, the size of the breach and the time in which itoccurs when the dam fails have to be assumed. Instantaneous faílure of the totaldam has be en used by a number of investigators. Such an assumption , however,appears to be rather unrealistic. .

When a dam fails, a positive wave propaga tes downstream, and a negatíve wavetravels upstream in the reservoír. Whíle using a finite-difference method, theboundary conditions at the dam site have to be developed. For thís purpose, therelationships (given in Chapter 15 of Ref. 7) for various sizes of the opening inthe dam rnay be used.

Tidal Oscillations

Ríverflows influenced by tides can be analyzed by soIving the S1. Venant equa-tion using the finite-difference methods. The main decisión to be made in suchinvestigations is whether a one-dimensional model can be used or noto Two-dimensional models are necessary if the flow conditions cannot be representedby one-dimensional Ilows, .

Various finite-difference schemes have been used successfully for the analysisof tidal flows: the U.S. Geological Survey!": I 9 used an explicit method based onthe characteristíc equations on a rectangular grid and an implicit method. Corn-parison of the cornputed results with those measured on the Three-Mile Slough,California, and on the Delaware River showed satísfactory agreement. TheNational Research Council of Cana da used an explicit leapfrog scheme and animplicit scherne'" to investiga te the effects of various channel ímprovements onnavigation in the S1. Lawrence River and estuary. The upstream boundary


was a discharge hydrograph, and the downstream boundary was a tidal stagecurve. Even though the variation of the channel geometry was rather large (thewid th varies by a factor of 75, and the cross-sectional area by a factor of 4000),it is c1aimed that the one-dimensional rnathernatical models based on the afore-mentíoned finite-difference schemes could be calibrated to yield results close tothe observed results.

Unlike the analysis of transients in power canals, rivers, and so on, it is notnecessary to determine the initial conditions while studying tidal oscíllations.Any initial conditions are assumed, and the system is subjected to tidal cycle anurnber of times until periodic flows are established. This is similar to the inves-tigation of the steady-oscíllatory flows in pipes by the method of characteristics(se e Section 8.4).

Secondary Oscillations or Favre's Waves

We discussed in Section 12.4 that the wave front of a positive wave becomessteep as it propagates in a channel. If the energy of the wave is large , then thewave breaks and beco mes a moving hydraulic jump or a bore (Fig. 12.16a).However, if the energy of the wave is not large, then the wave front assurnes anundular form as shown in Fig. 12.l6b. In other words, this is similar to a sta-tionary hydraulic jump in which a strong jump is forrned for Fraude numberF> 9 and an undular jump for F < 1.7. These secondary water-surface oscilla-tions are called Favre waves,52 as Favre described them first in 1935.

In the case of a bore, the discontinuity in the water surface occurs over a veryshort distance, and the flow may be assumed to be gradually varied in front ofand behind the bore. The length of this discontinuity is usually small comparedwith the reach length into which the channel is divided. Sin ce the Sto Venantequations are valid on both sides of the bore, quite satisfactory results are ob-tained using these equations, as long as one is not interested in the wave frontitself. However, the situation is quite different if the wave front has secondaryoscillations. In this case, the water surface has undulations for a long distance,the assumption of hydrostatic-pressure distribution is not valid, and the waterlevels computed by usíng the St. Venant equations are the average levels andnot the maximum levels. In addition, experience has shown that these oscilla-tions are higher near the banks than in the middle of the channel. Figure 12.17show the secondary oscillations of the water surface near the wave front. Thesephotographs were taken during the prototype tests conducted on the SetonCanal in British Columbia. (For a description of these tests, see Section 12.11.)

To design power canals or other channels in whích the wave fronts have secoondary oscillations, the maximum water level near the banks should be known.However, very Iittle data on these waves have been reported in the literature.


Groduolly voriedflow Bore

Groduolly variedflowI

I II I

Flow I I- I II I

7 7 l 7 l 7 7 ) / l ;, 7

( a)

llll77l

Groduolly voriedtlow I

Favre woves

Grodually variedflow

7 l l l l 7lll7l

III

7 } l l l 7 l 7 l

( b)

Figure 12.16. Variation of water surface at wave front.

Figure 12.18 taken from Ref. 42 may be used to determine the approximateheight of these waves.

Free-Surface-Pressurized Flows

Free-surface flows that may pressurize the conduit during the transient-stateconditions are called free-surface-pressurized flows. Such flows may occur insewers or in the tailrace tunnel of a hydroelectric power planto

Meyer-Peter+' and Calame i" studied this type of flow while investigatingsurges in the tailrace tunnel of the Wettingen Hydroelectric Power Plant. Theircomputed results were in close agreement with those measured on a hydraulic


..~

. >" ..~.)·h....!~'~':.;::{"\J~'~!::'.t~~;~~;jj ~""~....'. J '~':: "0" .. ;.;~ ':. '. ~ :-:-!~:~:'~'::""".l,~.·!.Jo •• .,. \ ..,:.:!pw" ,!~.~.,I :~"'~~

.• ·f_~~.C~·~~~,.,:.. ,~,.". ,,:"'~:~"'~:'.~'.t~::t/,..Figure 12.17. Seton Canal, secondary oscillations at wave front. (Courtesy of BritishColumbia Hydro and Power Authority, Vancouver, Canada).


_o. .~. ';"'~ .'~~._:~r'~3:f~~..,.,~..~.'l."H

Figure 12.17. (Continuedv


Steady stotewater level

(o) Nototion

i~¡¡;

I~~'1"

3,0 .------.-:--0..,-::-:------,----..-----.-1----,fO.. Wilhouf breakingPhysical model ~

Wilh bre,aking •" Canal d Oraison ,-,: Sogreohs

/Canal JouQue-SI. ¿;/ Met/Jod . -j

B "": 0'0:5(J;}_;;-D o 1_'·.'.-ii~ o" /' o

A o o o . ~:-;~ ...-000 °0

2,0 1- "---1_~~4i~ o(~ 4~>~'0. ~_o__o__ ......-."---1-----1

"~? o " ..iA~'

,.67~pV o o,. 00

1,5 I----<l--+----+------II----<>----"' ..+---.a----t·-+---,---i..

o2,5~~----

..

o

....

•.... ........ ..o ..

" "1,0 O 0,1 0,2 0,3 0,4 0,5 0,6

zy

( b)

Figure 12.18. Amplitude of secondary oscillations. (After Benct, F., and Cunge, J. A.42).

model. In 1937, Dríoli'" reported his observations on the translation of wavesin an industrial canal. Jaeger'" discussed this problem and presented a numberof expressions for various possible cases. Príessmann.i" Cunge,"? Cunge andWegner.P" Arnorocho and Strelkoff.t" and WiggertS9.60 studied sueh flows usingdigital computers.

To faeilitate eomparison, let us write the equations describing the transient-state flows in open channels and in cJosed conduits:


1. Open channels:a. Continuity equation

ay ay A av-+ v-+--=Oal ax B ax (12.28)

b. Dynamic equation

ay av avg-+-+ V-=g(So - Sr}ax at ax (12.44)

2. Closed conduits:a. Continuity equation

sn sn a2 av-+ V-+--==Oal ax g 3x(12.123)

b. Dynamic equation

sn av avg-+-+ V-=g(So - Sr)ax al ax (I2.124)

in which H = piezometeric head and a = waterhammer wave velocity.Comparison of Eqs. 12.28 and 12.123, and 12.44 and 12.124 shows that ~hese

equations are indentical if the depth of flow, y, is assumed equal to the piezo-metric head, H, and if a = YiATB == e, in which e == the celerity of surface waves.We can analyze pressurízed flow by solving the St. Venant equations by using anin teresting technique conceived by Príessmann.i" In this technique, a very nar-row slot is assumed at the top of the conduit (Fig. 12.19) such that this slot doesnot increase either the cross-sectíonal area or the hydraulic radius of the pressur-ized conduit. The width of the slot is selected such that e == a. Thus, the free-

HypolhtlficOIstot

I,II

R--I A. II ~-,~e \

\ )1//»>

A,R,e

Free flow Conduit cllorocteristics Pressurized flow

Figure 12.19. Hypothetical slot for analyzing free-surface-pressurized l1ows.

['f-


surface and the pressurized flows do not have to be ana1yzed separately. Oncethe conduit primes, then the depth.y, determined by using the St. Venant equa-tion, is the pressure head acting on the conduit walls at that location. Thís tech-nique has been successfully used for the analysis of storm sewers 7 and for theana1ysis of surges in the tailrace tunnels of a hydroelectric power plant (see Sec-tion 12.15).

Landslide-Generated Waves

If a landslide occurs into a body of water, waves are generated due to displace-ment of water and due to impact of the landslide. These waves, sometimesreferred to as impulse walles, have caused destruction"!" 6 5 and loss of humanlife. For example, the waves generated by the Vaiont slide, in Italy, killed about2300 people.

Landslide-generated waves have been studied on two-dimensional hydraulicmodels by Wiegel,66 Prins,67 Law and Brebner,68 Kamphuis and Boweríng.t"Noda,"? Das and Wíegel,"! and Babcock 72. Based on the model results. theseauthors presented empirical relationships or graphs for determining the charac-teristics of these waves such as initial wave height, wavelength, and type.

Three-dimensional hydraulic model studies were conducted to investigate thewaves generated by the movement of slides into reservoirs created by theMica,73 Líbby, 74 and Revelstoke 75-77 dams. The diffusive scheme of Section12.9 was used77,78 for the propagation of slide-generated waves approximately67 km along the reservoir in both the upstream and downstream directions fromthe slide site.

The empirical relationships given by Kamphuis and Boweríng'" are presentedherein (for similar relationships derived by others, see Refs, 66 through 72).These were derived from data for waves generated by loaded trays sliding downan inclined roller ramp into a 45-m-long, 1-m wide flume. The slides were simu-lated in the direction of the longitudinal axis of the flume from various heightsand various slide angles. The waves were measured at three locations on theflume. It was found that the waves became stable at or upstream of a pointlocated about 17-m from the point of slide impacto

The following equations were presented:

l. Maximum height 01 the stable wave:

H-1:==Fo.7(0.31 +0.210gq)d

(l2.125)

in which

He = maxímum stable wave height aboye the still water leve!d= water depth


VF=_s_.,[id

Vs = slide velocity upon irnpact with waterg = acceleration due lo gravity

1 hq = slide volume per unit width in dimensionless forrn, - . -

d d1 = length of slide up the slope

h = thickness of slide normal to the slope.

2. Wave-height attenuation:H H -O.08X-=~+0.35e dd d

(12.126)

in which

H = maximum wave height aboye still water level at distance x frorn the slíde;and

x = distance downstream frorn the point of slide irnpact.

For a given slide, the maxímum stable wave height, He, can be determined fromEq. 12.125, and height, H, at any point downstream from the slide impact canthen be computed by using Eq. 12.126.

3. Ware period:TI X

- = 11 + 0.225 -.,[id d(12.127)

in which

TI = period of the first wave (i.e., the time required by the wave to pass anyone point).

During the experiments, the waves were produced by small slides falling fromaboye the water level. These waves varied from apure oscillatory wave traín to awave approaching a solitary wave followed by an oscillatory wave train. Nobores were formed. The wave height became stable relatively quickly, Kamphuisand Bowering state that Eq. 12.125 gives a good estímate of the stable waveheight for 0.05 :¡;;; q :¡;;; 1.0, as long as the slide is thick (i.e., h/d > 0.5); the frontangle, ~, of the slide is 90° or greater; and the angle of the slide plane, 8 is about30°. However, the wave heights determined from Eq. 12.125 are higher for~< 90° and 8 > 30° but low for 8 < 30°.

Raney and Butler 79 have developed a two-dimensional mathematical model todetermine the characteristics of the slide-generated waves, Cornparison of thecomputed results with those measured on a three-dimensional hydraulic model?"


show satisfactory agreement. The finite elernent method+? is used to predictlandslide-generated waves in a one-dimensional reservoir.

12.15 CASE STVDY

In this section, studies carried out to determine the operatíng guidelines neces-sary to keep the surges in the tailrace system of the G. M. Shrum HydroelectricGenerating Station below a critical level'" are sumrnarized. A brief descriptionis presented of the mathematical model, its verification, and data for surgescaused by various loading and unloading operations. Based on the results of thismodel, the power plant operatíng guidelines are then summarized.

Project Details

The tailrace system of the G. M. Shrum Generating Station, owned and operatedby British Columbia Hydro and Power Authority, Vancouver, Canada, is corn-prised of two manifolds, two tunnels, a tailrace channel, and a rockfill weir.Units No. 1 to 5 discharge into Manifold No. 1, while Vnits No. 5 through 9discharge into Manifold No. 2 (as will Unit No. 10 when it is installed). Waterfrom each manifold is carried to the tailrace channel by a concrete-lined tunnel(see Fig. 12.20). Essential data for the tailrace systern are listed in Table 12.1.

Surge levels aboye the tailrace manifold deck (El. 510.5 m) would cause both awater-pressure loading and an air-pressure differentíal across the doors, whichmight lead to their collapse and the subsequent flooding of the powerhouse ,Therefore, a mathematical model was developed to determine what, if any,operating restrictions should be applied during periods of high tailwater levels.

Mathematical ModeJ

Description

The dynarníc and contínuíty equations derived in Section 12.5 were solvednumerically using Strelkoff's implicit finite-difference scheme (presented inSection 12.12). Because of the branch tunnel, the channels were nurnbered'i! asshown in Fig. 12.14 so that a banded rnatríx was obtained. The subroutineGELB, developed by 18M, was used to solve the systern of linear equations.Boundary conditions for the manifold, junctions of two or three channels, andthe downstream weír were developed and incorporated in the model. As dis-cussed in the last section, the primed tunnel was simulated by the open-channel-flow equations by assuming a very narrow vertical slot at the top of the tunnel.This slot allowed the depth of flow to exceed the tunnel height, thus represent-


I!~ .!::

~

~ ..~.. u.. I

" ~u".".

~2~!Ii ... u~ ..., 11)

2~

~~!r

I! . e., ,,",o... ._

~ Oi ....... ü ,.. ¡;; ~.. ~ 11)~ ...~ d!1 "~ .,~...

c::O

''¡::

:!CIlbO=;lO....=..

C,!)

e]CIl

::ác.:i...oes'"~..ulO

~:!

~ ...o<:> - ..""' e iE

el. o.... Po.

.s .."¡f ~'E..u c:i

N...¡...12.~¡¡.

I Transient Flows in Open Channels 435

Table 12.1. G. M. Shrum Generating Station, data fortailrace system.

GeneralNo. of tailrace tunnelsNo. of manifoldsUnits on Manifold No. 1Units on Manifold No. 2

Tailrace TunnelsShapeSizeLiningLengthManifold size

221 to 56 to 10

Modified horseshoe19.96 m high, 13.72 m wideConcreteTunnel No. 1, 405 m; No. 2,573 m13.71 by 99.67 m

Tailrace ChannelFor the length and cross sections of the channel, see Fig. 12.20.Weir length = 192 m .

Turbines

Unít No.Maxímum

Output'' (MW)Díscharge" per

Turbine (m3/8)

261275300

178190204

1 to 56 lo 89 and 10

3At a ne t head of 164.6 m.

ing the pressure on the tunne! crown due to priming, but at the same time didnot increase either the flow area or the hydraulic radius .

A new dam, called Peace Canyon, is being constructed 22.5 km downstream ofthe G. M. Shrum Generating Station. The reservoir created by this dam will extendto the rockfill weir. To simulate this reservoir, about 1200 m of its length down-stream of the existing rockfill weir was included in the analysis, and the waterlevels in this reservoir were varied by adjusting the height of a hypothetical weirat its downstream end. Sínce the retum wave-propagation time between themanifolds and this hypothetical weir is more than the time of the transíent-stateconditions of interest, this weir does not affect the manifold surge levels.

Load acceptance or rejection was simulated by varyíng the total inflow to eachmanifold at the rate at which the wicket gates open or close. Since total outflowfrom al! the turbines on each manifold is lumped as inflow to the manifold, loadvariation on individual units was not simu!ated.

Verificatton

The mathematical model was verified prior to the production runs by comparingthe computed results with those measured on the prototype or on the hydraulicmode!.


Comparison with Prototype Results. Load-rejection tests were conducted onthe prototype to obtain data lo verify the mathernatical mode!. Computed andmeasured water levels at various gauges for these tests were compared. The re-sults for the largest load rejection (1020 MW) are shown in Fig. 12.21. In thistest, the inflow to Manifold No. I was reduced from 810 to 133 m3/s in 8 sec-onds, and the inflow to Manifold No. 2 remained stcady al 240 m3/s.

As can be seen from this figure, there is good agreement between the com-puted and measured transient-state water levels, even though there was a slight

502

......s".~<, 501'"'"~...."~....~ 500

"~~

499

504_['ompUled sleady Tstat ,,~, 6. 1('- Meosured sleody stote

water level

f \ 1--- Compuled

/-,I \ Ij\I \I 1\ \I p\\ Sleody stat«I ~''tI \ f I Meosured_)III

,~

..-

I

\I

'_/\'-../ .- _. --- ..-'-' -~_._---. __ .- 1----

503

498

497o 107 8 9 1/2 4 6

Time (min)

(a) Manifold No.

Figure 12.21. Comparison of computed and measured prototype water levels followingreduction of inflow to manifold No. 1 from 810 to 133 m3/s in 8 s. Inflow lo manifoldNo. 2 remained steady at 240 m3/s.


503,---------,---------,---------,------- __ .- ~

'" "~ 502~~\~------i_--~~_t_,----------4_--------_+--------__J

499o 2 3 4 5 6 7 8 9 10

Time (min)(b) Gauge No. 2

503

...... --------1----1e:'-

-------~- - ..

&:::.~

502'":..~ »:":Meosured.."'"....?:; 501.,

"~500

O 2 3 4 5 6 7 8 9 10

Time (min)

(e) Gouga No. 5


difference between the computed and measured steady-state water levels. If thisdifference in the steady-state water levels is taken into consideration, the agree-ment between the computed and measured water levels would be further im-proved. The comparisons for the other tests were equally satisfactory.


6 7

(o)

3 4 5Time (min)Manifold No.

506 TM~ed\ "L'- ;::.:-"

\ I.... .... ""-\.....// ~

'... /

~ 7 compUled-/

_,

~ .t7 I

504

502

500

498 o 6 72 3 4 5

Time (min)(b) Tunnel I (3IE)

506

2 3 4 5Time (m/n)

(e) Tunnel 1(318)

6 7

506

~ Meosured---....""

"- r-. --- - _-"...... _--' ~ comaure«

I

504

502

500o 6 72 3 4 5

Time (min)

(d) Gauge 33

Figure 12.22. Comparison of computed and measured water levels on hydraulic modelCollowing simultaneous reduction of inflow to both manifolds from 990 to O m3/s in 8 s.Initial steady-state water level at Gauge No. 8 = El. 507.5 m.

Comparison with Hydraulic Model Results. Limited results were available ofthe tests conducted on an undistorted, 1 :96-scale hydraulic model. Computedand measured water levels are shown in Figs. 12.22 and 12.23.

As can be seen from these comparisons, the agreement between the hydraulic

3 4 5

Time (min)

Figure 12.23. Comparison of eomputed and measured water levels on hydraulic model.Initial steady-state water level at Gauge No. 8 = El. 507.5 m. loflow to manifold No. 2varied in 8 s as follows: (a) 990 to Om3/s, (b) 990 00 Om3/s and then O to 396 m3/s after20 s, and (e) 990 to Om3/s and then to 396 m3/s after 127 S.

