Hydraulics of pipeline systems

Hydraulics of Pipeline Systems Bruce E Larock Roland W Jeppson Gary Z Watters Library of Congress Cataloging-in-Publication Data Larock, Bruce E., 1940Hydraulics of pipeline systems / Bruce E Larock, Roland W Jeppson, Gary Z Watters p cm Includes bibliographical references and index ISBN 0-8493-1806-8 (alk paper) Pipe-Hydrodynamics Pipelines I Jeppson, Roland W II Watters, Gary Z III Title TC174.L37 1999 99-32568 621.8'.672-dc21 CIP This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale Specific permission must be obtained in writing from CRC Press LLC for such copying Direct all inquiries to CRC Press LLC, 2000 N.W Corporate Blvd., Boca Raton, Florida 33431 Trademark Notice: Product or corporate names may be trademarks or registered trademarks, only for identification and explanation, without intent to infringe Visit the CRC Press Web site at www.crcpress.com © 2000 by CRC Press LLC No claim to original U.S Government works International Standard Book Number 0-8493-1806-8 Library of Congress Card Number 99-32568 Printed in the United States of America 34567 Printed on acid-free paper 890 and are used Tbe AUTboRS Bruce E Larock, Ph.D., P.E., received his civil engineering degrees at Stanford University In 1966 he joined the faculty of the Civil and Environmental Engineering Department, University of California, Davis, where he is now a Professor He has pursued studies in fluid motion primarily by numerical methods, including finite element and finite difference analyses Dr Larock has published approximately 80 technical articles, including a co-authored book and several book chapters A member of the American Society of Civil Engineers, he is currently an associate editor of its Journal of Hydraulic Engineering Roland W Jeppson, Ph.D., is Professor of Civil and Environmental Engineering at Utah State University, Logan, with 33 years of teaching engineering courses there and at California State University, Humboldt, and conducting research at the Utah Water Research Laboratory A former department Chairman, he is author or co-author of over 70 technical articles and reports and two books He has served numerous organizations as a consultant and/or expert witness and also developed the USU-NETWK software package for pipe network analysis and design Gary Z Watters, Ph.D., P.E., completed his doctorate at Stanford University in 1963 After 17 years on the faculty at Utah State University, in 1980 Dr Watters became Dean of Engineering, Computer Science, and Technology at California State University, Chico, for 12 years He retired as Emeritus Professor in 1998 Dr Watters has consulted widely and given expert testimony on hydraulic transients for over 30 years He is the author, co-author, or contributor to, five books and two other book chapters He is a member of the American Society for Engineering Education and the American Society of Civil Engineers ACKNOWLEDGMENTS Each of the three authors has had a long and continuing involvement with some, indeed most, of the material in this book in one way or another since the time they first became acquainted with each other as graduate students at Stanford University in the 1960s At that time the teaching and enthusiasm of John K Vennard for the subject of flow in pipes made a positive impression on us, his then-young students In the years since then each of us has taught portions of this material to students in university courses, or in short courses, or in both This book is a long-delayed product of that early enthusiasm, now tempered by years of experience and aided by advances in knowledge about hydraulics and numerical methods and advances in computer hardware We acknowledge here his influence on our lives and on this book We thank Nicole Newman for her skill in preparing, and sometimes repairing, almost three hundred figures for this book, and we particularly appreciate her patience and tolerance