Water Hammer in Pipe-Line Systems, J. Zaruba

WAnR HAMMER IN PIPE = LINE SY!ITEMS

This Page Intentionally Left Blank

DEVELOPMENTS IN WATER SCIENCE, 43

PIPE - LINE SYSTEMS JOSEF ZARUBA

Vltava Basin, Prague, Czechoslovakia

ELSEVIER Amsterdam - London - New York - Tokyo 1993

Reoiwers Prof. Ing. Dr. Miroslav Nechleba, DrSc. Corresponding Member of the Czechoslovak Academy of Sciences Doc. Ing. Karel Haindl, DrSc.

Published in co-edition with Academia, Publishing House of the Czechoslovak Academy of Sciences, Prague

Exclusive sales rights in:

Academia, Publishing House of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia the East European Countries, China, North Korea, Cuba, Vietnam and Mongolia

all remaining areas Elsevier Science Publishers 25 Sara Burgerhartstraat PO. Box 21 1, 1000 AE Amsterdam, The Netherlands

Library of Congress Cataloging-in-Publication Data

Ziruba, Josef.

p. Water hammer in pipe-line systems / Josef Zaruba.

Translated from the Czech manuscript. Includes bibliographical references and index.

1. Water hammer-Mathematics-Data processing. 1. Title. 11. Series.

cm. - (Developments in water science; 43)

ISBN 0-444-98722-3 2. Water hammer-Mathematical models.

TC174.2374 1993 62 1.8'672-dc20 92-15017

CIP

ISBN 0-444-98722-3

0 Josef Zaruba, 1993

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form of by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright owner.

Printed in Czechoslovakia

DEVELOPMENTS IN WATER SCIENCE, 43

OTHER TITLES IN THIS SERIES

1 G. BUGLIARELLO AND F. GUNTER COMPUTER SYSTEMS AND WATER RESOURCES

2 H. L. GOLTERMAN PHYSIOLOGICAL LIMNOLOGY

3 Y. Y. HAIMES, W. A. HALL AND H. T. FREEDMAN MULTIOBJECTIVE OPTIMIZATION IN WATER RESOURCES SYSTEMS: THE SURROGATE WORTH TRADE-OFF-METHOD

4 J. J. FRIED

5 N. RAJARATNAM

6 D. STEPHENSON

7 V. HALEK AND J. SVEC

8 J. BALEK

9 T. A. McMAHON AND R. G. MEIN

GROUNDWATER POLLUTION

TURBULENT JETS

PIPELINE DESIGN FOR WATER ENGINEERS

GROUNDWATER HYDRAULICS

HYDROLOGY AND WATER RESOURCES IN TROPICAL AFRICA

RESERVOIR CAPACITY AND YIELD

SEEPAGE HYDRAULICS 10 G. KOVACS

I I w. H. GRAF AND w. c. MORTIMER (EDITORS) HYDRODYNAMICS O F LAKES: PROCEEDINGS O F A SYMPOSIUM 12-13 OCTOBER 1978, LAUSANNE, SWITZERLAND

CONTEMPORARY HYDROGEOLOGY: THE GEORGE BURKE MAXEY MEMORIAL VOLUME

13 M. A. MARIRO AND J. N. LUTHIN SEEPAGE AND GROUNDWATER

14 D. STEPHENSON STORMWATER HYDROLOGY AND DRAINAGE

15 D. STEPHENSON PIPELINE DESIGN FOR WATER ENGINEERS (completely revised edition of Vol. 6 in this series)

SYMPOSIUM ON GEOCHEMISTRY O F GROUNDWATER

TIME SERIE METHODS IN HYDROSCIENCES

HYDROLOGY AND WATER RESOURCES IN TROPICAL REGIONS

PIPEFLOW ANALYSIS

MORPHOMETRY OF DRAINAGE BASINS

HYDROLOGY OF THE NILE BASIN

12 W. BACK AND D. A. STEPHENSON (EDITORS)

16 w . BACK AND R. LETOLLE (EDITORS)

17 A. H. EL-SHAARAWI (EDITOR) IN COLLABORATION WITH s. R. ESTERBY

18 J. BALEK

19 D. STEPHENSON

20 I. ZAVOIANU

21 M. M. A. SHAHIN

22 H. C. RIGGS

23 M. NEGULESCU

24 L. G. EVERETT

STREAMFLOW CHARACTERISTICS

MUNICIPAL WASTE WATER TREATMENT

GROUNDWATER MONITORING HANDBOOK FOR COAL AND OIL SHALE DEVELOPMENT

GROUNDWATER MODELLING

KINEMATIC HYDROLOGY AND MODELLING

STATISTICAL ASPECTS O F WATER QUALITY MONITORING

WATER RESOURCES AND WATER MANAGEMENT

RESERVOIR SEDIMENTATION

MICROCOMPUTER PROGRAMS IN GROUNDWATER

HYDRAULIC PROCESSES ON ALLUVIAL FANS

ANALYSIS OF WATER RESOURCE SYSTEMS

WATER MANAGEMENT IN RESERVOIRS

WATER AND WASTEWATER SYSTEMS ANALYSIS

COMPUTATIONAL METHODS IN WATER RESOURCES, 1

25 W. KINZELBACH

26 D. STEPHENSON AND M. E. MEADOWS

27 A. M. EL-SHAARAWI AND R. E. KWIATKOWSKI (EDITORS)

28 M. JERMAR

29 G. W. ANNANDALE

30 D. CLARKE

31 R. H. FRENCH

32 L. VOTRUBA, Z. KOS, K. NACHAZEL, A. PATERA AND V. ZEMAN

33 L. VOTRUBA AND V. BROZA

34 D. STEPHENSON

35 M. A. CELIA ET AL. (EDITORS)

MODELING SURFACE AND SUB-SURFACE FLOWS 36 M. A. CELIA ET AL. (EDITORS)

COMPUTATIONAL METHODS IN WATER RESOURCES, 2 NUMERICAL METHODS FOR TRANSPORT AND HYDROLOGICAL PROCESSES

GROUNDWATER DISCHARGE TEST SIMULATION AND ANALYSIS MICROCOMPUTER PROGRAMMES IN TURBO PASCAL

GROUNDWATER RESOURCES ASSESSMENT

GROUNDWATER ECONOMICS

PIPELINE DESIGN FOR WATER ENGINEERS third revised and updated edition

41 D. STEPHENSON AND M. S . PETERSON WATER RESOURCES DEVELOPMENT IN DEVELOPING COUNTRIES

ESTIMATION THEORY IN HYDROLOGY AND WATER SYSTEMS

37 D. CLARKE

38 J. BALEK

39 E. CUSTODIO AND A. GURGUI (EDITORS)

40 D. STEPHENSON

42 K. NACHAZEL

43 J.ZARUBA WATER HAMMER IN PIPE-LINE SYSTEMS

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1 Water hammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.1 The origin of water hammer . . . . . . . . . . . . . . . . . . . . . . . 21 1.2 Physical principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.2.1 Incompressible liquid . . . . . . . . . . . . . . . . . . . . . . . 22 1.2.2 Compressible liquid - abrupt closing of the pipe-line . . . . . . . . . 23 1.2.3 Compressible liquid - linear closing of the pipe-line . . . . . . . . . . 26

1.3 Actualcases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4 Causes of water hammer . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.5 Examples of water hammer . . . . . . . . . . . . . . . . . . . . . . . 32

2 Basicequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2 Scope of application . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Solution of the basic equations . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1 Methods of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Application of the method of characteristics . . . . . . . . . . . . . . . . 45 3.3 Schematization of the pipe-line system . . . . . . . . . . . . . . . . . . . 47

4 Parameters of the basic equations . . . . . . . 4.1 Velocity of propagation of the pressure wave 4.2 Pipe-line diameter . . . . . . . . . . . . 4.3 Density of the liquid . . . . . . . . . . 4.4 Pressure losses due to friction . . . . . . 4.5 Velocity of the liquid . . . . . . . . . . 4.6 Solid particles in the pipe-line . . . . . . . 4.7 Gas in the pipe-line . . . . . . . . . . . 4.8 Pressure in the pipe-line . . . . . . . . . 4.9 Cavitation . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . 49

. . . . . . . . . . . . . . . . 49

. . . . . . . . . . . . . . . . 54

. . . . . . . . . . . . . . . . 55

. . . . . . . . . . . . . . . . 55

. . . . . . . . . . . . . . . . 57

. . . . . . . . . . . . . . . . 57

. . . . . . . . . . . . . . . . 62

. . . . . . . . . . . . . . . . 65

. . . . . . . . . . . . . . . . 66

5 Dampingdevices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.1 Junction without a damping device . . . . . . . . . . . . . . . . . . . . 68 5.2 Constant pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.3 Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.4 Airchamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7

Con tents

5.5 Surgetank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.6 Overtlow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.7 Air inlet valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.8 Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.9 Integrated damping device . . . . . . . . . . . . . . . . . . . . . . . . 79 5.10 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.11 Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 Pressure devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.1 Attachment without a pressure device . . . . . . . . . . . . . . . . . . . 89 6.2 Closed pipe-line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.3 Localloss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.4 Control valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.6 Butterfly valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.7 Condenser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.8 Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.9 Turbine with fixed characteristics . . . . . . . . . . . . . . . . . . . . . 106 6.10 Turbine with variable characteristics . . . . . . . . . . . . . . . . . . . . 108 6.1 1 Turbine controlled by a governor . . . . . . . . . . . . . . . . . . . . . 113

6.1 1.1 Guide blades . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.11.2 Action blades . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.5 Non-return flap valve . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7 Calculation of water hammer . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.1 Computer application . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.2 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.3 Input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.4 Data input and output of the results . . . . . . . . . . . . . . . . . . . 125 7.5 Solution for the sections . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.6 Solution for the junctions . . . . . . . . . . . . . . . . . . . . . . . . 129 7.7 The initial state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.8 Calculations for the damping devices . . . . . . . . . . . . . . . . . . . 135

7.8.1 Junction without a damping device . . . . . . . . . . . . . . . . . 135 7.8.2 Constant pressure . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.8.3 Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.8.4 Air chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.8.5 Surge tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.8.6 Overflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.8.7 Air inlet valve . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.8.8 Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.8.9 Integrated damping device . . . . . . . . . . . . . . . . . . . . . 139 7.8.10 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.8.11 Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.9 Calculations for the pressure devices . . . . . . . . . . . . . . . . . . . . 141 7.9.1 Attachment without a pressure device . . . . . . . . . . . . . . . . 141 7.9.2 Closed pipe-line . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.9.3 Local loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.9.4 Control valve . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.9.5 Non-return flap valve . . . . . . . . . . . . . . . . . . . . . . . 142 7.9.6 Butterfly valve . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8

Contents

7.9.7 Condenser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.9.8 Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.9.9 Turbine with fixed characteristics . . . . . . . . . . . . . . . . . . 145 7.9.10 Turbine with variable characteristics . . . . . . . . . . . . . . . . . 145 7.9.1 I Governor-controlled turbine . . . . . . . . . . . . . . . . . . . . 146

7.10 Calculation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.1 1 Iterative solution for the junctions . . . . . . . . . . . . . . . . . . . . . 150 7.12 Checking whether a steady state was attained . . . . . . . . . . . . . . . . 151 7.13 Maximum and minimum values . . . . . . . . . . . . . . . . . . . . . . 151 7.14 Submission for several calculations . . . . . . . . . . . . . . . . . . . . 152

8 Calculation of the steady state . . . . . . . . . . . . . . . . . . . . . . . . . 154 8.1 Abridged calculation of the steady state . . . . . . . . . . . . . . . . . . 154 8.2 Modification of the pipe-line sections . . . . . . . . . . . . . . . . . . . 155 8.3 Modification of the damping devices . . . . . . . . . . . . . . . . . . . 156 8.4 Modification of the pressure devices . . . . . . . . . . . . . . . . . . . . 157 8.5 Submitting the calculation . . . . . . . . . . . . . . . . . . . . . . . . 159

9 Inputdatafile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 9.1 Name of the calculation . . . . . . . . . . . . . . . . . . . . . . . . . 163 9.2 Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 9.3 Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 9.4 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

9.4.1 Accuracy of the calculation . . . . . . . . . . . . . . . . . . . . 167 9.4.2 Time interval of printing for the main calculation . . . . . . . . . . . 167 9.4.3 Parameters of the damping devices . . . . . . . . . . . . . . . . . 174 9.4.4 Parameters of the pressure devices . . . . . . . . . . . . . . . . . 178 9.4.5 Control valve - alternative method of submission . . . . . . . . . . . 185 9.4.6 Turbine with variable characteristics . . . . . . . . . . . . . . . . . 187 9.4.7 Governor-controlled turbine . . . . . . . . . . . . . . . . . . . . 193

9.5 Type of calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 9.6 Graphical and numerical outputs . . . . . . . . . . . . . . . . . . . . . 210 9.7 Subtitle of the calculation . . . . . . . . . . . . . . . . . . . . . . . . 215

10 WTHD program for creating the input data file . . . . . . . . . . . . . . . . . 216 10.1 Starting work with the WTHD program . . . . . . . . . . . . . . . . . . 216 10.2 Adding a data line to the end of the file . . . . . . . . . . . . . . . . . . 217 10.3 Inserting a data line into the file . . . . . . . . . . . . . . . . . . . . . 217 10.4 Deleting a data line from the file . . . . . . . . . . . . . . . . . . . . . 218 10.5 Modifying a data line . . . . . . . . . . . . . . . . . . . . . . . . . . 218 10.6 Adding a further data file . . . . . . . . . . . . . . . . . . . . . . . . 219 10.7 Formal checking of the file . . . . . . . . . . . . . . . . . . . . . . . . 219 10.8 Listing of a data file . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 10.9 Plotting of functions . . . . . . . . . . . . . . . . . . . . . . . . . . 220 10.10 Terminating the WTHD program . . . . . . . . . . . . . . . . . . . . . 222

1 1 Output of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11.1 Main output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11.2 Graphical output . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 11.3 Numerical output . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 11.4 WTHG program for plotting the numerical output . . . . . . . . . . . . . 226

9

Contents

12 Reduction of water hammer . . . . . . . . . . . . . . . . . . . . . . . . . . 229 12.1 Adjustment regime of a valve . . . . . . . . . . . . . . . . . . . . . . . 229 12.2 Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 12.3 Surge tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 12.4 Air chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 12.5 Other methods of protection . . . . . . . . . . . . . . . . . . . . . . . 232

13 WTHM program for the calculation of water hammer . . . . . . . . . . . . . . . 234 13.1 Basic layout of the program . . . . . . . . . . . . . . . . . . . . . . . 234 13.2 The main program . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 13.3 Subprograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 13.4 Errors in the calculation . . . . . . . . . . . . . . . . . . . . . . . . . 240 13.5 Denotation of the variables . . . . . . . . . . . . . . . . . . . . . . . . 243

14 Examples of the calculation of water hammer . . . . . . . . . . . . . . . . . . 247 14.1 Abrupt closing of a pipe-line without considering the effect of pressure losses due

to friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 14.2 Abrupt closing of a pipe-line with the effect of pressure losses taken into

account . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 14.3 Control valves in the delivery pipe-line of a pump . . . . . . . . . . . . . . 271 14.4 Calculation of the characteristics of a valve from pressure measurements . . . . 279 14.5 Calculation of the valve control rtgime producing a required pressure or discharge

curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 14.6 Calculation of the steady state of flow in a pipe-line network . . . . . . . . . 290 14.7 Periodic variations in pressure . . . . . . . . . . . . . . . . . . . . . . 294 14.8 Protection of a delivery pipe of a pump by an air chamber . . . . . . . . . . 301 14.9 Water hammer induced by cavitation after disconnection of a pump . . . . . . 306 14.10 Calculation of discharge from pressure measurements . . . . . . . . . . . . 310 14.1 I Calculation of the characteristics of a pump from the measured pressures and

speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 14.12 Starting-up of a pump with an electric motor, a butterfly valve and a condenser 316 14.13 Turbine with variable characteristics . . . . . . . . . . . . . . . . . . . . 321 14.14 Governor-controlled turbine . . . . . . . . . . . . . . . . . . . . . . . 327

Appendix A Subprograms for the damping and pressure devices . . . . . . . . . . . . 333 Appendix B List of files on WTHM diskette . . . . . . . . . . . . . . . . . . . . . 352 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Appendix of references . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

10

Preface

Anyone who has once manually calculated water hammer in a rather complex pipe-line system, would be loath to repeat the experience. It is a very tedious task demanding expert knowledge. TheLnext time, he would probably resort to a computer. However, he will have to face new difficulties. As a rule, the calculation programs available at present will prove unsuitable for the particular pipe-line system to be solved. One has either to write a new program, or to modify one of the ones available. This again is time consuming, requiring not only expert knowledge concerning water hammer, but also programming and computer skills. The frequently adopted outcome of these difficulties is to not calculate water hammer in complex pipe-line systems, but to estimate the results on the basis of the solution of a much simplified system and, at the same time, to rely on experience and good fortune.

The present work attempts to offer a more acceptable approach to the reader. It contains the description of a method of calculation and of a corresponding computer program, applicable to a broad variety of hydraulic systems encountered in practice, from hydraulic networks and cooling circuits of electric power plants to systems containing several governor-controlled turbines.

This method of calculation allows the hydraulic system investigated to be made up, as required, of individual pipe-line sections and various devices, such as reservoir, air chambers, surge tanks, air inlet valves, non-return valves, pumps, turbines and other devices, without any modification of the program as such, but merely through a suitable choice of input data. The calculation can take into account the separation of the fluid column (cavitation); the sections may also form closed circuits.

The present work explains the basic physical principles of water hammer, it introduces its most frequent causes and the methods aimed at limitating its effects. It contains a brief derivation of the fundamental equations describing water hammer, including the conditions under which they are valid, as well as examples of the calculation of water hammer.

The program may also be used for many other calculations: the calculations of the steady state of flow in pipe-line systems; the determination of the discharge in an operating hydraulic system on the basis of measurements of

11

Preface

pressure during unsteady flow; the determination of the characteristics of valves, pumps, turbines and other hydraulic devices; the calculation of the control regime of a valve in order to obtain a required pressure or discharge curve, etc. Examples of such applications of the program also form part of this publication.

The first version of the program for calculating water hammer originated in the Institute for Hydrodynamics of the Czechoslovak Academy of Sciences in 1969 [83]. Since that time, the program has been used extensively in practice in designing pipe-lines and it has been perfected and expanded according to the requirements of the users [17,85]. In the present work, we introduce the English version of the program modified for use with IBM PC/XT personal and other compatible computers. The program has also been used with IBM PC/AT, IBM PS/2 and other computers.

One of the new possibilities introduced in this publication is the solution of water hammer for a governor-controlled turbine, worked out with the valuable cooperation of Ing. J. JureCka, an employee of CKD Blansko [28].

A part of the publication is represented by the listing of the subprograms for the damping and pressure devices in FORTRAN language. The program for the calculation of water hammer and the programs for creating and checking the input data files and for plotting the results of the calculations are introduced on a diskette in translated form prepared for use with the MS-DOS system. The diskette also contains the input data files of all the examples dealt with in the text. This diskette may be ordered from the author*). The programs mentioned are copyright. All the rights are represented by Dilia, Czechoslovakia**).

The author is prepared to modify the programs according to the requirements of interested parties. The author would welcome suggestions for further addi- tions or improvements to the calculation procedures presented that would widen the sphere of their applicability.

The author would like to extend his gratitude to all who participated in working out the method of calculation and the program and in the preparation of the present publication.

J. Zaruba

*) Ing. Josef Zaruba, CSc., Fialova 3, 160 00 Praha 6, Czechoslovakia **) DILIA, P.O. Box 34, 128 24 Prague 28, Czechoslovakia

12

List of symbols

The denotation of the input data is included in Chapter 9. The denotation of the variables used in the subprograms for the damping and pressure devices is included in Sect. 13.5. Values for which no dimensions are indicated are dimen-

- velocity of propagation of a pressure wave (m s - l )

- velocity of propagation of a pressure wave used in the abridged calculation of the steady state (m s-

- cross-sectional area of a pipe-line (m ) - cross-sectional area of a pipe-line, variable as a function of

- cross-sectional area occupied by sediments in a pipe-line (m2) - constant of the characteristic of the electric motor of a pump

- constant of the moment characteristic of a pump or a turbine

- constants of the pressure characteristic of a pump or a turbine

- coefficient expressing the effect of the nature of the support of

~ refer to (7.30) (kg m-4 s-I) - refer to (7.31) (kg m-’ s - ~ ) - volume concentration of a liquid and of solid particles - coefficients for converting the parameters of a turbine and of its

model (dimensions depend on the type of parameter) - constant determining the effect of mutual friction between a li-

quid and solid particles - gain of the electric speed controller - time derivative constant of an electric speed controller (s) - time constant of a frequency converter (s) - constant of the feedback of the hydraulic amplifier of guide and

2 pressure in the pipe-line (m2)

(kg m2)

(kg m -4)

(kg m-’)

a pipe-line on the wave velocity

action wheels (V m-I)

13

- constant of the proportional feedback of the hydraulic am-

- standing static characteristic of an electric speed controller - gain of the distribution slide valve of a hydraulic amplifier - constant expressing the effect of the geometry of the slide valve

and the slave cylinder of guide and action wheels (kg m-’) - parameter of a control valve (dimension in agreement with the

type of parameter) - admissible difference in moment (N m) - admissible difference in pressure for the iterative solution of

- admissible difference in pressure at steady flow (Pa) - admissible difference in discharge for the iterative solution of

- admissible difference in discharge at steady flow (m3 s-I) - internal diameter of pipe-line (m) - diameter of an opening (m) - diameter of a pump or a turbine (m) - diameter of a butterfly valve (m) - thickness of pipe-line wall (m) - refer to (7.32) - modulus of elasticity of pipe-line material (Pa) - modulus of elasticity of rock (Pa) - detones a functional dependence (dimension according to the

- time constant of elastic feedback (s)

plifier of guide and action wheels (V m-’)

junctions (Pa)

junctions (m3s -’)

type of function)

-a 9

-B - maximum velocity of motion of the slave-cylinder piston during the opening and closing of the guide and action blades of a turbine (m s-I)

- static characteristic of a frequency converter (V) - maximum value of voltage (V) - voltage from a power regulator (V) - voltage determining the required speed (V) - force acting on the slave-cylinder piston induced by the pressure

- frictional force affecting the motion of turbine guide and action

- force induced by oil pressure on the slave-cylinder piston con-

- acceleration due to gravity (m s - ~ )

of water on the guide and action blades of a turbine (N)

blades (N)

trolling the guide and action blades (N)

14

List o l symbols

Gr output h hb

h c

hd

h f

h i

hl h,

htot

h U

H IT 1, 1, 1,

J J

J P

J S

K , K , K e

K 2

L

J

JJu, JJd

k

1

- denotation of changes in printing time interval for the graphical

- elevation of a pipe-line above a reference plane (m) - elevation of the lowermost interconnecting pipe of a condenser

- liquid level in the chamber of a damping device in its basic state

- level of the surface of the liquid in the downstream chamber of

- level of the bottom of the chamber of an integrated damping

- level of the i-th change of cross-section of an integrated damp-

- level of the liquid in the chamber of a damping device (m) - elevation of the uppermost interconnecting pipe of a condenser

- total height of the chamber of an integrated damping device

- liquid level in the upstream chamber of a condenser (m) - head (m) - number of stages of a pump or turbine - moment of inertia of the revolving part of an assembly (kg m2) - moment of inertia of a butterfly valve (kg m2) - number determining the type of damping or pressure device - number of junction - number of the junction to which a section is attached by its

- number defining the set of parameters of a damping or pressure

- number of a section - absolute roughness of a pipe-line (m) - bulk modulus of elasticity of a liquid (Pa) - constant expressing the effect of the elasticity of a liquid and a

- bulk modulus of elasticity of solid particles (Pa) - lenght of a pipe-line section (m) - modified length of a pipe-line section (m)

(4 (4 a condenser (m)

device (m)

ing device (m) ,

(m)

(4

upstream, downstream end

device

pipe-line (Pa)

L,, L,, L,

m - mass (kg) ma, m/9

- numbers defining the types of variable in the graphical and numerical outputs

- mass affecting the motion of turbine guide and action blades (kg)

15

M Min, Max - limiting value of a variable in the graphical output (dimensions

- torsional moment of a pump or a turbine (N m)

depend on the type of variable) ~ torsional moment of an electric motor or a generator (N m) - moment induced by the weight (N m) - moment induced by the hydraulic system (N m) - moment induced by inertia (N m) - moment induced by the liquid in a pipe-line (N m) - symbol used to denote a variable in the graphical and numerical

- speed of a pump or a turbine (s- l ) - synchronous or other constant speed of an electric motor or a

- rated or other constant speed of a pump or a turbine, for which

- initial speed of a pump or a turbine (s-') - number defining the type of integrated damping device - number of changes in the chamber of an integrated damping

- maximum number of iterations in the solution for junctions - number of junctions - number of changes in the time interval of printing for the main

output - number of disconnections and connections between an electric

motor or a generator and a network - number of switching off and on of the oil pump of a butterfly

valve ~ number of sections - denotation of the change in the printing time interval in the

numerical output - identification number of a calculation - pressure of a liquid converted to the level of the reference plane

~ atmospheric pressure, absolute (Pa) - absolute pressure (Pa) - pressure in the junction of a damping device in its basic state (Pa) - absolute pressure of air in the chamber of a damping device for

- constant pressure (Pa) - pressure at the downstream end of the interconnecting pipes of a

outputs

generator (s-')

characteristics are submitted (s-I)

device

(Pa)

the damping device in its basic state (Pa)

condenser (Pa)

List of symbols

PJ - pressure in a junction (Pa) Pmax - maximum pressure in a pipe-line section (Pa) Pmin - minimum pressure in a pipe-line section (Pa) Pr - actual pressure of a liquid (Pa) Pu - pressure at the upstream end of the interconnecting pipes of

Pv - pressure in a junction at cavitation (Pa) Po - pressure in a junction in the initial state (Pa) PI, P2, P3 - pressure in pipe-line (Pa) PIX - limiting value of pressure for t + 00 (Pa) Q, Q1, QZ, Q3 - discharge of a liquid through a pipe-line (m3 s - l )

a condenser (Pa)

- discharge of a liquid from a junction into a damping device

- steady state discharge (m3 s-') - initial discharge of a liquid through a pipe-line (m3 s-') - parameter of the function defining the moment induced by the

- coefficient of linear dependence for a damping device

- coefficient of linear dependence for a pressure device

- area of horizontal cross-section of a damping device (m2) - coefficient of linear dependence for a damping device (m3 s-') - area of the cross-section perpendicular to the direction of

flow, of the chamber of an integrated damping device (m2) - area of the horizontal section of a surge tank in its lower and

upper parts (m2) - coefficient of linear dependence for a pressure device (Pa) - subtitle of a calculation - time (s) - instant at which damping begins (s) - time at which the first period begins and ends (s) - time at which a change occurs in the time interval of printing

- time at which a change occurs in the time interval of printing

- maximum value of the calculation time (s) - instant at which an electric motor or a generator is disconnect-

ed or connected from or to a network (s) - instant at which the oil pump of a butterfly valve is switched

off or on (s) - period of shock wave (s)

(m3 s-I)

liquid in a pipe-line and acting on a butterfly valve

(kg-' m4 s)

(kg m - 4 ~ 1 )

for the graphical and numerical outputs (s)

for the main output (s)

17

List of symbols

Text outputs Title U

Uo to U ,

- verbal denotation of a variable in the graphical and numerical

- name of a calculation - point velocity (m s-I) - voltage (V)

U ~ m a x 3 U ~ m i n 9

‘23max 9

‘23min U

00

V Y4

‘b

Y o ,

v, VN, VNZ, VN2

- extreme values of voltage U, (V) - mean cross-sectional velocity of flow of a liquid in a pipe-line

- initial mean cross-sectional velocity of flow of a liquid in a

- volume of liquid (m3) - volume of air or of the void space at cavitation (m3) - volume of air in a damping device in the basic state (m3) - total volume of the chamber of an integrated damping devi-

- initial volume of a liquid in a damping device (m3)

- variables of pressure device (dimension in accordance with the

(m s-l)

pipe-line (m s-I)

ce (m3)

type of variable) ‘“max,

VNmin ,

‘Omax’ ‘Omin - extreme values of the parameters V N and VO (dimension in

VO, VZ, V2 - variables of damping devices (dimension in accordance with the

x, Y, - Cartesian coordinates (m)

xmina, xminB - extreme positions of the slide valve of the hydraulic amplifier of

xocc ’ xop - initial position of the slide valve of the hydraulic amplifier of

XCi, xp - position of the slide valve of the hydraulic amplifier of guide

agreement with the type of parameter)

type of variable)

‘ma, a 9 X m a x p

guide and action wheels (m)

guide and action wheels (m)

and action wheels (m) Ymax a 9 Ymaxp 7

Ymin.9 YminB - extreme extension of the slave-cylinder piston rod of guide and

J’a’ J’s - extension of the slave-cylinder piston rod of guide and action

2 - general variable (dimension according to the type of variable)

action blades (m)

blades (m)

18

List of symbols

- type of calculation - angle determining the pipe-line gradient (rad); parameter defin-

ing the position of turbine guide blades (dimension according to the type of parameter)

- angle of tilt at which the oil pump of a butterfly valve is switched off automatically (rad)

- angle of tilt of a butterfly valve (rad) - maximum angle of opening of a butterfly valve (rad) - initial angle of tilt of a butterfly valve (rad) - parameter defining the position of turbine action blades (di-

- polytropic exponent - damping exponent - difference in heights (m); head loss (m) - difference in pressures (Pa) - difference in the pressure of air in the chamber of a damping

- difference in pressure in front of and behind a pressure device

- initial excess pressure on a turbine (Pa) - difference in pressure between the downstream and upstream

- change in discharge of a liquid through a pipe-line (m3 s-') - time interval of the calculation (s) - time interval of printing for the graphical and numerical out-

- time interval of printing for the main output (s) - change in the mean cross-sectional velocity of flow of a liquid

- difference of the liquid volume in a damping device as com-

- difference in the liquid volume AVin the initial state (m3) - length interval (m) - length of printing interval for the main output (m) - coefficient of friction - modified coefficient of friction - shear viscosity (kg m-' s-') - Poisson's ratio

mension in accordance with the type of parameter)

device as compared with the basic state (Pa)

(in the positive sense of the attached section) (Pa)

ends of a pipe-line section in the steady state (Pa)

puts (s)

in a pipe-line (m s-')

pared with the basic state (m3)

[, [+, l-

rJi+, C J i -

- coefficients of pressure losses of a damping device

- coefficients expressing pressure losses at the inflow of a damp- [J, [J+, [ J - 9

ing device (kg m-7)

19

List of symbols

z a B

- density of a liquid (kg mP3) - density of air at atmospheric pressure (kg m-3) - density of a compressible liquid (a function of pressure) (kgm-3) - density of solid particles (kg m-3) - variable determining the pressure losses in a control valve - variable determining the pressure losses in the slide valve and the oil

distribution of the hydraulic amplifier of guide and action wheels (Pa) - shear stress (Pa) - subscript denoting the values pertaining to guide blades - subscript denoting the values pertaining to action blades - bar denotes the values pertaining to a turbine model - apostrophs denote the values determined in previous calcula-

tion steps or at previous calculation points of the grid of characteristics

20

1 Water hammer

1.1 The origin of water hammer

Let us consider a pipe-line, or a pipe-line system, filled with a flowing liquid. The system may contain various devices, such as valves, pumps, reservoirs, surge tanks, air inlet valves, etc., all of which may affect the flow.

The flow of liquid in a system may be steady or unsteady. At steady flow, the fundamental values defining the flow do not vary with time. From the point of view of the problem discussed, a flow in which pressure and velocity vary irregularly due to turbulence, is also steady. At unsteady flow, its fundamental values vary with time.

The pressure of a liquid in a conduit and its discharge are interdependent. Every change in discharge induces a corresponding change in pressure and vice versa. The changes in pressure caused by this dependence are called water hammer. In the present work, this term is used, for the sake of simplicity, also for cases, where the flowing liquid is not necessarily water, but may be any liquid having similar properties (refer to Sect. 2.2), possibly with an admixture of solid substances (refer to Sect. 4.6), or gases (refer to Sect. 4.7). The changes of pressure at water hammer may be insignificant, but could also be large, sometimes leading to the rupture of a pipe-line, or of other devices forming part of the pipe-line system. Such breakdowns are by no means rare.

Another danger of water hammer is the difficulty of estimating in advance, without detailed calculations, whether the water hammer imperils the system in a specific case, or whether its effects can be neglected. Such calculations are laborious, especially for more complex hydraulic systems; they require expert knowledge and such information about the system under consideration as may not always be available.

The problem of water hammer may be important in the design and operation of various types of hydraulic systems, such as water-supply networks, irrigation systems, industrial conduits, distribution systems and waste piping, cooling circuits of thermal and nuclear power stations and many other hydraulic installations in various branches of industry. Unsteady flow develops in all these systems, at least during their opening and closing, and is inevitably accompanied by water hammer, even if this may not be dangerous in every instance.

21

Water hammer

Water hammer may be the cause of other adverse effects besides an increased mechanical stress of the pipe-line and the attached equipment. It may affect the regulation of hydraulic systems or distort the results of measurements of hydraulic quantities (refer to Sect. 1.5).

On the other hand, the effect of water hammer is exploited in some devices: the hydraulic ram may serve as an example of such an application. It is a simple machine by means of which water can be pumped to considerable heights.

1.2 Physical principles 1.2.1 Incompressible liquid

In our discussion of water hammer, we can in many cases assume to be dealing with an incompressible liquid and a rigid conduit.

Consider the very simple case, where a liquid flows from a reservoir through a horizontal pipe-line. The discharge is controlled by means of a valve fixed at the downstream end of the pipe-line. The pressure losses along the pipe-line are neglected (Fig. 1.1). Qo is the initial discharge. A is the cross-sectional area of the pipe-line, I is the length of the pipe-line and Q is the density of the liquid.

I 0 - 1 Fig. 1 . 1 Outflow of an incompressible liquid from a reservoir through a pipe-line. t- 4 I

Now, the pipe-line is closed over an interval of time in such a manner that the discharge changes linearly with time. During this time interval, the pressure at the valve increases by a value Ap.

The change in momentum of the liquid in the pipe-line equals the momentum imparted to the liquid by the difference in pressure at the upstream and downstream ends of the pipe-line during the interval At.

This may be expressed by the equation

Qo A l e - = ApA At A

from which we may determine the increment in pressure

22

Physical principles

pipe-line. 0 Y

in pressure p for an incompressible liquid, at (a) the valve; (b) the midpoint of a

With the initial mean cross-sectional velocity

1 /

At t

QO Do = - A

it follows that

in the pipe-line

(1.3)

A diagram depicting the variation in pressure, as a function of time, at the valve and at the midpoint of the pipe-line is shown in Fig. 1.2.

Equation (1.4) can be employed to determine the increase in pressure during the closing of the pipe-line only if the closing interval is sufficiently long and the change in discharge is linear.

For a short closing interval (At + 0) the pressure, according to equation (1.4), would increase infinitely (Ap -+ a), which is not in accordance with reality. The main reason for this contradiction lies in the compressibility of the liquid and the elasticity of the pipe-line, both of which manifest themselves significantly during rapid changes in pressure and discharge.

Hence, water hammer has to be studied in more detail, taking into account the compressibility of the fluid and the elasticity of the pipe-line. More detailed knowledge enables us also to decide in actual cases, whether the change is sufficiently slow to allow equation (1.4) to be used to calculate the increase of pressure.

1.2.2 Compressible liquid - abrupt closing of the pipe-line

Let us consider the case described in Subsect. 1.2.1, but with the assumption of a compressible fluid and an abrupt closing of the conduit. If a small compressibility of the fluid is assumed, this leads to the conclusion (refer to Sect. 2.1) that

23

Water hammer

the changes in pressure and discharge travel through the pipe-line at a constant velocity, equal to the velocity of sound in a fluid in this case.

Closure of the pipe-line at time t = 0 causes an increase in pressure Ap at the valve and this increment travels at a velocity a in a direction opposite to that of the flow. The change in discharge from the initial value Qo to zero travels at the same velocity. Fig. 1.3 shows this state at time t . The longitudinal interval Ax travelled by the pressure wave in time t < I/a is determined by the relation

Ax = at (1.5)

11 I

I l

1 0, - Fig. 1.3 Outflow of a compressible liquid from a reservoir through a pipe-line and an abrupt closing of the pipe-line.

AX I

The change in momentum of the liquid in a section Ax of the pipe-line equals the momentum imparted by the difference in pressure Ap acting at the upstream and downstream ends of this section during time t . This may be expressed by the equation

Qo A Axe - = ApAt A

It follows from equations (1.5) and (1.6) that the increment in pressure is equal to

aeQo Ap = - A (1.7)

Substituting (1.3) into this equation, we obtain

Ap = q u o (1.8) The wave front progresses towards the upstream end of the pipe-line, reaching it at time t = l/a (Fig. 1.4a). At this instant, a condition of zero discharge and a pressure increased by Ap exists throughout the pipe-line. The liquid at the upstream end of the conduit, however, is not in a state of equilibrium. The pressure to which it is exposed from the left-hand side is lower by Ap than the right-hand pressure. A new pressure wave develops due to this difference and travels to the right towards the valve (Fig. 1.4b). In this wave, the pressure drops

24

Physical principles

to its original value at steady flow (p = 0) and the discharge is the same as during the initial steady flow, though in the opposite sense, that is, from the valve towards the reservoir (Q = -Qo). The wave reaches the valve at time t = 2l/a.

-AP 1- d l

-------- Fig. I .4 Propagation of an abrupt change in pressure and discharge in the liquid in a pipe-line.

At this instant, the discharge at the valve is again unbalanced. It is negative, directed towards the reservoir, while the valve is closed. Another pressure wave develops (Fig. 1.k) travelling towards the reservoir. The pressure drops to the value p = -Ap and the discharge is zero. This pressure wave reaches the reservoir at time t = 3 1 / a

The imbalance at the cross-section close to the reservoir creates another pressure wave travelling towards the valve (Fig. 1.4d) which it reaches at time t = 41/a. At this instant, the state of flow throughout the conduit is identical with that which existed at the closure of the pipe-line ( t = 0), p = 0 and Q = Qo*

25

Water hammer

The whole process is repeated with a period

41 a

T = -

Figure 1.5 illustrates the pressure at the valve and at the midpoint of the pipe-line, as follows from the discussion.

I

Fig. 1.5 Variations in pressure as a function of time, at (a) the valve; (b) the midpoint of a pipe-line for a compressible liquid and an abrupt closing of a pipe-line.

The development of water hammer as described above and the accompanying equations (1.7) to (1.9) satisfactorily reflect reality for cases of small losses of pressure and short closing intervals At, that is, for

At << T (1.10)

Equations (1.2) and (1.4) apply to cases of small pressure losses, linear changes in discharge and long closing intervals, that is, for

At >> T (1.11)

1.2.3 Compressible liquid - linear closing of the pipe-line

Let us again consider the case described in Subsection 1.2.1, but for a compressible liquid and a linear change in discharge, with the valve being closed over an interval At comparable with the duration of the period T. Hence, neither condition (1.10) nor (1.11) is satisfied.

26

Physical principles

This case is illustrated in Fig. 1.6, where At < l/u. The increase in pressure at the valve is given by

A p = -- ae AQ A

or Ap = - a e A v

(1.12)

(1.13)

which are analogous to the equations (1.7) and (1.8) respectively. The increments AQ and Av represent the changes in discharge through the valve and in the mean cross-sectional velocity at the valve, respectively. The negative sign expresses the fact that a reduction in discharge or in the velocity of flow corresponds to an increase in pressure. The pressure at the valve increases gradually in direct proportion to the change in the discharge up to the value Ap as given by equation (1.7). Every increase in pressure travels at a velocity a in the form of a pressure wave from the valve towards the reservoir accompanied by a drop in the discharge. This wave is reflected at the reservoir in the manner described in Subsect. 1.2.2. The reflected wave travels back towards the valve, where it is again reflected and once again progresses towards the reservoir. The individual pressure waves are added and complex variations in pressure and discharge develop throughout the pipe-line.

0, _c

Fig. 1.6 Outflow of a compressible Liquid through a pipe-line for a linear change in discharge during closing.

Fig. 1.6 illustrates the distribution of pressure along the pipe-line at time t = 0, corresponding to the steady state condition prior to any adjustment of the valve. The graph also shows the distribution of pressure at time t = At/2, when the first pressure wave has travelled a distance aAt /2 from the valve and a pressure Ap has appeared at the valve, corresponding, in accordance with (1.12), to a reduction in the initial discharge by half. The next line illustrates the distribution of pressure at time t = At, immediately after the pipe-line had been completely closed. The pressure at the valve has reached the value Ap as determined by equation (1.7), and the first pressure wave has reached a point at a distance a At from the valve. The last line shows the distribution of pressure

27

Water hammer

at time t > l/a, when the waves which have been reflected from the reservoir reduce the pressure waves progressing from the valve in the section at - 1.

The pressures at the valve and at the midpoint of the conduit corresponding to this description are shown in Fig. 1.7.

Fig. I .7 Variations in pressure as a function of time at (a) the valve; (b) the midpoint of a pipe-line for a compressible liquid, and a linear change in discharge during closing.

If the closing time of the valve is longer than half the period, At > T/2, the pressure at the valve does not attain the value Ap given by equation (1.7). The pressure at the valve is reduced by the waves which have already been reflected from the reservoir while the valve was being closed.

1.3 Actual cases

Actual cases of water hammer encountered in practice are much more complex than those presented in Sect. 1.2. There are many circumstances that can affect the course of water hammer. Usually, these cannot all be included in the calculations. The system to be analysed has to be simplified and only those factors can be considered which could substantially affect the results of the calculations for a given case. The most relevant problems in a calculation of water hammer are: choice of the most unfavourable cases of water hammer that could actually appear during operation; reliable data on the parameters of the various devices which form part of the system; correct schematization of the

28

Actual cases

hydraulic system analysed, and correct interpretation of the results obtained. Nowadays, advanced computer techniques allow us to calculate in detail even rather complex cases of water hammer, an achievement formerly impossible on account of the extensive mathematical calculations involved.

Apart from the compressibility of the fluid, the elasticity of the conduit and the nature of its support often significantly affect the wave velocity and, consequently, the development of water hammer as a whole. In many cases, one has to take into account the local pressure losses in addition to those which develop along the pipe-line.

The pipe-line need not be horizontal, its diameter, wall thickness and conditions of support may vary. It may be made up of an entire network of interconnected conduits. Such conditions also have to be included into the calculations, because the waves, travelling through the pipe-line, are partially reflected at every change in cross-section and at every division of the pipe-line, while partially continuing to travel in the original direction. The individual waves are mutually superimposed.

Various devices form part of a hydraulic system, for example, reservoirs, air chambers, surge tanks, air inlet valves, overflows, valves, turbines, pumps, etc., all of them affecting water hammer significantly. Therefore, their effects, including the effect of their manipulation in the course of water hammer have to be included in the calculation.

Cavitation (liquid column separation) is another phenomenon which sometimes has to be considered. It is possible that the pressure in the pipe-line drops to that of the vapour of the liquid during water hammer, so that the liquid column separates. This again may strongly affect the course of water hammer. Refilling of the developed voids could lead to a dangerous increase of pressure.

It is evident even from a preliminary list of fundamental factors and phenomena which may influence the course of water hammer under certain conditions that the equations introduced in Sect. 1.2 do not suffice for the analysis of every case which may occur in practice.

The following sections, therefore, describe a method of calculation of water hammer which allows us to take into account all the effects mentioned above.

There are yet other circumstances which may affect water hammer and which have not been incorporated into this method. Some of these, such as for example, a high velocity of flow relative to the velocity of the pressure wave, large deformations of the pipe-line, pressure losses not corresponding to the condition of steady flow, the presence of solid particles and gases in the pipe-line, etc., are considered in more detail in Chapter 4.

29

Water hammer

1.4 Causes of water hammer

It has been stated that water hammer develops due to any change in pressure or discharge in a conduit. The causes of such changes may be most varied. Let us mention at least some of those, which frequently induce large changes in pressure.

A liquid flows from a reservoir through a pipe-line provided with a valve at its downstream end. The discharge in the pipe-line varies as the valve is being closed. The variation in discharge induces water hammer (Figs. 1.6 and 1.7). If the closing process is not suitably controlled, the largest changes in pressure usually occur in the final stages of closing.

A reservoir may feed an entire pipe-line network, for example, a water-supply network with a number of offtakes, which may be adjusted independently. Every adjustment induces water hammer phenomena in the network and their effects are mutually superimposed.

Another relevant example is the pumping of a liquid into a reservoir. A pump is installed at the upstream end of a pipe-line followed by a check valve which prevents the outflow of the liquid from the reservoir when the pump is switched off. After the pump has been stopped, the liquid continues moving due to its own inertia and the pressure in the pipe-line drops. Sometimes, this drop may be so large as to cause cavitation. In the next stage, the liquid starts flowing back towards the pump. Its flow is, however, checked by the check valve, which closes. This produces an increase in pressure which may imperil the entire system. The increase and drop in pressure are repeated at regular periods until the entire phenomenon fades out.

A still more dangerous situation may develop, if the check valve does not prevent the backflow of the liquid through the pump in time. The liquid may then flow from the reservoir at a high velocity. If the check valve suddenly stops the backflow at this stage, the resulting effects of water hammer may be still more pronounced.

When the generator of a water turbine is disconnected from a power network, the turbine speed starts to increase. Consequently, the turbine controller closes the inflow to the turbine, thus creating water hammer in the penstock. Water hammer effects in the penstock are created by any changes in discharge through the turbine, caused by changes in the connected power network, by the opera- tors, or by breakdowns. Sometimes, the entire system becomes unstable due to the mutual influence of a turbine equipped with a controller and to an unsteady flow in the penstock. In such case, even small variations in pressure in the penstock may increase steadily and perilously.

Air entrapped in a pipe-line is a frequent cause of water hammer. Sometimes, it may reduce the effects of water hammer. If the air is in the form of minute bubbles dispersed throughout the stream of the fluid, it increase the compres-

30

Causes of water hammer

sibility of the fluid and reduces its average density, thus also reduces the effects of water hammer. Similarly, if air accumulates in the form of larger bubbles at suitable points of the conduit, it also has a damping effect on the development of water hammer. If, however, larger bubbles move through the pipe-line, they may reach, for example, the outflow which is restricted by the valve. While a bubble passes through the valve, the velocity of the liquid flowing in the pipe-line may increase considerably, because the pressure losses in the valve are much lower for the air than for the liquid. Once the bubble has escaped from the pipe-line, the liquid starts flowing through the valve again and the pressure losses in the valve are increased. This effect induces water hammer of a character similar to that caused by a rapid opening and a subsequent partial closing of a valve. A similar effect may take place not only at the outflow, but also at the point of any large source of a local resistance in a pipe-line system.

Air can enter the conduit in various ways. The entire pipe-line is filled with air before any liquid is let in. Air may also get into the pipe-line during operation, for example, by being additionally sucked in by the pumps, through air-inlet pipes or valves, or dispersed in the liquid, etc.

Solid substances may also be present in a pipe-line, either intentionally, for example in the case of hydralic transport, or accidentally. These solid substances may induce water hammer directly, for example, by the sudden clogging of some part of the pipe-line. Apart from this, they may also unfavourably affect water hammer produced by other causes. They may increase the density of the flowing mixture and reduce its compressibility thus increasing the effects of water hammer. Solid particles may settle in a pipe-line and reduce its cross-sectional area so that the velocity of the flowing liquid is higher for the same discharge and the pressure variations at water hammer are increased. The effect of solid particles in a pipe-line is described in more detail in Sect. 4.6.

There are many other causes of water hammer. In more complex systems especially, the cumulative effect of several types of devices which influence water hammer may have an adverse effect. However, even in simple cases, for example in pumping water into a reservoir, manipulations very unfavourable with regard to water hammer may take place. For example, after the failure of the pump, the operator may start it again. Much depends on the instant of this starting. If it is done at a time when the entire water hammer effect has died down, it is an operation for which the system must have been designed. If, however, the pump is started sooner than that, unfavourable effects may appear, due to the superimposition of water hammers produced by both the failure and the restarting of the pump, so that the pressure in the pipe-line may increase much more than when the pump is restarted during the steady state.

31

Water hummer

1.5 Examples of water hammer

In this section, some actual cases are described of the unsteady flow in hydraulic systems, where the values of the fundamental quantities were measured.

The first case is one of the automated closing of a valve following an interruption of the power supply to a pump feeding water from a lower basin to an upper one through a spherical valve. The layout of the system is shown in Fig. 1.8.

Fig. 1.8 Failure of hydraulic system.

a Basic layout of the

A steel conduit, 1500 m long, has a prevalent diameter of 3.8 m. The difference between the upper and lower water levels is 410 m and the discharge 20 m3 s-' at the time of measurement. The actual system was more complex: the cross- section of the conduit and the thickness of its wall were variable and the system contained various other devices. These conditions, however, did not greatly influence water hammer.

The variation with time of some of the parameters measured is shown in Fig. 1.9.

A steady state of flow had been established in the system. At a time t = 0, the power supply to the pump was disconnected. This resulted in the automatic closing of the spherical valve according to a predetermined closing rkgime. The time-dependent extension of the slave-cylinder piston rod operating the valve was measured and is illustrated by the curve y(t). The value y(0) = 1.35 m corresponds to a fully opened valve, and the value y(34) = 0.0 m to a closed valve. The process of closing was decelerated between t = 12.7 s and t = 33.5 s. Simultaneously with the closing of the valve, the speed of the pump, expressed by the curve n( t ) , decreased from the initial n(0) = 8.3 s-' to zero at t = 11.0 s. Then the pump started to revolve in the reverse sense attaining a minimum speed of n(21) = -4.6 s-'. Subsequently, the pump gradually came to a standstill.

The curves pl(t) and p 2 ( t ) represent the variation with time of the pressure in front of and behind the valve (Fig. 1.8). During the course of water hammer the pressure pl(t) in front of the valve did not exceed the pressure at the initial steady state.

32

Exumples of water hammer

The pressure p 2 ( t ) behind the valve attained its maximum value p,(36) = 4680 kPa after the valve had been completely closed. This pressure is approximately 8 Oh higher than the initial steady state pressure p2(0) = 4340 kPa. The minimum pressure p2(3) = 3240 kPa measured in the pipe-line behind the valve represents a drop of about 25 YO.

The measured pressures and speeds are satisfactory from an operational point of view and were achieved through a suitable choice of the closing regime of the spherical valve.

Fig. 1.9 Failure of a pump. Measured pressurep,(t) in the pipe-line between the valve and the pump; p2(f) behind the valve; extension y ( t ) of the slavecylinder piston rod of the valve; speed " ( 1 ) of the pump. (The values p = 2 x lo6 Pa; y = 1.35 m; n = 8.3 s-' equal 100%).

In the next case, an incorrect setting of the automatic closing of the turbine guide blades after its disconnection from the power network led to a breakdown of the turbine. The layout of the system is presented in Fig. 1.10.

Water flows from an upper basin through a steel penstock onto a turbine and, from there, into a lower basin. The approximate parameters of the system are: penstock diameter 1.0 m; penstock length 75 m; discharge 2.5 m3 s-'; difference in water levels of the basins 15 m.

Fig. I . 10 Disconnection of a turbine from a power network. Basic layout of the hydraulic system.

33

Wafer hammer

The time-dependent variations in the pressure p ( t ) measured in front of the turbine, the turbine speed n( t ) and the position y ( t ) of the slave-cylinder piston rod controlling the turbine guide blades are shown in Fig. I. 1 I .

- t (S I

Fig. 1 . 1 I Disconnection of a turbine from a power network. Measured time-dependent pressurep(f) in front of the turbine; turbine speed n(r); extension y(f ) of the slave cylinder piston rod of the guide blades. (The values p = 4 x lo5 Pa; n = 12.5 s-'; y = 0.3 m equal 100 %).

The pressure in the penstock prior to disconnecting the turbine from the power network, is p ( 0 ) = 121 kPa. The hydrostatic pressure after the fading of water hammer is p ( 3 ) = 150 kPa. The maximum pressure measured during the closing of the penstock is p(1) = 510 kPa, more than thrice the hydrostatic pressure. The measured closing time of the guide blades is 1.2 s, as may be seen from the curve y ( t ) . The value y(0) = 0.3 m corresponds to the open guide blades and the value y ( 1.2) = 0 to the closed ones. Any further reduction in the closing time would result in a still greater change in pressure. The course of water hammer is influenced by leakages in the penstock.

The turbine speed is portrayed by the curve n( t ) . After the turbine has been disconnected from the network, its speed gradually increases from the value n(0) = 12.5 s-' to the maximum value n(1.2) = 14.5 s-', that is, by about 16 % of the initial value. When the discharge is stopped, the speed gradually drops to zero.

The final example is based on laboratory measurements and emphasizes the importance of the manner in which the pressure is measured. An unsuitable type of manometer or its unsuitable connection with the pipe-line may result in completely distorted readings when the pressure variations are very rapid.

34

Examples of water hammer

The pressure in a pipe-line with a valve at its downstream end was measured after the flow was stopped very abruptly. The layout of the system is shown in Fig. 1.12.

The 17.5 m long glass pipe-line had a diameter D = 0.05 m. A constant water pressure was ensured through the use of an overflow placed at the upstream end. A specially designed valve at the downstream end could close the pipe-line within approximately 0.01 s.

Fig. I . 12 Rapid closing of a pipe-line at its downstream end. Layout of the hydraulic system.

The time-dependent variations in pressure p , ( t ) and p2( t ) , registered by capacitance transducers, installed at two points along the pipe-line are presented in Fig. 1.13.

Fig. I . I3 Rapid closing of a pipe-line at its downstream end. Pressure measured at two points along the pipe-line.

The pipe-line was closed so quickly that the maximum value of pressure p 2 ( t ) could not be influenced by the pressure waves which reflected from the upstream end. The maximum pressure calculated from equation (1.7) is shown by the dashed line in Fig. 1.13. The measured values Qo, Q and A were employed in the calculation; the density of water was assumed to be e = 1000 kg m-3.

Different methods of connecting a manometer to a pipe-line are shown in Fig. 1.14. The individual cases differ only in the shape of the feeding channel from the pipe-line to the measuring membrane. The arrangement portrayed in Fig. 1.14~ was used for the final measurements. Fig. 1.15 shows the oscillograph readings of the pressure p z ( t ) for measurements similar to those shown in

35

Water hammer

Fig. 1.13. Figs. 1.1 5a, b, c represent readings corresponding to the feeding chan- nels illustrated in Figs. 1.14a, b, c respectively. The pressure curves clearly show the effect of water hammer in the feeding channel connecting the pipe-line with the manometer, induced by a rapid change in pressure in the pipe-line.

I I I

Fig. 1.14 Attachment of a pressure-measuring capacitance transducer to a pipe-line with various arrangements of the feeding channel.

Cl a1

Fig. 1.15 Pressures measured for various arrangements of the feeding channel (denotation corresponds to that in Fig. 1.14).

Very rapid changes in pressure may also occur in present day hydraulic systems and not only under laboratory conditions. They may appear, for example, in the final closing stage of a valve, or due to cavitation. The results of pressure measurements obtained using a manometer with an inappropriate natural frequency can be seriously distorted. Besides the natural frequency of the manometer, the manner of its connection to the pipe-line, the natural frequency of the recorder and of other parts of the measuring system, are also important.

36

2 Basic equations

2.1 Derivation

The calculation of water hammer described in the present book is based on the solution of the following system of basic differential equations:

au aP @ - + - IuI u + - = 0

at 2 0 ax

au ap a2e- + - = 0 ax at

The constants e > 0, 1 2 0, D > 0, a > 0 are the density of the liquid, the coefficient of friction, the internal pipe-line diameter and the velocity of propagation of the pressure wave, respectively. x is the longitudinal coordinate along the pipe-line axis, t is the time, u(x, t ) is the mean cross-sectional velocity of flow of the liquid in the pipe-line through a cross-section with coordinate x at instant t. It is positive when the liquid flows in the direction in which the coordinate x increases. p ( x , t ) is the pressure of the liquid calculated with respect to the level of a reference plane according to the equation

P = Pr + esh (2.3)

p r ( x , t ) is the actual pressure in the pipe-line, and h is the elevation of the pipe-line above the chosen horizontal reference plane. By calculating the pressures with respect to a single reference plane the effect of the geodetic elevation of the pipe-line is eliminated.

The diagram in Fig. 2.1 a is employed to derive the equation of motion (2.1). A pipe-line of constant circular cross-section is considered. The equilibrium of forces acting along the x-axis on an elementary trolume of liquid in the pipe-line enclosed between two sections perpendicular to the pipe-line axis and spaced apart by a distance dx may be expressed by the equation

d(ju dY dz) dx (2.4) A

p,A - pr + -dx A - egAdxsinu - rnDdx = Q ( 2 ) dt

37

Basic equations

where 7CD2 A = -

4

is the cross-sectional area of the pipe-line, g is the acceleration due to gravity, the angle ct defines the longitudinal gradient of the pipe-line, 5 is the wall shear stress, u ( x , y, z , t ) is the point velocity of the liquid and x, y, z are Cartesian coordinates.

The first two terms of the equation express the effects of pressure, the other terms, in succession, the effect of gravity, of friction due to the pipe-line walls and the effect of inertial forces.

Fig. 2.1 Diagram for the derivation of (a) the equation of motion; (b) the continuity equation.

After rearrangement and using the relation

v = - udydz A ' S A

to define the mean cross-sectional velocity, Eq. (2.4) may be rewritten to the form av avdx T E D a p at ax dt A ax

Q- + Q-- + - + + Qgsincl = 0

Assuming that the velocity of flow is small compared to the velocity of the pressure wave, we may neglect the second term in Eq. (2.7) relative to the first term.

Expressing z with the aid of the relation

A 2 z = -Qv 8

38

Derivation

which corresponds to the pressure losses a t steady state turbulent flow [54,60], and using Eq. (2.5), we may arrange the third term in Eq. (2.7) to read

The absolute value means that the friction on the walls acts in a direction opposite to that of the velocity of flow. Differentiating equation (2.3) and using sin CI = dh/dx, we may write the last two terms of Eq. (2.7) to read

a P ax ax - + egsincr = - (2.10)

Equation (2.7) thus converts to Eq. (2.1). The continuity equation (2.2) may be derived with the aid of the diagram in

Fig. 2.lb. For its derivation one has to take into account the elasticity of the pipe-line and the compressibility of the liquid.

The difference in the mass of the liquid which passes through the end cross- section of an elementary volume of the liquid in a time interval dt equals the increment in the mass of the liquid in this volume. This may be expressed by the relation

ep AP Judydzdt - ( e p I u d y d z AP + ax

at (2.1 1)

where ep is the density of the liquid which, on account of its compressibility, is considered to be a function of pressure; A, is the cross-sectional area of the pipe-line and also considered to vary with pressure again owing to its elasticity.

Using equation (2.6), dividing by dx dt and rearranging equation (2.11) converts to the form

(2.12)

The first term in Eq. (2.12) may be neglected with regard to the third term, provided the velocity of flow is negligible compared to the velocity of the pressure. Let us assume a linear relationship between the mass of the liquid in an elementary volume of the pipe-line and the pressure in the pipe-line, of the

39

Basic equations

form

(2.13)

where K , is a constant expressing the effect of the compressibility of the liquid and the elasticity of the pipe-line. Substituting for pr from equation (2.3), dividing by dx and differentiating, we obtain relation (2.13) in the form

Denoting

we may write Eq. (2.12) thus:

(2.14)

(2.15)

(2.16)

With the assumption of a small deformation of the pipe-line and a small compression of the liquid, the coefficient of the second term is approximately equal to unity and Eq. (2.16) converts to the form (2.2).

The derivation of similar equations is presented, for example, in several references [20, 41, 54, 601.

2.2 Scope of application

The basic equations (2.1), (2.2) employed in this book for the analysis of water hammer in a conduit were derived using many simplifying assumptions. Some of these were mentioned in Section 2.1, whereas others are so common that they are usually not mentioned. Let us point out some of the more important ones: - The conduit has a constant circular cross-section. Its diameter is small in relation to the length of the conduit. - The conduit is completely filled with a homogeneous fluid. - The pressure in the conduit and the discharche of the liquid are functions of time and of the longitudinal coordinate along the axis of the conduit. - The density of the liquid and the cross-sectional area of the conduit are linear functions of pressure. The elasticity of the conduit and the compressibility of the liquid are small.

40

Scope o j application

- The velocity of flow is small with respect to the velocity of the pressure wave in the conduit. - The pressure losses due to friction in the conduit are proportional to the square of the discharge.

The assumptions used in the derivation of the basic equations are satisfied in many cases of water hammer; in some cases, however, they are not.

It follows from the solution of the basic equations (refer to Sect. 7.5) that the value a defined by relation (2.15) has the meaning of the velocity of propagation of the pressure wave in the pipe-line. It usually varies between 1000 and 1400 m s-I in steel pipe-lines filled with water, this value being sufficiently large as compared with the current velocity of flow of the fluid which is usually less than 5 m s-'. It may happen, however, that the wave velocity is much less, for example, due to the higher elasticity of the pipe-line, or due to the presence of air bubbles in the liquid. It is also possible that the velocity of flow could be much higher. The calculation of water hammer then would have to make use of the equations in their more general form.

The pipe-line need not have a circular cross-section. Changing the cross- section may significantly increase the elasticity of a pipe-line and, consequently, reduce the wave velocity. It may also affect the pressure losses. The shape of the cross-section of the conduit, however, is not important from the point of view of water hammer (refer to Sect. 4.2).

The density of the liquid also affects the wave velocity, but the deviations are not too significant for most liquids (refer to Sect. 4.3). On the other hand, an admixture of gases or solid particles in a flowing liquid may be significant. Gases may much reduce the wave velocity due to an increase in the elasticity of the mixture and a reduction of its mass. Moreover, the compressibility of the mixture need not be linear at all. This results, apart from other effects, in a deformation of the pressure waves travelling through the pipe-line. Provided the mixture is homogeneous and the variations in pressure in the pipe-line are so small that the compressibility of the mixture may be considered linear, the basic equations in the form introduced may be employed in the calculation. Their more general form would have to be used in other cases.

The admixture of solid particles in a liquid may affect the overall density and compressibility of the mixture and hence also the wave velocity (refer to Sect. 4.6). Solid particles may also influence pressure losses. As long as the particles are sufficiently small and the mixture behaves like a homogeneous fluid, application of the basic equations is possible. When the particles are not uniformly distributed in the liquid, or when they are not small enough, they, or parts of the liquid with higher and lower concentrations of solid particles, start moving relative to each other. In such a case, the entire phenomenon is complex and the basic equations can be employed only in some limiting cases. Water hammer is also affected by solid particles which have settled.

41

Basic equations

The quadratic relationship (2.8) between pressure losses due to friction and discharge was found to hold for the steady turbulent flow of water and many other fluids. In the analysis, however, its validity is assumed also to hold for unsteady flow. Application of the quadratic relationship is satisfactory for cases of slow variation in discharge, where the distribution of the velocity throughout the pipe-line cross-section is approximately the same as at steady flow. During rapid variations in discharge, however, this distribution, as well as the pressure losses, is different. These considerations have to be taken into account in the interpretation of the calculated results.

When the pressure in the pipe-line drops to that of the vapour pressure of the liquid, the liquid column separates. The vapour-filled cavity usually first expands and then contracts until it disappears entirely. I f any section of the pipe-line is not completely filled with fluid, the basic equations (2.1) and (2.2) do not apply. In spite of this, the course of water hammer may be studied, if, in the layout of the pipe-line, the points of potential liquid column separation (cavitation) are explicitly marked, and if the volume of the cavities developed in the pipe-line is not so large as to influence the length of the adjacent sections of the pipe-line. If these points of potential cavitation are not clearly defined, or if the cavities developed can move in the pipe-line during the course of water hammer, the calculations are neither reliable nor accurate. During a drop in pressure, gases dispersed in the liquid may separate and influence the wave velocity. This again reduces the accuracy of the results.

All the effects described above are discussed in more detail in Chapter 4. The nature of water hammer also depends on the characteristic of the devices

attached to the pipe-line. A precise evaluation of the influence of these devices and its representation by suitable equations is sometimes the most difficult problem in the calculation of water hammer.

42

3 Solution of the basic equations

3.1 Methods of solution

The basic equations (2. l), (2.2) represent a quasilinear system of partial differential equations of the first order with two unknown functions p and v and two variables x and t of the hyperbolic kind.

Several methods have been worked out for the solution of these equations, each with its advantages and shortcomings. Let us introduce at least the p in- cipal ones.

For slow variations in pressure (refer to Sect. 1.2), the elasticity of the liquid and pipe-line need not be taken into account [20,45,54]. Equation (2.1) suffices for the solution. Considering that in this case

-0 av ax _ -

and

the equation takes the form

dv Ae dt 2 0

e - + - 1111 v + f ( t ) = 0 (3.3)

The function f(t) is determined by the conditions at the upstream and downstream ends of the conduit. The equation may be solved graphically or numerically; for some cases, even the analytical solution is quite manageable. The calculations are relatively simple and do not demand much time, but they are not applicable to cases of rapid changes in pressure and discharge, where the elasticity of the liquid and the pipe-line become significant factors.

Neglecting the effect of pressure losses due to friction the system of basic equations (2.1) and (2.2) converts to

av ap at ax

. Q - + - = o

a e - + - = O 2 av aP ax at

(3.4)

(3.5)

For the solution of these equations, numerical methods have been worked out

43

Solution of the basic uquutions

which allow us to determine the pressure and the discharge even when these change rapidly [2,54]. Here, the elasticity of the fluid and the pipe-line is taken into account. The effect of various devices installed along the pipe-line can be included. The motion of the pressure waves through the pipe-line in both directions is followed numerically and the resulting state is given by their sum. The calculations are manual (a computer need not be used) and the results are tabulated. The disadvantages of these methods are: exclusion of the effect of friction; practical applicability only for very simple pipe-line systems; time- consuming, tedious and intricate calculations, which may lead to errors.

The Schnyder-Bergeron graphical method [IS, 20, 54, 601 is of a later date. Again, it is based on the solution of differential equations (3.4), (3.5), which do not take into account the effect of losses of pressure due to friction. The solution is achieved by graphical means by plotting either pressure versus velocity, or pressure versus discharge. The method allows for a simple inclusion of the effect of a variable conduit cross-section, local pressure losses, valves, pumps, surge tanks, air inlet valves and other similar devices. The effect of water hammer can be solved for ramified systems and corrections expressing the effect of friction can be included by substituting the pressure losses due to this effect at one or several points. For simple systems, this method gives a quick and neat solution, as compared with the numerical methods. Various boundary conditions are easily incorporated into the calculation, which can be carried out without a computer.

The disadvantages of the method are: it is unsuitable for complex pipe-line systems; the effect of pressure losses cannot be included with sufficient accuracy; a satisfactory precision of the results is difficult to maintain in more complex cases; the method requires much more time, especially for more complex cases and when repeatedly solving similar cases, when it is comprared with numerical methods making use of computers.

In the characteristics method [46, 59, 601 the system of basic partial differential equations (2.1), (2.2) is transformed into a system of ordinary differential equations valid along the characteristics (refer to Sect. 7.5).

A system of basic equations which include the effect of the pressure losses due to friction can be solved with the aid of this method. The solution can be worked out even for more general cases, for example, for a velocity of flow comparable to the wave velocity, for a very elastic liquid and pipe-line, for non-linear deformations of the pipe-line and the fluid, for pressure-loss functions other than (2.8) etc. Most variations in boundary conditions can be included. The solution can be transferred onto a computer without difficulty. It is carried out in difference form. In practice computers are necessary to perform these calculations.

There are other numerical methods [7, 37, 41, 51, 52, 58, 60, 66, 69, 70, 711 for solving the basic equations using computer techniques. Most of these are derived from the method of characteristics and are aimed at reducing the time needed for the solution and at reducing the necessary storage capacity of the

44

Application of the method of characteristics

computer. Such reductions are frequently only achieved at the cost of losing the general character of the method, or by reducing the accuracy of the results.

The impedance method [60, 761 is convenient for solving steady periodical motion. It was derived from methods employed for solving problems relating to electrical circuits. In this method, non-linear friction is assumed only for the basic steady flow. For its periodical component, the pressure losses are linearized. Computers are used in the solution which requires less time than the characteristics method. Fundamental and harmonic frequencies are determined. The method is suitable even for complex systems. Its disadvantage is that its application is limited to a periodically varying flow and that the pressure losses are partly considered, as being linearly dependent on the discharge. Analytical methods have also been employed for solving the basic equations assuming a linear relationship between pressure losses and discharge, but these have not been widely accepted, mainly because, in most cases, it was very difficult to satisfy all actual boudary conditions of the analysed system.

Graphs [14, 15, 32, 44, 49, 751 have also been worked out for some simple systems commonly encountered in practice. The basic values obtained for the solution of water hammer and required for the design or the checking of a hydraulic system can be read directly from these graphs. Although results are obtained very quickly and without the use of a computer, the diagrams can be applied only to a specific kind of hydraulic system and for a specifis type of operation.