I! Transient Flows in Open Channels 439

514

'J.II~ (a) Rejeelion only

/( ,'[\

H \VMeaSUred r>

""1\ /\\

f¡ \\ / \

e I \

'.\0 1/,......

~ V "compuledJ\\ l

'_ ~

5/2

5/0

-...508

~506

504

502

500o 2 3 4

Time (min)5 6 7

3 4

Time (min)6

5/ 8 ~!1, (e) Rejeefion followed

6'", by aíeePlonee _____

Compuled- ~I4

2 !I \\ I --Load ff \Y Measured I ,

rejeefion ,h.O ¡¡ \y /1 \~

8-t12?

Sees - ~~_\~ '/ \

6 I \\1 l'l -,

\ "-/--V,

.... -,4 \\ // <, - ------

Load occeprance2 ....... -::-VO

5/

5/

5/

5/

50

50

50

50

50 o 2 6 7


model and computed water levels is satisfactory but is not as good as that be-tween the prototype and computed results. The difference rnay be due to ap-proximation of the channel configuration. This was changed several times dur-ing testing, and no records are available of the configuration in use when thesurge results shown were obtained.

Results

Prototype transient-state conditions were computed for the following total andpartialload changes over a range of steady-state manifold levels:

1. load rejection2. load acceptance3. load rejection followed by load acceptance4. load acceptance followed by load rejection.

Although the load acceptance period is usually much more than 8 s, both theload acceptance and rejection periods were assurned to be 8 s because loadacceptance at this rate is possible. Slower operating rates were assurned onlywhile investigating the effects of increasing these periods on the maximum surgelevels. Since the units would be synchronized to the system prior lo load accep-tance, the wicket gates would be at speed-no-load (SNL) gate (about 6 percent).In the analysis, however, ít was assumed that the wicket gates are opened fromthe fully closed position lo their final steady-state values. The computed levelsare, therefore , slightly higher than would be expected from the actual proto-type operation.

For the multiple-turbine operations (i.e., load acceptance followed by loadrejection or vice versa), Ihe second operation was started at the most critica!time, which is defined as the time between the start of two operations thatproduces maximum upsurge . For example, for a load acceptance followinga load rejection, the load was accepted at points a, b, e, and d of Fig. 12.23a_ Itwas found that the maxirnum upsurge was produced when the load was acceptedat point e and not when it was accepted at point d. Tests on the hydraulicmodel gave the same result.

Operating Guidelines

The operating guidelines were formulated so that the maximum upsurge wouldnot exceed El. 511.5 m, í.e., the top of the manifold parapet wall. A largenumber of operating conditions were considered in order to identify those thatproduce the highest upsurges.


Th.e rate of loading, or reloading following a load rejection could be easilyre.stncted. F?r a load rejection, however , such restrictions could not be imposedwltho~t modlf~ing the existing control equipment. The unit goes to runawayspe~d if the ':lcket gates are closed slowly following a load rejection, and, as~ettlng. the unít operate at runaway speed for long periods was unacceptable,mcreasing the wicket-gate cJosing time did not appear to be attractive. A morereasonable alternative was to impose a Iimit on the initial steady-state load sothat, if the load was rejected, the upsurge would rernain within the allowablelimits.

For reloading following a load rejection, either the rate of reloading could berestricted or reloading could be done al the 8-s rate once the surges in themanifol~ due. to load rejection had subsided. Because of the inherent dangers,no consíderation was given to the possibility of reloading on the optirnum partof a cycle as a means of allowing more rapid reloading.

High upsurges may be produced even at low tailwater levels depending uponthe number , sequen ce, and size of the loading and unloading operations. There-fore, such multiple loading and unloading operations must be avoided , Forexample, in case of synchronízíng difficulties, or load rejection during a loadacceptance , the manifold water level should be allowcd to become steadybefore another attempt is made to load the units.

The operating guidelines showing the allowable amount of load that can beaccepted or rejected are presented in Fig. 12.24. This figure also shows, basedon the condition of total load rejection, the maximum permissible initial loadfor various initial manifold levels. Different operating conditions are designatedas follows:

Curve No. Description

I23456789

10II

Total load rejection; maximum allowable initialloadLoad acceptance from steady stateLoad rejection following 300-MW acceptanceLoad rejection following 600-MW acceptanceLoad rejection following 900-MW acceptanceLoad rejection following 1200-MW acceptanceLoad acceptance following 300-MW rejectionLoad acceptance folJowing 600-MW rejectionLoad acceptance following 900-MW rejectionLoad acceptance following 1200-MW rejectionLoad acceptance following 1500-MW rejection

I


5/2.-------r------,-------¡------~------~------,_------,___,

5/0

.....I¡

509'-,-~~~ 508

"-"la~ 507.~I¡

~."" 506~

200 400 600 800 /000 /200 /400

Maximum o!lowqb/~10101 load occ~ploncll or r~j"clion (MW) on oml manifold

Figure 12.24. Operating guidelines.

12.16 SUMMARY

In this chapter, transient flows in open channels were discussed. A number ofterms were first defined, the continuity and dynamic equations were derived,and numerical methods available for their solution were discussed. Details ofthe explicit and implicit finite-difference methods were presented. The chapterconcluded with a case study.

PROBLEMS

12.1. A 6.1-m-wide rectangular canal is carrying 28 m3/sec at a depth of 3.04m. The gates at the downstream end are suddenly closed. Determine theinitial surge height, z, and the velocity, Vw, of the surge wave.

12.2. An initial steady-state flow of 16.8 m 3 /sec in a 3-m-wide rectangularpower canal is suddenly reduced to 11.2 m 3¡sec at the downstream end.If the initial depth was 1.83 m, determine the height and the velocity ofthe initial surge wave.


12.3. A trapezoidal canal having a bottom width of 6.1 m and side slopes of 1.5horizontal to I vertical is carrying 126 m3/sec at a depth of 5.79 m. Ifthe flow is suddenly stopped at the downstream end , what would be thesurge height and the wave velocity?

12.4. Prove that if the surge height, z , is smaIl as compared to the initial flowdepth,yo, then

in which A o = initial steady-state flow area.

12.5. Develop the boundary conditions for the junction of three channels forthe diffusíve scheme based on the explicit finite differences. Neglect thefriction losses at the junction.

12.6. In Eq. 12.121, conditions irnposed by the boundary were written in ageneral formo Determine the values of coefficients for a reservoír, acontrol gate, and a rating curve. Linearize the relationships if they arenonlinear.

12.7. Plot the variation of water surface at the downstream end of a canal withtime foIlowing sud den closure of the control gates at the downstream end.Assume the canal is short, horizontal, and frictionless.

12.8. Derive the dynamic and continuity equations for nonprismatic channelshaving lateral outflow q per unit length of channel. Assume (l) gradualbulk outflow, e.g., over a side spillway ; (2) outflow has negligible velocity,e.g., seepage; and (3) gradual inflow, e.g., from tributarles, having velocitycomponent in the positive x-direction as u l.

Answers

12.1. z = 0.9 m; Vw = -5.13 m/sec.12.3. z = 0.93 m; Vw = -5.52 m/sec.12.8. l. Continuity equation:

a v ay ay aA (x y)A-+BV--+B-+V ' +q=Oax ax at ax

2. Dynamic equation:

a v a v ay- + V- +g -- =g(So - Sr) +D¡01 OX ox

in whích

D ¡= O for case 1


VDI = - for case 2

2A

(V- uI)qDI = for case 3

A

REFERENCES

1. Chow, V. T., Open-Channel Hydraulics, McGraw-HiIl Book Co., New York, 1959.2. Rouse, H. (ed.), Engineering Hydraulics, John WiJey & Sons, New York, 1961.3. Johnson, R. D., "The Correlation of Momentum and Energy Changes in Steady Flow

with Varying Velocity and the Application of the Former to Problems of UnsteadyFlow or Surges in Open Channels," Engineers and Engineering , The Engineers Club ofPhiladelphia, 1922, p. 233.

4. Rich, G. R., Hydraulic Transients, Dover Publications, Inc., New York, 1963.5. De Saint-Venant, B., "Theorie du movement non-permanent des eaux avec application

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6. Strelkoff, T., "One-Dimensional Equations of Open Channel Flow," Jour., Hyd. Div.,Amer. Soco of Civil Engrs., vol. 95, May 1969, pp. 861-876.

7. Mahrnood , K. and Yevjevich, V. (ed.), Unsteady Flow in Open Channels, Water Re-sources Publications, Fort Collins, Colorado, 1975.

8. Yen, B. C., "Open-Channel Flow Equations Revisited," Jour., Engineering Medl. Div.,Amer. Soco of Civ. Engrs., vol. 99, Oct. 1973, pp. 979-1009.

9. Arnein, M. and Fang, C. S., "Implicit Flood Routing in Natural Channels," Jour., Hyd.Div., Amer. Soco of Civil Engrs., vol. 96, Dec. 1970, pp. 2481-2500.

10. Vasiliev, O. F., Temnoeva, T. A., and Shugrin, S. M., "Chislennii Melod RaschelaNeustanoviv-shikhsia Techermi v Otkritikh Ruslakh" ("Numerical Method of Calculat-ing Unsteady Flow in Open Channels"), lzvestiia Akemii Nauk SS.S.R., Mekhanika,No. 2,1965, pp. 17-25 (in Russian).

11. Stoker, J. J., Water Waves, Interscience, New York, 1957.12. Forsythe, G. E. and Wasow, W. R., Finite-Difference Methods [or Partial Differential

Equations, John Wiley & Sons, New York, 1960.13. Courant, R. and Hilbert, D., Methods of Mathematical Physics, vol. Il, Interscience,

New York, 1962.14. Garabedian, P. R., Partial Differential Equations, John Wiley & Sons, 1964.15. Abbot, M. B., An Introduction lo the Method of Characteristics, American Elsevier,

New York, 1966.16. Lai, C., "Flows of Homogeneous Density in Tidal Reaches: Solutíon by the Method of

Characteristics," Open File Report, United States Department of the Interior, Geo-logical Survey, Washington, D.C., 1965.

17. lsaacson, E. J., Stoker, J. J., and Troesch, B. A., "Numerical Solution ofFlood Predic-tion and River Regulation Problems: Report 2, Numerical Solution of Flood Problemsin Simplified Models of the Ohio River and the Junction of the Ohio and MississippiRivers," New York University, Institute of Mathematical Sciences, Report No. IMM-205,1954.

·1


18. Isaacson, E. J., Stoker, J. J., and Troesch, B. A., "Numerical Solution of Flood Predic-tion and River Regulation Problems: Report 3, Results of the Numerical Prediction ofthe 1945 and 1948 Floods in the Ohio Rivers of the 1947 Flood through the J unctionof the Ohio and Mississippi Rivers, and of the Floods of 1950 and 1948 through Ken-tucky Reservoir," New York University, Institute of Mathematical Sciences, ReportNo. IMP.I-NYU-235, 1956.

19. Baltzer, R. A. and Lai, c., "Cornputer Simulation of Unsteady Flows in Waterways,"Jour., Hyd. Div., Amer. Soco of Ovil Engrs., vol. 94, July 1968, pp. 1083-1117.

20. Liggett, J. A. and Woolhiser, D. A., "Difference Solutions of the Shallow-Water Equa-tion," Jour. Engineering Mech. Div., Amer. Soco of Civil Engrs., vol. 93, April 1967,pp. 39-71.

21. Abbott, M. B. and Verwey, A., "Four-Point Method of Characteristics," Jour. Hydr.Div., Amer. Soco ofCivil Engrs., vol. 96, Dec. 1970, pp. 2549-2564.

22. Lax, P. D., "Weak Solutions of Nonlinear Hyperbolic Partial Differential Equations andTheir Numerical Computation," Communications on Pure and Applied Mathematics,vol. 7, 1954, pp. 159-163.

23. Lax, P. D. and Wendroff, B., "Systems of Conservation Laws," Communications onPure and App/ied Mathematics, vol. 13, 1960, pp. 217-237.

24. Priessmann, A. and Cunge , J. A., "Calcul des intumescences sur machines electron-iques," IX Meeting, International Assoc. for Hydraulic Research, Dubrovnik, 1961.

25. Richtmyer, R. D., "A Survey of Difference Methods for Non-Steady Fluid Dynamics,"NCAR Technical Notes 63-2, National Center for Atmospheric Research, Boulder ,Colorado, 1962.

26. Cunge , J. A. and Wegner, M., "Integration Numerique des equations d'ecoulement deBarre de Saint Venant par un schema Implicite de differences finies," La HouilleBlanche, No. 1, 1964, pp. 33-39.

27. Gelfand, I. M. and Lokutsievski, O. V., "The Double Sweep Method for Solution ofDifference Equations," in Theory of Difference Schemes, by Godunov, S. K. and Rya-benki, V. S., North HolIand Publishing Co., Amsterdam, 1964.

28. Richtrnyer, R. D. and Morton, K. W., Difference Methods for Initial- Value Problems,2nd ed.,lnterscience Publishers, New York, 1967.

29. Martín, C. S. and DeFazio, F. G., "Open Channel Surge Simulation by Digital Corn-puters," Jour., Hyd. Div., Amer. Soco 01 Civil Engrs., vol. 95, Nov. 1969, pp. 2049-2070.

30. Strelkoff, T., "Numerical Solution of Saint-Venant Equations," Jour., Hyd. Div., Amer.Soco of Civü Engrs., vol. 96, January, 1970, pp. 223-252.

31. Terzidis, G. and Strelkoff, T., "Computation of Open Channel Surges and Shocks,"Jour., Hyd. Div., Amer. Soco ofOvi/ Engrs., vol. 96, Dec. 1970, pp. 2581-2610.

32. Amein, M. and Chu, H. L., "Implicit Numerical Modeling of Unsteady Flows," Jour.,Hyd. Div., Amer. Soco o] Ovil Engrs., vol. 101, June 1975, pp. 717-731.

33. Chaudhry, M. H., "Mathematical Modelling of Transient State Flows in Open Chan-neis," Proc., lnternational Symposium on Unsteady Flow in Open Channels, Newcastle-upon-Tyne, 1976, published by British Hydrornechanic Research Assoc., Cranfield,Bedford , England , pp. CI-I-CI-18.

34. Cooley, R. L. and Moin, S. A., "Finite Element Solution of Saint-Venant Equations,"Jaur., Hyd. Div., Amer. Soco ofCivil Engrs., vol. 102, Sept. 1976, pp. 1299-1313.

35. Davis, J. M., "The Finite Element Method: An Alternative Subdomain Method forModelling Unsteady Flow in Coastal Waters and Lakes," Re! 33, pp. B4-41-B4-53.


36. Patridge, P. W. and Brebbia, C. A., "Quadratic Finite Elcments in Shallow Water Prob-lerns," Jour., Hyd. Div., Amer. Soco uf Civil Engrs., vol. 102, Sept. 1976, pp. 1299-

1313.37. Houghton , D. D. and Kasahara, A., "Non linear Shallow Fluid Flow over an Isolated

Ridge," Comm. on Pure and App. Math., vol. XXI, No. 1, Jan. 1968.38. Dronkers, J. J., "Tidal computations for Rivers, Coastal Areas and Seas," Jour., Hyd.

Div., Amer. Soc. ofCivil Engrs., vol. 95, Jan. 1965.39. McCracken, D. D. and Dorn, W. S., Numerical Methods and FORTRAN Programming,

John Wiley & Sons, New York, 1967.40. Stark, R. M. and Nicholls, R. L., Mathematical Foundations for Design, McGraw-Hill

BookCo.,1970,pp.l19-121.41. Courant , R., Friedrichs, K., and Lewy, H., "Uber die partiellen Differenzengleichungen

der Mathematischen Physik ," Math. Alln., vol. 100, 1928, pp. 32-74 (in German).42. Benet, F. and Cunge, J. A., "Analysis of Experiments on Secondary Undulations

Caused by Surge Waves in Trapezoidal Channels," Jour. Hyd. Research, InternationalAssoc, for Hyd. Research, vol. 9, No. 1,1971, pp. 11-33.

43. Vasiliev, O. F., et al., "Numerical Method of Computations of Wave Propagation inOpen Channels: Application to the Problem of Floods," Dokl. Akad. Nauk SSSR, 151,No. 3, 1963 (in Russian).

44. Vasiliev, o. F., et al., "Numerical Methods for the Calculation of Shock Wave Propaga-tio~ in Open Channels," Proc. 11th Congress, International Assoc. [or Hyd. Research,Leningrad, vol. Ill, 1965, 13 pp.

45. Kao, K. H., "Numbering of a Y-Branch Systern to Obtain a Banded Matrix for lmplicitFinite-Difference Scherne," B.C. Hydro interoffice memorandum to W. W. Bell,Hydraulic Section, November 1977.

46. Karnphuis, J. W., "Mathernatical Tidal Study of S!. Lawrence River ," Jour., Hyd.ot«, Amer. SOCoof Civil Engrs., vol. 96, No. HY3, March 1970, pp. 643-664.

47. Stoker, J. J., "The Forrnation of Breakers and Bares," Communications on Pure andApplied Mathemaücs, vol. 1, 1948, pp. 1-87.

48. Dronkers, 1. J., Tidal Computations in Rlvers and Coastal waters, North Holland Pub-lishing Co., Amsterdam, Netherland, 1964.

49. Amein, M., "Bore Inception and Propagation by the Nonlinear Wave Theory," Proc.,9th Conjerence on CoastaI Engincering, Lisbon, Portugal, June 1964, pp. 70-81-

50. Shen, C. C., "Selection and Design of a Bore Generator for the Hilo Harbar TsunamiModel," Research Report No. 2-5, United States Army Engr. Waterways ExperimentStation, Vicksburg, Miss., June 1965,94 pp.

51. Martín, C. S. and Zovne, J. J., "Fíníte-Difference Simulation of Bore Propagation,"Jour., Hyd. Div., Amer. Soco o] Civ. Engrs., vol. 97, No. HY7, July 1971, pp. 993-1010.

52. Favre, H., Ondes de translation dans les canaux de'couverts, Dunod, 1935.53. Meyer-Peter, E. and Favre, H., "Ueber die Eigenschaften van Schwallen und die

Berechnung von Unterwasserstollen," Schweizerische Bauzaitung, Nos. 4-5, vol. 100,Zurich, Switzerland, July 1932, pp. 43-50, 61-66.

54. Calame, J., "Calcul de ¡'ande translation dans les canaux d 'usines," Editions la Con-corde, Lausanne, Switzerland, 1932.

55. Drioli, C., "Esperienze sul moto pertubato nei canali índustriali," L'Energie E/ettrica,no. IV-V, vol. XIV, Milan, Italy, April-May, 1937.

56. Jaeger, C., Engineering Fluid Mechanics, Blackie & Son Limíted, London, 1956, pp.381-392.

67.

68.

i 69.

1 70.

f

( 71.

I 72.

73.

74.

i 75.

I


57. Cunge,1. A., "Cornparison of Physical and MathematicaJ Model Test Results on Trans-lation Waves in the Oraison-Manosque Power Canal," La HouilIe Blanche, no. 1,Grenoble, France, 1966, pp. 55-69, (in French).

58. Amorocho, J. and Strelkoff', T., "Hydraulic Transients in the California Aqueduct ,"Report No. 2, Departrnent ofWater Resources, State of California, June 1965.

59. Wiggert, D. C., "Prediction of Surge Flows in the Batiaz Tunnel," Researcñ Report,Laboratory for Hydraulics and Soil Mechanics, Federal Instítute of Technology, Zurich,Switzerland, no. 127,June 1970.