as the authors continued to revise some of them even after they were drawn Her careful attention to detail has resulted in a better book We appreciate the support our families have given each of us as we took time from our other activities to find time to pursue this project We want to express our heartfelt thanks and gratefully acknowledge the contribution of many, many students, really too numerous to mention individually here, at three They have listened universities and in numerous short courses over three decades attentively, questioned, probed, and otherwise interacted with the authors in classes, hallways, hydraulics laboratories, and offices over the years as we all sought to understand this subject better and as the materials in this book were developed, refined, and tested Thanks to all of you! Bruce E Larock Roland W Jeppson Gary Z Watters TABLE OF CONTENTS In trod ue ti on Review 2.1 2.2 2.3 2.4 2.5 Manifold 3.1 3.2 3.3 3.4 of Fundamentals The fundamental principles 2.1.1 The basic equations 2.1.2 Energy and Hydraulic Grade Lines Head loss formulas 2.2.1 Pipe friction 2.2.2 Darcy-Weisbach equation 2.2.3 Empirical equations 2.2.4 Exponential formula 2.2.5 Local and minor losses Pump theory and characteristics Steady flow analyses 2.4.1 Series pipe flow 2.4.2 Series pipe flow with pump(s) 2.4.3 Parallel pipe flow, equivalent pipes 2.4.4 Three reservoir problem Problems Flow Introduction Analysis of manifold flow 3.2.1 No frictIOn 3.2.2 Barrel friction only 3.2.3 Barrel friction with junction losses A hydraulic design procedure Problems Pipe Network 4.1 4.2 4.3 4.4 Analysis Introduc ti on 4.1.1 Defining an appropriate pipe system 4.1.2 Basic relations between network elements Equation systems for steady flow in networks 4.2.1 System of Q-equations 4.2.2 System of H-equations 4.2.3 System of L1Q-equations Pressure reduction and back pressure valves 4.3.1 Q-equations for networks with PRV's/BPV's 4.3.2 H-equations for networks with PRV's/BPV's 4.3.3 1Q-equations for networks with PRV's/BPV's Solving the network equations 4.4.1 Newton method for large systems of equations 4.4.2 Solving the three equation systems via Newton 7 9 10 12 14 16 18 20 20 22 25 26 28 33 33 33 33 34 35 45 50 53 53 54 55 56 57 60 62 65 66 72 74 78 78 94 4.4.3 Computer solutions to networks 4.4.4 Including pressure reducing valves 4.4.5 Systematic solution of the Q-equations 4.4.6 Systematic solution of the H-equations 4.4.7 Systematic solution of the L1Q-equations Concluding remarks Problems 98 103 107 II3 II7 124 126 Design of Pipe Networks 5.1 Introduction 5.1.1 Solving for pipe diameters 5.1.2 Solution based on the Darcy-Weisbach equation 5.1.3 Solution based on the Hazen-Williams equation 5.1.4 Branched pipe networks Large branched systems of pipes 5.2.1 Network layout 5.2.2 Coefficient matrix 5.2.3 Standard Linear Algebra Looped network design criteria Designing special components Developing a solution for any variables 5.5.1 Logic and use of NETWEQSl 5.5.2 Data to describe the pipe system 5.5.3 Combinations that can not be unknowns Higher order representations of pump curves 5.6.1 Within range polynomial interpolation 5.6.2 Spline function interpolation Sensitivity analysis Problems 141 141 141 141 145 145 150 151 154 155 158 175 183 183 185 187 196 196 198 203 216 4.5 4.6 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Extended Time Simulations and Economical Design 227 6.1 6.2 6.3 6.4 6.5 Introduction 7.1 7.2 7.3 7.4 Introduction Extended time simulations Elements of engineering economics 6.3.1 Economics applied to water systems 6.3.2 Least cost Economic network design 6.4.1 One principal supply source 6.4.2 Design guidelines for complex networks Problems to Transient Flow Causes of transients Quasi-steady flow True transients 7.3.1 The Euler equation 7.3.2 Rigid-column flow in constant-diameter pipes 7.3.3 Water hammer Problems 227 228 238 241 242 247 247 268 271 277 277 277 