3.2 Application of the method of characteristics

The method of characteristics has been chosen for the analysis of water hammer in this book. It is the most universal of all the methods considered, allowing us to analyse all cases which can be solved by any of the other methods, including very complex hydraulic systems. The effects of various devices which form part of the system, and of the pressure losses along the pipe-line, which need not be linearized in this case, can be included with comparative ease. Performing the calculations on a computer and attaining the required accuracy is not difficult. The results may be printed-out in the form of graphs or tables, etc. The initial steady state may also be determined by this method.

The disadvantage of the method is that the use of a computer is essential. This objection, however, has become irrelevant, in view of the present development of computer technology.

Other disadvantages are the necessity of knowing a suitable programming language and the time-demanding modification of programs required for actual hydraulic problems to be solved. In the method employed, this was eliminated by working out a very general program applicable to the most different hydraulic systems currently encountered without requiring any modification of the pro-

45

Solution of the basic equntions

gram. The type of hydraulic system analysed is defined only by the input data submitted. The worked-out program is complemented by sub-programs for calculating the effect of various devices, which form part of present-day hydraulic systems. The program needs to be complemented by a further sub-program, if an unusual device is included, the effect of which cannot be accounted for by any of the existing sub-programs.

The only actual disadvantage of the method of characteristics, as compared with other methods, seems to be the fact that there are quicker methods, or methods with smaller demands on computer time, for solving some water hammer problems. Thus, for example, for a system consisting of a conduit and a surge tank where the elasticity of the pipe-line and the liquid need not be considered, the calculation based on the solution of Eq. (3.3) will obviously be simpler than one employing the method of characteristics based on the solution of the systems of Eqs. (2.1), (2.2); the results obtained by either method will be practically the same. The second solution takes into account effects which are not significant in this particular case.

A similar example is represented by the solution for a periodically varying flow, when the prevailing part of the discharge is steady and the periodical variations are only small. Linearization of the pressure losses within the range of the variations of flow is admissible. A soution with the aid of the impedance method, and especially the determination of the natural frequencies of the system, will be clearly simpler than a solution applying the characteristics method, although the later method is able to deal even with such calculations.

If the solutions for a given case have already been determined previously and are available in the form of graphs or tables, the results should obviously be taken from these.

Hence, if special cases are to be solved repeatedly, and if a more convenient method exists, such a method should be used. In unusual cases, however, use of the characteristics method will probably be preferred, because it is general and ready to be used for the necessary calculations without any further preparatory work.

The method of characteristics employed solves the system of basic equations (2.l) , (2.2) arranged in the form (7.2), (7.3) of two differential equations for two unknown functions p ( x , t), Q ( x , t ) valid along the characteristics (7.4) and (7.6). The wave velocity is assumed constant. This allows us to predetermine all the intersections of the employed grid of characteristics used, on which the values of the pressure p and the discharge Q will be determined during the calculation. The initial state of flow is entered. The calculation proceeds step by step for multiples of the chosen time interval At of the calculation. The boundary conditions for determining p and Q at the end points of the pipe-line section are solved by interconnecting the sections and by the devices attached to the pipe-line network (refer to Sect. 3.3).

Using this method, the development of unsteady flow in a known pipe-line system can be calculated from a knowledge of the initial state of flow and a

46

Schematization of the pipe-line system

known adjustment regime of the devices affecting the flow. The steady state of flow in the pipe-line system may be also calculated with the aid of this method. The input values for the calculation are submitted in the form of a numerical table entered on a disk.

The results of the solution are: the values of discharge and pressure at all intersection points of the characteristics network considered and the variable parameters of all the devices which form part of the network analysed. These values can be displayed numerically or graphically on a screen, printed or stored on a disk for further processing.

3.3 Schematization of the pipe-line system

The analysed pipe-line system has to be schematized to facilitate calculations. The pipe-line is divided into individual sections along which the basic parameters of the pipe-line may be considered to remain constant. These parameters are: the velocity of propagation of the pressure wave a; the inside diameter D of the pipe-line; the coefficient of friction 1. Each section has to have a starting (upstream) and end (downstream) point. This also determines the sign convention for the section. The sections are interconnected at points denoted junctions. A single section or several sections may be connected to each junction. Each section has to be connected to a junction at both its ends. The dimensions of the junctions are neglected in the calculation relative to the length of the sections.

One damping device may be attached to each junction. A damping device is a device which can change the quantity of the liquid in the pipe-line system. It is, for example, a reservoir, a surge tank, an overflow, an air chamber, an air inlet valve, etc. Cavitation is also considered to represent a damping device, since the pressure remains constant at the separation point for a definite period of time while the volume of the cavity filled by the vapour of the liquid varies.

A pressure device may be installed between a junction and the section attached to it. It is a device which may induce a difference in pressure between the junction and the end of the attached section, but which does not change the quantity of liquid in the system. Such pressure devices include: pressure losses due to friction, a valve, a pump, a turbine, etc. The dimensions of pressure devices are neglected relative to the length of the pipe-line sections.

Schematization of a pipe-line system is one of the most important steps of the calculation. The precision of the results and the demands on computer time depend on its correct execution.

Several considerations have to be taken into account in the schematization of a hydraulic system.

The program described in the present publication has been worked out for a maximum number of 50 sections and 50 junctions. This number exceeds requirements for present-day cases of water hammer.

47

Solution of the basic eyuotions

The program can, however, be easily expanded to include a greater number of sections and junctions, the capacity of the computer memory forming the only limitation.

In the calculation, the actual lengths of the sections are usually not considered; they are instead rounded off to whole multiples of the values

where Ax and a are the longitudinal interval of calculation and the wave velocity, respectively, for the pipe-line section under consideration. At is the time interval chosen for the calculation, common for all sections. The length of all sections shorter than Ax is adjusted to the value Ax.

Certain inaccuracies in the calculation appear due to these approximations. The inaccuracies in the total pressure losses along the individual pipe-line sections due to the modified section lengths are automatically eliminated, when the program appropriately adjusts the coefficient of friction A. Deviations in the time needed by the pressure wave to travel the entire length of a section can be reduced by reducing the time interval of the calculation, or by schematizing the pipe-line system in such a way and choosing At so that the lengths of all sections differ little from whole multiples of Ax. In both methods, the modified section lengths should closely approximate the actual ones. Reducing the time interval of the calculation, however, leads to an increased use of computer time, apart from limits imposed by the program. The version, introduced in the present book, limits the total number of length intervals Ax, into which the sections are divided in the calculation, to 2000; this number is determined by the type of schematization and by the choice of the values At. The number of intervals increases with a decreasing value of At. The maximum admissible number of intervals is checked by the program.

In schematizing actual cases, one has also to consider the accuracy of the submitted parameters of the damping and pressure devices, the accuracy of other values used in the calculations and the degree to which the assumptions underlying the equations are satisfied in this particular case. Attempts to achieve a greater precision of calculation by disproportionately increasing the number of sections or decreasing the time interval of the calculation are usually unwar- ranted, because the inaccuracies inherent in the input data cause much larger deviations of the results from reality than the inaccuracies inherent in the method of calculation itself.

indeed, in many actual cases, the deviations of the results from reality do not depend so much on the method of calculation, as on the inaccuracy of the quantities submitted. It is often impossible to determine with satisfactory precision, for example, the discharge through a conduit, the velocity of propagation of the pressure wave, the coefficient of local losses, the characteristic of the valves, pumps, turbines, the control rkgime of these devices, as well as other values which affect water hammer.

A x = a At ( 3 4

48

4 Parameters of the basic equations

4.1 Velocity of propagation of the pressure wave

The velocity of propagation of a pressure wave in a homogeneous, little- compressible liquid and a rigid conduit is identical with that of sound in the liquid and is given by the relationship

a = &

Approximate values of the bulk modulus of elasticity K, the density p and the velocity of sound calculated according to relation (4.1) are given in Table 4.1 for various liquids [20, 54, 671.

Table 4.1 Approximate values of the bulk modulus of elasticity K, density p and the velocity of sound a (calculated from them), for various fluids

water 0 "C water 10°C water 20 "C water 40 "C sea water 0 "C 3.5 % salt

kerosene petroleum oil

1 . 8 9 ~ lo9 1 . 9 6 ~ 109 2.03 lo9 2 . 1 8 ~ 109 2.04 lo9

1.3 x 109 1.5 lo9

1.1 x 109-1.6x lo9

lo00 lo00 998 992

1028

670-760 830-840 855-963

1307-1393 1336-1344 1134-1288

Due to the elasticity of the pipe-line the wave velocity in a conduit is less than

For thin-walled pipe-lines of circular cross-section, the actual velocity may be that of sound [20, 22, 41, 54, 60, 65, 681.

found from the relation

(2.15)

49

Parameters of the hasic equations

where 1

1 - D K , =

- + - K eE

where e is the thickness of the pipe-line wall and E the modulus of elasticity of the pipe-line material.

Table 4.2 gives approximate values of the modulus of elasticity for various conduit materials [20, 54, 671.

Table 4.2 Approximate values of the modulus elasticity E for various materials

Conduit material

steel grey cast iron aluminium copper lead glass wood rubber concrete asbestos cement

E (Pa)

2.0 x 10"-2.2 x 10" 4.4 x 10"-1.2 x 10" 7.3 x 10'O 1.2 x 10" 5 x 109-1.7 x 10" 5 x 10"-8 x 10" 9 x 109-1.3 x 10" 2 x 105-6 x lo5 2 x 10''-3 x 10" 2.5 x 10"

The wave velocities for water (K = 1.96 x lo9 Pa, e = 1000 kg me3) in various types of circular conduit calculated with the aid of equations (2.15) and (4.2) are listed in Table 4.3.

Table 4.3 Propagation velocities a of a pressure wave calculated for water in various conduits of circular cross-section

Conduit material

steel steel steel steel steel steel steel cast iron asbestos cement

rubber

D (4 0.04 0.12 0.40 1.20 0.40 0.40 0.40 0.12 0.12

0.12

0.Gl 0.0 1 0.0 1 0.0 1 0.005 0.02 0.03 0.01 0.01

2.1 x 10" 2.1 x 10" 2.1 x 10" 2.1 x 10" 2.1 x 10" 2.1 x 10" 2.1 x 10" 1.0 x 10" 2.5 x 10"

1374 1328 1195 96 1

1059 1285 1320 1259 1004

0.01 I ].Ox 10' I 2.9

50

Velociry of propagation of the pressure wave

(4.4)

(4.5)

(4.6)

The wave velocity is also influenced by the support conditions of the pipe-line. When this factor is also taken into consideration, the value K, for a circular cross-section is expressed by the relationship

0.95

0.91

1 .oo

1 1 . Dc

K , =

- + - K eE

where the constant c depends on the support conditions of the pipe-line. Streeter [60] gives values of this coefficient for three types of pipe-line support.

With the pipe-line fixed only at the upstream end,

5 4

c = - - v (4.4)

v being Poisson's ratio.

ments are prevented, With the pipe-line fixed throughout its length so that all longitudinal displace-

(4.5)

c = l (4.6)

v 0.3 (4.7)

2 c = l - v

For a pipe-line provided with expansion bends throughout its entire length,

The approximate value of Poisson's ratio for structural steel is

Table 4.4 gives the velocity of propagation of the pressure wave for water (K = 1 . 9 6 ~ lo9 Pa; e = 1000 kgmP3) in a steel conduit ( D = 0.4 m; e =

= 0.01 m; E = 2.1 x 10'' Pa) for a variety of support conditions in accordance with the equations (2.15) and (4.3) to (4.7).

In thick-walled circular conduits, the stress on the pipe-line wall is not distributed uniformly, as has been assumed in deriving equations (4.2) to (4.6).

Table 4.4 Propagation velocity a of a pressure wave calculated for water in thin-walled circular steel conduits, for various support conditions

Support conditions

fixed at the upstream end anchored against longitudinal displacement expansion bends along the pipe-line

l c Equation employed

a (m s-I)

1203

1210

1195

51

Parnmeters of the basic equations

As long as the ratio D/e is less than roughly 25, it is recommended to calculate c with the aid of equations (4.8) to (4.10) [60] instead of (4.4) to (4.6).

For a thick-walled pipe-line fixed only at the upstream end, we have

For a thick-walled pipe-line fixed throughout its entire length so that longitudinal displacements are prevented,

2e D D D + e

c = - ( I + v) + -(1 - 2) (4.9)

For a thick-walled pipe-line provided with expansion bends throughout its entire length,

2e D D v, + D+e c = - ( I + (4.10)

For circular tunnels, the value K , (refer to 2.15) is given by the relation

I 1 2(1 + v) K , = -I

(4.1 1 )

K ' ER

which follows from equations (4.3) and (4.8) to (4.10) for large values of e. E R is the modulus of elasticity of the rock.

A steel-sheet tunnel lining, which is in direct contact with the rock increases the wave velocity. Neglecting the effect of the Poisson's ratio of the rock in which the tunnel has been excavated and of the steel lining, we may express the coefficient c by the simple relation [60]

2eE ERD + 2eE

C = (4.12)

where E and e are the modulus of elasticity and the thickness of the steel lining, respectively.

Table 4.5 gives the wave velocities for different types of conduit with circular cross-section, as calculated in accordance with different equations. The relevant parameters for water are: K = 1.96 x lo9 Pa, e = 1000 kg m-j; for concrete: D = 1.0 m, e = 0.1 m, E = ER = 2.5 x 10'' Pa, v = 0.15; for steel: D = 1.0 m, e = 0.01 m, E = 2.1 x 10" Pa, v = 0.3.

A pipe-line need not have a circular cross-section. Deviations may be caused

52

Velocity of propagation of the pressure wave

by inaccurate manufacture or subsequent deformation. Deviations from a circular cross-section may have a considerable effect on the wave velocity.

If the cross-section is other than circular, the constant K , is defined by the formula

which foll

1 1 A , - A

K, =

- + (4.13)

ws from equation (2.13) (with some small higher order terms neglected), and from the definition of the bulk modulus of elasticity of the liquid. A, is the cross-sectional area of the pipe-line at a pressure pr; A is the cross-sectional area of the pipe-line at a pressure pr = 0. The value (A, - A)/Ap, is to be considered constant and independent of pr. Without this assumption, the wave velocity would vary in the course of water hammer. The method of calculation is not applicable to such a case.

The velocity of propagation of the pressure wave through the pipe-line may be influenced not only by the elasticity of the pipe-line but also by other factors, such as cavitation due to a drop in pressure (refer to Sect. 4.9), the presence of gas (refer to Sect. 4.7) or of solid particles (refer to Sect. 4.6) in the pipe-line as well as other possible factors. In practice, it is usually difficult to reliably determine all the factors influencing the wave velocity. This leads to deviations

Table 4.5 Comparison of wave velocities a in different types of conduit of circular cross-section calculated according to different equations

Type of conduit

rigid pipe-line steel pipe-line D = 1.0 m, e = 0.01 m

concrete pipe-line D = I.Om, e = 0.1 m

tunnel in concrete without lining tunnel in concrete with steel lining D = 1.0 m, e = 0.01 m

Equation employed c

-

1 .00 0.95 0.91

0.80 1.017 0.798 1.015 1 .00 1.199 -

0.144

a (m s-I)

1400

1007 1019 1029

1098 1044 1098 1045 1048 1005 1264

1314

53

Parameters o j the basic equations

of the calculated wave velocity from the actual one. Apart from that, other inaccuracies appear: due to the velocity of flow being neglected in comparison to the wave velocity; to deviations from the quadratic relationship between the pressure losses and discharge; to the solution of the basic equations in their difference forms; to inaccuracies caused by the rounding-off of the section lengths; to neglecting the dimensions of the devices which form part of the pipe-line system, relative to the lengths of the section, etc. Hence, there often is no need to divide the pipe-line into a large number of short sections, when it is schematized. It is frequently possible to consider one longer part of the pipe-line as a single section, even if the values of a, D and Iz vary somewhat along it. The value of a for such a section may be determined using the relation

C ‘ i c-fi ‘ ai

(4.14)

where ai and I i are the wave velocity and the length of the i-th part of a section of the pipe-line, respectively. Equation (4.14) follows from the condition that the total period needed for a pressure wave to pass a section should remain constant.

The above procedure can only be used when the values of a, D and 1 vary only slightly along the section. If the differences are large even over a short portion of the pipe-line section, and the variations in pressure at water hammer are rapid, the above procedure may be the source of gross errors.

4.2 Pipe-line diameter

The pipe-line diameter appears in the basic equations (2. I ) , (2.2) only in the term expressing the effect of the pressure losses. Hence, it would seem that its exact value is not very important.

However, other terms of the basic equations also depend on its value as may be seen in the modified forms of these equations (7.2), (7.3), where the pipe-line diameter appears also in other terms. Therefore, its value is significant for the calculation.

For non-circular conduits, the value D should be determined from the relation

(4.15)

derived from the condition that the flow area of the conduit should remain constant in the calculation.

54

Pressure losses due to friction

If one section of the pipe-line is made up of several parts of slightly differing diameter (refer to Sect. 4.1), the effective pipe-line diameter in this section may be expressed by

(4.16)

where Di and I i are the diameter and length of the i-th part of the section, respectively.

Equation (4.16) follows from the condition that the value of Evili should

remain constant for the section, ui being the mean cross-sectional velocity in the i-th part of the section.

I

4.3 Density of the liquid

In the calculation of water hammer, it is assumed that the entire pipe-line system is filled with a homogeneous fluid. Its density e is considered constant in the course of water hammer.

Actually, the density of a fluid depends on a variety of factors. The variations induced by differences in pressure and temperature usually are not large; approximate values for water are listed in Table 4.1 [67].

When the pressure drops to a level close to the vapour pressure of the fluid, these variations become more significant and cavitation may occur. The effect of this phenomenon may be included in the calculations (refer to Sect. 4.9).

Under actual operating conditions, a mixture of liquids, gases and solid particles may flow through the pipe-line. In such cases the analysis of water hammer is much more involved. Some limiting cases for such flows can be solved by the method introduced in this book. Solid particles and gases may significantly influence the density of a flowing mixture (refer to Sects. 4.6, 4.7).

4.4 Pressure losses due to friction

The effect of the pressure losses due to friction along a pipe-line is included in the basic equations (2.1), (2.2). It is assumed that they are, as at the steady state, proportional to the square of the mean cross-sectional velocity, or to the square of discharge. Their effect is expressed by the relation

55

Parameters of the basic equations

Type of conduit

new cast iron rusted cast iron incrustated cast iron smooth cement rough cement

or

k (4 0.0005-0.001 0.001-0.0015

0.001 5-0.003 0.0003-0.0008 0.001-0.002

Relation (4.17) follows from Eq. (2.1) for steady flow, when av/dt = 0. Equation (4.18) is obtained by substituting in (4.17)

v = - 4Q RD2

(4.17)

(4.18)

(4.19)

valid for pipe-lines having a circular cross-section. The value of the coefficient 1 is approximately constant for a steady turbulent flow. It depends on the diameter and the roughness of the conduit, its value usually ranging between 0.015 and 0.030.

The coefficient may be calculated using the relation

a = 210g- + 1.138 ( : )-2 (4.20)

where k is the absolute roughness of the pipe-line wall. Approximate values of k for various types of conduit are given in Table 4.6

r w . Table 4.6 Approximate values of the absolute roughness k for various types of conduit

The local pressure losses in a pipe-line may be considered as being equivalent to a pressure device installed at the upstream or downstream end of a section (see Sect. 6.3). However, it is frequently more convenient to consider the local losses to be distributed throughout the section and to include their effect in the coefficient A, which is then determined from the relation

(4.21)

56

Solid particles in the pipe-line

derived from equation (4.18). Ap is the difference in pressure p between the upstream and downstream ends of a pipe-line section in the steady state; 1 is the length of this section.

The pressure losses expressed by relations (4.17) and (4.18) satisfactorily reflect reality for a steady flow in actual cases. In the course of water hammer, the pressure waves travelling in the pipe-line induce deviations of the velocity profile from the shape corresponding to the steady state. The different shapes of velocity profiles cause different values of pressure losses [23]. The developed deviations from the steady state velocity profile are gradually reduced. When the water hammer has died away, the flow reverts back to a steady one with the corresponding velocity profile.

The pressure losses calculated from equations (4.17) or (4.18) only agree with reality for cases of slow variations in discharge, when the shape of the velocity profile differs only slightly from the shape corresponding to the instantaneous discharge at the steady state, even during water hammer. During rapid changes in discharge, the actual pressure losses may differ considerably from the calculated ones. Significant deviations may also appear during periodic rapid changes in flow, because, in that case, the steady state is not established at all, the deviations appear all the time. In spite of these limitations, equations (4.17) and (4.18) are currently used to include the effect of the pressure losses in the calculation of water hammer. Frequently, the effect of the pressure losses is approximated with even lesser accuracy, for example, by concentrating them merely in a few profiles.

4.5 Velocity of the liquid

The velocity of a liquid differs from point to point throughout a cross-section. Time-dependent changes in velocity occur due to turbulence even during steady flow. These changes are not taken into account in the calculation of water hammer. Here, the critical quantity is the mean cross-sectional velocity in the pipe-line, determined by equation (2.6). In the method introduced in this book, the discharge of the liquid defined by (7.1) is employed instead of the mean velocity.

4.6 Solid particles in the pipe-line

Solid particles may appear inside a pipe-line during the hydraulic transport of material or perhaps in a contaminated liquid. The presence of such particles may affect water hammer, but in many cases it is difficult to include their effect in the calculation.

57

Paranzrters of the basic equations

Water hammer is affected by the quantity, density and compressibility of the solid particles, as well as by other conditions. Solid particles may be distributed uniformly or unevenly, or they may have settled. During water hammer, the particles move against each other owing to the difference between their density and that of the liquid. This motion is resisted by the viscosity of the liquid. Relative motion may occur between a liquid with a higher concentration of particles and a liquid with a lower concentration. Such relative motion is influenced by the friction between the particles and the pipe-line walls, especially in the case of settled particles.

Water hammer may be influenced by an uneven distribution of solid particles along the pipe-line, by the presence of gases which were captured in the pipe-line together with the particles, and by other circumstances.

A more detailded analysis of the effect of solid particles on water hammer can be found in the literature [84]. Here, only some of the results are presented. In practice it is frequently the case that not all the data are available necessary for taking into account the effect of solid particles in the calculation. The method of calculation employed in the present book allows us to solve some special limiting cases and to approximately assess the effect of the solid particles in the cases considered. Assuming a uniform distribution of the particles in the liquid and neglecting the friction between the particles and the pipe-line walls, we may consider two cases: in the first, the viscosity of the liquid prevents any relative motion between the liquid and the particles; in the second, the effect of the frictional forces between the liquid and the particles is neglected. The effect of friction increases in importance the smaller the particles, the higher the viscosity of the liquid and the slower the changes in pressure.

In the first case, when the frictional forces prevail, water hammer may be calculated as for a homogeneous liquid with a density equal to

e = cvlel + cDe2 (4.22)

where cy,, el, cv2, ez are the bulk concentrations and densities of the liquid and the solid particles, respectively. The concentrations satisfy the condition

Cy1 + cy;! = 1 (4.23)

The velocity of propagation of the pressure wave is defined by equation (2.15), where

1

C Y l cv2 Dc - + - + - K , K , eE

K , = (4.24)

Here, K , and K , are the bulk moduli of elasticity of the liquid and the particles, respectively; the discharge is expressed as the volume of the mixture flowing through the pipe-line per unit time.

58


In the second limiting case, when the frictional forces are neglected, the calculation proceeds in the same way, but the density is now defined by the rela tion

1

CVl cv2 - + - e = (4.25)

In intermediate cases where the viscosity of the liquid only partially prevents relative motion between the liquid and the solid particles, the solution is more complex. By way of illustration, the pressures calculated at the valve after its closure are shown in Fig. 4.1, as taken from the literature [84]. In the solution,

15x10'

1 N 4 AP

lo6

0.5xld

c

-0.5x10

-106

-1.5xld

Fig. 4. I Effect of friction between a liquid and the solid particles on water hammer induced by the closure of a pipe-line through which their mixture flows.

59

Parameters of the basic equations

the friction on the pipe-line walls was neglected and a linear relationship was assumed between the forces acting between the liquid and the solid particles and the difference in their velocities. The value C represents the effect of friction. At C = 0, friction prevents any relative motion between the liquid and the solid particles; at C = co, friction is neglected. It is evident from Fig. 4.1 that the forces acting between the liquid and the solid particles may even cause deformation of the pressure waves which travel in the pipe-line. The example in Fig. 4.1 was chosen to demonstrate the qualitative side of the problem, In reality, mutual deviations tend to be smaller because of the lower concentrations and smaller differences in density between the liquid and the solid particles.

In another limiting case, which may be solved as water hammer in a homogeneous fluid, all the solid particles have settled at the bottom of the conduit and the friction on its walls prevents their displacement. A pure liquid then flows through the pipe-line.

The water hammer is calculated in the same manner as in the preceding cases, but the density is defined by the relation

A

A - As e = el (4.26)

where As is the cross-sectional area of the pipe-line occupied by the settled particles. The value K, is given by equation (4.24), where cyl, cn are the mean bulk concentrations of the liquid and the particles respectively, in the pipe-line profile. Q is again the discharge (volume per unit time) of the ,,mixture", that is, in this case, of the pure liquid flowing through the conduit. Ascertaining the partial choking of a pipe-line is often difficult in practice; this condition may, however, significantly increase the pressure during water hammer.

In the most general case, when the concentration and the velocities of the liquid and the solid particles in a profile are distributed unevenly, and when part of the cross-section may be occupied by the settled material and there may be relative motion between the liquid and the solid particles, the solution is complex. As before, two limiting cases may be solved by the method presented.

In the first case, friction prevails over the forces induced by water hammer. The density is defined by the relation

(4.27)

where cyl, cn are the mean concentrations of the liquid and the solid particles, respectively, in the mixture flowing through the pipe-line. The value K, is defined by equation (4.24), where c y l , cn are the mean concentrations of the liquid and the solid particles of the mixture, respectively, in a pipe-line profile. It should be

60


Pure liquid

0.0

0.0

0.0

1000

1414 2.0 x lo9

4.5 x lo6

emphasized that these concentrations may differ significantly. Q is the discharge (volume per unit time) of the mixture flowing through the pipe-line. An uneven distribution of the concentrations along the pipe-line is not considered.

In the second case, where friction is neglected relative to the forces induced by water hammer, the calculation is similar to that of the preceding case. The density of the liquid is, however, defined as in equation (4.25). The values c y , , cy;! in this case, are the mean concentrations of the liquid and the solid particles, respectively, in the pipe-line profile. The concentration of the solid particles in the mixture flowing through the pipe-line and the quantity of settled particles are irrelevant in this case.

Even when these two limiting cases have been solved, it is by no means certain that the actual values will lie somewhere between the calculated results. Pheno- mena, similar to those portrayed in Fig. 4.1, occur in the intermediate cases. Forces acting on the solid particles may deform the pressure waves and damp the course of water hammer. In actual cases, however, the maximum and minimum values usually lie between those calculated for the limiting cases presented above. This is also the case portrayed in Fig. 4.1.

In most cases, the solid particles in a conduit increase the extreme values of pressure at water hammer, the reason being their higher density and lower compressibility compared to that of the liquid. This increase is generally not very significant on account of the usually low concentrations of solid particles. On the other hand, gases, which are frequently trapped in the conduit together with the particles, reduce the maximum and minimum values of the pressure - refer to Sect. 4.7.

. friction prevails

0.05

0.05

0.0

1075

1398 2.1 lo9

4.7 x 106

Table 4.7 Effect of the presence of solid particles on the increase in pressure resulting from the closing of a pipe-line

~

mean volume concentration of solid particles in flowing mixture mean volume concentration of particles in pipe-line cross-section cross-sectional area filled with settled material volume percentage) I e (kgm- Kc (Pa) a (m S K I )

AP (Pa)

Liquid with uniformly distributed solid particles

friction neglected

0.05

0.05

0.0

1031

1427 4.68 x lo6

2.1 x lo9

Liquid with settled

particles

0.0

0.15

20.0

1250

1365 2.33 x lo9

5.43 x lo6

61

Purumeters of the basic equutions

A substantial increase in pressure at water hammer may occur, when the pipe-line is partly choked by settled material. Then the velocity of flow increases as do the extreme values of pressure, which are a function of this velocity.

Table 4.7 gives the increase of pressure Ap induced by the closing of a pipe-line as determined in accordance with equation (1.12) taking into account the effect of the solid particles. The calculation was carried out for different limiting cases. In all the examples, the rigid pipe-line had a diameter D = 0.2 m, the initial discharge Q = 0.1 m3 s-', the density of the liquid and the solid particles e , = 1000 kg m-3 and e2 = 2500 kg m-3, respectively, and the bulk modulus of elasticity of the liquid and the solid particles and K2 = 4 x 10" Pa, respectively. These values correspond approximately to a mixture of water and sand. The increase in pressure was calculated for the flow of a pure liquid and for a liquid with uniformly distributed solid particles, in both cases for frictional forces prevailing (equations 4.22,4.24) and for frictional forces neglected (equations 4.24, 4.25). The last column contains the results for the case where the solid particles have settled on the bottom of the pipe-line (equations 4.24, 4.26).

K , = 2 x lo9 Pa

4.7 Gas in the pipe-line

Gas may be entrapped in a pipe-line while it is being filled, it may be sucked in at the inlet, through leakages in the pipe-line or in the devices attached to it, it may have separated from the flowing liquid, or it may be contained in the pores of solid particles transported hydraulically; it may also get into the pipe-line through other means.

The presence of gas in a pipe-line may contribute to a reduction of water hammer phenomena, but it may also induce a dangerous water hammer [20,33, 47, 60, 721.

Gas in a pipe-line can be present in a variety of forms. It may be dispersed in small bubbles, or have aggregated into large bubbles. Such bubbles may stay in one place, or they may travel along the pipe-line.

If the gas is dispersed uniformly in small bubbles, the mixture may be considered a homogeneous liquid. Its density is determined with the aid of equation (4.22) and the wave velocity by means of equation (2.15). K , is given by equation (4.24), as for a mixture of a liquid and solid particles. The quantities cn, e2 represent the bulk concentration and the density of the gas, respectively. The value K , corresponding to the bulk modulus of elasticity defines the compressibility of the gas and varies as a function of pressure. For the pressures and low gas concentrations found in actual practice, the density of the gas may be neglected in equation (4.22), relative to the density of the liquid. The relation defining the density of a homogeneous mixture then simplifies to

62

Gas in the pipe-line

e = e l C l (4.28)

The compressibility of the gas depends on the pressure and is defined by the polytropic curve

pabsVl = const (4.29)

where y is the exponent of the polytropic curve, and its value ranges from y = 1 for an isothermal process up to y = 1.4 for an adiabatic process. Va is the volume of the gas. Differentiating expression (4.29) we obtain

(4.30)

Comparing this with the definition of the bulk modulus of elasticity, it follows that

K 2 = ypabs (4.3 1) Substituting for K,, we obtain, instead of (4.24),

1 K _ = (4.32)

This formula may be used to calculate the wave velocity in a pipe-line at very low gas concentrations cn. For higher concentrations, the first and the last term in the nominator may be neglected in comparison to the middle term and equation (4.32) converts to

K , = - YPabs (4.33) cv2

Even small concentrations of gas in a flowing liquid significantly reduce the wave velocity. Table 4.8 lists the wave velocities for different pressures and concentrations of gas in water in accordance with equations (2.15), (4.28) and (4.32) or (4.33). The pipe-line considered was rigid, el = 1000 kg m-3, K , = 2 x 109Pa, y = 1.4.

Since the wave velocity is clearly a function of pressure, it will vary in the course of water hammer. The method of calculation employed in this book, however, considers this velocity to remain constant along a pipe-line section. In the case of a mixture of liquid and gas, this method is strictly applicable only in cases of small variations in pressure, when the variations in the wave velocity are not significant. In cases involving large variations in pressure the results of the solution are to be taken as approximate only. Changes in the wave velocity are more evident at lower pressures in the pipe-line than at higher ones.

63

Table 4.8 Effect of the concentration of gas and pressure on the wave velocity in a pipe-line

water water + 0.01 volume % gas

water + 0.1 volume % gas

water + 1.0 volume % gas

water + 10.0 volume YO gas

water + 20.0 volume YO gas

1 0.9999 0.9999 0.999 0.999 0.99 0.99 0.9 0.9 0.8 0.8

0 0.0001 0.0001 0.001 0.001 0.0 1 0.0 1 0.1 0.1 0.2 0.2

-

2.0 x lo6 2.0 x lo5 2.0 x 106 2.0 x lo5 2.0 x lo6 2.0 x lo5 2.0 x lo6 2.0 x lo5 2.0 x 106 2.0 105

1000 1000 1000 999 999 990 990 900 900 800 800

2.0 lo9 1 . 8 7 ~ lo9 1 . 1 7 ~ 109 1.17 lo9

2.76 lo7

1 . 3 9 ~ 107

2.40 x lo8 2.46 x lo8

2.71 x lo7 2.80 x lo6

1.40 x lo6

1414 1366 1080 1081 495 498 167 175 56

132 42

Pressure in the pipe-line

The wave velocity is much reduced at high gas concentrations of gases. In deriving the basic equations (refer to Sect. 2.1), it was assumed that the velocity of flow is negligible in comparison with the wave velocity. A reduction in the wave velocity leads to less accurate results.

If the gas is concentrated in one or more places in a pipe-line system, if it does not move and its quantity is known, its effect may be represented as similar to that due to a damping device, for example, an air chamber (refer to Sect. 5.4).

Large air bubbles moving in a system create an unfavourable situation. Their motion may by itself induce dangerous water hammer phenomena. The prin- cipal cause of such phenomena usually is the much lower resistance of the gas, compared to that of the liquid, during the flow through a restriction. The resistance is small when gas flows through the restriction and the velocity of flow in the system is increased. As soon as the bubble has passed the restriction, the resistance increases rapidly and this may induce dangerous changes in pressure in the system.

Such cases may occur, when a system is filled too quickly with a liquid and it is not sufficiently de-aerated. I t may, however, occur even during normal operation, when gas concentrates in one place and starts moving after having attained a specific volume or discharge.

The possibility of occurrence of such a water hammer should be taken into consideration as early as the design stage of the system and should, if possible, be avoided.

4.8 Pressure in the pipe-line

The actual pressure in a pipe-line depends on the condition of steady flow before the appearance of water hammer, on the changes induced by water hammer and on the geodetical elevation of the pipe-line. In the method employed in this book the actual pressures pr are recalculated to the pressures p at the level of a chosen horizontal reference plane; thus, individual phenomena can be better distin- guished and the submission of the input data is simplified. The recalculation is carried out with the aid of equation (2.3). In this way, the effect of the geodetical elevation of the pipe-line is eliminated. For the calculation of water hammer, the geodetical elevation of the pipe-line is not needed. It is introduced only at some points, where the course of water hammer depends on it. These are, for example, the points, where an air inlet valve, an air chamber, a surge tank, or some other similar device is attached to the pipe-line, or where cavitation may take place.

When the results of the calculation are interpreted and the actual pressure at a point along the pipe-line is to be determined, the pressure corresponding to the hydrostatic head of this point above the reference plane is to be deducted from

65

Purumetrrs of the basic eyuutbns

the calculated pressure. The actual pressure is determined in accordance with the rela tion

= P - esh (4.34)

The pressures p and pr usually have the meaning of an excess pressure as compared with the atmospheric pressure. But the pressure may be measured just as well from the absolute zero or from any other chosen zero value. The chosen system, however, has to be maintained throughout the entire calculation.

Some input data, particularly where the effect of the compressibility of the gas or of the separation of liquid column have to be considered, are expressed in terms of the absolute pressure. They are denoted pabs.

4.9 Cavitation

The pressure in a pipe-line changes in the course of water hammer. In some instances, it is higher, and in others, lower than at the steady state. When considering the possibility of damage to the pipe-line or the attached devices, it is not only the highest pressures that are important; a lowering of pressure may be just as dangerous.

During water hammer the pressure may even drop below the atmospheric pressure; the system is then subjected to external pressure. Such loading may be dangerous especially for thin-walled structures designed primarily for with- standing internal pressures. It may damage, for example, the packings, which sometimes are not designed for this kind of loading.

The pressure may drop to almost absolute zero, which can, however, never be reached since cavitation will occur. A cavity, filled with the vapour of the liquid, forms in the conduit [9, 63, 641.

When such a cavity develops in a pipe-line, its volume usually increases up to a maximum value and then it decreased again until it disappears. As long as the cavitation lasts, a constant pressure, equal to that of the vapour of the liquid, persists in that part of the pipe-line (refer to Table 4.9).

Table 4.9 Absolute pressure of water vapour at different temperatures [67]

Water temperature Saturated vapour pressure (Pa)

5.91 x lo2

7 . 1 4 ~ lo3 2.26 103

4.58 x lo4

66

Cavitation

The disappearance of the cavity may induce a sudden and dangerous increase in pressure. This phenomenon may be repeated several times in the course of water hammer (refer to Sect. 14.9).

Cavitation occurs most frequently at points with a large geodetic elevation, at points where there is a change in the pipe-line gradient or profile and at points where local pressure losses occur or a pressure device is installed. The general layout of the pipe-line system and the causes of water hammer are important.

The method of calculation introduced in the present book allows us to include the effect of cavitation even when this occurs at several points along the pipe-line simultaneously. I t is then necessary that in the schematized system (refer to Sect. 3.3), junctions provided with a special damping device have to be installed (refer to Sect. 5.8). Their position can be either estimated in advance or calculated. Using the latter approach, water hammer has to be calculated first without considering the effect of cavitation. Performing this calculation, we may find that the pressure at some point along the conduit drops below that of the liquid vapour. From the results of the calculation one has to determine at which point of the system this occurred first and then, in a new calculation, insert a junction with a damping device at this point in order to include the effect of cavitation. This device ensures that the pressure does not drop below that of the liquid vapour at this point. If the pressure at some other point now again drops below that of the liquid vapour in this repeated calculation, this means that cavitation will occur at yet another point along the system, that is, at the point where in this repeated calculation the pressure first dropped below that of the liquid vapour. The calculation has to be repeated again with the next junction provided with a damping device installed at the point thus determined. Theoretically, this procedure has to be repeated as long as the calculated pressure drops below that of the liquid vapour at any point in the system.

In some systems, the points, at which cavitation might occur, are not obvious. The points determined in the calculation may change even for small changes in the input data. In such cases, the place where cavitation might occur, cannot be determined reliably, bearing in mind the inaccuracies in the input data and in the method of calculation. In such cases the calculated pressures are then to be considered only approximate.

When the pressure in the pipe-line approaches that of the vapour, other phenomena appear, for example, the gases dispersed in the liquid separate; they may affect the wave velocity. There is not sufficient information available at present to include the effect of these phenomena in the calculation.

In the method of calculation employed, the effect of cavitation can be reliably included, provided the points of possible vapour separation are evident and the cavities developed are not so large as to influence significantly the lengths of the adjacent pipe-line sections.

67

5 Damping devices

5.1 Junction without a damping device

In schematizing the analysed pipe-line system (refer to Sect. 3.3), junctions have to be located at all the points where damping and pressure devices are installed, at all changes in the pipe-line cross-section which we wish to include in the analysis, at all bifurcations, at the ends of every pipe-line branch, at the points where cavitation is expected, and at all other points where the pressure and the discharge are to be studied in detail.

Each of these junctions which is not attached directly to any damping device is to be considered as a junction without such device. All the damping devices included in the version of the program presented are described in the subsequent sections.

The effect of a junction without a damping device is described by the relation

QJ = 0 ( 5 4

where Q, is the discharge from the junction into the damping device.

solution yields the pressure pJ in the junction. In the calculation no parameters are submitted of the damping device and the

5.2 Constant pressure

Constant pressure may be assumed in junctions, where the variations in pressure during water hammer are so small that they can be neglected. Constant pressure may be achieved by connecting the pipe-line to a large reservoir, where a change in the level of the liquid is practically unnoticeable, or the outflow may issue into a medium at atmospheric pressure.

The effect of a damping device characterized by a constant pressure is then described by the relation

The pressure pJ in the junction is constant and equal to the pressure po which exists in the junction in the initial steady state.

No parameter of the damping device is included in the calculation for the

68

Reservoir

pipe-line system. The pressure p,, is submitted together with the input data for the junction.

The calculation determines the discharge Q, from the junction into the damping device. In the case of a reservoir, it is the discharge from the pipe-line into the reservoir. The discharge from a reservoir into a pipe-line is considered to be a negative discharge. The effect of air, which penetrates into the pipe-line, is not taken into account with this kind of damping device.

If the effect of air is to be considered, an integrated damping device is considered to be attached to the junction for the purpose of the calculation (refer to Sect. 5.9).

The effect of the pressure losses of the intake into and the outlet from a reservoir is taken into account, when the damping device is a reservoir (refer to Sect. 5.3). For a damping device represented by a surge tank (refer to Sect. 5.5), an overflow (refer to Sect. 5.6) or an integrated damping device (refer to Sect. 5.9), the change in the level of the liquid is also included in the calculation.

5.3 Reservoir

If a sufficiently large reservoir is attached to the pipe-line, so that the variations in the level of the liquid need not be considered, but the pressure losses at both the inlet and the outlet have to be taken into account, we may assume the attachment of a damping device denoted “reservoir”. Here, the pressure losses are considered proportional to the square of the discharge. The coefficient of the pressure losses may differ for each direction of flow. The “reservoir” damping device is also suitable, for example, for describing a sprayer, where the liquid flows out into the atmosphere through a restricted cross-section. The “reservoir”, however, does not allow us to include the effect of the air sucked into the pipe-line during reverse flow.

A schematic representation of the “reservoir” damping device is shown in Fig. 5.1.

Its function is described by the relation

PJ = CJ IQJI QJ -k pb (5.3)

and for Q, < 0 CJ = CJ- (5.4)

for Q, 2 0 CJ = CJ+

Fig. 5.1 Schematic representation of a “reservoir” damping device.

69

Dumping diwiccs

where cJ, cJ+, cJ- are coefficients expressing the pressure losses due to friction at the intake into and the outlet from the damping device, pb is the pressure in the junction at the damping device in its basic state, that is, for the case where QJ = 0.

The values ofthe coefficients cJ+ 2 0, iJ- 2 0 and the pressurepb represent the parameters of the damping device, and all have to be submitted in the data input for the calculation.

The solution of the equations yields the pressure pJ in the junction and the discharge QJ from the junction into the damping device.

5.4 Air chamber

An air chamber is one of the devices used to reduce the effects of water hammer. It consists of a pressure vessel partly filled with air and attached by its base to the pipe-line.

When the pressure in the conduit increases, a portion of the liquid in the conduit flows into the air chamber, where it compresses the air. On the other hand, when the pressure in the conduit drops, some liquid flows from the air chamber into the conduit and the air pressure in the chamber is reduced. In this way, variations in pressure in the conduit induced by water hammer may be substantially lowered in many cases. The air chamber, however, has to be of adequate size and contain a sufficient volume of air.

The air chamber is often connected to the pipe-line through a restriction which produces different pressure losses during the inflow and the outflow of the liquid.

A schematic representation of an air chamber is shown in Fig. 5.2.

Fig. 5.2 Schematic rcprcscntation or an “air chamber” damping device.

The pressure pJ in the junction, to which the air chamber is attached, is defined by the equation

70

Air chamber

The pressure loss is proportional to the square of the discharge QJ into the air chamber. The coefficient CJ which determines the magnitude of this loss is determined by relation (5.4). The quantities Pbabs > 0 and Vb > 0 are the absolute pressure and volume of air, respectively, in the air chamber in its basic state. Any state of the air chamber, at which Qj = 0, may be chosen as the basic state. The values Pbabs and Vb determine the quantity of air in the air chamber. For an air chamber in its basic state, the difference in the volume of liquid in the chamber is A V = 0. For a higher level of the liquid, A V > 0 and for a lower level, AV < 0. The polytropic exponent y lies within the range 1 I y 5 1.4. The value y = 1 corresponds to isothermic change of pressure and the value y = 1.4 to adiabatic change of pressure. In the latter case, no exchange of heat takes place between the air in the air chamber and the ambient medium.

Values of y = 1.23 to 1.25 were found in measurements on air chambers having a total volume of 6 to 10 m3 [20]. The functioning of an air chamber depends also on its geodetical elevation. It is defined by the value pb, that is, by the pressure in the junction at the air chamber in its basic state. The following relation holds between the discharge QJ and the volume difference A K

where A h is the value of A V in the initial state, that is, at instant t = 0, when the calculation of water hammer is started. In including an air chamber in the calculation, the following values have to be submitted as the parameters of the damping device: C,+ 2.0, CJ- 2 0, Pbabs > 0, Vb > 0, 1 I y I 1.4. The value A h is not submitted: it is calculated from the pressure po existing in the junction in the initial state.

The solution yields the time-dependent values p,, QJ, A V and the absolute air pressure Pabs in the air chamber. The necessary size of the air chamber can be found from the values V, and A V

The calculation takes into account the effect of the variations in the air pressure in the air chamber and the variations in the pressure losses in the intake which are a function of the discharge into the air chamber. The calculation does not include the effect of the variations in the liquid level in the air chamber, the effect of the inertia of the liquid in the air chamber and the inlet piping, nor the effect of eventual variations in the pressure losses as a function of the level of the liquid in the air chamber. If these effects are substantial, it is necessary to consider, instead of an “air chamber” damping device, an “integrated” damping device (refer to Sect. 5.9), which allows these effects to be included in the calculation.

71

Damping devices

5.5 Surge tank

The damping device denoted “surge tank” corresponds to a reservoir attached by its base to a junction. It is assumed that the pressure in the junction depends on the level of the liquid in the reservoir, which may vary in the course of water hammer, and on the pressure loss in the intake to the reservoir.

This pressure loss is proportional to the square of the discharge QJ from the junction into the reservoir. The coefficients expressing this loss may differ for each direction of flow. The area of a horizontal cross-section through the tank may be different in its lower and upper parts. In the course of the calculation, the quantity of liquid flowing into and out of the tank is determined.

The damping device described need not only be used to express the effect of a surge tank. By choosing, for example, a sufficiently large area for the horizontal cross-section of the tank so as to render fluctuations in the liquid level practically insignificant, we may substitute a surge tank for a reservoir and observe the quantity of liquid flowing into and from the reservoir during water hammer. ”$7 _ _ _ _ --- AV=O

I Q.0

JJ Fig. 5.3 Schematic representation of a “surge tank” damping device.

A schematic represenlation of a surge tank is shown in Fig. 5.3. Its function is described by the following equations: for A V 2 0

PJ = CJ I Q J I QJ

and for A V < 0

PJ = CJ IQJI QJ + (5.7)

where pb is the pressure in the junction for the surge tank in its basic state, that is, for QJ = 0 and with the liquid surface at the transition level between the upper part of the tank having a horizontal cross-sectional area S,, and its lower part having an area S,. The symbol AVdenotes the difference in the volume of the liquid in the surge tank and in the tank in its basic state. AV > 0 holds when the surface of the liquid is in the upper part of the chamber, and A V < 0 when

72

OverJiow

it is in the lower part. The value of the coefficient CJ defining the pressure losses in the connection between the pipe-line and the surge tank is defined by equation

The data to be submitted for the calculation are the parameters of the damping device, that is, the values of the coefficients CJ+ 2 0 and [, - 2 0 expressing the pressure losses at the inlet into and outlet from the tank, respectively, the areas S , > 0 and S , > 0 of the horizontal cross-section of both the upper and lower parts of the tank, respectively, and the pressure pb in the junction of the tank in its basic state.

The value A V , is not required. It is derived from the pressure po existing in the junction in the initial state.

The calculation yields the time-dependent pressure pJ in the junction, the discharge QJ and the difference in the volume A V o f the liquid in the tank.

The program for the “surge tank” damping device does not check the possible overfilling of the tank or the lowering of the liquid level below its bottom. Only two different horizontal cross-section of the tank can, at most, be included in the calculation. The effect of the inertia of the fluid or the eventual dependence of the pressure loss on the liquid level is not considered. To include these effects in the calculation, one has to consider an “integrated” damping device (refer to Sect. 5.9) instead of the “surge tank”.

(5.4).

5.6 Overflow

The damping device denoted “overflow” corresponds to a chamber with a constant horizontal cross-section S. The lower part of the chamber is attached to the junction through an intake which produces a restriction of the discharge. The upper edge of the chamber forms the overflow. As long as the liquid spills over, the function of the damping device is similar to that of a “reservoir”-type damping device. The pressure in the junction is determined by the height of the overflow and by the pressure loss in the intake between the junction and the damping device. When the liquid flows from the damping device into the junction, the level of the liquid in the chamber falls. The level of the liquid then depends on the discharge from the chamber into the junction and the function of the overflow corresponds to that of a surge tank. The pressure in the junction is determined by the liquid level in the chamber and by the pressure loss in the intake. This is true until the liquid level reaches the level of the overflow. The liquid which spills over the overflow, does not return to the system.

The “overflow” damping device may be employed to describe a surge tank with overflow or the outflow from a pipe-line, where one has to include the effect of sucked-in air during reverse flow in the calculation, as well as other devices with a similar function, such as, for example, cooling towers.

13

Damping devices

A schematic representation of the “overflow” damping device is shown in

Its function is described by the relations: Fig. 5.4.

for A V = 0

PJ = CJ lQJl QJ pb

and for A V < 0

Fig. 5.4 Schematic representation of an “overtlow” damping device.

The value of the coefficient CJ is defined by equation (5.4). The overflow is in its basic state, with the pressure pb acting in the junction, when the discharge Qj from the junction into the damping device equals zero and the liquid is at the level of the overflow. For a lowering of the liquid level, A V < 0. The possibility of the liquid rising above the level of the overflow is not considered.

The relation t

A V = S Qj dt tl

(5.9)

holds between QJ and AV, where t , is the instant when A V = 0 and Qj > 0 for the last time.

The parameters of the damping device CJ+ 2 0, cJ- 2 0, S > 0 and pb have to be submitted for the calculation. The calculation yields the time-dependent values pJ, Qj and AV.

The program for the “overflow” damping device does not check for a possible lowering of the liquid surface below the level of the chamber bottom. It also does not allow us to include in the calculation a variable cross-section of the overflow chamber, the inertia of the liquid in the chamber or the eventual dependence of the pressure loss on the level of the liquid. To include these effects it is necessary to consider an “integrated” damping device (refer to Sect. 59) instead of an “overflow”.

74

Air inler valve

5.7 Air inlet valve

The air inlet valve is a device fitted on a pipe-line to de-aerate the conduit and to reduce the effect of water hammer. An air inlet valve allows air to be sucked into its chamber and to escape therefrom, while preventing the outflow of liquid. The outflow of air is usually restricted and the escaping air is compressed by the liquid in the chamber [21]’. With suitable chosen parameters, the “air inlet valve” damping device may be used to study other phenomena, for example, cavitation (refer to Sect. 5.8).

-I+ ”!” pabs+

! OJ’O Fig. 5.5 Schematic representation of an “air inlet valve” damping device. &

A schematic representation of the air inlet valve is shown in Fig. 5.5. The valve consists of a chamber attached by its lower part to a junction. The upper part of the chamber is connected with the atmosphere through a permanently open aperture 1 and by another aperture 2 which can be closed by a check valve. This valve permits the free flow of air into the air inlet valve, but prevents its flowing outward. The upper part of the air inlet valve is equipped with a partition provided with an aperture 3 controlled by a check valve with a float. This check valve allows the air to flow freely in both directions, but it closes when the space below the partition is filled with liquid. In the initial state, aperture 2 is open, aperture 3 is closed and the space below the partition is filled with liquid. When the pressure in the air inlet valve drops below the atmospheric pressure, aperture 3 opens and the air maintaining the atmospheric pressure in the air inlet valve flows freely under the partition. As soon as the liquid starts flowing back into the air inlet valve, aperture 2 closes, the air in the air inlet valve is compressed and starts escaping through aperture 1 under pressure. The effect of the volume of air above the partition is neglected. This condition persists, until the pressure in the air inlet valve drops to atmospheric pressure; then aperture 2 opens again and the previous condition in the air inlet valve is re-established. Alternatively the air keeps escaping through aperture I until it is all expelled from the air inlet valve. Then aperture 3 closes, aperture 2 opens and the air inlet valve is again in its initial state.

*) Czcchoslovak patent K. Haindl No. 104071

75

Dumping devices

The pressure in the junction to which the air inlet valve is attached is given

(5.10)

where Pabs is the absolute pressure in the chamber and Pubs is the absolute atmospheric pressure. The air inlet valve is in its basic state when Pabs = Pa& at which point the pressure in the junction is equal to pb The pressure is converted to the level of the reference plane using equation (2.3) and it depends on the elevation of the air inlet valve above the reference plane.

by the relation

PJ = Pb -k Pabs - Paabs

The volume of air in the air inlet valve is given by the relation

(5.1 1 )

If there is no air in the air inlet valve ( V , = 0) and if the pressure in it is higher than the atmospheric pressure ( pJ > Pb), then aperture 3 is closed and the function of the air inlet valve is described by the relation

(5.12)

The value of the pressure pJ in the junction may then be found from the calculation of water hammer in the pipe-line system, while the valuepabs can also be determined with the aid of equation (5.10).

Should the pressure pabs drop below the atmospheric pressure, the air inlet valve chamber would start filling with air (Q, c 0, V, > 0) and the pressure in the chamber would equal the atmospheric pressure

Pabs = Paabs (5.13)

The value pJ defined by equations (5.10) and (5.13) is used for the calculation of water hammer.

This state persists as long as Q, I 0. As soon as the air inlet valve chamber starts filling with the liquid again (Q, > 0), the check valve of aperture 2 encloses a quantity of air in the chamber with a mass

m = V.@, (5.14)

where e, is the density of the air at atmospheric pressure. The enclosed air is compressed by the liquid and the pressure in the air inlet valve increases. Simultaneously, air escaped through aperture I. The absolute pressure in the chamber is determined by the relation

(5.15)

76

cavitation

The mass of the enclosed air is obtained by solving the differential equation

- - dm - -- RD; 0.484 /F 0.85 /- (5.16) dt 4

which approximately determines [29] the quantity of air flowing through a circular sharp-edged aperture. The initial condition necessary for its solution is given by equation (5.14) valid at the instant when aperture 2 is closed.

The value p , found from equations (5.10) and (5.15) is then used in the calculation of water hammer in the conduit. The described state lasts until all the air is expelled from the air inlet valve (Va = 0) or the pressure in the air inlet valve drops to atmospheric pressure (pabs = In submitting the data of the air inlet valve, one has to introduce the following parameters of the damping device: the pressure pb in the junction at the air inlet valve in its basic state, the polytropic exponent 1 I y S 1.4 and the diameter Do 2 0 of the aperture through which the air escapes. In the calculation the following values are assumed for the standard atmospheric pressure and the density of air:

paabs = 1.01325 x lo5 Pa

ea = 1.22 kgm-3

(5.17)

(5.18)

The calculation yields the pressure p , in the junction, the discharge of the liquid Q, from the junction into the air inlet valve, the volume Va of the air in the air inlet valve, the mass m of the air in the air inlet valve and the absolute pressure pabs in the air inlet valve.

When the diameter of the aperture for the escaping air is infinitesimally small (Do = 0), air can flow into the air inlet valve chamber, but not out. On the other hand, at large values of Do, the pressure in the air inlet valve does not increase above atmospheric pressure as long as there is any air left in the air inlet valve.

The program for the damping device denoted “air inlet valve” does not check whether the air has not filled the entire chamber and is escaping into the pipe-line; neither does it take into account the changes in hydrostatic pressure induced by a change in the liquid level in the chamber, nor the pressure losses at the intake into and outflow of the liquid from the air inlet valve. If these effects are to be included in the calculation, an “integrated” damping device (refer to Sect. 5.9) is to be used instead of an “air inlet valve”.

5.8 Cavitation

Cavitation (separation of the liquid column) is not a damping device in the literal sense of the word, but it does affect water hammer in a similar way. Hence it is included in this section.

77

Damping devices

The basic equations (2.1), (2.2) for calculating water hammer in pipe-line sections do not take into account the fact that the pressure cannot drop below that of the vapour pressure of the liquid. If such a drop in pressure should occur at some point, a free space filled only with vapour would form and the pressure would not drop any further. In this way, the hydraulic system is divided into two separate parts, which do not influence each other until the free space disappears. However, according to the solution of the basic equations, no free space does develop and the pressure can actually drop below the vapour pressure of the liquid, and possibly even below absolute zero. The use of basic equations (2.1) and (2.2) in the calculation of water hammer may, therefore, in some cases yield unrealistic results. To include the effect of cavitation in the basic equations would complicate the calculation and the submission of data. Apart from other data, it would be necessary to submit the elevations of all parts of the pipe-line. Furthermore, the behaviour of the liquid under the conditions, when free spaces are formed is, even theoretically, not quite clear. Sometimes, especially in cases when the differences in elevation along the pipe-line are not large, it is difficult to determine the points where cavitation will occur with sufficient accuracy (refer to Sect. 4.9).

In the method of calculation introduced in this book, one has to select points in the schematized system, where cavitation might possibly occur, and then insert junctions with special damping devices at these points. If cavitation does not occur, the effects of these damping devices do not manifest themselves. If it does occur, these damping devices affect the course of water hammer in the same way as cavitation developed at the point where the device is installed. This also allows us to study the size of the cavity formed as a function of time. The calculation does not incorporate the effect of the cavities on the length of the pipe-line sections, nor that of any change in the position of the cavities during the course of water hammer.

At other points along the pipe-line system, one can only check, whether the pressure did not drop below that of the vapour pressure of the liquid. The minimum pressure, which appears in each section during the calculation, is registered, and this minimum pressure can be compared with the pressure at which cavitation would occur at the highest point along the section. If the calculated minimum pressure is higher, then cavitation did not occur in this section; in the opposite case, the pressure in the section has to be investigated in more detail.

The “air inlet valve” damping device can, with suitably chosen parameters, be used to incorporate the effect of cavitation in the calculation. A schematic representation of an air inlet valve is shown in Fig. 5.5.

The diameter Do of the air aperture of the air inlet valve has to be chosen sufficiently large, for example, Do = D, where D is the maximum diameter of any section of the pipe-line attached to the junction. This practically eliminates

78

Integrated damping device

the effect of the compression of air. Therefore, the value of the polytropic exponent is not important and may be chosen somewhat arbitrarely, for example, y = 1.

The pressure in the basic state is defined by the relation

(5.19)

wherepv is the pressure in the junction at cavitation. This is a pressure calculated relative to the reference plane. The pressure pv depends mostly on the elevation of the conduit above the reference plane (refer to 2.3) and on the nature of the liquid and its temperature. Values for the absolute vapour pressure of water at different temperatures are listed in Table 4.9.

The parameters of the “air inlet valve” damping device, Do 2 D, 1 I y I 1.4 and pb have to be submitted for the calculation.

The calculation yields the time-dependent pressure pJ, the discharge QJ (both in the junction) and the volume V, of the free space. QJ > 0 corresponds to a reduction of the free space. In performing the calculation, values for m and pabs are also found, but these do no have their original physical meaning.

5.9 Integrated damping device

An “integrated” damping device can be substituted for all the damping devices described earlier. In addition, it also permits us to include in the calculation some factors, which are not taken into account by the earlier devices at all, or only to a limited extent. These include the variable cross-section of the chamber of the damping device, the effect of the level of the liquid in the chamber on the pressure in the junction, the pressure losses along the chamber and the inertia of the liquid in the chamber. It is not advisable, however, to employ an integrated damping device in the place of the devices described earlier in all cases; this would only complicate the submission of data and the calculation.

An integrated damping device is particularly convenient in those cases, where various aeration pipes or narrow shafts are attached to the main pipe-line. Their effect cannot be neglected, but does not correspond to that of a surge tank. In such cases, the effect of the liquid level, of the inertia of the liquid and of the variations in pressure losses as a function of the level of the liquid in the damping device may be significant.

The schematic representation of an integrated damping device is presented in Fig. 5.6.

The device consists of a chamber attached by its lower part to the junction. The upper part of the chamber is either open when it is arranged as a chamber with an overflow, or closed when it is arranged like an air chamber, while alternatively, it may be arranged as an air inlet valve (refer to Sect. 5.7). In all

79

Damping devices

three variants, the chamber is composed of one or several interconnected parts which are situated above each other. Each of these parts has a horizontal cross-section of constant area S or Si and a cross-section perpendicular to the direction of flow of constant area S , or Sfi. The subcript i 2 1 denotes the sequence of the parts constituting the chamber, starting from the bottom part upwards. The values corresponding to the lowermost part have no subscript.

b)

r-w

Fig. 5.6 Schematic representation of an “integrated damping device” arranged as (a) a surge tank with overflow; (b) an air chamber; (c) an air inlet valve.

The areas S and Si are used to determine the relation between the liquid level and its volume in the chamber. The areas S,, Sfi are employed only when the effect of the inertia of the liquid is calculated. When the part of the chamber considered is vertical, then S = s, and Si = Sfi. When the liquid flows from the junction into the chamber, the discharge Q, is positive, for a flow out of the chamber, it is negative. Vis the total volume of the liquid in the chamber of the damping device, expressed by the relation

(5.20)

where V, is the initial volume of the liquid in the chamber. Equation (5.20) does not apply precisely to a damping device arranged as a chamber with an overflow, since, in this case, the liquid which flowed into the chamber during the time, when it was full, is not included in the volume K This liquid flowed off via the

80


overflow and does not return into the system. The volume V < 0 means that the chamber is completely empty and that air has penetraded into the conduit. Its volume is then defined by the value I VI . For V I 0, the level of the liquid is assumed to be at the bottom of the lowermost chamber. The volume V = 0 corresponds to the chamber filled with air and the pipe-line filled with liquid.

The pressure losses arising in the connections between all the individual parts of the chamber are taken into account. These pressure losses are proportional to the square of the discharge Qj. They may differ for either direction of flow. The total pressure loss is expressed by the relation

AP = CJIQJI QJ

where

and

The value cJ includes only the coefficients corresponding to the connections below the level of the liquid in the chamber at a particular instant. The coefficients cJ+ and cJ- apply to any level of the liquid. The coefficients CJo+, cJ+, CJo- and CJ- correspond to the connection between the lowermost part of the chamber and the junction. The coefficients cJi+ and cJi- apply to the connection at the level of the lower edge of the i-th part of the chamber.

A schematic representation of the i-th part of the chamber of an integrated damping device is portrayed in Fig. 5.7.

The equilibrium of forces acting along the axis in this part of the chamber, induced by the difference in pressure, by the inertia and the dead weight of the liquid may be described by the relation

(5.22)

where Api is the difference in pressure and Ahi the difference in height between the ends of the i-th part of the chamber.

81

Dumping devices

It follows from equation (5.22) that

Si dQj Api = Ahie - . - + eg Ahi sii dt

(5.23)

The total increment in pressure in the junction, induced by the inertia and the dead weight of the liquid, is determined by the sum of the pressure increments in the individual parts of the chamber that are filled with liquid.

The pressure pJ in the junction equipped with an integrated damping device is given by the equation

though, in some cases of the air inlet valve arrangement, it is calculated with the aid of equation (5.12).

The symbol Apa denotes the difference in air pressure on the surface of the liquid in the chamber as compared with the basic state; h, is the level of the liquid in the chamber measured from the bottom of the lowermost part of the chamber and h, is the same level but in the basic state.

The basic state of an integrated damping device, corresponding to a pressure pb in the junction, is characterized by the values V = 0 (the surface of the liquid is at the bottom of the lowermost part of the chamber) and Qj = 0, while the surface of the liquid is at atmospheric pressure.

In the air chamber arrangement (see Sect. 5.4), QJ = 0 and dQ,/dt = 0 but the volume of the liquid in the chamber may be different. It is determined by the submitted values of the volume of air Vb > 0 in the chamber and its absolute pressure Pbabs > 0.

The value Apa of the excess pressure varied for the different arrangements. For a chamber with overflow (refer to Sect 5.6) it reads

Apa = 0

For an air chamber, we have

(5.25)

(5.26)

where A Vis the difference in the volume of liquid in the chamber of the damping device as compared with the basic state, and 1 I y I 1.4 is the polytropic exponent.

For the air inlet valve arrangement, Apa is determined by equation (5.25) as long as the air flows into the chamber of the air inlet valve. If the air in the air

82


inlet valve chamber is compressed, Apa is defined by the relation

(5.27)

The absolute air pressure pabs in the air inlet valve chamber and the absolute atmospheric pressure are determined as for an “air inlet valve” damping device (refer to Sect. 5.7).

If there is no air in the integrated damping device arranged as an air inlet valve, but a surplus pressure exists there compared to atmospheric pressure, the pressure pJ in the junction is given by equation (5.12).

The parameters which have to be submitted for an integrated damping device are: the number 1 I Na 5 3 determining, whether the damping device is arranged like a chamber with overflow (N, = l), or an air chamber (Na = 2) or an air inlet valve ( N , = 3); the coefficients CJ+, CJ-, CJi+ and cJi-, expressing the pressure losses in the connections to the junction and between the individual parts of the chamber; the total height h,,, > 0 of the chamber; the pressure pb in the junction with the damping device in its basic state; the areas of the horizontal cross-sections of the chamber and of the cross-section perpendicular to the direction of flow for all parts of the chamber, that is, S, S,, Si and S,; the number N, 2 0 representing the number of changes in chamber cross-section; the heights 0 < hi < htot at which these changes occur relative to the bottom of the lowermost part of the chamber. In the case of an air chamber, one also has to submit the volume Vb of the chamber of the damping device, its absolute pressure for the basic state and the polytropic exponent y. For an air inlet valve ( N , = 3), the polytropic exponent y and the diameter Do of the opening for the escape of air have to be submitted together with the other input data.

The calculation yields the following time-dependent values; the pressure p , in the junction, the discharge Q, from the junction into the damping device, the volume V of the liquid in the damping device, the total mass rn of the air in the damping device, its absolute pressure pabs and the level of the liquid h, in the chamber.

If a damping device cannot be replaced with sufficient accuracy by an integrated damping device or by the damping devices “pressure” or “discharge” as described below, it may be replaced by a combination of several junctions with individual damping devices interconnected by sections across various pressure devices. If even such an arrangement would not yield a satisfactory result, the WTHM program would have to be complemented by another subprogram designed specifically to include the effect of the damping device considered (refer to Chapt. 13).

83

Damping devices

5.10 Pressure

The damping device denoted “pressure” may be substituted for any damping and pressure device, or for a large part of a pipe-line system, provided it is known in advance how the pressure in the junction varies as a function of time. Its applications are many; some of them are introduced in the following text. The device may be used to investigate the effect of arbitrary changes in pressure at some point along a pipe-line on other parts of the system. Abrupt, periodical and any other type of variation in pressure may be considered. The stability of a system may be estimated from the calculation of the response of the system to periodical variations in pressure with different frequencies and amplitudes, even with inclusion of the effect of non-linearities (refer to Sect. 14.7).

Sometimes, it is necessary to study water hammer in a small pipe branching off a large pipe-line system. The branch pipe is so small that it cannot significantly affect water hammer in the system. It suffices then to calculate water hammer in the entire system only once, to determine the pressure at the junction with the branch pipe, and then to calculate the effect for the branch pipe separately. The effect of the large pipe-line system is now effectively replaced by a damping device which produces the calculated pressure in the terminal junction. Water hammer may be then solved in greater detail for various arrangements of the branch pipe using shorter calculation steps.

The “pressure” damping device may be used to ascertain the adjustmen regime of an outlet valve which would produce a desired variation in pressure with time. The valve and the outlet have then to be replaced by a junction with a “pressure” damping device, which ensures the required pressure. To determine the opening-closing regime of the valve, we have at our disposal the pressure chosen in front of the valve, the constant pressure behind the valve and the calculated discharge. If the characteristics of the valve are known, these values suffice to define its adjustment regime (refer to Sect. 14.5).

The “pressure” damping device may be used to find the characteristics of valves on operating pipe-line systems. To achieve this, the pressure has to be measured in front of and behind the valve during the unsteady flow induced by the opening of the valve. Simultaneously, one has to measure the degree of opening of the valve as a function of time. In the calculation, the valve and the entire part of the system behind the valve are replaced by a “pressure” damping device, which ensures the pressure measured in front of the valve. The calculation, which starts with the initial steady state, when the discharge in the pipe-line is zero, determines the time-dependent discharge at the valve. For each state of opening of the valve, the discharge coefficient may be obtained, since the measured degree of opening of the valve, the pressure measured in front and at the back of the valve, and the calculated discharge, are known at every instant. If the part of the system in front of the valve, which influences the results of the

84

Pressure

calculation, is too complex, or its parameters needed in the calculation are not quite well known, its effect may be replaced by another “pressure” damping device attached to the pipe-line which brings the liquid to the valve. For this purpose, the pressure in the actual system has to be measured at another point along the feeding pipe-line, at a sufficient distance from the valve. Another “pressure” damping device ensures the pressure measured and thus replaces the effect of the part of the system in front of the point of measurement. The calculation is carried out only for that part of the system situated between the points where the pressure is measured. The parameters of this part of the system have to be known. If it is a conduit with a constant diameter, its inside diameter, the distance between the two measuring points and the difference in their elevation have to be known. The wave velocity and the coefficient of friction, on which the calculation depends, can be found from the measurements of the pressure.