60. Wiggert, D. C., "Transient Flow in Free-Surface, Pressurized Systerns," Jour., Hyd. Div.,Amer. Soco of Cipo Engrs., Jan. 1972, pp. 11-27.

61. Miller, D. J., "Giant Waves in Lituya Bay , Alaska," Professional Paper 354-L, U.S.Geological Survey, 1960.

62. Kiersch, G. A., "Viaont Reservoir Disaster ," Civ. Engineering, Amer. Soco of Civ.Engrs., vol. 34, no. 3, March 1964, pp. 32-47.

63. McCullock, D. S., Slide-Induced Waves, Seiching, and Ground Fracturing Caused by theEarthquakes of March 27, 1964, at Kenai Lake, Alaska," Professional Paper 543-A, U.S.Geological Survey, 1966.

64. Kachadoorian, R., "Effects of the Earthquake of March 27, 1964, at Whittier, Alaska,"Professional Paper , No. 542-B, U.S. Geological Survey, 1965.

65. Forstad, F., "Waves Generated by Landslides in Norwegian Fjords and Lakes," Publica-tion No. 79, Norwegian Geotechnicallnstitute, Oslo, 1968, pp. 13-32.

66. Wiegel, R. L., "Laboratory Studies of Gravity Waves Generated by the Movement of aSubmerged Body," Trans. Amer. Geophysica/ Un ion , vol. 36, no. 5, Oct. 1955, pp,759-774.Prins, J. E., "Characteristics ofWaves Generated by a Local Disturbance," Trans. A meroGeophysica/ Union, vol. 39, no. 5, Oct. 1958, pp. 865-874.Law, L. and Brebner, A., "On Water Waves Generated by Landslides," Third Australa-sian Conference on Hydraulics and Fluid Mechanics, Sidney, Australia, Nov. J 968, pp.155-159.Karnphuis, J. W. and Bowering, R. J., "Impulse Waves Generated by Landslides," Proc.,Twelfth Coast al Engineering Conference, Amer. Soco of Cipo Engrs., Washington, D.C.,Sept.1970,pp.575-588.Noda, E., "Water Waves Generated by Landslídes," Jour., Walerways, Harbors andCoastal Engineering Div., Amer. Soco of Civ. Engrs., vol. 96, no. WW4, Nov. 1970,pp. 835-855.Das, M. M. and Wiegel, R. L., "Waves Generated by Horizontal Motion of a Wall,"Jour., Waterways, Harbors and Coastal Engineering Div., Amer. Soco of Civ. Engrs., vol.98, no. WW1, Feb. 1972, pp. 49-65.Babcock, C. 1., "Impulse Wave and Hydraulic Bore Inception and Propagation asResulting from Landslides," a research problem presented to Georgia Insl. of Tech.,Atlanta, Georgia, in partíal fulfillment of the requirements for (he degree of M.Sc. incivil engineering, Oc!. 1975."Wave Action Generated by Slides into Mica Reservoir, Hydraulic Model Studies,"Report, Western Canada Hydraulic Laboratories, Vancouver, B.C., Canada, Nov. 1970.Davidson, D. D. and McCartney, B. L., "Water Waves Generated by Landslides in Reser-voirs," Jour., Hyd. Div., Amer. Soco ofCiv. El1grs., vol. 101, Dec. 1975, pp. 1489-1501."A Pilot Study of Water Surface Waves in Revelstoke Reservoir in the Event of DownieSlide Movement," Report, Narthwest Hydraulic Consultants, Vancouver, B.C., Canada,June 1976.


76. "Hydraulic Model Study of Waves from Downie Slide," Report, Northwest HydraulicConsultants, August 1976.

77. Mercer , A. G., Chaudhry, M. H., and Cass, D. E.,"Modelling of Slide-Generated Waves,"Fourth Hydrotechnical Conference, Vancouver, Canada (under preparation).

78. Chaudhry , M. H. and Cass, D. E., "Propagation of Waves Generated by Rapid Move-ment of Downie Slide into Revelstoke Reservoír," Report No. 809, HydroelectricDesign Divisíon, B.e. Hydro and Power Authority , Vancouver, B.C., Canada, Sept.1976.

79. Raney, D. C. and Butler, H. L., "A Numerical Model fOI Predicting the Effects of Land-slide-Generated Water Waves," Research Report H-75-1, U.S. Army Engineer Water-ways Experiment Station, Vicksburg, Miss., Feb, 1975.

80. Koutitas, C. G., "Finite Elernent Approach to Waves due to Landslides," Jour., Hyd.Div., Amer. Soco ofCiv. Engrs., vol. 103, Sept. 1977, pp. 1021-1029.

81. Chaudhry , M. H. and Kao, K. H., "G. M. Shrun Generating Station: Tailrace Surges andOperating Guidelines During High Tailwater Levels," British Columbia Hydro andPower A uthority , Vancouver, B.C., November 1976.

APPENDIX A

FORMULAS AND DESIGN CHARTS FORPRELIMINARY ANAL YSIS

Design charts and approximate formulas are presented in this appendix. Thesemay be used for quick computations during the preliminary design stages whena large number of alternatives are considered to have an economical design or toapproximately select the parameters of a system for a detailed analysis.

A-l EQUIV ALENT PIPE¡II!,

If the diameter, wall thickness, or wall material varies along the length of apipeline, then the pipeline rnay be replaced by an "equívalent pipe" for an ap-proximate analysis. By using an equivalent pipe, the partial wave reflections andthe spatial variation of the friction losses and of the elastic and inertial effectsare not correctly taken into consideration. The approximation is useful, how-ever, and gives satisfactory results provided that the changes in the properties ofthe original pipeline are minor.

The total friction losses, the wave travel time, and the inertial effects of theequivalent pipe should be equal to those of the pipeline. These characteristicsfor the equivalent pipe of a pipeline having n pipes in series may be determinedfrom the following equations:

¡i!!

J

LeA =--

e n L.¿-..!.;=1 A;

(A-l)

Lea =---e n L.¿-.!.

;=1 a;

(A-2)

(A-3)

I 449

450 Appendix A

in which a is the waterhammer wave velocity, and A, L, D, and f are the cross-sectional area, length, diameter, and Darcy-Weisbach friction factor for the pipe,respectively. The subscripts e and i refer to the equivalent pipe and to the ithpipe of the pipeline.

A-2 MAXIMUM PRESSURE DUE TO VALVE CLOSURE

Figures A-l and A-2 show the maximum pressure rise aboye the upstreamreservoir level at the valve and at the midlength of a pipeline caused by theclosure of a downstream valve díscharging into atmosphere. The valve closure isassumed to be uniform , i.e., valve-opening versus time curve is a straight lineo

The following notation is used:

,

// /.

/ V /

/ / / / // / / / / 1/ V

o/ / Y .J // / / /i¡Y /1,,/"

V Y / / V/ /I~~ / ./ /

/ Y / /"" .:/ / V ,IV /

// / */ /V V / / V /V

'--- ,/ '11' /' /

/ / / /

/ / ./ _V~ / .// .: // ~-, / /

,1/ V/ .> ~/ / // / V / _/1 --1

"j~./ /

//~ 0V ,.",..'" Y/ ,.",..

v/ V /' /v,.,. '" ///1 .: // /

,.,./

1.00.8

i'1 o0.6

<J :I:

0.4

0.2

0.1

0.08

0.06

0.04

0.02

0.010.05 0.1 0.2 0.3 0.4 0.6 0.8 1. 2. 4. 6. a 10.

.P-

(a) At valva

Figure A-l. 'Maximum pressure rise due to uniform valve closure: frictionless system(h = O).

ft

~I!'

l

I1t¡I1,~

I

Formulas and Design Charts for Preliminary Analysis 451

t1.00.8

El o0.6

:I: :I:<l

0.4

0.3

0.2

0.10.08

0.06

0.04

0.03

0.02

// ./

/ // v /

// V/

,/ VY y

0.1 0.2 0.4 0.6 0.8 l. 2. 4. 6. 8. 10.

.P-(b) At mid-Ienllth

Figure A-l. (Continueds

c.-f:~,.,t/~.) ;')./ft("L

+'euft.-.o (~,!(t:.~",,';:~I

aVo _p=--_

2gHo

k=_!s_(2L/a)

a = waterhamrner wave velocityg = acceleration due to gravity

Ha = static head (elevation of the reservoir level - elevation of the valve)L = length of the pipeline

Va = initial steady-state velocity in the pipelineTe = valve closure time

D.Hm = maximum pressure rise at the midlength above the reservoir levelD.Hd = maximum pressure rise at the valve above the reservoir level

h fa = initial steady-state head loss in the pipeline corresponding to velocityVa

::: Tr'2. L/:,

452 Appendix A

1.00.8

vI o 0.6~:r;

0.4

0.2

0.10

0.08

0.06

0.04

0.02

,lL t--~

'/_ L Át l,L. /;-1- 7rf,/ -;/ '/ , b -+-/¡ / /_ i

LI / V Vi;Vr

"....../ Y 1: //7'- --

1 1,/~

/! I!

Y'" l/ttJj /0 ~_/ ;J

j/-¡-- __o

V / , ,

I i // / .: I/_ .. ~

/ L t/ ,,_L ~¿0_-

/ I

-: i/ /-: / !Y V

V V ..,0, i

//' /V V ! i'L

/' /V VV ! I i

/ 11 i J I0.01

0.05 2. 4. 6. 8. 10.

.1'-0.2 0.4 0.6 0.8 1.0.1

(o) At Volva

Figure A-2. Maximum pressure rise due to uniform valvc closure; friction losses taken intoconsideration (h = 0.25).

h == h1alHoH m ax == maximum pressure head == H o + b. Hd at the valve

== H¿ + b.Hm at midlength of the pipeline.

A-3 MINIMUM PRESSURE DUE TO VALVE OPEN ING

Mínimum pressure head, Hmin, at the valve caused by uniformly opening thevalve from the completely closed position may be determined from the follow-ing eq ua tíon : 1

H . == H (- k + V+l)2rrun o VI<.-'1" 1 (A-4)

in which k = L V¡/(gHoTo); L = length of the pipeline; V¡= final steady-statevelocity in the pipeline; To = valve opening time; and H¿ == static head. Theminimum pressure occurs 2LJa seconds after the start of the valve movement.

El o~ :r; 0.6

0.4


1.0

0.8 /

I-f--!-¡,

u: ./ I ;L /1- • .: V / Vt : I

. ¿jI :/ / V v- 1 ''1 . .._.I /V / / /V

1- 0 •• _ c-, I1 /V V v' /~.J?t / V

/V '/

/ / V~-I-T .": .: L /

i -I;A- __c- -~ --- -¿- ---V b( .. /V

_

L Y ...-?/

/ / V ~.L_ /-j V V /r- - 11

yY ./ V V~_,-- IL

I_. __ ..

1, y/ /,! , ~)/, ,I V

0.2

0.10.08

0.06

0.04

0.02

0,010.05 0.1 0.2 0.4 0.6 0.8 1. 2. 4. 6. 8. 10.

./'--(b) At midlenoth

Figure A·2. (Continued)

Equation A-4 is applicable if t; > 2Lja. For t; ,¡;; u.»,a

Hmin == Ha - - b.Vg

in which b. V == change in the flow velocíty due to valve opening.

(A-5 )

A·4 POWER FAILURE TO CENTRIFUGAL PUMPS

Graphsi are presented in Figs, A-3 through A-7 fOI the minimum and maximumpressure heads at the pump, and at the midlength of a pipeline, and fOI the timeof flow reversal following power failure to the centrifugal pump units. Thegraphs are applicable to pumps with specific speed of less than 49 (SI units), í.e.,2700 (gprn units); they are not applicable to systems in which there is a valveclosure during the transient state or to systems that contain waterhammer con-trol devices other than large surge tanks. In the analysis, the latter are consid-ered as the upstream reservoirs.J

1

454 Appendix A

o.."..>

I

r-0.6

=:.I~~I·¡i-o':~WI 1;-0:4~V I !(~),AT.~ID-LENGTH OF DISCHARGEUNE

0.2 2 5 10 20

I1

50 100 200 400

Figure A-3_ Minimum head following power failure, (After Kinno and Kennedy+)


a = waterhammer wave velocityER = pump efficiency at rated conditions

g = acceleration due to gravityHR = rated head of the pumpHf = friction losses in the discharge line


V.loe 01 T

t

IIiI

Figure A-4. Minimum head following power failure, no friction losses, (After Kinno andKennedy é)

hf= Hf/HRHd = minimum transient-state head at the pumphd = HdlHR

Hm = minimum transient-state head at midlength of the discharge linehm = Hm/HR

Hmr = maximum transient-state head at midlength of the discharge linehmr = Hmr/HR

H, = maximum transient-state head at the pumph; = Hr/HR

892770 HRQRk = l l (in SI units)*

ER WR NR

*QR, HR, WR2, and N R are in m3/s, m, kg ml, and rpm, respectively ; and ER is in the frac-tional form, e.g., 0.8.I

456 Appendix A

Value 01 T

Figure A-S. Time of flow reversa) at pump following power failure. (After Kinno andKennedy+)

183200 HR QR----::-'~= (in English units)*

ER WR2 Nk,L = 1ength of the discharge line

NR = rated pump speedQR = rated pump discharge

t = timeto = elapsed time from power failure to flow reversal at the pump

VR = fluid velocity in the discharge line for rated pump dischargeWR2 = moment of inertia of rotating components of the pump and motor, and

entrained fluidp=aVR/(2gHR)

1r=--

2L-ka

A-S AIR CHAMBERS

Charts ' are presented in Fig. A-8 for the maximum upsurge and dow~su~ge atthe pump end, at the midlength, and at the quarter point on the reservoir side ofa discharge line following power failure to the pumps. These charts may be usedto determine the required air volume for a discharge lineo

*QR, HR, WR2, and NR are in ft3/sec, ft, Ib-ft2, and rprn, respectively; and ER is in thefractional form, e.g., 0.8.

¡I¡

!

~j

i~

!~tt

II


The charts are based on the following assumptions:

1. Air chamber is located near the pump.2. Check valve closes simultaneously with the power failure.3. Darcy-Weísbach formula for computing the steady-state friction losses is

valid during the transient state.4. The absolute pressure head, H*, and the volume of air, e, inside the air

chamber follow the relationship H*C 1.2 =: constant.


a =: waterhammer wave velocityVo = initial steady-state velocíty in the discharge pipe

g = acceleration due to gravityH¿ = sta tic head (Elevation of the reservoir - Elevation of the air charnber)

(o) Al mid-len~lh 01 dischcr qe line

Figure A-6. Maximum head following power failure. (After Kinno and Kennedy+)

458 Appendix A

I/alul 01 T

(b) Al pump

Figure A-6. (Continued)

1.4

J2 1.3<,

:r: 1.2o..";¡ l.l>

r---r-.~r--.

~ o.? ¡ ¡'''s

««r I f.-~~O'~l~ r--0.751.0

0.2 2 5 1005Value 01 T

Figure A-7. Maximum head at pump if reverse rotation of pump is prevented. (AfterKinno and Kennedy+)


.,N

II

1

Iif

If

1

I

,N

(o) Al pump

Figure A-B. Maximum upsurge and downsurge in a discharge line having an air chamber.Numbers on the curves refer to the maximum downsurge or maximum upsurge divided by~. (After Ruus3)

460 Appendix A

00

10

'0

'0

<.0

la~

z O

l.'

1.0

O.

0.8

o. 1

o .•

'02

1.,\

o ~\. "\

'.0\\\

~ f--',~

<.0

,o6.0

1.0

8.0

....

N

..1.

o.(b) At mid-Jength of discharge Jine

Figure A-S_ (Continued)


.o ·Vo.,

>-o •o 1

o •o •1.0

\

\

\

\_ \' 1\ \,\ ,,\ \', ' 1 \ \ 1', 1\ \..,

\ \ 1\ \ \ '~\ \ -i . 1',\ k\ \ l' \ '\ 1\ I "

(e) Al quarter paint of diseharge Jine (reservoir side 1

Figure A-S_ (Continued)

462 Appendix A

HÓ = absolute statie head = H o + 10.36 (in the English units, H o + 34)aVop* = ----''----

2g(HÓ + Hto)Hto = initial steady-state head losses in the diseharge line = fL V~/(2gD)Co = initial steady-state air volume in the ehamberQo = initial steady-state diseharge in the pipe

L = length of the diseharge line \D = diameter of the discharge lineo I

The maximum upsurge and downsurge are aboye and below the downstreamreservoir level, and the absolute pressure heads are obtained by subtraeting oradding the downsurge or upsurge to the reservoir level plus the barometrle head.

The air-chamber size for a pipeline may be determined as follows: For themaximum allowable downsurge at any critical point along the pipeline-e.g., avertical bend-determine 2Coa/{QoL) from Fig. A-8. Linear interpolation may beused if the bend is not loeated either at the midlength or at the quarterpoint.From the expression 2Coa/{QoL), compute the minimum initial steady-state airvolume, C¿ min- This volume eorresponds to the upper emergency level in theair ehamber. To this minimum air volume, add the volume of the ehamber be-tween the upper and the lower emergency levels-for example, 10 pereent ofC¿ min for large size chambersand 20 pereent for small ehambers. For this newair volume, C¿ rnax s determine the maximum downsurge at the pump end fromFig. A-8a, and then determine the absolute minimum head, Hmin, at the pumpend by subtracting the maximum downsurge at the pump from the absolutestatic head, H~. The maximum transient-state air volume, Cmax, may then bedetermined from the equation

(H* + H ) 1/1.2_ o [o

Cmax - Co max H*.mm

(A-6)

in whieh H~ + H[o is the absolute initial steady-state head. To prevent air fromenterlng the pipeline, a suitable amount of submergenee should be provided atthe ehamber bottom. For this purpose, the ehamber volume may be seleeted asabout 120 pereent of the maximum air volume, Cmax, for small air ehambersand about 110 pereent, for large air chambers.

A-6 SIMPLE SURGE TANKS

Figure A-9 shows the maximum upsurge in a simple surge tank following uní-form gate elosure from lOOpereent to Opereent, and Fig. A-lO shows the maxí-mum downsurge in a tank following uniform gate opening from O to 100 pereentand from 50 to 100 percent."

In these figures, in region A, only one maximum oeeurs and that also after theend of the gate movement; in region B, the largest of the maxima is the seeondone and that oeeurs after the end of the gate operation; in region C, the largest

,... Formulas and Design Charts for Preliminary Analysis 463

1.6

~, I V

~B,':>/

\;/

1\' A /

~

'\ / 'l..Q_,.¿VC\

1-

" \ ~J'" V"_""

1\<, f..-.--' I V

.~ VÁ

\ "r---- V //"V- I

I\.<, V Ir" p;"'-1---1'..... r-. ~

¡...--.35

r-,r-, -/ ·<10

f".- / I r--1"--.1'--.. " :L...., /lllt<15

~ 1-- -~'O ¡--....1-- ~50 r-,;.;,;;;,.._......._ -.._._ ~s" -, '\:-- ............... t-.....

~I'--- :---....<, ~ <, ~\

<, ". 6'",'" <, " .A"'" \ \ \A~'~" \ 1\(Po\~ \ \ \

1~\f,c.JI • \ \ 1

t•1- 1.4"-.,_!'

1.2

1.0

.8

.6

.4

.2

OO .4 .6 .8 1.0•ho/Z -

.2

Figure A-9. Maxunum upsurge in a simple surge tank for uniform gate closure from 100 toOpercent. (Arter Ruus and EI-Fitiany4)

of the two maxima is the first one and that oeeurs prior to the end of the gatemovement.