284 284 285 295 300 Elastic Theory of Hydraulic Transients (Water Hammer) 305 8.1 8.2 The equation for pressure head change L1H Wave speed for thin-walled pipes 8.2.1 Net mass inflow 305 307 307 8.3 8.4 8.5 8.6 308 308 315 315 316 316 318 319 320 322 324 Solution by the Method of Characteristics 329 9.1 9.2 9.3 9.4 10 8.2.2 Change in liquid volume due to compressibility 8.2.3 Change in pipe volume due to elasticity Wave speeds in other types of conduits 8.3.1 Thick-walled pipes 8.3 Circular tunnels 8.3.3 Reinforced concrete pipe Effect of air entrainment on wave speed Differential equations of unsteady flow 8.5.1 Conservation of mass 8.5.2 Interpretation of the differential equations Problems Method of characteristics, approximate governing equations 9.1.1 Development of the characteristic equations 9.1.2 The finite difference representation 9.1.3 Setting up the numerical procedure 9.1.4 Computerizing the numerical procedure 9.1.5 Elementary computer programs Complete method of characteristics 9.2.1 The complete equations 9.2.2 The numerical solution 9.2.3 The Lis-Lit grid Some parameter effects on solution results 9.3.1 The effect of friction : 9.3.2 The effect of the size of N 9.3.3 The effect of pipe slope 9.3.4 Numerical instability and accuracy Problems 329 329 332 334 336 338 347 347 349 351 352 352 354 354 354 356 Pipe System Transients 365 10.1 10.2 10.3 10.4 10.5 10.6 Series pipes 10.1.1 Internal boundary conditions 10.1.2 Selection of Lit 10.1.3 The computer program Branching pipes 10.2.1 Three-pipe junctions 10.2.2 Four-pipe junctions Interior major losses Real val ves 10.4.1 Valve in the interior of a pipeline 10.4.2 Valve at downstream end of pipe at reservoir 10.4.3 Expressing KL as a function of time 10.4.4 Linear interpolation 10.4.5 Parabolic interpolation 10.4.6 Transient valve closure effects on pressures Pressure-reducing val ves 10.5.1 Quick-response pressure reducing valves 10.5.2 Slower acting pressure-reducing or pressure-sustaining val ves Wave transmission and reflection at pipe junctions 10.6.1 Series pipe junctions 10.6.2 Tee junctions 365 365 366 369 372 372 375 376 378 378 379 380 382 383 385 386 386 388 388 388 389 10.6.3 Dead-end pipes Column separation and released air 10.7.1 Column separation and released air 10.7.2 Analysis with column separation and released air Prob lems 390 391 391 392 395 Pumps in Pipe Systems 399 399 401 403 404 406 408 412 10.7 10.8 II Pump power failure rundown 11.1.1 Setting up the equations for booster pumps 11.1.2 Finding the change in speed 11.1.3 Solving the equations 11.1.4 Setting up the equations for source pumps Pump startup , Pro b lem s Il.l 11.2 11.3 12 Transients Network 12.1 12.2 Introduction Rigid-column unsteady flow in networks 12.2.1 The governing equations 12.2.2 Three-pipe problem A general method for rigid-column unsteady flow in pipe networks 12.3.1 The method 12.3.2 An example Several pumps supplying a pipe line Air chambers, surge tanks and standpipes A fully transient network analysis 12.6.1 The initial steady state solution 12.6.2 TRAN SNE T Pro blems 12.3 12.4 12.5 12.6 12.7 13 417 417 417 417 418 422 422 423 427 429 435 435 437 451 Transient Control Devices and Procedures 463 13.1 Transient problems in pipe systems 13.1.1 Val ve movement 13.1.2 Check valves 13.1.3 Air in lines 13.1.4 Pump startup 13.1.5 Pump power failure Transient control 13.2.1 Controlled valve movement 13.2.2 Check val ves 13.2.3 Surge relief valves 13.2.4 Air venting procedures 13.2.5 Surge tanks 13.2.6 Air chambers 13.2.7 Other techniques for surge controL Pro blem s 13.2 13.3 463 463 464 464 465 465 466 466 467 467 469 469 473 484 486 R e fe re n c e s 489 A ppe ndi c e s 493 14 A Numerical A.l A.2 Methods Introduction Linear algebra 493 493 493 A.2.1 Gaussian elimination A.2.2 Use of the linear algebra solver SOLVEQ A.3 Numerical integration A.3.1 Trapezoidal rule A.3.2 Simpson's rule AA Solutions to ordinary