On a completed pipe-line system the calculation may be carried out even for the unsteady flow induced by closing the valve. The calculation, however, has to be repeated several times for different values of the initial discharge. The correct calculation will result in a zero discharge at the valve at the instant of its complete closure (refer to Sect. 14.4).

The characteristics of operating pumps, turbines and other devices may be found in the same way (refer to Sect. 14.1 1 ). Instead of the opening of the valve, one measures the speed (for a pump) and possibly the position of the guide and action blades (for a turbine). The unsteady flow at which the pressure is measured, may be induced by starting or switching off the pump or the turbine, closing or opening the valve, or through a combination of factors. One has to know, however, the discharge at one point, at one instant at least, so that the initial steady state of flow can be determined. If the moment of inertia of the device is known, and if it is disconnected from the power network during the measurements, its moment characteristic, as well as its pressure characteristic, may be found from the measured quantities (refer to Sect. 14.11). If the device is connected to a power network during the measurements, the electric quantities have to be known or measured to determine the moment of the electric motor or generator.

The procedure described may be also employed to determine the discharge at steady flow in an actual pipe-line system (refer to Sect. 14.10). It is sometimes difficult to measure this by other methods, especially for large-diameter pipelines. The measurements may be carried out in various ways. A pipe-line having a long section with a constant diameter and ending with a valve is the most convenient. To determine the discharge in such a case, it suffices to continuously measure the pressure, at the upstream and downstream ends of this section, while the discharge is reduced by closing the valve, from a steady flow to zero. The closing rtgime and the characteristics of the valve are not important in this

85

Damping devices

case. To calculate the initial discharge, one has to know the length of the pipe-line between the two points, where the pressure is measured, the difference in their elevations and the inside diameter of the pipe-line. All other quantities required can be calculated from the pressures measured.

A schematic representation of a “pressure” damping device is shown in Fig. 5.8.

Fig. 5.8 Schematic representation of a “pressure” damping device.

The “pressure” damping device ensures the following pressure in the junction:

(5.28)

where fp,(t) is the submitted function of time. This curve may be submitted in a variety of ways (refer to Sect. 9.4).

The calculation yields the value of Qj as a function of time. In some cases, a combination of the “pressure” and “discharge” damping

devices has to be employed, for example, when the regime of a valve is to be determined, which would ensure a desired pressure curve, when the valve is not installed at the outlet of the pipe-line. The pressure behind the valve then is not constant, but is a function of the discharge through the valve. Such cases can be solved using a suitable combination of the damping devices “pressure” and “discharge” (refer to Sect. 5.1 I ) .

5.1 1 Discharge

The damping device denoted “discharge” ensures a time-dependent discharge from a junction into the damping device which is independent of the pressure in the junction.

This damping device is convenient for the schematic inclusion of the effect of gear, piston and other kinds of pumps, where the discharge is practically independent of the pressure in front of and behind the pump. No pressure device described in Chapter 6 can be used to represent such pumps.

If a gear pump is installed at the end of a pipe-line, and if it pumps a liquid from a reservoir, a junction with a “discharge” damping device is substituted for the reservoir including the pump. Only one section, corresponding to the delivery pipe, is attached to the junction. The discharge - QJ, that is, the discharge from the damping device into the junction, is equal to the discharge through the pump.

86

Discharge

If the pump is not installed at the end of the pipe-line, the conduit has to be divided into two branches at the point where the pump is installed, and each branch has to be provided with a junction connected with a “discharge” damping device at this end. A discharge - Q, occurs in the junction on the delivery branch, and + Q, on the suction branch. The discharge QJ corresponds to that through the pump.

It is also necessary to use a “discharge” device when the adjustment regime of the valve is calculated in order to obtain the required pressure in front of the valve, when it is not installed at the outflow. The hydraulic system has to be divided into two parts at the point where the valve is situated; the calculation is then performed for each part individually. First, the unsteady flow is calculated in the part of the system located in front of the valve. A junction with a “pressure” damping device is substituted for the remaining part of the system including the valve (refer to Sect. 5.10). Only the section corresponding to the conduit bringing the liquid to the valve, is attached to this junction. The “pressure” damping device ensures that the required pressure in front of the valve is achieved and then the corresponding discharge is calculated. Then the flow is calculated for the system behind the valve. A junction with a “discharge” damping device is substituted for the valve and the entire part of the system situated in front of the valve. Only the section corresponding to the off-take pipe is attached to the junction. The damping device ensures an inflow into the junction equal to the discharge determined in the foregoing calculation. In the solution for the other part of the system, the pressure behind the valve is determined. The necessary adjustment regime can be found from the known pressure in front of the valve, the calculated discharge through the valve, the calculated pressure behind the valve and the known characteristic of the valve.

The “discharge” damping device may be used to ascertain the stability of a hydraulic system in the same manner as the “pressure” damping device.

A schematic representation of a “discharge” damping device is shown in Fig. 5.9.

It ensures a discharge Q, from the junction into the damping device expressed by the relation

QJ = f Q d t ) (5.29)

wherefQJ(t) is a function of time. This curve may be submitted into the entry data file in various ways.

87

Damping devices

In the calculation the pressure pJ in the junction is determined. Sometimes, a case has to be solved, in which the pressure in the junction is

known up to a definite instant, as well as the subsequent discharge. Hence, neither the “pressure” nor the “discharge” damping device can be used for the calculation. The problem of this nature is, for example, to determine the adjustment regime of a valve up to its complete closing (refer to Sect. 14.5). In the initial stage of closing, the variation in pressure may be controlled. With the valve closed, the discharge in the pipe-line is zero. The calculation then has to be carried out twice. First, a “pressure” damping device is substituted for the valve and the calculation is carried out up to the point of closing of the valve. In the second calculation, a “discharge” damping device is substituted for the valve. In the first stage of this second calculation, up to the point of closing of the valve, the discharge submitted is that determined in the first calculation and, after the closing of the valve, Q, = 0. The second calculation leads to approximately the same results, as if a “pressure” damping device were attached to the junction up to the closing of the valve, and then a “discharge” damping device. The above procedure may be applied even in combination with other damping or pressure devices.

88

6 Pressure devices

6.1 Attachment without a pressure device

One pressure device may be inserted between each junction and the end point (upstream or downstream) of a section attached to this junction (refer to Sect. 3.3). The attachment of a section to a junction without a pressure device is used in cases, where the pressure in the junction and at the end of the attached section has to be equal.

The calculation uses the relation

Ap,, = 0

where App is the difference in pressure in front of and behind the pressure device in the positive sense of the attached section.

No parameter of the pressure device is needed for the calculation. In the calculation the pressure p and the discharge Q at the end point of the

section are determined. The discharge is positive provided the liquid flows in the positive direction of the section, without considering, whether the section is attached to the junction by its upstream or downstream end.

When an attachment without a pressure device is used, the local pressure losses, due, for example, to the effect of the inlet into the conduit, or due to a change of the conduit cross-section, due to branch pipes, etc., may be neglected, or they may be incorporated in the coefficient of friction rZ for the attached section (refer to Sect. 4.4).

6.2 Closed pipe-line

The pressure device denoted “closed pipe-line” expresses the fact that the discharge between a junction and an attached section is prevented throughout the calculation.

The relation employed reads

Q = O (6.2) No parameters are submitted for the pressure device in the calculation which

determines the pressure p at the end point of the section.

89

Pressure devices

Application of the “closed pipeline” pressure device yields the same results as if the section was not attached to the junction at all.

6.3 Local loss

The pressure device denoted “local loss” may be employed everywhere, where the loss in pressure at the point where a section is attached to a junction, is proportional to the square of the discharge at the end of the section, and the coefficient of loss remains constant throughout the calculation. This coefficient may be different for either direction of flow.

Local losses at points other than the ends of sections may be included either in pressure devices located at the ends of the sections, or in the coefficient of friction A (refer to Sect. 4.4). For a more accurate calculation, a junction with a “local loss” pressure device would have to be inserted at every point along the schematized pipe-line where such local losses occur.

The effect of a local pressure loss is expressed by the relation

where the coefficient of loss [ is

r = c+

c = c-

for Q 2 0 and

for Q < 0 (6.4)

D is the inside diameter of the attached pipe-line.

[ in the relation The coefficient of loss c in equation (6.3) corresponds to the coefficient of loss

where Ah is the loss of head (elevation) and u is the mean cross-sectional velocity of flow at the end point of a section.

The parameters of the pressure device submitted for the calculation are

In the calculation, the pressurep and the discharge Q at the end point of the section are determined. For [+ = [- = 0, the pressure device “local loss” leads to the same results as an attachment without a pressure device.

5, 2 0, 5- 2 0.

90

Control valve

If the coefficient of loss varies during the course of the calculation, the “control valve” pressure device (refer to Sect. 6.4) has to be used in place of the “local loss” device.

6.4 Control valve

The pressure device denoted “control valve” ensures a pressure loss proportional to the square of the discharge which exists at the starting or end point of a section, at the point where the section is attached to a junction. The coefficient of loss varies in accordance with a known function of time. It is the same for either direction of flow.

This pressure device may be used to include the effect of a control valve and of any other pressure device which induces pressure losses corresponding to the following relations:

for0 < D I 1

and for D = 0

Q = O

where D is the inside diameter of the attached section and o(f) is a variable determining the pressure loss in the valve.

The relation, at any particular instant, between the variable 0 and the coefficient of loss [ in formulas (6.3) and (6.5) reads

1

D = J r + l The variable D is introduced instead of the coefficient [ to avoid the difficulties

connected with submitting large values of [ during the final closing stage of the valve, when ( + co. The value of the variable D varies within the interval (0, 1) with the value u = 0 corresponding to a closed valve and (T = 1 to an open one, when no local loss is induced.

In submitting the data of the control valve, it is necessary to define the variable D as a function of time

An example of such a function, corresponding to the closing of the valve, is illustrated in Fig. 6.1.

91

Prrssure deviees

The function (6.9) is a curve which can be determined in advance from the characteristic of the valve and from its adjustment regime; this has to be entered into the calculation with the aid of values representing several points on the curve. This is a rather tedious procedure. To simplify this, especially when more variants of adjusting the same valve are being investigated, an alternative approach is possible. Here, the characteristic of the valve

and the adjustment regime

(6.10)

(6.1 1)

are submitted separately, d being any parameter defining the state of the valve, for example, the extension of the piston rod of the slave cylinder, the angle of tilt, etc.

Fig. 6. I Example of a function corresponding to the closing rCgime of a control valve.

The characteristic (6.10) of the valve, normally a smooth curve, is submitted only once for all variants of the calculation. The adjustment regimes (6.1 1) are submitted for each variant, however, they are usually expressed by straight or broken lines, composed of a few sections (Fig. 6.2), so that their submission is simpler. Moreover, the necessity to calculate curve (6.9) manually is avoided.

The parameters which have to be submitted for the pressure device are either the curve f,, or the curves fud and fd.

a) b )

Fig. 6.2 Alternative method of determining the adjustment regime of a control valve: (a) adjustment curve; (b) characteristic of the valve.

92

Butterfly valve

The calculation yields the pressure p and the discharge Q at the end point of a section as well as the variation with time of the variable c which determines the opening of the valve.

6.5 Non-return flap valve

The pressure device denoted “non-return flap valve” permits the flow only in one direction at the point, where a section is attached to ajunction. If the liquid should flow in the opposite direction, the flap valve closes and prevents any discharge. Simultaneously, a surplus pressure (in the opposite sense of the direction of the original flow) develops on the flap valve. The flap valve remains closed until a surplus pressure develops in the opposite sense. Then it opens again. The calculation does not take into account the inertia of the flap valve or any other forces which may affect its motion. The flap valve closes and opens abruptly. It is assumed that the valve does not induce any pressure loss while it is open.

Two kinds of flap valve are considered. Each of them permits flow in one direction.

The following relations apply for a flap valve which permits flow in the positive sense of the attached section. For an open flap valve, when Q 2 0

App = 0

and for a closed flap valve, when App I 0

Q = O

(6.12)

(6.13)

For a flap valve which allows the liquid to flow in the negative sense of the attached section, relation (6.12) is used for an open flap valve, when Q I 0, and relation (6.13) for a closed flap valve, when App 2 0.

No parameters need to be submitted for the “non-return flap valve” pressure device. The calculation yields the pressure p and the discharge Q at the end point of the attached section.

If the calculation has to include the effect of the inertia of the flap valve, of the forces induced by the flowing liquid and acting on the flap valve, of the weight which closes the flap valve, and of any other forces, a “butterfly valve” pressure device (refer to Sect. 6.6) would have to be used.

6.6 Butterfly valve

The pressure device denoted “butterfly valve” induces a pressure loss in the conduit which is proportional to the square of discharge. The coefficient of loss is a function of the angle of tilt of the valve. This coefficient may be different for either direction of flow.

93

Pressure devices

The angle of tilt of the flap is influenced by the weight, which closes the valve; by the hydraulic dash-pot, which retards the closing; by the oil pump, which opens the valve, when it is turned on; by forces induced by the flowing liquid and acting on the valve, and by the inertia of the valve. The forces through which the flowing liquid acts on the valve, may depend on the angle of tilt of the flap, on the direction of flow and on the pressure in the pipe-line with regard to potential cavitation.

Mt.j>O. M,>O, MH>O, M,>O 2 n -

I 13

Fig. 6.3 Schematic representation of a butterfly valve: I flap; 2 weight; 3 hydraulic slave cylinder; 4 oil pump; 5 dash-pot; 6 hydraulic valve.

A schematic representation of a butterfly valve is shown in Fig. 6.3. The valve tilts inside the pipe-line and its position is determined by the angle

of tilt av . The valve is closed for av = 0, and while it is open a, > 0. The valve induces a pressure loss in the pipe-line expressed by the relation

(6.14)

for O < o < l

For o = 0, the calculation uses the relation

Q = O (6.15)

For Q r O

Q = fo+ (av)

Q = fo-(Uv)

and for Q < 0

Dv is the diameter of the butterfly valve.

(6.16)

94

Butterfly valve

The torsional moment M, induced by the weight is expressed by the relation M G = fG(‘V) (6.17)

The relation depends on the geometry of the valve and on the mass of the weight. For the arrangement illustrated in Fig. 6.3, the curve for M, as a function of the angle of tilt is showh in Fig. 6.4.

Fig. 6.4 The moment M, of a butterfly valve as a function of the angle of tilt av of the flap. M,< 0

The moment MQ induced by the liquid in the pipe-line and acting on the valve is defined by the relation

(6.18)

(6.19)

0.1 .’

Fig. 6.5 Example of the relationship determining the moment, with which the liquid in a pipe-line acts on the flap of a butterfly valve: (a) for flow in the “positive” direction; (b) for flow in the “negative” direction.

I )

fa,(md) 0

95

Pressure devices

The dimensionless parameter of the functions fQ + and fQ - reads

P - Pv r = - IAPpl

(6.20)

being an analogy of the cavitation number. The symbol p denotes the lower of the pressure behind the valve or in front

of it, pv is the pressure in the junction at cavitation. The functions fQ +(ay r ) and fQ - (av, r ) depend on the geometry of the valve.

An example is shown in Fig. 6.5. The hydraulic slave cylinder induces a torsional moment MH acting on the

butterfly valve. During the closing of the valve (dav/dt < 0), when the oil pump is switched off and the hydraulic valve (6 in Fig. 6.3) is open this moment is defined by the relation

2

MH = fH('V) (2) (6.21)

Equation (6.21) follows from the assumed proportionality between the moment MH and the square of the discharge of oil through the dash-pot. An example of the shape of the function fH(aV) is presented in Fig. 6.6.

Fig. 6.6 Example of the relationship between the moment induced by the hydraulic slave cylinder at the closing of the valve, and the angle of tilt of the flap of a butterfly valve. f o l , (md)

During the opening of the valve (dav/dt 2 0) the hydraulic losses of oil are

MH= 0 (6.22)

The moment MI acting on the valve as a result of its inertia is defined by the relation

neglected and then

da$ dt2

M - -I,- r - (6.23)

where I, is the moment of inertia of the valve (including all attached parts) with respect to the axis of rotation. It is considered constant.

96

Butterfly valve

When the oil pump is switched off, the hydraulic valve opens. The motion of the butterfly valve is then determined by the equilibrium of the moments to which it is subjected.

(6.24)

When the oil pump is switched on, the hydraulic valve closes automatically and the butterfly valve opens at a velocity determined by the relation

M , + M Q + M , + M , = 0

(6.25)

I_ Fig. 6.7 Example of the relationship between the rate of opening of the flap by the oil pump, and the angle of tilt of the flap of a butterfly valve. 4 adrad)

The function fv(av) depends on the output of the oil pump and on the geometry of the butterfly valve. An example of the function is shown in Fig. 6.7.

If the opening velocity resulting from equation (6.24) is higher than that yielded by (6.25), the motion of the butterfly valve obeys equation (6.24) even when the hydraulic pump is switched on. The pump is switched off automatically when the angle of tilt of the flap attains the value

av = ap (6.26)

with the hydraulic valve remaining closed. This valve opens only when the pump is switched off as the result of outside intervention. Moreover, the motion of the butterfly valve is limited within the range

(6.27)

The parameters of the damping device which have to be submitted for the analysis are: the diameter Dv of the butterfly valve, the angle of tilt avo for the initial position of the butterfly valve; the angle of tilt ap at which the oil pump is switched off automatically; the angle of tilt aVmax for the maximum opening of the butterfly valve; the pressure pv in the junction at cavitation; the moment of inertia I, of the butterfly valve; the instants tp of switching the oil pump on and off through outside intervention; the functions f, + (av) and f, -(uv) which determine the pressure loss due to the butterfly valve; the function &(av) defining the moment induced by the weight; the functions fQ+(av, r) and fQ - (av, r) defining the moment induced by the liquid flowing in the pipe-line; the function fH(av) defining the moment due to the slave cylinder; the function

0 I av I aVmax

97

Pressure devices

fv(av) defining the closing velocity of the butterfly valve with the oil pump switched on.

The calculation yields the pressure p and the discharge Q at the end poind of the section; the degree of opening av of the butterfly valve; the velocity dav/dt of the motion of the butterfly valve; the variable o which determines the pressure loss; and the moment MQ acting on the butterfly valve and induced by the liquid in the pipe-line.

The “butterfly valve” pressure device may also be used to calculate the effect of simpler flap valves; in such a case, some effects are omitted from the calculation through a suitable choice of parameters. This procedure may also be used for more complex flap valves. Thus, for example, if the switching-on of the oil pump depends on a specific pressure being attained at a point along the pipe-line system, an initial calculation is carried out with the pump switched off so as to determine the instant at which the critical pressure has been reached. Since the point at which the pump is switched on is now known, a second, final, calculation can be performed.

6.7 Condenser

When analysing the unsteady flow in some cooling circuits, it does not suffice to substitute the “local loss” pressure device for the effect of a condenser. The liquid level in the chamber of a condenser may drop to such an extent during the operation, that the discharge through the condenser is also affected directly. The schematic representation of the condenser considered is shown in Fig. 6.8.

A condenser consists of two chambers interconnected by a number of horizontal pipes. The chambers are attached to the pipe-line by their bottom parts. When the pressure drops to that of the vapour pressure of the liquid, the level of the liquid in the chambers can fluctuate. If the upper parts of the chambers are arranged similar to air inlet valves, the level of the liquid in them can fluctuate, when atmospheric pressure has been reached.

I Fig. 6.8 Schematic representation of a condenser: I upstream chamber; 2 downstream chamber;

4 ___ .dr- - - - - - - -A

98

3 interconnecting pipes; 4 reference plane.

Condenser

The discharge through the interconnecting pipes is inflluenced by the difference in pressure in the two chambers and by the liquid levels. The mouths of the interconnecting pipes may emerge above the surface of the liquid. This eliminates any discharge of the liquid through the upper pipes and the pressures in the space above the liquid surfaces are balanced.

-- - 4J

2 2

Fig. 6.9 Schematic diagram for the calculation of the function of a condenser: I junctions; 2 integrated damping devices; 3 section of pipe-lines; 4 “condenser” pressure device.

’JU ’Jd

The height of the top interconnecting pipe is denoted h, and that of the bottom one h,. The pipes are assumed to be equally spaced between these two limiting heights. The bottoms of both the chambers, from which the heights h, and h,, as well as the heights of the liquid levels in the upstream h, and downstream chamber hd are measured, are considered to be identical. It is assumed that the pressure losses in the interconnecting pipes are proportional to the square of the discharge while the pressure losses in the chambers are negligible.

In the calculation, the schematic representation in Fig. 6.9 is used. The condenser is replaced by two junctions with integrated damping devices

interconnected by a section. The junction to which this section is attached at its upstream end bears the number JJu. It is attached without a pressure device. The section is attached at its downstream end to the junction bearing the number JJd through a condenser-type pressure device, in which all the pressure losses in the interconnecting pipes are concentrated. The section has a length 1 = 0 and a diameter D of such a magnitude that its cross-sectional area is equal to the sum of the cross-sectional areas of the interconnecting pipes. The section also has a coefficient of friction I = 0, while the value of the wave velocity is to be chosen very low and the initial discharge Qo is equal to that through the condenser. The introduced areas S, and SIi of the connected integrated damping devices have to be large. The calculation scheme then corresponds satisfactorily to the condenser under consideration (refer to Sect. 14.12).

The pressure loss in the “condenser” pressure device is determined through the equations which take into account the level of the liquid in the chambers of the integrated damping devices at the junctions J, , and .JJ&

99

Pressure devices

If the mouths of all the interconnecting pipes are submerged below the liquid surface, the pressure loss is defined by the relation for h, 2 h, and h, 2 h,

ApP = C - 8e IQIQ (6.28) D4n2

corresponding to the local pressure loss (refer to Sect. 6.3). The coefficient of loss [, however, is the same for both directions of flow. D is the diameter of the pipe-line section between the junctions J,, and JJd. Q is the discharge at the downstream end of this section.

The pressure losses for other levels of the liquid relative to the position of the interconnecting pipes are calculated in accordance with the following relations:

for h, 2 h, and h, < h, < h,

for h, 2 h, and h, I h,

for h, < h, < h, and h, < h, I h,

for h, < h, < h, and h, I h,

for h, < h, < h, and h, I h,

8e h, - ht ApP = C - IQI Q + es -

D4n2 2

for h, I h, and h, 5 h,

8e App = C - IQI Q + es D4n2

(6.29)

(6.30)

(6.31)

(6.32)

(6.33)

(6.34)

Conderser

for h, < h, < hd and hb < hd < h,

(6.35)

for h, I h, and h, < hd < h,

for h, 5 h, and hd 5 hb

Q = O (6.37)

Equations (6.28) to (6.37) have only an approximate validity. They are derived by assuming an approximate equality of the difference in pressure in front of and behind the "condenser" pressure device so that the difference in pressure between the junctions .IJu and JJd is equal to

(6.38)

If the mouths of the interconnecting pipes are submerged below the surface of the liquid, the pressures at their ends differ by the hydrostatic pressure corresponding to the difference in the heights of the pipes. In the calculation, however, a mean pressure is assumed for all pipes having their mouths below the surface of the liquid. If all the interconnecting mouths of the pipes on the inlet side are submerged below the surface of the liquid (refer to Fig. 6.8), then the following approximate relation holds between the pressure in the junction .IJu and the pressure pu at the inlet into the pipes:

'Pp = PJ (JJu) - PJ (JJd)

(6.39)

where h, is the height of the bottom of the integrated damping device above the reference plane.

I t all the outlet mouths of the interconnecting pipes are situated above the liquid surface, then the following relation holds between the pressure in the junction JJd and the pressure p d at the downstream end of the interconnecting pipes:

The magnitude of the pressure loss due to the discharge through the interconnecting pipes is given by the relation

PJ(JJd) = P d + eg(hd + hf) (6.40)

(6.41)

101

Pressure &vices

Substituting (6.39), (6.40) and (6.41) into (6.38), we obtain equation (6.30). Equations (6.28), (6.32), (6.34) and (6.36) are obtained in similar fashion. In equations (6.32) and (6.36), the effect of the reduction in the flow area due to the fact that some pipes are situated above the surface of the liquid, is taken into account. Equation (6.37) results from the fact that all pipes are situated above the liquid surface. Equations (6.29), (6.31), (6.33) and (6.35) were derived by using the method of linear interpolation.

The “condenser” pressure device also ensures a balancing of the air pressure in the chambers of the attached integrated damping devices, provided the top interconnecting pipe on the higher pressure side is above the level of the liquid. For this reason, it is necessary for the upper parts of both the chambers to have the configuration of an air inlet valve. Aeration or cavitation can be taken into account in either chamber. However, the same conditions must be considered in both chambers.

The parameters of the damping device which have to be submitted for the calculation are: the coefficient of loss 2 0, the height h, > h, > 0 of the topmost and lowermost interconnecting pipe above the bottom of the integrated damping devices, and the numbers J,, and JJd of the junctions at the upstream and downstream ends of the interconnecting pipes. Besides this, all the parameters of the integrated damping devices for these junctions (refer to Sect. 5.9) are required.

The calculation determines the pressure p and the discharge Q at the downstream end of the section at the “condenser”. Other values are obtained through further calculations for the attached integrated damping devices.

6.8 Pump

The pressure device denoted “pump” replaces a centrifugal pump driven by an electric motor in the calculation. It may also be used to represent other types of pumps, or other types of driving mechanism, provided their function can be described by the relations used. It is unsuitable, for example, for representing gear pumps and other similar kinds of pumps, where the discharge is practically independent of the difference in pressure (refer to Sect. 5.1 I ) .

In the calculation, the function of the pump is defined by the pressure and the moment characteristic for one rated speed of the pump [l 1, 35, 391. For other speeds, the characteristics are converted on the basis of similarity of flow. The function of the electric motor is defined by its characteristic. The effect of the inertia of the revolving parts of the entire assembly is taken into account.

The pressure characteristic of the pump reads

ApP = f p ( Q 3 n) (6.42)

I02

Pump

the torsional moment characteristic

M = fdQ, n ) (6.43)

and the characteristic for the torsional moment of the electric motor

The positive sense of the individual quantities is chosen so that, during usual steady state operation of the pump, the speed n > 0, the discharge through the pump Q > 0, the difference in pressure in front of and behind the pump (in the positive sense of the attached section) Ap, -= 0, the torsional moment of the pump M < 0 and the torsional moment of the electric motor ME > 0.

For the calculation, the characteristics (6.42) and (6.43) of the pump for n = n S , * n = - n , and n = 0 have to be known, where n, > 0 is the rated or any other constant speed.

An example of the characteristics used to describe the operation of the pump and the electric motor is presented in Fig. 6.10. The directions of the coordinate axes were chosen so that the shapes of the curves are in agreement with common usage.

The characteristics of the pump are converted to those at an arbitrary speed n by using the following relations which follow from the similarity of flow [39]. For n > 0

and for n < 0

For n = 0, the characteristics are assumed to take the form

for Q 2 0

AP, = Bp,Q2

M = BM+Q2

(6.45)

(6.46)

(6.47)

(6.48)

(6.49)

(6.50)

103

Pressure devices

C ) ME>O

0

Pump operation

ME=fEln)

c

nE n >O

and for Q < 0 Ap, = B,-QZ (6.51)

M = B M - Q 2 (6.52)

The symbols B p + ; B,-; EM+; BM- denote constants defining the characteristics of the pump for n = 0.

104

Pump

The speed of the pump is determined from the dynamic equilibrium of an assembly of machines [39]

dn dt

M + ME = 2x1- (6.53)

The symbol I denotes the moment of inertia of the revolving part of the

When the effect of inertia is neglected, that is, for I = 0 the relation assembly.

M + M E = O (6.54)

The torsional moment of the electric motor is defined by its characteristic

ME = 0 (6.55)

The case may be also considered, where the motor maintains a constant pump

is used instead of (6.53).

(6.44) and, when disconnected from the power network, by the relation

speed regardless of its loading. This is determined by the relation

n = nE (6.56)

where nE is the synchronous or other constant speed of the electric motor. The parameters of the pressure device to be submitted for the solution are: the

number N , 2 0 representing the number of times that the electric motor was disconnected and connected and the instants t , at which this occurred; the moment of inertia I 2 0 of the revolving parts of the assembly; the rated speed n, > 0 of the pump; the initial speed no of the pump; the constant speed nE of the electric motor; the characteristics f,(Q, n) and fM(Q, n) of the pump for n = n S’ n = - n, and n = 0, and the characteristic fE (n) of the electric motor. If the electric motor is disconnected from the network during the entire period over which the unsteady flow is observed, its characteristic need not be submitted. If the motor ensures a constant speed of the pump, neither the initial speed nor the moment characteristic of the pump have to be introduced.

The calculation determines the pressurep and the discharge Q at the end point of the section at the pump, the speed n and the moment A4 of the pump, and the moment M E of the electric motor.

For some pumps, the characteristics vary, for example, with a change in the setting of the guide blades. Such a pump should be treated in the analysis by the method described for a “turbine” pressure device. The same subprogram is used for the calculation for a pump and a turbine, hence all the possibilities described for a turbine are applicable to a pump. They are not discussed in the present section, because such cases are rather exceptional.

In the present method of calculation, a pump and a turbine differ merely in the sign convention used to introduce some quantities. In the calculation for

105

Pressurc~ devices

a pump, the positive senses of the quantities are chosen either as described above, or as described in the section dealing with turbines (refer to Sect. 6.9) and these sign conventions have to be strictly maintained.

6.9 Turbine with fixed characteristics

The problems of water hammer in systems with water turbines are very nume- rous. This results from the complexity of operation of a turbine, which moreover, may be influenced by the operation of the governor, the generator and the entire power network connected to it, as well as from strict demands on the accuracy of the calculation. Individual types of turbine differ greatly in design and in the method of control.

The solution described here may be applied to Francis turbines, propeller, Kaplan and other turbines, whose operation can be described with the aid of relations (6.42) to (6.56).

In such cases, App is the difference in pressure in front and behind the turbine, Q is the discharge through the turbine, n is' the speed of the turbine, M is the torsional moment of the turbine, ME is the torsional moment of the generator and n, is the rated speed of the turbine.

For turbines, in contradistinction to pumps, adjustment of the guide and action blades during unsteady flow is quite common. The characteristics are changed by changing the setting of the blades. Calculations with variable characteristics are discussed in Sections 6.10 and 6.1 1. The present section deals with the solution for invariable turbine characteristics which is similar to the solution for pumps. However, the sign convention used for some quantities is different. For a usual steady state of operation of a turbine, its speed is n > 0, the difference in pressure in front and behind the turbine (in the positive sense of the attached section) App > 0, the discharge through the turbine Q > 0, the torsional moment of the turbine M > 0 and the torsional moment of the generator ME c 0. When the turbine operates as a pump, we have for the steady state n < 0, App > 0, Q < 0, M > 0, ME < 0.

In the calculation, the pressure characteristic of the turbine is considered to have the form (6.42) and the moment characteristic the form (6.43). The calculation allows us also to use the characteristic of the generator in the form (6.44). However, when the generator is connected to the power network, a constant speed of the turbine is usually ensured in accordance with relation (6.56), while when the generator is disconnected, the moment of the generator is zero as given by (6.55).

The characteristics of the turbine have to be known for the speeds n = n,; n = -n, and n = 0. Equations (6.45) to (6.48) are employed, when the characteristics of the turbine have to be converted to those for another speed.

106

Turbine with jixed characteristics

Fig. 6.1 1 Characteristics of a turbine for one setting of the guide and action blades: (a) pressure characteristic; (b) moment characteristic.

An example of the turbine characteristics for one setting of the guide and action blades is presented in Fig. 6.1 1.

The speed of the turbine is defined on the basis of equation (6.53) or, neglecting the effect of inertia, equation (6.54).

In submitting the data for a turbine with a generator, the parameters are analogous to those for a pump with an electric motor (refer to Sect. 6.8). The same values are obtained.

Section 6.10 describes the calculation for the case where the setting of the guide and action blades changes in the course of the calculation. Instead of the characteristics of the actual turbine, one may introduce the characteristics of a geometrically similar turbine, its model. These characteristics are converted

107

Pressure devices

automatically to those of the actual turbine during the calculation. Therefore, it may be convenient to use the procedure described in Section 6.10 even for the case when the setting of the blades is not changed, if only the model characteristics are known and the conversion would have to be done manually.

The effect of a Pelton turbine may be treated as that of a “control valve” pressure device when calculating the flow in the penstock. This device describes the operation of the needle valve at the outlet of the pipe-line. The operation of other parts of the turbine do not affect the calculation of the flow.

6.10 Turbine with variable characteristics

In this section, the solution is discussed for a turbine, where the setting of the blades changes during the calculation of the unsteady flow and the adjustment regime of this setting is known in advance. The setting of the guide blades is defined by the parameter a and that of the action blades by the parameter B. The parameter may be represented by the angle of tilt of the blades, the magnitude of the gap between the blades, the extension of the piston rod of the slave cylinder controlling the blades, or by any other quantity unequivocally determining the position of the blades. If some of the blades are immovable, the value of the corresponding parameter is constant.

The adjustment regime is defined by the functions

(6.57)

(6.58)

For every setting of the blades, that is, for every combination of the parameters a and p, the characteristic of the turbine is different. The characteristics of the turbine for all the combinations of the parameters a and B, which may appear in the calculation, have to be known. In the method of calculation introduced, the characteristics are submitted only for some combinations, and for other combinations, the required values are determined by linear interpolation.

The turbine characteristics are frequently known for the model of the turbine and their conversion into those of the actual turbine is tedious. Therefore, the method of calculation is arranged so that it allows us to submit the model characteristics directly while their conversion is carried out automatically. The characteristics are submitted in the form

(6.59)

(6.60)

for the speeds 5 = fi,; f i = -iis and ii = 0. The bar denotes the values corresponding to the model.

108

Turbine with variable characteristics

The characteristics of the model for one pair of parameters 6, B have a curve similar to that of the characteristic of the turbine shown in Fig. 6.1 1.

The model characteristics are converted to the characteristics of the actual turbine at a given speed, on the basis of similarity of flow [39]. The following relations are employed for the conversion

for n > 0

for n < 0

(6.61)

(6.62)

(6.63)

for n = 0, Q 2 0

4 .

Ap = (2) ? E p + Q 2 IT

P

and for n = 0, Q < 0

App = (2T: - y B p - Q 2 -

(6.65)

(6.66)

(6.67)

(6.68)

where DT is the diameter of the turbine. The number of stages in a turbine usually is iT = i;. = 1. Its introduction into the calculation has a practical meaning only for multistage pumps.

109

Pressure devices

The following relations are used to obtain the parameters a, of the actual turbine, used in equations (6.57) and (6.58) from the parameters Cr, /J of the turbine model

for c, # 0

Cr = c,a

for c, = 0

Cr=a

(6.69)

(6.70)

for ca # 0

B = C B B (6.71)

for cp = 0

B = B (6.72)

The coefficients c, and cp express the rations of the corresponding parameters of both the model and the actual turbine. They may be used, for example, when the parameters depend on the dimensions of the turbine, when they are introduced in differing units, or when they approximately express the effect of a difference in the number of blades between the actual turbine and the model.

Equations (6.70) and (6.72) apply to those cases, when same for the actual turbine and the model, for example, blades.

2.

+-

0

the parameter is the the tilt angle of the

Fig. 6.12 Diagram illustrating method of interpola-

I L, & 4 - - p ; & 3

a tion between the characteristics. I I

The method of interpolation between the characteristics submitted is presented in Fig. 6.12. The combination of the parameters E, B, for which the characteristics are submitted, are marked with circlets. The point go, Po corresponds to the characteristic sought. The plane E , B is divided by lines parallel to the coordinate axes into four quadrants. One characteristic is chosen from each quadrant and the point E, B corresponding to it is one nearest to the point go, flo, that is, the characteristic for which the value

110


(6.73)

is the smallest. Quantities corresponding to the chosen characteristics are marked with sub-

scripts which correspond to the numbers of the quandrants. Points situated on the boundaries of the quadrants, are included in both quadrants. If no characteristic exists in a quadrant, a chosen characteristic is taken from another quadrant and the error is pointed out in the calculation.

The value of an arbitrary variable zo corresponding to the characteristic Eo, Po is determined, in a general case, with the aid of the interpolation formulas

- -

(Bo - B34) ,712 - z34 Z, = 234 + 812 - B34

where

Z2 - Zl P o - 61) Z12 = Z, + - a2 - E,

(Eo - 51) B2 - B l

B12 = Bl + - u2 - E l

B3 - B4 (io - i4) B34 = $4 + -

u3 - 64 (6.74)

Here, Zo to Z4 are the values of the variable Z corresponding to the characteristics

If the nominator in any one of equations (6.74) equals zero, this equation is El), Bo to E4, B4.

replaced by the corresponding one among those listed below

(6.75)

Actually, this is interpolation between corresponding characteristics.

111

Pressure devices

If the turbine is completely closed for some combination of the parameters ti, the relation

Q = O (6.76)

is used instead of the pressure characteristic (6.59) for the calculation of the unsteady flow.

I - I - 0' 0 0' (1

Fig. 6.13 Diagram illustrating method of interpolation between the characteristics: (a) none of the characteristics corresponds to a closed turbine; (b) one of the characteristics corresponds to a closed turbine.

In such a case, the interpolation equations (6.74) or (6.75) cannot be used for the values resulting from the pressure characteristics of the turbine. As long as both combinations of the parameters E, B, between which one has to interpolate, correspond to a closed pipe-line, the resulting combination corresponds to it as well. If only one combination, for example, til, PI corresponds to a closed turbine, and the second combination, ti2, p2, does not, App is calculated with the aid of the pressure characteristic given by

(6.77)

This is the pressure characteristic (6.59) for the parameters E2, P2, where the value Q (ti2 - El) / (Eo - til) is substituted for Q. A similar procedure is adopted, when another combination of parameters corresponds to a closed turbine. The method of interpolation between two Characteristics is schematically shown in Fig. 6.13.

The speed n of the turbine is constant while the generator is connected to the power network. It is defined by equation (6.56), where nE is the constant speed of the generator.

112

Turbine controlled by a governor

When the generator is disconnected, the speed of the turbine is determined from the dynamic equilibrium of the entire assembly with the aid of equation (6.53). The moment ME of the generator is then zero (6.55).

While the generator is connected, its moment characteristic is given by equation (6.44) and its speed by equation (6.53) or (6.54).

In submitting the data of the turbine, for a known adjustment of its blades, the following parameters have to be introduced: the number NN 2 0 representing the number of times that the generator and the network are disconnected and connected; the instants tN at which this takes place; the moment of inertia I 2 0 of the rotating parts of the assembly; the rated speed n, > 0 of the turbine (an approximate value suffices); the initial speed no; the diameter DT > 0 of the turbine; the number of stages iT 2 1 of the turbine; the coefficients c,, cB which determine the ratios of the parameters of the model and the turbine; the constant speed nE of the generator; the functions f,(t) and fB(t) which determine the setting of the blades, and the characteristic fE (n) of the generator. Other parameters concern the model of the turbine: the diameter DT 2 0 of the model; the number of stages i;- 2 I ; the rated or other constant speed ii, > 0, and a system of characteristics for different combinations of the parameters a, p. Each of these combinations includes the functions 4 (Q, i i) and 3M (Q, i i ) for ii = ii,,

The calculation yields the pressure p and the discharge Q at the end point of the section which includes the turbine, the speed n, the moment of the turbine M and the moment of the generator ME.

The procedure described can also be applied to turbines with fixed characteristics as well as pumps.

If the adjustment regime of the turbine blades is not known in advance, but is controlled by a considered type of governor, the calculation may be carried out as in Sect. 6.1 I .

n = -is, n = 0.

6.11 Turbine controlled by a governor

There are many kinds of governors for the control of turbines. For large assemblies, electrohydraulic governors are used most frequently. One of them, developed in Czechoslovakia [36], is considered in the present chapter.

The calculation for a turbine controlled by a governor [28, 361 differs from that of a turbine with variable characteristics introduced in Sect. 6.10 in the following respects:

(a) The moment of inertia Iconsidered in Eqs. (6.53) need not be constant. During the time when the generator is connected to the power network, this moment is defined by the relation

(6.78)

113

Pressure devices

and while it is disconnected by the relation

I = IT (6.79)

The value IT is the moment of inertia of the revolving parts of the turbine and the generator, while the function f,(t) expresses the effect of the power network (the effect of the machines connected to the network).

p-! Turbine

I

I I I Governor

I I I * lu, ! Action I blades

L----- ----- 1 - A _ - - _ 1 I I I IH ;A%

Fig. 6.14 Block diagram of a turbine speed governor.

(b) The torsional moment ME of the generator may depend on the speed of the turbine, but also on the connected power network. The effect of the network may change with time. The value of the torsional moment employed in Eqs. (6.53) and (6.54) is defined by the relation

(6.80)

The functions fn(t) and fc(t) express the effect of the connected power network (the effect of self-regulation and the moment of loading). When the generator is disconnected ME = 0.

(c) The control rbgime of the turbine blades is not submitted with the aid of equations (6.57) and (6.58), but it is defined by calculating the function of the governor of the turbine speed.

The block diagram of the governor considered is shown in Fig. 6.14. The governor determines the setting a of the turbine guide blades, and sometimes also the setting /? of the action blades, on the basis of the speed n of the turbine, of the surplus pressure App on the turbine, and on the basis of a number of parameters submitted. Some of these are introduced in the diagram.

ME = (n - nE)fn(t) -k fdt)

114


If the governor controls only the setting of the guide blades (refer to Subsect. 6.1 l . l ) , only that part of the diagram drawn in full lines is considered. If it also controls the setting of the action blades (refer to Subsect. 6.1 1.2), the part of the diagram drawn by dashed lines has to be considered as well.

6.11.1 Guide blades

The calculation of the function of the governor is determined from the speed n( t ) of the turbine. In the frequency converter unit (Fig. 6.14) a voltage Uo(t) is generated as defined by the equation

(6.81)

where C,is a time constant andfhn) is the static characteristic of the frequency converter.

The voltage U , ( t ) is found from the voltage Uo(t ) using the relation

u, = uo + fr(t) (6.82)

where the function &(t ) describes the voltage determining the required speed. In the electric-speed-controller unit, the voltage U2(t ) is determined from the

voltages Uo( t ) and U , ( t ) with the aid of the equation

fb('0) -.- d U 2 + c u = f ( u b 0 )' d . - d2U1 + f ( U ) dU1 + U , (6.83) C, dt P 2 C, dt2 O d t

Here the symbols C,, Cp and c d represent the controller gain, the standing static characteristic of the controller, and the time derivative constant of the controller, respectively; fb(U0) is the time constant of the elastic feed-back. Its value may depend on the voltage U,.

The voltage U2 resulting from Eq. (6.83) is limited to satisfy the condition

(6.84)

The voltage U,( t ) is determined from U2 and the voltage from the power

(6.85)

In calculating the voltage U,, the values of U, are considered to be within the interval

'23min '2 '23max (6.86)

U2min 5 '2 5 U2max

where U2min and U,,,, are the extreme values of U,.

regulator expressed by the function f p ( t ) as

u, = u, + f p ( t )

115

Prrssrrre devices

In the opening-limiter unit, the voltage U,(t) is determined from U , and the function fmax(t):

for U, < 0

for O I U, I fmax(t)

for u, ’ fm,,(t)

u, = 0

u, = u,

U, = fmax(t) (6.87)

The function fmax(f) determines the maximum value of U,. In the hydraulic-amplifier-of-the-guide-wheel unit, the extension y,(t) of the

piston rod of the slave cylinder controlling the guide blades is calculated from U, and the excess pressure Ap,(t) on the turbine:

(6.88)

This relation defines the feedback effect. The subscript ct denotes the values appertaining to the guide blades. The symbols C,,, C,,, C,, represent the feedback constant, gain of the

distribution slide valve and the constant of proportional feedback of the hydraulic amplifier, respectively. The function x,(t) defines the position of the slide valve of the hydraulic amplifier of the guide wheel.

The position of the slide valve is mechanically limited so that

Xmina I ‘ a I Xmaxa (6.89)

where xmina and x,,,, define the extreme positions of the slide valve. Apart from the above, one also assumes the validity of the equation of

dynamic equilibrium of the forces acting on the piston of the slave cylinder for

in the form

dya - F,(y,, Ap,) + F, sign - = 0 dt

(6.90)

116


and for

in the form

a = o dY dt

(6.91)

The first term in Eq. (6.90) expresses the effect induced by the inertia of the guide blades, of the piston of the slave cylinder and of the motion of the mechanism which connects them; ma 2 0 has the meaning of mass; Foa(x4() is the force through which oil acts on the piston of the slave cylinder when it is stationary. Its magnitude depends on the position of the slide valve.

The third term in Eq. (6.90) expresses the reduction of the forcefoa(xa), during movement of the piston of the servomotor due to the pressure loss due to the discharge of oil through the slide valve and in the distribution pipes. It is assumed that this reduction is proportional to the square of the discharge of oil and its dependence on the position x, of the slide valve is defined by the function a,(x,) for which we have

0 I Oa(Xa) I 1 (6.92)

The symbol C,, denotes a constant which expresses the effect of the geometry of the slide valve and of the slave cylinder; F,, is a force acting on the piston of the slave cylinder induced by the pressure of water on the turbine guide blades. It is assumed that its value is expressed by the relation

(6.93)

The function f,(y,, App) depends on the geometry of the turbine and the governing mechanism of the guide blades. Its values are a function of the extension of the piston rod of the slave cylinder and of the surplus pressure on the turbine.

The value F , 2 0 is a force acting on the piston of the slave cylinder due to the friction of the piston and other parts of the governing mechanism. It is assumed that its values is constant as long as the piston moves and that it acts against the direction of the motion. As long as the sum of other forces expressed by their absolute values is less than the force F,, the piston is stationary; this is expressed by Eq. (6.91).

The velocity of the motion resulting from equations (6.88) to (6.91) is limited so that

(6.94)

117

Pressure devices

or

fdy - aha) dt d y a (6.95)

Here,fdy + a(yg) andfdy - g(ya ) are functions defining the maximum velocity of the piston of the slave cylinder during the closing and opening of the guide blades. Their value depends on the extension ya( t ) of the slave cylinder piston rod.

Besides this, the extension cannot exceed its extreme values ymina and ymaxa so that

(6.96) Ymina 5 Y a 5 Ymaxa

In the mechanical-transmission-of-the-guide-wheel unit (Fig. 6.14) the parameter a(t) of the position of the guide blades is determined. It is a function of the extension of the piston rod of the slave cylinder

a = f a b a ) (6.97)

The functionfa(ya) depends on the geometrical arrangement of the control of the guide wheel blades and on the nature of the parameter a(t).

The data which have to be submitted for the analysis of a governor- controlled turbine (when only the position of the guide blades is variable) comprise the following quantities: N , 2 0; t , ; n, > 0; no; DT > 0; i, 2 1; ca; c8; n,; D, > 0; iT 2 1; fi , > 0. Their meaning is described in Sect. 6.10. Additional information required includes the values I , 2 0; Cs; C,; C,; cd;

-

u 2 min ; U Z m a x ; u 2 3 min ; u 2 3 max ; C F a ; C s a ; ‘,a ; Xmin a ; x m a x a ; ma ; C n a ; ‘ f a ; Ymina ; Ymax a and the functions fi(t); fn(t); fc(t); fAt); f,(t); fb(U0); F O a ( x a ) ; fp(t); a a ( x a ) ;

f , a ( Y a , ‘Pp); f d y + g(Ya); f d y - a(Ya); fa (Yg); fmax( t )* Their meaning has been described in the preceding subsections. Also required are, the initial position xOa of the slide valve of the hydraulic amplifier of the guide wheel, and the initial surplus pressure Ap,, on the turbine. Moreover, one has to submit the system of characteristics of the turbine model for different combinations of the parameters 5 and p.

The characteristics have to be introduced for such combinations of parameters, which allow us to find by interpolation (refer to Sect. 6.10) the characteristics for all the combinations of parameters, that may appear in the calculation.

All the values and functions introduced above.are the parameters of the pressure device “governor-controlled turbine”.

The calculation yields the pressure p, the discharge Q at the end of the pipe-line section in which the turbine is located, the turbine speed n, the torsional moment M of the turbine, the torsional moment ME of the generator, and the values U,; U,; U , ; U , ; U, ; Ap,; xa; y a ; a.

118


6.11.2 Action blades

When the governor controls the adjustment of both the guide and action blades of the turbine, the whole of the block diagram shown in Fig. 6.14 has to be considered.

The part of the diagram represented by full lines, has already been described in the preceeding subsection (6.1 1.1).

In the correction unit, the voltage U,( t ) is defined as a function of the voltage U4(t) and the head H (which is considered constant throughout the calculation). The voltage U, is calculated using the relation

U , = f H ( U 4 , H ) (6.98)

One may also consider a case, where the extension y , ( t ) of the slave cylinder represents the input into the correction block, instead of the voltage U4(t). Then equation (6.98) is replaced by

u, = fH(Ya, H) (6.99)

The following calculation of the adjustment of the action blades is similar to that of the guide blades. In the hydraulic-amplifier-of-the-action-wheel unit, the voltage U,( t ) is used as input, in contradiction to the hydraulic-amplifier-of-the- guide wheel. The relation

(6.100)

is substituted for equation (6.88) and in equations (6.89) to (6.96) the subscript B which denotes the values appertaining to the action blades, is substituted for the subscript 01.

The output is represented by the extension yp(t) of the slave cylinder piston rod which controls the action blades.

In the mechanical-transmission-of-the-action-wheel unit, the parameter p(t) of the setting of the action blades is defined with the aid of the relation

P = fp(Ya) (6.101)

Apart from all the parameters introduced in Subsect. 6.11.1, the following additional values have to be submitted for the calculation H ; CFp; Csp; CGp; xming; xmaxB; mp; Cop; Ffp; yminp; Y , , , ~ ; xos and the functions Fop; gp(xp);

parameters of the pressure device “governor-controlled turbine”. Their meaning corresponds to similar parameters bearing the subscript 01 for the guide blades, introduced in Subsect. 6.1 1.1.

The calculation yields the values U,, x p , yp, 8, in addition to the values introduced in Subsect. 6.1 I . 1.

fes(Yp, Ap,); fdy + p(Yp); fdy - p(Yp); fp(Yp); fH( U4 3 H ) . All these values represent the

119

7 Calculation of water hammer

7.1 Computer application

A set of programs suitable for use on an IBM PC/XT or any compatible computer was written for calculating water hammer using the method introduced in this book.

The WTHM program is used for the calculation; it is written in the IBM FORTRAN 2.0 language, described in Chapter 13 and the listing of subprograms for the damping and pressure devices is presented in Appendix A. On the WTHM diskette, the program is presented in compiled form under the name WTHM.EXE.

The WTHD program was worked out in order to create and check the input data file needed for the calculation using the WTHM program. The WTHD program is written in TURBOPASCAL 5.0 language, it is described in Chap- ter 10. On the WTHM diskette, it is presented under the name WTHD.EXE in compiled form.

The last program of the set is the WTHG program which is used for the graphic processing of the numerical results calculated with the aid of the WTHM program. The WTHM program is written in TURBOPASCAL 5.0 language, it is described in Section 11.4. On the WTHM diskette, it is presented under the name WTHG.EXE in compiled form.

Apart from these programs, the WTHM diskette also contains the input data for the examples introduced in Chapter 14.

In the following description of the calculation, we have considered a computer configuration cantaining one 5 %" disk drive, a hard disk, a screen, a keyboard and a matrix printer for 80 characters per line, permitting printing on continuous feed paper.

The computer programs use the MS DOS 3.2 operating system.

7.2 Preparation

The first step in the calculation of water hammer is the submission of the following data: the layout of the pipe-line system; the initial state of flow; the initial state and the adjustment regime of all the devices forming part of the system.

120

Input data

The layout of the pipe-line system has to be schematized, that is, it has to be replaced by a hypothetical system composed of sections, junctions and damping and pressure devices (refer to Sect. 3.3).

The damping and pressure devices included in the computer program have already been discussed in Chapters 5 and 6.

A positive direction has to be chosen for each section in the schematized system. Although any direction may be chosen as positive, the choice does affect the sign of the discharge values, and consequently, in the case of some types of pressure devices, it also affects the data submitted. Particular attention is to be given to the butterfly valve (refer to Sect. 6.6), the pump (refer to Sect. 6.8) and the turbine (refer to Sects. 6.9 to 6.1 I). It is assumed in the quoted descriptions that the liquid flows in the positive direction through an operating butterfly valve, pump and turbine. It is convenient to adjust the choise of the positive direction in the sections accordingly. A reverse choice of the positive direction would necessitate changes in the form of some of the functions submitted.

The sections and the junctions have to be numbered. The sections are given natural numbers, 1, 2,. . . ., N , , where N < 50 is the total number of sections. No number may be omitted or repeated. Similarly, the junctions are numbered 1, 2,. . . ., N, where N < 50 is the total number of junctions. The sequence in which the sections and junctions are numbered is arbitrary but does affect the order in which the results are printed.

Next, a reference plane has to be chosen. This is a horizontal plane, common to the entire pipe-line system. All pressures are converted relative to this plane according to equation (2.3). For some devices though, as on exception, it is necessary to use absolute pressures, not converted to the level of the reference plane. Any horizontal plane may be chosen as the reference plane. It is convenient, however, to choose a plane passing below the entire system or at least through its lower part. Otherwise, some of the converted pressures may be negative; this impairs a clear arrangement of the input data and the results.

A zero pressure also has to be chosen. As a rule, one chooses the atmospheric pressure, but a pressure equal to absolute zero or any other value may be used instead.

Various methods of schematization and of further proceeding with the calculation are evident from the examples in Chapter 14.

S T

J T

7.3 Input data

Prior to performing the calculation, the input file has to be prepared. It constains all the data needed for the calculation. In the calculation proper, only the name of the input file is submitted together with the data determining the required type of output of the calculated results (refer to Sect. 7.4).

121

Calculation of water hammer

The layout of the input file is described in detail in Chapter 9. The input data are divided into several groups. In the first group, the iden-

tification number Num of the calculation and its name Title are presented. They merely denote the calculation and have no meaning as far as the solution itself is concerned.

The next group is formed by the data relating to the sections of the pipe-line. For each section, one submits its number J s , the number J,, of the junction to which the section is attached by its upstream end and the number JJd of the junction to which the section is attached by its downstream end.

This determines the interconnection of the pipe-line sections in the system and the positive direction of the sections. The system may be ramified in various ways and may even form closed circuits.

A section may be attached by both its upstream and downstream ends to a junction through a pressure device. The nature of pressure device is determined by its number J for which values are presented in Section 9.2.

One pipe-line system may contain several pressure devices of the same kind, but with different parameters. A set of parameters of a pressure device is defined by the number J p .

One may choose J p = 0, I , . . . ,9, hence a system may at the most contain ten pressure devices of the same kind and with different parameters.

In case of need, other values J p may be chosen, but always such ones that would exclude mutual conflict between the parameters submitted (refer to Chapter 9). If all the parameters of the pressure devices of the same type are identical, the same number J p may be assigned to them. In such a case the parameters are submitted only once. The number of such identical devices in the system is not limited. A turbine equipped with a governor (refer to Sect. 6.1 1) is an exception. Each turbine of this kind in a system must have a different number J,,.

The numbers J and J p are submitted for the upstream and downstream ends of a section.

Besides this, the length I of the section, its internal diameter D, the coefficient of friction A, the wave velocity a and the initial discharge Q, are submitted for each section. The length of the section submitted is its actual length; its conversion to whole multiples of the interval Ax and the corresponding modification of the coefficient 3, (refer to Sect. 7.5) is carried out automatically in the course of the calculation. The determination of the values D, 3, and a is described in Sects. 4.2, 4.4 and 4.1.

It is assumed that at the initial state, that is, at instant t = 0, the discharge is constant in each section and equal to Qo . The value of Q, for each section have to be determined beforehand, for example, by manual calculation or any other method, and then submitted. One may also determine the values Q, by means of the abridged calculation of the steady state described in Chapter 8. In that case the values Q, need no be submitted.

122

Input data

The next set of data concerns the junctions. For each junction, one submits its number J J , the number Jdefining the type of damping device attached to it, the number J p defining the set of parameters of this damping device and the initial pressure po in the junction. These numbers are described in Sect. 9.3. The set of parameters may be denoted by the numbers J p = 0, 1,. .., 9. The same limitations that apply to the numbers of the pressure devices also apply to the numbers of the damping devices.

The number of identical devices with the same parameters is not limited and the maximum number of identical devices with different parameters is 10; this could, however, be increased, if necessary (refer to Chapter 9).

The pressures po have to be determined in advance, for example, by manual calculation or with the aid of the abridged calculation of the steady state as described in Chapter 8. In the latter case the pressurespo need not be submitted, except for the “constant pressure” damping device.

The parameters of the damping and the pressure devices, which are also included among the input data, have been described in Chapters 5 and 6.

The next group of parameters defines the nature of the calculation. These parameters are: the total number of sections Ns; the total number of junctions N,; the number 2 denoting the type of calculation; the time interval At of calculation; the time interval of printing At,,., for the main output; the longitudinal interval of printing Axm for the main output; the wave velocity a, for calculating the steady state, and the maximum value t,,, of the computing time.

One input file may contain the data for one or several consecutive calculations. Each of the sections may concern the calculation of water hammer or the steady state (refer to Chapter 8). An iteration procedure may or may not be employed in solving the equations for the junctions (refer to Sect. 7.11). The number Z indicates which of the procedures mentioned above are to be used. Its values are presented in Section 9.5.

The choice of the time interval of calculation At may significantly affect the accuracy of the calculation and its duration. It determines the grid of characteristics for the calculation (refer to Fig. 7.1). A suitable choice may reduce and, in some cases, fully, eliminate the effect of inaccuracies resulting from the rounding-off of the lengths of the sections to the multiples of A x (refer to Sect. 7.5). The choice of A t also has to be adjusted with regard to the damping and pressure devices forming part of the system. It has to be sufficiently small for the calculation to follow the function of these devices satisfactorily (refer to Sects. 7.8, 7.9). No enequivocal rule can be formulated for determing the value of At. In principle, a reduction of At leads to a more accurate calculation and to an increase in the use of computer time. This is, however, also limited by the program which, in the version introduced here, allows us to divide the pipe-line system into a maximum of 2000 parts of length Ax. The program automatically checks whether this condition is satisfied. Reducing the value At need not always

123


lead to more accurate results, since the assumptions in deriving the equations employed (refer to Chapter 2) may not be satisfied exactly whilst the accuracy of the input data is also limited.

Both the time and length intervals of printing do not affect the calculation proper. They only affect the main output of the results (refer to Sect. 11.1). If the value At, is not submitted, the results for all 0 I t I t,,, are presented for time intervals At. Otherwise they are presented for intervals At,. In all cases, the results are presented for t = 0 and t = t,,,. The value At,,, may be changed several times within one calculation (refer to Sect. 9.4). The time data are rounded-off to whole multiples of At.

If the value Ax, is not submitted, the pressure and the discharge in the results are presented for intervals of whole multiples of Ax from the upstream end of a section. Otherwise, they are presented for intervals of multiples of Ax,. The value Axm is common to all sections. The lengths are rounded-off to whole multiples of Ax for each section. Both the pressure and the discharge are presented for any Ax, at the upstream and downstream ends of the section.

In practice, values for At and Ax usually have to be submitted in order to "! m. reduce the number of data in the main output.

The wave velocity a, has a meaning only for the abridged calculation of the steady state (refer to Chapter 8). Otherwise, it need not be submitted.

The calculation starts with the initial state, when t = 0 and it ends when t = tmax.

The next group of data does not affect the calculation proper of water hammer. It defines the values and the manner in which they will be printed in the graphical (refer to Sect. 11.2) and numerical (refer to Sect. 1 1.3) outputs. If neither of these two outputs is required, this group of data need not be introduced at all. It contains the numbers L,; L, and L, which determine the variables included in the graphical and numerical outputs; the denotation Murk of the variables; the verbal description Text of the variables; the limiting values Min and M a x of the variables in the graphical output; the denotations Gr, Nu of the changes in the time interval of printing in the graphical and numerical outputs; the time tgn at which a change in the time interval occurs and its new value Atg.

The number L, is that of the section Js or of the junction J,, which relates to the given variable. For a damping device, one introduces the number of the junction, to which the device is attached. For a pressure device, the number of the section attached to a junction through this device is introduced. In the case of a governor-controlled turbine (refer to Sect. 6.1 I), it may be another number. The numbers L,, L,, L, denoting the individual variables, are listed in Table 9.10. The number L, gives the distance of the point considered from the upstream end of a section for the variables pressure and discharge. It is introduced in multiples of the longitudinal interval Ax for the modified length I,

124

Data input and output of the results

of the pipe-line (refer to Sect. 7.5). Thus, L, = 0 correponds to the upstream end of a pipe-line section whilst for the downstream end L, = IJAx, and for intermediate points, 0 < L, < 1JAx. L, is always an integer. For variables relating to junctions and damping devices, the value L, is not introduced. For variables relating to pressure devices, L, = 1 for a pressure device located at the upstream end of the pipe-line, and L, = 2 for one located at its downstream end. The meaning of L, for a governor-controlled turbine is different in some cases.

Mark represents any chosen symbol. It denotes the values corresponding to the variable in the graphical and numerical calculations. In the graphical output, this symbol also denotes the curve of the variable as a function of time. A selected verbal description of the variable denoted Text, is introduced in the numerical and graphical outputs.

Min and Max are the values of the variable corresponding to the left-hand and right-hand margins of the graph in the graphical output. They define the scale used for plotting the variable as well as the range of the plotted values (refer to Sect. 11.2).

In the graphical and numerical outputs, the values of a variable are presented for a time interval At. If this interval is to be changed, one has to-submit t , defining the instant at which this change is to occur, and AtB? giving the new value of the time interval. If the change concerns the graphical output only, Gr = 1, Nu = 0; if it concerns only the numerical output, Nu = 1, Gr = 0; in other cases, both Gr and Nu = 1.

The last information to be entered consists of the secondary name of the calculation denoted Subtitle. It is introduced in both the graphical and numerical outputs.

7.4 Data input and output of the results

After starting the calculation with the aid of the WTHM program (refer to Chapter I3), the basic information relating to the program appears on the screen together with a request to submit the name of the input data file. This has to be done via the keyboard. The format of the name has to correspond to the requirements of the operating system employed. The submission of the name is terminated by pressing the RETURN key. If the file with the introduced name is not found, the calculation is terminated by an error message. If the name of the input file has not been submitted, a request as to whether the calculation is to be continued appears after pressing the -RETURN key. After pressing the Y (yes) key, the request for the name of the input file reappears. Otherwise, the calculation is terminated.

When one of the possible answers introduced after the request appears in

125


brackets, this means that pressing the RETURN key suffices for its selection. When the input file with introduced name is found, a table appears on the

screen giving the numbers denoting the inidividual output devices and their combinations, and the request to submit the number of the main output.

The output of the results of the calculation may be realized in three forms: the main, the graphical and the numerical outputs. The main output (refer to Sect. 11.1) contains a complete listing of the input data for the calculation; all the computed values of the pressure and discharge in the pipe-line; nearly all the computed values of the variables describing the state of the damping and pressure devices; the maximum and minimum values of some of these variables; the maximum and minimum values of the pressures in the individual pipe-line sections and information on the accuracy of the calculated results. The results are presented in a numerical form and arranged in tables. The number of results presented in the main output can be very extensive and has to be restricted in most cases.

In the graphical output (refer to Sect. 11.2), the values of some selected variables are presented. They are shown in a graphical form so that the position of a symbol along a line corresponds to the value of the corresponding variable. Each line appertains to a specific time. All the computed values of the variables introduced are plotted graphically, but their number may be restricted. The graphical output allows us to obtain an immediate survey of the variation with time of the selected variables and thus also a rough survey of the progress of the entire calculation. From the graphical output, the values of the variables can be read off only approximately. Their precise values have to be obtained from either the numerical or the main output.

In the numerical output (refer to Sect. 11.3), only the values of selected variables are presented, as for the graphical output. The numerical output is convenient for determining accurately the values found only approximately from the graphical output and for further processing of the results.

Each of the three outputs (main, graphical and numerical) may be directed to any combination of the following output devices: the screen, the hard disc or the printer. The use of the various output devices and their combinations is denoted by the numbers 0 to 7. First, one is prompted to direct the main output by entering one of these numbers (and then to press RETURN). If no number is specified (only RETURN is pressed), the default is 0 and the main output will not be written anywhere. If one chooses the numbers 2, 4, 6 or 7 (the main output will in all cases be entered on the hard disk), one is prompted to enter the name of the output file. If no name is specified (only RETURN is pressed), the default is MOUTWTH.

Similarly, one is prompted to choose, through successive requests on the screen, the desired combination of output devices for the graphical and numerical outputs. It may be necessary to submit the names of the output files. The

126

Solution for the sections

name of GOUTWTH is pre-defined for the graphical output and NOUTWTH for the numerical output.

In choosing the output devices and the names of the output files, care should be taken that main, graphical and numerical outputs are not entered simultaneously on the screen, the printer or stored into the same file on the disk.

In practice, it is convenient to store the main output on the disk (choice 2), to enter the graphical output on the screen and to store it on the disk (choice 4), to store the numerical output on the disk (choice 2) and to use the pre-defined names of the output files. The progress of the calculation can then be followed on the screen with the aid of the graphical ouptut, and the results may, in addition, be printed when necessary.

After having chosen the output devices, a request as to whether the choice should be changed appears on the screen. If Y (yes) and RETURN are pressed, the entire above procedure is repeated. If RETURN is pressed, the calculation is started.

7.5 Solution for the sections Let us substitute

XD2 Q(x, t ) = -u(x, t ) 4

where Q(x, t) is the discharge of the liquid through a cross-section having the coordinates x at an instant t; the discharge is positive for a flow in the positive direction of the coordinate x. We obtain the system of Eqs. (2.1), (2.2) converted to the form

4e aQ 8Je aP -- + -IQI Q + xD2 at x2Ds

0 ( 7 4

The system of equations is solved in the difference form by the method of

Along the characteristics characteristics [46].

x - at = const (7.4)

the following relation holds:

127


Similarly, along the characteristics

x + at = const

the following relation applies:

T

Fig. 7.1 Grid of characteristics for the solution of a pipe-line system.

The grid of characteristics for solving one section of the pipe-line is shown in Fig. 7. I . The grid is defined by the time interval At of the calculation, which is common to all sections, by the length 1 of the section and the velocity a. This also determines the longitudinal interval of the calculation for the section

Ax = aAt (7.8) Bearing in mind that the length of the section need not be exactly a whole

multiple of the values Ax, we consider a modified section length I , , which is the nearest whole positive multiple of Ax. By analogy, instead of the actual coefficient of friction A, the modified coefficient of friction L, is used, in order to eliminate the effect of the rounding-off of the section length on the total pressure loss along the section. The conversion is effected using the relation

1 1,=1-

4 (7.9)

The solution of the analysis of the section starts from the values p and Q for t = 0, determined by the initial conditions. It is assumed that the initial flow in the individual sections is steady, hence

_ - a p _ - - aQ - 0 at at

(7.10)

128

Solution for he junctions

It follows from equations (7.2) and (7.3) for the initial values that

(7.1 1)

Q(x, 0) = Qo (7.12)

where Qo is the discharge of the liquid through the pipe-line section in the initial state. Then the values p and Q are calculated successively at all points of intersection of the characteristics marked out from equations (7.5) and (7.7) converted to the difference form.

Thus, for example, to calculate the pressure pc and the discharge Qc at point C on the basis of the pressures pA , p B and the discharge QA , QB at points A and B, the following equations are used:

QA) + $ I Q ~ I Q~ 2 Ax = 0 X D

Q ~ ) - - I Q B I Q B - ~ - = 8 k Q Ax 0 x2D5

(7.13)

(7.14)

Only one equation is available for the calculation of the unknown values of p and Q at each of the two end points of a section. Thus, for example, for points E and G (Fig. 7.1), we have equations analogous to equations (7.14) and (7.13), which determine the relationship with the values at points D and E The other conditions necessary to calculate the values at E and G are provided by the solution for the junctions to which the section is attached (refer to Sect. 7.6).

7.6 Solution for the junctions

It is assumed in the calculation that the effect of all damping devices is described by a function in the form

PJ = ~ ( Q J ) (7.17)

129

Calculation of water hummer

or by Qj = const (7.18)

where pJ ( t ) is the pressure in the junction and Qj ( t ) is the discharge of the liquid from the junction into the damping device. The function fJ may have various forms. It may be different in each step of the calculation depending on the time and other parameters. This complicates the numerical solution of the calculation for the junctions. For this reason, instead of equation (7.17), we consider the linear relation

(7.19)

obtained after expanding the function fJ into a Taylor series at the point QJ = Q; and neglecting the non-linear terms. Q; is the value of Qj as determined in the preceding step. The coefficients Rd and s d are defined by the relations

QJ = RdpJ + sd

(7.20)

and

(7.21)

If

(7.22)

we now use, instead of the equation (7.19), the relation

PJ = Pc (7.23)

where pc represents a constant pressure. The error caused by linearization is eliminated with the aid of iteration. Here,

the calculation for each junction is repeated for the same time t and for series of new values of Qj , until a solution is reached which corresponds to that of the non-linearized relation (7.17) with satisfactory accuracy.