The following notation ís used in these figures:

A t = cross-sectional area of the tunnelAs = eross-seetional area of the surge tank

g = aceeleration due to gravityho = head losses plus velocity head in the tunnel eorresponding to a steady

flow of QoL = length of the tunnel from the upstream reservoir to the surge tank

Te = gate-elosing timeTo = gate-opening timeT* = 21TviLAs/(gA t) = perlod of surge oscillations following instantaneously

stopping a flow of Qo in a eorresponding frictionIess system

464 Appendix A

1.6 r r I

1 I I

\ IIB :\• 1.4 f-' 1- rt1-..... I

1-0I

\ I

~ I1.2-

i I A

(o) 50 to 100 PercentFigure A·IO. Maximum downsurge in a simple surge tank CoruniCorm gate opening CromOto 100 percent and Crom50 to 100 percent, (After Ruus and EI·Fitiany4)

Zmax = maximum upsurge (or downsurge) aboye (or below) the upstreamreservoir level

Z* = Qo v'L/(gAtAs) = maxirnum surge followíng instantaneously stopping aflow of Qo in a corresponding fríctionless system.

A-' SURGES IN OPEN CHANNELS

The height and the celerity of a surge in a trapezoidal or rectangular open chan-neIs produced by instantaneously reducing flow at the downstream end of thechannel may be computed from Fig. A-Il. The height of this wave is reduced asit propagates upstream. Figure A-12 may be used to determine the wave heightat any location along the channel.

I I EU~, ~ , .'. i.....~.,~.: .. ,...•. _.~_, .... ~,,; "_"'''''~_'-'-''''''''_-'' ...~

Formulas and Desígn Charts for Preliminary Analysis 465

1.6

JI/(¡ I I f.f-

fI ,~\ ~ B

\,

'\~. f AI

r ~ \)S\ in 1// )<1

¿ / DI I~ ºI- / JI V i-J:2,........._7" '"-¡.....- ~

y v.V.I ~~~ Iro--- ~ :::::: .,.,...VV~~fJoJ V I-~

¡.....- r- qV ,........ ...- ~~i--:....-_.. ...-V /~- ~ .~,.~---- _.. 11: ofJ_.. .......~~~'/i

/ »: ....... V V~

.",.

/' V /

1.2

1.0

.8

.6

.4

.2

oO .4.2

(b) O to 100 PercentFig. A·IO. (Continued)

For the selection of the top elevation of the channel banks, the water surfacebehind the wave front may be assumed horizontal (see Section 7.2).

The following notation is used in Figs. A-ll and A-12:

b¿ = bottom width of channele = celerity of surge wave

Fo = Froude number corresponding to initíal steady-state conditíons, Vo/ygy;;g = acceleration due to gravityk = dimensionless parameter = bo/(rnyo)K = dimensionless parameter = 1 + 1/(1 + k)m = channel side slope, m horizontal to 1 vertical;

Qo = initial steady-state dischargeQ¡ = final steady-state dischargeSo = channel bottom slope

o o 0.5 0.6

(a) K = 1.75 xs,Yo

t1.0

I.08 1.o

466 Appendix A

(a) E!.. 05Q. .

(b) ~. 0.0Q.

Figure A-tI. Height and absolute velocity of a surge in a rectangular or trapezoi~al open-channel caused by instantaneous Ilow reduction at the downstream end, (ACterWu )

10


I 1.0

\ ,

0.6

(b) K= 1.50 XSo-_Yo

Figure A-l2. Variation of wave height of a positive surge propagatíng in a trapezoidalopen channel, (ACterWus)

Vo = initial steady-state flow velocityVw = absolute wave velocity = V + e

x = distan ce along the channel bottom from the control gatesYo = initial steady-state flow depth

z = surge wave heíght at distance xZo = initial surge wave height at downstream end

{3 = dimensionless parameter = zo/YoA = dimensionless parameter = Vwo/Vo.

468 Appendix A

REFERENCES

1. Parrnakían, J., waterhammer Analysis, Dover Publications, Inc., New York, 1963, p. 72.2. Kinno, H. and Kcnnedy, J. F., "Water-Hammer Charts [or Centrifugal Pump Systems,"

Jour., Hyd. Div., Amer. Sac. ai Civ. Engrs., vol. 91, May 1965, pp. 247-270.3. Ruus, E., "Charts for Waterhammer in Pipelines with Air Chamber," Canadian Jour. Civil

Engineering, vol. 4, no. 3, Septernber 1977.4. Ruus, E. and El-Fitiany, F. A., "Maximum Surges in Simple Surge Tanks," Canadian

Jour. Civil Engineering, vol. 4, no. 1, 1977, pp. 40-46.5. Wu, H., "Dimensionless Ratios for Surge Waves in Open Canals," thesis presented to the

University of British Columbia in partial fulfillment of the requirements for the degreeaf master of applied science, April 1970.

I APPENDIX B*

8-1 PROGRAM LISTING

C*** ANAlYSIS OF TRANSIENTS IN A PIPELINE CAUSED BY OPENING ORC ClOSING OF A VAlVEC

REAL lO I ME NS ION C 110,100) ,H I 10.100) ,CP 110,1001 ,HP I 10 tl 00 I ,C A 110 1, F 11 ü I ,

1 C FIlO 1, AR I 10 I ,A I 101 ,l( 10 I ,NIlO I ,O 110) ,Y I20 1, HM,\X I 10, 100 1,2 HMIN I1 0,100 I

OATA G/9.611CC REAOING ANO WRITING OF INPUT DATACC GENERAL DATA

REA015,101 NP,NRlP,IPkINT,CO,HRES,TlAST10 FORMATI313,5F10.Z)

WRITEI6,20) NP,NRLP,CO,HRES,TlASTZO FORMATIPX,'NUMBER OF PIPES =',13/8X,'NUMBER OF ~EACHES ON lAST PIP

lE =',13/8X, 'STEAOY STATE OISCH. ',F7.3,' M3/SEC' /OX,'RESERVOIR LE2VEl =',F7.1, ' ~'/BX,'TIME FOR WHICH TRANSIENTS AKE ro BE COMPUTEJ3 =',F7.1, ' SEC'/)

C DATA FOR VALVEREAOI5,Z5) M,TV,OXT,TAUO,TAUF,IYIII,I=l,MI

25 FORMATI12,F6.2,3FIO.2/12F6.21~RITEI6,271 M,TV,DXT,IYIII,l=l,MI

27 FORMAT(BX,'NUMBER OF POINTS ON TAU VS TIME CURVE =',12/BÁ,l'VALVE OPERATION TIME =',F8.2,' SEC'/6X,'TIME INTERVAl FOR STURING2 TAU CURVE ·',F6.3,' SEC'/8X,'STOREO TAV VALUES'/~X,15F6.3/1

C DATA FOR PIPESREAOI5,30) ILlII,OIII,Alll,FIII,I=l,NPI

30 FORMATI4FIO.31IIRITElb,401

40 FORMATI/8X,'PIPE NO', 5X, 'LENGTH',5X,'OIA',5X,'WAVE VEL.',5X,'Fkl2C FACTOR'/21X,'IMI',7X,'II'II',6X,'IM/SECI'/1

W R IT E I 6, 5 o I I 1,L I I I ,o I I 1 ,A I I 1 ,F I I I , I= 1 ,NP 150 FORMAIIIOX,13,6X,F7.1,3X,F5.2,5X,F7.1,11x,F5_31

OT=lINPI/INRLP*AINPIIWRlTEI6,511

51 FORMATI/6X,'PIPE NO',5X,'ADJUSTEO wAVE VEL'/25X,'IM/SEtl'/1C CALCVLATION OF PIPE CO~STANTS

00 60 I=l,NPAklll=0.7654*OIII**2AUNAOJ=AIIIAN=L IIII IDT*AI 11)NI II=ANANl=NlllIFIIAN-ANll.GE.0.5) NI I I=NI I 1+1AIII=LIII/IOT*NIIIIIIRITEI6,551 I,AII)

55 FORMATIIOX,13,12X,F7.1156 CAIll=G*ARIIIIAIII

C F I I 1= F I I 1*OT / I2. *0 I 1 I*AR I 1 I IF I 1 I=F I 1I *L I II ti 2 •*G*O I 1 I'ioNI I I*AR I 1 1**21

*The author or publisher shall have no liability, consequential or otherwise, of any kindarising frorn the use of the computer programs or any parts thereof presentad in Appen-dixes B through D.

469

470 Appendix B

60CC

CONTINUE

CAlCUlATION OF STEAOY STATE CONOITIONSHll,lI=HRES

C

70

DO 80 I=I,NPNN=NIII+lDO 70 J=l,NNHII,JI=HII,ll-IJ-l'·FIII*OO*·Z011 ,J 1 =00CONTlNUEHI l+l,ll=HI I,NNICONTINUENN=NINP'+1HS=HINP,NNI0$=00DO 85 1'= 1, NPNN=NIII+100 B5 J=I,NNHMAXII, J 1= H( I , J 1HMINII ,JI=H( I,JICONTINUET=O.OTAU=TAUOWRITEI6,881FORMAT (/ BX, I TIME' , ZX, '1AU' ,Z X, ' PIPE' , 7 X, 'HEAD 1M 1 ' ,7X, ' DI SCH •

l'IM3/S"/ZOX,'NO' ,5X,'Il1' ,5X, '(1'4+1" ,5X,'lll',5X,'IN+ll'/1K=O1= 1NN=NIII+lWRlTElb,1001 T,TAU,I.HIl.lI,HII,NNI,OII,lI,Oll,NNIFORMATIF12.1,Fó.3,14,2F9.Z,F9.3,FI0.31IF INP.EO.ll GO TO 150DO 14Q 1= 2, NPNN=NIII+lWRITElb,12011,HII,1I,HII;NNI,01I,lI,OII,NNI~OPMATIZOX,IZ,2F9.Z,F9.3,FI0.31CONTlNUET= T+DTK=K+lIFIT.GT.TlASTI GO TO 240

BO

85

B8

90

100

IZO140150

CCC

UPSTREAM RESEPVOIR

HP 11 oll=HRESCN=OI l,21-HI I,ZI*CAIlI-CF( 1 '.011, lI*ABSIOI 1,11 IQP[l,ll=CN+(.A[II"HRES

CCC

INTERIOR POINTS

00 170 l=l,NPNN=N[ I 100 160 J=2'NNCN=O II ,J + 1 I -CA [ II *H [ I ,J + lI-CF [ 1 1*0 [ 1 , J+ 11 "AB S [ O[ 1, J+ 11 ICP=OII,J-ll+CAI [I*HI [,J-l'-CFI 1'''01 [,J-ll*ABSICI I,J-lllOPI I,JI=O.S*ICP+CNIHP[ 1 ,J I=ICP-OPI I,J 1 l/CAl 1I

I'!

175CCC178

.,180

190

200

CCC210

Appendíx B 471

160 CONTINUE170 CONTlNUECC SERIES JUNCTIONC

NPl=NP-lIFINP.EO.ll GO TO 178DO 175 l=l,NPlNl= NIl INN=NIII+lCN=QII+1.ZI-CA(I+11*HII+l,ZI-CFI!+11*011+!,21*ABSI011+1,21

11C P=Q 11, N 1 I +CA I 11 *H I 1, NI 1-(. F I 1 I *0 I l. Nll*A&S [O 1 [ ,N 11 IHPI I.NNI=ICP-CNII ICAI II+CAI [+1 IIHPII+l,II=HPII,NNIOP 1 1 ,NNI=CP-CA 1 1 I *HPII, NNIOP 11 + 1,1 I =CN +C Al 1+ 11 *HP 1 1 + 101 ICONTINUE

VAlVE AT OOWNSTRcAM ENO

NN=NINPI+1CP=QINP,NN-ll+CAINPI*HINP,NN-ll-CF[NPI*QINP,NN-ll*ABSI~INP,

1 NN-tllIFIT.GE.TVI GO ro 180CAll PARABIT,DXT,y,TAUIGO TO 190TAU=TAUFIF ITAU.lE.O.OI GO TO 200CV=IOS*TAUI**?/[HS*CAINPIIQP[NP,NNI=0.5*I-CV+SORTICV*CV+4.*CP*CVIIHPINP,NNI=ICP-~PINP,NNIJ/CAINPIGO TO 210QP(!>¡P,NNI=O.OHPINP,NNI=CP/CAINPI

STORING VARIABLES FOR NEXT T[ME STEP

DO 230 1=I,NPNN=NINPI+lDO 220 J=l,NN011 ,JI=OPI 1 ,JIH [ 1 ,J 1 =H P I I , J IIF IH[ I,JI.GT.HMAXII,JII H~lAXI I.JI=HI I,JIIF IHI 1 ,JI.lT .HI4INI I.JII HI4IN( I ,JI=H( I,J I

220 CONTINUE230 CONTINUE

IFIK.FO.IPRINTI GO TO 90GO TO 150

240 WRITE16,2501250 FORUAT(/8X,'PI~E NO'.3X,'SECTION NO',jX,'MAX PRESS.',)X,

1 'MIN. PRESS.'/IDO 270 1=1,NPNN=N 111 + 1DO 270 J;l,NNWRITEI6,260J I,J,HMAXII,JI,HMINII,JJ

260 FORMATI9X, 12ol3X, 12.¿F13.21

472 Appendix B Appendix B 473

270 CONT INUE TIME TAU PIPE HEAD 11'11 O ISCH. (1'13/SISTOP NO 111 IN+lI 111 IN+11END

PARABIX,DX,V,ZISUBROUTINE 0.0 1.000 1 67.70 65.78 1.000 1.000DIMENSION YI201 2 65.78 60.05 1.000 1.0001=X/DX 0.5 0.963 1 67.70 65.78 1.000 1.0DOR= IX-I*DX 1/0X 2 65.78 63.46 1.000 0.989lFll.EQ.OI R=R-l. 1.0 0.900 1 67.70 68.73 1.000 0.9881= 1+1 2 66.73 69.78 0.988 0.970lFII.LT.211=2 -11111 1.5 0.813 1 67.70 74.16 0.977 0.967Z=VIlI+0.5*R*IV1I+II-VI ¡-lI+R*IVI I+lI+YI I-11-2.*{ 2 74.16 79.88 0.967 0.937RETURN 2.0 0.700 1 61.70 79.92 0.935 0.922ENO 2 79.92 95.83 0.922 0.884

2.5 0.600 1 67.70 88.25 0.867 0.8472 86.25 110.41 0.647 0.614

B-2 INPUT DATA 3.0 0.500 1 67.10 94.95 0.161 0.7552 94.95 125.13 0.155 0.722

3.5 0.400 1 61.10 99.18 0.643 0.6332 99.18 139.20 0.633 0.609

002002002 l. 67.7 10. 4.0 0.300 1 61.70 104.40 0.506 0.49607 6. l. l. O. 2 104.40 149.14 0.496 0.413

l. .9 .7 .5 .3 .1 .0 4.5 0.200 1 67.70 10B.47 0.350 0.344550. •75 UOO • .010 2 108.47 158.61 0.344 00325450. •60 90D • .012 5.0 0.10D 1 67.70 111 .20 0.183 0.177

2 lU.20 165.65 0.177 0.1665.5 0.038 1 67.70 113.07 0.006 ü.004

B-3 PROGRAM OUTPUT 2 113.07 149.46 0.004 0.0596.0 0.0 1 67.10 96.01 -0.17:;' - 0.106

2 96.01 114.28 -0.106 0.06.5 0.0 1 67.70 63.25 -0.217 -0.157

2 63.25 61.19 -0.151 U.ONUM6ER OF PIPES = 2 7.0 0.0 1 67.70 34.'<5 -0.139 -0.085NUMBER OF REACHES ON LAST PIPE = 2 2 34.25 12.33 -0.085 0.0STEAOV STATE DISCH. 1.000 M3/StC 1.5 0.0 1 67.70 23 .55 0.tl47 0.035RESERVOlf<.LEVEL = 61.1 M 2 23.55 6.75 0.035 0.0TIME FOR WHICH TRANSIENTS ARE TO BE COMPUTEO 10.O SE!:

B.O 0.0 1 67.70 47.63 0.208 0).1262 47.63 34.76 0.126 0.0

NUMBER OF POINTS ON TAU VS TIME CURVE = 1 8.5 0.0 1 67.70 82.89 0.205 0.148VAlVE OPERATlON TIME = 6.00 SEC 2 82.89 88.45 0.148 0.0T1M[; INTERVAL FOR STORING TAU CURVE = 1.000 SEC 9.0 0.0 1 67.70 10~.95 0.U86 0.054STOREO TAU VALUES 2 105.95 130.93 0.054 0.0

1.000 0.300 0.700 0.500 0.300 0.100 0.09.5 0.0 1 67.70 IJIl.OI -0.0'17 -O.OH

DIA i'I<c~~ 2 108.01 123.42 -0.071 U.OPIPE NO LENGTH WAVE '1EL. FRIe FACTO~

10.0 0.0 1 61.70 78.38 -0.229 -0.13'111'11 1M1 IM/SEC) 2 78.38 115.12 -o .139 U.O

1 550.0 0.750.L¡42 1100.0 0.010 PIPE NO SECTION NO MIl.XPKESS. MIN. PRESS.2 450.0 O.600?i:' 900.0 0.012

1 1 61.70 et , 70PIPE NO ADJUSTEO WAVE VEL 1 2 91. IR 44.05

(M/SECI 1 3 113.07 23.552 1 113.07 2;;.55

1 1100.0 2 2 140.26 9.542 900.0 2 3 165.t5 ~.42

APPENDIX e

C-l PROGRAM LISTING

Cc*** ANALYSIS OF TRANSIENTS IN A PIPELINE CAUSEO BY PUMPSC

REAL L,NR,NOOIHENSION O( 10,20) ,H( 10,20) ,OP( 10,20) ,HP( 10,ZO) ,CAl 10) ,F( 10),

1 CF(101,ARI101,A(10),l(101,N(10),OI10),FHI601,FB(601,HMAX(lO),2 HHIN(lO)

COMMON /CP/ALPHA,QR,V,CN,OALPHA,OV,BETA,CS,C6,NPP,TCOHHON /PAR/FH,FB,OTHDATA G/9.81/

CC REAOING ANO WRITING OF INPUT DATACC GENERAL DATA

PEAO(S,10) NP,NRLP,IPR[NT,NPP,QO,NO,TLAST10 FORMAT(4[2,5F10.2)

WRITEI6,20) NP,NRlP,QO,NO,TLAST,NPP20 FORMAT(8X,'NUHBER OF PIPES =',I3/8X,'NUMBER OF REACHES ON LAST PIP

lE =' ,13/BX,' STEAOY STATE OISCH. =' ,F6.3,' H3/S'/8X,' STEAOY STATE2 PUMP SPEEO =', F6.1,' RPM'/8X,'T[ME FOil.WHICH TRANS. STATE CONO.3ARE TO BE COMPUTEO =',FS.l, ' S'/8X,'NUM8ER OF PARALLEL PUMPS ='4 ,[311

CC REAO[NG ANO WRITING OF PUMP DATAC

REAO(S,21) NPC,OTH,QR,HR,NR,ER,WR2,IFH([),I=1,NPC)21 FORMAT(I2,6F10.2/17FIO.311

READ(S,22) (FBIII.!=l,NPC)22 FORMAT(7FIO.3)