differential equations AA.l Introduction AA.2 Runge- Kutta method AA.3 Use of the ODE solver ODESDOL 493 494 496 496 498 501 501 501 507 B Pump characteristic C Valve loss coefficients 521 C.l Globe curves 513 and angle valves C.2 Butterfly valves C.3 B all val ve s D 521 523 523 Answers to selected problems 525 Index 533 CHAPTER INTRODUCTION Pipeline systems range from the very simple ones to very large and quite complex ones They may be as uncomplicated as a single pipe conveying water from one reservoir to another or they may be as elaborate as an interconnected set of water distribution networks for a major metropolitan area Individual pipelines may contain any of several kinds of pumps at one end or at an interior point; they may deliver water to or from storage tanks A system may consist of a number of sub-networks separated by differing energy lines or pressure levels that serve neighborhoods at different elevations, and some of these may have pressurized tanks so that pumps need not operate continuously So these conveyance systems will adequately fulfill their intended functions, they may require the inclusion of pressure reducing or pressure sustaining valves To protect the physical integrity of a pipeline system, there may be a need to install surge control devices, such as surge relief valves, surge tanks, or air-vacuum valves, at various points in the system How these systems work? What principles are involved, and how are the systems successfully analyzed and understood? How can the behavior of a preliminary design be evaluated, and how can the design be modified to correct deficiencies? These are some, of many, questions that immediately confront any engineer who is involved in creating the physical infrastructure to satisfy a basic need of mankind: the delivery of water when and where it is wanted at a price that is affordable It is the primary objective of these engineers to develop and apply their knowledge to make the system work Success at this task first requires an adequate knowledge of some fundamental principles of fluid mechanics Some experience with the solution of hydraulic flow problems is certainly desirable, and it will come with time and effort These days an understanding of some particular numerical methods and the ability to implement them on a computer, sometimes for the solution of very large problems, is also a vitally needed skill Computations associated with engineering practice have changed dramatically in the past quarter century from the estimation of a few key values by using a slide rule to the generation of pages of computer output that are the result of detailed simulations of system performance in response to various alternative designs, so that the consequences of various ideas can be ascertained quantitatively The volume of computer output can overwhelm one's ability to glean the most pertinent information from the numbers The purpose of this book is to empower the reader with the knowledge, experience, and tools to accomplish this objective This book will present to the reader a comprehensive and yet relatively practical study of pipeline hydraulics, with a substantial component being the use of computers for detailed computations that are not practical to perform by hand The intent of the authors was to create a book, and an accompanying CD, that will serve well any of the following roles: (1) as a text for senior-level courses for BS students electing to specialize in fluid mechanics, hydraulics, water supply and distribution, and/or water resources; (2) as a text for graduate engineering courses in the same subject areas; (3) to provide instructional material for professional practicing engineers who wish to update their knowledge of specialties associated with the distribution, conveyance, and control of fluids in pipelines; (4) to provide resource material for engineers in governmental agencies at all levels who have