Further, it is also assumed that the effect of all pressure devices is described by a function in the form

AP, = f ( Q ) or by the relation

Q = O

(7.24)

(7.25)

130

Solution for the junctions

where Q ( t ) is the discharge through the pipe-line at the point where the pressure device is located.

The function fmay have various forms and may be different for each step of the calculation.

-.Upstream end of section m q Q y w r e equipment

Section Jc=2 %itive direction of

Downstream end of%&on \- section . Fig. 7.2 Schematic representation of a junction.

The calculation for the junctions is simplified when, instead of equation (7.24),

(7.26)

we use the linear relation

AP, = RpQ + S ,

where the coefficients R, and S , are defined by the relations

and

S , = f ( Q ' ) - Q'-

(7.27)

(7.28)

where Q' is the value of Q as determined in the preceding step. Equation (7.26) was obtained by expanding it into a Taylor series and neglecting the non-linear terms.

The errors due to the linearization of equation (7.24) are eliminated with the aid of iteration, in a similar manner as used for equation (7.17).

The effect of the damping and pressure devices may also depend on other variables, besides the discharge. These variables are considered to remain constant during each calculation step and to change abruptly at intervals At i.e. between steps. The inaccuracy introduced in the calculation in this way can be limited by reducing the interval Af.

A junction with a damping device and three attached sections is shown in Fig. 7.2.

Three equations analogous to equation (7.13) or (7.14) and three equations similar to (7.26) are available in the general case for calculating the pressure p ,

131

Culculution of water hummer

and the discharge QJ, the pressures p , , p2 and the discharges Q1, Q2 at the downstream ends of sections 1, 2, and the pressure p3 and the discharge Q3 at the upstream end of section 3. Besides this, there is also one equation corresponding to (7.19) as well as the condition that the net quantity of liquid flowing into the junction equals zero. These equations form a system of eight linear equations with eight unknowns; they may be written in the form

P I - PJ = Rp,Qi + Sp,

PZ - PJ = Rp2Q2 + Sp2

PJ - ~3 = Rp3Q3 + Sp,

QJ = R ~ P J + s d

Qi + Q2 - Q3 - QJ = 0 (7.29)

where the numerical subscripts denote the numbers of the sections. The values pi and Qi (i = 1, 2, 3) have the meaning of the pressure and the

discharge at the upstream or downstream end of the section at time t (they correspond to points E and G in Fig. 7.1), pf and Qf (i = 1, 2, 3) denote the pressure and the discharge at points at a distance AxJ2 from the upstream or downstream end of the section at time t - At/2 (points D and F i n Fig. 7.1).

Eliminating the unknowns QJ and p i (i = 1, 2, 3) and using the notation

(i = 1, 2, 3) (7.30) 4a&? 710;

cRi = rpi + -

(i = 1, 2, 3) (7.31)

132

The initial stale

and e i = + 1 (i = 1, 2, 3) (7.32)

where the positive sign applies to the sections with their downstream ends in the analysed junction, and the negative sign to the sections with their upstream ends in this junction we obtain a converted form of the system;

PJ ejfRiQi = ‘si (i = 1 , 2, 3) (7.33)

whence it follows that the pressure in the junction is given by

(7.34)

(7.35)

The values Qi (i = 1 ,2 ,3 ) are determined from equations of the type (7.33), the value Q, from equation (7.19) and pi from the relations

pi = pJ + eiRpiQi + Spi (i = 1, 2, 3) (7.36)

which are derived from equations (7.26) and (7.32). The system of equations for junction (7.33) and (7.34) cannot be solved with

the aid of the above method, when the determinant of the system equals zero. The calculation proceeds in a similar manner when a different number of

sections is attached to the junction. When the effect of the damping device is described by relation (7.18), Rd = 0

and s d = const. When the damping device ensures a constant pressure in the junction, pJ is

defined by equation (7.23) instead of (7.35). When the pressure device completely closes the section, equation (7.25) ap-

plies and the analysis of the junction is solved as if the closed section were not attached to the junction. The values p and Q at the end point of the closed section are calculated using equations (7.25) and (7.13) or (7.14).

7.7 The initial state

The first step in the solution of water hammer is the reading of the input data for one calculation from the input data file. The input data are entered into the beginning of the main output. The data determining the form of the graphical and numerical outputs are entered into the beginning of these two outputs.

133


Then, the initial pressure and discharge in the entire pipe-line system and the initial state of all damping (refer to Sect. 7.8) and pressure (refer to Sect. 7.9) devices is determined. The method of their determination depends on the type of calculation (refer to Sect. 9.5). For Z I 4, the submitted initial pressure po in the junctions is used, and for Z > 4, the pressure determined in the preceding abridged calculation of the steady state (refer to Chapter 8) is taken.

In the initial state the discharge along each section is assumed to be constant. For Z 5 4 it is taken as equal to the submitted value Qo, whilst for Z > 4 it is calculated by the abridged calculation of the steady state.

A similar procedure is adopted for some parameters of the pressure devices, for example, for the initial speeds of pumps and turbines.

The initial pressure at the individual points along the pipe-line sections is defined by the relation

(7.37)

where po, in this case, is the initial pressure in the junction, to which the section is attached by its upstream end, App is the difference in pressure on the pressure device, through which the upstream end of the section is attached to the junction, and Qo is the initial discharge through the pipe-line section. The distance x from the upstream end of the section and the coefficient of friction 1, are considered after adjusting the lengths of the sections (refer to Sect. 7.5).

If the pipe-line is closed by a pressure device at the upstream end of the section, one starts from the initial pressure in the junction in which the section ends, in order to determine the pressure at the individual points along the pipe-line section.

The initial pressure at the individual points is then defined by the relation

(7.38)

In this case, po is the initial pressure in the junction at which the section ends and App is the difference in the pressure through the pressure device at the downstream end of the section.

When a pressure device closes the pipe-line at both ends of the section, the initial pressure is taken to be constant throughout the length of the section and equal to the initial pressure in the junction at the upstream end of the section:

P ( X 9 0 ) = Po (7.39)

In the calculation of water hammer, the program does not check whether the initial state of flow is steady. This may be used to simplify the submission for some calculations of water hammer induced by an abrupt change. Instead of

134

Calculations for the damping devices

submitting data for the steady state before the change and the change inducing water hammer, one may directly submit the data for the unsteady state of flow which occurs immediately after the change.

7.8 Calculations for the damping devices

In solving the equations for the junctions (refer to Sect. 7.6), it is assumed that the effect of all the damping devices is described by either function (7.17) or (7.18), hence, that the pressure in the junction is a function of the discharge from the junction into the damping device, or alternatively, that this discharge is constant. For some damping devices this assumption is correct, but the effect of some of them also depends on other variables, such as the quantity of liquid or the air pressure in the chamber of the damping device, etc. Consequently, it is assumed that these other variables change abruptly after each step and that they remain constant during a step for a period At.

7.8.1 Junction without a damping device

In the case of such a junction (refer to Sect. 5. l), equation (5.1) corresponds to equation (7.18). The calculation uses the coefficients of linear dependence (7.19)

R , = 0 (7.40)

s, = 0 (7.41)

7.8.2 Constant pressure

For the damping device “constant pressure” (refer to Sect. 5.2), equation (5.2) corresponds to equation (7.17). The pressure in the junction is calculated using the relation (7.23), where

r.

Pc = Po (7.42)

7.8.3 Reservoir

In the case of a reservoir (refer to Sect. 5.3), equation (5.3) corresponds to equation (7.17). In the calculation, we consider:

for CJ # 0, Q; > 0 1

(7.43)

135


1 (7.45) R - _ -

d - 25; - Q;

(7.46)

and for lJ = 0 or Q; = 0, we make use of equation (7.23). For a reservoir,

pc = Pb (7.47)

7.8.4 Air chamber

The function of an air chamber (refer to Sect. 5.4) depends on the quantity of liquid in its chamber which varies during the course of water hammer. This quantity is denoted by the variable A V and is considered to remain constant during each step of the calculation so that equation (5 .5 ) formally corresponds to equation (7.17). In the calculation, the values Rd and s d are determined with the aid of equations (7.43) to (7.46), or else equation (7.23) is used. The value pc is defined by:

The initial value of the variable A V is defined by: 1 f --\

(7.48)

(7.49)

where po is the initial pressure in the junction to which the air chamber is attached. Equation (7.49) follows directly from (5.5) for QJ = 0. Values of A V for other values of the time t are determined successively using the relation

(7.50) A V = A V + Q; At

136

Calculations for the damping devices

7.8.5 Surge tank

The function of a surge tank (refer to Sect. 5.5) depends on the quantity of liquid in the chamber which is variable during the course of water hammer. It is, however, considered to remain constant during each step of the calculation. Equations (7.43) to (7.46), or (7.23) are employed. In this case, the valuep, reads:

for AV 2 0

pc = esv+ pb S U

for A V < 0

For the initial value Al/, we have:

for PO 2 Pb

(7.51)

(7.52)

(7.53)

(7.54)

both these relations follow directly from equation (5.7) for QJ = 0. Other successive values of A V are obtained using equation (7.50).

7.8.6 Overflow

The function of an overflow (refer to Sect. 5.6) depends on the quantity of liquid in the chamber which is considered to remain constant during each step of the calculation. Equations (7.43) to (7.46) or (7.23) are used where:

for A V = 0

(7.55)

Pc = - e g A v + Pb S

(7.56)

137


For the initial value of the variable A V we have: for PO 2 Pb

AV, = 0

for P O < Pb

(7.57)

(7.58)

both these relations are derived from the function of the overflow and from equation (5.8) for Q, = 0. Further values of A V are determined successively with the aid of equation (7.50), which, however, is considered to apply only as long as A V < 0. Otherwise, A V = 0.

7.8.7 Air inlet valve

The function of an air inlet valve (refer to Sect. 5.7) depends on the volume Va of air and on the absolute air pressure pabs in the chamber of the valve. These values are considered to remain constant during each single step of the calculation.

When there is no air in the chamber of the air inlet valve (Va = 0) and when there is an excess pressure in it relative to atmospheric pressure (pabs > paabs), the state of flow in the junction is not affected by the air inlet valve and equations (7.40), (7.41) are employed.

If there is air in the chamber of the air inlet valve (V, > 0), the calculation of the pressure in the junction proceeds using equation (7.23), where, making use of equation (5.10):

pc = p b -k Pabs - Paabs (7.59)

When the chamber is being filled with air, it is at atmospheric pressure

pabs = paabs (7.60)

and the mass of the air is given by equation (5.14). During the escape of air from the chamber, its pressure is determined by equation (5.1 5) and its mass is found by solving equation (5.16). To simplify the calculation, the mass of air in the chamber is determined in successive steps with the aid of the approximate relation

(7.61)

on the basis of the mass of air in the chamber in the preceding step and its actual

138

Calmlatwns for the damping devices

volume. The variation -

with equations (5.15) and (5.16) for m = m'.

first derivatives for At = 0 and At = 00.

of the mass of air is determined in accordance dm dt I t = t '

The relation (7.61) has the same values as the solution of Eq. (5.16) and the

The volume of air in the initial state is assumed to equal to

v , = o (7.62)

Further values of the volume of air are then determined successively from the

= v,l - Q;At (7.63)

state existing in the preceding step according to the relation

7.8.8 Cavitation

The same procedure as used in the calculation for the air inlet valve (refer to Art. 7.8.7) is applied in the case of cavitation (refer to Sect. 5.8).

7.8.9 Integrated damping device

The function of an integrated damping device (refer to Sect. 5.9) depends on the variable values of the volume V of the liquid in the chamber of the damping device, on the surplus air pressure Ap, on the liquid surface as compared with the basic state and on the rate of change of the discharge dQJ/dt of the liquid into the chamber of the damping device. These values are considered to remain constant during each step of the calculation.

Then, one may start from equation (5.24), which corresponds formally to equation (7.17). Alternatively, if the upper part of the damping device has an arrangement similar to an air inlet valve, if its chamber is filled with liquid and has a surplus pressure as compared to atmospheric pressure, equation (5.1) corresponding to (7.18) may be used as a starting point.

In the calculation, the following coefficients for CJ # 0 and Q, > 0 are employed in the linear relation (7.19):

(7.64)

(7.65)

139


and for CJ # 0 and Qj < 0

(7.66)

(7.67)

Equation (7.23) is used for iJ = 0 or Q'J = 0. The value cJ is determined by equation (5.21) and pc is given by the expression

The sum in the last term includes only those parts of the chamber, which are filled with liquid.

The value Apa depends on the arrangement of the damping device; it is defined by relations (5.25), (5.26) or (5.27).

The initial level of the liquid in an integrated damping device and the corresponding volume of liquid Vo are determined in such a manner that the pressure in the junction corresponding to the steady state of the integrated damping device is equal to the initial pressure in the junction. If the input values for the calculation are such as to make this impossible, the error will be pointed out, and, in some cases, the calculation is terminated.

Further values of V are then found successively with the aid of the relation

V = V' + Q;At (7.69)

The value Vb - AV in equation (5.26) has the meaning of the volume of air in the chamber of the damping device. In the calculation, it is determined as the difference of the total volume of the chamber obtained from the data submitted, and the volume V of the liquid.

7.8.10 Pressure

The ,,pressure" damping device (refer to Sect. 5.10) ensures a pressure in the junction as defined by equation (5.28). It is assumed that the pressure in the junction remains constant during each step of the calculation. Equation (5.28) then corresponds to equation (7.17). Relation (7.23) is employed, where

140

(7.70)

Calculations for the pressure devices

7.8.1 1 Discharge

The ,,discharge" damping device (refer to Sect. 5.1 1) ensures a discharge Q,, as determined by equation (5.29), from the junction into the damping device. It is assumed to remain constant during each step of the calculation. Equation (5.29) then corresponds to equation (7.18). In the calculation, relation (7.19) is employed, where

R , = 0 (7.71)

s, = fQJ(t) (7.72)

7.9 Calculations for the pressure devices

I t is assumed in performing the calculations for the junctions (refer to Sect. 7.6) that the effect of any pressure device is described by either the function (7.24) or (7.25), hence, that the difference in pressure Ap, in front of and behind the pressure device is a function of the discharge Q through the pressure device, or that the discharge Q is zero. For some pressure devices this assumption is valid but the effect of others depends also on additional variables. The effect of the control valve and the butterfly valve depends on their opening, the effect of the condenser on the liquid level in the chambers, and the effect of the pump and the turbine on the speed, or also on the setting of the guide and action blades. Consequently, it is assumed that these values change abruptly after each step of the calculation, while remaining constant during this step.

7.9.1 Attachment without a pressure device

For an attachment without a pressure device (refer to Sect. 6.1), relation (6.1) corresponds to relation (7.24). The following coefficients of the linear function (7.26) are used in the calculation

R , = 0 (7.73)

s, = 0 (7.74)

7.9.2 Closed pipe-line

A closed pipe-line (refer to Sect. 6.2) ensures zero discharge. Relation (6.2), which is identical to relation (7.25), is used in the calculation.

141


7.9.3 Local loss

For the pressure device ,,local loss" (refer to Sect. 6.3) equation (6.3) corresponds to relation (7.24). For the calculation, the following coefficients are employed in the linear relation (7.26),

s = - - 're IQ'I Q' x2D4 P

(7.75)

(7.76)

where r is defined by relation (6.4).

7.9.4 Control valve

The function of a ,,control valve" (refer to Sect. 6.4) depends on the extent of its opening, defined by the variable 6, which may vary with time. It is, however, considered to remain constant during each step of the calculation. Relations (6.6) and (6.7) then correspond to relations (7.24) and (7.25). In the calculation the linear function (7.26) is used for Q # 0 with the coefficients

R, =

s = P

When Q = 0, relation (7.25) The value 6 is defined by

7.9.5 Non-return flap valve

(7.77)

(7.78)

is applied. function (6.9), or by relations (6.10) and (6.1 1).

For the non-return flap valve (refer to Sect. 6.5) relations (6.12) and (6.13) are employed, corresponding to relations (7.24) and (7.25).

142


7.9.6 Butterfly valve

The effect of a butterfly valve (refer to Sect. 6.6) depends on the degree of opening of the flap as determined by the angle of tilt u, which is considered to remain constant during each step of the calculation. In this case relations (6.14) to (6.16) correspond to relations (7.24), (7.25). In the calculation, the linear function (7.26) is employed for Q # 0 with the coefficients

(7.79)

(7.80)

while relation (6.15) is used for Q = 0. The value CJ is defined by relation (6.16). The value of the initial angle of tilt avo of the flap is either submitted or

determined from the abridged calculation of the steady state (refer to Chap- ter 8).

The subsequent motion of the flap is calculated in successive steps by numerically solving Eq. (6.24) on the basis of the state existing in the preceding two steps.

For the solution equation (6.24) is arranged in difference form by substituting equations (6.17) to (6.23) and introducing, instead of the derivatives, the following approximate relations

(7.81) duv . UV - U; -- - dt At

(7.82) d2uv dt2 At2

uv - 24, + u'; -- -

If uv denotes the angle of tilt of the flap at instant t , then u; denotes this value at instant t - At, and u'; at instant t - 2At. For the first step the values u'; = a; = uvo are used.

Moreover, the motion of the flap is limited in accordance with condition (6.27) and also by (6.25) and (6.26) when the oil pump is switched on.

7.9.7 Condenser

A condenser (refer to Sect. 6.7) which forms part of a hydraulic system is replaced in the calculation by the pressure device denoted ,,condenser" and a section with two junctions to which two integrated damping devices are at-

143


tached. The calculations pertaining to this section, the junction and the integrated damping devices are described in Sections 7.5 and 7.6 and in Subsection 7.8.9. The function of the ,,condenser" depends on the levels of the liquid in the two integrated damping devices. These levels are considered to remain constant during each step of the calculation. Equations (6.28) to (6.36) then correspond to function (7.24) and condition (6.37) to condition (7.25). For the calculation either linear function (7.26) or (7.25) is used. The value of the coefficients in function (7.26) depends on the level of the liquid in the integrated damping device. These coefficients are determined with the aid of relations (7.75) and (7.76) for h, 2 h, and hd 2 h,. The value [ is constant and is submitted. For h, 2 h, and h, < hd < h,, the coefficient Rp is defined by relation (7.75) and S, by the relation

s = - 8h ht - hd - I Q ' I Q' + e s 7 n2D4 P (7.83)

The coefficients R, and S for other liquid levels are determined with the aid of relations (7.27) and (7.28) from relations (6.30) to (6.36) in a similar manner.

7.9.8 Pump

The effect of a pump (refer to Sect. 6.8) is described by relation (6.42). It depends on the variable speed n which is considered to remain constant during each step of the calculation. Relation (6.42) then corresponds to relation (7.24). The linear relation (7.26) is employed in the calculation. Its coefficients are determined from the'submitted function f,(Q, n) with the aid of the relations (7.27) and (7.28). . The initial speed no of the pump is either submitted or determined from the abridged calculation of the steady state (refer to Chapter 8). Further speeds are calculated successively by solving Eq. (6.53) arranged in the difference form:

n = n' + -(M' At + Mk) 2nz

(7.84)

When Z = 0 the speed is determined by numerically solving Eq. (6.54). When the electric network guarantees a constant speed of the pump, this

speed is defined by relation (6.56). The torsional moment of the pump M' which is required for calculating the

speed from equation (7.84) is obtained from the discharge and the speed in the preceding step using relation (6.43) in the form

144

(7.85)


The functionfM is submitted. The torsional moment M i of the electric motor is obtained from the speed in the preceding step with the aid of relation (6.44) in the form

ME = fE(n’) (7.86)

The function fE is also submitted. If the electric motor is disconnected from the network, ME = 0.

When the speed is calculated by solving Eq. (6.54), the moments are determined directly by relations (6.43) and (6.44). The discharge found in the preceding step is used only in relation (6.43). The accuracy of the solution of Eq. (6.54) depends on the value of the difference of the moment d,, which is neglected.

7.9.9 Turbine with fixed characteristics

The effect of this type of turbine (refer to Sect. 6.9) is calculated in a similar manner as that of a pump (refer to Par. 7.9.8).

7.9.10 Turbine with variable characteristics

The effect of a turbine with variable characteristics (refer to Sect. 6.10) on water hammer depends on the variable values of the speed n and of the parameters a and which determine the setting of the guide and action blades. These values are considered to remain constant during the course of each step of the calculation. Then, equations (6.61) (6.63), (6.65) and (6.67) correspond to relation (7.24) and relation (6.76) is identical to (7.25). The values E and fl are determined from the submitted functions (6.57) and (6.59 by conversion in accordance with relations (6.69) to (6.72). The function J(Q, ii, E, B) is determined from the submitted characteristics by interpolation as described in Sect. 6.10. The speed of the turbine is calculated similar to that of a pump in accordance with relations (6.54) and (6.56), or successively with the aid of equation (7.84).

The initial speed no can be either submitted, or found from the abridged calculation of the steady state (refer to Chapter 8), when it is calculated with the aid of equation (7.84).

The torsional moment M of the turbine is defined by equations (6.62), (6.64), (6.66) or (6.68), where the speed n and the discharge Q are taken from the preceding step. The values of the function A(Q, ii, E, 8) again are obtained by interpolation as described in Sect. 6.10. The relations (6.74) or (6.75) are also used for the case of a conipletely closed turbine. The torsional moment of the generator operating at a constant speed is found by using relation (6.54), for a disconnected generator according to relation (6.55), or with the aid of the submitted function (6.44).

145


7.9.1 1 Governor-controlled turbine

The calculations for this type of turbine (refer to Sect. 6.1 1) are similar to those of a turbine with variable characteristics (refer to Subsect. 7.9.10), the only difference being that the opening of the guide and action blades is not determined by the submitted functions (6.57) and (6.58), but has to be determined in successive steps by calculating the function of the governor. Besides this, the variable value of the moment of inertia I of the generator and the dependence of the torsional moment ME of the generator not only on the speed, but also on time, are taken into account. The values I and ME are considered to remain constant during each step of the calculation.

The parameters ct and /3 are determined for each step in the calculation of water hammer. The starting point is the turbine speed together with the values defining the state of the governor in the preceding two steps. The initial speed may be either submitted or determined in the preceding abridged calculation of the steady state. It is assumed that the initial state is steady. The voltage Uo is defined in accordance with relation (6.81) arranged in difference form:

(7.87)

The function fr (n) is submitted. In this arrangement, the derivative has been replaced by a term similar to that of relation (7.81).

In the initial state, dUo/dt = 0 and equation (6.81) converts to

uo = fj (no) (7.88)

The voltage U, is obtained by means of relation (6.82). The function fr(t) is

The voltage U, is determined from equation (6.83) arranged in difference form submitted.

by using relations similar to (7.81) and (7.82):

(7.89)

The function fb(Uo) is submitted. The calculated voltage is limited so as to satisfy condition (6.84).

146


i n the initial state, all derivatives in equation (6.83) are zero and it converts to

Ul u, = - cP

(7.90)

In the first step of the calculation, the values U‘,‘ = U ; = U , are used. The voltages U, and U4 are determined with the aid of relations (6.85) to

(6.87). The functions fp( t ) and f,,,(t) are submitted. The position of the slide valve in the hydraulic amplifier of the guide wheel is

defined by the relation

(7.91)

which was obtained by rearranging relation (6.88). The extension of the piston rod y’ in the preceding step is used in relation (7.91).

The initial position xoa of the slide valve is either submitted or obtained from the abridged calculation of the steady state. The calculated values of x , are limited to satisfy condition (6.89).

When the absolute values of the sum of other forces acting on the piston of the servomotor is larger than the force due to friction F, , the position ya of the piston rod governing the guide blades is determined by the numerical solution of Eq. (6.90) arranged in difference form

- Fea(ya, A P ~ ) + F f a sign (Ya - Y;) = 0

When friction prevails, it follows from Eq. (6.91) that (7.92)

Y a = Y ; (7.93)

Relations similar to (7.81) and (7.82) were employed in the above arrangement

The functions Foa(xa) and oa(xa) are submitted. The values of the function F,(ya, Apk) are determined from relation (6.93).

The functions f , , (ya, Ap,) are submitted for one or several values of Ap, and linear interpolation is then carried out among them.

of relation (7.92).

In the initial state, the value ya is determined from the relation

(7.94)

147


which is obtained by rearranging relation (6.88). In solving Eq. (7.92), the values y," = y ; = y, are used in the first step of the calculation.

The calculated velocity of the slave cylinder piston is limited to satisfy the inequality

or

(7.95)

(7.96)

resulting from relations (6.94) and (6.95). The functions fdy+,(ya) and fdy-,(ya) are submitted.

In addition, the calculated value y , is limited by condition (6.96) which determines its maximum and minimum values.

The parameter a is determined in accordance with (6.97) on the basis of function f,(y,) which is submitted.

When the governor merely controls the manipulation of the guide blades, one uses the parameter

p = 0 (7.97)

When the governor also controls the setting of the action blades, the voltage U, is determined using either relation (6.98) or (6.99) depending on the type of governor. The function fH(U4, H) or fH(ya, H) is submitted for one or more values of H, and a linear interpolation is carried out.

The position of the slide valve in the hydraulic amplifier of the action wheel is defined by the relation

(7.98)

which was obtained by re-arranging equation (6.100). The initial position xoF of the slide valve is either submitted or obtained from

the abridged calculation of the steady state. The continuation of the calculation proceeds as for the parameter a described

by relations (7.92) to (7.96) including the limiting condition (6.96). The only differences are that the voltage U , is substituted for U4 and the subscript B for the subscript a. The value of the parameter /? is found from the calculated position yF of the slave-cylinder piston controlling the action blades and from the function fs(yF) according to function (6.101).

When the generator is connected to the network, its moment of inertia I is

148

Calculation procedure

determined with the aid of the relation (6.78) and the torsional moment of the generator in accordance with relation (6.80). The functions f , ( t ) , fn(t) and &(t) are submitted. When the generator is disconnected, I = I, and ME = 0.

7.10 Calculation procedure

The calculation of water hammer proceeds as follows: the state of flow and of the damping and pressure devices in the entire system at time t + At is calculated on the basis of their known state at time t.

In this way, the state at time t = At is calculated in the first step of the calculation from the known initial state of the system at time t = 0. In the next step, the state of the system at time t + 2At is determined from the state at time t = At. The calculation proceeds in stages in this manner up to time t = tmax, when it is terminated. The results are entered into the output files immediately after each step. The results of the preceding steps are not stored in the working storage of the computer. If an output of the results for some instants of time is not required (refer to Sect. 7.3), this output is not effected and the results which are not recorded can be obtained only by repeating the entire calculation.

In each step of the calculation, the new state of all damping and pressure devices at time t + At is determined from their known state at time t. First the pressure and the discharge at the points of the grid of characteristics for time t + At/2 are determined for all sections from the corresponding values for time t, and then these values (for time t + At/2) are used to determine the values for time t + At. The pressure and discharge at both ends of each section are determined together with the solution for the junctions. In the solution for the junctions (refer to Sect. 7.6), the pressure in all the junctions, the discharge from the junctions into the damping devices and the pressure and the discharge at the ends of all the sections are determined for time t + At. The solution for each junction leads to the solution of a system of linear equations. If the determinant of this system of equations equals zero in any one step, an error message appears on the screen, but the calculation does not stop. In this way, all the values sought for time t + At are determined. Now, a more accurate calculation for the junctions by means of iteration (refer to Sect. 7.11), the checking whether a steady state has been attained (refer to Sect. 7.12), the printing of the results, the subsequent steps of the calculation, or the termination of the calculation (refer to Sect. 7.14) can follow. The subsequent procedure depends on the type of calculation requested and on the nature of the further data submitted.

149


7.11 Iterative solution for the junctions

An iterative solution for the junctions is carried out when this type of solution is requested by submitting the value Z = 2, 4, 6 or 8 (refer to Sect. 7.3).

In solving the equations for the junctions without iteration, as explained in Sections 7.6 and 7.10, the linear relations (7.19) and (7.26) are substituted in each step for the actual, often non-linear relations between the pressure and the discharge for the damping and the pressure devices. Such a simplification is acceptable, provided the changes in the pressure and discharge are slow and the time interval At of the calculation is sufficiently short. In the case of rapid changes and a longer calculation time interval, such a simplification might cause unacceptable inaccuracies in the results. Therefore, in analysing the junctions, an iterating procedure may be chosen, in which these inaccuracies are eliminated. Here, the equations for each junction are solved repeatedly. In each solution, new values for the discharges Q’ and Q; are used, equal to the values Q and Q, ‘calculated in the previous step. The calculation is repeated until the discharges Q’ and Q and Q; and QJ satisfy the agreement required. The linearization of the relation Ap, = f ( Q ) for a pressure device during one iteration step is shown in Fig. 7.3.

Fig. 7.3 Linearization of the curve representing the difference in pressure as a function of discharge for a pressure device

Q during an iteration step.

In solving the equation for a junction at time r without iteration, the tangent Ap, = RbQ + Sb is substituted for the relation Ap, = f(Q), the point of tangent being defined by the discharge Q’. The solution for the junction then yields the values Q and Ap,. The point corresponding to them is located on the tangent Ap, = RkQ + Sb. The difference Ap defines the inaccuracy due to the linearization of the relation Ap, = f(Q).

Using iteration, the calculation procedure is repeated with the difference that the value Q and, hence, also the tangent App = RpQ + Sp are substituted for Q‘ and Ap, = RbQ + Sb, respectively. A similar procedure is adopted for all pressure and damping devices attached to the junction. For damping devices, QJ is substituted for Q;.

The iteration is repeated automatically until the following condition is met for two consecutive calculations

150

Maximum and minimum values

and (7.99)

(7.100)

where dp and dQ are the permissible differences in pressure and discharge, respectively.

The maximum number of iterations is N i . If inequalities (7.99) and (7.100) are not satisfied even after Ni iterations, the following values are used in the subsequent calculation:

(7.101)

(7.102)

as obtained in the last iteration step. At the termination of the calculation, the highest values dp and dQ are

presented in the main output. They can be used to check the accuracy achieved in the iterative solution.

7.12 Checking whether a steady state was attained

Attainment of a steady state is verified only in the case of the abridged calculation of the steady state, that is, for the type of calculation selected by the value Z = 3, 4, 7 or 8 (refer to Chapter 8).

After each step of the calculation, the program checks for each section, whether the discharges at the ends of the section do not differ mutually by more than dQs, where des is the admissible difference in discharge for the steady state.

Subsequently, each section is checked as to whether the pressures at the ends of the section do not differ from those corresponding to the steady state by more than dps, where dps is the admissible difference in pressure for the steady state.

When these conditions are satisfied for all the sections, the calculation of the steady state is terminated, even if the maximum calculation time t,,, has not been reached. Should t = t,,, be reached first, the calculation is also terminated, even if the steady state has not been attained with the accuracy required.

7.13 Maximum and minimum values

In the course of the calculation, the maximum and minimum values of the pressure in the sections of the pipe-line are calculated for each individual section. The pressures are considered at all points of the grid of characteristics used

151

Cukulution of' water hammer

(refer to Fig. 7.1). The maximum and minimum values of the pressure are presented at the end of the main output for each individual section.

Next, the maximum and minimum values of some parameters of the damping and pressure devices are determined: the difference A V in the volumes of liquid in the air chambers, surge tanks, overflows, the volumes V, of air in the air inlet valves, the volumes V , of void space during cavitation, the volumes V of liquid in the integrated damping devices, the variables r~ of the control valves, the angle of tilt aV of the butterfly valves and the speed n of the pumps and the turbines. The maximum and minimum values are presented at the end of the main output. Apart from this, the maximum and minimum values of all the variables presented in the graphical and numerical outputs are determined. They are presented at the ends of these outputs.

In searching for these extremes, all the calculated values of the quantities investigated are considered without regard to the chosen time interval of printing.

7.14 Submission for several calculations

The data file (refer to Sect. 7.3) may contain the data for one or more consecutive calculations of water hammer. In Sections 7.2 to 7.13 the procedure for a single calculation was described. After a calculation has been completed, a request appears on the screen, as to whether the input or output files should be changed for the next calculation, or whether the present calculation has been completed. In such a case, the calculation has to be terminated by pressing the S key, upon which we return to the DOS operating system. The next calculation may be carried out quite independently by repeating the preceding procedure.

If the calculations are carried out for similar pipe-line systems, or if they concern changes in flow conditions in the same pipe-line system, or if the steady state is required for the subsequent calculation of water hammer, it is worth while to carry out several calculations with a single set of input data. In most cases, the submission of the calculation is then much simplified.

The full submission is necessary only for the first calculation. Immediately afterwards, the next calculation may be submitted in the same form as the first one. However, all lines which are identical to the preceding submission may be omitted, except for the line beginning with the number 7. This line must be included anew in each submission.

Now, the third calculation can be submitted; it will contain, similar to the submission of the second calculation, the line beginning with 7 and, apart from this, only those lines which differ from the preceding submission (the second calculation in this case). The number of successive calculations is not limited by the program. Calculations presented in different data files may be also submitted in succession.

152

Submission for several calculations

When the calculation of the steady state has to be used in order to ascertain the initial state in the calculation of water hammer, and if both these calculations were to be carried out independently, it would be necessary to read the resulting pressures in all the junctions, the discharges in all the sections, off the results of the calculation of the steady state, as well as perhaps yet other values. These values would then have to be entered into the data file for the calculation of water hammer proper. However, provided the calculation of the steady state is followed by that of water hammer within one submission, the calculated values of the steady state are automatically used in the subsequent calculation. When the type of calculation is submitted by means of the value 2 = 5 to 8, the values determined in the preceding abridged calculation of the steady state are used for the calculation, instead of the submitted values of the pressure in the junctions, of the discharges in the sections and the values of some parameters defining the initial state of the pressure devices. The calculation of the steady state need not immediately precede that of water hammer. When the calculation of water hammer is preceded by several calculations of the steady state, the values determined by the last such calculation are used. One calculation of the steady state may also be employed for several calculations of water hammer. It is also possible to use the results of the abridged calculation of the steady state as input values for another such calculation, say, for different conditions of flow.

After completion of the first calculation, the answer N may be given to the request whether the input and output parameters are to be changed. Now, the next calculation presented in the input file, is started. Its results are entered after the results of the preceding calculation into the same files. If the answer is 0, a table denoting the individual output devices and their combinations appears on the screen and the further procedure continues as described in Section 7.4. If the results are to be written onto a hard disk, however, output files other than those used in the preceding calculations, have to be employed. Otherwise, the existing data in the files are overwritten by the new results. When the answer is I, the name of the input file may again be submitted in addition to the output files. Otherwise, the calculation proceeds in the same manner as when the answer is 0. With the answer C, the new calculation is started from the beginning as at the start of the WTHM program.

The request for a change of the input and output files reappears at the termination of each subsequent calculation. The answer S terminates the calculation carried out with the aid of the WTHM program and control returns to the DOS system.

153

8 Calculation of the steady state

8.1 Abridged calculation of the steady state

The calculation procedure introduced in the present book may be employed for studying water hammer in various pipe-line systems, and also for calculating the steady state of flow. If other methods are used, calculation of the steady state of flow can be a tedious operation, especially for more complex hydraulic systems.

The steady state may be calculated either as an independent problem or it may be used to determine the initial state of flow before the calculation of water hammer.

In calculating the steady state, one submits any state of flow in the pipe-line system investigated as the initial state and the subsequent course of the unsteady flow in the system is calculated repetitively until such time as the flow becomes steady. The boundary conditions now are identical to those for the required steady state.

The procedure described in Chapter 7 can be applied in practice for this purpose, but it is complicated by two problems. In some systems, the time for the flow to become steady is too long; moreover, it is rather difficult to check with sufficient accuracy, whether the steady state has been reached.

For these reasons, the method of calculation was modified to allow us to use an abridged calculation of the steady state in conjunction with the calculation of water hammer. The basic procedure remains unchanged, but, instead of the hydraulic system submitted, another system which results in the same steady state is considered in the calculation.

The modifications of the system are carried out automatically during the calculation. They are described in Sections 8.2 to 8.4. They lead to a reduction on the time needed for the steady state to be established, to a simplification of its verification and to a simplification of the submission. The course of the unsteady flow in the system substituted does not correspond to that in the original one, but this has no adverse effect, considering the purpose of the calculation.

154

Mod@cutiun of the pipe-line sections

8.2 Modification of the pipe-line sections

For the steady state (dQ/dt = 0, dp/& = 0), the system of equations (7.2), (7.3) converts to

- = o dQ dx

It follows from the solution of Eq. (8.2) that

Q = Q, where Q, is the constant discharge at the steady state. It follows from Eq. (8. l), after substituting (8.3) for the difference in pressure between the upstream and downstream ends of the pipe-line sections that

8 1 ~ 1

n2DS Ap, = - - IQ,I Q,

Consequently, neither the discharge through a pipe-line section nor the difference in pressure for the steady state depends on the wave velocity or on the length I of the section, provided the coefficient of friction 1 is also modified so that the value of the product I1 remains unchanged.

In the abridged calculation of the steady state, one uses the same wave velocity a, > 0 and the same modified length of the sections

1, = a, At

for all sections. The velocity a, may be chosen. The modified coefficient of friction I , is

determined in accordance with equation (7.9), so that the value 1I is maintained. To now ascertain the steady state of flow in the sections, it suffices to check

for all sections, whether the discharge at their upstream and downstream ends is the same and whether the difference in pressure at their ends corresponds to the equation (8.4). During the course of the calculation, the program checks after each calculation step, whether the absolute value of the difference of the discharge at the end points of all sections is less than the admissible difference dQ,. The difference in pressure at the upstream and downstream ends of the pipe-line are found from the calculation of water hammer. Then, the difference in pressure at the upstream and downstream ends of the pipe-line is calculated with the aid of equation (8.4). These two differences are subtracted and the result

Calculation of the steady state

is taken as an absolute value. This absolute value is then compared with the value d p . In addition, after each step, the program ascertains whether the pressure devices, the butterfly valve, the pump and the turbine, provided they form a part the system investigated, are in a steady state. This checking in not required for other damping and pressure devices in the light of their modification (refer to Sects. 8.3 and 8.4).

After all the conditions introduced above have been satisfied, the state of the hydraulic system is considered steady.

8.3 Modification of the damping devices

Some damping devices extend the time needed to attain the steady state. Therefore we consider, in the abridged calculation, modified devices which lead to the same steady state of flow. They are modified so as to correspond to the steady state during each step of the calculation.

Of these devices, the junction without a damping device, the constant pressure and the reservoir remain unchanged. However, the air chamber, the surge tank, the air inlet valve and cavitation are replaced by a junction without damping device. At the steady state, the discharge from the junction into these damping devices is zero (Q, = 0), as for the case of a junction without a damping device (refer to Sect. 5.1). The volume of the liquid in the air chamber and in the surge tank is not calculated, it is derived from the resulting pressure in the junction. It is only determined as late as at the beginning of the calculation of water hammer. For the air inlet valve, it is assumed that it is full of liquid at the steady state. Similarly, it is assumed that there is no cavitation. The submitted parameters of these damping devices are irrelevant for the abridged calculation: they are not used in it.

The damping device denoted “overflow” is replaced by a junction without damping device only in case the pressure in the junction is lower than the pressure p,, submitted for the junction with the overflow in the basic state. Otherwise, the “overflow” damping device is considered to have the same function as in the calculation of water hammer (refer to Sect. 5.6). The parameter S submitted is irrelevant in the abridged calculation of the steady state.

An integrated damping device is replaced by a junction without damping device, if it is arranged as an air chamber or an air inlet valve. If it is arranged as a chamber with overflow, it is replaced by a junction without damping device only in case the pressure in the junction is lower than at the steady state with Q, = 0 and with the liquid surface at the level of the overflow edge. Otherwise, it is considered as described in Sect. 5.9. The calculation does not include the effect of the inertia of the liquid in the chamber of the damping device; it has no effect in the steady state. The pressure in the junction is then determined by

156

Modijcation of the pressure devices

the relation

PJ = Pb + CJ I Q J I QJ + Qgh,, (8.6)

which follows from relations (5.24) and (5.25). The quantity CJ is defined by relation (5.21).

In the abridged calculation of the steady state, only the submitted value N , is of importance for an integrated damping device; it defines the type of arrangement of the device. For chamber with overflow, when N , = 1, further values which affect the steady state are also significant.

The damping device denoted “pressure” is treated as described in Sec- tion 5.10. In the junction, however, the pressure determined by function (5.28) for t = 0 is taken to persist indefinitely.

Similarly, the damping device “discharge” is treated as described in Sec- tion 5.1 1. The discharge from the junction into the damping device takes the permanent value resulting from function (5.29) for t = 0.

Of the parameters submitted for the damping devices pressure and discharge only those values are important in the abridged calculation, which determine the pressure or the discharge at time t = 0.

8.4 Modification of the pressure devices

The function of some pressure devices depends on time. In the abridged calculation of the steady state, the steady state corresponding to the conditions valid for t = 0 is sought. Hence, .the presssure devices were modified to correspond to these conditions all the time. Some effects which depend on derivatives of time, are not considered, since the resulting steady state does not depend on them. The effect of inertia, though, is considered for the butterfly valve, pumps and turbines, because it favourably influences the attainment of the steady state. For these devices it is necessary to check whether a steady state has been attained. The remaining pressure devices correspond permanently to the steady state. Pressure devices such as the attachment without a pressure device (refer to Sect. 6.1), the closed pipe-line (refer to Sect. 6.2), local loss (refer to Sect. 6.3) and non-return flap valve (refer to Sect. 6.5) are included in the calculation in a similar manner as in the calculation of water hammer. The control valve is considered as described in Section 6.4, but the quantity r~ takes permanently the value resulting from relations (6.9) or (6.10) and (6.1 1 ) for t = 0.

The effect of the butterfly valve is concidered in like manner as in the calculation of water hammer (refer to Sect. 6.6). The oil pump, however, is either permanently switched on or off, as may be seen from the submission for t = 0.

In each step of the calculation, the program checks, whether the butterfly valve is in the steady state, i.e. whether

157


or whether the flap is in an extreme position and whether it is kept there by the resulting moment.

The effect of the condenser is considered to permanently remain that given by relation (6.28), corresponding to full chambers. Of the parameters submitted, only the value [ is important.

The pump and the turbine with fixed characteristics are treated as in the calculation of water hammer (refer to Sects. 6.8 and 6.9).

The electric motor or the generator is either permanently connected to or disconnected from the network in the manner corresponding to the submission for t = 0.

In each step of the calculation, the attainment of the steady state is checked, i.e., whether

This checking is unnecessary when the electric motor or the generator is connected to the network, thus ensuring a constant speed.

A turbine with variable characteristics is treated in a similar manner as in the calculation of water hammer (refer to Sect. 6.10).

The connection of the generator to the network or its disconnection from it and the setting of the guide and action blades is considered as corresponding permanently to the submission for t = 0. The attainment of the steady state of the turbine is checked in agreement with relation (8.8) in each calculation step.

For a governor-controlled turbine (refer to Sect. 6.1 I) , the effect of the network is considered to correspond permanently to that which follows from the submission for t = 0. The setting of the guide and action blades is determined by the governor.

The function of the governor is modified to correspond permanently to the steady state for t = 0. All derivatives with respect to time are zero.

For the calculation of the voltage Uo(t) , the relation

IM + M,I 5 d M (8.8)

is used instead of equation (6.81). To determine the voltage U,( t ) , relation (6.82) is used, but the value of the function &(t) for t = 0 is considered to remain constant, so that

To determine the voltage U,(t) , we use the relation

(8.10)

Ul u, = - cP

158

(8.1 1)

Submitting the caIcuIation

which follows from equation (6.83) for the steady state. The voltage U,( t ) is limited to satisfy condition (6.84).

The voltage U , ( t ) is determined using relations (6.85) and (6.86). The function f,(t) is considered permanently to retain its value for t = 0, so that

u, = u, +fp(O) (8.12)

The value U , ( t ) is limited in accordance with relation (6.87). The value f,,,(O) is considered permanently instead of the value of the funciton f,,,(t).

The position x, of the slide valve of the hydraulic amplifier of the guide wheel and the extension y, of the piston rod of the slave cylinder are determined by numerically solving the system of equations formed by equations (6.88) and

(8.13)

Equation (8.13) resulted from Eq. (6.90) for the steady state by assuming that

F , = 0 (8.14)

If a solution compatible with conditions (6.89) and (6.96) is not found, the maximum or minimum values of x, and y, are used.

The parameter a defining the position of the guide blades is determined from relation (6.97).

If the governor also controls the action blades of the turbine, the voltage U , ( t ) is calculated using functions (6.98) or (6.99). The subsequent procedure is similar to that determining the position of the guide blades. Relations (6.100) and (6.101) are substituted for relations (6.88) and (6.97); the subscripts are substituted for the subscripts o! in the other relations used.

The steady state of the governor-controlled turbine is checked in each step of the calculation with the aid of relation (8.8).

In the abridged calculation of the steady state, submission of the following values is irrelevant: xoa; xop; Cf; C,; C , ; ma; Cop; F,; ms; C, * F f p and the

because none are employed in the calculation. For the functions f;6f;f$);fc(); f r ( t ) ; f p ( t ) ; fmax(t), only their value for t = 0 is relevant.

F,, + F,, = 0

functions: f b ( uO) ; aa(Xa); f d y + a(ya) ; f d y - a(Ya) ; ap(xfi); f d y + ) * f d - p(y 1,

8.5 Submitting the calculation

The form of submission of the abridged calculation of the steady state does not differ from that of the calculation of water hammer. The difference lies only in the meaning of some quantities.

The submitted values of the wave velocity a in the sections are irrelevant, since they are not used in the calculation. The initial discharge Qo in the sections may be chosen arbitrarily. I t is convenient to estimate, at least approximately, dis-

159


charges corresponding to those for the steady state sought, since this shortens the calculation. The initial pressures in the junction may be chosen arbitrarily. For the same reason, it is again convenient to use values estimated to approximate the pressures corresponding to the steady state sought. An exception is the junction with the “constant pressure” damping device. In this junction, the pressure has to be given accurately, since this value represents one of the boundary conditions for the calculation.

The accuracy of the calculation of the steady state may be determined by submitting the values d,, > 0; d , > 0; d , > 0.

Notes concerning the submission of the parameters of the damping and pressure devices are introduced in Sects. 8.3 and 8.4.

The abridged calculation of the steady state is realized, when Z = 3, 4, 7 or 8 is submitted for the type of calculation. The values corresponding to the steady state are then calculated, viz., the pressures in all the junctions and the discharges in all the sections. When the system contains the pressure devices butterfly valve, pump or turbine, additional values are determined for each of these devices, that is, the angle of tilt of the flap, the pump or turbine speed and the setting of the slide valve of the hydraulic amplifier of the guide and action blades. All these values are stored in the memory of the computer after the calculation has been completed and they may be used as the initial values for subsequent calculations of water hammer or abridged calculations of the steady state.

For 2 = 3 and 4, the submitted values are used as the initial values for the abridged calculation of the steady state.

For Z = 7 and 8, the values determined in the preceding abridged calculation of the steady state are used as the initial values. These values are the discharges, the pressure and, as the case may be, other quantities described above. The submitted values of these quantities are irrelevant in this case, since they are not used in the calculation.

With 2 = 3 and 7, the calculation of the junctions is carried out without iteration, with Z = 4 and 8, iteration is applied (refer to Sect. 7.11).

For the abridged calculation of the steady state, the wave velocity a, has to be submitted; it has the same value for all the sections (refer to Sect. 8.2). Its value need not correspond to the actual wave velocity in the pipe-line of the system investigated. Nevertheless, the wave velocity has to be chosen suitably, since it substantially affects the duration of the calculation. A reduction of the value a, up to a specific optimum value reduces the duration of the calculation. Further reduction, however, prolongs the duration and, with excessively low values of as, the calculation ceases to converge.

An example of the relationship between the calculation time (expressed by the number of calculation steps t /At ) and the value a, for the hydraulic system presented in Sect. 14.2 is shown in Fig. 8.1.

160

Submitting the calculation

The optimum value a, for a given system can only be found by trial-and-error. In many cases, safisfactory results are obtained with the smallest a, possible, though still such that the following condition is satisfied for all sections:

(8.15)

Fig. 8.1 Example of the relationship between the duration of the abridged calculation of the steady state, and the wave velocity.

The value At > 0 may be chosen arbitrarily. The duration of the calculation does not depend on it. It is relevant only in some cases, when the hydraulic system includes a pump or a turbine for which a constant speed is not ensured by an attached network, or when the system contains a butterfly valve. An increase of At in such cases leads to a shortening of the duration of the calculation up to a limit. A further increase of At prolongs the duration and, with an excessively large At, the calculation ceases to converge.

The optimum value At is to be found by trial-and-error for every system. The abridged calculation of the steady state is terminated when the steady

state has been reached with the required accuracy. Another limit on the duration of the calculation is the attainment of the maximum calculation time tmm. When this value is reached, the calculation is terminated, even if the steady state required has not been attained. Consequently, the value t,,, has to be chosen sufficiently large. A first estimate, with a suitable choice of a, and At, may be made using the relation

t,,, = 40NsAt (8.16)

In the abridged calculation of the steady state, it is usually convenient to make

161


use of the graphical output of certain parameters on the screen; thus, the steadying of the flow can be observed and the quantities a,, At and tmax modified, if necessary.

_t A t

150

100

50

- OD1 062 0.03 0.06 A t k i 01

Fig. 8.2 Example of the relationship between the duration of the calculation of the steady state, the time interval At of the calculation and the wave velocity a,.

The influence of the choice of the values At and a, on the duration of the calculation of the steady state, which is established after the fading of water hammer, for the hydraulic system presented in Sect. 14.12, is shown in Fig. 8.2. In the calculation, we used the submission contained in Table 14.24 and in the D12.WTH file on the WTHM diskette, where the folloving data were changed. We estimated the initial discharge Qo = 5 m s-'; the initial opening avo = 1.425 rad of the butterfly valve and the constants B,, = 6000 kg m-7, B,- = -4000 kgm-', B,, = 1300 kg m-4 and B,- = - 1000 kg m-4 of the pressure and moment characteristics, respectively. The calculation denoted 2 = 4 was used and the graphical output was realized for each calculation step At. The values a, and At were chosen within the intervals 25 I a, I 150 m s-' and 0.001 I At I 0.046 s, respectively.

When the abridged calculation of the steady state serves to determine the initial values for the subsequent calculation of water hammer, the complete submission of data for the calculation of water hammer may already be introduced for the abridged calculation. The subsequent submission for the calculation of water hammer will then consist only of two entries, viz., the lines beginning with numbers 5 and 7, containing the parameters defining the type of calculation and its verbal denotation.

162

9 Input data file

The input data are entered into the data file before commercing the calculation. The WTHD program, described in Chapter 10, may be employed for the preparation and formal checking of this file. The data in the file are arranged into lines. Each line contains 79 characters at the most. All the data are entered into the appointed columns, as far to the right as possible. Numbers may contain decimal points. If this is not the case, a decimal point is assumed behind the column designated for the respective data. Decimal numbers may be also entered in the exponential form. Blank spaces may be substituted for zeros. The submission of the calculation contains data lines numbered 1 to 7 in the first column. The line beginning with the number 1 may be omitted from the submission. The submission has to contain one line beginning with the number 2 for each section and one beginning with the number 3 for each junction. Moreover, the file must contain one line beginning with the number 5. The number of lines beginning with the numbers 4 and 6 may differ. Every submission has to be terminated by a line beginning with the number 7. The sequence of the remaining lines is irrelevant, it merely affects the clarity of arrangement of the data file.

The data file must contain neither blank lines nor any other lines not appertaining to this file. The arrangement of the input data is discussed in Section 9.1 to 9.7.

9.1 Name of the calculation

The type of data and their location in a line are presented in Table 9.1.

Num - the identification number of the calculation (0 5 Num 4 9999), Title - the name of the calculation (maximum of any 74 characters).

The first column contains the number 1. There follows:

163

Table 9. I Data for name of the calculation

Numberoffirstcolumn Number oflast column

Name of calculation

I Type of data 11 Integer I Characters I 1 2 6 1 5 79

1 Num Title

Number of last column

Junctions

Table 9.2 Data for the sections

1 5 10 12 79

3 J1 J JP Po

Table 9.3 Data for the junctions

Integer II Type of data I I I, I I

Number of first column 11 1 1 2 1 1 9 1 1 1 1 I 70

Junctions

9.2 Sections

The nature of the data and their location in a line are presented in Table 9.2. The first column of each line contains the number 2. Subsequent columns

contain the following data: Js - number of the section corresponding to this data line (1 5 J s 5 50). J,, - number of the junction to which the section is attached by its upstream

end (1 I J,, 5 50). J - number defining the type of pressure device. In columns 9 and 10 J

determines the pressure device located at the upstream end of a section; J in columns 16 and 17 the pressure device at the downstream end of a section. The pressure devices are denoted by the numbers

J = 51 Attachment without pressure device; J = 52 Closed pipe-line; J = 53 Local loss; J = 54 Control valve; J = 55 Non-return flap valve permitting discharge in the positive direction

of a section; J = 56 Pump or turbine; J = 57 Non-return flap valve permitting discharge in the negative direction

of a section; J = 58 Condenser; J = 59 Butterfly valve.

J, - number defining the set of parameters of a pressure device. Columns 1 1, 12 relate to the device at the upstream end of a section; columns 18, 19 to that at the downstream end (0 5 J, I 9). Some values of J, may also be chosen within the interval 10 I J , I 99, but only when the conditions introduced at the beginning of Section 9.4 are satisfied.

- number of the junction to which a section is atttached by its downstream

- length of a pipe-line section (decimal, I 2 0; m).

- internal diameter of pipe-line (D > 0; m). - coefficient of friction (A 2 0, dimensionless). - wave velocity (a > 0; m s-l).

- initial discharge of liquid through a pipe-line section (m3 s-').

JJd

1

D 1 a

Qo

end (1 5 Jjd I 50).

The length need not be rounded off to multiples of Ax.

9.3 Junctions The layout and nature of the data for a junction in a pipe-line is presented in Table 9.3.

The first column of each line contains the number 3, followed by the data:

165

Input dato file

J ,

J

- Number of the junction to which the line of data corresponds

- Number defining the type of damping device attached to the junction. (1 I J , 5 50).

The damping devices are denoted by the following numbers: J = 1 Junction without damping device; J = 2 Constant pressure; J = 3 Reservoir; J = 4 Air chamber; J = 5 Surge tank; J = 6 Overflow; J = 7 Air inlet valve and cavitation; J = 8 Integrated damping device; J = 9 Pressure and discharge.

J , - number defining the set of parameters of the damping device (0 I J, I 9). Some values of J , may also be chosen within the interval 10 I J , I 99, though only when the conditions introduced at the beginning of Section 9.4 are satisfied.

po - pressure in junction in the initial state (Pa).

9.4 Parameters

The nature of the parameters of the damping and pressure devices, the location of the data in the lines, together with some other data are presented in Tables 9.4 to 9.7. Changes in the printing interval and the functions may be submitted by various number of values. In the tables, their location is given for the maximum admissible number of values. Values that will not be used in the calculation, need not be submitted. All data lines containing no (or only zero) values in columns 6 to 79 may be also omitted.

The numbers in columns 2 to 5 of the submission of one calculation may not be repeated and the highest number in these columns can be 2000. This has to be taken into account in choosing the numbers Jdc; Jdm; Jd, (Table 9.4), J,, (Table 9.5), J,, (Tables 9.6 and 9.7) and Jg (Table 9.7). Choosing J, within the interval 10 I J , I 99, it should also be remembered that zero values of some numbers in columns 6 to 10 are significant in the calculation, while the data lines with these numbers need not be introduced in the submission. For an integrated damping device, this concerns columns 2 to 5 of the data line with number 400 + J,; line 80 + J , for a valve; lines 160 + J,, 170 + J,, 180 + J,, 240 + J,, 250 + J,, 290 + J, and 330 + J, for a pump and a turbine, and line 440 + J , for a butterfly valve. These lines cannot be utilized for the submission of data for devices other than those to which they correspond according to Tables 9.4 to 9.7 for 0 I J , I 9. The value 10 I J , 5 99 cannot be used for submitting data for either a pump or a turbine.

166

Parameters

The values Jd,, J+Q, J,, , Jd,, J,, , J , may be identical for several devices. The corresponding functions are then submitted only once. When several calculations are submitted within one submission (refer to Sect. 7.14), all the data lines introduced in the submission of the preceding calculations have to be considered as forming part of the new submission, with the exception of the lines which have been changed. The submission used in the calculation is introduced at the beginning of the main output (refer to Sect. 11.1).

9.4.1 Accuracy of the calculation

The data which determine the accuracy of the calculation are listed in the first line of Table 9.4. Besides the numbers 4 and 0, this line also contains the following values:

Nl - maximum number of iterations in the solution for junction ( N , 2 I ) ; d p - admissible difference in pressure in the iterative solution for junctions

d , - admissible difference in discharge in the iterative solution for junctions

d , - admissible difference in pressure in the abridged calculation of the

d,, - admissible difference in discharge in the abridged calculation of the

d , - admissible difference in moments ( d , > 0; N m); e - density of the liquid (e> 0; kg ~n-~) .

If the line with the data determining the accuracy of the calculation is not submitted, the values Nl = 10; dp = 10 Pa; dQ = lo-' m3 s-'; dP = 100 Pa; dQs = m3 s-I; dM = lo-' N m and e = 1000 kg m-3 are automatically used.

(dp > 0; Pa);

(d , > 0; m3 s-');

steady state (dp, > 0; Pa);

steady state (dQs > 0; m3 s-');

9.4.2 Time interval of printing for the main calculation

The time interval of printing for the main output is determined by the value At, submitted in the line beginning with the number 5 (refer to Sect. 9.5). Changes in the time interval of printing of the main output are submitted in accordance with the 2nd and 3rd lines of Table 9.4. Besides the numbers 4, and either 1 or 2 these lines also contain the following values:

N m - number of changes in the time interval of printing for the main output (1 I N , I 6);

167

Table 9.4 Data determining the accuracy of the calculation, the time interval of printing of the main output and the parameters of the damping and pressure devices

Integer II Kind of data Real I

e main ou

(Continued Table 9.4) 0

Type of data

Number of first column

Number of last column ~~ ~

Pump or turbine with fixed characteristics J = 56 (for alternative method of submission for a turbine with variable characteristics and a governor controlled turbine see subsect. 9.4.6, 9.4.7)

Parameters

171

Input data fire

n

a" * v

-

a"

n

a" * v

a"

m

+ - >

4"

172

L 4 w

4

4

4

4

r3 I

Input datafi/e

‘mi

Atrni

- the instant at which the i-th change of the interval of printing occurs

- time interval of printing or the main output after the i-th change

The time data are rounded off to whole multiples of At in the calculation.

(1 I i I N,; fmi < f m ( , + y s);

( - 1 < i I N,; Atmi 2 0; s).

When submitting Atmi = 0, the value At,, = A f is used.

9.4.3 Parameters of the damping devices

Parameters of the damping devices are listed in Table 9.4. The meaning of the variables presented is as follows:

JP - number defining the set of parameters of the damping device. It is

C J + , CJ- - coefficient of loss (TJ+ 2 0; cJ- 2 0; kgmP7); p b ‘b Y Pbabs

S , , S ,

S

DO

Nii

identical to the number J p submitted in Table 9.3.

- pressure in a junction for the damping device in the basic state (Pa); - volume of air in the damping device in the basic state ( v b > 0; m3); - polytropic exponent (I I y I 1.4; dimensionless); - absolute pressure of air in the chamber of the damping device in the

- area of the horizontal section through a surge tank in its lower and

- area of the horizontal section through the chamber of the damping

- diameter of the opening for the air in the chamber of the damping

- number defining the type of arrangement of an integrated damping

basic state (pbabs > 0; Pa);

upper parts (s, > 0, S , > 0; m2);

device (S > 0; m2);

device (Do 2 0; m);

device. The following values relate to individual arrangements: N , = 1 chamber with overflow N , = 2 air chamber N , = 3 air inlet valve

C J + , C,- - coefficients expressing the pressure losses in the connection between the chamber of an integrated damping device and the junction (CJ+ 2 0, CJP 2 0; kgmP7). These coefficients are always considered, whether there is any liquid in the chamber or not;

- total height of the chamber of an integrated damping device

- area of the section through the lowermost part of the chamber of an integrated damping device; the section is perpendicular to the direction of flow (S, > 0; m2). The effect of the inertia of the liquid in the chamber is eliminated from the calculation by submitting large values of S, and S,;

h o t

SI (h,,, > 0; m);

174

Parameters

number determining the lines with input data for an integrated damping device (450 I Jdc I 2000, refer to the conditions at the beginning of Sect. 9.4). When the chamber of the integrated damping device has a constant cross-section, neither the value nor the subsequent data lines containing Jdc are submitted;

coefficients expressing the pressure losses in the connection between an integrated damping device and the junction for QJ 2 0 and Q, < 0 (cJo+ 2 0, cJ0- 2 0; kgm-7). These coefficients are considered only in the case where there is liquid in the chamber of the integrated damping device; number of changes in the cross-section of the chamber of an integrated damping device ( N , 2 0);

coefficients expressing pressure losses for Qj 2 0 and Qj -= 0 at the point of the i-th change in the cross-section of the chamber of an integrated damping device (1 I i I N , ; CJi+ 2 0, C J i - 2 0; kg m-7). The losses are considered only when the i-th change of the cross-section is below the surface of the liquid; elevation of the i-th change in the cross-section of the chamber of an integrated damping device (1 5 i I N , ; 0 < hi < h,,,; hi < h i + , ; m); area of the horizontal section through the chamber of an integrated damping device above the i-th change of the cross-section ( I I i I N , ; si > 0; m2); area of the section through the chamber of an integrated damping device, perpendicular to the direction of flow, above the i-th change of the cross-section (1 I i I N , ; S,i > 0; m2); number determining the lines with input data for the damping devices “pressure” and “discharge” (450 5 JdpQ I 2000. Refer to the conditions at the beginning of Sect. 9.4); beginning and end of the period of the functionfpQ(t) (s). When the values tbp and leP are not submitted, or tb, 2 t,,, the function fpQ(t) is not considered to be periodical; starting time of the damping of the function tpQ(t) (s); coefficient of damping (s-I). In the case of damping, 6 > 0; for an undamped function, 6 = 0; number defining the type of damping device. For a “pressure” damping device, N Q = 0; for a “discharge” damping device,

limit to which the functionfpJ(t) orfQ,(t) tends for t 4 00 and 6 > 0 (for N = 0, it has the dimension Pa, for N Q = 1, its dimension is m3 s-f;

N Q = 1;

175

Input data file

- number of points of the junction f'(t) submitted (NpQ 2 1). The value NpQ, moreover, is limited by the conditions introduced at the beginning of Sect. 9.4;

- time for which the value of the function f p Q ( t ) is submitted

- value of the function &(t ) (1 I i I N d ; for NQ = 0, Pa; for

Different methods of submitting the functionfpl(t) are illustrated in Fig. 9.1. For an aperiodic function, the values tb,, t,,, tba, 6 and p , need not be submitted. Fig. 9.la presents the example of an aperiodic function submitted by using 7 points (Npe = 7). The values of the functionfpQ(t) for t , _< t I t7 are determined by linear interpolation between the submitted values. For t < tl ,

A periodic function can be submitted in the same way as an aperiodic one. To simplify the submission, one may also use the method shown in Fig. 9.1 b. Here, it suffices to submit a periodic function, or a function which is periodic from a specific instant, in the same manner as in the case of an aperiodic function, but only up to the end of the first period. One submits the value tbp corresponding to the beginning of the first period and the value tep corresponding to the end of this period. The values t,, 6 and p , need not be submitted. For t I t,, , the value of the functionfpe(t) is determined as in the case of an aperiodic function. For t > tep, periodic behaviour with a period identical to the first period submitted, is assumed.

Some submitted functions are damped. To simplify the submission in such cases, the following relations are employed to convert the submitted function fpe(t) to the function fpJ(t) (refer to 5.28), or to fQj(t) (refer to 5.29):

for NQ = 0 and t I tb,

N P Q

ti

fpQ(ti)

(1 I i I N p Q ; ti I t i + l ; s);

N Q = 1, m3 s-I).

f p Q ( l ) = fpQ(tl)' for > t79 f&(l) = f&(t7)-

fp&) = f&) (9.1)

( 9 4

fQJ(') = (9.3)

(9.4)

for N Q = 0 and t > tba

fpJ(l) = P, + (fpQ(') - P ~ o ) ~ - ~ ( ~ - ~ ~ )

for N Q = 1 and t I tba

for NQ = 1 and t > t ,

fQJ(t) = P, f ( f&c t ) - Pa) e-d(r-tba)

For 6 = 0, relation (9.2) agrees with (9.1) and relation (9.4) with (9.3). The functions fpJ(t) or &(t) agree with the function f&(t) for all t.

176

Parameters

An example of the submission of a damped function is presented in Fig. 9. lc. The function is submitted by using four points (iVH = 4) as periodic from instant tbp = t 2 . The end of the first period is defined by the value t, = t,. The damping is accounted for by using relation (9.2) from time t = tb, . The example corresponds to the submitted values 6 > 0 and p , > 0.

Fig. 9.1 Examples of different methods of submission for the function f@(t) for a “pressure” damping device: (a) aperiodic function; (b) periodic function; (c) damped function.

177

Input data file

9.4.4 Parameters of the pressure devices

Parameters of a pressure devices are submitted in accordance with Table 9.4. An alternative method of submitting data for a control valve is described in Subsect. 9.4.5, for a turbine with variable characteristic in Subsect. 9.4.6 and for a governor-controlled turbine in Subsect. 9.4.7. The meaning of the variables presented below is as follows:

J P - number determining the set of parameters of the pressure device. It

[+, [- - coefficients of loss ([+ 2 0, c- 2 0; dimensionless); Nu, - number of submitted values for the function fut(t) (1 I Nut I 24); ti - instants for which the values of the function fut(t) are submitted

fut(ti) - values of the function (1 I i 5 Nut; 0 I !,,(ti) I 1 , dimensionless) An example of the submission of the function f,,(t) is presented in Fig. 9.2.

The function is submitted by using four points (Nut = 4).

is identical to the number submitted according to Sect. 9.2;

(1 I i I Nut, ti 5 t i + l ; s);

t t ( t 1

Fig. 9.2 Example ol'the submission of the functionf,,(t) corresponding to the closing of a regulating valve.

The values of the function fgt(t) for Z, < t < t4 are determined by linear interpolation. For t I t , , tut(t) = fut(t,) and for t > t,, fut(t) = fut(t4).

NN - number of disconnections and connections of an electric motor or generator from and to a network (0 I NN I 3).

For NN = 0, the electric motor or generator is connected to the network.

- tN, and tN3 determine the time of disconnection, tN2 the time of connection of the electric motor or generator (1 I i I 3,

- moment of inertia of the revolving parts of the assembly (I 2 0;

- rated or other constant speed of the pump or turbine, for which the

tNi

tNi < tN(i+l); I

n, kg m2);

characteristics are submitted (n, > 0; s-');

178

Parameters

no - initial speed of a pump or turbine (s-');

- constants defining the pressure characteristic of a pump or turbine Bp+ 3

BP-

B M + ,

BE

for n = 0 (Bp+ 2 0, Bp- 50; kgm-7);

BM- - constants defining the moment characteristics of a pump or turbine for n = 0 (BM+ 2 0, B,- I 0; kgm-4);

- constant of the characteristic of the electric motor of a pump (BE 2 0; kgm s );

- synchronous or other constant speed of an electric motor or generator (SKI);

- number of points submitted of the pressure characteristicf,(Q, n) of a pump of turbine for n = +n, (1 5 N,+ I 12);

- discharges for which the values of the characteristics of the pump or the turbine are submitted (1 5 i 5 12, Qi I Qi+l; m s ). The discharges Qi may differ in the submission for each characteristic. The values of the functions always correspond to the values of the variables which occur to the left from them in Table 9.4;

fp+(Qi) - values of the pressure characteristicfp(Q, n) of a pump or turbine for n = +n, (1 I i I N p + ; Pa). For these characteristics (6.42) we have:

2 -4

f lE

N P +

Q i 3 - I

fp(Qi 3 ns) = fp+ (Qi) (9.5 1 An example of the submission of the pressure characteristics of a pump and a turbine for n = n, and n = 0 is shown in Fig. 9.3. The direction of the positive axes is chosen so that the curve of the characteristics corresponds to current practice. For zero speed, the pressure characteristics are determined by relations (6.49) and (6.51).