WRITE(6,23)NPC,OTH,OR,HR,NR,ER,WR2,IFH(I),1=1,NPCI23 FORMAT(BX,'NUMBER OF POINTS ON CHARACTEIl.ISTIC CURVE =' .14/

1 8X,'THETA INTERVAL FOil.STORING CHARACTEIl.ISTIC CURVE =',F4.0/8X,2 'RATEO OISCH. =',FS.2,' H3/S'/8X,'RATEO HEAO =',F6.1,' M'/IIX,3 'RATEO PUMP SPEEO =',"6.1,' RPH' /BX,'PUMP EFFICIENCY =',F6.3/8X,;'4'WR2=',F7.2,' KG-H2'//RX,'POlNTS ON HEAO CH"R"C'/(8X.lOF7.31)

WRITE(6,2S) (FB((I,I=l,NPCI25 FOPMAT(/8X,'POINTS ON TORQUE CHARACTERISTIC'/ 18X,lOF7.3)1C DATA FOil.PIPES

REAOIS,301 (L(II,O( [1 ,AIII,F(II,I=l,NPI30 FORMAT(4FIO.31

WRITE(6,40140 FORMAT(/BX,'DIPE NO', 5X, 'LENGTH',SX,'OIA',5X,'WAVE VEL.',5X,'FRI

2C FACTOR'/22X,'IMI',6X,'(MI',7X,'(M/SI' 1WRITE(6,5011 I,L(I),O(II,A(II ,F(II,I=l,NP)

50 FORMAT(10X,13,6X,F7.1,3X,FS.2,SX,F1.1,11X,F5.31OT=LINP)/INRLP*A(NPI)WRITEI6,SlI

51 FOPMAT(/8X,'PIPE NO',5X,'AOJUSTEO WAVE VFL'/26X,'IM/S)'1C CALCULATION OF PIPE CONSTANTS

DO 60 l=l,NPAR 11 1=0. 7B54*01 II**2AUN"OJ=" 1 I1AN=U I 11( OT*A( 11)NII) =AN

474

Appendix C 475

55

ANl=N([)[F((AN-ANll.GE.O.S) NI[)=N(I)+lAl 11 = LI 1 1/1 OT*N 11) )WRITE(6,S5) I,AII)FORMAT( 10X, 13.lZX,F1.11CA(I)=G*AR(I)/A(IICF(II=F(II*OT/(Z.*O(I)*AR(I))F( I)=F( I I*L ([)/ (2.*G*0( I)*N( I )*IIR(I )**2)CONTINUE60

CCCC

COMPUTATION OF CONSTANTS FOR PUMPTHE FOLLOWING CONSTANTS ARE FOR SI UNITS. FOil.ENGLISH UNITS,REPLACE 93604.S9 8Y 59S.875 ANO 4.77S BY 153.744TR=193604.99*HR*QR)/(NR*ERIC5=CAIll*HRC6=-14.77S*TR*OT)/(NR*WR2)ALPHA=NO/NRV=QO/ (NPP*QR)OV=O.OOALPHA=O.O

CCC

CALCULATION OF STEAOY STATE CONO[TIONS

IFIV.EO.O.OI GO TO 6STH=ATANZ(ALPHA,V)TH=57.Z96*THGO TO 68TH=O.OCALL PARABITH,l,Z)HO=Z*HR*(ALPHA**2+V**2)HI 1,1I=HOCALL PARAB(TH,Z,Z)BETA=Z*(ALPHA**Z+V**2)DO 80 [=l,NPNN=NII)+1DO 10 J=l,NNH( I,J)=HI [,I)-(J-ll*F( [)*00**2IF(I.NE.NP.ANO.J.EQ.NN) H([+I,l)=HII,NN)Q(I,J)=QOCONT[NUEHMAX( II=H( Iol)HMIN( 1)=H([.11CONT INllENN=N(NP)+lHRES=H(NP,NNIT=O.OWRITE(6,85)FORMAT(/8X, 'TIME' ,ZX, 'ALPHA', 4X, 'V' ,4X, 'PIPE' ,7X,

1 'HEAO (MI',1X,'OISCH. (M3/SI'/29X,'NO.',SX,'(ll',SX,'(N+l)'l,5X,' (1)' ,5X, '( N+ll'/)

K=O1=1NN=N (1) + 1WRITEI6,86) T,ALPHA,V,I,HIl,lI ,H( 1 ,NN) ,QIl, 11 ,O( I,NNIDO 89 1=2,NPNN=NI 1)+1WRITEI6,971 1, H([.lI,!-f(I,NNI,Q(I,ll,Q([,NN)FORMAT(FIZ.l.F7.Z,F1.Z,I5,F9.1,F9.1,F9.3,FlO.31FORMAT (26X,15,2F9.1,F9.3,FI0.31CONTINUET=T+OTK=K+l

6568

70

80

85

90

8687B9150

11 160170

ill CC

IrlC

,~I(Ii

476 AppendixC

CCC

lFIT.GT.TL~STI GO TO 240

PUMP AT UPSTREAM ENO

CN~QI1,21_Hll,21*CAI1¡-CFI11*011,ll*ABSI011,lll

CAll PUMPOPIl,ll~NPP*V*QRHPI1,ll=IOPIl,ll-CNI/CAI1'

CCC

INTERIOR POINTS

00 170 l=l,NPNN=NI(I00 160 J=2.NNCN=O I [ ,J + L '-CAl II *H 1 1 • J+ 11-C F (1 '*0 ( 1 ,J + 11 *ABS 10 I 1, J + 11 ICP=Qll ,J-lI +CAI II*H( I.J-lI-CFI II *0 11. J-11 *ABSI 011 ,J-ll 1

OPIl,JI=0.5*ICP+CNIHP I 1 ,J I= (C P- 01'( 1, J I 11 CA I 1 ICONTINUECONTINUE

SERIES JUNCTlON

175CCC178

Npl=Np-llFINp.EO.lI GO TO 11800 175 I = l. N P 1Nl=NIIINN=Nlll+1CN=011+1.2 ¡-CAl 1 +11 *HI 1+1.21-CFI 1+11*QI 1+1,21 *....BSIQII+l,ZI

1¿P=Q I 1,N 11 +C A I 1 )*H I1, NU -CF I t J*0 11 ,Nl1 * ABS I01 1, N 111HP ( I .NN I = le P-CN )1( c« ( [ I..CA I 1+ 11 )HPIl+l,II=HPII,NNIOPII.NNI",CP-CAIII*HPII.NNJOPIl+l,11=CN+CAI1+11*HPI1+I,llCONTINUE

RESERVOIR AT DOWNSTREAM END

eee210

NN=NINPI +lHPINP,NNI=HRESCP=0INP,NN_11+CAINPI*HINP,NN-II-eFINP)*QINP,NN-II*ABSIQINP,

1 NN-lllOPINP,NNI=CP-CAINPI*HPINp,NNI

STORING ~AX. ANC MIN. PRESSURES ANO V ....RIABlES FOR NEXT TIMF STEP

00 230 l=l,NpNN=NINPI+100 220 J=I.NNOll,JI=OpII,JIH I I , J I =H p I I , J 1CONT INUEIF IHI I, 1l.GT .HMA)(I 1" HMAX[ I)=HII ,11t F [H II , 11 • lT •H M II'II 1 1 1 HM 1N I 1 1=1'1I1 .11CONTINUEIFIK.EQ.lpRINTI GO TO 90GO TO 150

220

230

Appendix C 477

240250

WRITEI6,2S01FORMATI//IOX,'PIPE NO.',5)(,'MAX. PRESS.'.5x,'MIN. PRESS.'/27X

1 .'M' , ló X, 'M' 1IWRITEló,26011I,HMAXII'.HMINIII,I=1,NPIFORMATI12X,13.7X,F7.1,9X.F7.11STOPENO

260

SUBROUTT NE PUMPDIMENSION FH1601,FB160JCOMMON ICp/AlpHA,OR,V,CN,DAlPHA,DV,BETA.C5.C6.NPp.T

1 IpAR/FH,FB,OTHKK=OJJ=O

CCe

COMPUTATION OF PUMp OISCHARGE

58la

VE=V+OVALPHAE=AlPHA+DAlpHAJJ=JJ+lIF IVE.EQ.O.O.AND.AlPHAE.EQ.O.OI GO ro 20TH=ATAN2IAlPHAE,VEITHl=THTH=TH*57.296IF ITH.lT.O.OI TH=TH+360.IF ITHl.lT.O.OI TH1=THl+6.28318GO TO 30TH= O. oTHl=O.OM"TH/DTHt-l.Al=FH(MI*M-FH(M+ll*IM-llA2=IFH(M+l'-FHIMII/IDTH*0.0174531A3=FBIMI*M-FB(M+IJ*IM-11A4=IFB(M+II-FBIMII/IOTH*0.0174531....LpSO·~lpH ....E*AlpH ....EVESO=VE*VEAl pV=AlP SO+VESOFl=eS*Al*AlPV+C5*A2*AlPV*THI-0R*VE*NPP+CNF2=AlpHAE-C6*A3*Alpv-e6*A4*AlPV*THI-AlPHA-C6*SETAFIAl=C5*12.*Al*AlpHAE+A2*VE+2.*A2*AlpHAE*THllFIV=C5*12.*Al*VE-A2*AlPHAE+2.*A2*VE*THll-OR*NPpF2Al=1.-C6*(Z.*A3*AlPHAE+A4*VE+2.*A4*ALPHAE*TH11F2V=C6*1-2.* ....3*VE+ ....4* ....lPH ....E-2.*A4*VE*THIJDENOM=FIAl*F2V-FIV*F2AlOAlPHA=(FZ*FIV-Fl*F2YI/DENOMDY"IFl*F2Al-FZ*FlAlI/OENOMAlPHAE=AlPHAE+OAlpHAVE=VE+DVIF (ABSIDVI.lE.0.001 .....NO .....BS(DAlPHAI.LE.0.OOll GO TO 50IF IJJ.GT.301 GC TO 70GO TO 8TH= ....TAN2IAlPHAE,VEITH=57.296*THIF ITH.l T.O.ol TH·TH"360.CAll PARABITH.2,BETAIIF IMB.EQ.MI GO ro 60

20

30

50

478 Appendix C

MB=TH/oTH+1IF IMB.EO.MI GO ro 60GO TO 8oALPHA=AlPHAE-AlPHAOV=VE-VALPHA=ALPHAEV=VEBETA= BETA * IALPHA*ALPHA+V*VIRETURNWR ITE 16,801 T ,AlPHAE, VEFoRMATI8X,'***ITERATIONS IN PUMP SUBROUTINE FAIlEo' /8X,'T=',F8.2

2/8X,'AlPHAE =' ,F6.3/8X,'VP =',F6.31STOPENo

SUBROUTINE PARABIX'J,ZICOMMON /PAR/FH,FB,oXolMENSloN FH1601,FB16011=X/OXR=(X-I*oXI/oXlFll.EO.OI R=R-1.1"1+1lFl1.LT .21 1=2GO rn 1l0,201,J

10 Z"'FHI11+0.5 *R*I FH 11+1I-FHI 1-11 +R*I FH 11+ 11+FHC [-11-2. *FH 111I1RETURN .

20 Z=FBC 11+0.5*R*1 FBII+l I-FBII-ll+R*IFBI [+ll+FBI 1-1'-2 ••FBI I1I1RETURNENO

60

1080

C-2 INPUT DATA

02020202 •'1·10 1100. 1~.55 :: . • :?5J 60.00 1100. .84 16.850

-. )~ -.I,7! -.392 -.291 - .15 o -.037 .075

.201) .,,,, .500 •655 •777 .900 1.007

1.115 1.1 ¡:;ó 1.245 1.278 1.29 o 1.~q1 1.269

1.~4'_' 1.201 1.162 l. 115 l. Cl69 1.025 .992

.945 .9'J8 .875 .848 .819 .7AQ .755

.723 .690 .65ó .619 .583 .55'i .531

.51'J .502 .500 .505 .52 o .539 .565

.593 .615 .634 .640 .li3E! .630

-.350 -.474 -.180 -.062 .037 .135 •228

.320 .425 .500 .54~ .588 • f, 12 .615

.000 .5t9 .530 .479 .440 .402 .373• 3~'J .340 .340 .350 .::80 .437 .520

.Ó05 .683 .750 .802 .845 ."177 .883

.878 .8FoD .823 .780 .725 .660 .580

.490 .197 .:110 .230 .155 .085 .018

- ..) ')~ -.123 -.220 -.348 -.490 -.680

45J.0 .FO 900.0 .01550.0 .750 1100.0C' .012

Appendix C 479

C-3 PROGRAM OUTPUT

NUMBER OF P[PES" 2NUMBER OF REACHES ON lAST PIPE = 2STEAoy STATE oISCH. = 0.500 M3/SSTEAoy STATE PUMP SPEEO =1100.0 RPMTIME FOR WHICH TRANS. STATE CONO. ARE TO BE COMPUTEONUMBER OF PARAllEL PUMPS = 2

15.0 S

NUMBER OF POINTS ON CHARACTERISTIC CURVE = 55THETA lNTERVAl FOR STORING CH~RACTERI STIC CURVE 5.RATEO OISCH. " 0.25 143/5RATEO HEAO = 60.0 MRAT!:O PUMP SP!:!:O"1100.0 RPMPUMP EFFICI ENCY = 0.840WR2" 16.85 KG-M2

POINT5 ON HEAO CHARAC-0.530 -0.416 -0.392 -0.291 -0.150 -0.037 0.075 0.200 0.345 0.500

0.655 0.771 0.900 1.007 1.115 l. 188 1.245 1.278 1.290 1.2R11.269 1.240 1.201 1.162 1.115 1.069 1.025 0.992 0.945 0.9080.815 0.B48 0.819 0.188 0.755 0.723 0.690 0.656 0.619 0.5830.555 0.531 0.510 0.502 0.500 0.505 0.520 0.539 0.565 0.5930.615 0.634 0.640 0.638 0.630

PO[NTS ON TOROUE CHARACTERISTIC-0.350 -0.474 -O.lBO -0.062 0.031 0.135 0.228 0.320 0.425 0.500

0.548 0.588 0.612 0.615 0.600 0.5ó9 0.530 0.479 0.4~0 0.4020.313 0.350 0.340 0.340 0.350 0.380 0.437 0.520 0.ó05 0.óil30.150 0.802 0.845 0.812 0.883 0.878 0.860 0.823 0.780 0.7250.660 0.580 0.490 0.397 0.310 0.230 0.155 0.0B5 0.018 -0.052

-0.123 -0.220 -0.348 -0.490 -0.680

.1

II

f

PIPE NO lENGTH OlA WAVE velo FRie FACTORIMI IMI 1M/SI

1 450.0 0.75 900.0 0.0102 550.0 0.75 11OO. o 0.012

PIPE NO AOJUSTEO WAVE VF'L(MI S 1

1 900.02 1100.0

TIME AlPHA V PIPE HEAD 1MI DI5CH. IM3/SINO • 111 'N+ 11 111 1"1+11

0.0 1.00 1.00 1 6C.0 59.6 0.500 0.5002 5S.Ó 59.0 0.500 0.500

0.5 0.72 0.12 1 30.7 59.6 0.359 0.5002 5S.6 59.0 0.500 0.500

1.0 0.56 0.59 1 11.7 21.5 0.297 0.3742 21.5 59.0 0.374 0.500

1.5 0.46 0.51 1 9.1 13.2 0.287 0.3182 13.2 59.0 0.318 0.248

2.0 0.39 0.5ó 1 5.3 36.3 0.219 0.1582 3é. 3 59.0 0.15B a.131

2.5 0.34 0.05 1 9.1 45.8 0.027 0.()842 45.8 59.0 0.084 0.068

3.0 0.32 -0.18 1 S.3 2ó.9 - O.091 -0.0592 26.9 59.0 -0.059 8.')32

480 Appendix e

3.5 0.30 -0.28 1 10.1 17.7 -0.139 -0.1312 17.7 59.0 -0.131 -0.185

4.0 0.26 -0.34 1 10.1 37.4 -0.168 -0.2702 37.4 59.0 -0.210 -0.294

4.5 0.15 -0.68 1 22.6 46.5 -0.341 -0.3422 4t.5 59.0 -0.342 -0.355

5.0 -0.05 -0.85 1 29.0 40.6 -0.425 -0.4262 40.6 59.0 -0.426 -0.391

5.5 -0.30 -0.93 1 32.4 38.6 -0.464 -0.4702 38.6 59.0 -0.470 -0.497

6.0 -0.55 -0.95 1 37.0 48.0 -0.476 -0.5382 48.0 59.0 -0.538 -0.548

6.5 -1).79 -1.04 1 51.3 55.1 -0.520 -0.5612 55.1 59.0 -0.561 -0.579

7.0 -1.01 -1.04 1 6~.4 61.3 -0.519 -0.5662 61.3 59.0 -0.566 -o. 51ft

7.5 -1.18 -1.00 1 74.8 67.7 -0.499 -0.5372 67.7 59.0 -0.537 -0.554

8.0 -1.30 -0.93 1 82.5 73.9 -0.464 -0.4932 73.9 59.0 -0.493 -0.500

8.5 -1.37 -0.86 1 87.1 76.1 -0.428 -0.4312 76.1 59.0 -0.431 -0.433

9.0 -1.39 -0.76 1 86.6 74.9 -0.379 -0.3682 74.9 59.0 -0.368 -0.363

9.5 -1.37 -0.66 1 82.2 72.2 -0.332 -0.3092 72.2 59.0 -0.309 -0.305

10.0 -1.33 -0.59 1 75.4 68.6 -0.293 -0.2662 68.6 59.0 -0.266 -0.257

10.5 -1.27 -0.53 1 68.3 63.8 -0.267 -0.2372 63.8 59.0 -0.237 -0.228

11.0 -1.21 -0.49 1 61.7 59.7 -0.247 -0.2252 59.7 59.0 -0.225 -0.218

11.5 -1.15 -0.48 1 56.3 57.1 -0.241 -0.2252 57.1 59.0 -0.225 -0.222

12.0 -1.11 -0.49 1 52.5 55.3 -0.247 -0.2362 55.3 59.0 -0.236 -0.232

12.5 -1.08 -0.52 1 50.4 53.8 -0.259 -0.2522 53.8 59.0 -0.252 - 0.250

13.0 -1.06 -0.55 1 49.4 53.2 -0.273 -0.2732 53.2 59.0 -0.273 -0.272

13.5 -1.05 - O.58 1 49.6 53.6 -0.290 -0.2932 53.6 59.0 -0.293 -0.295

14.0 -1.06 -0.61 1 50.7 54.4 -0.306 -0.3122 54.4 59.0 -0.312 -0.313

14.5 -1.07 -0.64 1 52.6 55.3 -0.320 -0.3272 55.3 59.0 -0.327 -0.330

15.0 -1.09 -0.66 1 54.7 56.5 -0.329 -0.3382 56.5 59.0 -0.338 -0.341

PIPE NO. MIIX. PR E55. MIN. PRE5S.M M

I 87.4 5.32 76.1 13.2

APPENDIX D

0-1 PROGRAM LlSTlNG

CeC

FREQUENCY RFSPONSf lF A StRJE~ P1PING SYSTEM HAV1Nb RESERVOJh ATTHE UPSTRlAM ENO ANU AN L5C1LATING VALVE Al THE DCWNSTRE~M ENe

CJMPLEX A.~.t.HV.~V.CC.CMPLXREAL LDIMlNSIDN L(~0),WV'¡D'.L'¿OI.~RI2U'.PI¡O,.CP(;Ol A(' ., "11 .,

1 C(2',2),F(ZOJ ' "i 't. "-,"- .,~EAD(5,lo) N,Ml,M2,M3,f~A(FURMATI413.FtO.~'R~Al (~.20) TAU.'.I-tO,Qt ... AMP.THPE"FIjPI4A TI 7HO .31WRITf 16.30) TAIJ",H(l.~"',t~p,THf'~R.NF;f(l'l~T~~X:'~HN"V!LVE ~~'l:.NIN;;, =' .... '.2 Ir.X, '~l.AllC Ht/í, =' ,~7.:',! "/~x, "'tAN ,ILOIAH·c =',F/.? ' M~/S' I~X,'AM~LITLlt:f ~t- VALvE

~('SCILATION<'~'.F5.! Ib~, 'lH1cUf'.l:TICAL fl"llC['l"t- lHl HPfLIr.JE 00'., Fb.3,' S'/(X.'NUMblf, CF PIrES ='. l?/IfEAC(5.~ul ILIII.QIII.WVIII.I~I.NIFL:f<M.ATI'fl¡'.21W" ITI:: 11>.<,ulFCRMATluX.'U·lc,H: 1"11'.:,1(,",1.6 IMI'.:'X.'WAVl VEL. IM/~")DO e o 1= I.N~1<lTE(6.~~I LIII,tlll I.WVIIIf~R"'AT(r16.2.fll.?.1~.~Irll'=U Il/WVII I(.D r r I =1.'1:)47"'1) ( II'.?/I<V(1)IN FNI.-Ll~HlJNIT~" H't-'LAL.~7.7(,4'( bY ~"'.'~<'fCúNTlNUfVC=-I¿.*HO*"MP JlT lo uoTW=o.;?H~.?/THPfR~"_lTEI6.6"If-'OK~Al r zex , 'wF/~T' ,~X., '1 ../HI~· ,(JX," I~.!I' ,4Y, ' ...·Ht~~t. ti' ,~~'W:, '¡'>t;A~E "'/)[in ~~(, J=M.l,~;·t~'PAJ:.J"'=~",I.C,,"tJ~TwAl 1,11=(MfoL ~ 1l. ,':•I~I 1.2 I:.('~f'LXIO."'.".I"Ih( 2.,1 );:Ctwlt-LX(l, .,;,41.(1)

A 1 ~ • ~ loo e ~p L ~ ( 1 .1; , c , ( I(;1' 70 1=1."(,=~.P(IID(l,ll=LMPL"(L~'GI.c.GIrll,21~CM~LXf0.G'-'1./(fll».!JNlb»)" f 2.11 =L"1f'L~ ("''''-o 11 I"~,lN' el )b , ¿ , ;' I =e Mf' L x I e e ~ I ( 1 • o • (J ICALL ""LTlh,A.e,¿.;n[tLL tOPYIC,~.Z.21C'lNT H;UEce- 'ICI ICI1,Z1-2.*Hf'''C12,::)/1,1')

HV=(.C'( 1 l. ~ I",V=I:(*CI2.:' I;'¡f:"Iof/T'~H=LM:,S Io-JVI/H('

{,"_.