responsibilities to design and/or approve plans for pipeline systems; and (5) to provide reference material for consultants who are asked to solve problems, review plans, or suggest project alternatives in the subject areas of this book 533 INDEX Adiabatic process, 430 Affinity laws, pumps, 18-19,399-400 Air chambers, 429-432, 473-484 sizing, 479 Air entrainment, wave speed, 312, 318319,391-394 Air in lines, 464-465 Air venting, 469 Air, polytropic process 477 Algorithm pressure control, 229 tank level, 229 Analysis of sensitivity, 203-215 Angle valve, 523-524 Annual operating costs, 242-245 Back pressure valve (BPV), 65-78, 175, 242 and L1Q-equations, 74-78 and H-equations, 72-74 and Q-equations, 70-72 Ball valve 525 Band width, 120-122 Benefit/cost ratio, 54 Bernoulli equation, 7-8 Body forces, Booster pump, 175, 181 equations, 401-403 Boundary conditions, 335-336, 352 check valve, 336 constant-speed pump, 335-338, 344347 internal, 352, 365-369 reservoir, 335 velocity, 335 Branched system, 55, 145-157 Branched system design, 145-157,249-267 coefficient matrix, 154-155 network layout, 151-154 standard linear algebra, 155 Branching pipes four-pipe junctions, 375-376 three-pipe junctions, 372-375 Bulk modulus of elasticity, liquid, 308, 312,318-319 Butterfly valve, 430-431, 525 Bypass, pump, 401, 405-409, 465, 469 Capital investment, 238-240 Capital recovery factor, 238-242 Cascade, two-tank, 301 Cavitation, 20 Characteristic curves, pumps, 19-25, 196202,515-522 Characteristic equations, 330-332, 347-348 Characteristic grid, 333, 349, 367-369 Characteristic lines, 331-332, 348-349, 365-369 Check valve, 401-402, 464, 467 Colebrook-White equation, 12-16,89, 142-145,158-166 Column separation, 391-394, 408-409, 466,473-476,485 Complex networks, design, 268-270 Compressibility, fluid, 277,308 Conservation of mass, 7, 53-55, 307, 319321,329,347-348 Continuity, 7, 53-55, 279-280 Contraction coefficient, 47 Control devices and techniques, transient, 463-487 Controlled valve movement, 466-467 Convective acceleration, 323 Corrective loop discharge, 62-65, 74-78, 80-82,88-92,97-98, 102-103, 117124 Cost/benefit ratio, 54 Darcy-Weisbach equation, 10-16,20-26, 55,89, 141-145, 158-166,285 Demand schedule, 228-229,232,240-241, 263 Design point, pumps, 18 Design, complex networks, 268-270 DIAPIP, 142-143, 155 DIAPIP2,141-144 DIAPIP3, 141-145 DIAPIPA,141-144 DIAPIPH, 145 Differential head device, 175-180,229 Differential-algebraic equations, 3, 417435 Discharge coefficient, 47 pumps, 18-19 Discharge rules, 229 534 Index Dissolved air, 463 Dominant path, network, 269 L1Q-equations, 62-65, 74-78, 80-82, 8892,97-98, 102-103, 117-124,422424 DVERK, 421, 434, 501-502, 513 Economic analysis, 238-270 Economic network design, 247-270 Efficiency, pumps, 8, 18-20 Elastic analysis, see water hammer Elasticity fluid, 277 liquid, 308 pipe, 277, 308-310 ELECECG,202 Elevation head, 8-9 Elliptic integral, 280 Empirical head loss formulas 12-16 Energy grade line, 7-9, 34-36 55, 279, 285,352-353,388-391,417-418, 466,470-471 Energy line, see Energy grade line Energy loop equation, 55, 60, 68, 71, 89 Energy loops, 55-65 Energy principle, see work-energy principle Enlargement, head loss, 16-17 Equivalent pipe, 25-26 EQUSOLl, 86-93, 98-102, 183 Euler equation, 284-285, 321-322, 329, 347-348 Euler predictor, 501-502 Exponential formula, head loss, 14-16,27, 55ff.,159 Extended time simulation, 212-215, 227237, 247-270, 426 Fire flow, 249, 261-262 Flow establishment, 285-290 Flow rules, 228 Fluid acceleration convective, 278 temporal, 278 Fluid elasticity, 277 Fluid friction, transient, 333 Friction factor, 10-16 Friction, effect on water hammer pressure, 352-353 FUNCT, 86-93 GAUSEL, 84, 493-494 Gauss-Seidel iteration, 93, 142, 184 Gaussian elimination, 493-496 Globe valve, 523-524 H-equations, 60-62, 72-74, 80-82, 88-92, 96-97, 10 1-102, 113-117 Hardy Cross method, 53 Hazen-Williams equation, 13-16,55, 145, 158-159 Head