The characteristics for n = n, are determined by using four points ( N p + = 4) in both cases. Intermediate values along the curve are obtained by linear interpolation between the values fp+(Q1) to fp+(QNp+) for Q1 I Q I QNp+ . Outside this interval, the characteristic is defined by the following relations:

for Q < Q1

ApP = f J Q , 0) - &(QI 9 0) + fP+(Q1)

App = fJQ, 0) - f p ( Q ~ , + 9 0) + f p + ( Q ~ ~ + )

( 9 4

(9.7 )

for Q > QNpt

These relations correspond to a vertical shifting of both ends of the characteristics for n = 0 so that they pass through the submitted end points of the characteristic for n = n,.

179

Input data file

- number of points submitted for the pressure characteristicf,(Q, n) of

fp- (Qi) - values of the pressure characteristicsf,(Q, n) of a pump or turbine for

NP - a pump or turbine for n = - n, (1 I N,- I 12).

n = -n, (1 I i I N p - ; Pa). For these characteristics (6.42) we have:

fp(Qi, -ns) = fp-(Qi) (9.8 1 The pressure characteristic of a pump or turbine for n = - n, is submitted as

for n = +ns (refer to Fig. 9.3). The values Np- and fp-(Qi), however, are substituted for N , , and fp+(Qi).

I '\

b)

i Fig. 9.3 Example of lhe submission of the pressure characteristic for speeds n = nB and n = 0 (a) pump; (b) turbine.

180

Parameters

N M + >

NM- - number of the points submitted for the moment characteristic fM(Q, n) of a pump or turbine for n = +n, and n = -n, (1 I N M + I 12, 1 I N & I 12).

fM+ (QiL fM-(Qi) - values of the moment characteristicf,(Q, n) of a pump or turbine for

n = +n, and n = -n, (1 I i I NM+, 1 I i I NM-; Nm). For this characteristic (6.43) we have:

f d Q i 9 ns) = f~ + (Qi) (9.9) and

Fig. 9.4 Example of the submission of the moment characteristic for speeds n = n, and n = 0: (a) pump; (b) turbine.

An example of the submission of the moment characteristics of a pump and a turbine for n = n, and n = 0 is shown in Fig. 9.4. The values of the moment characteristicf,(Q, n) are determined in a similar manner as the values of the

181

Input data file

pressure characteristicf,(Q, n). For n = 0 they are determined in accordance with relations (6.50) and (6.52). For n = n, they are found by linear interpolation

for Q < Q 1

between the valuesf,+(Ql) tof,+(QNM+) for Q1 5 Q 5 Q N M + y

M = f ~ ( Q 9 0) - f ~ ( Q i 9 0) + f ,+(Qi) (9.1 1)

(9.12)

Fig. 9.5 Example of the submission of the moment characteristic of an electric motor.

The moment characteristic of a pump or turbine for n = - n, is submitted as for n = +n, (refer to Fig. 9.4), but the values N , - andfM-(Qi) are substituted

N E - number of the points submitted for the moment characteristicfE(n) of an electric motor (0 5 N , I 12). For NE = 0, a constant speed n = nE of the pump or the turbine is assumed, provided the electric motor or the generator is connected to the network. The values ni and fE(ni) are not substituted in that case.

- speed for which the values of the moment characteristic fE(n) (1 I i I N , ; ni 5 n,+l ; s-') of an electric motor are submitted. One has to choose nNE = nE.

- values of the moment characteristicfE(n) (1 5 i I N , ; N m) of an electric motor.

An example of the submission of the moment characteristic of an electric motor is presented in Fig. 9.5. The characteristic is submitted by using seven points ( N , = 7).

for N , + and f~ + (Qi).

ni

fE(ni)

182

Parameters

The values of the characteristic for nl I n I nNE are determined by linear interpolation between the submitted values fE(nl) to fE(nN,). For n < nl , they are defined in accordance with the relation

(9.13)

which corresponds to a vertical displacement of the parabola plotted by the dashed line in the figure, so that it passes through the first submitted point.

For n > n E , the values of the characteristic are determined from the relation

(9.14)

It may be observed that the characteristic considered is centrally symmetrical

ME = B E ( ~ E - n)2 - B E ( ~ E - n1)2 + fE(n1)

kf, = -fE(2n, - n)

with the centre of symmetry at point (nE, 0).

c h,, h,

- coefficient of loss (( 2 0, dimensionless). - level of the uppermost and lowermost interconnecting pipes of the

condenser above the bottom of the chambers of an integrated damping device (h, > h > 0; m).

JJ,, JJd - number of the junctions at the upstream and downstream ends of a condenser (1 I J , , I 50, 1 I JJp I 50).

JdV - number defining the lines with input data for a butterfly valve (450 I Jdv I 1979, refer to the conditions at the beginning of Sect. 9.4).

b 7

*V - diameter of a butterfly valve (Dv > 0; m). avo - initial angle of tilt of valve flap of a butterfly valve (avo 2 0; rad). aP - angle at which the oil pump of a butterfly valve is switched off

aVmax - maximum tilt angle of the flap of a butterfly valve (0 < ctVmax. rad); Pv - pressure in the junction at a “butterfly valve” pressure device at

I V - moment of inertia of the flap of a butterfly valve (Iv 2 0; kg m2). NP - number of times of switching off and on of the oil pump of a butterfly

‘pi - instant at which switching off or on of the oil pump of a butterfly

NU+ - number of points submitteh for the function f,+(av)

‘ i - angle, for which the value of the function is submitted (1 I i I 12,

f,+(ai) - value of the functionfu+(av) (1 I i I N,, 0 I f,+(ai) I 1; dimen-

An example of the submission of the functionf,, (av) is shown in Fig. 9.6. The function is submitted by using four points (Nu+ = 4). Its value for al 5 av I a4

automatically;

cavitation (Pa).

valve (0 I N < 6). For N p = 0, the pump is switched on.

valve occurs (1 5 i I 6 , tp i < tp i + l); s).

P -

(1 I Nu+ I 12).

ai I ai+ ; rad).

sionless).

183

Input data file

is determined by linear interpolation between the values submitted. For av < al we have f,+(av) = f,+(al), and for av > a ~ * + We use fD+(av) = fU+(a,vu+). Other functions are submitted in a similar manner.

'I P-

Fig. 9.6 Example of the submission of the function fu +(aV) for a buttefly valve.

No-; N,; NH ; N, - number of points by which the functions f,- (av); fG(av); fH(av);

fv(av), respectively, are submitted (1 I Nu- I 12; 1 5 N , 5 12; 1 I NH I 12; 1 I Nv I 12).

f,- (ail; fG('i);

fH(.i);

&(ai) - function values submitted (0 I &-(ai) I 1; dimension of the functions: dimensionless; N m; N m s2 rad-*; rad s-l).

Within the range of the submitted values ai, the values of the functions are determined by linear interpolation, and outside this range, they are equal to the values of the function for the maximum and minimum ai.

* r - number of the values r for which the functions fQ+(av, r ) and fQ-(av, r ) are submitted (N, 2 1; the maximum value of Nr is limited by the conditions introduced at the beginning of Sect. 9.4).

- values r for which the functions fQ+ (av, r) , f Q - (av, r) are submitted (1 i j I N,, dimensionless).

'i

NQi+ ; NQI- - number of submitted points of the functions fQ+(av, rj);

fQ-(aV,rj); (1 I j I Nr; 1 I N Q ~ + I 12; 1 I N Q ~ - I 12). fa+ (4; fa-(ai) - values of the functions fQ+(av, r); fQ-(av, I ) for r = rj and

av = ai (1 I i I NQi+ or 1 I i I N a - , Nm).

The values of the functionsfQ+(av, r ) andfQ- (av, r ) are determined as for the functionsf,, (av) (refer to Fig. 9.6). Within the range of the submitted values ai, these values are determined by linear interpolation, and outside this range, they

184

Parameters

are taken as constant and equal to the value for the extreme ai. For intermediate values of rj , the values of the functions are also determined by interpolation and outside the submitted range of r j , they are also taken as equal to the submitted values of the function at the maximum and minimum values of rj respectively.

9.4.5 Control valve - alternative method of submission

In the alternative method of submission, the corresponding data lines in Table 9.5 have to be substituted for the lines in Table 9.4, which are numbered 70 + J p to 140 + J p in columns 2 to 5. The values presented in Table 9.5 have the following meaning:

number of the points submitted for the function determining the control regime (1 I Nd I 24); instants for which values for the function fd(t) are submitted

values of the function submitted (1 I i I N,; dimension the same as of parameter d); number defining the lines with input data for the valve (450 I Jdu I 2000, refer to the conditions at the beginning of Sect. 9.4). For Jda # 0, the submission of the control valve is treated as for Table 9.5); number of points submitted for the characteristicfud(d) of the valve (1 I Nod I 24); parameters d for which the values of the characteristic are submitted (1 I i I Nod); di I dimension depends on the type of parameter d); values submitted for the characteristic &(d) of the valve (1 I i I Nud; 0 I fU&) I 1; dimensionless).

(1 < - i I Nd, ti I t i+* ; S);

Fig. 9.7 Example of an alternative submission for a control valve: (a) closing rkgime; (b) characteristic.

185 .

Table 9.5 Data for an alternative method of submission for a control valve

Type of data Integer

Number of first column 1 2 6 20

Number of last column 1 5 10 29

Control valve J = 54 4 7 0 + J p N d ‘I

4 80+ J p J,+, t4

4 9 0 + J p t7

(alternative method of submission)

Real

Pururneters

An example of the alternative submission for a control valve is shown in Fig. 9.7.

The function fd(t) for t , c t < t,, is determined by linear interpolation between the submitted vahesf,(t,); for t < t , we havefd(t) = fd(t1); for t > t,,

The functionfod(d) is determined in a similar manner. Within the range of the values of the parameters submitted, it is determined by linear interpolation, and outside this range, the values of the function is equal to the values for the maximum and minimum parameters submitted.

Within the submission of one calculation, one may, for different parameters J p , employ Table 9.4 for some control valves, and for others, the alternative method of submission according to Table 9.5.

we use f,(t) = fd(t,Vd)’

9.4.6 Turbine with variable characteristics

In submitting data for a turbine with variable characteristics, Table 9.6 has to be substituted for the lines in Table 9.4 which have the numbers 150 + J p to 360 + J p in columns 2 to 5.

The meaning of the values NN; tNi; I; n,; no; BE; nE; NE; ni andfE(ni) is explained in Subsect. 9.4.4. The meaning of the other values is as follows:

- number determining the lines with input data for the turbine characteristics (450 I Jch 5 1980; refer to the conditions at the beginning of Sect. 9.4). If Jch # 0 and Jg = 0 (refer to Table 9.7), the data are submitted in accordance with Table 9.6.

- turbine diameter (D, > 0; m); - number of turbine stages (iT 2 1 dimensionless). For a turbine,

i, = 1 in most cases; - coefficients for converting the parameters a, of the actual turbine

to the parameters ci, B of the turbine model (ca 2 0; cB 2 0; dimension depends on the type of parameter);

- number of points submitted for the functions f,(t) and fB(t) (1 I N, I 18; 1 I Ng I 18);

- instants for which the values of the functions fa(t) and fp(t) are submitted (1 5 i I Nu; 1 I i I Ng; t i 2 t i + , ; s);

- values of the functions (1 I i I N,; 1 5 i 5 NB; the same dimension as that of parameters a or p).

An example of the submission of the adjustment rkgime of the turbine action blades is shown in Fig. 9.8.

187

Input dota file

r"

i

N

z

...

-

...

-

...

-

...

-

... -

... -

-

-

...

-

...

Param

eters

-

n

N

.5

iw

189

Input data file

c

W

N

+ 2

Param

eters

-

...

-

...

-

...

-

...

-

...

-

...

-

-

-

...

-

...

-

-

191

input data file

I Fig. 9.8 Example of the submission of the adjust- I I

The values of the function fa(t) for t l I t I tNa are determined by linear interpolation between the submitted values of the function; for t c t , , we have f,(t) = f,(t,), and for t > t ~ ~ , we have fa@) = f,(t~.).

The functionf@(i) is defined in the same way, but the subscript pis substituted for the subscript a.

JI

b, i;.

ii,

- number introduced in columns 2 to 5 of the last line of the submission

- diameter of the model of the turbine (DT > 0; m); - number of stages of the model of the turbine (G 2 1; dimensionless).

- rated or other constant speed, for which all the following characteristics

The characteristics of the turbine model for one setting of the guide and action blades defined by the parameters ii, 4 are introduced in groups of 25 lines. The first group for the parameters iil , @, is introduced in the lines having the numbers Jch + 1 to Jch + 25, a further group for the parameters ii2, has the numbers Jch + 26 to Jch + 50, etc. Generally, the j-th group has the numbers Jch + 25j - 24 to Jch + 25j in columns 2 to 5 . The last line of the submission of the characteristics of the turbine model has the number J, in columns 2 to 5 .

In the calculation, only the groups having the number 1 in the first line of column 10 are considered.

The symbols ij, 4 are the parameters determining the setting of the guide and action blades of the turbine model, for which thej-th group of characteristics is submitted (j 2 1, the maximum value of the subscript j, is limited by the conditions introduced at the beginning of Sect. 9.4; the dimension of the parameters depends on their nature). The sequence, in which the groups of characteristics for the individual combinations of the parameters aj, are submitted, is irrelevant.

The meaning of the other values of the characteristics was described in Subsect. 9.4.4. The values provided with a bar correspond to the model of the turbine. The submission in Table 9.6 differs from the description in Subsect. 9.4.4 in that the number of points submitted for the characteristics is limited to 18 (1 I Np+ I 18; 1 I N p - I 18; 1 I NM+ I 18; 1 I NM- I 18).

for the Characteristics of the turbine model (470 I J, 5 2000);

For a turbine, 4 = 1 in most cases;

of the turbine model are submitted ( f i , > 0; s-I).

192

Parameters

If closed turbine corresponds to some combination of the parameters ij, 4, then the pressure characteristics are not submitted for this combination and the values NM+ and NM- are introduced with negative sign.

Within one calculation, it is possible, for different values of J p , to use the submission according to Table 9.4 for some turbines or pumps, together with submissions as in Table 9.6 for others.

9.4.7 Governor-controlled turbine

In submitting data for a governor-controlled turbine, Table 9.7 has to be substituted for the lines in Table 9.4 which have the numbers 150 + J p to 360 + J p in columns 2 to 5.

The meaning of the values N , ; tNi ; n, ; no and nE is described in Subsect. 9.4.4. The meaning of the values J,, ; DT ; iT ; c, ; cp and of the values in the lines having numbers Jch to J, in columns 2 to 5 , has also been described in Subsect. 9.4.6.

IT - moment of inertia of the revolving parts of the assemblage (IT 2 0 kg m2);

- number defining the lines with data for the turbine governor (450 5 Jg 5 1920; refer to the conditions at the beginning of Sect. 9.4). The submission of data for a governor-controlled turbine according to Table 9.7 is followed, when J,, # 0 and Jg # 0;

Other values introduced in Table 9.7 have the following meaning:

Jg

I I 0' t1= t2 t (sl

Fig. 9.9 Example of the submission of the functions determining the effect of the electric network on a turbine generator.

193

Input data file

h

u

w

.n

S y1

u

h

VI

u

w

S -

VI

u

h

v t

S

194

Param

eters

Input data file

I

5

- '& Y

...

b" v

* 3

...

h

VI

... b"

VI

*" ..*

h

2 ...

2

v

b"

...

rn I

+ .'.

P

* ... I-

196

Param

eters

I

197

(Continued Table 9.7) 00

Type of data Integer Real

End of input data for the case rnor controls only

which also controls action blades

Param

eters

...

...

-

... -

...

... -

...

...

...

I..

199

(Continued Table 9.7)

Type of data It Integer I I Real I 2

5

J , + 147

J , + 148

J , + 149

J , + 150

J , + 151

J , + 152

J,+ 153

J ,+ 154

J , + 155 4- J , + 156

~~

J , + 159

J , + 160

J , + 161

I I %a1

Param

eters

... -

... -

... -

...

-

...

-

... -

-

...

-

...

-

...

-

-

201

Input data file

H - head (m); - standing static characteristic of the electric speed controller (di-

- initial position of the slide valve of the hydraulic amplifier of the

- initial excess pressure on the turbine (Pa);

- number of points submitted for the functions f , ( t ) ; f , ( t ) ; f c ( t )

- instants for which the values of the functions are submitted

mensionless);

guide and action wheels (m);

cP

xOa; xos

N,; N,; N C

ti

(I I N , 16; 1 I N, I 6; 1 I N , 5 6);

( I 4 i 5 6 ; t i 5 t i + I ; s); h(ti); &(ti); &(t i ) - values of the functions determining the effect of the electric net-

work on the generator of the turbine ( I I i I 6; &(ti) > 0; kg m2; kg m2 s-'; N m).

An example of the submission of the functions f,(t), f,(t) and fc(t), which correspond to an abrupt partial unloading of the electric network, is shown in Fig. 9.9. The values of the functions are determined by linear interpolation when lies within the range of the submitted values t i , while outside this range they are taken as constant, equal to the submitted values for the highest and lowest t i . In the case presented in Fig. 9.9, where several different values of the function are submitted for one instant t i , a value of the function denoted by the lowest subscript i is considered to correspond to t = t i .

Nmax

ti

N,; Np; - number of points submitted for the functionsf,(t); f p ( t ) ; f,,,(t)

- instants for which the values of the functions are submitted (1 I N, I 12; 1 5 N, I 12; 1 I N,,, I 12);

(1 < - i 5 12; t i I t i + s); &(ti); f p ( t i ) ;

fmax(ti) - values of the functions affecting the operation of the governor

An example of the submission of the function fmax(t) for the starting of a turbine is presented in Fig. 9.10.

The values of the functions &(t); fp(t) and fmax(t) are determined by linear interpolation within the range of the submitted values t i ; outside this range, they are constant and equal to the values of the functions for the maximum and minimum values of t i submitted.

- number defining the type of governor employed as denoted below: N , = 1 governor controlling only the guide blades of the turbine;

(fmax($) 2 0; v; v; v).

N ,

202

Parameters

f f ' 0 (v I

0

N , = 2 governor controlling both the guide and action blades of the turbine. Relation (6.98) is used in the calculation; N , = 3 governor controlling both the guide and action blades of the turbine. Relation (6.99) is used in the calculation;

- time constant of the frequency converter (s); - gain of the electric speed controller (C, # 0; dimensionless); -time derivative constant of the electric speed controller (s V-I);

- extreme values of the voltage U, (U2min I UZmax; V).

CC ' d

UZmax U2min ;

L X > O t

'

c

n (i' 1

ff (n,)

Fig. 9.10 Example of the submission of the function Jmx(f) for the starting of a turbine.

203

Input datafile

The values of the function are determined by linear interpolation within the range of a variable submitted; outside this range, they are constant and equal to the values for the submitted maximum and minimum values of the variable.

ni f / ( n i ) - static characteristic of the frequency convertor (V).

-speed of the turbine (s-I),

An example of the submission of the function fh) is shown in Fig. 9.11.

uOi -voltage (V); fb( Uoi)- time constant of elastic feedback (s).

I I I I I

01 UmU02 Urn Uo4 UOR) Fig. 9.12 Example of the submission of the curve of the time constant of elastic feedback as a function of the voltage U,.

An example of the submission of the curve of the time constant of elastic

- constant of feedback of the hydraulic amplifier of the guide and

- gain of the distribution slide valve of the hydraulic amplifier of the

feedback as a function of the voltage is presented in Fig. 9.12.

CFa; CFD

C,, ; CsD

action wheels (V m-I);

guide and action wheels (dimensionless);

I I I 1 I I _

Fig. 9.13 Example of the submission of the function o.(x,) defining the pressure losses due to friction of the slide valve and of the oil distribution of the hydraulic amplifier of the guide wheel.

204

Parameters

CGa; CGp - constant of proportional feedback of the hydraulic amplifier of

ma; mp - mass affecting the motion of the guide and action blades (kg);

U23max - maximum and minimum values of the voltage U2

Cca; Cop - constant expressing the effect of the geometry of the slide valve and of the slave cylinder of the guide and action wheels (kg m-');

Ffa ; FfB - force due to friction affecting the motion of the turbine guide and action blades ( F , 2 0; Ffp 2 0; N);

the guide and action wheels (V m-');

'23min ;

( U 2 m i n 5 U23min 5 U23max 5 U 2 m a x ; V);

Ymin a ; Ymax a ; Yminp;

Ym,xp - extreme positions of the piston of the slave cylinder of the guide and action wheels (Ymina 5 Ymaxa; ymiop 5 ymaxp; m);

Xmin a ; 'maxa ; XminB;

Xmaxp

x a i ; xBi

- extreme positions of the slide valve of the hydraulic amplifier of the guide and action wheels (xmina I xmaxa; xminB 5 xmaxp; m);

- position of the slide valve of the hydraulic amplifier of the guide and action wheels (m);

ca(Xai);

0,dxpi) - function determining the pressure losses of the slide valve and of the oil distribution of the hydraulic amplifier of the guide and action wheels (0 < ca(xai) I 1 ; 0 c op(xpi) 5 1; dimensionless).

An example of the submission of the function oa(xa) is shown in Fig. 9.13.

- extension of the piston rod of the slave cylinder of the guide and yai ; ypi action wheels (m);

fdy + a b a i h

fd,+B(yBi) - values of the functions limiting the extension rate of the piston rod of the slave cylinder of the guide and action wheels ( fdy+a(yai) ' O; fdy+j(Ypi) ' O ; '-'I;

fdy - a(yai);

fdy-p(ypi) - values of the functions limiting the insertion rate of the slave- cylinder piston rod controlling the guide and action wheels

An example of the submission of the functions limiting the velocity of the (fdy-a(Yai) < O ; fdy-B(Ypi) < O; '-')*

slave-cylinder piston rod controlling the guide wheels is shown in Fig. 9.14.

205

Input data file

0

-L(yai); f s ( yp i ) - values of the functions expressing the geometrical relations bet-

ween the extension of the slave-cylinder piston rod and the parameter CI or /3 defining the position of the turbine guide or action wheel (dimension depends on the type of parameter).

$y.a(YrU I \fdy+ a (Ya 1

" fdy t a (Yal

I I

= Ymin ly a1 = Y a 2 Ymax Y M h )

f d y - a ( Y a l ) fdy - a lya 1

" I

Yd2 Ye Ya&=Ymox ymin a = YUI

Fig. 9.15 Example of the submission of the relationship between the parameter a determining the position of the turbine guide wheel, and the extension y, of the slave-cylinder piston rod.

An example of the submission of the function f,(rxi) is shown in Fig. 9.15.

NApa; NAPp - number of the excess pressure values Ap, on the turbine, for which the functions fe,(y,, Ap,) or Lp(yp, Ap,) are submitted

Appj; Appllj- values of the excess pressure Ap,, for which the function (1 I NApa I 10; 1 I NA,p I 10).

L a ( Y a , A ~ p j ) or fep(Yp, AP&) is submitted ( A ~ p j < Appa(j+ 1) ; Appb < Appb(j+ 1) ; Pa)*

feaj(yai); fepi(ypi) - values of the function f a ( Y , i APpaj) or feP(Yji Appgi); (m2).

206

Parameters

An example of the submission of the function f,(y,, Ap,) for NAP, = 2; Neal = 4, Nfa2 = 3 is presented in Fig. 9.16.

For App within the range of the submitted values Ap . and ApPB the value of the function f , , (ya, Ap,) or LP(yP, App) is determinerby linear interpolation between the submitted values of the functions. Outside this range, the values of the functions are considered to be equal to those for the maximum and minimum Appaj or Apppj submitted.

Fop(xpi) Foa(xai);

- values of the function determining the force by which oil acts on the slave-cylinder piston of the guide and action wheels at the steady state (N).

Fig. 9.16 Example of the submission of the function f,(ym, App) defining the force by which the liquid flowing through the pipe-line acts on the turbine guide blades.

I I

I Fig. 9.17 Example of the submission of the function F&(x.) defining the force by which oil acts on the slave-cylinder piston rod of the guide wheel in the steady state.

207

Input data file

An example of the submission of the function Foa(xa) is shown in Fig. 9.17. The lines of the input data having numbers Jg + 85 to Jg + 214 in columns 2 to 5 are submitted only in the case where the governor controls both the guide and action wheels of the turbine, that is, for N g = 2 or N g = 3.

N" - number of values of the head H for which the functionf,( U , , H) (for N g = 2), or the function fH(ya, H) (for N g = 3), is submitted.

- head for which the functionfH(U4, Hi) (for N g = 2), or the function fH(ya, H j ) (for N g = 3) is submitted (Hi < H j + , ; m);

- for N g = 2, voltage U,, for which the value of the function fH( U,i, Hi) is submitted (V). For Ng = 3, instead of the values U,i the values yai are presented, for which the values of the functions fH(yai, Hj) are submitted (m).

For N = 3, the valuesfHj(yai) are submitted instead of the values fHj(U43. They are the values of the function fH(yai, Hi) (V).

For H within the range of the values Hi submitted, the value of the function fH( U, , H) orf,(y,, H) is determined by linear interpolation between the submitted values of the functions. Outside this range, the values of the functions are taken as equal to the values for the maximum and minimum Hi submitted. For intermediate values of U, or y, the values of the functions are determined by linear interpolation, and outside the submitted range of U, or y they are taken as equal to the values for the extreme values of U, or y , submitted.

The method of submission of the functions fH(U4, H) and fH(y,, H) is the same as of the functions f,(y,, App) (refer to Fig. 9.16).

(1 I N, I 10); Hi

'4i

fHi(Uj i ) - for Ng = 2, the values of the functionfH(U4i, Hi) (V).

a.'

9.5 Type of calculation

The layout of the lines containing the data determining the type of calculation is shown in Table 9.8.

The first column of the line contains the number 5 and the following data:

Ns - total number of sections in the pipe-line system analysed (1 I Ns 5 50).

N, - total number of junctions in the pipe-line system analysed (1 I N , I 50).

2 - number defining the type of calculation (1 I 2 5 8). The individual calculations are denoted by the following numbers: Z = 1 calculation of water hammer without an iterative solution for

the junctions;

208

Table 9.8 Data defining the type of calculation

Input data file

2 = 2 calculation of water hammer using an iterative solution for the junctions;

2 = 3 abridged calculation of the steady state without an iterative solution for the junctions;

2 = 4 abridged calculation of the steady state using an iterative solution for the junctions;

2 = 5 calculation of water hammer without an iterative solution for the junctions. The initial values employed are those determined in the preceding abridged calculation of the steady state;

2 = 6 calculation of water hammer using an iterative solution for the junctions. The initial values employed are those determined in the preceding abridged calculation of the steady state;

2 = 7 abridged calculation of the steady state without an iterative solution for the junctions. The initial values employed are those determined in the preceding abridged calculation of the steady state;

2 = 8 abridged calculation of the steady state using an iterative solution for the junctions. The initial values employed are those determined in the preceding abridged calculation of the steady state;

At - time interval of the calculation (At > 0, s); At, - time interval of printing for the main output (At, 2 0, s); Ax, - longitudinal interval of printing for the main output (Ax, 2 0, m); a, - wave velocity for the abridged calculation of the steady state (a, > 0,

m s-I). The value a, need not be submitted for 2 = I , 2, 5 and 6; t,,, - maximum value of the calculation time (tmax > 0, s).

9.6 Graphical and numerical outputs

The layout of the lines containing the data for the graphical and numerical outputs is shown in Table 9.9.

The first column of each line contains the number 6, in the second column are the numbers 1 to 9. Each data line having the numbers 1 to 5 in the second column, determines one variable presented in the graphical and numerical outputs. The number in the second column indicates the sequence of the variables in the graphical and numerical outputs.

‘x i

L,i, L,i, - numbers defining the variables introduced in the graphical and

numerical outputs (1 5 i I 5) . Their values are listed in Table 9.10;

210

Table 9.9 Data for the graphical and numerical outputs

Type of data I Number of first column

Number of last column

Graphical and numerical outputs

Integer Real I Characters

Graphical and numerical outputs

Mark, - symbol defining a variable in the graphical and numerical outputs (1 I i I 5; any symbol);

Text, - verbal designation of a variable in the graphical and numerical outputs (1 I i I 5; up to 45 symbols);

Mini, Max, - values of the variables related to the left-hand and right-hand margins of the graph in the graphical output (1 I i I 5, Mini c Maxi , dimension is that of the variable plotted). Mini and Max, need not be submitted if a graphical output is not used.

Each data line having the number 6 in the first column and the numbers 6 to 9 in the second column determines a change in the time interval of printing of the graphical and numerical outputs.

Gr,; Nui - number defining whether the change concerns the graphical or the numerical output (1 I i I 5; 0 5 Gri 5 1; 0 I Nu, 5 1). For Gri = 1 or Nui = 1, a change will be effected for the graphical or the numerical output. For Gri = 0 or Nui = 0, no change will be made;

- instant at which a change in the time interval of printing will occur (I I i I 5; s);

- new value for the time interval of printing (1 I i I 5; s). In submitting Atgn, < At, Atp , = At will be assumed.

Each variable in Table 9.10 is defined by the values LSi, LKi , &, which appear

tgni

Atwi

next to it. The meaning of the quantities listed in Table 9.10 is as follows:

J S x /Ax

P J ; QJ

JJ AV

V

vo

- pressure and discharge at a definite point along a section of the

- number of section (1 5 Js I 50); - distance of a point from the upstream end of a section in multiples

of Ax = aAt (0 I x/Ax I 2000; integer, dimensionless). The distan- ces are considered after modification of the length 1 of the section to the value 1, (refer to Sect. 7.5). Hence, for the upstream end of a section x/Ax = 0 and for the downstream end x/Ax = 1JAx;

- pressure in, and discharge from a junction into a damping device (Pa; m3 s-I);

- number ofjunction (1 I JJ I 50); - difference in the volume of liquid in the chamber of an air chamber,

- volume of liquid in the chamber of an integrated damping device

- volume of air in the chamber of an air inlet valve, or the volume of

- denotation of the variables A K V and V, in the main output (m3);

pipe-line (Pa; m3 s-');

a surge tank or an overflow relative to the basic state (m3);

(m3);

the empty space at cavitation (m3);

213

Input datafile

m

V1

v 2 pabs

4 d

UV n VN

1 or2

duv/dt

M VN1

MQ

ME

MH

ApP

VN2

- mass of the air in an air inlet valve of an integrated damping device

- denotation of the variable m in the main output (kg); - absolute pressure in the chamber of an air inlet valve (Pa); - denotation of the variable Pabs in the main output (Pa); - level of the liquid in an integrated damping device (m); - variable defining the pressure loss in a control valve (dimensionless); - angle of tilt of a flap of a butterfly valve (rad); - speed of a turbine or pump (s-I); - denotation of the variables 0, uv, n in the main output (dimension of

the respective variable); - the value Lxi = 1 is introduced for a pressure device installed at the

upstream end of a section, Lxi = 2 for one installed at the downstream end of a section;

- rate of change of the angle of tilt of the flap of a butterfly valve (rad s-I);

- torsional moment of a pump or turbine (N m); - denotation of the variables duv/dt and M in the main output (dimen-

- moment by which the liquid flowing through a pipe-line acts on a

- torsional moment of the electric motor of a pump or the generator

- denotation of the variables Mo and M E in the main output (N m); - moment by which the hydraulic system acts on a butterfly valve

- difference in pressure in front of, and behind a butterfly valve (Pa).

arranged as an air inlet valve (kg);

sion of the respective variable);

butterfly valve (N m);

of a turbine (N m);

(N m);

The variables listed in the right-hand part of the table relate only to a turbine controlled by a governor.

Ub; Ui; u;; y; V,; V,

x a ; xB

y,; ys

a; B

Appg

214

- voltage (V). The apostrophe indicates that the values are introduced

- position of the slide valve of the hydraulic amplifier of the guide and

- extension of the slave-cylinder piston rod of the guide and action

- parameters determining the position of the guide and action blades

- difference in pressure in front of and behind a governor-controlled

for time t - At.

action wheels (m);

blades (m);

(dimension depends on the type of parameter);

turbine (Pa).

Subtitle of the calculation

9.7 Subtitle of the calculation

The layout of this line is shown in Table 9.1 1. The first column contains the number 7; the remainder of the line consists of the subtitle of the calculation (a maximum of 45 arbitrary symbols).

Table 9.1 I Subtitle of the calculation

Number of first

215

10 WTHD program for creating the input data file

The input data file may be created with the aid of various editors on the basis of the description given in Chapter 9. A special program named WTHD was prepared to simplify the work of assembling the data and to make checking easier. The program is written in Turbopascal 5.0 language. It appears on the WTHM diskette in translated form.

The WTHD program allows us to either establish a new data file or to modify an already existing file.

With the aid of the program, additional data lines may be added at the end of a file. The data are entered automatically in the correct columns. The data lines may be inserted at any place in the file and any data line may be deleted or changed. Another data file may be added at the end of a data file. With the aid of the WTHD program, the data file may be checked formally, displayed on a screen, printed by a printer or stored on a disk.

The WTHD program also enables us to plot the submitted functions. This assists further checking of the input values submitted.

10.1 Starting work with the WTHD program

The WTHD program is called up by entering the statement WTHD. Following this, preliminary information on the program appears on the screen, together with the request to submit the name of the new input file. The name submitted has to comply to the requirements of the DOS operating system and must not exceed 14 characters (the maximum length is indicated on the screen). When the input file has been created, it will be stored under this name. If the name of the file is not submitted, the name DATA.WTH will be used.

After pressing the RETURN key, a query appears on the screen as to whether the file will be read from the disk. If the answer is Y, the submission of the name of this file is requested. The maximum length of the name again is 14 characters. If no name is submitted, the name of the new input file is used. After pressing the RETURN key, the chosen file is read from the disk and the main menu of the program appears on the screen.

216

Inserting a data line into the j l e

If the answer to the query concerning the reading of the file from the disk is N or RETURN, the main menu of the program is called up directly.

The functions of the program described in the following sections are called up by the keys identified in the main menu. After a function has been completed, the program returns the main menu.

The program is terminated only after pressing Q or in the case of certain types of error.

10.2 Adding a data line to the end of the file

From the main menu, the function of adding a data line to the end of a file is selected by pressing A or RETURN.

A request to specify a number in column 1 then appears on the screen. Any one of the numbers 1 to 7 may be entered (refer to Sects. 9.1 to 9.7). After a number has been specified, a request to submit other data corresponding to Tables 9. I to 9.1 1 appears on the screen. Simultaneously with this request, basic information about the type of data and the admissible values is presented. The maximum permissible length of the data is also marked out on the screen. Decimal data may contain decimal points. If this is not the case, a decimal point is assumed behind the last numeral submitted. The exponential form may be also used. The submission of each of the data is terminated by pressing the RE- TURN key. If any of the data is not submitted, it suffices to press RETURN. When all the data in a data line have been submitted, the whole line appears on the screen together with a query as to whether it should be added to the data file. The data submitted are located in the line in columns corresponding to Tables 9.1 to 9.1 1. If the answer is Y or RETURN, the line submitted is placed at the end of the input file and the program returns to the main menu. If the answer is N, the new data line is deleted and the program again returns to the main menu.

10.3 Inserting a data line into the file

From the main menu, a data line may be inserted into the data file by pressing the I key.

The first line of the data file appears on the screen together with instructions on further procedure. The individual lines of the data file may be successively read with the aid of the up and down cursor keys. The lines read appear on the screen. If a data line in front of which a new line should be placed, appears on the screen, RETURN has to be pressed. The further procedure for inserting a new data line is the same as described in Sect. 10.2 for adding a data line at the

217

WTHD program for creating the input data file

end of a file. The new data line, however, will not be placed at the end of the file, but in front of the line selected.

Prior to returning to the main menu, a query appears on the screen, as to whether a further data line should be inserted. If the answer is Y or RETURN, an instruction on how to proceed further again appears on the screen together with a data line. This is not, however, the first line, but the line chosen in the preceding cycle. The procedure described in this section is then repeated.

When the answer is N, the program returns to the main menu.

10.4 Deleting a data line from the file

From the main menu, the function of deleting a line from the data file is selected by pressing D.

The first line of the data file then appears on the screen, together with instructions on further procedure. The lines of the input file are read and written successively with the aid of the cursor keys.

If a line which should be deleted has appeared on the screen, D is pressed. This deletes the line and the next line of the file appears. Further lines may be selected by means of the cursor keys and then deleted with D. The program returns to the main menu after pressing the Esc key.

10.5 Modifying a data line

From the main menu, the function of modifying a line in the data file is selected by pressing M.

Instructions on further procedure, together with the first line of the data file appear on the screen. The individual data lines are read and written successively on the screen with the aid of the cursor keys. When a line that requires modification appears on the screen, RETURN is pressed. Then, new instructions on further procedure, the original data line and a new data line coincident with the original one, appear on the screen. The arrangement of the data in the line, corresponding to Tables 9.1 to 9.1 1 and expressed graphically, appears below the data lines. The arrangement is governed by that of the original data line.

The cursor may be moved along the new data line to the left and right with the aid of the cursor keys. The characters on which the cursor is placed may be transcribed from the keyboard. The formal correctness of the changes made is not checked by the program at this stage.

When the changes in the data line have all been made, RETURN must be pressed. In this way, a new line is substituted for the original line in the input file and a query appears on the screen, whether a further data line requires modifica-

218

Listing of a data file

tion. When the answer is Y or RETURN, the instructions again appear on the screen together with a further data line of the file. The procedure described above is repeated. When the answer is N, the program returns to the main menu.

10.6 Adding a further data file

From the main menu, it is possible to add a further data file to the end of the original data file by pressing E A request for submitting the name of the file to be added, appears on the screen. The name of the existing data file must be entered. The maximum length of the name is 14 characters. It is displayed on the screen. After pressing RETURN, the file entered is read and added to the end of the data file. The program returns to the main menu.

10.7 Formal checking of the file

Formal checking of a data file is carried out from the main menu after pressing C.

Instructions on further procedure and a description of errors found in the data file appear on the screen. Each description of an error is accompanied by the first five characters of the data line, where the error was found. After each error listing, RETURN should be pressed to continue the checking. After the entire input file has been checked, the total number of errors found appears on the screen. The program returns to the main menu after pressing Esc.

10.8 Listing of a data file

A data file may be listed on the screen or printed out. The first is realized by pressing L, the other by pressing €?

For listing on the screen, ENTER has to be pressed after each line has been displayed. The program returns to the main menu after pressing Esc.

After listing the data file on the printer, the program returns to the main menu.

The printing is interrupted by pressing Esc. The program then returns to the main menu.

219

WTHD program lor creating the input data file

10.9 Plotting of functions

From the main menu, the function of plotting the graphs of the submitted functions on the screen, is selected by pressing G.

Instruction on further procedure and the first line of the data file appear on the screen. The individual lines of the data file are read and displayed with the aid of the cursor keys. When the first line of the function to be plotted appears on the screen, RETURN is pressed.

A request now appears on the screen asking for the values MuxX; M i n x MuxFx; MinFx, which determine which part of the graph should be plotted and at what scale. This is followed by a request for entering the values B+; B- and Xb which determine the function outside the range of the submitted values of the variable.

The dimensions of the area on which the graph is plotted are always the same. The values of the variable are plotted along the horizontal axis and denoted X and the values of the function are plotted along the vertical axis and denoted Fce( X).

MuxX and MinX are the values which the variable will have at the right- and left-hand margins of the graph, respectively. MuxFx and MinFx are the values of the function Fce(X) at the top and bottom margin of the graph, respectively.

If there are no special demands concerning the form of the graph, the values MuxX, Minx, MuxFx and MinFx need not be submitted. The program will specify them automatically. The values submitted must satisfy the conditions MuxX > MinX; MuxFx > MinFx. Within the range of the values Xsubmitted, the function is determined by linear interpolation between the values Fce( X ) submitted.

The values B+; B-; Xb need not be submitted either; it suffices to press RETURN. Their values are then taken to be equal to zero and the value of the function outside the submitted values X is regarded as constant.

It is convenient to submit the values B+, B-, Xb when plotting the characteristics of pumps, turbines and electric motors (refer to Subsect. 9.4.4), because the function Fce(X) outside the range of the submitted values X is determined from the relation

for X < X,

Fce(X) = Fce(X,) + [ ( X b - X)2 - (Xb - X,)’] B -

and for X > X,

Fce(X) = Fce(X,) + [ (Xb - X)2 - (Xb - X, ) ’ ]B+ (10.1)

X , to X , are the submitted values of the variable and Fce(X,) to Fce(XN) are the

220

Plotting of functions

submitted values of the function. In plotting the pressure characteristic of a turbine or a pump, B+ = Bp.; B- = Bp- and Xb = 0. In plotting the moment characteristic of a turbine or a pump, B+ = BM+; B - = B M- and Xb = 0. In plotting the moment characteristic of an electric motor, B+ = B- = BE and Xb = n E .

When the submission of the values introduced above is completed, the graph of the function as submitted, appears on the screen together with a query, whether the graph of a further function should be plotted in the same screen image. After answering N or RETURN, any text may be written under the graph using the keyboard; this is mentioned in a note which appears on the screen. This note is overwritten by the new text. Adding an explanatory text to the graph may be useful when the graph is printed. Printing is effected by using the PrtSc key, provided the GRAPHICS statement had been used prior to commencing work with the WTHD program.

After pressing RETURN, the graphs of the functions disappear from the screen and query appears as to whether further graphs will be plotted. If the answer is N or RETURN, the program returns to the main menu. If the answer is Y, an instruction on further procedure appears on the screen together with a data line, as after pressing G in the main menu. However, in this case, the line, which follows the entry for the plotted function appears. The further procedure is the same as described earlier in this section.

If, after plotting the graph on the screen, the question on plotting another graph in the same image is answered by X the data line which follows the entry for the plotted function, appears below the graph. Further data lines from the file may be successively written on the screen using the cursor keys. If the first line of a further function, which should be plotted on the graph, appears on the screen, RETURN is to be pressed. The requests for submitting the values B+; B- and Xb then appear successively on the screen. If these values are not submitted, they are taken as equal to zero. The values MaxX; M i n x MuxFx; MinFx are considered to be the same as in the graph plotted earlier. The submission is terminated by pressing RETURN.

Now follows a query whether the new graph should actually be plotted in the original image. If the answer is Y or RETURN, the graph of the new function is superimposed on the preceding image. With the answer N, the image of the function is not plotted. In either case, a query appears on the screen, whether the graph of a further function should be plotted on the original image. The subsequent procedure is similar to that already described in this section. One image may thus contain the graphs of several functions.

22 1

WTHD program for creating the input data file

10.10 Terminating the WTHD program

In working with the program, the data file may be stored on the disk at any time by pressing S while the main menu is on the screen. The file stored will have the name chosen at the beginning of the work with the WTHD program.

From the main menu the WTHD program is terminated by pressing Q. A query appears concerning the storage of the file. With the answer Y or RE- TURN, the new data file is stored. If the answer is N, it is not stored. If it had not been stored previously by pressing S, it is deleted after the answer N.

The control then reverts to the DOS system.

222

11 Output of the results

The results of the calculation of water hammer may be obtained in three forms: main output, graphical output and numerical output. Each of these outputs can be displayed on a screen, printed by a printer or stored on a disk. Individual parts of the numerical output stored on a disk can, in addition, be plotted in the form of a graph on the screen and, if necessary, copied by using the printer (refer to Sect. 1 1.4). Examples of the different types of output are presented in Chapter 14. The type of output is selected at the beginning of the calculation of water hammer. Several types of output may be chosen simultaneously (refer to Sect.

The size of each individual'output is determined by the input data file (refer to Sect. 7.3). The individual types of output are discussed in the subsequent sections.

In view of the limited possibilities of the programming language employed, it was impossible to use exactly the same symbols to denote the values in the output files as is used in the text of the present book. Where the symbols employed differ they are shown in parentheses.

7.4).

11.1 Main output

In the main output, all the input data are presented first, with the exception of the data concerning the graphical and numerical outputs only; then follow the results of the calculation and, finally, the maximum and minimum values and data on the accuracy of the calculation.

At the start of the main output the following information is presented: the name of the method of calculation employed, the name of data file used; the number of the calculation (Num), the name of the calculation (Title), and the secondary name of the calculation (Subtile).

The basic data concerning the calculation follow: the number of sections N s (Ns); the number of junctions N , (N'); the type of calculation 2; the time interval of the calculation At (Dt); the time interval of printing of the main output At , (Dtrn); the longitudinal interval of printing of the main output

223

Output of the results

Ax, (Dxrn); the wave velocity a, (us) for the abridged calculation of the steady state and the maximum calculation time t,,, (trnax).

The next part contains the data on the sections. They are arranged after the numbers J , (Js) which denote the individual sections. Each section is characterized by the folowing quantities: the number JJ (Jj) of the junction to which the section is attached by its upstream and downstream ends; the number Jdefining the type of pressure device through which the section is attached; the number J p (Jp) defining the set of parameters of the pressure devices; the submitted length of the section; the modified length I, (k) of the section; the inside diameter D of the pipe-line; the coefficient of friction Iz (lambda); the wave velocity u and the initial discharge Q, (Qo).

Then follow the data on the junctions. They are arranged according to the numbers JJ (Jj) of the junctions. Each junction is characterized by the following quantities: the number JJ (Jj) of the junction; the number J defining the type of damping device attached; the number J p (Jp) determining the set of the damping device, and the initial pressure po (po) in the junction.

The submitted parameters of the damping and pressure device and additional data given in the input file on the lines beginning with the number 4 are introduced next. They are arranged according to the numbers in columns 2 to 5 and denoted by the numbers of the columns corresponding to Tables 9.4 to 9.7.

Next are the results of the calculations, presented in groups. Each group contains the results for one time t.

The data appertaining to the sections are denoted by the number J , (Js) of the section. Values for the pressure p and the discharge Q are presented for each section at its upstream and downstream ends as well as at all points corresponding to the longitudinal interval Ax, (Dxrn) of printing submitted for the main output. The distance x of point from the upstream end of a section is always presented before the respective values of p and Q.

The group of results includes the data for the junctions denoted by the number JJ (Jj) of a junction. The results include values for the pressure pJ (pj) in the junction and the values of the parameters VO, VI or V2 as the case may be. The values of the parameters are given only for the junctions to which damping devices are attached and to which these parameters apply (refer to Sect. 7.8).

If the hydraulic system analysed includes pressure device to which the parameters VN, VNZ or VN2 apply (refer to Sect. 7.9), their values are also included in the results. In front of the values of these parameters, the number Js (Js) of the section, to which the pressure equipment is attached, is shown together with information on whether the pressure device is attached to the upstream or downstream end of the section.

In the final part of the main output, the maximum and minimum values for pressure pmin (prnin) and pmax (prnax), which appeared in the section during the calculation, are presented for each section Js (Js). For any junctions JJ (a), to

224

CraphicaI output

which damping devices, to which the parameter VO applies, are attached, the extreme values VOmin ( VOmin) and VO,,,,, (VOmax) of this parameter are included. For pressure devices to which the parameter V N applies, the extreme values VNmin ( VNmin) and VN,,,,, (VNmax) of this paramater are shown. In front of the extreme values, the number Js (Js) of the section to which the pressure device is attached, is given, together with the information on wether it is attached to the upstream or downstream end of the section.

The main output is concluded by the values dp (dp); do (dQ) and dM (dM), that is, by the values of the neglected differences in pressure, discharge and moment, respectively, that had not been exceeded during the calculation (refer to Sect.

The main output often is rather extensive. Hence, it is convenient to limit its size by a suitable choice of the values At,,,, Ax,,, (refer to Table 9.8) and, if need be, the values t,, to tm6 and At,,,, to Atm6 (refer to Table 9.4).

7.1 1).

11.2 Graphical output

The graphical output contains all the input data which determines which variables will be plotted and in what form. This is followed by a graphical plot of the variables and their maximum and minimum values.

The graphical output begins with the name of the method of calculation employed, the name of the data file, the number of the calculation (Num), the name of the calculation (Title) and the secondary name of the calculation (Subtitle). Then follows the verbal description of the variables denoted “Text ”, the symbol “Mark” denoting the variable and used in the plotting of the variable, and the values Min and Max of the variable corresponding to the margins of the graph. This is followed by the time interval At (Dt) of calculation; the numbers Gr and Nu defining the changes in time interval of printing of the graphical and numerical outputs; the instant t , ( tgn) of the chage in this time interval of printig; the new time interval At,, (Dtgn) of printing; the denotation “Mark” of the variable, and the values Ls (Ls), L, (Lk) and L, (Lx) defining the type of variable (refer to Sect. 9.6).

Then follow plots of the variables in graphical form. Each line of the output corresponds to time t, which is always presented on the left-hand margin. The values of the variables are expressed by the position of the symbols denoting the variable in the line. 66 columns are reserved for plotting the variables. The first column corresponds to the value Min and the last to the value Max. The position of a symbol in a line is determined by linear interpolation between these two extreme values. The number of a column is determined by rounding off the values calculated. The reserved field is divided by vertical dotted lines into five equal parts for easier orientation. The values of the variables corresponding to

225


these lines are shown at the beginning of the graph. They are presented individually for each variable plotted. The variables are denoted by their symbols.

When the value of a variable is outside the range of the maximum and minimum values submitted, the symbol of the variable is plotted either in the left of right-hand margin of the plot as the case may be.

The size of the graphical output can be limited by submitting the values Gr,; tFni and Atgni (refer to Sect. 9.6). Lines corresponding to the instants at which the time interval of printing changes are marked with dots.

11.3 Numerical output

The numerical output begins with the input data defining which variables will be presented and in what time interval. The next part of the output contains calculated values of the variables, while their extreme values are shown at the end of the output.

The first part of the numerical output up to the printing of the results is identical to the graphical output. Only the values Min and Max which have no meaning for the numerical output, are omitted.

The calculated values of the variables are printed in numerical form. Each line contains the variables corresponding to one instant t shown along the left-hand margin. The variables are denoted by their symbols a t the beginning of the columns.

At the end of the numerical output, the extreme values of the variables are presented in the same way as in the graphical output.

The size of the numerical output can be limited by submitting suitable values for Nu,; tgni and At,,, (refer to Sect. 9.6).

11.4 WTHG program for plotting the numerical output

The graphical output described in Sect. 11.2 is convenient for following the progress of the calculation and for readily obtaining an idea of how the individual variables behave during the course of the calculation. The values of the functions can be read from this output only approximately. A change in the scale used for plottini the variables and a change in the time scale requires a new calculation; in addition the possibilities of choosing the time scale are rather limited. In practice the graphical form is not always very convenient for all applications, particularly on account of its size.

In view of this, a special program called WTHG was written. I t allows us to plot individual variables using the numerical output store on a disk while calculating water hammer is in progress.

226

WTHG program for plotting the numerical outpur

In the DOS system, the WTHG program is called-up using the WTHG statement. The WTHG program plots the graphs of the functions on the screen. If the graphs are to be copied from the screen to the printer, the statement GRAPHICS has to be submitted prior to the statement WTHG in the DOS system.

After the statement WTHG has been submitted, the basic information on the program appears on the screen together with a request for introducing the name of the input file. I t is necessary to give the name of the file in which the numerical output of the results of the calculation was entered by the WTHM program. The results of only one single calculation may be entered in the file. The name of the file has to comply with the DOS operating system employed and should not be longer than 14 characters. The maximum length of the name is marked on the screen. If no name is introduced and only RETURN is pressed, the program assumes that the name of the file is NOUT.WTH as presented in brackets.

Then, the input file is read and the verbal denotation of the variables entered in the file appears on the screen. They are marked with the numbers 1 to 5. If the file contains less than five variables, some of the numbers are not used. A variable may be selected by pressing the corresponding number. This request also appears on the screen.

Now, successive requets appear on the screen for submitting the numbers MinT; MaxT; MinF and MaxF, determining which of the values of the chosen variable will be plotted and at what scale. Time is plotted along the horizontal axis, the values of the variables along the vertical axis. The overall dimensions of the graph are always the same. MinT and MaxT are time values corresponding to the left-hand and right-hand margins of the graph. MinF and MaxF correspond to its lower and upper margins. For the values submitted, it is necessary that MinT < MaxT and MinF < MaxF.

If the required values are not submitted and only RETURN is pressed, the values in brackets will be used. These are the maximum and minimum values taken from the input file so that the entire graph of the variable is plotted.

When the requested values have been entered, a query appears on the screen, whether the maximum and minimum values submitted should be changed. If the answer is Y, the submission of the extreme values is repeated. However, the values from the preceding submission are predefined. When the answer is N or RETURN, the graph of the chosen variable is plotted on the screen.

In the graph, time is denoted t and the values of the variable F(t). The values of the variable are determined by linear interpolation with respect to time between the values introduced in the input file. In the lower part of the screen the verbal denotation of the variable appears and a query whether the graph should be modified. I t may be modified by adding graphs for further variables and by complementing the text. With the answer N or RETURN, the graphs are deleted and a query appears on the screen whether another graph will be plotted.

221


When the answer is again N or RETURN, the WTHG program is interrupted and control is returned to the DOS system.

If the query concerning the plotting of another graph is answered by Y, a request appears on the screen for submitting the name of the new input file. The resulting situation and further procedure are identical to those already described in the present section.

After the answer Y is given to the query concerning a modification of the graph, a query appears on the screen, whether another graph should be plotted on the same image. If the answer is Y, a request appears on the screen for submitting the name of the input file. This again is the situation described earlier with the difference that the image will contain both the newly chosen variable as well as the variable plotted in the preceding cycle. In this way, one image may contain the graphs of several variables.

After the answer N is given to the query on plotting a further graph in the same image, a further query appears, whether the text will be modified. With the answer N or RETURN, the situation described earlier arises, namely one where a query appears on the screen whether another graph will be plotted.

The answer Y to the inquiry about the modification of the text may be useful before the graph is copied to the printer. This allows us to complement or change the text anywhere on the screen (with the exception of the right-hand bottom corner), although in a rather complicated manner. First, the position of the first symbol of the new text is to be submitted by means of four figures. The first two figures, that is 01 to 80, indicate the column and the next two figures, 01 to 25, the line of the symbol on the screen. The columns and the lines are counted from the upper left-hand corner of the screen, which is identified by the number 0101. The right-hand bottom corner is defined by the number 8025. The number which has been entered appears in the lower part of the screen on the left; RETURN has to be pressed and the writing of the text can start. The text which is written in the chosen line begins in the chosen column. The text has to be terminated by pressing RETURN before the end of the line. An asterisk then appears at position 0124. This means that another text may be written on the screen applying the same method. The four figures denoting the column and the line of the screen, RETURN, the text and again RETURN, have to be entered anew. The procedure may be repeated indefinitely. If the procedure is not followed exactly, the plotted image is deleted and an inquiry appears on the screen, whether another graph will be plotted. This situation may be selected at any time, for example, by pressing RETURN repeatedly. The resulting situation and further procedure were described earlier.

The graph may be copied by means of the printer at any time by pressing PrtSc, provided the statement GRAPHICS had been entered before starting work with the WTGH program.

228

12 Reduction of water hammer

The design of a hydraulic system usually has to be modified to satisfy several demands. These include the demand that the water hammer phenomena which appear during the operation of a system should not impair its safety. The effect of water hammer is often not considered in the first of the design, other demands receiving priority. Only in second instance is the effect of water hammer calculated and the hydraulic system modified, if necessary, in the light of the results of this calculation. The method introduced in the present book, corresponds to such an approach.

The modifications leading to a reduction of the effects of water hammer may be of the most varied kind. No unequivocal rule exists for the metod of modification. We may, for example, merely change the operation regime of the hydraulic system so as to exclude operations dangerous with regard to water hammer. Another possibility is to modify the operating regime or install some device which reduces the effect of water hammer. In some cases the effects of water hammer may necessitate a modification of the entire system. Some means employed in limiting effects of water hammer are discussed in the following sections.

12.1 Adjustment rCgime of a valve

The effects of water hammer may be reduced by slowing down the opening or closing of valve. In some cases, however, such a measure is impossible.

Frequently, a substantial reduction of the effect of water hammer may be obtained by adjusting the opening-closing regime without increasing the total time involved. After completing some adjustments, especially after a complete closure of the valve, significant changes in pressure may develop in the pipe-line, threatening the system more than the changes in pressure which occurred during the adjustment. The pressure may drop to such an extent that cavitation is induced with all its adverse consequences. Even such changes in pressure may be eliminated or at least considerably limited by suitable adjustment regime. The effects of the adjustment regime on water hammer has been studied by many authors. For the simplest hydraulic systems, where the liquid flows from a

229

Reduction ($ water hammer

reservoir through a pipe-line provided with a valve at its end, operating regimes were proposed, where minimum changes in pressure are attained. This, however, requires abrupt changes in the position of the valve, exactly coordinated with the movement of the pressure waves along the pipe-line [ 13,791. In other proposed operating regimes, the variations in pressure are somewhat greater, but the proposed adjustments to the valves are continuous and not so Sensitive to a strict time coordination [55,60,81]. The disadvantage of all these proposals is that the operating regime depends on the initial and the final steady states, which complicates the design of suitable valves. Closing regimes for a valve, which are independent of the initial state, were also worked out. They mainly limit the variations in pressure after the closing of the pipe-line [80, 811.

Most of the valves used in practice are designed rather unfavourably from the point of view of water hammer. The largest changes in discharge occur during the final closing stages of the valve [3,20,75]. Valves producing a linear change in discharge with time would be more suitable. A linear relatioship between the area of flow and time, however, does not necessarily mean that the discharge through the valve will also change linearly with time.

To limit the variations in pressure after the final closing stage of the valve, it is important to reduce the closing speed in this final stage.

Some authors have also studied proposals for suitable adjustment regimes for more complex hydraulic systems [60]. One of the methods of solving the problem is by controlling the adjustment of the valve in accordance with the measured pressure in the pipe-line. It is possible to choose a suitable pressure curve, to measure the deviation from the chosen pressure, of the actual pressure in the pipe-line during the closing of the valve, and to adjust the adjustment regime accordingly. A similar principle was also employed in Sect. 14.5.

12.2 Pump

In operating a pump, the most important stages with regard to water hammer are its starting and stopping. The excessively rapid changes in discharge which occur during the starting of a pump may be eliminated, for example, by starting the pump with the valve on the delivery pipe at the pump closed, and by afterwards opening this valve so that the induced water hammer does not exceed acceptable limits. A more complex condition arise, when the pump is switched 0% this may occur even unexpectedly, for example, when there is an electric power failure.

The changes in pressure induced by the fall-out of a pump have been studied by several authors. Graphs have been constructed for the most common cases; the maximum and minimum pressures as well as other important quantities can be determined with the aid of these graphs [ 1 1, 32,441.

230

Surge tank

A sufficient reduction of the variations in pressure may be sometimes obtained, when the moment of inertia of the pump and the electric motor is increased through an added flywheel. This reduces the variations even during an electric power failure.

A non-return flap valve is frequently installed in the delivery pipe behind the pump so as to prevent backflow through the pump. However, this valve may remain open after a fall-out of the pump by an accidental error and it closes rapidly only after a strong backflow. The water hammer induced in this way may be dangerous. Its development is not prevented even by an increased moment of inertia of the pump and motor. Sometimes non-return flap valves are used which incorporate a hydraulic mechanism designed to retard their motion.

An air chamber (refer to Sect. 12.4) is often employed to protect the delivery pipe of a pump. When correctly designed, it may solve the problem of starting, stopping, damping of vibrations and the possible incorrect functioning of the non-return flap valve.

A detailed analysis of the problems connected with the unsteady flow induced by the operation of pumps and turbines may be found in the literature [20, 56,601.

12.3 Surge tank

A surge tank is a reservoir with a free fluid level which is attached to the pipe-line which has to be protected against the effects of water hammer. The level in the reservoir corresponds to the pressure in the pipe-line at the steady state. In the course of water hammer, the reservoir fills and empties. A pressure roughly corresponding to the instantaneous level in the reservoir is ensured in the pipe-line close to the surge tank. With a correctly designed surge tank, the variations in pressure are much smaller than in pipe-line without surge tank. The required dimensions of the surge tank have to be determined from the calculated water hammer.

In some cases, such as that of the protected penstock of a governor-controlled turbine, for example, undamped fluctuations of the level in the surge tank may occur. Attempts to eliminate such phenomena may affect the required dimen-

Fig. 12.1 Various typcs of surge tank: (a) Tank with constant cross-section; (b) Tank with restriction; (c) Tank with variable cross-section; (d) Differential surge tank.

23 1

Reduction 01 woter hammer

sions of the surge tank. The reservoir of a surge tank may have either a constant cross-section or a variety of shapes (refer to Fig. 12.1). Various shapes of the reservoir are designed mainly to reduce the required volume of the surge tank, to achieve the highest damping effect and to ensure stability of the level in the reservoir [8, 18, 26, 27, 54,601.

Surge tanks are not very suitable for protecting hydraulic systems operated under higher pressures, because they would have to be very deep. In such cases, other solutions an air chamber, for example. may be used.

12.4 Air chamber

An air chamber is a pressure vessel attached to the pipe-line which is to be protected. The vessel is partly filled with compressed air or other gas, and partly with liquid. The function of an air chamber is similar to that of a surge tank, but the pressure in the pipe-line is determined mainly by the pressure of the air in the air chamber rather than by level of the liquid. To increase the damping effect, the air chamber is frequently attached to the pipe-line through a restriction which has a higher coefficient of loss for the intake into the air chamber, as compared to the outflow. The air chamber should be installed as close as possible to the protected pipe-line in order to reduce the unfavourable effect of the inertia of the liquid in the connection. The length of the pipe-line between the air chamber and the pump, or other device inducing water hammer, should also be short so that even this part of the pipe-line is well protected.

An air chamber is frequently employed as means of protection for the delivery pipe-lines of pumps, water mains and similar hydraulic systems.

More detailed information on the function and design of air chambers may be found in the literature [3. IS, 20, 331.

12.5 Other methods of protection

Many other devices are employed to reduce the effects of water hammer. A one-way surge tank or air inlet valve may be used to prevent cavitation

[20,60]. A one-way surge tank is a reservoir with a free fluid level which is attached

to the pipe-line trough a non-return flap valve. As long as the pressure in the ‘pipe-line is higher than that corresponding to the level in the reservoir, the flow in the pipe-line is not affected in any way. When the pressure in the pipe-line drops below this value, the flap valve opens, the liquid from the reservoir flows into the pipe-line, which prevents any further drop in pressure. When the pressure rises, the valve closes again.

232

Other methods of protection

The function of an air inlet valve is similar, but, instead of a liquid, air is sucked into the pipe-line when the pressure drops below the atmospheric pressure. This prevents a further drop in pressure in the pipe-line. During a subsequent rise in pressure, the sucked-in air is compressed and contributes to the damping of water hammer. The air may, however, also create difficulties. It may move through the pipe-line and induce water hammer at a later stage.

Some air inlet valves are designed so that they permit the sucked-in air gradually to escape from the pipe-line, but prevent the liquid from doing so. Air inlet valves of this type may in some cases represent a very efficient and cheap method of protection for a system. At the same time, they deaerate the pipe-line.

Safety valves are another device used for protecting a pipe-line against water hammer [60]. They permanently close an opening in an operating pipe-line. When the pressure in the pipe-line rises to a predetermined limit, the safety valve opens; when the pressure drops, it closes automatically. Sometimes, the closing process is retarded so as to avoid changes in pressure induced by it. The valve is kept in closed position by means of a weight or a spring. The limiting pressure at which the safety valve opens, should not be exceeded at the point where it is installed. During very abrupt changes in pressure, such as may occur during water hammer, the safety valve sometimes does not react quickly enough due to its inertia.

Openings in pipe-line are sometimes closed by means of thin steel membranes, which burst when a limiting pressure is exceeded [60]. Their advantage is their small inertia. A variety of other devices may be employed to limit the effects of water hammer [60]. Various elastic elements inserted into the pipe-line walls or directly into the liquid reduce the wave velocity. The flow velocity can be modified to some extent by increasing the pipe-line diameter.

An attached blind pipe-line may damp periodic variations in pressure, if its length is such that it returns the reflected pressure waves in a phase opposite to that of the original presure variations.

233

13 WTHM program for the calculation of water hammer

13.1 Basic layout of the program

The block diagram of the WTHM program is shown in Fig. 13.1. This includes the main program, the sub-programs and the inputloutput devices. The basic control of the WTHM program is effected by the main program. The possible paths through the main program are shown by thick solid lines in the diagram. Some of the blocks of the main program utilize the subroutines, and some of these subroutines use other subroutines and functions. Their interconnection is shown by thin solid lines in the diagram. Control always returns from the subroutine into the block of the main program from which the subroutine was called-up. The block diagram in Fig. 13.1 is drawn so that control of the calculation in calling-up the subprograms is handed over from the main program to further subprograms which are always located to the right, and then returns to the main program by the same path.

The left-hand part of the diagram shows the input/output devices. Their use from the various blocks of the main program is indicated by dashed lines.

The use of the input/output devices and the termination of the calculation may also be requested from other parts of the main program and from some subroutine and functions due to errors which originated during the course of the calculation (refer to Sect. 13.4). These possibilities are not shown in the diagram.

13.2 The main program

The calculation begins and ends at the points denoted START and STOP, respectively (refer to Fig. 13.1). The Declarations and Formats used in the calculation are introduced at the beginning of the program. In the “Constants” block, the initial values of some constants are entered and the fields for loading the input data are cleared down. In the next block denoted “I/O units”, the basic information on the WTHM program is presented on the screen, the name of the input data file is determined, and the output files as well as the respective output devices are defined. If a disk is going to be used for the output, the names of the output files are also determined. All devices and files to be used are prepared for

234

1/0 UNITS c

Formats

WBROUTI NES

1 I PREP I- . '~esu~ts writing -HGRAPH u

i 0 of one calculation

FUNCT I0 NS

'4 CD EQUIP ]

DAMP WPQTIME I

h)

Fig. 13.1 Block diagram of the WTHM program.

WTHM program for rhe calculation of water hammer

the calculation. All the requests for submitting data are presented on the screen and the data are submitted via the keyboard.

In the “Input file reading” block, the submission for one calculation is read from the data file on the disk. The input data read are stored in the fields of the computer main storage.

In the “Input data modification” block, the values calculated earlier are substituted for some input data, when the result of the preceding calculation of the steady state is used as the initial state. In calculating the steady state, the velocities a, are substituted for the wave velocities a. Beside this, the submitted section lengths are rounded off to multiples of Ax for each calculation and the coefficients h are modified accordingly.

In the subsequent “Input data writing” block, the submitted input data are entered in the output files. They are entered only in those output devices, that have been chosen in the “I/O units” block. All input data are entered in the main output file, except the data exclusively relating to graphical and numerical outputs. These data are entered in the graphical and numerical outputs.

In the “Preparation” block, the initial values of the variables for one calculation are determined. The initial values of the parameters of the damping and pressure devices are determined by means of the PREP, DAMP, and PRESS subroutines and with the aid of the subroutines for the damping and pressure devices; the initial values of the variables for the graphical and numerical outputs are found with the aid of the GRAPH subroutine.

In the “Results writing” block, the results of the calculations are written in the output files, always for the single relevant value of time t. The values for the printing of the graphical and numerical outputs are determined with the aid of the GRAPH subroutine. The results are printed in the output files only in the case where the instantaneous value t corresponds to the submitted time interval of printing. At the end of the “Results writing” block, the time is determined for which further results will be printed.

Then follows the “End of one calculation check”. The calculation is terminated when either the maximum submitted calculation time t,,, was reached, or the steady state was attained with sufficient accuracy in the abridged calculation. If neither of these conditions has been satisfied, the calculation continues to the “Time change” block.

In the “Time change” block, the value of the calculation time t is increased by At.

In the “New values for junctions and sections” block, new parameters of the damping and pressure devices and the coefficients of the system of linear equations for the calculation of all functions are calculated with the aid of the subroutines for the individual damping and pressure devices. The pressure and discharge at all points of intersection of the characteristics for times t-At/2 and t are calculated with the aid of the BODY subroutine, the maximum and

236

The main program

minimum values of pressure at all these points of intersection are checked, and the pressure in the junctions and the discharge from the junctions into the damping devices for time t (refer to Sects. 7.5 and 7.6) are calculated.

In the “Iteration in junctions” block, the error is eliminated which appears in the solution for the junctions and which is caused by linearizing the equations which determine the effect of the damping and pressure devices, if such a procedure has been requested in the submission of the calculation (refer to Sect. 9.5). The ITER, DAMP, and PRESS subroutines are employed in this calculation, together with the subprograms for the individual damping and pressure devices.

In the “Extremes” block, the maximum and minimum values of the parameters of the damping and pressure devices and the pressure at the end points of the sections are checked.

In the “Steady state check” block it is ascertained with the aid of the STEADY subroutine whether, for the abridged calculation of the steady state (refer to Sect. 9.5), the steady state of flow has been already reached with the required accuracy.

After the calculation of the “Steady state check block” has been completed, the calculation returns to the “Results writing” block. The calculation is then repeated, as described earlier. At each repetition, the calculation time t is increased by a value At and a check is made, whether the end of one calculation has not been reached.

As soon as one of the conditions determining the termination of one calculation has been attained, the cycle is interrupted and the calculation changes over to the “End of one calculation” block. For the abridged calculation of the steady state, the calculated values needed are stored in this block so that they can be used as input values for the subsequent calculation (refer to Sect. 9.5) and the extreme values found are printed in the output files.