481

482 Appendix D

(=CAb~II.'VI/U'A~GH=~7.2~57~.ATAN2IAIMAG(HVI.R[ALlhVIIANl,(=~,7.~"~ ¡t.ATAN~IAIMA(;II;;V' .REALf~VI I~~ll[lb.¿~1 w~.~,~,ANGH,ANGCeN11 NUF.,<:~.MATI:;X,~Fl{t.:'1STúP[NO~11~PI]IJT INE !'!lJL 1 IA ,E .c ,N,MIeúMfLtX A,B,e,CMPLX~IMEN5IUN AIN,NI.bIN.NI,CIN,N'DI_' 6 1=I.No:] " J=I.NCII,JI=CMPLXln.Ct,u.UI[~1 e "'=l.NCII,JI=AII,""*ijIK,JI + (II,JIf(-TlIfNFNO~UfkLUTINi (OPyIC,A.N.~Ier"'f-LlX "',eUIMfN~_Hjll: AIN,N',(II""",[JI] " 1=I,N[.;:'] f. J=I,NAl I,JI=CII,JIRITUkN~NO

0-2 INPUT OATA

002001020001 .~l. 30.4~

/,0".5 .61228.6 .3

.00é9121<;1.914.4

• ?

Appendix D 483

0-3 PROGRAM OUTPUT

MEAN VALVE OPENIN3 =1.000SrA1IC HEAD = 30.48 MMEAN DISCHARGE = 0.009 M3/SAMPLITUDE oF VALVE nSCILATIoNS =0.200THEORETICAL PEtHOD OF THE PIPELINE = 3.000 SNUMBER oF PIPES = 2

3 •

LENGTH (M I DIA (MI WAVE VElo (1.1/51609.<;1) 0.61 1219.00228.60 0.30 914.40

I'lF/WI H/HO 0/00 PHASE H PHASE O

0.')00 0.03 T 0.199 -9').2')7 -"J.2,)71.1)00 0.123 0.190 -107.887 -17.'3871.500 0.076 0.196 100.897 10.8972.000 0.046 0.199 -96.552 -6.')')22.500 0.149 0.186 -111.940 -21.9403.000 0.400 O.COO 179.999 89.9993.500 0.141,1 0.186 111.940 21.9404.000 0.046 0.199 96.,)"Jl 6.5514.')00 0.076 0.196 -100.898 -10.898':l.OCO 0.123 0.190 107.886 17.881'iS.500 0.037 0.199 9').2'17 5.2'176.000 0.000 0.200 -90.000 -o.nOO6.500 0.031 0.199 -95.2')8 -5.2587.000 0.123 0.190 -107.8813 -17.8887.')00 0.')76 0.196 100.'396 10.8968.000 0.046 0.199 -96.')'12 -6.5"J28.')00 0.141,1 0.18t- -111.941 ,-¿ 1.9419.000 0.400 0.000 179.99"J f30.9959."J00 0.149 0.1 BI'i 111.9,10 21.'-,1.19

10.000 0.046 0.199 96.')51 6.551

APPENDIX EAppendix E 485

REFERENCES

PUMP CHARACTERISTIC DATA* L Thomas, G., "Determination of Pump Characteristics for a Co~puterized TransientAnalysis," PrOC., First In ter. Conference on Pressure Surges, British Hydromech. Re-search Assoc., England, Sept. 1972, pp. A3-21 to A3-32.

2. Donsky, B., "Complete Pump Characteristics and the Effect of Spccific Speeds on Hy-draulic Transients," Jour., Basic Engineering ; Trans., Amer. Soco of Mech. Engrs., Dec.

1961,pp.685-699.

-1 N = 25 N, = 147 N. = 261f---:-s

9 = tan -7- h (J h

0(6h _L__

(Degrees) i.; ...Z:-l s:» O(Z-;-;Z 0(2 + l,

O - 0.530 - 0.350 - 1.560 - 1. 560 - 1.000 - 0.5605 - 0.476 - 0.47~ - 1. 290 - 1.200 - 0.948 - 0.600

10 - 0.392 - 0.180 - 1.035 - 0.895 - 0.892 - 0.60515 - 0.291 - 0.062 - 0.795 - 0.600 - 0.820 - 0.58020 - 0.150 0.037 - 0.540 - 0.355 - 0.665 - 0.50325 - 0.037 0.135 - 0.308 - 0.135 - 0.475 - 0.35530 0.075 0.228 - 0.032 0.060 - 0.275 - 0.16035 0.200 0.320 + 0.122 0.235 - 0.055 + 0.07040 0.345 0.425 0.310 0.380 + 0.200 0.32045 0.500 0.500 0.500 0.500 0.500 0.50050 0.655 0.548 0.635 0.51l0 0.7G5 0.62055 0.777 0.588 0.745 0.645 1.035 0.70860 0.900 0.612 0.860 0.695 1.280 /0.82565 1.007 0.615 0.932 0.755 1.508 0.95570 1.115 0.600 1.140 0.850 1.730 1.15075 1.1B8 0.569 1.365 0.970 1.970 1.41380 1.245 0.530 1.595 1.115 2.225 1.60885 1.278 0.479 1.790 1.300 2.485 1.78090 1.290 0.440 1.960 1.485 2.740 1.96095 1.287 0.402 I 2.048 1. 518 2.980 2.150

100 1.269 0.373

I2.110 1.540 3.195 2.345

105 1.240 0.350 2.158 1.545 3.380 2.525110 1. 201 0.340 I 2.203 1.560 3.515 2.710115 1.162 0.340 I 2.258 1.592 3.572 2.900120 1. 115 0.350 I 2.315 1.642 3.570 3.000125 1.069 0.380 , 2.390 1.720 3.490 3.010130 1.025 0.437 I 2.495 1.900 3.350 2.925135 0.992 0.520 2.630 2.090 3.140 2.760140 0.945 0.605

I2.735 I 2.315 2.875 2.500

145 0.908 0.683 2.905 2.530 2.570 2.245150 0.875 0.750 3.000 ! 2.650 2.300 1.990155 0.848 0.802 I 3.020 2.720 2.065 1.750160 0.819 0.845 2.975 2.740 1.840 1.518165 0.788 0.872 2.825 2.635 1.633 1.30017J 0.755 0.883 2.652 2.535 1.440 1.085175 0.723 0.878 2.442 2.310 1.260 0.870180 0.690 0.860 I 2.135 2.090 1.0BO 0.660185 0.656 0.823 i 1.890 1.850 0.920 0.500190 0.619 0.780 1.525 1.570 0.780 0.505195 0.583 0.725 1.195 1.250 0.710 0.555200 0.555 0.660 0.935 0.955 0.670 0.615205 0.531 0.580 0.695 0.730 0.660 0.630210 0.510 0.490 0.500 0.530 0.555 0.500215 0.502 0.397 0.374 0.350 0.410 0.315220 I 0.500 0.310 0.277 0.175 0.265 0.100225 0.505 0.230 0.190 0.000 0.065 - 0.075230 I 0.520 0.155 I 0.114 - 0.160 - 0.140 - 0.315235 0.539 0.085 0.058 - 0.295 - 0.345 - 0.515240 0.565 0.018 - 0.015 - 0.425 - 0.550 - 0.715245 0.593 - 0.052 -0.110 - 0.550 - 0.745 - 0.880250 0.615 - 0.123 - 0.220 - 0.670 - 0.960 - 1.030255 O.63~ - 0.220 - 0.334 - 0.820 - 1.200 - 1. 225260 0.640 - 0.34:1 - 0.449' - 0.992 - 1.480 - 1.450265 0.638 - 0.490 - 0.5,,0 - 1.213 - Ull O - 1.~6()

270 I 0.630 - O.GUO - 0.670 - 1.500 - 2.200 _ 2.200

*These pump characteristic data are based on data presented by Thomas1 and Donsky.2For notation. see Section 4.3.

APPENDIX F SI and English Units and Conversion Factors 487

SI AND ENGLISH UNITS ANDCONVERSION FACTORS

F-2 CONVERSION F ACTORS

To Convert

SI (Systérne Internationale) units for various physical quantities are listed inSection E-I, and the factors for converting thern to the English units are pre-sented in Section E-2.

Quantity From SI unit To English unit Multiply by

Acceleration m/s2 ft/sec2 3.28084m2 ft2 10.7639Area

kg/m3 Ib/ft3 62.4278 X 10-3Densitykg/m3 slug/ft3 1.94032 X 10-3

Discharge m3/s ft3/sec 35.3147m3/s gal/min (U.S_) 15.8503 X 103

m3/s gal/min (1 mperial) 13.1981 X 103

Force N Ibr 224.809 X 10-3Length m ft 3.28084

kg lb 2.20462Mass68.5218 X 10-3kg slug

Moment of inertia kg m2 lb-ft2 23.7304Momentum (Angular) kg m2/s lb-ft2/sec 23.7304

(Linear) kgm/s lb-ft/sec 7,23301Power W ft-lbr/scc 0.737561

W hp 1.34102 X 10-3Nm lbr-ft 737.562 X 10-3Torquemis ft/sec 3.28084VelocitymIs mile/hr 2.23694

Volume m3 rt3 35,3147m3 yd3 1.30795m3 in.3 61.0237 X 103

Specific weight N/m3 Ibr/ft3 6.36587 X 10-3Ternpera ture oC °F 1.8; and

add 32

I H F :: 55(1 lb H/5

E-I SI UNITS

Physical Quantity Name of Unit Syrnbol DefinitionLength Meter mMass Kilogram kgForce Newton N 1 kg m/s2Energy Joule J 1 NmPressure, stress Pascal Pa 1 N/m2Power Watt W 1 J/sBulk modulus

of elastici ty Pascal Pa 1 N/m2

The multiples and fractions of the preceding units are denoted by the follow-ing letters:

10-3 milli m10-1 deci d103 kilo k106 mega M109 Giga G

For exarnple , 2.1 GPa = 2.1 X 109 Pa; 1.95 Gg m2 = 1.95 X 106 kg m2.

486

r=-::"~~".,_,-...','_-....j =¡::~:L ¡.,,~._.,~., ".~.'"

AUTHORINDEX

AUTHORINDEX

Aanstad, 0.,181,182,185,187

Abbot, H. F., 208, 271,444

Abbot, M. B., 42,72, 188Albertson, M. 1.,189, 329Allievi, L., 4, S, 23, 24,

42,206,271,307,329Almeras, P., 156Amein, M., 423, 444-446Amorocho, J., 429, 447Amsden, A. A., 188Anderson, A., 4, 23Andrews, J. S., 189, 329Angus, R. W., 6, 24, 307,

329Araki, M., 157Arker, A. J., 189

Babcock, C. 1.,431,447Bagwell, M. U., 200Baltzer, R. A., 279, 283,

292,299,445Banerjee, S., 179, 180, 188Barbarossa, N. 1.,370,381Bates, C. G., 157Baumeister, T., 43Bechteler, W., 330Bedue, A., 301Bell, P. W. W., 107,330Belonogoff, G., 72, 187Benet, F., 411, 446Bergeron, L., 6, 24,42,

206,271, 299Bernardinis, B. D., 278,

299Binnie, A. M., 7, 25, 200Blackwell, W. A., 271Blade, R. J., 273Boari, M., 42

Bonin, C. C., 17, 19, 26Bordelon, F. M., 187Borot, G., 329Boure, J., 189Boussinesq, J., 300Bowering, R. J., 431, 432,

447Bradley, M. J., 331Braun, E., 5, 23, 24Brebbia, C. A., 446Brebner, A., 431, 447Brekke, H., 157Brown, E. A., 187Brown, F. T., 73Brown, R. J., 108, 282,

283,293,296,297,299Bryce, J. B., 381Burnett, B. R., 199Butcher, J. C., 188, 380Butler, H. 1.,432,448

Cabelka, J., 7, 25Carnichel, C., 6,24,206,

252,271Carlton, J. L., 187Carpenter, R. C., 4,23Cass, D. E., 107,448Chan, T. C., 189Chaudhry, M. H., 26, 60,

63,72,107,111,124,135, 155-157, 188, 210,252,272,273,324,330,331,337,342,366,380,408,415,445,448

Chen, W. L., 187Chow, V. T., 444Chu, H. L., 445Clame, J., 7, 25,426,446Collatz, 1.,59,72Combes, G., 73, 329

489

Concordia, C., 156Constantinescue, G., 5, 6Contractor, D. N., 73Cooley, R. L., 445Cooper, K. F., 189Courant, R., 72,165,404,

446Crawford, C. C., 329Cunge, J. A., 411, 429,

445-447Cunningham, W. J., 380

Daily, J. W., 301Das, M. M., 431, 447Davidson, D. D., 447Davis, J. M., 445Davis, K., 301D'Azzo, J. J., 155De Fazio, F. G., 445De Saint-Venant, B., 444Den Hartog, J. P., 208, 271Dermis, N. G., 156Deriaz, P., 272Díjkrnan, H. K. M., 283,

300Donelson, J., 273Donsky, B., 106Dorn, W. S., 155, 188,

272,380,446Dresser, T., 200Driel, M., 330Drioli, C., 429, 446Dronker, J. J., 423,446D'Souza, A. F., 273Duc, J., 108, 301Duc, F., 6, 24Duncan, J. W. L., 381

Eagleson, P. S., 42,72Edwards, A. R., 177. 188

490 Author lndex Author Index 491.1

Elder, R. A., 381 Gray, C. A. M., 7, 25, 72 Joukowski, N. E., 4, 5, 6, Lescovich, J. E., 329 Moshkov, L. V., 273 Raney, D. e., 432, 488El-Fitíany, F. A., 468 Graze, H. R., 309, 329 43 Lewís, D. G., 189 Mosonyi, E., 379 Rateau, A., 7,25Enever, K. J.,42 Green, J. E., 200 Lewis, W., 273 Mueller, W. K., 273 Rayleigh, J. W. S., 5, 23Engler, M. L., 307, 329 Grolmes, M. A., 189 Kachadoorian, R., 447 Lewy, H., 446 Reed, M. B., 272Escande, L., 7, 2S Gromeka, 1. S., 4, 23 Kadak, A., 330 Li, W. H., 301, 380 Nakamura, S., 188 Resal, H., 4, 23Esposito, V. J., 188 Guerrini, P., 42 Kalinin, A. V., 189 Liebermann, P., 187 Newey, R. A., 157 Rich, G. R., 6,24,42,107,Euler, L., 4, 23 Guymer, G., 38, 43 Kalkwijk, J. P., TH., 281, Ligget, J. A., 445 Newton, L, 3, 23 379,444Evangelisti, G., 7, 25, 42, Gwinn, J. M., 187 299.300 Líndros, E., 330 Nicholls, R. L., 446 Richard, R. T., 6,24,301

44,48. 63, 71. 73, 195, Kamphuis, J. W., 431, 432, Lindvall, G. K. E., 43 Noda, E., 431, 447 Richtmyer, R. D., 188,200.272,314.329 de Haller, P., 301 446,447 Linton, P., 107 O'Bríen, T. P., 59, 72,177, 300,445

Evans, W. E., 329 Kao, K. H., 446, 448 Lister, M., 42, 44, 48, 72, 188 Riemann, B., 3, 23Eydoux, 0.,24, 271 Ha11iwell, A. R., 35, 37, 42 Kaplan, M., 60,72, 199, 200 Okrent, D., 187 Ripkin, J. F., 299

200 Lokutsievski, O. V., 445 Roark, R. J., 43Fabic, S., 189 Hamill, F., 107

Kaplan, S., 72 Lowy, R., 6, 24 Oldenburger, R., 156,273Roberts, W. J., 273

Fahlberg, E., 380 Hammitt, F. G., 301Kasahara, A., 446 Ludwig, M., 199, 200 O'Leary, J. R., 187 Rocard, Y., 26, 209, 271

Fang, e. S., 444 Hancox, W. T., 170, 188Kasahara, E., 276, 299, Luk, e. H., 187 Olsen, R. M., 299 Rose, R. P., 189

Fashbaugh, R. H., 271 Harding, D. A., 42301 Lundberg, G. A., 200 Padmanabhan, M., 188, Rosl, G., 250, 272

Fauske, H. K., 189 Harlow, F. H., 188Katto, Y., 273 Lundgren, C. W., 330 300 Rossí, R., 42

Harper, C. R., 187Favre, H., 269, 272, 446 Helmholtz, H. L., 3 Kawa, D., 188 Lupton, H. R., 6, 24, 301 Parmakian, J., 6,25,37, Reuse, H., 23, 41, 444Federici, G., 299 Henry, R. E., 189 Kennedy,J. F., 108,468 42,72,106,107,155, Ruus, E., 6. 25, 157,329,Fett, G. H., 157 Henson, D. A., 73 Kennison, H. F., 43 Mahmood, K., 444 156,271,299,330,468 330,342,380,381,459,Físcher, S. G., 72, 155