coefficient, pumps, 18-19 Head loss, 8, 376-378 coefficient, transient, 380-385 formulas, 9-18 see Darcy-Weisbach equation, HazenWilliams equation, Manning equation enlargement, 16-17 valves, 376-388 HYDEQS, 173 Hydraulic grade line, 7-9, 55, 352-353, 417-418.466.470-471 Inertia control, pumps, 485 Initial conditions, 333 Internal boundary conditions, one-way surge tanks, 471 Internal head loss, 376-378 Interpolation of characteristics, 349-352, 367-369 Interpolation of data, 381-385 INVERM, 84 Irrigation manifold, 33, 52 Irrigation system, 146-149, 155-157,217, 241-242 Isothermal process, 430 Jacobian, 81, 87-88, 95, 97-102, 105-108, 113-124, 142-143, 187.420-421 Junction, 55 continuity equation, 55, 59-62,67 70,89,99, 150, 158-165 176 supply source 55, 60 Lagrange multiplier, 329-330, 347-348 Lagrangian interpolation, 23, 196-198, 246 Laminar flow, 11-12 Least cost, 238, 242-247 Linear algebra, 493-496 Linear interpolation 382-383 Linear momentum conservation, 8, 305307 Local losses, coefficients, 9, 16-18, 21-22, 37-41 Loop corrective discharge, 62-65, 74-78, 80-82, 88-92, 97-98, 102-103, 117124 535 Looped network design, 158-174, 181, 227-237, 247-270 Loops, 55-65, 158-175 Loops, pseudo, 56,60,68, 78,107,158159 164, 175 Loss coefficients, valves, 523-525 Manifold flow, 33-49 applications, 33, 45-52 barrel friction, 34-35 junction losses, 35-45 no friction, 33-34 Manifold hydraulic design, 45-49 Manning equation, 13-16 MathCAD, 4, 24, 59, 155, 183,501 Mathematica, MATLAB,155 MCBRAN 249-256 MCOST, 246 MCOSTl, 249 Method of characteristics, approximate, 329-347 complete, 329, 347-352 Midpoint method, 501 Minor losses, coefficients, 9, 16-18, 2122,37-41 Modulus of elasticity, pipe 308-312 Momentum principle, 8-9 Moody diagram, 11-14, 26 Net Positive Suction Head, 20 NETWEQS 1, 183-196, 206 NETWEQST, 169-174, 189-190, 195-196 NETWK,68-72, 156-157, 169-171, 177182, 190-195, 204-215, 231-232, 249,258-267,429-431.435-450 Network analysis 53-140 design, 141-226,268-270 economic, 247-270 design criteria, 158, 165, 175 equation systems, 56-78 80-82 88124 number of loops, 56 pseudo loops, 56, 60, 68, 78 reduced, 152 sol ution methods, 78-124 transients, see Transients, network verification, 54 Newton method, 54, 78-124, 142-143, 177, 198,419-423 Newton's second law, 284-285 Node, 55 Numerical accuracy, water hammer, 354-355 Numerical instability, water hammer, 354-355 Numerical integration, 280, 493, 496-501 Numerical methods, 493-513 Numerical solution, approximate method of characteristics, 334-342 complete method of characteristics, 349-352 9DESOL, 501-502, 507-513 ODESOLS, 508 Ordinary differential equations, 493,501513 Orifice equation, 280,471 Parabolic interpolation, 383-385 Parallel pipe flow, 25-26 Partial derivative, 203-204 Peaking factor, 204-206, 212-214, 226, 230 271 PGRAPH, 369,405-408 Piezometric head, 9-10, 285, 296-299, 322, 330, 348 Pipe collapse, 463 diameters determining, 141-145 using Darcy-Weisbach, 141-145 using Hazen-Williams, 145 flow problem types, 20 material, brittle, 463 network analysis, 53-140 design, 141-226 reinforced concrete, 463 strain, 308-310 stress, 308-314 thick-wall,315-316 thin-wall, 308-312, 316 vibration, 463 PIPK_N,90 PIPST AND, 434-435 Pitot tube, Poisson ratio, 308, 312 Polynomial interpolation, 196-202 Polytropic process, air, 477 Power coefficient, pumps, 18-19 Power failure, pump, 465-466, see Pump rundown Power, 536 Index Present worth factor, 238-240, 259 payment, 238-240 Pressure head, 8-9 increase, 305-307, 313 Pressure zones, 76-78 Pressure-reducing valve (PRV), 65-78, 103-107,112-113,175-176,242, 376, 386-388 and L1Q-equations, 74-78,105-107, 175-176 and H-equations, 72-74, 105 and Q-equations, 66-72, 103-105, Il2-113 PROFILE, 369 PROGI,339-343 PROGl,369 PROGIP, 344-347 PROG2, 369-373 PROG3,407-409 PROG4, 405-406 PROG7,480 PROG8, 393, 409-410, 465 Pseudo loops, 56, 60, 68, 78, 107, 158159, 164, 175,424 Pump booster, 401-406, 