The further course of the calculation depends on instructions from the keyboard. Possible choices are shown on the screen. One may change over to the “Constants block” and start a new calculation from the beginning with new input data. Another possibility is to go to the “I/O units” block. In this case, new output and/or input files are chosen, but the input data read for the preceding calculation remain at one’s disposal. The third possibility is a direct change-over to the “Input file reading” block. In this case, the input and output files remain. The final possibility is to go to the “End of all calculations” block. The calculation is now terminated and control reverts to the DOS operating system.

The calculation also proceeds to the “End of all calculations” block when certain errors are found in the input of the data. Then, the calculation may either be terminated or a new calculation started by going to the “Constants” block.

237

WTHM program for the calculation o j water hammer

13.3 Subprograms

The PREP subroutine determines the initial values of the pressure and discharge along the sections, of the pressure in the junctions and of the discharges from the junctions into the damping devices as well as the initial values of the parameters of the damping and pressure devices. It checks the maximum and minimum values, determines the values of some constants used in subsequent calculations, and it checks the total number of the parts into which the sections are divided for the calculation. The PREP subroutine is called-up from the main program through the “Preparation” block. In the calculation, the subroutines DAMP and PRESS are employed as well as the subroutines for the individual damping and pressure devices.

The GRAPH subroutine determines the values of the variables presented in the graphical and numerical outputs. It determines, which values will be entered in the graphical and numerical outputs and for which time, it defines the position of the symbols of the variables in the lines of the graphical output, and the maximum and minimum values of the variables. The input into GRAPH subroutine is effected from the main program through the “Preparation” and “Results writing” blocks.

The BODY subroutine carries out the calculation of the pressures and discharges along the sections, the pressures in the junctions and the discharges from the junctions into the damping devices at time t on the basis of the known pressures and discharges at time t - At and of the known functions of the damping and pressure devices. Besides this, it determines. the maximum and minimum values of the pressure in the individual sections. The input into the BODY subroutine is effected from the main program through the “New values for junctions and sections” block.

By applying iteration, the ITER subroutine eliminates the inaccuracies induced in the solution of the junctions by linearizing the equations which express the effects of the damping and pressure devices. In this subroutine, the solution for the junctions is recalculated repeatedly using newly calculated values for discharges at the end points of the sections, of discharges from the junctions into the damping devices, and of some parameters of the damping and pressure devices, until the required precision of the solution for the junctions or the chosen maximum number of iterations is attained. The ITER subroutine is called-up from the main program through the “Iteration in junctions” block. The ITER subroutine uses the DAMP and PRESS subroutines, as well as the subprograms for individual damping and pressure devices.

The EXTREME subroutine determines the maximum and minimum values of some parameters of the damping and pressure devices, and includes the effect of the pressure at the ends points of the sections on the maximum and minimum values of the pressure in the sections. It is called-up from the main program through the “Extremes” block.

238

Subprograms

The STEADY subroutine ascertains, for the abridged calculation of the steady state, whether the flow has become steady to within the required accuracy. The input for this subroutine is realized from the main program through the “Steady state check” block.

The DAMP subroutine determines the parameters of the damping devices and the coefficients of the linear equations for the junctions. These coefficients which express the effect of the various damping devices are determined, with the exception of the simplest cases, by employing the subprograms for the individual damping devices. The DAMP subroutine is called from the PREP and ITER subroutines and from the main program through the “New values for junctions and sections’’ block.

The PRESS subroutine determines the parameters of the pressure devices and the coefficients of the linear equations for the junctions. These coefficients express the effect of the pressure devices. The PRESS subroutine is called from the PREP and ITER subroutines and from the main program through the “New values for the junctions and sections” block.

The damping device subprograms, RES for the reservoir, ACHAMBER for the air chamber, STANK for the surge tank, AVALVE for the air inlet valve, CDEQUIP for the integrated (combined) damping device and PQTIME for the time dependence of pressure and discharge, determine the coefficients of the linear equations for the junctions or, alternatively, the pressure in the junctions directly. Some of subprograms also determine the parameters of the damping devices. The relations employed for calculating these quantities have been presented in Sect. 7.8. The subprograms for the damping devices are called-up only from the DAMP subroutine.

The subprograms for the pressure devices, LOSSES for the local pressure losses, VALVE for the control valve, NRFV for the non-return flap valve, PUMP for the pump and the turbine, COND for the condenser and BVALVE for the butterfly valve determine the coefficients of the linear equations for the junctions and also, in some cases, the values of the parameters of the pressure devices. The relations employed for calculating these values have been presented in Sect. 7.9. The subprograms for the damping devices can be called only from the PRESS subroutine. The PUMP subprogram uses the subprogram GOVERN for calculations pertaining to a governor-controlled turbine; this subprogram determines the opening of the guide and action blades. The GO- VERN subprogram simulates the function of the governor. The relations used in the calculation have been introduced in Subsect. 7.9.1 1. Some of the subprograms for the pressure devices employ the functions FCE, FCEl ands FCE2 in the calculation, as may be seen from Fig. 13.1.

The function FCEl determines the values of the characteristics of a turbine for a specific setting of the guide and action blades by means of interpolation between submitted four characteristics.

239

WTHM program for the calculation of water hammer

The function FCE2 determines the values of a function dependent on one parameter and on the sign of another parameter by means of interpolation.

The function FCE determines the values of a function submitted by individual points. The values of the function are found by linear interpolation within the range of the points submitted, while a parabolic curve is assumed for the function outside this range.

13.4 Errors in the calculation

In the calculation various errors may appear, usually caused by errors in the data file used. Formal errors may be found by the checking procedure which is included in the WTHD program (refer to Sect. 10.7). Other errors may appear during the calculation.

Error messages appear only on the screen and are not incorporated in the output files on the disk and the printer. Each error message begins with the letters WTH Err., followed by the number of the error and a brief description. Some errors result in the termination of the calculation, while others merely cause an interruption. The calculation continues after pressing the RETURN key as stated in the error messages. The causes of errors in the calculation are as follows:

WTH Err. I

WTH Err. 2

WTH Err. 3

WTH Err. 4

240

The error appeared during the reading of the input file. In the error message, the instantaneous values CZ and C5 are presented. They are the last values read in columns 1 and 2 to 5. They may correspond to the data line in which the error appeared or the preceding line. The error terminates the calculation. The error was caused by an excessively fine division of the sections for the calculation so that the total number of points for which the pressure and the discharge are calculated, is greater than 2000. The error may be eliminated by increasing the time interval At of the calculation. The error terminates the calculation. The error was caused by employing the number J of a damping device not included in the WTHM program. In the 2.0 version, only the values J = I , 2,. . ., 9 can be used. After pressing RE- TURN, the calculation continues, the junction with the incorrect J being considered as one without a damping device. The number of the junction together with the required number of the pressure device is given in the error message. The error was caused by using the number J of a pressure device not included in the WTHM program. In the 2.0 version, only the

Errors in the calculation

values J = 51, 52,. . ., 59 can be used. After pressing RETURN, the calculation continues, the pipe-line being considered closed at the point where the pressure device had been incorrectly submitted. The number of the section to which the pressure device is attached and the required number J are included in the error message.

WTH Err. 5 - The error message is introduced in a calculation not using

WTH Err. 6

WTH Err. 7

WTH Err. 8

iteration, when a system of equations having zero determinant appears in the solution for a junction. Such a system may appear, for example, when analysing a junction without a damping device separated from all the attached sections by means of the pressure device “closed pipe-line”. The number J, of the junction, in which this occurred, is included in the error message which also contains the value t /Az, denoted t /Dt , representing the instant, at which the above error appeared for the first time. The error message is introduced only once for each junction. After pressing RETURN, the calculation continues, since it is not necessary that the pressure in this junction be given correctly. The error message is induced by the same conditions as pertaining in the case of WTH Err. 5, provided they developed in the iterative solution of a junction. The message is induced by an error in the calculation of the function of an air chamber. It may be caused by incorrectly submitting the parameters of the air chamber (the first five symbols of the relevant data line are given in the error message) or by employing an excessively large value At, which does not permit variations in the volume of air in the air chamber to be followed with sufficient accuracy. The error terminates the calculation. The message is induced by an error in the calculation for an air inlet valve. (The first five symbols of the respective data line are given in the error message). It is caused by an excessively large value At for the time interval of the calculation, which does not permit changes in the quantity of air in the air inlet valve to be followed with sufficient accuracy or by an error in the parameters of the air inlet valve submitted. The error causes the calculation to terminate or the calculation continues with the error caused by neglecting the rest of air in the chamber. When air- inlet valve is used as a substitution for the separation of the liquid column, the message is groundless. It may be eliminated by submitting the diameter Do of the aperture with the negative sign.

24 1


WTH Err. 9 - The message is induced by an error in the calculation for an integrated damping device. It may be due to various causes. The type of error is indicated by a number in the error message. If this number is negative, the error terminates the calculation. If this number is positive, the calculation continues after pressing RETURN. The error message also contains the numbers in the first five columns of the first data line of the corresponding damping device.

An error of type - 1 is reported, when any of the conditions S > 0; S , > 0; Si > 0; S,i > 0; V, > 0; Vtot > V,; y # 0 is not satisfied in the submission for the integrated damping device with an air chamber arrangement. Here, Vtot is the total volume of the chamber of the integrated damping device. A type - 1 error is also reported when the submitted initial absolute pressure in the chamber is not positive.

An error of type - 2 is reported, if, for any reason, it is not possible to calculate the initial volume of the liquid in the chamber of the integrated damping device with the air chamber arrangement.

An error of type - 3 is reported, when, according to the calculation, the air chamber or the air inlet valve without orifice was overfilled with liquid.

An error of type 1 is reported, when the calculated level of the liquid in the initial state is below the bottom of the chamber for an integrated damping device arranged as an overflow or an air inlet valve. In the calculation, the initial volume Vo of the liquid is assumed to be zero.

An error of type 2 is reported, when the pressure loss at the outlet of an overflow is zero and the initial pressure submitted is higher than that corresponding to the level of the overflow edge at the steady state. In the calculation, the initial level of the fluid is considered to be at the overflow edge.

An error of type 3 is reported, when, according to the calculation, the air chamber is overfilled with liquid. This may occur at an exceedingly large value of the time interval At, which does not allow the function of the air chamber to be followed with sufficient accuracy. The remainder of the calculation proceeds with an error caused by neglecting the volume of the liquid causing the overfilling of the air chamber.

The error of type 4 is reported, when, in accordance with the calculation, the air inlet valve with the orifice is full of liquid before all the air has escaped. The rest of air is neglected. This may occur when At is too large and the orifice is too small.

242

Denotation of the variables

WTH Err. 10

WTH Err. 11

WTH Err. 12

WTH Err. 13

WTH Err. 14

WTH Err. 15

This message concerns an error in the submission of data for the turbine. It appears whenever a zero value is entered for any of the variables D,, iT, DT, rT or 3,. The error message also contains the numbers in the first five columns of the first line of the submission for the turbine.

The calculation terminates when this type of error has been found. This message denotes an error in the submission for the turbine governor. It appears when the conditions C, # 0; Cj # -At; C, # 0; Cp C, Af # fb( Uo); C,, # 0; C,, # 0; Cs, # 0 and in some cases, CGs # 0; CFs # 0; Csp # 0 are not satisfied. The error terminates the calculation. The message denotes an error in the calculation for the turbine governor. It is presented in each calculation step, when the program does not succeed in solving the equation which defines the position of the servomotor piston rod. This may be caused by the value of the function oa(xd) or np(xp) equalling zero.

The error message indicates an attempt to perform a calculation when the name of the input file has not been submitted. After pressing Y, the entire calculation is restarted. Otherwise, the calculation is terminated. The message is induced by the incorrect submission of a function, where the values of a variable decrease with an increasing subscript i. The message includes the first five numbers of the data line in which the submission of the function begins. The message stops the calculation. The message points out an attempt to interpolate the turbine characteristics for the parameters a, /I (denoted A , B), for which the characteristics are not submitted in all the quadrants. The interpolation for such values a, /I will not be realized quite correctly. If no characteristic has been submitted, the calculation terminates.

The calculation continues after pressing RETURN.

WTH Err. 16 - The error is caused by an erroneous submission of the parameters of the “condenser”. I t terminates the calculation.

13.5 Denotation of the variables

The denotation of the quantities in the source text of the WTHM program differs from that in the text of the present publication. This is due to the limited possibilities of the programming language, to different demands on the denota-

243


tion in the program and the description, and also, because the program was originally run to correspond to the Czech language. This section contains the meanings of some basic variables and constants employed, to facilitate the following of the listing of the subprograms for the damping and pressure devices (refer to Appendix A).

When the denotation employed in the source text of the program corresponds at least approximately to some denotation in the text of this book, the latter is presented in parenthesis.

- angle of tilt of a butterfly valve - steady state angle of tilt of a butterfly valve

- parameters defining the setting of the guide and action blades

- number in columns 2 to 5 of the first data line for a damping of the turbine model (refer to Subsect. 9.4.6);

or pressure device;

- admissible difference in moment; - admissible difference in speed of a pump or a turbine; - admissible difference in pressure for the iterative solution of

junctions, entered and employed in the calculation; - admissible difference in discharge for the abridged calculation

of the steady state; - internal diameter of pipe-line; - rate of rotation of a flap; - derivative; - difference in pressure in front of and behind a pressure device; - time interval of the calculation; - time; - acceleration due to gravity; - elevation of the lowermost interconnecting pipe of a condenser; - elevation of the uppermost interconnecting pipe of a con-

- liquid level in the downstream chamber of a condenser; - liquid level in the upstream chamber of a condenser; - the array in which the input data from the line of the submis-

sion beginning with 4, from columns 6 to 10, are entered. The data in columns 2 to 5 are the subscripts of the array;

denser;

- number of the type of an error; - variable determining from where the DAMP and PRESS

subprograms are called. In calling from the PREP subroutine ZZ = -1, from the ITER subroutine ZZ = 0, and from the “New values for junctions and sections” block, IZ = 1;

244

Denotation of the variables

QO' (Qi)

- calculation time in multiplex of At. In the abridged calculation of the steady state, ZTV = 0;

- number determining whether a pressure device is at the upstream (J = 1) or downstream (J = 2) end of a pipe-line section;

- moment induced by the weight; - moment induced by the hydraulic system; - moment induced by inertia; - moment induced by a liquid in a pipe-line; - pressure in a pipe-line section; - pressure in a junction; - atmospheric pressure, absolute; - the ratio of the circumference of a circle to its diameter; - discharge of a liquid through a pipe-line; - discharge of a liquid from a junction into a damping device; - . -

R; S (Rp; Sp) - coefficients of linear dependence for a pressure device; RO; SO ( R d ; 'd) RC

RO (e) ROA (4 U

uo

UPREP

UR VT

USETRO

UTURB

UUST

UVNK

- coefficients of linear dependence for a damping device; - the array in which the input data are entered from the lines of

the submission beginning with 4, from columns 20 to 79. The data in columns 2 to 5 are its first subscript. The second subscript determines the sequence of the numbers of the line;

- density of a liquid; - density of air at atmospheric pressure - variable determining, whether a pressure device closes the

pipe-line. For a closed pipe-line; U = TRUE; - variable determining whether a damping device ensures a con-

stant pressure in a junction. If the pressure is constant, UO = TRUE;

- variable determining whether the calculation of an overflow is to be carried out;

- variable determining whether a turbine is controlled by a governor;

- variable determining whether the moment of inertia of the revolving part of an assembly is equal to zero;

- variable determining whether the characteristics of a turbine are variable;

- variable determining whether the values calculated in the preceding abridged calculation of the steady state are used as input values;

- variable determining whether the speed of a pump or a turbine is constant;

245

WTHM program for the calculation 01 water hammer

UZAP

UZKR

VO to V5 (VO; VZ; V2) - variables of damping devices; VC (V,,,) VN; VNZ to

V N I ; VN2) VNU; VNUZ; VNU2; VNU3- variables of the pressure devices, determined in the abridged

- variable determining whether a pump or a turbine is connect-

- variable determining whether the abridged calculation of the ed with the electric-supply network;

steady state is to be carried out;

- total volume of the chamber of an integrated damping device;

VN.5 (VN; - variables of pressure devices;

calculation of the steady state.

246

14 Examples of the calculation of water hammer

The following sections contain several examplex of the calculation of water hammer and the steady flow in different hydraulic systems. The examples were chosen to illustrate a variety of possible applications of the described method of calculation. Some of the examples represent calculations for actual case his- toires. The procedures once used, are usually not explained again in subsequent examples.

The input data files are presented in the text and also on the WTHM diskette. The output files are presented only in some cases. The results of the calculation are in most cases rendered in graphical form.

Section 14.1 and 14.2 deal with the simplest cases of the calculation of water hammer; the abrupt closing of a pipe-line with and without consideration of the effect of pressure losses. Here the basic procedure for submitting the data for the calculation are presented, as well as different types of output for the results. In each of these sections, a different method for submitting the sudden changes in discharge is employed. The procedure in calculating the steady state is demonstrated in Sect. 14.2

Section 14.3 contains an example of submitting data for local losses, different procedures for submitting the operating regime of the control valve, and of one of the ways of submitting the parameters of a pump.

In section 14.4 the “pressure” damping device is used to submit the measured pressures in the calculation of valve characteristics.

Section 14.5 demonstrates possible procedures for determining the operating rCgime of the control devices needed to attain a required pressure or discharge. The “pressure” damping device is used in the calculation.

Section 14.6 presents the calculation of the steady state of flow in a pipe-line network composed of 32 branches interconnected to form several circuits.

Section 14.7 demonstrates some possibilities of the calculation of the unsteady flow in a hydraulic system during periodical variations in pressure. In this case, resonance occurs between the frequencies of the pressure variations and the natural frequency of the hydraulic system. The periodical pressure variations are induced with the aid of a “pressure” damping device.

Section 14.8 contains the calculation for the protection of a delivery mains by

241

Examples of the calculation of water hammer

means of an air chamber. The effect of the size of the air chamber and of various restrictions of the inflow into the air chamber on the course of water hammer for a sudden fall-out of the pump are studied.

The example in Sect. 14.9, solves water hammer induced by the fall-out of a pump which forms part of the cooling circuit in a thermal power station, with the accompanying cavitation behind the condenser. The latter phenomenon is observed with the aid of the damping device “air inlet valve”. The damping device “overflow” is substituted for the cooling tower in the calculation. Non- return flap valves are installed behind the pumps. The pressure device “local losses” is substituted for the condenser.

Section 14.10 introduces the calculation of the discharge steady state flow. The discharge was calculated on the basis of measurements of the pressure at two points along the pipe-line at water hammer induced by the fall-out of the pump and the closing of the non-return flap valve.

In Section 14.1 1, we determine the pressure and the moment characteristics of a pump. During water hammer induced by the fall-out of the pump and the closing of the non-return flap valve, we measured the pressures in front of and behind the pump together with its speed. The discharges were determined as in Sect. 14.10. These values are sufficient for calculating the characteristics, when the moment of inertia of the pump and the electric motor are known.

In Section 14.12, water hammer induced by starting a pump equipped with an electric motor is solved. A butterfly valve and a condenser form part of the system. An integrated damping device and the pressure devices “butterfly valve” and “condenser” were employed in the calculation.

In Section 14.13, we solve water hammer at the starting of a water turbine with variable characteristics. The water hammer is induced by adjusting the turbine guide blades. The adjustment regime is known in advance. The characteristics of the turbine model are submitted for the calculation.

The last calculation, presented in Sect. 14.14, deals with water hammer induced by disconnecting the generator of a governor-controlled turbine from the electric network. The governor controls the adjustment of the turbine guide blades as a function of the instantaneous speed of the turbine and other input data.

The precision of the calculated results remains an open problem. On the basis of experience gathered to date, the method of calculation has little influence on the precision of the results in practice (refer to Sect. 3.1). The decisive factors usually are: the accuracy and the completeness of the data available for the calculation, a suitable schematization of the actual pipe-line system, a careful checking of the input data file and a correct interpretation of the results.

The characteristics of some devices which form part of the hydraulic system and affect water hammer are not always at hand. Sometimes, they are not known with sufficient accuracy or they may not cover the entire range of the flow

248


conditions that may appear during water hammer. Sometimes, we do not have at our disposal even sufficiently accurate data on the propagation velocity of the pressure waves in the pipe-line, the discharges, the adjustment regime of the valves, etc. The missing data have to be estimated and this often leads to much greater inaccuracies than those inherent in the method of calculation employed.

In schematizing the pipe-line system, one has to estimate correctly which effects have to be included in the calculation and which may be neglected. An automatic inclusions of all the details of an actual hydraulic system results in excessively complex calculation schemes, and the necessary choice of very short time intervals of calculation, long calculation times, difficult checking of the calculation and not necessarily, in more accurate results. Such an approach is even impossible in the case of more complex hydraulic systems. A computer ensures a rapid execution of the calculation, but the submission of the problem always has to be carefully thought over. For more complex systems, good results were obtained by calculating water hammer several times, first, for a greatly simplified system and subsequently, by making the scheme more accurate.

The submission of the calculation may contain numerical errors that cannot be found either by a formal checking carried out by the WTHD program, or by checks effected during the calculation. Sometimes, an error in a single datum suffices for the entire system to change. In view of this, it is not reliable enough to rely only on the maximum and minimum values presented at the ends of the outputs, but the values of all the variables for the initial state and their changes during the calculation have to be examined in detail and verified. A detailed logical checking of the calculated results is very important for assuring their reliability.

The results of the calculation, their accuracy and reliability in particular, have to be evaluated for every individual case separately. They depend on many circumstances, as has been explained above.

The currently used method of characteristics, in its simplest form possible (refer to Sect. 7.5), has been chosen for solving the equations describing water hammer. The precision of this method in some cases, with regard to reality, is testified to, for example, in Sect. 14.10. Simultaneously with the calculation of the discharge based on the measurements of the pressure at two points along the pipe-line, the discharge was also measured directly with the aid of twenty calibrated hydraulic measuring vanes installed in the pipe-line. The measurements were repeated several times for different discharges. The differences in the calculated and measured results never exceeded 2 %. In other cases, when the effect of pressure losses during unsteady flow, the effect of air in the hydraulic system or other effects may be more significant, the deviations from reality may be greater. The accuracy of the results also depends greatly on the extent to which the devices forming a part of the actual hydraulic system correspond to the damping and pressure devices employed in the program.

249


14.1 Abrupt closing of a pipe-line without considering the effect of pressure losses due to friction

The calculation deals with the simple case described in Subsect. 1.2.2, where water flows from a reservoir through a horizontal pipe-line of constant cross- section, with an inside diameter D = 0.4 m and a length 1 = 357 m. The outflow of the liquid is restricted by a valve at the downstream end, resulting in a discharge Qo = 0.15 m s at the initial steady state. The calculation determines water hammer induced by a sudden closing of the valve. The wave velocity considered is a = 1195 m s-’ and pressure losses due to friction in the pipe-line are neglected.

The hydraulic system and the calculation scheme employed are portrayed in Fig. 14.1.

3 - 1

a1 +210 m

PY

J= 51 J =51

J S I I I O l h l a l Q ~ 1 I357 10.4 I 0 1119510.15

Fig. 14.1 Abrupt closing of a pipe-line without considering the effect of pressure losses due to friction: (a) hydraulic system; (b) calculation scheme.

The reference plane was chosen at the level of the pipe-line axis. The atmospheric pressure is taken as zero. The unsteady state is submitted as the initial state; it develops immediately after the closing of the valve. Another possibility would be to submit the steady state prior to the closing of the valve, together with the closing rkgime of the valve in the manner employed in Section 14.2. The submission then would be more complex. The chosen approach relates to the calculation scheme, in which a “control valve” pressure device need not be considered. The calculation scheme consists of two junctions connected by one section. It includes the numbers of the junctions, the number of the section and

250

Abrupt closing of a pipe-line without considering the effect of pressure losses due to friction

the chosen positive direction of the section. The points corresponding to the junctions are also marked on the longitudinal profile. The type of damping device attached to the junction is submitted for each junction. A “constant pressure” (J = 2) damping device is attached to junction I while junction 2 is without a damping device (J = 1, refer to Sect. 9.3). At the upstream and downstream ends of the section, the type of pressure device, through which the section is attached to the junction, is submitted. In the case discussed, it is an attachment without a pressure device (J = 51, refer to Sect. 9.2). None of the damping and pressure devices employed requires the submission of parameters, hence, the values J,, are not given for them. The initial pressurePo in the junction is calculated in accordance with equation (2.3). In this case, the elevation of the pipe-line above the reference plane is h = 0 and

(14.1)

The input file used for this case is presented in Table 14.1. It was compiled with the aid of the WTHD program. It is also entered on the WTHM diskette under the name D 1. WTH.

po = p , = 9.81 x 1000 x 210 = 2 060 100 Pa

Table 14.1 Input data file for calculating water hammer due to an abrupt closing of a pipe-line without considering the effect of pressure losses due to friction

2 1 151 251 357 . 4 1195 * 15 1 1 SUDDEN FLOW INTERRUPTION - FRICTIONLESS 3 1 2 2060100 3 2 1 2060100 5 1 2 1 * 03 .225 1 . 5 6 1 1 1 l o p Pressure at the valve (Pa) -939900 14060100 62 1 1 5x Pressure at midpoint of section (Pa) -3939900 11060100 63 1 2 50 Discharge at midpoint of section (m3 /s ) -1.2 . 0

7 64 1 2 OQ Discharge from the tank (m3/s) - 1 . 6 . 4

In the calculation, iteration (2 = 1, refer to Sect. 9.5) need not be used, since there are no non-linear relations. We chose a time interval of calculation At = 0.03 s, which leads to the pipe-line being divided into 10 parts of length

(14.2)

From the point of view of the calculation, the division of the pipe-line in such a number of parts is not necessary. It was, however, used to clearly demonstrate the propagation of the pressure waves in the pipe-line.

Ax = a At = 1195 x 0.03 = 35.85 m

The length of the pipe-line, modified for the calculation, then will be

1, = IOAx = 358.5m (14.3)

The maximum calculation time was chosen so as to calculate the course of water hammer during 1.25 period

= 1.5 s 41, 4 x 358.5 a 1195

t,,, = 1.25 - = 1.25 ( 14.4)

25 1


Table 14.2 Water hammer due to an abrupt closing of a pipe-line without considering the effect of pressure losses - main output

WATER HAMMER IN PIPE-LINE SYSTEMS Input file Num WTHM Version 2.0 Copy 13 dl.uth 1

SUDDEN FLOW INTERRUPTION - FRICTIONLESS

Ns Nj 1 2

Js 1

Z 1

1 357.00

Columns of input file 2-5 6-10 20-29

t Js .ooo 1

Jj 1 2

t Js .210 1

JJ 1 2

t JB .420 1

X .oo

71.70 143.40 215.10 286.80 358.50

X . 00 71.70 143.40 215.10 286.80 358.50

X . 00 11.70

Dt Dtm Dxa as tmax .03000 .22500 . 00 .oo 1.5000

SECTIONS

Upstream end Downstream end

1 1 51 0 0 J8 Jj J Jp

lc D lambda a Qo 358.50 ,4000 .OOOOO 1195.00 .15000

JUNCTIONS

J;: . J JP PO 2 0 2060100.

2 1 0 2060100.

PARAMETERS'

30-39 40-49 50-59 60-69 70-79

P 2060100. 2060100. 2060100. 2060100. 2060100. 2060100.

P 2060100. 2060100.

Q .15000 .15000 .15000 .15000 .15000 .15000

X 35.85 107.55 179.25 250.95 322.65

P 2060100. 2060100. 2060100. 2060100. 2060100.

RESULTS

Q .15000 .15000 I 15000 .15000 .I5000

P Q X P 9 2060100. .15000 35.85 2060100. .15000 2060100. ,15000 107.55 2060100. ,15000 3486526. .OOOOO 179.25 3486526. .ooooo 3486526. . O O O O O 250.95 3486526. .ooooo 3486526. .OOOOO 322.65 3486526. .ooooo 3486526. .ooooo

P 2060100. 3486526.

P Q X P 9 - .15000 - .15000

2060100. -.15000 35.85 2060100. 2060100. -.15000 107.55 2060100.

252

Abrupt closing of a pipe-line without considering the eflect of pressure losses due to friction


Jj 1 2

t Js . 630 1

Jj 1 2

t Js .840 1

Jj 1 2

t Js 1 .050 1

Jj 1 2

t Js 1 .260 1

Jj 1 2

t Js 1.470 1

1 4 3 . 4 0 215.10 286 .80 358 .50

X . 00

71 .70 143 .40 215.10 286 .80 358 .50

X

. 00 71 .70

143.40 215 .10 286.80 358 .50

X . 00

71.70 143 .40 215 .10 286 .80 358 .50

X . 00 71.70

143 .40 215.10 286.80 358 .50

X . o o

71 .70 143 .40 215 .10 286.80 358 .50

3486526; 3486526. 3486526. 3486526.

P 2060100. 3486526.

P 2060100. 2060100. 2060100. 2060100. 2060100.

633673.

P 2060100.

633673.

P 2060100. 2060100.

633674. 633674. 633674. 633674.

P 2060100.

633674.

P 2060100. 2060100. 2060100.

633674. 633674. 633674.

P 2060100.

633674.

P 2060100. 2060100. 2060100. 2060100. 2060100. 3486527.

P 2060100. 3486527.

P 2060100. 3486527. 3486527. 3486527. 3486527. 3486527.

.ooooo ,00000 .ooooo .ooooo

Q -. 15000 -. 15000 - . 15000 - . 15000 - . 15000 ,00000

4 -. 15000 - . 15000 .ooooo * 00000 * 00000 .ooooo

Q .15000 .15000 .15000 .ooooo .ooooo .ooooo

Q . 15000 .15000 .15000 .15000 .15000 .ooooo

9 . 15000 .ooooo .ooooo .ooooo .ooooo * 00000

179 .25 250.95 322.65

X 35.85

107.55 179.25 250.95 322.65

X

35.85 107.55 179.25 250.95 322.65

X 35.85

107 .55 179.25 250.95 322 .65

X

35.85 107.55 1 7 9 . 2 5 250.95 322.65

X

35.85 107.55 179 .25 250.95 322.65

3486526. 3486526. 3486526.

P 2060100. 2060100. 2060100. 2060100. 2060100.

P 2060100.

633673. 633674. 633674. 633674.

P 2060100. 2060100.

633674. 633674. 633674.

P 2060100. 2060100. 2060100. 2060100. 3486527.

P 2060100. 3486527. 3486527. 3486527. 3486527.

.ooooo

.ooooo * 00000

4 - .15000 - .15000 - .15000 - .15000 - .15000

Q -. 15000 .ooooo .ooooo .ooooo .ooooo

4 . 15000 .15000 .ooooo .ooooo .ooooo

4 .15000 . 15000 .15000 .15000 .ooooo

Q . 15000 .ooooo .ooooo .ooooo . 00000

253

Examples OJ’ the calculation of water hammer


Jj 1 2

P 2060100. 3486527.

t Js X P Q X P e 1.500 1 .OO 2060100. .15000 35.85 3486526. .ooooo

71.70 3486527. .OOOOO 107.55 3486527. .ooooo 143.40 3486527. . O O O O O 179.25 3486527. .ooooo 215.10 3486527. .OOOOO 250.95 3486527. .ooooo 286.80 3486527. .OOOOO 322.65 3486527. .ooooo 358.50 3486527. .ooooo

t Js X P Q X P e 1.500 1 .OO 2060100. .15000 35.85 3486526. .ooooo

71.70 3486527. .OOOOO 107.55 3486527. .ooooo 143.40 3486527. . O O O O O 179.25 3486527. .ooooo 215.10 3486527. .OOOOO 250.95 3486527. .ooooo 286.80 3486527. .OOOOO 322.65 3486527. .ooooo 358.50 3486527. .ooooo

Jj P 1 2060100. 2 3486527.

Js 1

EXTREMES

p i n pmax 633673. 3488527.

ACCURACY

dp d e dM .1000Et02 .1000E-04 .1000Bt00

END

The size of the main output was limited by the choice of the time interval of printing At, = 0.255 s.

In the graphical and numerical outputs, the pressure at the valve is denoted by p , the pressure at the midpoint of the pipe-line by x, the discharge at the midpoint of the pipe-line by o and the discharge at the reservoir by Q. The values L,, L,, L, to which these variables correspond, were determined with the aid of Table 9.10. The maximum and minimum values of the variables were chosen so as to avoid superimposition of the curves of the variables in the graphical output. Thus, for example, the initial value 2 060 100 Pa for the pressure is located in the second dotted column from the left (refer to Table 14.3). The distance between the columns corresponds to a diference in pressure of 3 000 000 Pa. The value

(14.5) Min = 2060100 - 3000000 = -939900Pa

corresponds to the left-hand margin of the graph, and the value

Max = 2 060 100 + 4 x 3 000 000 = 14 060 100 Pa (14.6)

The main, graphical and numerical outputs are presented in Tables 14.2, 14.3 to the right-hand margin.

and 14.4. They are described in Sections 11.1, 1 1.2 and I I .3.

254


Table 14.3 Water hammer due to an abrupt closing of a pipe-line without consideration of the effect of pressure losses - graphical output

WATER HAMMER IN PIPE-LINE SYSTEMS WTHM Version 2.0 Copy 13

Input file Num dl . uth 1

SUDDEN FLOW INTERRUPTION - FRICTIONLESS VARIABLES

Text Mark Min Max

Pressure at the valve (Pa) p -.9389OE+O6 .14060Et08 Pressure at midpoint of section (Pa) x -.393983+07 .11060E+08

Discharge at midpoint of section (m3/e) o -.12OOOE+O1 .80000Et00 Discharge from the tank (m3/s) Q -.16000E+01 .40000EtOO

.00000E+00 .00000Et00

Time interval Changes of time interval of printing Codes of variables of calculation

Dt Gr Nu tgn Dtgn Mark Ls Lk Lx

.0300 0 0 .oooo .oooo 0 0 .oooo . 0000 0 0 .oooo .oooo 0 0 a 0000 .oooo

p 1 1 10 X 1 1 5 0 1 2 5 Q 1 2 0

0 0 0

RESULTS Values of yariables in graph (dotted columns)

p -.9399E+06 .2060Et07 .5060Et07 .8060Et07 .1106Et08 .1406Et08 x -.3940Et07 -.9399Et06 .2060Et07 .5060Et07 .8060Et07 .1106Et08 0 -.1200E+01 -.8000E+00 -.4000Et00 -.2980E-07 .4000EtOO .8000Et00 Q -.16OOE+Ol -.1200Et01 -.8OOOEtOO -.4OOOEtOO .0000Et00 .4000EtOO

.0000Et00 .0000E+00 .0000Et00 .0000Bt00 .0000Et00 .0000Et00

Time Min Max

.ooooo

.03000

.06000

.09000

.12000

.15000

.18000

.21000

.24000 ,27000 .30000 .33000 .36000 .39000 9 42000 .45000 .48000 .51000 .54000 .57000 .60000 .63000 ,66000 .69000 .72000 .75000

............ .p............ x.................o............Q........ P X . o * Q P X . o * Q P X . o * Q P X . o * Q P X . o * Q P X 0 - 4 P X 0 * Q P X 0 * Q P X 0 * Q P X 0 * Q P X 0 Q - P X 0 Q . P X 0 0 . P X 0 Q . P X 0 Q . P X 0 . 0 . P X 0 . Q . P X 0 . Q . P X 0 . Q . P X 0 . 4 .

X 0 . 0 . X 0 . 4 . X 0 . Q . X 0 . 4 . X 0 . Q .

255



.78000

.El000

.84000

.87000

.90000

.93000

.86000

.99000 1.02000 1 .05000 1 .08000 1 .11000 1 .14000 1.17000 1 .20000 1 .23000 1 .26000 1 .29000 1 .32000 1 .35000 1 .38000 1 .41000 1 .44000 1 .47000 1 .50000

Minimum Maximum

P X P X

P X

P X

P X P X

P X

P X

P X

P X P X

P X

P X

P X

P X

P X

P X

P X

P X

P X P X

P X

P X

P X

P X

0 Q . 0 Q . 0 Q . 0 9 . 0 Q . 0 . Q 0 . Q 0 . Q 0 . Q 0 . Q . o . Q . o . Q . o . Q . o . Q I 0 * Q . o * Q . o * Q . o - Q . o . Q . o . Q 0 . Q 0 . Q 0 . Q 0 . Q 0 . Q

EXTREMES

P X 0 Q .63367E+06 .63367Et06 - .15000Et00 -.15000Et00 .00000Et00 .34865Et07 .34865EtO7 .15000Et00 .15000Et00 .00000Et00

END

Table 14.4 Water hammer due to an abrupt closing of a pipe-line without consideration of the effect of pressure losses - numerical output

WATER HAMMER IN PIPE-LINE SYSTEMS WTHM Version 2 . 0 Copy 13

Input file Nurn d l . wth 1

SUDDEN FLOW INTERRUPTION - FRICTIONLESS

VARIABLES

Text Mark

Pressure at the valve (Pa) p Pressure a t midpoint of section (Pa) x

Discharge at midpoint of section (m3/s ) o Discharge from the tank (m3/s ) Q



.0300 0 0 .oooo .oooo 0 0 .oooo .oooo 0 0 .oooo .oooo 0 0 .oooo .oooo

p 1 1 10 X 1 1 5 0 1 2 5 Q 1 2 0

0 0 0

RESULTS

256



Time t

.ooooo

.03000

.06000

.09000

.12000

.15000

.l8000

.21000 ,24000 .27000 .30000 .33000 .36000 .39000 .42000 .45000 .48000 .51000 .54000 .57000 .60000 .63000 .66000 .69000 ,72000 .75000 .78000 .81000 .84000 .87000 ,90000 .93000 .96000 .99000

1 * 02000 1.05000 1.08000 1.11000 1.14000 1.17000 1.20000 1.23000 1.26000 1.29000 1.32000 1.35000 1.38000 1.41000 1.44000 1.47000 1.50000

Minimum Maximum

Values of variables P

.20601E+O7

.34865E+07

.34865E+O7

.34865EtO7

.34865E+07

.34865E+07

.34865E+O7 ,34865Et07

.34865E+O7

.34865E+O7

.34865E+O7

.34865E+O7

.34865Et07

.34865E+O7

.34865Et07

.34865E+O7

.34865E+O7

.34865E+O7

.34865E+O7

.34865E+07

.63367E+06

.63367E+06

.63367E+06

.63367E+06 ,63367Et06 .63367E+06 .63367E+O6 .63367E+O6 .63367EtO6 '. 63367Et06 .63367E+06 .63367Et06 .63367EtO6 .63367E+06 .63367Et06

.63367E+O6

.63367E+06

.63367E+O6

.34865E+07

.34865Ei07

.34865E+O7

.34865E+07

.34865E+O7

.34865E+O7

.34865E+O7

.63367E+06

.63367E+06

.34865E+O7

.34865Et07 ,34865Et07 .34865E+07

P .63367Et06 .34865E+07

X

.20601E+07

.20601E+O7

.20601E+07

.2060113+07

.20601E+O7

.20601E+07

.34865E+07

.34865E+O7

.34865E+O7

.34865E+07

.34865Et07

.34865E+07

.34865E+07

.34865E+O7

.34865E+O7

.34865E+07

.20601E+O7

.20601E+07

.20601E+O7

.20601E+O7

.20601E+07

.20601E+O7

.20601E+07

.20601E+07

.20601E+O7

.63367E+06

.63367E+O6

.63367E+O6

.63367E+O6

.63367E+O6

.63367E+06

.63367E+O6

.63367E+06

.63367E+O6

.63367E+O6

.20601E+O7

.20601E+07

.20601E+O7

.20601E+O7

.20601E+O7

.20601E+07

.20601E+07

.20601E+07

.20601E+07

.20601E+07

.34865E+07 ,34865EtO7 .34865E+07 .34865E+07 .34865E+O7

.20601E+07

X

.63367E+06

.34865E+07

0

.15000E+00

.15000Et00

.15000E+00

.15000E+00

.15000E+00

.15000E+00

-.16155E-O7 -.16118E-07 -.16057E-07 -.15975E-07 -.158753-07 -.15762E-07 -.15638E-O7 -.155063-07 -.15369E-O7 -.15227E-07 -.15000Et00 -.15000E+00 -.15000E+00 -.15000E+00 -.15000E+00 -.15000E+00 -.15000E+00 -.15000E+00 -.15000E+00 -.15000E+00 .19225E-O7

-.50033E-09 -.51089E-09 -.5227OE-O9 -.53522E-O9 -.54809E-08 -.56111E-O9 -.574158-09 .12561E-07 .12555B-07 .15000E+00 .15000E+00 .15000Et00 .15000E+00 .15OOOE+OO .15000Et00 .15000E+00 .15000E+00 .15000E+00 .15000E+00 .46719E-09 .43921E-09 .41319E-09 .39368E-O9 .38191E-O9

0

-.15000E+00 .15000E+00

e .15000E+00 .15000E+00 .15000Et00 .15OOOXtOO .15000E+00 .15000E+00 .15000Et00 .15000E+00 .15OOOE+OO

.15000E+00 -.15000Et00 -.15000E+00 - .15000E+00 -.15000E+00 -.15000E+00 -.15000E+00 -.15000E+00 -.15OOOEtOO -.15OOOEtOO -.15OOOE+OO -.15000E+00 -.15000E+00 -.15000Et00 -.15000E+00 -.15000E+00 -.15000E+00 -.15000E+00 -.15OOOEtOO -.15000Et00 -.15000Et00

.15000E+00

.15000E+00

.15000E+00

.15000E+00

.15000E+00

.15000E+00

.15000E+00

.15000E+00

.15000E+00

.15000E+00

.15OOOE+OO

.15000E+00

.15000Et00

.15000E+00

.15000E+00

.15000E+00

.15000E+00

.15000Et00

.15000E+00

.15000E+00

.15000E+00

Q -.15000E+00 .15000E+00

.00000E+00

.0000OE+00

.00000E+00

.00000Et00

.00000E+00

.00000Et00

.OOOOOE+OO

.00000Et00

.OOOOOE+00

.00000E+00

.00000E+00

.00000E+00

.00000E+00

.00000Et00

.00000E+00

.00000E+00

.OOOOOE+OO

.00000E+00

.00000E+00

.00000E+00

.OOOOOE+OO

.00000E+00

.00000E+00

.00000E+00

.00000E+00

.00000E+00

.00000E+00

.00000E+00

.00000E+00

.00000E+00

.OOOOOE+OO

.00000E+00

.00000E+00 . OOOOOEt 00

.00000E+00

.00000E+ 00

.00000E+00

.00000E+00

.00000Et00

.00000E+00

.OOOOOE+OO

.00000E+00

.00000Et00

.00000E+00

.00000E+00 . OOOOOEt 00

.00000E+00

.00000E+00

.00000E+00

.00000E+00

.00000E+00

EXTREMES

.00000E+00

.00000E+00

END

257


Fig. 14.2 presents a plot of the pressure at the midpoint of the pipe-line; it was drawn with the aid of the WTHG program on the basis of the numerical output contained in Table 14.4.

Fig. 14.2 Graph showing the pressure at the midpoint of the pipe-line plotted with the aid of the WTHG program.

The fronts of the waves in the graph are not exactly perpendicular, because the values of the pressures are determined by interpolation from the values in the numerical output.

Similarly, one may also plot diagrams for the other variables presented in the numerical output. The calculated results correspond to the analysis in Subsect. 1.2.2.

14.2 Abrupt closing of a pipe-line with the effect of pressure losses taken into account

We consider a pipe-line consisting of several parts having different cross- sections. The pipe-line is supported at different elevations. The central part is bifurcated and forms a closed circuit. The downstream end contains a valve restricting the outflow in the initial steady state. The reference plane is at the elevation of 400 m and the atmospheric pressure was taken as zero pressure. First, using the WTHM program, we calculated the pressure and the discharge in the entire pipe-line for the initial steady state with a known water level in the reservoir and a known closing of the valve. Then, the water hammer induced by an abrupt closing of the valve was calculated. The necessary data for the pipe-line system are given in Fig. 14.3. The input data used in the calculation are listed in Table 14.5.

258

Abrupt closing of a pipe-line with Ihe effect of pressure losses taken into account

Table 14.5 Input data file for calculating water hammer resulting from the abrupt closing of a pipe-line with the effect of pressure losses taken into account

1 2 SUDDEN FLOW INTERRUPTION - FRICTION INCLUDED 2 1 151 251 4803 .2 .03 1250 .025 2 2 251 351 1352 * 15 . 0 2 5 1324 2 3 351 451 2436 .1 .025 1330 2 4 251 451 6430 .15 ,025 1324 2 5 451 554 0 3014 .2 .03 1250 3 1 2 3 2 1 3 3 1 3 4 1 3 4 1 3 5 2 4 70 2 5 5 5 4 61 5 1 lp 6 2 5 4 Q 7 4 1 1 5 5 5 6 6 1 2 3 2 6 2 3 3 3 6 3 4 3 4 64 5 1 125 66 1 7

0 .031 0 0 .1 20 1500

pressure at the valve (Pa) Discharge through the valve (a3/s)

Steady state calculation 6 40 .2 2 * 600

Pressure in junction 2 (Pa) Pressure in junction 3 (Pa) Pressure in junction 4 (Pa) Pressure at the valve (Pa)

Water hammer calculation

0 0

.Ol

.01 .015 .025

1353780 1000000 1000000 1000000 1000000 627840

20 1500000

* 05

40 2500000 2500000 2500000 2500000

.4

with the effect of pressure losses taken into account: (a) hydraulic system; (b) calculation scheme.

259


The data on the sections and the junctions, introduced in the lines 2 and 3, correspond to Fig. 14.3. The initial discharges Q, were estimated. Their exact values were found from the abridged calculation of the steady state. The initial pressure in junction I was determined with the aid of equation (2.3) from the elevation of the water surface in the reservoir above the reference plane.

p , = (538 - 400)x 1000~9.81 = 1 353 700 Pa (14.7)

The initial pressure in junction 5 was determined similarly from the elevation of the outflow above the reference plane.

The initial pressures in other junctions were estimated. Their correct values were found from the calculation of the steady state.

A control valve (J = 54) is installed at the downstream end of section 5; it requires the submission of parameters (refer to Sects. 6.4 and 9.4). The number J , = 0 defines the set of these parameters. The type of submission presented in Table 9.4 was used.

The number 70 + J , = 70 + 0 = 70 is introduced in line 4 in columns 2 to 5. The functionf,, (t) is submitted by two points (NDt = 2). The first point with the coordinates t l = 0 andf (0) = 0.031 defines the values (i for t I 0 and the second point with the coordinates t2 = 0 andf,, (0) = 0 defines the values Q for t > 0.

First, the abridged calculation of the steady state was carried out. The equations for the junctions were solved by iteration, because the inclusion of the pressure losses in the pipe-line leads to non-linear equations and the changes in the pressures and discharges are sudden. This is expressed by the value 2 = 4 (refer to Sect. 9.5).

The steady state was calculated for a chosen time interval of calculation At = 0.1. The value of the time interval is not important in the present case (refer to Sect. 8.5). The wave velocity chosen for the abridged calculation of the steady state was a, = 1500 m s-'. It was determined so as to make inequality (8.15) valid for each section. Section 4 is critical, with

P'

(14.8)

The maximum value of the calculation time was chosen in accordance with relation (8.16)

(14.9)

The chosen time interval of printing for the main output was At, = 20 s, the same as tmax. The main output then contains only the initial and the final states of flow. The longitudinal interval of printing Axm need not be given in the abridged calculation of the steady state, because, in this case, only the pressure

t,,, = 40Ns At = 40 x 5 x 0.1 = 20 s

260

Abrupt closing of a pipe-line with the eflect of pressure losses taken into account

and the discharge at the upstream and downstream ends of the sections are printed.

In order to observe the gradual steadying of the initial state submitted, the graphical output includes the pressure in front of the valve, denoted p and the discharge through the valve denoted Q. The values submitted were defined in agreement with Section 9.6. For the pressure at the downstream end of section 5 ,

L,, = 5, L,, = 1, L,, = 1 (14.10)

The value L,, results from the modified length I , = Ax = a,At which applies to all the sections in the abridged calculation of the steady state. Consequently, L,, = 0 for the upstream end, and L,, = 1 for the downstream end of a section. The discharge through the valve is considered equal to the discharge from junction 5 into the damping device. According to Table 9.10, L,, = 5 and LK2 = 4, while the value L,, is not submitted.

Data line 7 starts the abridged calculation of the steady state (refer to Sect.

Further data lines concern the calculation of water hammer. Only data lines which differ from the preceding submission (except for line 7),

have to be submitted. For this reason, the data for the calculation of the steady state were submitted, as far as possible, in the form corresponding to the calculation of water hammer; only lines 5 and 6 need to be changed.

We chose the type of calculation which includes iteration, where the previously calculated values of the steady state are employed as the initial values (Z = 6). We used a time interval of calculation At = 0.2 s, which leads to some differences between the submitted and the modified values of the section lengths (refer to Tdbk 14.7). One could also employ another value At, for example, At = 0.2025 s. This would reduce the differences in the section length, however, at the cost of certain manual calculations and a less clear arrangement of the time data. It has to be decided whether there is any point in striving for such an accuracy considering other inaccuracies in the calculation, the accuracy of the results required and the accuracy of the input data employed.

The chosen value t,,, = 40 s makes it possible to calculate approximately one period of water hammer. The values At, = 2 s, Ax,.,, = 600 m and the change At, at line 4 were submitted to limit the size of the main output.

In the graphic and numerical outputs, the pressures in junctions 2, 3 and 4 were presented and denoted 2, 3, 4, and the pressure in front of the valve was denoted 5. All the pressures were plotted at the same scale in the graphic output. The value Min = 0 corresponds to the pressure at the level of the reference plane, and the value Max = 2 500 000 Pa to a level 2 500 000/8910 = 255 m above the reference plane, that is, on the elevation 400 + 255 = 655m. The chosen time interval for the graphic output Atm = 0.4 s. The graphical outputs

9.7).

26 1


Table 14.6 Graphical output for the abridged calculation of the steady state for the abrupt closing of a pipe-line with the effect of pressure losses included


Input file Num d2. wth 2

SUDDEN FLOW INTERRUPTION - FRICTION INCLUDED Steady state calculation

VARIABLES

Text Mark Hin Max

Pressure at the valve (Pa) p .00000E+00 .15000E+07 Discharge through the valve (m3/s) Q .00000Et00 .50000E-01

.00000Et00 .00000Et00

.00000Et00 .OOOOOE+OO

.00000Et00 .00000Et00


Dt Gr Nu tgn Dtgn

.loo0 0 0 ,0000 .oooo 0 0 .oooo .oooo 0 0 .oooo .oooo 0 0 .oooo .oooo

Hark

P Q

Values of variables in graph (dotted columns)

p .OOOOE+OO .3000Et06 .6000E+06 .9000E+06 .1200E+07 Q .OOOOE+OO .lOOOE-Ol .2000E-01 .3000E-01 .4000E-O1

.0000E+00 .0000Et00 .0000Et00 .0000Et00 .0000E+00

.0000Et00 .0000Et00 .0000Et00 .0000Et00 .0000E+00

.0000E+00 .OOOOEtOO .0000Et00 .0000Et00 .0000E+00

Time Min

.OOOOO ................................. 8. ..p.................

.10000 . * Q .P ,20000 . .30000 . .40000 . ,50000 . .60000 . ,70000 . .80000 . .90000 .

1.00000 . 1.10000 . 1.20000 . 1.30000 1.40000 . 1.50000 . 1.60000 1.70000 1.80000 . 1.90000 . 2.00000 . 2.10000 . 2.20000 . 2.30000 . 2.40000 . 2.50000 . 2.60000 . 2.70000 .

8. .Q 4 * Q * Q .Q Q Q. Q. Q Q Q Q 4 Q Q Q Q Q Q Q Q Q Q 4 Q

Ls Lk Lx

5 1 1 5 4 0 0 0 0 0 0 0 0 0 0

RESULTS

.1500E+07

.5OOOE-01

.0000E+00

.0000E+00

.0000E+00

Max

..... I

262

Abrupt closing of a pipe-line with the effect of pressure losses taken into account


2.80000 . Q P . 2.90000 . Q P . 3.00000 . Q P . 3.10000 . 9 P . 3.20000 . Q P . 3.30000 . Q P .

EXTREMES

P Q Mini mum .81681E+06 .18942E-01 .00000E+00 . O O O O O E t O O .00000Et00 Maximum .92203E+06 .25000E-01 .00000E+00 .00000E+00 .00000Et00

END

for the calculation of the steady state and the main and the graphical outputs for the calculation of water hammer are given in Tables 14.6, 14.7 and 14.8.

A graph showing the pressure at the valve after the closing of the pipe-line, plotted with the aid of the WTHG program on the basis of the numerical output is presented in Fig. 14.4.

It is obvious from Table 14.6 that the steadying of the flow to within the required accuracy (and consequently, also the termination of the calculation) occurred much sooner than the chosen t,,, = 20 s. The submission oft,, only serves as a limitation of the length of the calculation, if an unsuitable value a, had been employed.

In Table 14.7, the initial values of the discharge Qo in the sections and the pressures po in the junctions in the parts SECTIONS and JUNCTIONS are not the submitted values, but those determined in the preceding abridged calculation of the steady state. The initial steady state of flow considered for the calculation is presented at the beginning of the part denoted RESULTS for t = 0. The value VN = 0.031, which has the meaning of the parameter Q for the valve in this case, also corresponds to this steady state. In listing further results, one can clearly observe the front of the pressure wave travelling through the pipe-line away from the valve. For t = 2 s the pressure wave hit only a part of section 5. For t = 4 s, it also affected section 3 and 4 and is partially returning through section 5.

In the part denoted EXTREMES, the maximum and minimum pressures in the individual sections are given. They are important for the proportioning of the pipe-line and for checking whether cavitation did not occur during water hammer. These pressures have been converted to the level of the reference plane. The actual pressures in the pipe-line may be found using relation (2.3). Thus, for example, the actual minimum pressure in the pipe-line in section 3 (refer to Fig. 14.3) was

p r = p - g g h = 592797 - 1OOOx9.81(437-400)=229827Pa (14.11) The values given at the end of the file under the heading ACCURACY

indicate that the required precision in the iterative solution of the junctions was always attained.

263


Table 14.7 Main output for the calculation of water hammer after the abrupt closing of a pipe-line with the effect of pressure losses included


Input file Num d2 .wth 2

Ns Nj 2 5 5 6

Js 1 2 3 4 5

SUDDEN FLOW INTERRUPTION ~ FRICTION INCLUDED Water hammer calculation

Dt Dtm Dxm as tmax .20000 2.00000 600.00 .OO 40.0000

SECT IONS

Upstream end Downstream end Js Jj J Jp J j J Jp 1 1 51 0 2 51 0 2 2 51 0 3 51 0 3 3 51 0 4 51 0 4 2 51 0 4 51 0 5 4 5 1 0 5 54 0

1 lc D lambda a Qo 4803.00 4750.00 .2000 .03000 1250.00 .01998 1352.00 1324.00 .1500 .02500 1324.00 .00725 2436.00 2394.00 .loo0 .02500 1330.00 .00725 6430.00 6355.20 ,1500 .02500 1324.00 .01274 3014.00 3000.00 .2000 .03000 1250.00 .01999

JUNCTIONS

Jj J Jp PO 1353780. 1 2 0

2 1 0 1208044. 1189136. 3 1 0

4 1 0 929765. 627840. 5 2 0

PARAMETERS

2-5 6-10 20 -29 30--39 40-49 50 59 60- 69 70-79 1 1 .60000Et01 .40000Et02 .00000Et00 .00000Et00 .00000Et00 .00000E+00

70 2 .00000Et00 .31000E-01 .00000Et00 .00000Et00 .00000Et00 .00000E+00

Columns of input file

RESULTS

t Js .ooo 1

X

* 00 1000.00 2000.00 3000.00 4000.00 4750 .OO

2 .oo 1059.20

3 . 00 1064.00 2128.00

4 . 00 1058.20 2118.40 3177.60 4236.80 5296.00

P 1353780. 1323104. 1292427. 1261751. 1231075. 1208068. 1208044. 1192868. 1189136. 1073836. 958536. 1208044. 1161658. 1115272. 1068886. 1022500. 976114.

Q .01998 .01998 .01998 .01998 .01998 .01998 ,00725 .00725 .00725 .00725 .00725 .01274 .01274 .01274 .01274 ,01274 .01274

X

500.00 1500.00 2500.00 3500.00 4500.00

529.60 1324.00 532.00 1596 .OO 2394.00 529.60 1588.80

3707.20 4766.40 5825.60

2648.00

P 1338442. 1307766. 1277089. 1246413. 1215737.

1200456. 1189074. 1131486. 1016186. 929711. 1184851. 1138465. 1092079. 1045693. 999307. 952921.

Q .01998 .01998 .01998 .01998 .01998

.00725

.00725

.00725

.00725

.00725

.01274

.01274

.01274

.01274

.01274

.01274

264

Abrupt closing of a pipe-line with the efect of pressure losses taken into account


6355.20 929728. 5 .OO 929765.

1000.00 899269. 2000.00 868773. 3000.00 838276.

Jj P 1 1353780. 2 1208044. 3 1189136. 4 929765. 5 627840.

J S 5 Downstream

t J 2.000

IS X

1 .oo 1000.00 2000.00 3000.00 4000.00 4750.00

2 .oo 1059.20

3 . 00 1064.00 2128.00

4 .oo 1059.20 2118.40 3177.60 4236.80 5296.00 6355.20

5 * 00 1000.00 2000.00 3000.00

P 1353780. 1323104. 1292427. 1261625. 1230993. 1207982. 1207982. 1192779. 1189043. 1073762. 958587.

1207982. 1161601. 1115173. 1068886. 1022495. 976108. 929732. 929732. 1667782. 1667766. 1667805.

Jj P 1 1353780. 2 1207982. 3 1189043. 4 929732. 5 627840.

Js 5 Downstream

t Js X

4.000 1 . 00 1000.00 2000.00 3000.00 4000.00 4750.00

2 .oo 1059.20

3 .oo 1064.00 2128.00

4 .oo 1059.20 2118.40

P 1353780. 1322985. 1292350. 1261649. 1230958. 1207943. 1207943. 1192766. 1188964. 1869876. 1828105. 1207943. 1161567. 1115191.

.01214 ,01999 500.00 914517. .01999 1500.00 884021. .01999 2500.00 853525. .01999

VN VN 1 .3100E-01 .0000E+00

Q .01998 .01998 .01998 ,01998 .01998 .01998 .00725 .00725 .00725 .00725 .00725 .01274 .01274 .01273 .01274 .01274 .01274 .01274 .01999 .00077 .00038 .ooooo

X

500.00 1500.00 2500.00 3500.00 4500.00

529.60 1324.00 532.00

1596.00 2394.00 529.60 1588.80 2648.00 3707.20 4766.40 5825.60

500.00 1500.00 2500.00

P 1338442. 1307766. 1276968. 1246283. 1215652.

1200362. 1189043. 1131405. 1016240. 929732. 1184791. 1138361. 1092079. 1045693. 999302. 952915.

914475. 1667771. 1667805.

VN VN 1 .0000Et00 .0000Et00

Q .01999 .01998 .01998 .01998 ,01998 .01998 .00725 .00725 .00725 .00269 .00256 .01273 .01273 .01273

X 500.00

1500.00 2500.00 3500.00 4500.00

529.60 1324.00 532.00

1596.00 2394.00 529.60 1588.80 2648.00

P 1338327. 1307691. 1276992. 1246310. 1215615.

1200365. 1188964. 1891154. 1848820. 1817906. 1184755. 11 38380 I 1092018.

?01999 .01999 .01999

VN2 . OOOOEtOO

Q .01998 .01998 .01998 .01998 .01998

.00725 ,00725 .00725 .00725 ,00725 .01274 .01273 .01274 .01274 .01274 .01274

.01999

.00057

.00019

VN2 .0000Et00

Q .01998 .01998 .01998 .01998 .01998

.00725 . .00725 .00276 .00263 .00253 .01273 .01273 .01274

265

Examples of the calculation of water hamme r


3177.60 4236.80 5296.00 6355.20

5 . 00 1000.00 2000.00 3000.00

J J 1 2 3 4 5

1068828. 1022410. 1818987. 1817906. 18 17906. 18 10 1 9 1.. 1705804. 1705797.

P 1353780. 1207943. 1188964. 1817906. 627840.

Js 5 Downstream

t Js X 6.000 1 .oo

1000.00 2000.00 3000.00 4000.00 4750.00

2 .oo 1059.20

3 .oo 1064.00 2128.00

4 . 00 1059.20 2118.40 3177.60 4236.80 5286.00 6355.20

5 .oo 1000.00 2000.00 3000.00

JJ 1 2 3 4 5

P 1353780. 1323078. 1292420. 1261634. 1452970. 1439899. 1439899. 1645799. 1644405. 1628994. 1613947. 1439899. 1161557. 1879182. 1877415. 1875802. 1874280. 1872745. 1872745. 1865405. 1957965. 1958135.

P 1353780. 1439899. 1644405. 1872745. 627840.

JS 5 Downstream

t Js X

40.000 1 .oo 1000.00 2000.00 3000.00 4000.00 4750.00

2 .oo 1059.20

3 .oo 1064.00 2128.00

P 1353780. 1419764. 1469041. 1413354. 1363005. 1374648. 1374648. 1382293. 1338382. 1302409. 1225112.

.01274 3707.20 1045646.

.01273 4766.40 1819477.

.00171 5825.60 1818452.

.00141 ,00394 500.00 1814049. .00338 1500.00 1806337. .00038 2500.00 1705805. .ooooo

VN VN 1 .0000E+00 .0000Et00

Q .01998 .01998 .01999 .01998 .01440 .01429

- .00142 . 001 37 .00137 .00135 .00134 .01571 ,01273 ,00261 .00232 .00202 ,00173 ,00144 ,00434 ,00379 .00073 * 00000

X 500.00 1500.00 2500.00 3500.00 4500.00

529.60 1324.00 532.00 1596.00 2394.00 529.60 1588.80 2648.00 3707.20 4766.40 5825.60

500.00 1500.00 2500.00

P 1338419. 1307767. 1277076. 1246282. 1444255.

1435519. 1644405. 1636729. 1621428. 1872745. 1407694. 1138365. 1878261. 1876586. 1875030. 1873523.

1869048. 1861814. 1958071.

VN VN 1 .0000B+00 .0000E+00

Q .00947 .00914 .01054 .00961 .01033 .01132 .00267 .00231 .00289 ,00371 ,00347

X 500.00 1500.00 2500.00 3500.00 4500.00

529.60 1324.00 532.00 1596.00 2394.00

P 1347690. 1389068. 1464698. 1343331. 1391167.

1394303. 1338382. 1372076. 1315212. 1206048.

.01274

.00186

.00156

.00366

.00310

.00019

VN2 .0000Et00

Q .01998 .01999 .01999 .01998 .01433

-.00145 .00137 .00136 .00135 .00290 .01567 .01273 .00246 ,00217 .0018'7 .00158

.00406

.00351

.00037

VN2 .0000E+00

Q .00887 .0092 1 .01052 .01098 ,01169

,00272 .00289 .00396 .00423 .00345

266

Abrupt closing of a p ip l ine with the effect of pressure losses taken into account

.d Table 14.7)

A. . 00 1059.20 2118.40 3177.60 4236.80 5296.00 6355.20

5 . oo 1000.00 2000.00 3000 .OO

J J 1 2 3 4 5

1374648. 1280164. 1119661. 1159264. 11351 49. 1151723. 1206048. 1206048. 1109994. 1016384. 757915.

P 1353780. 1374648. 1338382. 1206048. 627840.

Js 5 Downstream

.00865 I 00701 .00422 .00393 .00100 .00201 .00205 .00550 .00284

- .00253 .ooooo

529.60 1588.80 2648.00 3707.20 4766.40 5825.60

500.00 1500.00 2500.00

1287909. 1094399. 1107941. 1236673. 1069766. 1199317.

1127108. 1056776. 871513.

.00752 * 00439 .00318 ,00249 .00083 .00234

.00337

.00108

.00225

Js 1 2 3 4 5

VN VN 1 VN2 .0000E+00 .0000Et00 .0000Et00

EXTREMES

pmin pmax 825032. 2118115. 711415. 2210053. 592759. 2472896. 604288. 2347685. 556160. 2393695.

Js VNmin VNmax 5 Downstream * 0000 .0310

ACCURACY

dP d 9 dM 1000Et02 .1000E-04 .1000Et00

END

2,5BE(B6 F(t) (Pa)

2,BBEflS

1,5BE(Bs

I M(B6

5,8BE15

e.aa t 5 ) e,mm t.mm I,EBEII 2,mei 3,111~tei vami

Fig. 14.4 Graph of the pressure at the valve alter an abrupt closing of the pipe-line with the effect of pressure losses included.

267


Table 14.8 Graphical output for the calculation of water hammer after the abrupt closing of a pipe-line with the effect of pressure losses included

WATER HAMMER IN PIPE-LTNE SYSTEMS WTHM Version 2.0 Copy 13


SUDDEN FLOW INTERRUPTION - FRICTION INCLUDED Water hammer calculation

VARIABLES

Text Mark Min Max

Pressure in junction 2 (Pa) 2 .00000Et00 .25000Et07 Pressure jn junction 3 (Pa) 3 .00000Et00 .25000Et07 Pressure in junction 4 (Pa) 4 .00000Et00 .25000Et07 Pressure at the valve (Pa) 5 .00000Et00 ,25000Et07

.00000Ei00 .00000Ei00

Time interval Changes of time interval of printing Codes oP variables of calculation


.2000 1 0 .oooo .4000 0 0 .OD00 .oooo n o , nooo * 0000 0 0 .oooo ,0000

2 2 3 0 3 3 3 0 4 4 3 0 5 5 1 12

n o o

RESULTS Values of variables in graph (dotted columns)

2 .0000Et00 .5000Ei06 .1000Et07 .1500Ei07 .2000Et07 .2500Et07 3 .0000Et00 .5000Et06 .1000Et07 .1500Et07 .2000Et07 .2500Et07 4 .0000EinO .5000EtOtj .1000Et07 .1500Et07 .2000Et07 .2500E+07 5 .0000Et00 ,500OEt06 .1000Et07 .1500Et07 .2000Et07 ,2500Et07

.0000Ei00 .0000Et00 .0000Et00 .0000Et00 .0000Et00 .0000E+00

Time Min Max

.ooooo

.40000 .

.8oooo . 1.20000 .

2.00000 . 2.40000 . 2.80000 . 3.20000 . 3.60000 . 4.00000 . 4.40000 .

5.20000 . 5.60000 . 6.00000 .

. . . . . . .

i.6nooo .

4 . 8oonn .

6.40000 . 6.8onoo . 7.60000 . 7.20000 . 8.00000 . 8.40000 . 8.80000 . 9.20000 . 9.60000 . 10.40000 . 10.80000 . io.oooon .

. . . . . . . . . 5.4 . . . . . . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . 2 . 5 4 . 2 . 5 4 . 2 . 5 4 . 2 . 5 4 . 2 . 5 4 . 2 . 5

. 2 . 5 4

. 2 . 5 4 .

. 2 . 5 4 . - 2 . 5 4 . . 2 . 3 5 4 . . 2 . 3 5 4 . . 2 , 3 4 5 .

2 . 3 4 5 . 2 . 3 4 5 .

3 2. 4 5 . 3 2. 4 5 3 2. 4 5 3 . 2 4 5

3. 2 4 . 5 2 4 .3

2 4 .3 2 4 . 3

42. - 3 42. 3 52 3 4

5 2 3 4

268

Abrupt closing of a pipe-line with the efect of pressure losses taken into account


11.20000 . 11.60000 . 12.00000 . 12.40000 . 12.80000 . 13.20000 . 13.60000 . 14.00000 . 14.40000 . 14.80000 . 15.20000 . 15.60000 . 16.00000 . 16.40000 . 16.80000 . 17.20000 . 17.60000 . 18.00000 . 18.40000 . 18.80000 . 19.20000 . 19.60000 . 20.00000 . 20.40000 . 20.80000 . 21.20000 . 21.60000 . 22.00000 . 22.40000 . 22.80000 . 23.20000 . 23.60000 . 24.00000 . 24.40000 . 24.80000 . 25.20000 . 25.60000 . 26.00000 . 26.40000 . 26.80000 . 27.20000 27.60000 28.00000 28.40000 28.80000 29,20000 29.60000 30.00000 30.40000 30.80000 31.20000 31.60000 32.00000 32.40000 32.80000 33.20000 33.60000 34.00000 34.40000 34.80000 35.20000 35.60000 36.00000 36.40000 36.80000 37.20000 37.60000

.5

5 .23 4 5 .2 34 35 .2 4

3 4 . 2 5 . 24 3. 5 . 24 3 5 . 32 4 . 5 . 3 2 4 . 5

f . 534 . 2

. 2

. 2 23 2

2. 3 2 . 3 2 . 3 2 . 3 2 . 3

3 2 . 3 2 . . 3 2 .