Higikata, K., 189 Kephart, J. T., 301 Marchal, M., 75, 17, 106 Parrnley, L. J., 107 463,464,468Flammer, G. H., 42 Hilbert, D., 444 Kerensky, G., 26 Marey, M., 4,23 Papadakis, C. N., 329Flesch, G., 106 Hill, E. F., 157 Kerr, S. L., 7, 24, 200,330 Marris, A. W., 7, 26, 342, Parzany, K., 157 Safwat, H. H., 43, 299Florio, P. J., 273 Kersten, R. D., 200 352,367,380 Patel, Y. A., 187 Saito, T., 208, 271Forrest, J. A., 329 Hirose, M., 27,42

Kiersch, G. A., 447 Martín, e. S., 73, 188, 189, Patridge, P. W., 446 Salmon, G. M., 330Forstad, F., 447 Hold, A., 189

Kinno, H., 108, 331,468 300,329,424,445,446 Paynter, H. M., 7,25,26, Sehleif, F. R., 157Forster, J. W., 330, 366- Holland, L. K., 187

Kirchmayer, L. K., 156 Mathers, W. G., 188 135,147,156,209,261, Schnyder, O., 6,24,107369,373,380 Holley, E. R., 273

Kittredge, J. P., 106 Matthias, F. T., 381 271,342,360,367,370, Schüller, J., 7, 25, 341,

Forsythe, G. E., 444 Houghton, D. D., 446 Klabukov, V. M., 155 MeCaig, 1. W., 208, 271 380 346,379Fox, J. A., 73 Houpis, C. H., 155 Knapp, R. T., 106, 301 McCalla, T. R .• 380 Pearsall, 1. S., 42, 279, 299, Sharp, B. B., 299, 300Fox, P., 72 Hovanessian, S. A., 271 Kobori, T., 299, 330 MeCartney, B. L., 447 330 Sheer, T. J., 73Franc, 1., 7, 2S Hovey, L. M., 135, 140, Komine, A., 189 McCracken, D. D., 155, Penzes, L. E., 187 Shen, e. C., 423, 446Frank, J., 7, 25, 341, 346, 146,156 Korteweg, D. J., 4, 5, 23 188,272,380,446 Perkíns, F. E., 42,59, 72, Shin, Y. W., 187

379 Hsieh, J. S., 187 Koutitas, e. G., 448 McCulloek, D. S., 447 111, 155 Shugrin, S. M., 444Friedricks, K., 446 Hsu, S. T., 329 Kranenburg, C., 188, 281, McDonald, B. H., 188 Pestel, E. e., 272 Siccardi, F., 281, 299Frízell, J. P., 4, 23 Hyrnan, M. A., 72 282,288-290,292,299, Meeks, D. R., 331 Phillips, R. D., 200 Sideriades, L.. 142, 380Funk, J. E., 43 300 Menabrea, L. F., 4, 23 Pickford, 1.,42,379 Siemons, J., 283, 300

Ikeo, S., 330 Krivehenko, G. l., 111, Mercer, A. G., 448 Pipes, L. A., 271 Silberman, E., 299Gaden, D., 8, 25 Inee, S., 23 155 Meyer-Peter, E., 426, 446 Pool, E. B., 187 Solbrig, C. W., 187Gadenberger, W., 329 Ippen, A. T., 42,72 Krueger, R. E., 156, 380 Miehaud, J., 4,23 Porschíng, T. A., 189 Spencer, e. A., 188Garabedian, P. R., 444 Isaacson, E. J., 444,445 Kuwabara, T., 157 Mikielelewicl., J., 189 Portfors, E. A., 72,155, Standart, G., 189Gardel, A., 7,25 Isbin, H. S., 187 Miller, D. J., 447 330 Stark, R. M., 446Gariel, M., 24, 271 Lackíe, F. A., 272 Mises, R., 72 Porwit, A. J., 187 Stein, T., 156, 157Gear, e. W., 380 Jacobson, R. S., 330 Lagrange,1. L., 3, 23 Miyashiro, H., 73,107, Prasil, F., 7, 25 Stepanoff, A. J., 107, 300Gelfand, 1. M., 445 Jaeger, e., 6,24,25,107, Lai, e., 8, 25,42,60,72, 108,299 Priessrnann, A., 429, 430, Stephenson, D., 107Gibson, N. R., 6, 24 248,252,272,366,370, 73,444,445 Moek, F. J., 43 445 Stoker, J. J., 71, 423,444,Gibson, W. L., 208, 271 379,380,381,429,446 Laplace, P. S., 3, 23 Moffette, T. R., 187 Prins, J. E., 431, 447 446Goldberg, D. E., 42 Jenker, W. R., 38, 43 Larsen, P. S., 43, 299 Moin, S. A., 445 Pulling, W. T., 26, 184, Stoner, M. A., 42, 72Goldfish, A. L., 189 Jeppson, R. W., 42 Lathi, B. P., 155, 271 Molloy, e. T., 272 187 Streeter, V. L., 7, 25, 26,Goldwag, E., 157 Johnson, G., 330 Law, L., 431, 447 Monge, G., 3, 23 41-44,48,60,72,73,Gonzalez, S., 189 Johnson, R. D., 7, 25, 367, Lax, P. D., 164, 165, 188, Moore, J. S., 187 Rachford, H. H., 72 106,107,155,199,200,Goodykootz, J., 273 381,388,444 288,289,300,445 Mori, Y., 189 Rahrn, S. L., 43,381 271,282,299,300,329Gordon, J. L., 135, 151, Johnson, S. P., 199 Léaute, A., 7, 25 Morton, K. W., 188,300, Raiteri, E., 281, 299 Strelkoff, T., 423,429,

156 Joseph, l., 107, 300 Lein, G., 157 445 Ralston, A., 71, 200 444,445

492 Author Index

Strowger, Eo s., 6, 24,330Swaffield, Jo Ao, 43,282,

300Swarninathan, K. v; 43Swanson, Wo Mo, 107, 300Sweicicki, 1., 156Suter, Po, 106

Tacho, Ro, 200Tadaya, l., 273Tanahashi, r., 276, 299,

301Tedrow, s: c., 42, 72Temnoeva, r. s., 444Terzidis, Go, 423,445Thorna, Do, 7, 25, 341,

367,380Thornas, c., 77, 107Thomson, Wor., 271Thorley, Ao Ro Do, 38,39,

43,272Thorne, Do n., 157Tirnoshenko, So, 41Tison, c., 43Todd, Do, 72Travers, r. Jo, 381Trikha, s: x., 43Troesch, Bos; 444, 445Tucker, Do Mo, 329Twyrnann, Jo Wo Ro, 39, 43

Vasiliev, 00 r.. 444, 446Vaughan, Do Ro, 157Verwey, Ao,445Vogt, r., 7, 25Vreugdenhil, c. s., 283,

300Vries, s: a., 300

Walker, Ro Ao, 381Waller, z. Jo, 200, 209, 271Wallís, c. Bo, 188Walsh, Jo Po, 301Wasow, Wo Ro, 444Watters, o. Zo, 42Weber, Wo, 5,23Webster, Ao o., 71Wegner, Mo, 429, 445Wender, Po Jo, 187Wendroff, n., 165, 188,

288,289,300,445Weng, c., 273Wentworth, Ro c., 72Weston, E. Bo, 4, 23Weyler, Mo s., 43, 276,

278,283,299Whiternan, x. Jo, 299,330Widrnann, Ro, 330Wiegel, Ro L., 431, 447Wiggert, Do c., 188,429,

447Wilf, n. So, 71, 200

Wilson, D. c., 189Winn, c. s., 330Wood, Ao s., 299Wood, Do Jo, 43,331Wood, r, Mo, 4, 6, 22,42Woolhiser, Do Ao, 445Wostl, Wo Jo, 200Wozníak, Lo, 157Wu, a., 466, 467, 468Wylie, c. Ro, 42,156,188,

271,380Wylie, s. s., 25,42, 44,

48,60,72,73,107,199,200,209,252,264,271,272,282,300,329

Ybarrondo, Lo Jo, 187Yen, B. eo, 444Yevjevich, Vo, 444Yokoyarna, So, 299Young, o. s. Jo, 329Young, r., 3, 23Yow, Wo, 60, 61, 72

SUBJECT INDEX

Zaoui, Jo, 73Zielke, Wo, 27, 41,250,

272Zolotov,L. Ao, 155Zovne, Jo Jo, 424, 446Zuzak, WoWo, 188

SUBJECT INDEX

Accumulator, 161Actuator, 121, 122Adiabatic process, 296, 309Air cavity, 283, 295Air chamber, 4-6, 95, 198, 292, 303, 306-

308,310,311,328,456,457charts for, 459-461boundary conditions for, 308-311

Air compressor, 306, 307Air pocket, 186,295,296,315Air, entrapped, 161, 168, 195

boundary conditions for, 168Air release, 162Airvalve, 95-97, 104, 186, 208, 293, 312,

315, 320, 322, 323boundary conditions for, 320-323

Amplitude, 201-203,206,209, 237, 239,241,245,248,252,257,264,270,338,339,360

Antinode,203,248,249Arithmetical method, 337Area, Thoma, 7, 342, 353, 357,366

Barometric pressure head, 309, 322, 323,462

Block diagram, 120,121,210,212,216,241,245,247,278

fOI branch system, 243for parallel system, 224for series systern, 247

Booster pumping station, 100-102Bore, 17,382,384,389,394,395,397,

424,425,432Boundary conditions, 34,44,46,48,62,

70,86,96,100,109-111,126,153,158,186,209,210,231,232,246,

257,282,303,304,308,328,399,404,405,408,433

for air chamber, 308for air-inlet valve, 320for branching junction, 57for centrifuga! pump, 58, 74for condenser, 165for dead end, 53for downstream boundary, 428fOI downstream reservoir, 52, 93, 98, 400,

401for entrapped air, 168for Francis turbine, 5S, 114for junction of three channels, 433for orifice, 55for pressure-regulating valve, 67for pressure-regulating valve and centrifu-

gal pump, 317for pressure-regulating valve and Francis

turbine, 325for surge tank, 304for seriesjunction, 55, 93, 98, 400f402,

433for upstream boundary, 418, 424for upstream reservoir, 51, 98, 399, 400for valve, 54for valve at intermediate location, 70, 71

Boyle's law, 3

Branch channe1, 420,421Branch lines, 4Branch with dead-end, 232. with oscillating valve, 233with reservoir, 233

Bulk modulus of elasticity, 10, 32, 280, 284of common liquids, 37

493

494 Index

Bypass valve, 198, 208Bubbles, 162, 275, 279, 280, 284-286

Canals, 18, 109, 110, 192,410,411pcwer, 17,382,425navigation,3B4

Cavitation, transient, 274, 275, 281, 2B2,292

Cavitating flow, 274-276, 278, 281, 282,285,298

Cavity, 2, 274-276, 282, 283,285,292,296

Celerity, 3, 383, 384, 387,465Characteristíc curves, 172, 173Characteristic directions, 290Characteristic equations, 49, 56, 57,144,

167,174,296,304,30B,31B,320,326,395-397,399,418,419,424

Characteristíc grid, 100Characterístic Unes, 46, 48Characteristic method (see Method of

characteristics)Characteristic impedance, 219Characteristic roots, 343Characteristics, pump, 6, 74, 75, 77,79,80,

89, IOO, 103,104,162,293,295,4B4

Check valve, 95, 96,102,104,105,161,182,183,198,208,307,312,315,317,457

Chezy 's formula, 392Closed surge tank, 334, 377, 379Coefficient of discharge, 54, 227, 364Column separatíon, 2, 102-104,274-276,

281,282,290-293,298,302,303,306, 315 (see also Water-columnseparation)

Column vector, 213, 222, 229,420Compatibility equations, 46, 47, 173,290Complex variable, 250Computer

analog,377,342,366,370digital, 34, 74,104,135,190,337,366,

370,423Cornputer analysis, 208, 394Computer program, 63,93,97,100,104,

129for transients caused by opening or

closing valve, 469-473for transients caused by power failure,

474-480

fOI frequency response of series system,481-483

Condenser, 160, 164, 165, 166, 186boundary conditíons for, 165

Conduits, 16, 27, 33,109, no, 129, 332noncircular, 38rigid, 35, 334thick-walled, 35thin-walled, 36

Continuity equation, 4,27, 30, 33, 34,40,44,100,169,170,195,209,216,229,283,284,288,298,334,337,362,363,372,382,386,388-390,393,394,433,442,443

Contraction losses, 407Control devices, 20, 95, 100, 198,292,453Control gates or valves, 18, 161,334,384,

401,467Control systems, 120, 208Control volume, 7,9,30,284,385,386,

389,390-393Controlling transients, methods oí, 302, 303Convergence of finite-difference scheme, 44,

58,59,70Conveyance factor, 417Coolants, reactor, 159-161Cooling systems, ernergency core, 160, 161Courant's stability conditíon, 59-61,71,

129,165,170,404,405,408,422Cushioning stroke, 130, 133

Dam-break, 19, 384,423,424Darcy-Weisbach friction factor, 29,30,217,

457Dashpot, 122Dashpot spring, 119Dashpot time constant, 129, 137,152Dead band, 137Dead end, 53, 237, 238, 250

boundary condition for, 53Degrees of freedom, 203Delay time, 315Density, 171, 191,278,323

of common Iíquíds, 37Design charts, 449Design crítería

for penstocks, 130for pipelines, 74

Differential equationsordinary, 16, 34,45,120,123,137,139,

142, 143, 16~ 164, 165, 172,173,

203,209,290,337,338,341,343,346,395,406

partial, 6, 8,16,33,34,40,44,45,162,164,172,203,209,222,257,288,394

Differential orífice, 306Dífferential surge tank, 7, 333, 334, 362,

366,367Diffusion coefficient, 286Disc, rupture, 169Discharge

fluctua ting, 238instan taneous, 211mean, 211

Discharge fluctuations, 252Díscharge line, 74, 75, 94, 100, 105, 206,

238,249,454,456,457,462Discharge valve, 74, 81,97,102,103,208Distributed system, 163Distributed-system approach, 162, 164Distribu ting valve, 122Downsurge, 366, 367, 376, 440,456,459,

460,462"Draft tube, 110, 114, 129, 150,376Drainage, surge tank, 340, 365, 366, 372,

373,374Dronker's scheme, 397Dynamic equation, 4, 5, 27,33,34,40,44,

100,169,195,209,216,217,283,334,335,361,363,372,382,389,390,392,394,433,442,443

Elasticity, 6bulk modulus of, 10,32,280,284Young's modulus of, 31, 36, 280, 284

End conditions, 50, 62,186,250Energy

kinetíc, 2, 11elastic, 2, 11

Energy, dissipation or, 278, 282Energy equation, 163, 170,402,406,407Enthalpy, 171, 177Entrance losses, 399.450Equivalent pipe, 127, 165, 171,449,450Exciter, 206, 207Expansion losses, 407Explicít finite-dífference method, 164,

290,291,382,395,397,414,415,422-424,433,442

Explicit leapfrog scherne, 424

I

Index 495

Factor of safety, 95, 96,130,133,190,302Favre's waves, 411, 425Feedwater line, 178, 179, 181, 182, 185Field matrix, 201, 214-216, 231, 241, 242,

244,245,247,270for conduit having variable characteristics,

221, 223for parallel loops, 225for single conduit, 216, 218-220

Field tests, 257, 294 (see also Prototypetests)

Finite-difference method, 44, 48,164, 177,209,283,287,337,395,404,406

Firrn capacity, 375Floods, 382, 384Flow

combined free-surface pressurized, 16, 17,426,430,431

gradually-varied, 17,382,406,423,425laminar, 27nonuniform, 1ene-dimensional, 27, 284, 383, 389periodic, 203,425rapidly varíed, 17,382steady, 1,46, 385, 386steady-oscillatory, 1, 203, 204-206, 208-

210,216,264,425subcrítícal, 389, 424supercritical, 389, 424transient, 1, 2, lB, 44,127, 195,201,

382,442turbulent, 1,27uniform , 1

Flowchart, 63, 64for boundary condition fOI Francis

turbine, 125for pump-pressure regulating valve, 321for series piping system, 64

Flowmeter, 127Fluid

compressible, 3, 16density, 9, 36incompressible, 3, 4

Flyballs, 119Flywheel,292Forcing functions, 206, 209, 227, 234,

237,241,245,250,257fluctuating discharge, 237, 238lluctuating pressure head, 237oscíllating valve, 237, 240, 241

496 lndex

Fourieranruys~, 209, 227, 235, 239,240,242

Francis turbine, 58, 67,71,109,111,114,124,126,127,135,147,153,208

boundary conditions for, 58, 114Freebody diagram, 28, 304, 361, 362, 390Frequency, 2, 17,203, 211,236,239,240,

242,247narurru, 201, 203,206, 270resonant, 210, 212, 250, 252, 257, 269system, 109, 133,341

Frequency domain, 208, 209Frequency response, 201, 210, 235, 237,

238,241,242,250,252,257,259-262,267-270

procedure for determining, 241Frequency response diagram

of branch system, 244, 262of parallel system, 261of series system, 260

Friction factor, 29, 33,68, 103, 129, 171,190,367,449

Darcy-Weisbach, 29, 30, 217,457Hazen-Williams, 30

Friction losses, 2, 4, 6, 30, 97, 103, 111,113,195,209,389,411,449

frequency-dependent, 27steady-state, 27,451,454, 462

Froude number, 389,425,465

Governing characteristics, 110, 133, 325,361,364

Governing stabilíty, 109Governor, 109, 110, 118, 119, 130, 135,

138,153,208,352,356,360,379accelometric, 119, 135dashpot, 119, 121, 124, 135isochronous, 118proportional-integrru-derivative (PIO),

119,135,153,154Governor-settings, 129, 135, 152

optimum, 109, 135, 146, 152, 153Graphical method, 3,6, 34, 282, 337Grid, characteristics, 100Grid points, 100, 290Grid system, 315, 366

Harmonics, 203, 206, 212, 227, 238, 239,240, 242, 248, 269

even, 205odd,205,252,253

Headdeviation,211instantaneous,211mean, 211piezornetric, 9, 27,114,304,318,430

Heat exchanger, 159, 160, 170Helmholtz resonator, 270Henry 's law, 286Hill charts for turbine, 111Homologous relationships, 76Hydraulic gradeline, 8,12-14,28,51,52,

54,81,88,93,101-104,115,277,298,305,318,335,362,363

Hydraulic jump, 425Hydraulic model, 111,431,432,435,

438-440Hydraulic radius, 417, 430, 435Hydraulic transients, 1, 7, 16, 99,109,153

causes of, 18, 124, 161, 194in closed conduits, 16,382,429in hydroelectric power plants, 109in nuclear power plants, 158in oil pipelines, 190in open channels, 16, 382,384,415,429methods of controlling, 302, 303

Hydroelectric power plants, 17, 109, 110, •133,137,155,208,315,332,333,364,426,431

transien ts in, 109Hydraulíc turbine, 16, 111, 145, 147, 153,

324,332,334,352,363,365characteristics of, 114, 130,326,348efficiency of, 366, 372motoring of, 113operations, 109

load acceptance, 17, 18, 109,384,401load rejection, 17, 18, 109, 130,384,

401start-up, 109

rated head, 129,148,341rated output, 129, 148runner, 18, 129, 148speed rise, 150-154, 302, 303, 324types

Franc~,58,67, 71, 109, 111,114, 124,126,127,135,147,153,208

Kaplan, 111, 126, 135, 155Pelton,114

1\1\,j: ~L/.