465 bypass, 401, 405-409, 465, 469 change in speed, 403-404 characteristic curves, 19-25, 196-202, 336, 399-404, 515-522 curves, 19-20, 196-202, 399-404, 428, 434, 515-522 curves, interpolation, 19-23, 196-202 efficiency, 18-20 in network, 67-78 99-107, 111-117 inertia, 403 inertia control, 485 power failure, 465-466 rules, 228 rundown, 399-408 schedule, 228-229, 232,236 similarity, 18-19,399-400 source, 406-408 startup, 409-411, 465 types, 18-19 PUMPP AR, 428-429 Pumps, 8, 18-20 pipe systems, 399-411 unsteady flow, 427-429 Q-equations, 57-60, 66-72, 80-82, 88-91, 94-96,99-101,107-113,422 Quasi-steady flow, 227-237, 277-283, 472 Real loops, see Energy loops or Loops Reduced network, 152 Released air, 391-394 Reliability, 241 Reynolds number, 1O-12 Rigid column theory, 417-435 governing equations, 417-418, 422 Rigid-column flow, 277, 285-295 Roughness pipe, 10-12 relative, 11-12 RUKU4, 502 RUKU4A, 502 RUKU4S, 502-505 RUKUST, 283 RUKUST, 501, 504-507, 510-513 Runge-Kutta integration, 281, 290-292 method, 501-507 Sand-grain roughness, 10 Scaling laws, pumps, 18-19 Sensitivity analysis, 203-215 SENSITV, 204-207, 212 Series payment, 238-240, 259 pipe flow, 20-25 Shear stress, 9-10 SHPT ANK, 283 Similarity laws, pumps, 18-19 SIMPR, 202, 498-501 Simpson's rule, 280, 288, 498-501 Skeletonization, 54-55, 179, 248 Sloping pipes, effect on water hammer accuracy, 354 SOLBRAN, 151-157,249 SOLBRAN2,151-155 SOLBRAN3, 151-155 SOLDQBAN, 122 SOLDQEQI,122 SOLDQEQS, 117-124 SOLHEQS, 114-117, 122-123 SOLQEQS, 108-112, 118, 122-123 SOLVEQ, 59, 86-88, 122, 185, 190,421, 494-496 SOLVR, 144 Specific speed, pumps, 19 Spline interpolation, 198-202,290-294 SPLINESU, 201-202, 283 Standpipe, 429, 432-435 Steady pipe flow, 20-27 Storage requirements, 241 Subnetwork, 76-78 Surface forces, 537 Surge anticipation valves, 485 relief valves, 467-469 tanks, 429-430, 469-473, 480-484 one-way, 470-473, 480-483 open-end,470 vented, 484 SURGNET, 431-432 SYMMAT, 120-121 Taylor series, 78-79 Thorley formula, 403 Three reservoir problem, 26-27, 80-83 THREPIP, 421 Time increment selection, 366-369 TK-solver, 24, 59, 155, 161-163, 183, 501 Transient control, 466-487 flow, 277-487 types, 3, 277 pressure wave, 296-299 Transients branching pipes, 372-376 network,417-450 pipe length increment, 366-369 series pipes, 365-373 systems, 365-394 TRANSNET, 435-450 Trapezoidal rule, 280, 496-498 TRAPR,497-498 Turbine, 8, 115 Turbulent flow, 11-12 Two-tank cascade, 301 UNSTPIP, 435-436, 441 Vacuum valve, air, 464-465, 469, 484-485 VALCLO, 294 VALCLO 1, 293 Valve air-vacuum, 409 check,401-402 closure, transient, 380-386 head loss, 376-388 loss coefficients, 523-525 movement, 463-467 rules, 229 schedule, 229 surge anticipation, 485 surge relief, 467-469, 485 Vapor cavity, 463 Velocity head, 8-9 Vented surge tank, 484 Vibration, pipe, 463 Water hammer equations, 320-323 linearized, 322-323 Water hammer numerical accuracy, 354355 Water hammer numerical instability, 354355 Water hammer programs, approximate, 338-347 Water hammer, 277, 295-299, 305-411, 435-450, 463-485 networks, 417-450 Wave equation, 322-323 Wave speed, 296-299, 305-323, 330-331, 348 adjustment, 368 concrete pipe, 316-318 constraint cases, 310-313, 321 tunnels, 316 Wave transmission and reflection, 388-391 Work-energy principle, 7-8, 53-55, 165, 279 Zones in network, 76-78 ... The presentation of a technique for the solution of these systems of equations is one of the contributions of this book As the future requires more sophisticated simulations of engineering problems,... the hydraulics of pipeline systems, an important secondary objective is to describe with care, and to present examples of the application of, some reliable numerical methods for the solution of. .. and study of this book that the solution of pipeline hydraulics problems, especially as the systems become larger, can require substantial computational effort The routine computation of solutions