. 3 2 . 4 3 2 * 5 4

2 .3 4 2 . 3 4 2 . 3 4

2 4 3 . 5 2 4 . 3 . 5 2 . . 5 342 . . 5 3 4 2 . . 5 3 4 2 .

5. 3 4 2 . 45 . 3 2 . 4 .5 3 2 .

45 32 . 4 . 5 2 . 4. 5 2 4.5 32 .

5 4. 32 . .4 2 3

5 .4 2 3 5 . 34 2

5 .34 2 5. 3 2

4 5. 2 4 .2 4 .2 4 32 5 43 2 5 3 4 2 . 5 3 42.

3 2 4 . 5 2 4 . 5 2. 4 5 2 3

5 24 3 52 . 3 52 . 3 3 . 2 3 52 4 5.32

543 2 543 2

4 5 . 3 4 5 3

2 2

3 2 4 . 3 5 . 4 35 . 4 5 . 4

5 . 4 5. 4 45 4. 5

4 . 5 4 . 5 4 . 5 4 . 5 4 . 5

.5



38.00000 . 45 32 38.40000 . 45 2 38.80000 . 4 . 5 2 3 39.20000 . . 4 5 2 3 39.60000 . 5 . 4 2 3 40.00000 . 5 . 4 32

2 3 4 Minimum .82503Et06 .75064Et06 .71044Et06 Maximum .21181Et07 .22101Et07 .22645Et07

EXTREMES

5 55616Et06 .00000E+00 23937Et07 .00000E+00

END

In Table 14.8, the selected pressure curves are plotted as functions of time. Their extreme values appear at the end of the table under the heading EXTREMES. The extreme values of the pressures at individual points of the system may be used to complement the data on the extreme values of the pressure in the sections mentioned in Tables 14.7

In Fig. 14.4, the curve for the pressure at the valve has again been plotted as in Table 14.8 (symbol 5), but in greater detail.

b) q J = 2 eJ=2

Fig. 14.5 Control valves installed in the delivery pipe-line of a pump: (a) hydraulic system; (b) calculation scheme.

270

Control valves in the delivery pipe-line of a pump

14.3 Control valves in the delivery pipe-line of a pump

This is a case where the delivery pipe-line from a centrifugal pump is divided into three branches and a control valve is installed at the end of each branch (refer to Fig. 14.5). In the initial steady state, all the valves are partly open and the pump is switched on. The first part of the calculation deals with the determination of the initial steady state. The operating condition of the pump and the distribution of pressures and discharges in the pipe-line system are found by calculation, assuming that the speed of the pump is known.

Then water hammer induced by the simultaneous manipulation of all three valves is calculated. The control rkgime of each valve differs. For the sake of illustration, different types of submission of the control r&me are presented.

At the upstream end of section 3, local pressure losses are taken into account.

Table 14.9 Input data file for calculating water hammer induced by adjusting the valves on the delivery pipe-line of a pump

1 3 VALVES ON DELIVERY PIPE-LINE OF PUMP 2 1 156 0 251 312 1 . 4 . 0 3 921 2 2 251 351 134 1 . 4 . 0 3 921 2 3 353 0 454 2 2 50 1 . 2 . 0 3 953 2 4 251 554 0 426 . 7 5 . 0 3 1030 2 5 351 654 1 67 . 7 5 . 0 3 1030 3 1 2 3 2 1 3 3 1 3 4 2 3 5 2 3 6 2 4 60 4 70 3 4 80 450 4 71 2 4 81 450 4 450 8 4 451 4 452 4 72 7 4 82 4 92 4 150 4 160 4 170 9 4 180

4 210 8 4 220 4 230 5 5 6 4 61 1 1 Op 62 1 2 OQ 7 5 5 6 6 6 3 4 11 2 s 64 3 1 4 + 65 4 1 6 t 66 1 7

4 190

2 8 . 5 2 8 . 5 4 17 8 6

0 1 3 1 4 100

0 0 1 0 . 0 6 30 . 2 8 40 . 4 5 80 . 8 6 100 .96

0 .156 5 . 1 . 1 1 2 . 3 . 0 3 1 4 . 8 .Ol

17 0

27778 -40000 - 1 . 5 3.15115 -1 -2 .7685

. 7 5 - 2 . 4 8 1 5 1 . 5 -2 .2585 3 - .95E5 3 . 7 5 .05E5

- 2 . 5 -2.5E5 - 1 . 7 5 - 1 . 7 5 1 5 - . 2 5 -1.SE5 . 2 5 -1.15E5

1 . 5 -.15E5 2 . 2 5 .8E5 1 200 50

Pressure at the pump (Pa) Discharqe through the pump ( m 3 / s )

Steady state calculation

Parameter sigma of the valve at junction 5 Pressure at the end of section 3 (Pa) Pressure at the end oP section 4 (Pa)


. 0 6 8 1 0 500

1 2

20 60

9 . 1 16

8 . 0 8 3

- . 2 5 2 . 2 5 4 . 2 5

- 1 . 75

0 1

0 0 0 0

2 . 5 2

1 . 5 .5 .5 0

120000 120000

20601 61803 77499

0

. 1 4

.?1

. 0 6 .004

8 -2 .5285 - 1.73E5

1.B5 -1 .4485 - . 8 3 8 5

200 2.5E5

3 . 5

20 . 2 5

2.5E5 2.5E5

.27

27 1


The pressure devices, for which J = 53, 54 and 56, required the submission of parameters, hence the value J, was entered for them. The control rCgime of each valve (J = 54) is different; therefore, different values of J , were chosen.

The initial pressures in junctions I , 4, 5 and 6 were determined from the known elevations of the outflows; in junctions 2 and 3, they were estimated. The initial discharges were also estimated.

The input file is shown in Table 14.9. The parameters of the pressure devices are introduced in the lines beginning with number 4.

The line heaving number 60 in columns 2 to 5 contains the parameters of the local losses (refer to Sects. 6.3 and 9.4). The lines having the numbers 70 to 92 and 450 to 452 in columns 2 to 5 contain the parameters of the control valves (refer to Sects. 6.4 and 9.4). For the valve at the downstream end of section 3 (J, = 2), the parameters were submitted in accordance with Table 9.4, and for

Fig. 14.6 Control regimes of the valves: (a) at the downstream end of section 3 ( J , = 2); (b) at the downstream end of section 4 (J, = 0) and 5 (J, = I ) .

212


the valves at the downstream ends of sections 4 and 5, an alternative method of submission of the parameters was employed according to'Table 9.5. The control regimes of the valves are presented in Fig. 14.6. The valves at the downstream ends of sections 4 and 5 are identical and differ only in their control rkgimes. The same characteristic was used for both of them (refer to Fig. 14.6b); it was submitted only once in the lines with numbers 450 to 452 in columns 2 to 5 .

Fig. 14.7 Pressure characteristic of a pump.

The parameters of the pump are given in the lines with numbers 150 to 230 in columns 2 to 5. We used the method described in section 6.8 and Subsect. 9.4.4 (Table 9.4). It is assumed that the pump is permanently connected with the electric supply network which maintains a constant speed of the pump, n = 8.0 s-'. To establish this condition, N , = 0 and N , = 0, in other words, these values are not submitted. The pressure characteristic of the pump for the speed n, = 8.083 s-' is shown in Fig. 14.7. Other parameters of the pump listed in Table 9.4 need not be submitted in the present case.

The subsequent data lines beginning with numbers 5, 6 and 7 determine the type of calculation and the printing of the results for the steady state. The submitted values were determined as in Section 14.2.

The calculation of water hammer is submitted by means of the next data lines. The values resulting from the preceding calculation are employed as the initial ones. Only lines which differ from the preceding submission are submitted for the calculation.

273

Examples of the calculation of wuter hummer

Table 14.10 Main output for the calculation of water hammer induced by adjusting the valves on the delivery pipe-line of a pump

WATER HAMMER IN PIPE-LINE SYSTEMS Input file N um WTHM Version 2.0 Copy 13 d3. wth 3

Ns Nj 5 6

Js 1 2 3 4 5

z Dt 6 .06800

JS 1 2 3 4 5

1 lc 312.00 313.14 134.00 125.26 250.00 259.22 426.00 420.24 67.00 70.04

Columns of input file 2-5 6-10 20-29 30-39 60 0 70 3 71 2 72 7 80 450 81 450 82 0

150 0 160 0 170 9- 180 0 190 0 210 8 - 220 0- 230 0 450 8 451 0 452 0

92 n

VALVES ON DELIVERY PIPE-LINE OF PUMP Calculation of water hammer

Dtm Dxm as tmax 10.00000 500.00 .oo 20.0000

SECTIONS

Upstream end Downstream end Jj J Jp Jj J Jp 1 56 0 2 51 0 2 51 0 3 51 0 3 53 0 4 54 2 2 51 0 5 54 0 3 51 0 6 54 1

I) 1 ambda a Qo 1.4000 .03000 921.00 2.62523 1.4000 .03000 921.00 2.12935 1.2000 .03000 953.00 1.80573 ,7500 .03000 1030.00 .49595 .7500 ,03000 1030.00 .32369

Jj J Jp 1 2 0 2 1 0 3 1 0 4 2 0 5 2 0 6 2 0

JUNCTIONS

I' 0 0.

118741. 115995. 20601. 61803. 77499.

PARAMETERS

40-49 50-59 60-69 70 79 .. ~

.285OOE+O2 .28500E+02 .00000Et00 .00000E+00 .00000Et00 ,00000EtOn

.40000E+01 .17000Et02 .80000Et01 .60000E+01 .12000Et02 .00000Et00

.00000E+00 .13000Et02 .14000Et02 .10000E+03 .00000Et00 .00000Ei00

.00000E+00 .15600Ei00 .51000E101 .10000Et00 .91000Et01 .60000L-U1

.00000E+00 .00000E+00 .00000Et00 .00000Ei00 .00000Et00 .00000Et00

.12300E+02 .30000E-01 ,14800Et02 .10000E 01 .16000EtO:! .40000E-02

.17000Ei02 .00000Ei00 .00000Et00 .00000[.:t00 .00000Et00 .00000Et00

.00000E+00 .00000E+00 .00000Et00 .00000Et00 .80830Et01 .00000E+00

.27778E~05-.40000Et05 .00000Et00 .00000Et00 .00000Et00 .80000Et01

.15000E~01~.31500EtO6~.lOOOOEtOl-.276OOEtO6-.25OOOEtOO~ .25200Et06

.75000E+00-.24800E+O6 .15000Et01-.22500EtO6 .22500Et01-.17300Et06

.30000E+01-.95000EtO5 .3750OE+Ol .50000E+04 ,42500Et01 ,10000Et06

.25000E+01-.25000E+06-.17500E+01-.17500Et06~ .10000Et01-.14400Et06 ~.25000E+00-.13000E+06 .25000E+00-.11500E+O6 .75000Et00-.83000Et05 .15000E+01-.15000E+O5 .225OOE+Ol .80000Et05 .00000Et00 .00000B+00 .00000E+00 .00000Et00 .10000E+02 .60000E-01 .20000Et02 .14000E+00 .30000E+02 .28OOOEi00 .40000Et02 .45000Ei00 .60000Ei02 .71000E+00 .80000E+02 .86000E+00 .10000Et03 .96000Et00 .00000Et00 .00000E+00

.000OOE+00 .00000Et00 .00000Et00 .00000Et00 .00000Et00 .00000Ei00

274



RESULTS

t Js .ooo 1

2 3 4 5

Jj 1 2 3 4 5 6

Js 1 3 4 5

t Js 9.996 1

2 3 4 5

Jj 1 2 3 4 5 6

Js 1 3 4 5

t Js 19.992 1

2 3 4 5

Jj 1 2 3 4 5 6

JS 1 3 4 5

X

. o o

.oo

.oo

. o o * 00

P 129464. 118741. 79669. 118741. 11 5995.

P 0.

118741. 115995. 20601. 61803. 77499.

Ups t r earn Downstream Downstream Downstream

X P 00 133326. 00 129534. 00 113750. 00 129534. 00 123026.

P 0.

128534. 123026. 20601. 61803. 77499.

Upstream Downstream Downstream Down 6 t ream

X P 00 139477. 00 126781. 00 124659. 00 126781. 00 124659.

P 0.

126781. 124659. 20601. 61803. 77499.

Upstream Downstream Downstream Downstream

Q X P Q 2.62523 313.14 118742. 2.62523 2.12935 125.26 115994. 2.12935 1.80573 259.22 71702. 1.80573 .49595 420.24 108004. ,49595 .32369 70.04 115275. .32369

VN VN 1 VN 2 .8000Et01 .0000E+00 .0000E+00 .1560Et00 .0000E+00 .0000E+00 .1160E+00 .0000E+00 .0000E+00 .8400E-01 .0000E+00 .0000E+00

Q X P Q 2.57800 313.14 129534. 2.57748 2.45039 125.26 123026. 2.45028 .91246 259.22 141923. .91026 .12711 420.24 189610. .12741

1.53782 70.04 80377. 1.53782

VN VN 1 V N 2 .8000E+01 .0000Et00 .0000E+00 .5160E-01 .0000E+00 .0000E+00 .1804E-01 .0000Et00 .0000E+00 .8234Et00 .0000E+00 .0000E+00

Q X P Q 2.51823 313.14 126781. 2.51948 2.52005 125.26 124659. 2.52194 -.00632 259.22 127202. .ooooo -.00057 420.24 123320. .ooooo 2.52926 70.04 78892. 2.52820

VN VN 1 V N 2 .8000E+01 .0000Et00 .0000E+00 .0000E+00 .0000E+00 .0000E+00 .0000Et00 .0000E+00 .0000E+00 .9600Et00 .0000E+00 .0000B+00

215



EXTREMES

JS

I 2 3 4 5

pmin pmax 108166. 144349. 103709. 144161.

83884. 213079. 78532. 138817.

71702. 173824.

Js VNmin VNmax I Upstream 8.0000 8.0000 3 Downstream .oooo .1560 4 Downstream .oooo .1160 5 Downstream . 0840 ,9600

ACCURACY

dp dQ dM .1000Et02 .1000E-04 .1000Et00

END

The value At = 0.068 s was chosen so as to reduce the difference between the actual and the modified section lengths. The lengths may be compared using Table 14.10.

In this table, the values of the parameter VN are also presented under the heading RESULTS. This parameter has the meaning of the speed n for the pump, and it equals the value o (refer to Sect. 9.6) for the valves. Its maximum and minimum values obtained in the calculation of water hammer can be found at the end of the table under the heading EXTREMES.

Table 14.1 1 gives the graphical output of the calculation of water hammer. The size of the output was limited by changing the time interval of printing. The change is submitted by means of the data line beginning with numbers 66 (refer to Table 14.9). This affects only the graphical output.

U,BBE+W s'BBEM 9 , M a 4 8 4,MM 8.WM 1.28E1I I.EBEM 2,M+tlI

Fig. 14.8 Graphs showing variations in pressure due to the adjustment of the valves: a pressure/time curve at the downstream end of the section 4, and b at the pump.

276

Control valves in the delivery pipe-line of a pwnp

Table 14.1 1 Graphical output for the calculation of water hammer induced by adjusting the valves on the delivery pipe-line of a pump

WATER HAMMER IN PIPE-LINE SYSTEMS Input file Num WTHM Version 2.0 Copy 13 d3. uth 3

VALVES ON DELIVERY PIPE-LINE OF PUMP Calculation of water hammer

Text Mark

Pressure at the pump (Pa) p Discharqe through the pump (m3/s) Q

Parameter sigma of the valve at junction 5 s Pressure at the end of section 3 (Pa) * Pressure at the end of section 4 (Pa) t

Time interval Changes of time interval of printing of calculation

Dt Gr Nu tgn Dtgn

.0680 1 0 .oooo .2700 0 0 .oooo .oooo 0 0 .oooo .0000 0 0 .oooo .oooo

Values of variables in graph (dotted columns)

p .0000Et00 .500OE405 .1000Et06 .1500Et06 Q .1000Et01 ,1500Et01 .2000Et01 .2500E*01 6 .0000Et00 .5000E-01 .1000Et00 ,150OE400 it .0000Et00 .5000Et05 .1000Et06 .1500Et06 t ,000OEt00 .5000Ei05 .1000Et06 .1500E+06

Time Min

.ooooo . .

.27200 .

.54400 . ,81600 .

1.08800 . 1.36000 . 1.63200 . 1.90400 . 2.17600 . 2.44800 . 2.72000 . 2.99200 . 3.26400 . 3.53600 . 3.80800 . 4.08000 . 4.35200 . 4.62400 . 4.89600 . 5.16800 . 5.44000 . 5.71200 . 5.98400 . 6.25600 . 6.52800 . 6.80000 . 7.07200 . 7.34400 .

, . ........................ t . s ..p........ Q. * . + s p * . t s p * . t s p * . t s p * . t s p * . t s p * . t S p * . t s p * . + s p * .t s p * . t s p * . i s p * . t s p * . i s p * .t s p *. s4 p *.s +p s* P

s * p t s . * p + .

S .* P t . S .* P t

* s . * P . s . * P . s . * P S * * P

6 . ' + P

Q Q Q Q Q Q Q Q Q Q

. VARIABLES Min Max

.00000Et00 .250006+06

.10000Et01 .35000E+01

.00000Et00 .25000E+OO

.00000Et00 .25000B+06

.00000Et00 .25000E+06

Codes of variables

Mark

P Q S * t

.2000Et06 ,3000Et01 .2000Et00 .2000E+06 .2000E+06

..........

Q Q 9 Q Q Q Q 9 Q 9 Q Q

+ Q Q Q +

Q + * Q t .

Ls Lk Lx

1 1 0 1 2 0 4 11 2 3 1 4 4 1 6

RESULTS

.2500Et06

.3500Et01

.2500Et00

.2500E+06

.25OOE+O6

Max

. .

277


(Continued Table 14.1 1)

7.61600 . s . 7.88800 8.16000 8.43200 8.70400 8.97600 9.24800 9.52000 9.79200 10.06400 10.33600 10.60800 10.88000 11.15200 11.42400 11.69600 11.96800 12.24000 12.51200 12.78400 13.05600 13.32800 13.60000 13.87200 14.14400 14.41600 14.68800 14.96000 15.23200 15.50400 15.77600 16.04800 16.32000 16.59200 16.86400 17.13600 17.40800 17.68000 17.95200 18.22400 18.49600 18.76800 19.040OO 19.31200 19.58400 19.85600

s s

S

S S

s S

. R

. s * s . s . s . s . 6

. s S

S

S

S S

S

S

S

S

S

S

S S

S

s S

S

S

S

8

S

S

S

S

S 5

S

s 5

S

t . t

* P . Q * P . Q *P . Q P . Q P . Q p* . Q p* . Q p * . Q p* . Q p * . Q p *. Q p *. Q P "9 P * Q p .Q* 13 .Q* P Q * P Q *

+ p Q * P Q * P Q *

P Q * p Q + *

+ PQ. * t pQ. *

+ P Q p*Qt P Q+

t PQ. * I P.* t P*

+*pQ. * P Q

* P Q * + p Q +* p Q

t * p Q * ' p Q * + p Q * pt Q * tp .Q * P Q

t* p Q

t p Q * p .Y*

t P Q *

t

. t

. t

. t

t

t .

4 .

t . t . t . t . t . t .

t . t . t t

EXTREMES

* i

Mini mum Maximum

P Q S

.12067EiOR .24709Et01 .00000Ei00 .71702Ei05 .83884E+05

.14435EtO6 ,27009EtOl .11600E+00 .17382Et06 .21308Et06

END

The plots of some of the variables contained in the graphical output are presented in greater detail in Fig. 14.8. The figure was drawn with the aid of the WTHG program on the basis of the numerical output for the calculation of water hammer.

278

Calculation of the characteristic of a valve from pressure measurements

14.4 Calculation of the characteristic of a valve from pressure measurements

The characteristics of valves, pumps, turbines and other parts of hydraulic circuits are usually measured on special hydraulic circuits designed for this purpose. The characteristics of the larger elements are frequently measured using specially made models. In the present example, the characteristic of a large valve that has been already installed, is found. The valve forms part of the hydraulic system portrayed in Fig. 14.9. To determine the characteristic, measurements carried out for other purposes were employed. During these measurements, the valves at junctions 5, 6 and 7 (refer to Fig. 14.9a) were permanently closed. In the initial steady state, a pump located at junction 8 was pumping water through an open valve from a lower into an upper reservoir. After disconnecting the pump, the pressures at the points of junctions 2,4 and 8 were measured as well as the closing rCgime of the valve at junction 1.

a )

m

b)

J = 9 i2G(q/i Jp-2 6 5 Jp=l Jp=3

J = 1

J-1

Fig. 14.9 Calculation of the discharge through a valve: (a) hydraulic system; (b) calculation scheme.

219


Thus, the mutually corresponding values for the difference Ap, in pressure in front of and behind the valve, and the values of the parameter d defining the position of the valve, could be determined from the measurements for every instant. The value o and hence also one point of the characteristic of the valve (6. lo), could be found from the value Ap, with the aid of relations (6.6) and (6.7). Further points on the characteristic were found from the values measured for other instants.

To calculate the values d from the difference in pressure Ap, one has to know the value of the discharge for every instant considered. The discharge was not measured and such measurement is frequently virtually impossible under actual operating conditions. Therefore, the instantaneous values of the discharge were found by calculating the water hammer.

The calculations was carried out in accordance with the scheme presented in Fig. 14.9b. Only a part of the hydraulic system was analysed. The pressure measured in junction 4 was substituted for the effect of the upper part of the system starting at junction 4. and the pressure measured in junction 1 was substituted for the effect of the lower part, including the effect of the valve and the pump. The solution for the central part is relatively simple from the point of view of water hammer.

Junction 8 which is not connected to the analysed part of the system, was considered only to enable us to enter the values of the pressure at the junction in the same output files as the other results. It is convenient for further calculations.

A damping device “pressure” (refer to Sects. 5.10 and 9.4) was attached to junctions 1.4 and 8; this device ensures the measured pressure in the junctions.

The input data file employed is presented in Table 14.12 and also on the WTHM diskette in the D4.WTH file.

The coefficient of friction 1 was found from the measurement of the pressure in junctions I and 4 for the initial steady state and from the calculated initial discharge through the pipe-line.

The wave velocities for the individual sections were calculated and modified so as to correspond to the measured periodical variations in the pressure in the pipe-line system which occurred after closing the valve.

The initial discharge through the pipe-line was determined from the condition of zero discharge at the upstream end of section I after closing the valve. The calculation was repeated several times for different initial discharges to satisfy this condition.

The initial pressures in junctions I, 4 and 8 were measured. The initial pressures in the remaining junctions were determined by manual calculation.

The data lines beginning with number 4 contain the measured pressures in junctions I , 4 and 8. They are introduced as the parameters of a “pressure” damping device.

280

Calculation of the characteristic of a valve from pressure measurements

Table 14.12 Input data file for calculating the discharge through a valve

1 4 2 1 1 5 1 251 2 2 251 3 5 1 2 3 351 451 2 4 251 551 2 5 251 6 5 1 2 6 2 5 1 7 5 1 3 1 9 1 3 2 1 3 3 1 3 4 9 2 3 5 1 3 6 1 3 7 1 3 8 9 3 4 411 500

4 501 4 602 4 503 4 504 4 505 1 506 4 507 4 50R 4 412 550 4 550 30 4 551 4 552 4 553 4 554 4 555 4 556 4 557 4 550 4 559

4 500 26

4 41.1 f;no 4 con 2 4

4 603

4 601 4 602

4 604 4 605 4 606 4 607 5 6 8 2 6 1 1 2 OQ 6 2 1 3 1 6 3 4 3 4 I 3 4 9 3 8 66 1 67 1

DETERMINATION OF VALVE CHARACTERISTIC FROM MEASURED PRESSURE 1227 22 .7 1229 22.7

80 -.

172 988

8 0 80 73

. 7 4 8

1 4 . 5 19 .2 25 .2 30 .5 32 .6 3 4 . 8

1 . n 3

4.7 6 . 7

1 4 1 9 27

31.9 3 3

31 .5

. 7 4 7

12 1 7 . 5

22. 3 1

i n . 1

..