Ir :

I

l·I

impulse, 111, 114propeller, 135reaction, 348, 410

lmpedance, characteristic, 219Impedance diagram, 265, 266lmpedance method, 201, 208-210, 256,

260,269Impedance, terminal, 210Implicit fmite-difference method, 34, 164,

165,170,175,382,414-416,422-424,442

Inertiagenerator, 133, 149normal generator, 148pump-motor, 75,81, lOO, 103,276,293,

456turbine and generator, 117, 129, 133, 327

Inflow, lateral, 394Initial conditions, 142, 144,405,425Intake, 336, 365

pipe, 206power, 127, 147

Integrationarithmetic, 366, 370graphical, 366, 370

Interpolationparabolic, 114, 317

Interior sections, 63, 399,420Isoclines, 347, 348Isothermal process, 279, 280, 296, 309,

320,322

Johnson's charts for dítferential tank, 367Junction

of two channels, 402, 406-408of two pipes (see Series junction)of three pipes, 57series, 55,214,216,226,402

Kaplan turbine, 111,126,135, 155

Lake, 109,384Landslide-generated waves, 431, 432

height, 431period, 432

Lax Wendroff fmite-difference scherne, 164,165, 288-290,423

Lirnit cycle, 358, 360Line packing, 192, 193, 198

Index 497

Liquid-column separation (see Water-column separation, Column separa-tion)

Liquid-vapor mixture, 158Load, 117, 118, 134, 137, 140, 145

base, 373Load acceptance, 126,366,371,373,375,

410,411,435,440,441Load rejection, 126, 130-133, 153,315,

327,346,366,367,374-376,410,411,435,436,440

Long pipeline, 191, 192, 199Loop

prirnary, 159, 169, 170secondary, 159, 170

Loss.of-coolant-accident (LOCA), 160, 169.186

Lumped-system approach, 162-164

Manning's formula, 392, 411, 417Manning'sn, 371Mathematícal model, 96, 97, 109, 110,129,

135,153,162,163,169,283,295,325,326,408,425,432,433,435

Maximum pressure, charts for, 450-453,457,458

Matricesbanded,177,420,433field (see Field matrices)point (see Point matrices)transfer (see Transfer matrices)overall, 212, 213, 214, 223, 225, 226,

231,233,242,250,256,270unity, 223

Mechanical starting time, 134, 136, 148Method

bisection, 169explicit finite-difference, 164, 290,291,

382,395,397,414,415,422-424,433,442

fmite-element, 394,433impedance, 201, 208-210, 256, 260, 269implicit fmite-difference, 34, 164, 165,

170,175,382,414-416,422-424,442

phase-plane, 333, 342, 360, 376Newton-Raphson, 83, 91, 195, 251, 306,

310,311,402Runge-Kutta, 124, 222,223,338,362,

406

498 Index

Method (Continued)predictor-corrector, 48, 195, 196transfer matrix, 201, 208, 210, 235, 238,

250,256,257,259,263,264,267,269

Method of characteristics, 7, 34,44, 74,93,9~ 11~ 164,165, 17~ 174, 177,191,195,199,201,208,209,256,257,259,263,269,282,287,295,296,394,395,423,425

Modelhomogeneous-flow, 163hydraulic, 111,183,431,432,435,

438-440lumped-system, 163separated-flow, 163, 283

Momentum equation, 170, 285, 288, 298Moody formula, 111

Net head, 111, 114Newton's second law of motion, 336, 361,

386Node, 203,205,248,249,250, 252Normal depth, 417Nuclei,275Number

Froude, 389,425,465Reynolds, 33, 171

Oil-hamrner, 2,190ou pipeline, 190, 191, 194One-dimensional flow, 27,284,383, 389One-way surge tank, 102, 104,292,315,

333,334Open channels, 17, 382, 384, 406

transients in, 16, 382, 384,415Operation, 193, 194,325

isolated, 325float-tank, 193put-and-take, 193tightline,194

Operations-research techniques, 324Operating conditions, 19, 21, 158, 190

catastrophic, 95, 96, 130, 133critícal, 366emergency, 95, 96, 103, 130, 133, 197normal, 20, 95, 130, 197

Operating guidelínes, 191,440-442Optimal control of transient flows, 303,

323

Optirnurn valve closure, 7, 324Orifice, 216, 250, 306, 308

boundary conditions for, 55differential, 306losses, 306, 371surge tank, 333, 334, 335,360-363,365-

367Oscillating valve, 227, 240, 246, 252, 257,

261,262,267transfer matrix for, 229

Oscillationsauto- or self-excited, 206, 207free-damped, 250perpetual, 360, 377stable, 340, 360unstable, 340, 360

Outflow,lateral,394

Parallelchannels, 420, 421loops, 216, 223, 225, 226pumps, 21, 86, 93,98,317system, 223, 224

Pelton turbine, 114Penstock, 67, 110, 127, 133, 135,145,

147, 148,326,33~365,366design criteria, 13O

Perfect gas law, 168,287,296,308,309Period, 2,238, 239,258,338,339,376

natural, 194,203,206,238offundamental,212, 239, 252,253, 256,

257,264,338of higher harmonícs, 257of surge-tank oscillatioris, 339,463theoretical, 15, 211, 212, 267

Periodic flow, 203,425Phase angle, 239, 241, 246,257,261,263,

264,270Phase portraits, 342,347,348,350,351,

354,355,358,359,372Pilot valve, 118, 119Pipe

concrete, 9, 27, 38metal, 9, 27PVC,38rigid, 10water supply, 16

Pipelinewith variable characteristics, 221, 222,

267,268,269rupture, 161, 169, 170, 178, 194

1,

Piping systemsbranch,212parallel, 216, 223, 225, 226series, 63, 64, 65, 253,260, 263

Point matrices, 201,214,215,226,231,240,241,242,247

for air chamber, 270fOIbranch junction, 233, 235, 245for orífice, 225, 226, 270for oscillating valve, 229for series junction, 225, 226for simple surge tank, 270for valve, 225, 226, 229, 230

Poisson's ratio, 30, 36Positive surge, 467Potential surge, 191, 192, 195Power faílure to pumps, 21, 74, 75, 77, 97,

98, lOO, 102-104, 161, 164,193,194,197,198,276,302,307,456

charts for maximum and minimum pres-sures, 454-458

Power intake, 127, 147Power plants, 109°hydroelectric, 17, 109, 110, 133,137,

155,208,315,332,333,364,426,431

nuclear, 158, 162, 186Pressure

charts for maximum and minimum, 450-458

fluctuating, 257partial, 276, 282saturatíon, 286vacuum, 75

Pressure controlIers, 197, 198Pressure regulating valve, 67, 70, 105, 130,

133,153,292,293,312-315,317,325

Pressure regulator, 365Pressure relief valve, 96,197,198,293,

312-314,317Pressure rise, 4, 5, 10,21,95, 102-104,

151, 15~ 191,274,30~30~325,332,450,451

Pressure wave, 3, 4, 7, 11,46, 191,281,303,332

velocity of (see Waterhammer wavevelocity)

Pressurizer, 159Primary loop, 159, 169,170Principie of superposition, 227

Index 499

Propeller turbine, 135Prototype, 414, 440Prototype tests, 35, 109, 126, 153,408,

436,437 (see also Field tests)Pulsation darnper, 198Purnp, centrifugal, 6,18,74,75,197,198,

208,295,307,315,334boundary conditions for, 74, 77, 79characteristícs, 6,74,75,77,79,80,89,

100,103,104,162,293,295,484discharge line, 74, 75, 94, lOO, 105, 206,

238,249,454,456,457,462events folIowing power failure, 75impeller, 18, 76inertia, 75, 81, 100, 103, 276, 293, 456instability of, 161power failure, 21, 74, 75, 77, 97, 98,100,

102-104, 161, 164pressure characteristics, 77, 80,94, 318rated condítions, 76

head, 103, 104,293,454speed, 74-76,93, 103, 293,456torque,76

runaway speed, 75, 100shutoff head, 197specific speed, 77, 97, 293,453start-up, 74,94,95start-up time, 100stoppage, 75, 95torque characteristics, 77, 80, 94, 318zones of operation, 295

energy dissipation, 75, 77pump operation, 75,77, 100turbine operation, 77

Pumpscentrifugal (see Pump, centrifugal)multiplex, 198parallel, 21, 86, 93, 98, 317reciprocating, 194, 198, 206, 238series, 88starting and stopping, 74, 161, 164, 194,

384,401Pumping mode, 324Pumped-storage scheme, 324, 364Pumping head, 190,276,318Pumping stations, 190, 193, 194, 197Pumping systems, 332Pyramidal effect, 193

Rapidly-varied flow, 17,382Rarefaction control, 194, 198

500 Index

Rarefaction waves, 276Reactor coolants, 158-160, 169Reaction turbine, 348, 410Reactors, 158, 159, 179, 183

boiling water, 160liquid-metal fast breeder, 160pressurized water, 159

Regulating characteristícs, 332, 363Regulation

line, 194station, 194

Reínforced-concrete pipe, 38Reservoir

constant-level, 187,204,237,245,249,250,252,257,267,295,399,400,408,411

downstream, 52,110,129upstream, 2, 10, 11, 14, 15, 51, 109, 110,

203,246,332,450,463Residual, 252, 256Resonance,6, 194, 198,201,206,250Resonant frequency, 210, 212, 250, 252,

257,269Resonating characterístícs, 216, 257, 267Resonating conditions, 201, 252Resonator, Helmholtz, 270Reynolds number, 33, 171Rigid water-colurnn theory, 137Riser, surge tank, 363, 364, 367Rivers, 18, 110Root locus technique, 210Runner, turbine, 111, 129,208Runge-Kutta method, 124,222,223,338,

362,406Rhythrnic opening and closing of valve, S,

6,204,227

Safety valve, 4, 5, 312, 313, 314, 317Saínt-Venant equations, 389, 394, 395,

397,398,405,414,415,424,425,430,431

Scroll case, turbine, 114, 133Secondary water-surface fluctuations, 411,

425,427,429Secondary loop, 159, 170Self-excited oscillations, 206, 207Self-regulation constant, 129, 135,137,

140, 152Series junctíon, 55, 214, 216, 226,402

Series system, 63, 64,249,252,253,257,259,260,263,265,271

Servomechanism, 118Sewers, 17, 18,384,426Singular point (or singularity), 342, 343,

348,349,353,355-357,359cornpound, 357nonsimple, 343,347simple, 343types

focus, 344, 345, 349, 356, 357, 359node,344,345,349,353,356,357,

359saddle, 344, 352, 353, 356, 357vortex, 344, 345

virtual, 348,352Slide velocity, 432Sluíce gate, 192,385Smoothing operator, 290Solution trajectory, 347, 348Specific speed

pump, 77,97,293,453turbine, 147

Specific weight, 28, 167, 264, 304, 322,336,390

Speed droop, 118, 119ternporary, 119, 120, 129,137,152permanent, 119, 120, 122,129,135,

137,140,152Speed-no-load gate, 111, 126, 131,410,440Speed ríse, 133, 150-153,365Spríng-mass systern, 201-203Stability diagram, 342, 360Stability límit curve, 135, 136,139-141Stability of fínite-difference scherne, 44,58,

59,70,404,415,417,422Standpipe, 304, 305, 310, 312, 371State vectors, 212, 223, 238, 240,252

extended,213,229Static head, 206, 227, 325, 326, 348,457Starting time

mechanical, 134, 136, 148purnp, 94, 100water, 134, 136, 149,332

Steady flow, 1,46,385,386Steady-oscillatory flow, 1,203,204-206,

208-210,216,264,425Steam-generator, 179, 183Steamhammer, 2

Steel-lined tunnel, 37, 40Strain, 27, 31Stress, 27, 279Subcritical flow, 389,424Suction line, 74, 79, 82, 86, 87, 98,100,

206,238Supercritical flow, 389, 424Surface tension, 275, 279, 281, 285Surge, 195,382,384,426,431,433,464-

467absolute velocity of, 383, 386, 387,466,

467charts for surges in open channels, 464-

467free (for surge tank), 339, 376,377height, 384, 388,443,466,467potential, 191, 192, 195suppressor, 95, 96

Surge tanks (or surge chambers), 5, 7, 16,95,96, 110, 130, 131, 199,248,292,303-305,307,325,332-334,337,341,360,362,364,365,376,453

closed, 333, 337duferential, 7, 333, 334, 362,366, 367downstream,368-371double,370inclined, 376, 377multiple, 378one-way, 102, 104, 292,315,333,334orifice, 333, 334, 360-363,365-367simple, 333-335, 360-363, 365, 366,

369,462,463-465system of, 333, 364tailrace, 376upstream, 369-371virtual, 328

Surge tankcharts fOI upsurges and downsurges in,

463-465dynamic and continuity equations

for differential tank, 363for orífíce-tank, 361for simple tank, 335, 336

gallery, 334, 367period of oscillations, 339, 374,463

Surge Waveabsolute velocity of, 383,386,387,466,

467celerity of, 3, 383, 384, 387, 389,465

Index 501

negative, 384, 387positive, 384, 387,425

Synchronous operation, 315, 325Synchronousspeed, 112, 118, 129, 134,

135,148System response, 236, 240Systerns

branch, 252, 257, 262, 263, 265, 271distributed, 16, 17, 203, 220lurnped, 16, 17,203,219,220,269,270,

303,310parallel, 223, 224, 252, 257,261,263pressure-regulatíng valve-centrifugal pump,

318,321series, 63, 64, 249,252,253,257,259,

260,263,265,271

Tailrace channel, 127,433-435Tailrace manifold, 129,368,371,376,433Tailrace tunnel, 17,364,368,426,431,

433-435Tailwater level, 433Thick-walled pipe, 35Throughput, 190Thoma area, 7, 342, 353, 357, 366Tides, 382, 383, 384,424Tidal oscíllations, 424Time constant

actua tor, 120dashpot, 120distribu ting valve, 120

Time dornain, 208Trajectory, solution, 358, 360Transfer function, 120Transfer matrices, 212-214, 216, 238, 245;

250,252extended, 235-237, 242, 246extended overall, 240, 241field (see Field matrices)overall, 213, 214, 223, 225, 226,231,

233,242,250,256,270point (see Point matrices)

Transfer rnatríx method, 201, 208, 210,235,238,250,256,257,259,263,264,267,269

Transient analysis, 39, 162,186Transient cavitation, 274, 275, 281, 282,

292

502 Index

Transient flow, 1,2,18,44,127,195,201,282,442

Transients, 1, 7, 16,99,109,153causes of, 18, 124, 161, 194caused by centrifugal pumps, 74, 93in closed conduits, 16,382,429in long oil pipelines, 190in hydroelectríc power plams, 109in nuclear power plants, 158in open channels, 16,382,384,415,429in power canals, 425in rivers, 425methods of controlling, 302, 303

Transmission line, 109Traveling waves, 16Tuner, 270, 271Tunnels, 110,335,336,340,435,463

concrete-lined, 368,433free-flow, 127rock, 9, 27, 37steel-lined, 37, 40tailrace, 17,364,368,426,431,433-435unlined,37,368

Turbine, 16, 111,145,147, 153,324,332,334,352,363,365

characteristics of, 114, 130, 326, 348efficiency, 366, 372motoring, 113operations

load acceptance, 17, 18, 109,384,401load rejection, 17, 18, 109, 130, 384,

401start-up, 109

rated head, 129,148,341rated load, 366rated output, 129, 148runner, 18, 129, 148scroll case, 11Ospeed rise, 150, 153, 302, 303, 324unit flow, 111, 114unit power, 111,112,114unit speed, 111-113

Turbine flow-demand characteristics, 341,344,345

constant flow, 341, 344constant gate opening, 341,348,357constant power, 341, 352, 357constant power combined wíth full-gate,

341,356,357

TurbinesFrancis, 58, 67, 71, 109, 111, 114, 124,

126,127,135,147,153,208impulse, 111, 114Kaplan, 111, 126, 135, 155Pelton, 114propeller, 13 5reaction, 348, 410

Turbogenerator, 109, 111, 117, 118, 127,363,364

Two-phase flows, 158, 164, 170, 186, 195homogeneous, 162, 164separated, 162, 164

Undíssolved gases, 35Unsteady flow, 1,7,27,382,394Upsurge, 367,440,442,456,459'-463

Valve, 2,6,7,10,11,18,185,193,194,203,205,206,208,216,228,276,278,303,312,450

boundary conditions for, 54characteristics, 206control, 161, 182, 183, 185,194optimum closure, 7, 324rhythrníc or periodic movements, 5, 6,

204,227stroking, 324uniform closure, 450

Valvesah,95-97, 104, 186,208, 292,293,312,

315,320,332,323by-pass, 198, 208check,95, 96, 102, 104, lOS, 161, 182,

183,198,208,307,312,315,317,457

discharge, 74, 81, 97, 102, 103,208leaking, 207, 208pressure-regulating, 67, 70, 105, 130, 133,

153,292,293,312-315,317,325pressure-relief, 96, 197,198,293,312-

314,317safety, 4, 5,312-314,317

Vapor pressure, 2,35,274-277,279,281,282, 284, 286, 287

Velocity potential, 3Velocity of pressure waves tsee Water-

hammer wave velocity)

1

~I Vibrations

forced, 206mode of, 203mechanical,210self-excited, 206, 208steady, 201-203

Void fraction, 163, 164, 171, 278, 280-284,286

i'

¡"I

Water-boxes, 165Water-column separation, 6, 75, 96 (see

also Column separation)Water passages, 134Waterhammer, 2,4,6, 178, 182

pressures, 1, lID, 134, 137, 150, 162,364,365,371

waves, 33, 86, 332Waterhammer wave velocity, 5,7, lO, 11,

17,33-35,39,60,68, lOO, 129, 163,164,171,206,212,220,267,278,281,285,287,288,298,302,328,430,451,457

in gas-liquid mixture, 274, 279in noncircular conduits, 38in PVC pipes, 38in reinforced-concrete pipe, 38in rock tunnels, 37in steel-líned tunnels, 37in thick-walled conduits, 35in thin-walled conduits, 36in woodstave pipes, 38

Water starting time, 134, 136, 149,332Wave

absolute velocity of, 383, 386, 387,466,467

amplitude, 383attenuation, 193celerity, 3, 383, 384, 387, 389, 465propaga tion, 1, 2, 5, 11reflection, 1,4, S, 11,282,449

Index 503

Wavefront, 7,9,193,283,286,388,389,425,465

Wavelength, 383,384Waves, 237, 382, 384, 386,431

deep-water, 383, 384elastic, 3impulse, 431landslide-generated, 431, 432negative, 384, 387positive, 384, 387,425pressure, 3, 4, 7, 11,46, 191, 281, 303,

332shallow-water, 383, 384solitary, 384,432standing, 206, 249sta tionary, 384sound, 2,4surface, 206translatory,384water, 8

Weh,433-435Wicket gates, 70, 109,lIl,114, 118,119,

126,127, l30, 315, 324, 327, 372,435,440,441

breakaway gate, 126,410effective closing time, 123, 130, 136,

150,365,410effective opening time, 123, 131, 135,

136,150,410speed-no-load gate, 111,126, 131, 41D,

440Windage losses, 111-113, 130

Young's rnodulus of elasticity, 31, 36, 280,284

Zone of energy dissipation, 75, 77Zone of pump operation, 75, 77, 100Zone of turbine operation, 77

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