. O f 3 3

2 .2 3.6 3 . 8 2 .2 2 .2 2 .2

4 39 5000 3200000 4050000 4456000 4577000 4334000 4415000 4577000 4 0 9 i o o n

4354000

4144000

4374000

4414000

4279000

4 3 9 4 on0 3203000

3603000 3 n n 2 no o 1361000 1000000

8 80OOO i n

4244000

4287000

4379000 4374000

4349000

1923000

.032

.032

.032

.032

.032

.032

2 5 .6

9 15 .2 20.8 2 7 . 7 3 1 . 2 33 .2

35

2 4.2 4.9

7 1 5 . 2

23 31

3 2 . 1 33.5 34 .9

2 5 . 5 7 . 7

11 1 3

1 9 . 5 25 32

1000

~~~

1216 1227 1227 1227

3726000 3443000 4091000 4375000 4537000 4456000 46 18000 4252000 4172000

4229000 4 2 6 4 0 0 0 4134000 4359000

4384000 4364000

4349000 4294000

3723000 3402000 3923000 3443000 2362000

961000 961000 921000

4 3 4 4000

4422000

3 6 . 6

13 .5 1 6 . 1 22.8

30 3 2

3 3 . 9

2 . 2 4 . 4 5 .5 8.4

1 6 25

31 .5 32.5 33.9

35

3 6.2

9 1 1 . 4

1 6 2 1 3 0 35

22.7 0 0 0

4395000 4374000 4370000 4354000 4374000 4374000 4374000 4395000

3362000 3969000 4496000 4375000 4577000 4375000 4658000 4050000

4214000 4194000 4274000 4294000 4344000 4364000 4379000 4414000 4324000 4344000

3362000 3762000 3803000 3282000 1601000

961000 921000 841000

35 .... _. _.

Discharge through the valve [ m 3 / s ) - 2 0 30 Pressure at junction 1 [Pa) 0 5000000 Pressure at junction 4 (Pa) 0 5000000 Pressure at junction 8 [Pa) 0 5000000

0 1 0 . 1 . . . _

7 Calculation of discharge through the valve

The calculated curve for the discharge through the valve was plotted with the aid of the WTHG program in Fig. 14.10. At time t = 31.5 s approximately, the closing of the valve was terminated and the calculated discharge started to vary about a constant value of zero. This variation obviously is due to inaccuracies in the measurements and the calculation, since the actual discharge through the closed valve is of course nil. For this reason, the similar fluctuations which are also apparent in the preceding curve of the discharge were also neglected. The

28 1


3,BBEtBI F(t)

z@#

1 ,BBE+el

e m w

- I ,BEtlll

- Z & + e I t

e , o w 7.uiw i.acei z.imi z . m i 3 . 5 8 ~ 4 Fig. 14.10 Graph showing the calculated discharge through a valve as a function of time.

Fig. 14. I 1 Graph showing the pressure in junctions I , 4 and 8 as a function of time.

values of the discharge required to calculate the characteristic of the valve were taken as corresponding approximately to the zero amplitude of these vibrations.

The values for the pressure in junctions I , 4 and 8 used in the calculation are shown in Fig. 14.11 plotted with the aid the WTHG program.

The measured closing rtgime and the calculated characteristic of the valve are presented in Fig. 14.12.

The values u were calculated with the aid of the relation for Q # 0

(14.12)

obtained by rearranging relation (6.6). For the closed valve, u = 0. The values

282

Calculation of the valve control rkgime producing a required pressure or discharge curve

Q and App were obtained by measuring them from Figs. 14.10 and 14.1 1. The values d for the same times t were found in Fig. 14.12a. The points of the characteristic obtained in this way are marked by circlets in Fig. 14.12b.

The part of the characteristic appertaining to a large opening of the valve, not portrayed in Fig. 14.12b, has no practical meaning for the system in question. To find it, one would have to ensure a more precise measurement of the pressure in front of and behind the valve.

Fig. 14.12 Valve: (a) measured valve closing regime; (b) calculated characteristic of the valve.

The method employed converts the measurement of the instantaneous discharge, which tends to be difficult in practice, into the much simpler measurement of pressure.

14.5 Calculation of the valve control rkgime producing a required pressure or discharge curve

The calculation of the control regime of a valve was carried out for the hydraulic system described in Sect. 14.3 and portrayed in Fig. 14.5. It is, however also applicable, to a variety of other systems.

We determined a control rtgime for the valve at junction 4 which established the required pressure/time curve in the pipe-line in front of the valve, and then a further control regime which produced the required discharge/time curve through this valve. Apart from this, we also determined a control regime which ensured a required pressure/time curve up to the moment of closure of the valve. All other conditions, including any adjustment of the valves at junctions 5 and 6 and the operation of the pump remain as in Sect. 14.3.

283


At the end of section 3, a pressure curve as represented by curve a in Figure 14.13, was required.

For the calculation, a “pressure” damping device at junction 4 was substituted for the damping device “constant pressure” at this junction and the valve at the end of section 3 (refer to Fig. 14.5b). First, the steady state and then the discharge from junction 4 into the damping device were calculated with the aid of the WTHM program. The input data employed are listed in Table 14.13. They are presented under the name D5A. WTH on the WTHM diskette.

The calculated discharge into the damping device at junction 4, which is equal to the discharge through the valve, is represented by curve a in Fig. 14.14.

Consequently, we know the pressure in front of the valve (required), the pressure behind the valve (constant and corresponding to the elevation of the outflow opening) and the discharge through the valve (calculated). The values

2.88Et85 F(t) (Pa)

1,EXtBs

1 ,at05

B,BBE+M

4,M*

t 5)

e,mtm emtm 4 . w ~ 8 , ~ t m I.E#I I,KEMI 2 . 8 8 ~ ~ 1

Fig 14.13 Pressure in a pipe-line in front of a valve at junction 4 a required pressure curve, b corresponding to the required discharge, c required unrealistic curve.

5,M-U

-5,BE-BI

Fig. 14.14 Discharge through a valve at junction 4: a corresponding to the required pressure/time curve, b required discharge, c corresponding to the required unrealistic pressure curve.

284

Calculation of the valve control regime producing a required pressure or dircharge curve

Table 14.13 Input data file for calculating the control rkgime of a valve necessary to attain the required pressure/time and discharge/time curves

1 5 VALVE CONTROL FOR REQUIRED CURVES FOR PRESSURE AND DISCHARGE 2 1 156 0 251 312 1.4 .03 921 2.5 2 2 251 351 134 1.4 .03 921 2 2 3 353 0 451 250 1.2 .03 953 1.5 2 4 251 554 0 426 a75 .03 1030 . 5 2 5 351 654 1 67 .75 .03 1030 . 5 3 1 2 3 2 1 3 3 1 3 4 9 0 3 5 2 3 6 2 4 60 4 70 3 4 80 450 4 71 2 4 81 450 4 450 8 4 451 4 452 4 150 4 160 4 170 9 4 180 4 190 4 210 8 4 220 4 230 4 410 500 4 500 4 4 501 5 5 6 4 61 1 1 Op 62 1 2 04 7 5 5 6 6 6 1 4 4 e 6 2 4 3 p 66 1 1 7 4 410 500 4 500 2 7 4 410 500 4 500 2 7

28.5 28.5 4 17 8 6

0 13 14 100

0 0 10 .06 30 .28 40 9 45 80 .86 100 .S6

27778 -40000 -1.5 -3.1535 -1 -2.7685 .75 -2.4835 1.5 -2.2585 3 -.95E5 3.75 .05E5

-2.5 -2.585 -1.75 -1.7585 - . 2 5 -1.385 .25 -1.1585 1.5 -.1585 2.25 .8B5

0 71701 4 150000

1 40 50 Pressure at the pump (Pa)

Discharqe through the pump (m3/s) Steady state calculation

Discharge in junction 4 (m3/s) Pressure in junction 4 (Pa)

Required pressure/time curve

0 1.80571 15 0 Required discharge/time curve

0 71701 4 150000 Unattainable pressure/time curve

12.5 90000

.068 10 500

12

20 60

8.083

- .25 2.25 4.25

-1 .75

8 . 5

0 0

- . 5 0 0

1

0 120000 120000 71701 61803 77499

0

.14 * 71

8 -2.5265 -1.73E5

-1.44E5 - .83E5

1.E5

b

15OOOO

40 2.5E5

3.5

20 2

200000 .272

of time, discharge through the valve and pressure in front of the valve are included in the numerical output (refer to Table 14.14). These values were used to calculate the valve control regime represented by the value 0, applying relation (14.12). The calculate control regime is portrayed in Fig. 14.15 by curve a.

To determine the control regime of a specific valve, we have to know its characteristic (6.10), which allows us to convert the values 0 to the corresponding values of the parameter d which determines the position of the valve.

In the subsequent part of the calculation (refer to Table 14.13), we determined the pressure in junction 4 for a chosen discharge from junction 4 into the damping device. The discharge chosen is represented by curve b in Fig. 14.14.

285


Table 14.14 Numerical output for the calculation of the discharge ( Q ) at junction 4 for the required pressure/time curve ( p ) in the pipe-line in front of the valve

kAI'ER IIAMMIIH IN PIPE-LINE SYSTEMS WTHM Version 2.0

Input file Num d5a.wth 5

VALVE CONTROL FOR REQUIRED CURVES FOR PRESSURE AND DISCHARGE Required pressure/tirne curve

VARIABLES

Text Mark

Discharge in junction 4 (rn3/s) Q Pressure in junction 4 (Pa) p

'T i me i titer v a 1 of calculation

Dt

.a680

Time t

.ooooo

.27200

.54400

.81600 1.08800 1.36000 1 ,63200 1.90400 2.17600 2.44800 2.72000 2.99200 3.26400 3.53600 3.80800 4.08000 4.35200 4.62400 4.89600 5.16800 5.44000 5.71200 5.98400 6.25600 6.52800 6.80000 7.07200 7.34400 7.61600 7.88800 8.16000 8.43200 8.70400 8.97600 9.24800

Changes of time interval of print

Gr Nu tgn

1 1 .oooo 0 0 .oooo 0 0 .oooo 0 0 .oooo

Values of variables 9

.18057E+O1

.17994E+Ol

.17878Et01

.17687E+O1 ,17449Et01 .17178Et01

.16516E+O1

.16117E+01

.15681E+01

.15232Et01

.147658+01 ,14248Et01 .13706EtOl .13130E+01 ,12571Et01 ,12024Et01 .11458E+01 .10923E+Ol .10413Et01 .99131Ei00 .94662Et 00 ,90096EtOO .85750E+00 .81642Et00 .777138+00 .74141E+00 .70629E+00 .67062Et00 ,63827Et00 .60769EiOO .57958Et00

.50839E+OO

.16875Ei01

.55599E+00

.53038Et00

P

.71701E+O5

.77025Et05

.82350Et05

.87674E+O5

.92998Et05

.98323E+05

.10365Et06

.10897E3+06

.1143OE+O6

.11962E+O6

.12494E+06

.13027E+O6

.13559EtO6

.14092E+O6

.14624E+06

.15000E+06

.15000Et06

.15000E+06

.15000Et06

.15000Et06 ,15000Et06 .15000Et06 .15000Et06 .15000E+06 .15000E+06 .15000E+Ob: .1500OE+06 .15000E+06 .15000E+06 .15000E+06 .15000E+06 .15000E+06 .14694Et06 .14286E+06 .13878Et06

Dtgn

.2720

. oooo

.oooo

.oooo

.00000Et00

.00000Et00

.00000Et00

. OOOOOE tOO

.00000Ei00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.000OOEt00 . OOOOOEI 00

. OOOOOE +00

.00000Et00

.00000Et00

.00000E+00

.00000E+00

.00000E+00

.00000Et00 . OOOOOE t O O . O O O O O E i 00

.00000Et00

.00000Et00

.00000E*00

.00000E+00

.00000E+00

.00000Et00

.00000Et00

.00000Et00 . O O O O O B t 00

.00000EtOO

.00000Et00

.00000Et00

.00000Et00

Codes of variables

Mark

9 P

.00000Et00

.00000Et00

.00000E+00

.OOOOOEtOO

.00000Et00

.00000Ei00

.00000Et00

.00000Et00

.00000EI 00

.00000Et00

. 0 0 0 0 0 8 i 00

.00000Et00

.00000Et00

.00000E+00

. OOOOOEI 00

.00000Et00

.00000Et00

.00000Et00

.00000E+00

.00000E+00

.00000Et00

.00000E+00

.00000Ei00

.00000Et00

. OOOOOEi 00

.00000Et00

.00000Et00

.00000Et00

.00000E+00

.00000EtOO

.00000Et00

.00000E+00

.00000Et00

.00000E+00

.00000Et 00

Ls 1.k Lx

4 4 0 4 3 0 0 0 0 0 0 0 0 0 0

RESULTS

.00000Et00

.00000Et00

.00000E+00

.00000E+00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000E+00

.00000E+00

.00000Et00

.00000Et00

.00000E+00

.00000Et00

.00000E+00

.00000E+00

.00000E+00

.00000Et00

.00000Et00

.00000E+00

.00000Et00

.00000Et00

.00000E+00

.00000Et00

.00000Et00

.00000Et00

.00000E+00

.00000Et00

.00000Et00

.00000Et00

286

Calculation of the valve control regime producing a required pressure or discharge curve


9.52000 9.79200 10.06400 10.33600 10.60800 10,88000 11.15200 11.42400 11.69600 11.96800 12.24000 12.51200 12.78400 13.05600 13.32800 13.60000 13.87200 14.14400 14.41600 14.68800 14.96000 15.23200 15.50400 15.77600 16.04800 16.32000 16.59200 16.86400 17.13600 17.40800 17.68000 17 I 95200 18.22400 18.49600 18.76800 19.04000 19.31200 19.58400 19.85600

Mini mum Maximum

.49128Et00

.47051Et00 ,47957EtOO

.46506EtOO

.46056EtOO

.46254EtOO

.46669EtOO

.47404EtOO

.48456E+OO

.49718EtOO

.51441Et00

.533OOEtOO

.54970Et00

.55269Et00

.55984Ei00

.57029Et00

.59617Et00

.61161Et00

.62603EtOO

.63138EtOO

.63854E+00

.64439Et00

.65301EtOO

.66580Et00

.68307Et00

.70000Et00

.69928EtOO

.70395EtOO

.71269Et00

.72497Ei00 ,73410Et00 .74335E+00 .74728Et00 .74976Et00 .75036Et00 .75598E+00 .76667Et00 .77590Et00

,69478EtOO

Q .46056EtOO .18057Et01

.13470E+06 ,13062EtO6 .12654EtO6 .12246Et06 .11838Et06 .1143OEtO6 .11022Et06 .10614Et06 .10206EtO6 .97980Et05 .939OOEtO5 .90000Et05 .90000Et05 .90000E+05 .90000Et05 .90000E+05 .90000Et05 .90000Et05 .90000Et05 .90000Et05 .90000Et05 .90000Et05 .90000Et05 .90000Et05 .90000Et05 ,90000Et05 .9OOOOEt05 ,900OOEt05 .90000Et05 .90000Et05 .90000Et05 ,90000Et05 .90000Et05 .90000Et05 .90000Et05 .90000Et05 .90000Et05 .90000Et05 .90000Et05

P .71701EtO5 .15000Et06

.00000E+00

.00000Et00

.00000E+00

.00000Et00

.00000E+00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000E+00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000E+ 00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000E+00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.OOOOOEtOO

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000E+00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000E+00

.00000E+00

.00000Et00

.00000E+00

.00000Et00

.00000Et00

.00000Ei00

.00000E+00

.00000E+00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.OOOOOEtOO

.00000Et00

.00000E+00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000E+00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000E+00

.00000Et00

.00000Et00

.00000E+00

.00000E+00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000E+00

.00000Et00

.00000Et00

.00000Et00

.00000Et00

.00000E+00

.00000E+00

.00000Et00

.00000E+00

.00000Et00

.00000E+00

.00000Et00

.00000E+00

.OOOOOEtOO

.00000Et00

.00000E+00

.OOOOOEtOO

.OOOOOEtOO

.00000E+00

.00000E+00

EXTREMES

.00000Et00

.00000Et00

.00000Et00

END

The pressure corresponding to it is shown by curve b in Fig. 14.13. The value Q,

represented by curve b in Fig. 14.15, was calculated in a similar manner to the preceding case.

In principle, one may choose any pressure curve and calculate the corresponding discharge curve, and vice versa. Some choices are, however, unrealistic in practice for various reasons. Thus, for example, the value CT may be changed by means of a control valve only within a limited range. If the value IJ corresponds to a desired pressure and a calculated discharge falls outside this range, the desired pressure curve cannot be achieved. There may be yet other practical reasons which render this impossible, for example, cavitation.

An unrealistic pressure curve of this type is represented by curve c in Fig. 14.13. The discharge curve corresponding to it was calculated (refer to Table

287


14.13) and is represented by curve c in Fig. 14.14. Curve c in Fig. 14.15 represents the calculated values cr for 0 I t 5 I 5 s. For t = 15 s, however, the valve is completely closed and the required pressure curve can no longer be established with the aid of the valve.

The curve which the pressure actually follows in such a case, was determined in a separate calculation. The input data for this calculation are contained in Table 14.1 5 and are listed on the WTHM diskette under the name D5B. WTH.

Fig. 14.15 Calculated valve control curves: a for attaining the desired pressure/time curve; b for attaining the required discharge/time curve; c for attaining the required pressure/time curve up to the closing of the pipe-line.

E,WE+M 4,BBEM 1,BBEI I,aBEtEl 1.68E91 2,M81

Fig. 14.16 Graph showing the calculated pressure in a pipe-line in front of the valve at junction 4 corrcsponding to the required pressure curve up to the closing of the pipe-line.

288

Calculation 01 the valve control rigime producing a required pressure or discharge curve

Table 14.15 Input data file for calculating the control regime of a valve necessary to attain the required pressure/time curve up to the complete closing of the valve

1 5 2 1 2 2 2 3 2 4 2 5 3 1 3 2 3 3 3 4 3 5 3 6 4 60 4 70 4 80 4 71 4 81 4 450 4 451 4 452 4 150 4 160 4 170 4 180 4 190 4 210 4 220 4 230 4 410 4 500 4 501 4 502 4 503 4 504 4 505 4 506 4 507 4 508 4 509 4 510 4 511 5 5 61 4 62 4 66 1

VALVE CONTROL FOR REQUIRED CURVES FOR PRESSURE AND 156 0 251 312 1.4 .03 921 251 351 134 1.4 .03 921 353 0 451 250 1.2 .03 953 251 554 0 426 .75 .03 1030 351 654 1 67 .75 .03 1030

2 1 1 9 0 2 2

3 4 17 8 6 12

2 0 13 14 100

8 0 0 10 .06 20 30 .28 40 .45 60 80 .86 100 .96

8.083

9 -1.5 -3.15E5 -1 -2.7635 -.25 .75 -2.48E5 1.5 -2.2585 2.25 3 -.95E5 3.75 .05E5 4.25

8 -2.5 2.5E5 -1.75 -1.7535 -1 -.25 -1.3E5 .25 -1.15E5 .75

500 1 35 0 1.8057 .544 1.7878 1.088

1.632 1.6875 2.176 1.6117 2.720 3.264 1.4240 3.808 1.3130 4.352 4.896 1.0923 5.440 .9913 5.984 6.528 .8164 7.072 .7414 7.616 8.160 .6077 8.704 .5524 9.248 9.792 .4403 10.336 .3889 10.880 11.424 .2951 11.968 .2521 12.24 12.512 .2128 12.784 .1966 13.056 13.328 .1278 13.600 .0992 13.872 14.144 .0739 14.416 .0552 14.688 14.960 .0032 15.000 0

28.5 28.5

450

450

27778 -40000

1.5 -.15E5 2.25 .8E5

6 2 .068 10 500 4 Q Discharge in junction 4 (m3/s) -.5 3 P Pressure in junction 4 (Pa) 0 1 0

DISCHARGE 2.62527 2.12930 1.80567 .49594 ,32364

0 118736 11 5988 71701 61803 77499

0

14 71

8 -2.5235 -1.7335

1.E5 -1.44E5 - .83E5

1.7449 1.5232 1.2024 .g010 .6706 .4927 .3400 .2325 .1568 ,0923 .0268

20 2

200000 .27

7 Pressure/time and discharge/time curves

The initial values of pressure and discharge corresponding to the steady state were determined in the preceding calculation (Table 14.13) and taken from the main output. The discharge curve for 0 5 t I 15 s was also determined in the last part of the preceding calculation. This can be calculated from the required pressure curve up to the closing of the pipe-line, and was read from the numerical output. For t > 15 s, we took the discharge as equalling zero corresponding to a closed valve. The discharge used for 0 I; t I; 15 s is represented by curve c and for t > 15 s by curve b in Fig. 14.14. The calculated pressure curve is portrayed in Fig. 14.16.

289


14.6 Calculation of the steady state of flow in a pipe-line network

In this section, we present an example of the application of the WTHM programs for the calculation of the steady state in a pipe-line network without a subsequent calculation of water hammer. We chose a network consisting of 32 sections interconnected by means of 24 junctions and forming several closed circuits. Four reservoirs, one pump and three outflows are included in the network. The calculation scheme for the network is shown in Fig. 14.17. The elevations above sea level of the surfaces of the liquid in the reservoirs (J = 2) and the elevations of the mouths of the outflows into the atmosphere (J = 3) are given in the scheme (Fig. 14.17). The atmospheric pressure is taken as the zero pressure and the elevation of the reference plane is 165 m a.s.l., that is, at the

J = 3 Jp.1

Fig. 14.17 Calculation scheme for a pipe-line network.

290

Calculation of the steady state of flow in a pipe-line network

- 0 . 5 ~ 1 0 ~

Fig. 14.18 Characteristic of a pump for

pipe-line network. calculating the steady state of flow in a

0

level of the surface of the liquid in the reservoir at junction I . The junctions and the section ends, at which no number for a damping or pressure device is given, are treated as junctions without a damping device (J = I), or attachments without a pressure device (J = 51).

The wave velocity in the pipe-line need not be given, since it is not used in the abridged calculation of the steady state. In the formal checking of the input file carried out with the aid of the WTHD program, the omission of this value induces an error message. If the input file is used only for the abridged calculation of the steady state, this message can be disregarded. The initial discharges through all the sections and the initial pressures in all the junctions were estimated to be the same in order to simplify the submission. Accurate values, which correspond to the liquid levels in the reservoirs have to be introduced only for junctions 1, 6, 12 and 21, at which the pressure device “constant pressure” has been installed.

I I

I I I I

1.96 I 0.1 0.2 Clms’]

The input file is shown in Table 14.16 and appears on the WTHM diskette under the name D6.WTH. The parameters of the “reservoir” damping device (refer to Sects. 5.3 and 9.4) are introduced in the input file in the lines beginning with the numbers 4, 10 and 4, 12. The “reservoir” damping device was used to describe the outflow of the liquid through a restriction. The pressure pb corresponds to the elevation of the outflow above the reference plane.

The parameters of the pump (refer to Sects. 6.8 and 9.4) are given in the data lines beginning with the numbers 4, 150 to 4, 180. The characteristic of the pump was submitted only to the extent shown in Fig. 14.18 by a solid line. Other parts of the characteristic do not affect the resulting steady state of flow. The values used for the calculations outside the range of the points submitted, corresponding to the values Bp+ = Bp- = 0 is plotted by a dashed line.

29 1

Exumples of the calculation of water hammer

Table 14.16 Input data file for calculating the steady state of flow in a pipe-line network

1 6 2 1 156 0 251 2 2 251 2 3 351 2 4 1051 2 5 451 2 6 551 2 7 251 2 8 351 2 9 351 2 10 1051 2 11 1151 2 12 1351 2 13 851 2 14 751 2 15 851 2 16 951 2 17 951 2 18 1151 2 19 1151 2 20 1551 2 21 1651 2 22 1451 2 23 1451 2 24 1551 2 25 1951 2 26 1951 2 27 2051 2 28 1851 2 29 2251 2 30 2351 2 31 2351 2 32 2051 3 1 3 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 3 10 3 1 1 3 12 3 13 3 14 3 15 3 16 3 17 3 18 3 19 3 20 3 21 3 22 3 23 3 24 4 10 4 1 1 4 12 4 150 4 160 4 170

2 1 1 3 0 1 2 1 1 1 1 1 2 1 1 1 1 3 1 1 1 1 2 1 1 3 2

5

351 451 551 551 651 851 951 1051 1151 1251 651 751 1451 951 1451 1551 1551 1351 1651 1351 1751 1851 1951 2251 2051 1651 2251 2351 2451 2051 2151

4 180 5 32 24 4 61 1 1 Op 62 1 2 OQ 66 1 1 7

379 224 359 275 310 396 280 292 330 226 327 246 332 408 306 240 308 136 568 485 314 441 2 30 143 188 327 272 260 235 222 219 334

.3

.3

.2

.3 15 .3 .2 .3 .3 .3 .3 .2 15 15 15 15 .3 .3 15 . 2 1 5 .1 15 .3 15 .3 15 15 .2 .1 15 .3

57.8e6 57.8e6 255. 5e6 255.5e6 122.7e6 122.7e6

STEADY STATE CALCULATION .025 .025 .03 .025 .03

.025

.027

.025

.025 .03 .03 .02 .03

,025 .03 .03

.025

.025 .02 .02 .02 a03 so3

.025 * 02 .02 .02 * 02 .02 .02 .02 .02

216700 91200

193200

092 -114e4 .12 -108e4 -18 -90.5e4 .207 --79e4

1 1000 600 500 Pressure at the pump (Pa)

Discharge through the pump (m3/s)

8.04

.151

500000 . 0 5 0

.2

.2

.2

.2

.2

.2

.2

.2

.2

.2

.2

.2

.2

.2

.2

.2

.2

. z

.2

.2

.2

.2

.2

.2

.2

.2

.2

.2

.2

.2

.2

.2 0

500000 500000 500000 500000 522100 500000 500000 500000 500000 500000 709000 500000 500000 500000 500000 500000 500000 500000 500000 657000 500000 500000 500000

8.04 - 100e4

i ono

20

1500000 .3

292

Calculation of the steady state of flow in a pipe-line network

The wave velocity a, = 500 m s-l chosen for the calculation of the steady state is much lower than that corresponding to equation (8.15), from which the condition a, 2 5292 rn s-' results for section 22. The discrepancy was caused by the very rough estimate of the initial discharges in the system. The lower value was chosen to reduce the duration of the calculation of the steady state. The maximum calculation time tmax = 1000 s was chosen on the basis of trial calculations performed for other values of a,.

The resulting values of the discharges and the pressure in the system at the steady state of flow are listed in Table 14.17. It contains a part of the main output.

The value t indicates the calculation time, in which the steady state was reached with the required accuracy. One line of the results corresponds to each section. It contains, in succession, the number J , of the section; the distance x = 0 from the upstream end of the section; the pressure p in the pipe-line at the upstream end of the section; the discharge Q at the upstream end of the section; the modified length x = 500 of the section, which does not have any effect on the resulting steady state of flow, the pressure p at the downstream end of the section, and the discharge Q at the downstream end of the section. The discharges at the upstream and downstream ends of a section differ by at the most

Table 14.17 Part of the main output for the results obtained in the abridged calculation of the steady state of flow in a pipe-line network

t Js 582.000 1

2 3 4 5 6 7 8 9

10 1 1 12 13 14 1 5 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

X

* 00 .oo .oo * 00 .oo .oo .oo .oo . 00 .oo .oo 100 .oo .oo . o o .oo . 00 . oo . 00 . 00 . 00 . 00 . 00 . 00 . oo .oo .oo .oo . 00 . 00 . 00 . oo

P 834891. 712900. 662357. 644948. 561358. 587977. 712900. 662357. 662357. 644948. 658732. 569009. 693340. 6731 50. 693340. 660937. 660937. 658732. 658732. 658343. 647178. 652480. 652480. 658343. 657325. 657325. 656749. 651462. 650700. 647430. 647430. 656749.

9 .19646 ,16450 .06085 .15762

.14124

.03196

.02410

.07955 - .07810 - . 12395 .06135 .01378 .01378 . 01818 .01049 ,03183 .01869 .02720 .02131 .03415 .02056 .00371 .02910 .01285 ,01635 .01283 ,00371 .01657 .03068

-.01412 - .01070

-.01638

X

500.00 5O0.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00 Fjoo.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00 500,OO 500.00 500.00

P 712900. 662357. 561358. 587977. 587977. 522100. 693340. 660937. 644948. 658732. 709000. 522100. 673150. 652480. 660937. 652480. 658343. 658343. 569009. 647178. 569009. 199196. 651462. 657325. 650700. 656749. 647178. 650700. 647430. 308690. 656749. 657000.

Q .19646 .16450 ,06084 .15761

.14124

.03196

.02414

.07952 ,07806

- . 12393 .06135 .01378 .01378 .01818 .01049 .03179 .01863 .02720 .02131 .03415 .02056 .00371 .02920 .01285 .01625 .01284 .00371 .01656 .03068

- .01637

-.01411 - .01060

293


d,, = 10-4m3 s-I, the admissible difference in the discharge for the steady state; the pressure at the upstream and the downstream ends of the section differ from the value resulting from equation (8.4) by, at the most d,, = 100 Pa, the admissible difference for the steady state.

The discharge and pressure at the upstream end of section 1 correspond to the operating state of the pump as marked in Fig. 14.18.

14.7 Periodic variations in pressure

A system portrayed in Fig. 14.19 was chosen to demonstrate the calculation of the effect of periodic pressure variations on the flow.

A pump delivers water from a lower to an upper reservoir through a pipe-line with a blind branch pipe attached to it. It is assumed that the pressure in the delivery part of the pump is not constant, but varies periodically, with a period equal to a whole multiple of the time required for the pressure wave to pass through the individual sections of the pipe-line. The pressure losses at the upstream ends of all the sections are considered proportional to the square of the discharge. The pressure losses along the pipe-line sections are neglected.

The input data file is shown in Table 14.18. The calculation starts from the steady state of flow resulting from the constant

pressurepo = 2 271 180 Pa in the delivery part of the pump. For the calculation, a “pressure” damping device attached to junction I was substituted for the effect

a1

Fig. 14.19 Calculation of the effect of periodic variations in pressure: (a) hydraulic system; (b) calculation scheme.

294

Periodic variations in pressure

Table 14.18 Input data file for calculating the effect of periodic variations in pressure

1 7 PERIODIC VARIATIONS IN PRESSURE 2 1 153 0 251 86.4 .12 1200 .028 2 2 253 0 351 21.6 .12 1200 .028 2 3 253 0 451 21.6 .12 1200 0 3 1 9 0 2271180 3 2 1 2117947 3 3 2 19647 14 3 4 1 2117947 4 60 50 50 4 410 450 0 .012 4 450 12 0 2271180 .006 2396180 .012 2487686 4 451 .018 2521180 .024 2487686 .03 2396180 4 452 .042 2146180 . 0 4 8 2054674 . 0 5 4 2021180

.066 2146180 .072 2271180 4 453 .06 2054674 4 1 1 1.428 0 5 3 4 2 .006 1.5 0 1.5 6 1 1 3 1 Pressure in junction 1 (Pa) 0 5000000 6 2 2 3 2 Pressure in junction 2 (Pa) 0 5000000 6 3 4 3 4 Pressure in junction 4 (Pa) 0 5000000 64 1 2 12Q Discharge at the end of section 1 (m3s-1) 0 .05 65 2 2 O* Discharge at the start of section 2 (m3s-1) 0 .05 7 Resonance

of the lower reservoir and the pump. This damping device ensures a constant pressure in junction 1 up to instant t = 0, and periodical variations in pressure with a period of 0.072 s for t > 0. The pressure curve was submitted, in accordance with Sect. 9.4, in the data lines with the numbers 4, 410 to 4, 453. We only submitted the values for the first period of the curve together with the instant tbp = 0 when it commences and the time t,, = 0.072 s marking the end of the first period. This is portrayed in Fig. 14.20 by the curve denoted 1.

The variations in pressure in junction 1 induce variations in pressure and discharge throughout the entire system, which, in view of the chosen period of the pressure variation, begins to resonate. The variations in pressure and discharge gradually increase, until they attain a roughly periodical character. The changes in pressure that occur in junction 4, are represented by curve 4 in Fig. 14.20.

5MtB6 F(t)

4 m e 6

3,BEtBS

2.BEt86

I ,BBEtB6

U,BBE+M t

Resm in th jvrctim I nd 4 (Pa)

Fig. 14.20 Graph showing pressures at resonance.

295

Examples of the calculafion of water hammer

Table 14.19 Main output for the calculation of the effect induced by periodic variations in pressure at resonance

WATER HAMMER IN PIPE--LINE SYSTEMS WTHM Version 2.0


Ns Nj 3 4

JS 1 2 3

2 2

1 86.40 21.60 21.60

Columns of input fi 2 - 5 6-10 20-29

1 1 .14280EtOl 60 0 .50000Et02

410 450 .00000Et00 450 12 .OOOOOE+00 451 0 .18000E-01 452 0 .42000E-01 453 0 .60000E-01

PERIODIC VARIATIONS IN PRESSURE Resonance

Dt Dtm Dxm as tmax 00600 1.50000 . oo .oo 1.5000

.l 6

1 2 3

lc 86.40 21.60 21.60

SECT IONS

Upstream end Downstream end J j J Jp Jj J Jp 1 53 0 2 51 0

2 53 0 4 51 0

D 1 ambda a Qo

.1200 . oonoo 1200.00 ,02800

.1200 .ooooo 1200.00 .ooooo

2 , 53 0 3 51 0

.1200 .ooooo 1200.00 ,02800

Jj J Jp 1 9 0 2 1 0 3 2 0 4 1 0

t Js .ooo 1

2

3

Jj 1 2 3 4

X

. 00 14.40 28.80 43.20 57.60 72.00 86 .40

.oo 14.40

. 00 14.40

le 30-39 40-49

.00000Et00 .00000Et00

.50000E+02 .00000E+00

.72000E-01 .00000E+00

.22712Et07 .60000E-02

.252121+07 .24000E-01

.21462E+07 .48000E-01

.205478+07 .66000E--01

P 2117947. 2117947. 2117947. 2117947. 2117947. 2117947. 2117947. 1964714. lYfi4714. 2117947. 2117947.

P 2271180. 2 1 1 7947. 1964714. 2117947.

9 .02800 .02800 .02800 . 02R00 .02800 .02800 .02800 .02800 .ooooo * 00000

.0280n

50-.59 .00000E+00 .00000Et00 .00000Et00 .23962E+07 .24877E+07 .20547E+07 .21462E+07

X

7.20 21.60 36.00 50.40 64.80 79.20

7.20 21.60 7.20

21.60

60-69 .00000Et00 .00000Et00 .00000Et00 .12000E-01 .30000E-01 .54000E-01 .72000E-01

P 2117947. 2117947. 2117947. 2117947. 2117947. 2117947.

1964714. 1964714. 2117947. 2117947.

JUNCTIONS

PO 2271180. 21 17947. 1964714. 2117947.

PARAMETERS

70-79

.00000Et00

.00000Et00

.24877E+07

.23962E+07

.20212Et07

.22712Et07

. o n o o o ~ t o o

RESUI.1S

Q

.02800

.02800

.02800

.02800

.n2aoo

.0280n

.02800

.02800

.ooooo

.ooooo

296



t J s 1.434 I

2

3

Jj 1 2 3 4

t Js 1 .440 1

2

3

Jj 1 2 3 4

t Js 1.446 1

2

3

Jj 1 2 3 4

t J s 1.452 1

X

.oo 14 .40 28.80 43.20 57.60 72 .00 86 .40

.oo 1 4 . 4 0

* 00 14 .40

X . 00 14 .40 28.80 43.20 57 .60 72 .00 8 6 . 4 0

* 00 14 .40

.oo 1 4 . 4 0

X

.oo 14 .40 28.80 43.20 57 .60 72 .00 86.40

,oo 1 4 . 4 0 . 00 1 4 . 4 0

X

. oo 14 .40 28 .80 43 .20 57 .60

P 2076661.

671700. 86303Y.

21 35919. 3375715. 3469609. 2098180. 1950357. 1879490. 2112116.

804857 .

P 2146182. 2098180. 1964714.

608652.

P 2117949.

565877. 565877.

2 1 17949. 3615552. 3615552. 2117947. 1964714. 1964715. 2117947.

608655.

P 2271183. 21 17947. 1964714.

393565.

P 2140499.

863046. 671704.

2076952. 3469604. 3375711. 2136384. 1987687. 2049941, 21221 33.

804863.

P 2396181. 21 36384. 1964714.

608658.

P 2159109. 1448778. 1235058. 2027419. 2910310.

Q .01886 .02233 ,03030 .03613 .03110 .02356 .01906 ,02750 .02656

- .00844 - ,00388

Q- . 02800 .02599 .02598 .02800 .02718 .02718 .02800 .02800 ,02628 * 00000 .ooooo

Q .03617 .03030 .02233 .O le86 .02356 .03110 .03612 .02758 .02656 .00854 .00388

Q .04100 .03431 .01968 .01221 .02053

X

7 .20 21.60 36 .00 50 .40 64 .80 79 .20

7 .20 21.60

7.20 21.60

X 7 .20

21 .60 36 .00 50.40 64 .80 79 .20

7 .20 21.60

7.20 21 .60

X 7 .20

21 .60 36 .00 50 .40 64 .80 79.20

7 . 2 0 21.60

7 .20 21.60

X

7 .20 21.60 36.00 50 .40 64.80

P 1235050.

565874. 1448770. 2823188. 3615554. 2910315.

187 3 3 15. 1964714. 1363299.

608652.

P 1190318.

344428. 1190316. 3022556. 3822763. 3022557.

1964717. 1964714. 1216155.

393565.

P 1448775.

565878. 1235056. 2910312. 3615549. 2823183.

2056114. 1964714. 1363302.

608658.

P 1813226. 1190322. 1558341. 2524001. 3022552.

e .01968 .02599 .03431 .03465 .02718 .02053

.02714

.02628 - . 0 0 7 1 1

.ooooo

e . 02721 .02541 .02721 .02777 ,02688 .02777

.02737

.02576

.ooooo

.ooooo

Q .03431 .02598 .01968 .02053 .02718 .03465

.02714

.02628

.00711 * 00000

Q .03925 .02721 .01397 .01464 .02777

297

Exaniples of the calculation of water hammer


2

3

JJ 1 2 3 4

t Js 1 .458 1

2

3

Jj 1 2 3 4

t Js 1 . 4 6 4 1

2

3

Jj 1 2 3 4

t Js 1 .470 1

2

3

72 .00 8 6 . 4 0

.oo 14 .40

.oo 14.40

X . 00 14.40 28 .80 43.20 57 .60 7 2 . 0 0 8 6 . 4 0 . 00 14 .40

.oo 14 .40

X

. oo 14 .40 28 .80 43 .20 57 .60 72 .00 8 8 . 4 0

. o o 14 .40

.oo 14 .40

X . oo

14 .40 28 .80 43 .20 57 .60 72 .00 8 6 . 4 0

. o o 14 .40 . 00 14 .40

2823181. 2154615. 2014769. 2056113 . 2 1 15210. 1363306.

P 2487687. 2154615. 1964714 . 1216162.

P 2166736. 2140503. 2076958. 2005389. 2076948. 2136379. 2162414. 2026767 I 1987684 . 2 1 1 1 195. 2122140 .

P 2521 179 . 2162414. 1964714. 2117953.

P 2159109 . 2828280. 2910320 . 2027419. 1235048. 1449372. 2154614 . 2014769. 1923369. 2115210. 2869861.

P 2487685. 2154614 . 1964714. 3028118.

P 2140497 . 3380649. 3474187. 2076954 .

672164 . 863651 .

2136383 . 1987686. 1918571. 2122133. 3428471.

.03465 ,04095 .02875 .02714 .01420 . 0 0 7 1 1

Q . 04258 .03617 .01886 ,00976 .01886 ,03612 .04253 .02634 .02758 .01619 .00854

Q .04100 .03469 .02053 . 0 1 2 2 1 .01968 .03430 .04095 .02675 .02761 .01420 .00709

9 .03617 .03114 .02360 .01886 .02232 .03029 .03612 .02758 ,02736 .00854 .00377

79.20

7.20 21 .60

7 . 2 0 21 .60

X

7 . 2 0 21.60 3 6 . 0 0 5 0 . 4 0 64 .80 79.20

7.20 21 .60

7.20 21 .60

X 7 . 2 0

21.60 36 .00 5 0 . 4 0 64 .80 7 9 . 2 0

7 .20 21 .60

7 .20 21 .60

X 7 . 2 0

21 .60 36 .00 50 .40 6 4 . 8 0 78 .20

7 . 2 0 21 .60

7 .20 21 .60

2489538.

2072911. 1964714 . 1710841 . 1216162 .

.03945

.02678

.02737

.01242

.ooooo

P 9 2159112. .04100 2117956. .02800 2027422. . 0 1 2 2 1 2027417. , 0 1 2 2 1 2117941. .02800 2154612. ,04095

2014768. .02675 1964714 . . 0 2 8 0 0 _.._ -~ . . .

2 1 15213. .01420 2117953. .ooooo

P 2494012. 3027139 . 2524007. 1558337. 1190775. 1809255.

1941541. 1964714 . 2522493. 3028118.

P 2828287. 3620654. 2910317. 1235051.

566480. 1449374 .

1923370 . 1964714. 2869857. 3621769.

Q . 03950 , 0 2 7 8 1 .01464 .01397 . 0 2 7 2 1 .03920

.02715 , 0 2 7 8 0 . 0 1 2 3 1 .ooooo

Q .03469 . 0 2 7 2 3 ,02053 .01968 .02598 .03430

. 0 2 7 6 1

.02722

.00709

.00000

298



Jj 1 2 3 4

t Js 1.476 1

2

3

J j 1 2 3 4

t. Js 1.482 1

2

3

Jj 1 2 3 4

t Js 1.488 1

2

3

J j 1 2 3 4

X

.oo 14.40 28.80 43.20 57.60 72.00 86.40

.oo 14.40

.oo 14.40

X . 00 14.40 28.80 43.20 57.60 72.00 86.40

. o o 14.40

. o o 14.40

X

.oo 14.40 28.80 43.20 57.60 72.00 86. 40

. oo 14.40

. o o 14.40

P 2396178. 21 36383. 1964714. 3621768.

P 2117946. 3620651. 3620651. 21 17949. 566482. 566482.

2117947. 1964715. 1964715. 211 7947. 3621764.

P 2271177. 2117947. 1964714. 3828825.

P 2076698. 3474182. 3380645. 2140962. 863658. 672168.

2084739. 1936568. 2010859. 2099704. 3428465.

P 2146179.

1964714. 3621 761.

2084739.

P 2025611. 2910307. 2828280. 2159714. 1449382. 1235057. 2039536. 1901O30.

2080763. 2869851.

P 2054674. 2039536. 1964714. 30281 05.

2006059.

Q X P .02800 7.20 3027133. .02723 21.60 3827697. .02723 36.00 3027135. .02800 50.40 1190781. .02598 64.80 345040. .02598 79.20 1190779. .02800 .02800 7.20 1964716. .02722 21.60 1964714. ,00000 7.20 3028111. ,00000 21.60 3828825.

Q ,01885 .02360 .03114 ,03616 .03029 .02232 .Ole78 .02753 .02736 .00875 .00377

(d .01219 .02053 .03469 .04100 .03430 ,01968 .01210 .02662 ,02761 ,01452 .o0709

X 7.20

21.60 36.00 50.40 64.80 79.20

1.20 21.60 7.20

21.60

X

7.20 21.60 36.00 50.40 64.80 79.20

7.2n 21.60 7.20

21.60

P 2910309. 3620648. 2828283. 1449380. 566484.

1235055.

2006060. 1964714. 2869855. 3621761.

P 2523746. 3027129. 2494472. 1813838. 1190785. 1566128.

1982711. 1964714. 2500057. 3028105.

Q .02781 ,02693 .02781 .02721 .02540 .02721

.02780

.02693

.00000

.ooooo

Q .02053 .02723 .03469 ,03430 .02598 .01968

.02761

.02722

.00709

.ooooo

Q .01464 .02781 .03950 .03925 .02721 . 01 390 .02710 .02780 .01252 .ooooo

299



t Js 1 . 4 9 4 1

2

3

Jj 1 2 3 4

t Js 1.500 1

2

3

Jj 1 2 3 4

X

.oo 1 4 . 4 0 2 8 . 8 0 43 .20 5 7 . 6 0 7 2 . 0 0 8 6 . 4 0

1 4 . 4 0 .on

.on 1 4 - 4 0

X

. o o 1 4 . 1 0 2 8 . 8 0 43 .20 5 7 . 6 0 7 2 . 0 0 8 6 . 4 0

1 4 . 4 0

1 4 . 4 0

.no

.on

P 2002675. 2076694. 2140957. 2167348. 2140966. 2084745.

1885112. 1936566. 2072608. 2099697.

P 2021 1 8 0 . 201 9061. 1964714. 211 7942.

2019061.

P 2025613. 1233241. 1449370. 2159713. 2828289. 2922436. 2039534. 1901029. 1859685. 2080762. 1328850.

P 2054675.

196471 4 . 1171290.

2039534.

Q .00973 .01885 .03616 .04258 .03616

.00963

.02618

. 0 2 7 5 3 - .01655 - , 0 0 8 7 5

.n1878

9 .01219 .01966 .03430 .04100 .03469 .02042 .01210 .02662 ,02701

- .01452 - .00744

J.; 1 2 3

X 7 . 2 0

2 1 . 6 0 3 6 . 0 0 50 .40 6 4 . 8 0 7 9 . 2 0

7 . 2 0 2 1 . 6 0

7 . 2 0 2 1 . 6 0

X

7 . 2 0 21 .60 36 .00 50 .40 6 4 . 8 0 7 9 . 2 0

7 .20 21.60

7 . 2 0 2 1 . 6 0

P P 20256n9. .n1219 2117938. . nzaoo 2159711. .041oo 2159716. .04100 2117955. .02800 2039538. . 01210

1801029. , 0 2 6 6 2

2080760. -- .01452 1964714. .n28oo

2117942. .noono

P 9

1190521. , 0 2 7 2 1 1813833. .03925 2493476. .03950 3034926. , 0 2 7 7 4

.01451 2537679.

1838967. . 0 ? 6 6 1 196471 4 . .02727

1555623. . n i 3 9 4

1672248. - .01278 1171290. . oonon

EXTREMES

1m i n pmax 04441.8 . 3827697.

1812485. 2105490. 385855. 3828825.

ACCURACY

d il dQ dM . ~ o o n ~ t n z .ionor 04 . i o o o E + o o

END

The distribution of the maximum and minimum values of pressure and discharge along the pipe-line sections occuring during the last period calculated (1.428 s I I 5 1.5 s) is portrayed in Fig. 14.21. The extreme values were determined from the main output presented in table 14.19. The extent of the main output was limited by the data (refer to Table 14.18 and Sects. 9.4 and 9.5) so that only the results of the last period would be printed in detail.

Protection of a delivery pipe of a pump by an air chamber

- 0.025

Fig. 14.21 Distribution of the maximum and minimum values of pressure and discharge along the pipe-line sections at resonance.

14.8 Protection of a delivery pipe of a pump by an air chamber

This section deals with the calculation of the protection of the delivery pipe of a pump with the aid of an air chamber. The longitudinal profile of the delivery pipe-line is shown in Fig. 14.22a.

The pump is provided with a non-return flap valve preventing reverse discharge. The moment of inertia of the pump and the electric motor is so small that the pump stops, after disconnection of the electric motor, in a time much shorter than the period of the pressure wave. The pressure wave, induced by disconnecting the electric motor, might lead to a drop in pressure, which could induce cavitation, and excess pressures much higher than those corresponding to the steady state (refer to Subsect. 1.2.2). For this reason, it is proposed to protect the delivery pipe-line by an air chamber, which prevents cavitation and excessively high pressures.

The calculation was carried out according to the scheme shown in fig. 14.22b. We considered a sudden closing of the pipe-line at its upstream end. The unsteady state, which developed immediately after the closing of the pipe-line, was submitted as the initial state. The input data are presented in Table 14.20 and on the WTHM diskette in the D8.WTH file.

30 1

Examples of the calc~clation of nwer hummer

a)

, a l s s l I I

J - 2

1

J s I I I D I hl a 100 1 132641 0.14 10.08511181 10.01

Fig. 14.22 Protection of the delivery pipe of a pump by an air chamber: (a) longitudinal profile; (b) calculation scheme.

Table 14.20 Input data file for calculating the effect of protecting the delivery pipe of a pump by an air chamber

1 8 2 1 3 1 3 2 4 20 5 1 61 1 62 1 63 1 64 1 65 1 66 1 7 4 20 7 4 20 7

7

7 4 2 0 7

4 zn

4 20

302

1 5 1 251 4 0 2

2 2 1 00 1 1 1 1 33 1 55 5 v 1

PROTECTION OF PIPE-LINE BY AIR CHAMBER 3264 .14 .025 1181 * 01

1144835 1021853

1.2 1054531 .1 1021853 .46 10 100

Pressure in km 0.0 (Pa) 500000 1500000 Pressure in km 0.544 (Pa) 500000 1500000 Pressure in km 1.632 (Pa) 500000 1500000 Presure in km 2.720 (Pa) 500000 1500000

0 1 Change of air volume in air-chamber (m3) -.2 .1

1

Initial volume of air 0.1 m3 .45 1021853

Initial volume of air 0.45 m3

Initial volume of air 1.0 m3

Friction 15000, volume 0.1 m 3

Friction 150000, volume 0.1 m3

Friction 1500000, volume 0.1 m3

1.2 1054531

1. 1021853 1.2 1054531

15000. .1 1021853 1.2 1054531

150000. .1 1021853 !.2 1054531

L 500000. .1 1021853 1.2 1054531


The submission comprised a total of 6 calculations. In the first three calculations, we considered different quantities of air in the air chamber and we did not consider any pressure losses in the connection between the air chamber and the pipe-line. In the next three calculations, we considered the same quantities of air in the air chamber and different pressure losses along the connection with the air chamber.

The parameters of the air chamber are given in the data lines beginning with numbers 4,20 (refer to Sects. 5.4 and 9.4). For the basic state of the air chamber, we chose the state corresponding to the hydrostatic pressure in the reservoir at the downstream end of the pipe-line

(14.13)

The polytropic exponent y = 1.2 and the absolute air pressure in the air chamber in its basic state was

(14.14)

where Pabs = 101 325 Pa is the normal atmospheric pressure. The elevation of the air chamber is 477 m above sea level.

The calculated curve for the pressure at the upstream end of the pipe-line for different volumes of air (Vb = 0. I m3; V, = 0.45 m3 and Vb = I .O m3) in the air chamber in its basic state, without considering the effect of the pressure losses during the inflow into the air chamber, is illustrated in Fig. 14.23. The maximum and minimum pressures along the pipe-line for these three cases are shown in Fig. 14.22 and the calculated curves for the volume of air in the air chamber are plotted in Fig. 14.24.

Figures 14.23 and 14.24 were plotted with the aid of the WTHG program based on the numerical output. Consequently, each numerical output had to be entered on the disk in a separate file. In the calculation, we determined the variation in the volume A V, which represents the difference in the volume of the liquid in the air chamber as compared with the basic state. The scale in Fig. 14.24 was modified to correspond to the volume of air in the air chamber.

The curves of the maximum and minimum values plotted in Fig. 14.22 were determined directly from the extreme values presented in the numerical outputs. The curves denoted a correspond to a volume v b = 0.1 m3 of air in the basic state, curves b to Vb = 0.45 m3 and curves c to Vb = 1.0 m3.

It is evident from Figs. 14.22 and 14.24 that the volume of air Vb = 0.45 m3 is sufficient to prevent cavitation during a fall-out of the pump and also ensures that the maximum pressure does not significantly exceed the pressure at the initial steady state. At the same time, the total volume of air in the air chamber does not exceed the value of 0.52m3 (refer to Fig. 14.24). This value may be determined by subtracting the minimum value for AV, given in the numerical, graphical and the main outputs, from the volume V, of air in the air chamber

pb = (574.2 - 470.0) e g = I 021 853 Pa

Pabs = (574.2 - 477.0) e g + 101 325 = I 054 531 Pa

303


in the basic state. In practice the actual volume of the air chamber has to be larger, considering the inaccuracies in the calculation and also in order to prevent the escape of air from the air chamber during operation.

A further reduction in the required volume of the air chamber may be achieved by restricting the inflow into the air chamber. In such a case the minimum pressure in the pipe-line is not decreased while, by using a suitable restriction, the time needed for damping the water hammer is reduced.

RWSW at la e.a (Pa)

Fig. 14.23 Pressure curves for an air chamber without considering the effect of pressure losses at the inlet into the air chamber. (Volume of air in the air chamber in its basic state: a v b = 0.1 m3; b V,, = 0.46 m3; c Vb = 1.0 m’).

a,mm f(t)

2,NE81

4,E-BI

l,i€%l

9,R81

I , E l t

B.BBE+m 2,88E@I 4.BBE4I 6,BBItBI 8,BBItBI I , d l Vclw c4 air in air- (n3)

Fig. 14.24 Graphs showing the volume of air in an air chamber without considering the effect of pressure losses at the inlet into the air chamber. (Volume of air in the air chamber in its basic state: a V, = 0.1 m3, b V, = 0.45 m3, c v b = 1.0 m3).

304


The effect of such a restriction is demonstrated in the last three calculation included in the submission given in Table 14.20. These three calculations were carried out for the same volume of air V, = 0.1 m3 in the basic state and for different restrictions of the inflow. The resulting pressures in the air chamber are plotted in Fig. 14.25. The maximum drop in the pressure in the pipe-line is the same in all the cases.

I ,5BEt86 F(t)

I ,38Et86

I ,lBEt86

9,BBE15

7,BBEt8s

t 5)

e,mm Z,~BE~UI ~ . B B E + ~ I 6 . ~ 1 1 I,BBE+~I I,BBE(BZ

Resm at la 9,9 (Pa)

s m t e s

Fig. 14.25 Pressure curves for an air chamber with an initial volume of air Vb = 0.1 m3 and for restrictions of the inlet into the air chamber: a c,+ = 0, d cJ+ = 1.5 x lo4 kg m-’, e CJ+ = 1.5 x 10’ kg m-’,fCJ+ = 1.5 x lo6 kg mP7.

It is evident from Fig. 14.25 that a restriction with a value CJ+ = 1 . 5 ~ lO4kgmP7 (curve d) suffices for the pressure in the pipe-line to remain below the pressure at the initial steady state. A further increase in the restriction of the inflow into the air chamber (CJ+ = 1.5 x lo5 kg m-7, curve e) leads to a steady state being attained more rapidly, but the maximum pressure is higher. If the restriction is excessive (CJ+ = 1.5 x lo6 kg m-7, curvef), the time needed to achieve the steady state is again prolonged while the maximum pressure increases yet further.

The value CJ+ = 1.5 x lo4 kg m-7 corresponds approximately [54] to a sharp-edged opening of a diameter

I

A 0.015 m 8e TC* 0.972 x lo6 Cj+

Do = ‘J (14.15)

305


14.9 Water hammer induced by cavitation after disconnection of a pump

The calculation of water hammer induced by disconnecting a pump, when the effect of the inertia of the pump and the electric motor is taken into account, is demonstrated for the hydraulic system portrayed in Fig. 14.26a.

In the initial steady state, two pumps are drawing water from a reservoir through non-return flap valves into a common delivery pipe-line. A valve is installed at the upstream end of this pipe-line. The central part of the pipe-line contains a condenser which produces considerable pressure losses. A cooling tower is placed at the downstream end of the pipe-line. We calculate the water hammer caused by the simultaneous disconnection of both the pumps.

The calculation scheme employed is shown in Fig. 14.26b. The two pumps (J = 56) at the upstream ends of sections I and 2 are considered to be identical. Since we suppose that both pumps experience the some adjustment rkgime, we

a1 ,, A 1 . 6 m

Ref. plane c; - J-2

Jp=l

24000 4 140000 El

ALsrn

J =6 Jp=O

Fig. 14.26 Fall-out of a pump resulting in cavitation: (a) hydraulic system; (b) calculation scheme.

306

Water hammer induced by cavitation after disconnection of a pump

Table 14.21 Input data file for calculating water hammer after the disconnection of a pump (cavitation) 1 9 2 1 156 2 2 256 2 3 353 2 4 451 3 1 2 3 2 2 3 3 1 3 4 7 3 5 6 4 40 4 50 4 60 4 61 4 150 1 4 160 4 170 8 4 180 4 190 4 250 9 4 260 4 270 5 4 5 4 61 1 1 1 62 1 1 63 1 2 64 4 5 65 4 3 66 1 1 7 5 4 5 6 7

0 355 0 3 5 5 0 453 1

551

0 0

In OP OQ

V 4

1 4 3

134 275

15.64 46.469

0

- .70

4 . - . 4 7 2.25

3.375 .05

2 .7

1.5 1.5

2 2

15.64 46.469

-361000

DISCONNECTION OF PUMP WITH CAVITATION 0 921 0 921

025 856 169 856

6.6 112776. 25506.

366.25

1.5 -307100 -2 41 300 3.25 -188400

-8400 1.25 -13540 -15450 2.55 -15650 -15160 3.9 -14520

8 50 Speed of the pump (s-1)

Pressure at the Dump (Pal

-58900 4.4 80000

Discharge through the pump’ (m>/s-l) Empty volume in junction 4 (103)

Pressure in junction 4 (Pa)

Steady state calculation

Water hammer calculation . I l l 5 4 .75 300

1.2

8.0833

2.25 3.75

1.75 3.0 4.8

-15 600000

-9 0

200000 0

2.8 2.8 5.6 5.6 0 0

240000 140000 130000

1

8,0833 8.0833 -272700 - 117700

-14720 -15550 -12400

8 10

400000 6

2.5 800000

.1

7.5

could choose the same values for the set of parameters ( J p = 0). The downstream ends of sections I and 2 are provided with non-return flap valves (J = 55) which allow water to flow in the positive sense of the section. All the pressure losses, including those along section I and 2, in the non-return flap valves and in the valve, are considered concentrated at the upstream end of section 3 (J = 53, J, = 0). The valve is not adjusted during the course of water hammer. A pressure device “local losses”, located at the downstream end of section 3 (J = 53, J, = I ) was employed to represent the effect of the condenser. Cavitation is expected at junction 4, hence an “air inlet valve” damping device J = 7) was attached to this junction. Its parameters were chosen to simulate the conditions at cavitation. An “overflow” damping device (J = 6) was substituted for the cooling tower.

The input file employed for the calculation is presented in Table 14.21 and is included on the WTHM diskette under the name D9.WTH.

In the data lines beginning with numbers 4,40, the parameters of the damping device “overflow” are introduced (refer to Sects. 5.6 and 9.4). The pressure in junction 5, with the damping device in the basic state, was calculated from the elevation of the edge of the overflow above the reference plane

P , = 1 1 . 5 ~ g = 112776Pa (14.16)

307

Examples of the calculation of wafer hammer

The data line beginning with numbers 4, 50 contains the parameters of the “air inlet valve” damping device (refer to Sects. 5.7, 5.8 and 9.4). The pressure in junction 4, with the “air inlet valve” in the basic state, was calculated from the elevation of the pipe-line above the reference plane, and the pressure which exists in the pipe-line at cavitation

(14.17)

Here it is assumed that cavitation occurs at a negative pressure equivalent to a water column of 9 m height in the pipe-line. The submitted value y is not very relevant. The diameter Do = 1 m was chosen sufficiently large to maintain a constant pressure pb, to within acceptable limits in the pipe-he at cavitation.

Pb = 11.6 eg - 9 ~ g = 25 506 Pa

Fig. 14.27 Pressure and moment characteristics of a pump.

The parameters of the pumps (refer to Sects. 6.8 and 9.4) are given in the data lines beginning with numbers 4, 150 to 4, 270.

It is assumed that in the initial steady state the electric motors of the pumps are connected to an electric supply network, ensuring a constant speed. In the course of the calculation, the electric motors were disconnected once (N, = 1) at instant tN, = 0. While they remain connected, the network ensures a speed nE = 8.0333 s-’ (The value N, = 0 or it is not submitted). The speed n, = 8.0333 s-’ is the one, for which the pressure and moment characteristics were submitted. The initial speed no = 8.0333 s-’ did not have to be submitted in this case, because it is determined by the value n,.

The submission of the pressure and moment characteristics of the pumps is contained in lines 4, 170 to 4, 270. We submitted only those parts of the characteristics which are necessary for the calculation, as is evident from Fig. 14.27.

308

Water hammer induced by cavitation ajier disconnection of a pump

The submitted parts of the characteristics are plotted as a thick solid line. The margins of the regions to which the submitted portions of the characteristics apply in the calculation, are drawn by a thin dashed line. On the dotted curves are located the points corresponding to the excess pressure on the pumps, the moments and the discharges through the pumps, which appeared simultaneously during the calculation. They are located within the regions corresponding to the parts of the characteristics submitted. The values Bp+ ; Bp- ; BM+ ; B,- were not submitted; therefore, outside the region for which data was submitted the characteristics were assumed to have a constant value as represented by the thick dashed line. It was also not necessary to submit the characteristics for n = -ns, since negative speeds for the pumps did not appear

Fig. 14.28 Graph showing the discharge Q through a pump, the speed n of the pump and the pressure p behind the pump after a fall-out of the pump (100 % correspond to the values e = 4 m 3 s - ' ; n = I O S - ' ; ~ = ~ . O X I O ~ P ~ ) .

4,EKtBS p( t )

Z,M185

0,Ml

;;:;;; e,mtm t

5) 0.BBEtm 1.sBEtm 3Mtm 4 , I M G . B B E l 7 , I t m

RKsw p (Pa) and wid wluy U (a31 in jmtion 4

Fig. 14.29 Diagram depicting the pressure p behind a condenser and the void volume V caused by cavitation after disconneting both pumps.

Examples of’ the calculation of water hammer

First, we determined the conditions at the steady state during which the electric pumps were connected to the supply network maintaining the constant speed nE. The discharges and the pressures in the junctions were estimated approximately. Only the constant pressure in junctions 2 and 2 has to be determined accurately.

Subsequently, we calculated the water hammer induced by switching off both pumps. Some of the results of these calculations are shown in Figs. 14.28 and 14.29. These figures were drawn with the aid of the WTHG program based on the numerical output of the water hammer calculation.

14.10 Calculation of discharge from pressure measurements

The WTHM program may also be used to calculate the discharge through a pipe-line from the pressure curve measured during unsteady flow. This approach was employed to determine the steady state discharge in the system portrayed in Fig. 14.36a. We used the pressures measured at points 2 and 3 during water hammer induced by disconneting the pump and closing the non- return flap valve. Water hammer was calculated only in the pipe-line between points 2 and 3. The scheme of the calculation appears in Fig. 14.30.

Jp.2

Fig. 14.30 Scheme for calculaling discharge from pressure measurements.

& I l I D l A l a 1 @ 1 1 I2701 2 W 670 16.41

Junctions 1 and 2 in Fig. 14.30 correspond to points 2 and 3 in Fig. 14.36a. The effects of the parts of the system in front of, and behind these points were replaced by “pressure” damping devices attached to junctions 2 and 2 (refer to Sect. 5.10). The measured pressures, converted to the level of the reference plane, are shown in Fig. 14.3 1.

A steady state exists in the system until the instant t = 4.15 s. At this instant, the pump is switched off and the non-return flap valve starts to close. It is completely closed at time t = 19.6 s and, from then on, the discharge at the upstream end of section I is zero. The course of water hammer was calculated several times, for different values of the initial discharge Q,. The correct initial discharge, which is also the required discharge at the steady state, was determined from the condition that a discharge Q, = 0 exists at the upstream end of the section for t > 19.6 s. The calculated curve of the discharge which satisfies

310

Calculation of discharge from pressure measurements

Table 14.22 Input data file for calculating discharge from pressure measurements

1 10 2 1 3 1 3 2 4 411 4 500 4 501 4 502 4 503 4 504 4 505 4 506 4 507 4 508 4 509 4 510 4 412 4 600 4 601 4 602 4 603 4 604 4 605 4 606 4 607 4 608 4 609 4 810 5 1 61 1 62 2 63 1 66 1 7

CALCULATION OF DISCHARGE FROM MEASURED PRESSURES 151 251 270 2 .03842 670 6.47 9 1 410000 9 2 398000

33 4.15 4ionoo 4.75 295000 5.27 240000 5.62 i6800n 5.70 161000 5.75 150000 6.00 144000 6.01 136000 6.21 131000 6.47 133000 6.70 128000 10.72 128000 12.1 147000 15.1 150000 15.9 156000 16.4 167000 17.2 168000 18.0 189000 18.4 209000 19.0 247000 19.4 282000 19.7 311000 19.9 316000 20.3 310000 20.8 324000 21.0 324000 21.3 310000 21.9 300000 22.7 252000 23.0 231000 23.8 226000 24.3 229000 24.6 238000

33 4.52 399000 5.0 335000 5.61 263000 6.19 199000 6.32 183000 6.85 171000 7.31 164000 7.62 161000 8.43 148000 9.03 150000 9.40 157000 8.85 171000 10.9 184000 11.7 197000 12.7 184000 13.8 178000 14.9 178000 17.1 188000 17.8 199000 18.2 206000 19.3 243000 19.9 284000 20.2 314000 20.6 318000 20.9 312000 21.4 316000 22.9 243000 23.4 230000 24.1 228000 24.4 235000 25.1 238000 25.8 250000 26.7 271000

2 2 .05757 25 300 25 3 u Pressure at the upstream end (Pa) 0 500000 3 d Pressure at the downstream end (Pa) 0 500000

0 .25

500

600

2 OQ Discharge at the upstream end (11138-1) -1 8

Disronnection of the pump

Is! e.wtea S.BBEM I , B B E ~ B I I . $ E M I ~ , ~ B E Q I Z,EZ(BI

Pressw at the wtrearr ad danstren end of sect ion (Pa)

Fig. 14.31 Measured pressure curves: u -Junction I ; d -Junction 2.

this condition, is presented in Fig. 14.32. It corresponds to an initial discharge Q, = 6.47 m3 s-'. The small periodical fluctuations in this curve are not significant. They are obviously caused by inaccuracies in the measurements and in the calculation.

31 1


The input data employed in the calculation are presented in Table 14.22 and on the WTHM diskette in the DIO.WTH file.

The coefficient of friction 1 was determined from the difference in the pressure at the upstream and the downstream ends of section I during the initial steady state, and from the initial discharge (refer to Sect. 7.5). We used the relation

- A = n2D5 [p (O , 0) - p ( l , 0)1 - 8eQ$

- 7 ~ ~ 2 ~ (410 OOO - 399 OOO) = o.03842 - 8 x 1OOO x 6.472 x 270

(14.18)

resulting from equation (7.1 I).

-2.BEtBB I I I I I 13 e,wtM S , B B E ~ I,BBEIBI I.~~EIBI Z.BBE(BI 2.58~1~1

Fig. 14.32 Graph showing the calculated discharge at the upstream end of section 1.

A wave velocity a = 670 m s-’ was calculated from the measured pressures. The parameters of the “pressure” damping device are introduced in lines

4, 500 to 4, 610. They are arranged in accordance with Sect. 9.4. The values Jdpe = 500 and Jdpe = 600 were chosen within the interval 450 do 2000 to avoid the same numbers appearing in columns 2 to 5. The submitted times and corresponding pressures were obtained from the measurements.

The time interval At = 0.05757 s of calculation was chosen to make the length of the section equal to a whole multiple of the longitudinal interval of calculation, Ax = a At.

312

Calculatwn of the characteristic of a pump from the measured pressures and speed

14.11 Calculation of the characteristics of a pump from the measured pressures and speed

We calculated the pressure and moment characteristics of a pump forming part of the system portrayed in Fig. 14.36a. During a fall-out of the pump and the closing of a non-return flap valve, we measured the pressures in front of and behind the pump, at points 2 and 3 as well as the speed of the pump. The relationship between the discharge and time was defined by the method presented in Sect. 14.10. The pump is installed in the immediate vicinity of the non-return flap valve, hence the discharge through the pump equals that through point 2. These values for one instant t are sufficient to determine one point of the pressure characteristic. A substantial part of the pressure characteristic was calculated from such measurements, namely that part which includes all the states that appeared in the course of the water hammer studied.

[J=9 Jp.1 l J = 9 Jp=2 l J = 9 Jp=3 [J=9 J p 4 l J = 9 Jp.5

Fig. 14.33 Scheme for calculating the characteristics of a pump from measurements of pressure and speed.

In a similar manner the moment characteristics of the pump were determined. The moment of the pump was calculated for every instant from the known moment of inertia of the pump (including the electric motor), and from the rate of change of the speed as determined by measurements.

The calculation scheme applied is shown in Fig. 14.33. Junctions 2 and 2 correspond to points 2 and 3 in Fig. 14.36a. Junctions 3, 4 and 5 are included in the calculation to enable us, with the aid

of the WTHG program, to plot the variations in the pump speed, the pressure in front of, and behind the pump. The measured curves of these quantities were submitted as the parameters of the “pressure” damping device attached to the junctions.

The input data employed in the calculation are listed in Table 14.23 and on the WTHM diskette in the D1l.WTH file. Compared to the data file contained in Table 14.22, the input data were expanded by including junction 3 and the

313


values for the speed curve submitted in lines 4,413 to 4,705, junction 4 and the values for the pressure curve in front of the pump submitted in the lines 4,414 to 4, 802 and junction 5 with the values for the pressure curve behind the pump submitted in lines 4, 415 to 4, 907.

Table 14.23 Input data file for calculating the characteristics of a pump from measurements of pressure and speed

1 1 1 CHARACTERISTICS OF PUMP FROM MEASURED PRESSURES AND SPEED ,03842 670 6.47

410000 399000 6.188 149551

2 1 151 251 3 1 9 1 3 2 9 2 3 3 9 3 3 4 9 4 3 5 9 5 4 411 500 4 500 33 4 501 4 502 4 503 4 504 4 505 4 506 4 507 4 508 4 509 4 510 4 412 600 4 600 33 4 601 4 602 4 603 4 604 4 605 4 606 4 607 4 608 4 609 4 610 4 413 700 4 700 17 4 701 4 702 4 703 4 704 4 705 4 414 800 4 800 7 4 801 4 802 4 415 900 4 900 22 4 901 4 902 4 903 4 904 4 905 4 906 4 907 5 1 5 2

270

4.15 5.62 6.00 6.47 12.1 16.4 18.4 19.7 20.8 21.9 23.8

4.52 6.19 7.31 9.03 10.9 13.8 17.8 19.9 20.9 23.4 25.1

4.37 6.00 7.56 9.00 12.7 23.1

4.5 9.7 50.0

4.37 5.82 7.40 10.2 11.5 16.9 20.0 28.9

.05757

2

4inooo 168000 144000 133000 147000 167000 209000 311000 324000 300000 226000

399000 199000 164000 150000 184000 178000 199000 284000 312000 230000 238000

6.188 4.063 3.000 2.500

1.5 .625

149551 150042 151219

410000 239000 180000 162000 171000 159000 147000 146000

5

4.75 5.70 6.01 6.70 15.1 17.2 19.0 19.9 21.0 22.7 24.3

5.0 6.32 7.62 9.40 11.7 14.9 18.2 20.2 21.4 24.1 25.8

5.0 6.56 8.00 10.0 15.0 30.0

7.1 10.7

4.92 5.91 8.32 10.6 12.5 18.3 20.0

300

295000 161000 136000 128000 150000 168000 247000 316000 324000 252000 229000

335000 193000 161000 157000 197000 178000 206000 3 14000 316000 228000 250000

5.125 3.563 2.875 2.188 1.25 .3125

150532 150042

309000 230000 168000 166000 171000 156000 151000

61 1 2 OQ Discharge through the pump (013s-1) 6 2 3 3 n Speed of the pump (s-1) 63 4 3 u Pressure at the upstream side of pump (Pa) 64 5 3 d Pressure at the downstream side of pump (Pa) 66 1 7 Disconnection of the pump

5.27 5.75 6.21

10.72 15.9 18.0 19.4 20.3 21.3 23.0 24.6

5.61 6.95 8.43 9.85 12.7 17.1 19.3 20.6 22.9 24.4 26.7

5.55 7.00 8.52 11.0 20.0

8.1 13.1

5.32 6.55 9.20 11.2 15.2 19.7 22.1

-1 0 0 0 0

410000

240000 150000 131000 128000 156000 189000 282000 310000 310000 231000 239000

263000 171000 149000 171000 184000 199000 243000 319000 243000 235000 271000

4 * 50 3.281 2.625 1.938

0.8125

150532 151022

262000 204000 159000 166000 159000 147000 150000

25 9 10

500000 500000

.25

314

Calculation of the characteristic of a pump fiom the measured pressures und speed

The calculated discharge through the pump agrees with that plotted in Fig. 14.32. The values of the discharge for the calculation of the points of the Characteristics were also taken from this figure after eliminating the small periodic fluctuations.

The measured pressures in front of and behind the pump, and the speed of the pump are shown in Fig. 14.34. The values for calculating the characteristics were taken directly from the original measurements in order to attain the highest possible accuracy.

The calculated characteristics of the pump are presented in Fig. 14.35. Along the smooth curves the actual calculated points are also indicated.

1 ,ate2

4M+6I

Fig. 14.34 Measured pressure curves: u - in front of the pump; d - behind the pump, and the speed curve n (100 YO correspond to the values u = d = 5.0 x lo5 Pa; n = 10 s-I).

% (Pal

3x10'

h

-2xld

-10'

- c

M ( N m ) -7.5 104

I I

Fig. 14.35 Calculated pressure and moment characteristics of a pump for the speed n = 6 s-'.

315


The values of the moment of the pump were determined using equation (6.53), where ME = 0 and I = 3 722 kg m2.

The discharge Q, the difference in pressure App and the moment M were converted from the actual speed n to the speed n, = 6 s-' on the basis of model similarity corresponding to relations (6.45) and (6.46).

14.12 Starting-up of a pump with an electric motor, a butterfly valve and a condenser

We calculated the water hammer induced by starting-up a pump driven by an electric motor in the system presented in Fig. 14.36a. This system was modified, compared to the actual one, in order to emphasize the effect of the condenser.

In the initial steady state, the speed of the pump was zero, the butterfly valve behind the pump was closed and zero discharge existed in the entire system. The hydrostatic pressure in front of the butterfly valve corresponded to the water level in the suction tank and the hydrostatic pressure behind the butterfly valve

d 2 5 . 1 m

I m

.;., p;:02JfE14 p" Jp :1

J=56 J.59 J.51 J.51 J=51 J.58 J-51 J.51 Jp.0 Jp.0 Jp.0

Fig. 14.36 Starting-up of a pump with an electric (a) hydraulic system; (b) calculation scheme.

motor, a butterfly valve and a condenser:

316

Starting-up of a pump with an electric motor

Table 14.24 Input data tile for calculating water hammer induced by the starting-up of a pump with an electric motor in a hydraulic system containing a butterfly valve and a condenser

1 12 START-UP OF PUMP WITH ELECTRIC MOTOR, BUTTERFLY VALVE AND CONDENSER 2 1 156 259 2 2 251 351 2 3 351 458 0 2 4 451 551 3 1 2 3 2 1 3 3 8 0 3 4 8 0 3 5 8 1 4 150 4 160 4 170 12 4 180 4 190 4 200 4 250 6 4 260 4 330 11 4 340 4 350 4 360 4 430 500 4 500 8 4 501 4 502 4 504 8 4 505 4 506 . ...

4 508 5 4 5 0 9 . . ~ .

4 512 2 4 516 8 4 517 4 518 4 520 1 4 521 9 4 522 4 523 4 525 9 4 526 4 527 4 390 3 4 400 4 420 4 391 1 4 401 600 4 600 1 5 4 5 2 61 1 1 1 62 1 12 63 1 13 64 1 2 65 1 1 66 1 1 7 61 I 1 1 62 1 12 63 1 13 64 65 7

61 3 3 62 4 3 63 3 9 64 4 9 65 3 2 7

In 1M 1E OQ OP

2a 2d 2M

3 4

d 1Q

U

18 270 0

602

-2 2.75 5.5

7 -2

6.5 0

3.7 5.81 6.17 1.6 0

.785 1.414

0 .785

1.414

1.05 .143

0 .74

1.24

.73 1.23

.73 1.23

0

15.38

.025

1.6 2

1.43 2

-385 -1.97E5 -2.59E5 -1.9685 -66000 -56000 58160 63110 92800 79200

0 0

.27

.67 0

.27

.67 -3114 -4021

2,4486 .0336 .0293 .0358

.0471 .121 .16

.0471 .121 .16 0

5

0 .0342

0 .0305

. 5 4 6

7.5 2

7.5 1.23 5.06 5.93 6.25 1.425

* 393 .943

1.571 .393 .943

1.571 .35

1.43 * 143 .25 .92

1.43

.38

.91 1.35 .38 .91

1.35 18.1

17.9 3.6

2.5 100

721 870

1 670

3722

-2.24E5 -2.3885 -2.51E5 -1.5385 -60100 -52000 56930 73010 107700

0 1.43 .095 .36 .7

.095 .36 .7

-3899 -3251 6336 .0307 .0303 .0372

.O637 .164 .0112 .0637 .164

.0112 82376

11.5 216728

Speed of the pump (s-1) Moment of the pump (Nm)

Moment of the electric motor (Nm) Discharge through the pump (m3s-1)

Presxure at the pump (Pa)

Pump Position of the butterfly valve (rad)

Rate of change of valve position (rad 6-1) Moment caused by water (Nm)

Butterfly valve

Pressure in junction 3 (Pa) Pressure in junction 4 (Pa)

Wat,er level in junction 3 ( m ) Water level in junction 4 ( m )

Discharge through condenser (1036-1) Condenser

6

2 5

6.5 9

4.5 9

2.49 5.61 6.05

57859 * 55

1.257

.55 1.257

.7

.50 1.11

.56 1.08 1.43 .56

1.08 1.43 7.5

1 3

5.73

28.27

-2 ~. 9e5 -.le5

-2 0

0 - .05

-5000

le5 le5 15 15 -1

0 0 0 0

151022 252031 252031 252031 252031

0. 6.25

-1.9485 -2.5885 -2.3585 0.05E5 -59000 -30000 58160 82300 116300

537 .15 .59

.15

.59

-4214

.0293

.0327

.084

.183 .0044 .084 .183

.0044 1000

10 4

5.73

28.27 40 8

. le5

.9e5 8

500000 .2

1.5 .05

5000

6e5 6e5 20 20 9

Examples of the ciilculution (?I’ w t e r huninirr

corresponded to the elevation of the edge of the overflow installed at the downstream end of the pipe-line. Both chambers of the condenser were equipped with an air inlet valve.

Water hammer was induced by switching on the electric motors of the pump and the butterfly valve which started opening the valve.

The input data employed in the calculation are presented in Table 14.24 and on the WTHM diskette in the D12.WTH file.

e.mta w ~ ~ ~ t a 4,ma ~.BBBBE~BB t,wmta ~ , B B B B E ~ B I Ilonent of electric notor (h) as function of rwolutions ( 5 - 1 )

Fig. 14.37 Moment characteristic of the electric motor of a pump.

In the schematized system (refer to Fig. 14.36b), two integrated damping devices attached to junctions 3 and 4, section 3 and a “condenser” pressure device installed at the downstream end of section 3, were substituted for the condenser. The length of section 3 was chosen 1 = 0 and the wave velocity a = 1 m s - I to minimize the difference in pressure at the points of the attachment of the interconnecting pipes to the chambers of the condenser and the pressure at the damping device “condenser”. A diameter D = 1.43 m of the pipe-line was chosen to ensure that its cross-sectional area is equal to the sum of the cross-sectional areas of the interconnecting pipes. All the pressure losses in the condenser were included in the “condenser” pressure device.

The data lines 4, 170 to 4, 260 contain the pressure and moment characteristics of the pump for n, = 6 s - ’ . The characteristics used were determined in Sect. 14.11 and they are portrayed in Fig. 14.35. The characteristics for the negative speed need not be submitted, since they did not appear in the present case. The moment characteristic of the electric motor is introduced in lines 4,330 to 4, 360 and in Fig. 14.37. The speed n = 6.25 s-l, for which the last point of the characteristics is submitted, agrees with the synchronous speed nE = 6.25 s - ’ of the electric motor submitted in data line 4, 160.

318

Slarting-up of a pump with an electric motor

The parameters of the butterfly valve (refer to Sects. 6.6 and 9.4) are given in lines 4, 430 to 4, 527. Line 4, 440 was not submitted, because the oil pump of the butterfly valve is switched on throughout the calculation. Lines 4, 500 to 4, 506 contain the characteristic of the butterfly valve. It is taken to be the same for both flow directions. The data in lines 4, 508 and 4, 509 define the curve of the moment induced by the weight. The data in line 4, 512 define the moment by which the hydraulic system decelerates the closing of the butterfly valve. In the present case, these data did not need to be introduced, as the valve is not being closed. Lines 4, 516 to 4,518 contain the data defining the rate of opening of the butterfly valve by the oil pump. Lines 4,520 to 4,527 give the data defining the moment induced by the water in the pipe-line acting on the butterfly valve. These data are given only for one value of the parameter r.

Lines 4, 390 and 4, 400 present the data for the integrated damping device attached to junctions 3 and 4 (refer to Sects. 5.9 and 9.4). We chose the coefficients cJ+ = c,- = 0 and the area S, = 1000 m2 to minimize the difference in pressure at the point of interconnection of the cooling pipes to the chambers of the condenser and the difference in pressure in front of and at the back of the “condenser” pressure device. The effect of the inertia of the water in the chambers of the condenser is practically eliminated by choosing a large value for S,. The value j+, = 82 376 Pa is equal to the pressure in junctions 3 and 4 at a water level equal to the level of the bottoms of the chambers of the integrated damping devices. In the chambers, the air pressure is considered equal to the atmospheric pressure. The choice of a large value for the diameter of the opening for the air, D, = 10 m2, ensures an atmospheric pressure in the chambers throughout the calculation, as long as the chambers are not full of water. The value of the parameter y = 1 does not affect the actual calculation, it must, however, be submitted.

t 5)

e ,mm a,mm I,E(BI Z , ~ I 3 . 2 8 ~ ~ 1 4,m~l

Fig. 14.38 Graphs showing the calculated speed n of a pump; the pressurep behind the pump; the moment ME of the electric motor (ME in the figure); and the discharge Q through the pump (100 % correspond to the values n = 7 s-’; p = 5 x lo5 Pa; ME = 2 x lo5 N m; Q = 10 m3 s-I).

319


Line 4,420 contains the parameters of the pressure device “condenser” (refer to Sects. 6.7 and 9.4). The coefficient = 15.38 includes all the pressure losses in the condenser. The numbers JJ, and JJd are introduced in decimal form, although they are integers.

Lines 4,401 to 4,600 present the parameters of the integrated damping device attached to junction 5, arranged as an overflow. Its chamber has a variable cross-section.

Fig. 14.39 CalcL.ded curves for the opening av of the butterfly valve (aVin t..e figure) and of the moment Mp (MQ in the figure) induced by the liquid in the pipe-line (100 % correspond to the value av = 1.5 rad; Mp = 3 x los N m).

Fig. 14.40 Calculated curves for the liquid level u in front of the condenser; the liquid level d behind the condenser; the pressure pu in front of the condenser; the pressure pd behind the condenser; the discharge Q through the condenser (100 % correspond to the values u = d = 20 m; pu = pd = 8 x 10’ Pa; Q = 40 m3 s-I).

320


The time interval At = 0.025 s of calculation was chosen rather short with respect to the maximum value t,,, = 40 s of the calculation time, in order to not only maintain a satisfactory agreement between the submitted and modified lengths of the pipe-line sections, but, above all, to be able to accurately follow the characteristic of the electric motor (refer to Fig. 14.37) in the calculation; this characteristic is very steep in some parts.

The calculation of water hammer was carried out thrice, and different variables were entered in each graphical and numerical output as can be seen from the data in the lines beginning with the numbers 6 and 7.

The calculated speed, the pressure behind the pump, the moment of the electric motor and the discharge through the pump are shown in Fig. 14.38.

The calculated curve for the opening of the butterfly valve and the moment by which the water in the pipe-line acts on the valve, are portrayed in Fig. 14.39.

Fig. 14.40 presents the calculated curves of the levels in the integrated damping devices attached to junctions 3 and 4, of the pressures in junctions 3 and 4 and of the discharge through the “condenser” pressure device.

14.13 Turbine with variable characteristics

In this section, we discuss the calculation of water hammer induced by adjusting the guide blades of a turbine. The adjustment rkgime of the blades is known in advance.

The pipe-line system considered and the calculation scheme employed are presented in Fig. 14.41. The hydraulic system consists of an upper reservoir, from which water flows through a pipe-line of variable cross-section and through a turbine into a lower reservoir. A shaft with valves is located close to the upper reservoir and a pipe, closed at its end, branches off in front of the turbine. The calculation deals with water hammer due to the starting-up of the turbine which had been disconnected from the network.

The input data employed are given in Table 14.25 and on the WTHM diskette in the D13.WTH file.

In the initial steady state, the turbine blades are closed, the discharge is zero and the pressure is equal to the hydrostatic pressure.

In the first section of the pipe-line, the actual concrete rectangular cross- section was replaced in the calculation by a circular pipe-line with the same cross-sectional area, the same pressure losses (friction) and the same wave velocity.

An integrated damping device (J = 8) was substituted to represent the effect of the valve shaft. The damping device “surge tank” (J = 5) was not used, since the calculation for this device does not take into account the effect of the inertia of the liquid in the chamber of the damping device, this effect being manifest in

32 1


r

I *- .

b

Fig. 14.41 Starting-up of a turbine with variable characteristic: (a) hydraulic \! \iLm: (h) c;ilctilation scheme.

322


Table 14.25 Input data file for calculating water hammer induced by the starting-up of a turbine with variable characteristic 1

3 3 3

13

4 5 6

151 251 351 451 451

2 8 1 1 2 1

4 0 10 4 390 1 4 150 1 4 160 500 4 180 12 4 190 4 200 4 210 4 500 770 4 501 1 4 502 1 1 4 503 4 504 4 505 4 508 1 4 514 8 4 515 4 516 4 520 1 4 526 1 4 527 10 4 528 4 529 4 530 4 533 1 4 539 9 4 540 4 541 4 545 1 4 551 1 4 552 11 4 553 4 554 4 555 4 558 1 4 564 9 4 565 4 566 4 570 1 4 576 1 4 577 10 4 578 4 579 4 580 4 583 1 4 589 8 4 590 4 591 4 595 1 4 601 1 4 602 10 4 603 4 604 4 605 4 608 1

251 351 451 556 0 651

67.8 1206.8 203.6 79.8 79.8

1000 127 - 1

2.6 0

5.5 10.4 21.0 .55

1670000 -.l

.44

.84

-.04 .252 .57

2010000 -.I

.42 1.1

- .04 .252 .46

2600000 -.l

.32 ,764

-. 04 . 2

.46

3430000 -.l

.26 .4 96

-. 04 . 2

.4090

5390000 -.l

. 1 A ,433

4.28 3.8 3.6 2.2 2.2

.OOl 127

1 0

,0278 .0278 .0151

1 .8370000 410000 530000 700000 1390000 205000 -1190 -200 2310 1030

-8440000 410000 545000 765000

2540000 272500 -1140

-80 1510 960

9450000 410000 640000 720000 1770000 320000 -1080 -170 2050 900

- 11 660000 410000 585000 720000 1270000 292500 -1040

-40 2050 840

4 10000 605000 720000

1440000 302500

-17250000

TURBINE WITH VARIABLE CHARACTERISTICS .02 1 1246 0 .021 1294 0 .02 1 1294 0 .021 1167 0 * 021 1167 0

4229608 4229608 4229608 4229608 185346

4229608 1000 35.5

211.53 .3

7.0 12.3 28.0

25 14000 -.055

.12

.56 1.1

.32

.72

15770 -.055

.14

.62

.320 .6

18040 - .055

.14

.42 1.05

.32

.52

20890 -.055

* 14 .34

.28 .4960

26640

.08

.26

-.055

.001 4093000 265600

.0011 ,0315 .0229 ,0095

-42900 485000 540000

2220000

-1030

845000

180 4440

-48270 490000 565000

1110000

--960 350 3470

-55210 520000 595000 865000 3000000

-900 610

2890

-63940 525000 625000 870000

-840 560

3380

-81510

630000 540000

88nooo

1000 1000 7.46 7.46

0

4.5 .0257 9.7 .0321

15.5 .0229 38.0 .0078

30 - . 02 515000 .27 580000 .72 1110000

.1 -720 .43 910

26 - . 02 535000 .25 600000

.853 1670000

.1 -640 .380 800 .74 5800

22 - . 0 2 610000 .22 625000 .54 1110000

.1 -560

.4 1340 .64 5060

18 - . 02 575000

. 2 655000 .42 1060000

.1 -480 .34 1150

14 -.02 6onooo .14 655000 .34 1120000

323



4 614 4 615 4 616 4 620 4 626 4 627 4 628 4 629 4 633 4 639 4 640 4 641 4 645 4 651 4 652 4 653 4 654 4 658 4 664 4 665 4 670 4 701 4 714 4 715 4 720 4 726 4 727 4 728 4 729 4 733 4 739 4 740 4 741 4 745 5 5 61 4 62 4 63 4 64 4 65 2 66 67 1 7

7

1 1 7

1 7

1 1 7

1 6

1 1

- 5

- 1 1 7

1 8

1 6 2 1 12 2

1 1 5 1

- .04 -990 .17 -50

.433 3360 790

8860000 -27500000 -.l 410000

650000 .21 950000

325000 -.04 -940

.1 -280 .28 1660

720 21900000 -65700000

-.l 410000 705000

.14 1010000 352500

-.04 -900 .06 -400

660

-. 04 - 800 .06 110

490 13000000 -35000000

-.l 410000 670000

.21 1160000 342000

-. 04 -920 .06 -450 .21 1100

690 .052 10

.24

33600 - .055

.08

.152

48400 - , 0 5 5

.04

.1

128400

.1

39000 - * 055

.08

.1 .28

2000

~ 790 520

-102800 560000 685000

-720 60

-148100 580000 740000

-660 -110

-392900 -490 765

-1 10000 570000 730000

-690 -200 2450

1P Pressure at the turbine (Pa) 2M Moment of the turbine (Nm) 1Q Discharge through the turbine (013s-1) 2n Speed of the turbinc ( s - 1 ) V Volume of water in the t.ank (1x13)

StArt

. I .34

10 - .02 .14

.06 -21

6 - .02 .08

.03

.14

0 .03

8 - .02 .14

.03

.14

3 0 0 0 0 0 0 0 0 0

80

400 1810

630000 755000

-490 610

680000 820000

-550 350

235

640000 830000

520 160

40 55oonno 1000000

2 5 10

130 .2 . 5

not be submitted. The value Fz, = 25 s-' introduced in line 4,500 is used for the conversion of the characteristics. The value no = 0 represents the speed at t = 0. The number Jch = 500 is introduced in columns 2 to 5 of the first line of the characteristics of the turbine model. The diameter D, = 2.6 m and the number iT = 1 for the stages of the turbine are used in the conversion of the model characteristics to those of the actual turbine.

The coefficient c, = 21 1.53 is used to obtain the parameters 6 of the model of the turbine by converting the parameters 01 which define the opening of the guide blades of the actual turbine. The values BE, nE and the characteristics of the generator need not be submitted, because the turbine is permanently disconnected from the network during the calculation.

The curve of the opening a(f) of the turbine guide blades is defined in the data lines beginning with the numbers 4, 180 to 4, 210. They contain the times and


e.BBBBE+m '

the corresponding values of the parameter a. The curve of the parameter a, which defines the gap between the guide blades in this case, is shown in Fig. 14.42. The figure was plotted with the aid of the WTHD program using the D13.WTH data file.

The parameter /I need not be submitted, since the action blades of the turbine are not adjustable. The value of this parameter is taken as permanently equal to zero.

X

The characteristics of the model of the turbine are given in the lines beginning with the numbers 4, 500 to 4, 745. The first line of the submission of the characteristics has the number Jch in columns 2 to 5; this number agrees with Jch

in line 4, 160. Number J, = 770 limits the data lines destined for submitting the characteristic of the model of the turbine. The diameter BT = 0.55 m of the model, its number of stages i;- = 1 and its rated speed ns = 25 S - ' are used for converting the characteristics of the model to those of the actual turbine.

25 data lines are assigned for submitting the characteristics corresponding to one combination of the parameters c i and 8. Thus, for example, the data lines beginning with the numbers 4,501 to 4,525 were used for ii = 30 and B = 0. The first of these lines always has the number 1 in the tenth column; this number indicates that the submission of the characteristics follows. The same line contains the values of the constants B p + ; ZIP-; B,, and BM-, and, of the parameters Cr and f i The parameter /I = 0 need not be submitted. In the subsequent line follow the pressure characteristics for the speeds ii = ii, and ii = -i i , and the moment characteristics for ii = ii, and ii = -i is. The lines beginning with the numbers 4,526 to 4,550 contain the characteristics for the combination of the parameters ii = 26 and B = 0, and the some procedure continues, for other combinations. The numbers in columns 2 to 5 must correspond exactly to

- -

325


Table 9.6. The sequence used for the combinations of the parameters Or, B is irrelevant.

If a completely closed turbine corresponds to a combination of the parameters 5, B the pressure characteristics are not submitted and the values NM+ and NM- are introduced with a negative sign. Other sections of the characteristics are submitted in the same way as in the other cases. The parameters Or = 0 and B = 0 correspond to closed guide blades. The characteristics corresponding to these values are submitted in the lines beginning with the numbers 4, 701 to 4, 720.

Fig. 14.43 System of pressure characteristics of a turbine model for A = 25 s- ' and parameters 6 = 30, 26, 22, 18, 14, 10, 6.

Fig. 14.44 System or moment characteristics of a turbine model for R = 25 s - I and parameters ti = 30, 26, 22, 18, 14, 10, 6, 0.

326

Governor-controlled turbine

The pressure and moment characteristics for fi = +ii, are shown in Fig. 14.43 and 14.44. They were plotted with the aid of the WTHD program from the D13.WTH data file.

Some results of the calculation are presented in Fig. 14.45 drawn with the aid of the WTHG program. It contains the curves for the turbine speed n, the pressure p in front of the turbine, the discharge Q through the turbine, the moment M of the turbine and the volume V of water in the valve shaft.

I \ I I I

H n

M H, speed n, pmsur p, dischaw P nd uolune U

Fig. 14.45 Results of the calculation for the starting-up of a turbine with variable characteristics (100 % correspond to the values n = 10 s-'; p = 7.5 x lo6 Pa; Q = 20 m3 s-'; M = lo6 Nm; V = 250m3).

14.14 Governor-controlled turbine

The calculation of water hammer induced by disconnecting a governor- controlled turbine from the electric network relates to the hydraulic system described in Sect. 14.13 and portrayed in Fig. 14.41a. In this case the calculation scheme shown in Fig. 14.41 b was again used, apart from a difference in the initial discharge in sections 1 to 4, which, in this case is Q, = 30 m s-'.

First, we calculated the initial steady state with the generator of the turbine connected to the electric supply network. We defined the pressures and discharges in the hydraulic system, the speed of the turbine and the values of the decisive parameters of the governor, all of which correspond to this steady state. Then, we calculated the progress of the water hammer induced by disconnecting the electric network. The adjustment of the guide blades was controlled by the governor as a function of the instantaneous speed of the turbine, the discharge through the turbine and the difference in pressure in the pipe-line in front of, and behind the turbine, the values for a11 of which were obtained from the solution of water hammer.

327

Examples of the calculution of water hammer

Table 14.26 Input data file for calculating water hammer induced by the fall-out of a governor- controlled turbine

4 180 4 200 4 220 4 240 4 280 4 320 4 500 4 501 4 502 4 503 4 504 4 505 4 508 4 514 4 515 4 516 4 520 4 536 4 527 4 528 4 529 4 530 4 533 4 539 4 540 4 541 4 545 4 5 5 1 4 552 4 553 4 554 4 5 5 5 4 558 4 564 4 565 4 566 4 570 4 576 4 577 4 578 4 5 7 9 4 5R0 4 583 4 5119 4 590 4 591 4 595 4 601 4 602 4 603

328

14 1 151 251 2 251 351 3 351 451 4 451 556 0 5 451 651

3 1 2 3 2 8 3 3 1 3 4 1 3 5 2 3 6 1 4 0 10 4 390 1 4 150 4 160 500 4 170 1200

L

2 2 1 2 1

770 1

1 1

1 8

1 1

10

1 9

1 1

1 1

1 9

1 1

10

1

0

1 1

10 0

n

1000 127

2.6 412.4

1.6 1.6 1.6

0 1.6

.55 1670000

-.l

.44

.84

- .04 .252 .57

2010000 -.l

.42 1.1

. n4 , 2 5 2 .46

2600000 -.l

.32 .764

-.04 . 2

.46

3430000 -.l

.26 .496

- .04 .2

.4090

5390000 -.l

* 001 127

1 0.04 10000

-25600 -2 188000

.06 10.4

10 1

~ 8370000 410000 530000 700000

205000 -1190

-200 2310 1030

-84 4 0000 4 10000 545000 765000

254 0000 272500 -1140 -80 1510 960

-9450000 410000 640000 720000 1770000 320000 -1080 -170 2050 900

11660000 410000 585000 720000 270000 292500

i39nooo

TURRINE WITH GOVERNOR 67.8 4.28 .021 1246 30

1206.8 3.8 .021 1294 30 203.6 3.6 .021 1294 30 79.8 2.2 .021 1167 30 79.8 2.2 .021 1167 0

4229608 4229608 4229608 4229608 185346

4229608 1000 7.46

8.258 8.333 400000

1000 7.46

8.333

1

-1040 -40

2050 840

250000 410000 605000

1000 35.5

211.53

1.6 1.6 1.6

5

25 14000 ~. 055

.12

.56 1.1

.32

.72

15770 -. 055

.14

.62

,320 .6

18040 -.055

.I4 * 42

1.05

* 32 .52

20890 -.055

.14

.34

.28 .4960

26640 - ,055

I 08

.001 4093000 265600

0 0 0

0

-42900 485000 540000 845000

2220000

-1030 180

4440

-48270 490000 565000

1110000

- 960 350 3470

-55210 520000 595000 865000

3000000

-900 610

2R90

-63940

625000 870000

525000

-840 560

3380

-81510 540000 630000

30 .02 515000 .27 580000 .72 1110000

.1 -720 .43 910

26 - . 02 535000 .25 600000

.853 1670000

.1 - 640 .380 800 .74 5800

22 - . 0 2 610000 .22 625000 .54 1110000

.1 -560

.4 1340 .64 5060

18 - . 02 575000

. 2 655000 .42 1060000

.1 - 480 * 34 1150

14 - . 0 2 600000 .14 655000



4 604 4 605 4 6OR 4 614 4 6 1 5 4 616 4 620 4 626 4 627 4 628 4 629 4 633 4 639

4 641 4 645 4 651 4 652 4 653 4 654 4 658 4 664 4 665 4 670 4 701 4 714 4 715 4 720 4 726 4 727 4 728 4 729 4 733 4 739 4 740 4 741 4 745 41200 41201 41206 41207 41210 41211 41212 41213 41214 41215 41216 41217 41221 41225 41226 41227 41230 41231 11232 41235 41236 41237 41240 41241 41242 41245 41246 41247 4 1250 41251

4 640

0

I 7

n

1 1 7

0 1 7 0

1 1 7

1 6

1 1

- 5

1 1 7

1 8

1 1 2 4

1 5

2 2 7

5 4

4

4

4

4

.19 .433

~. 04 .17

.433

886nooo - . I

.21

-. .04 .1

.28

21 900000 - .1

..14

.04

.06

‘I 2 00 00 1440000

-990 -50

3360 790

-27500000 41Q000 650000 950000 325000

302500

-940 - 280 1660 720

41 0000 705000

1010000 352500

- 6 6 7fJOOOO

- w n -400

660

.04

.06

13000000 - -.1

.21

-.04 .06 .21

.2 7.5 -1

ROO 110 490

35000000 410000 670000 1160000 342000

-920 -450 1100 690

1 .2

1 .2 200 3.4

4onooooo .Ol .65

- .002 .I54

.0003 .0083 - .0002 . no4 1

. no4 .313 .00714 .0179 .00714 .0179

n n .15 .0525 ,357 .15

-.133 ,357 .on545

-.133 .357 .00833

-.133 .357 ,0117

- . 133 .357 .0151

- . 133

-26

.24

33600 - * 055

.08

.152

48400 -.055

.04

.1

128400

.1

39000 - ,055

.08

.1 . 2 8

2 9.167

1

28.01

.OO6 . . on05

. 0005 .006

.00714

.00714 ,0338 .225

.179

* 179

,179

.179

.I79

880000

- 790 5 2 0

-102800 560000 685000

-720 60

148100 580000 74onoo

660 -110

-392900 -490 765

-110000 570000 730000

690 -?OO 2450

2 -1 8

103700

. 4 5 3 ,0265 .0015 .02c5 ,453

.0238

.32n

,0338 .on81 .086

.00909

.0166

.025

.0368

.0486

.34

.1 .34

10 - .02 .14

.06

.21

6 - .02 .08

.03

.14

0 .03

8 .-. 02 .14

.03

.14

1.5

1

-.015 - .004

- .0003 .0002

.Ol . no2

.075 .3

.2R6

.268

.25

.25

.25

1120000

-400 1810

630000 755000

-490 610

680000 820000

-550 350

-235

640000 830000

-520 160

12

8

10 .015 .313

.0083 ,0041 . I54

.65

.023 .1225

55onnno

4800ono

428nooo

~ Y O O O O O

.00909

.0166

.025

.0368

3500000 ,0486

329



41252 41280 8 41281 41282 5 5 6 4 61 4 1 l p 62 4 1 2 2M 6 3 4 2 1Q 64 4 11 2n 65360 17 5y 7 5 5 6 6 66 1 67 1 7

. 3 5 7 . 0? . 0005 -1174000 - . 0 0 0 3 -1056000

- . O O O l -516400 .0001 516400 .O003 105fi000 , 0 0 0 5 1174000

' 0 5 2 10 2000 25 Pressure at thr turbine (Pa)

Moment of the turbine ( N m ) - Discharge t,hrough the turbine (rn3s- 1 )

Speed of the turbine (s-1) Position of blades ( m )

Steady state . 0 5 2 1 0 . 2000

Disconnection of the electric network

- .0002 -845000 .0002 845000

20 3500000 6000000

-1000000 4000000 - 60 40

3 1 3 0 . 4

75 . 5 . 2

The input data used in the calculation, are presented in Table 14.26 and on the WTHM diskette in the D14.WTH file.

The initial values of the pressure and discharge given in the lines beginning with numbers 2 and 3 were estimated, with the exception of the pressures in junctions 1 and 5. The estimated values are employed only as the initial values for the abridged calculation of the steady state.

The data lines beginning with the numbers 4, 150 to 4, 1282 contain the parameters of the turbine and the governor. They were submitted in the manner described in Subsect. 9.4.7 (refer also to Sect. 6.11). The initial speed no = 8.258 s-l was estimated. The actual speed was found from the abridged calculation for the steady state. The characteristics were submitted as in Sect. 14.13. The quantity H = 4 12.4 m is the difference in the water levels between the upper and lower reservoirs. It affects only the function of the correcting element of the governor. The values xoa = 0 and Appo = 0 were estimated. The correct values were obtained by calculating the steady state.

The effect of the electric network is represented by the data in the lines beginning with the numbers 4, 180 to 4,220. The data correspond to a complete disconnection of the network at instant t = 1.6 s. The same result could be obtained by applying the procedure described in Sect. 14.13 using the data contained in line 4, 150. The procedure employed in Sect. 14.14 enables us to consider even a change in the loading of the network, not merely its complete connection or disconnection.

The data lines numbered 4, 240 to 4, 320 contain the data which relate to the required speed, the effect of the power governor and the voltage limiter.

Other parameters of the governor are introduced in the lines beginning with the numbers 4, 1200 to 4, 1282. They contain data which do not change frequently and are usually common to more turbine governors working in one hydraulic system. These data, like the characteristics of the turbines, need to be submitted only once if several identical turbines form part of a system, even if the adjustment regime of the turbine differs.

330


The number Jg = 1200 in data line 4, 1200 has to correspond to the value Jg in data line 4, 170.

The subsequent lines contain the data which describe the parameters of the governor, various numerical and mechanical limitations on the calculated values, the effect of inertia, friction and the forces by which the flowing water acts on the blades, as well as the effect of the geometric arrangement of the mechanisms controlling the turbine blades. The meaning of the introduced data is explained in Sect. 6.1 1 and Subsect. 9.4.7.

The value a, = 25 m s- I introduced in the data line beginning with the numbers 5, 5, 6 ,4 was determined experimentally so as to reduce to a minimum the time needed for calculating the steady state. The values Lsi; LKi and LXi given in lines 6, 1 to 6, 5 were determined in agreement with Sect. 9.6.

Fig. 14.46 Results of the calculation of water hammer induced by disconnecting a governor- controlled turbine from the power network (100 % correspond to the valuesn = 14 s-'; y = 0.4 m; M = 4.0 x lo6 N m; p = 1.2 x 10' Pa; Q = 80 m3 sCI).

The dynamic equilibrium of the initial steady state was disturbed by disconnecting the network. The speed started to rise and the governor reacted by closing the guide blades. This stopped the discharge through the turbine and the moment by which water acted on the turbine changed sign, reducing the speed of the turbine. When the required speed was reached, the governor started opening the guide blades (with some delay) to establish the new steady state with the network disconnected and the desired turbine speed.

Some results of the calculation are presented in Fig. 14.46. It shows a plot of the turbine speed n, the extension y of the slave-cylinder piston rod governing the setting of the guide blades, the moment M with which water acts on the turbine, the pressure p in front of the turbine and the discharge Q through the turbine.

33 1

This Page Intentionally Left Blank

Appendix A

Subprograms for the damping and pressure devices

SUBROUTINE RES(UO,QO,RO,SO,PO,B) _ - _ _ _ - - _ - _ _ _ _ _ _ calculates parameters of reservoir

COMMON /PAR/ IC(2000),RC(2000,6)

INTEGER B LOGICAL UO

IF (90) 1,4,2 1 EE=-2.O+RC( B, 2 )*QO GOTO 3

2 EE: 2.0*RC(B,l)*QO 3 IF(EE.EQ.O) GOTO 4 UO=.FALSE. RO=l./EE SO=QO/2. - RC(B,I)/EE GOTO 5

4 UO=.TRUE. PO=RC(B,4)

5 RETURN END

SUBROUTINE ACHAMBER(II,UO,QO,RO,SO,PO,VO,B) calculates parameters of air chamber _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _

COMMON /GOV/ CHM1,CHN,CHP1.CHPU,CHQl,~HQU,DT,G,IS(50,2),IT,ITISKT, * ITT,ITTOT,ITTZ,ITV,ITZ,M(5O),N(5O),NIl,NS,NU,PI,RO, * UBEZIT,UNT,UU,UUST,UVT,UZKR,X(5O),Xl(5O),Y(lOO) COMMON /PAR/ IC(200O),RC(2000,6)

INTEGER B,X,Xl,Y LOGICAL UO,UBEZIT,UNT,UU,UUST,UVT,UZKR

IF (UZKR) GOTO 8 IF(I1) 1,3,2

Preparation

1 EEl=( PO-RC( B, 4 )tRC( 8.6 ) )/RC( B, 6 ) IF( EE1. LE. 0. )GOTO 9 VO=RC(B,3)-(EEl)+*(-l./RC(B,5))*RC(B,3) GOTO 3

Change in volume of liquid

2 VO=VOtQO*DT

333

Appendix A

Pressure in air chamber

3 IF(RC(B,3).LE.VO) GOTO 9 EE2=RC( B ,3 ) / ( RC( B , 3 ) -VO ) IF(EE2.LE.O.) GOTO 9 EEl=RC(B,6)*EE2**RC(B,5)-RC(B,6)tRC(B,4)

Parameters of linear equations or pressure in junction

IF(Q0) 5,7,4 4 EE=2. *RC( B, 1 )*QO GOTO 6

5 EE=-2.*RC(B,Z)*QO 6 IF(EE.EQ.0.) GOTO 7

UO= . FALSE. RO= 1. /EE S0=40/2. - EEl/EE GOTO 10

7 UO=.TRUE. PO=EEl GOT0 10

Steady state

8 IF(II.GE.0) GOTO 10 UO=.FALSE. RO=O. so=o. GOTO 10

Error

9 WRITE(*,’(”OWTH Err.7 - Error in air chamber calculation ”, * ” ( data line 4”,14,” ) ” ) ’ ) B STOP

LO RETURN END

SUBROUTINE STANK(II,UO,UPREP,QO,RO,SO,PO,VO,B) calculates parameters of surge tank and overflow _ _ - _ _ - _ _ _ _ _ _ _ _ _ _

COMMON /GOV/ CHM1,CHN,CHP1,CHPU,CHQl,CHQU,DT,G,IS(50,2),IT,ITISKT, * ITT,ITTOT,ITTZ,ITV,ITZ,M(5O)~N(5O),NIl~NS,NU,PI,RO, * UBEZIT,UNT,UU,UUST,UVT,UZKR,X(5O),Xl(5O),Y(lOO) COMMON /PAR/ IC(200O),RC(2000,6)

INTEGER B,X,Xl,Y LOGICAL UO,UBEZIT,UNT,UPREP,UPROM,UU,UUST,UVT,UZKR

IF (UZKR) GOTO 14 UPROM=.NOT.UPREP.AND.(RC(B,6).GT.O.O) IF(I1) 1,3,2

Preparation

1 VO=RC( B,3)*( PO-RC( B,4 ) )/RO/G IF (UPROM.AND.(VO.GT.O.Ol) VO=VO*RC(B,6)/RC(B,3) GOTO 3

Change in volume of liquid

2 VO=VOtQO+DT

3 34

Appendix A

C PPeSBUre in surge tank

3 IF(UPREP) VO=AMINl(VO,O.O) EE=RC( B, 3 ) IF( UPROM. AND. ( VO . GT. 0.0 ) ) EE=RC( B, 6 ) EEl=RC(B,4)tRO*G*VO/EE

C Parameters of linear equations or pressure in junction

4 IF(Q0) 11,13,10 10 EE=2*RC(B,l)+QO

GOTO 12 11 EE=-2.+RC(B,2)*QO 12 IF(EE.EQ.0) GOTO 13

UO=.FALSE. RO=l./EE SO=QO/L.-EEl/EE GOTO 20

13 UO=.TRUE. PO=EEl GOTO 20

C Steady state

14 IF(UPREP) GO.TO 16

15 UO=.FALSE. IF(II.GE.0) GOTO 20

RO=O. so=o * GOTO 20

EE1 =RC( B, 4 ) GOTO 4

20 RETURN END

SUBROUTINE AVALVE(II,UO,QO,RO,SO,PO,VO,Vl,V2,V3,B)

16 IF(QO.LT.(-CHQU).OR.PO.LT.RC(B,4)) GO TO 15

calculates parameters of air valve - _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ C

COMMON /GOV/ CIWl,CHN,CHP1,CHPU,CHQl,CHQU,DT,G,IS[5O,2),IT,ITISKT, * ITT,ITTOT,ITTZ,ITV,ITZ,M(5O),N(5O),NIl,NS,NU,PI,RO, it UBEZIT,UNT,UU,UUST,UVT,UZKR,X(5O),Xl(5O),Y(lOO) COMMON /PAR/ IC( 2000 ) , RC( 2000,6 )

INTEGER B,X,Xl,Y LOGICAL UO,UBEZIT,UNT.UU,UUST,UVT,UZKR CHARACTER CH

PA= 101 325. ROA=1.22 ICH=O IF(UZKR.OR.II.LT.0) GOTO 4

C Change in volume of liquid

1F( II.GT.0) V3=VO IF(II.GT.0) VO=VO-QO+DT IF(U0) GOTO-1 IF(PO.GE.RC(B,Q)) GOTO 5 GOTO 6

IF(VO.LE.O..AND.QO.GT.O.) GOTO 3 IF(VO.LE.O..AND.QO.LE.O.) GOTO 6

1 IF(VO.LE.O..AND.Vl.GT.CHQl*DT.AND.RC(B,6).EQ.O.) GOTO 11

335

Appendix A

EE =ROA*VO IF(V1.LE.EE) GOTO 6

C Leakage of air

IF(II.EQ.0) GOTO 2 EEl=RC(B,L) IF(EEl.EQ.0.) EE1=1. EE2 = PA* ( V1 /EE )*ME 1 EE2=2.*Vl+EE2/VO+(l-(PA/EE2)**2) IF(EE2.LT.O.) GOTO 11 EE2=-3.1687*RC(B.6)**2*SQRT(EE2) V1=EE+(V1-EE)**2/(V1-EE-DT*EE2) VB=PA*(Vl/EE)+*EEl IF(V1.LE.EE) GOTO 6

2 UO=.TRUE. PO=VZ-PA+RC(B,I) GOTO 10

EE=ROA*V3 EEl=RC(B,6) IF(EEl.EQ.0.) EE1=1. EE2 =PA* ( V1 /EE ) **EE1 EE2=2.*Vl*BE2/V3*(1-(PA/EE2)**2) IF(EE2.LT.O.) GOTO 11 EE2=3.1687*RC(B,6)**2*SQRT(EE2) IF(2.*DT+EE2.LT.Vl.AND.RC(B,6).GT.O.) ICH=1 GOTO 5

3 IF(V1.LE.O..OR.~0.LE.CHQl.OR.II.LT.l.OR.V3.LE.O.) GOTO 5

C Air valve is full of liquid

4 v0:o.o 5 Vl=O.O VZ=PO+PA-RC(B,Q) UO=.FALSE. RO=O. 0 so=o. 0 GOTO 10

C Level falls

6 Vl=VO*ROA V2-PA UO= .TRUE. PO=RC( B.4) GOTO 10

C Error

10 IF(ICH.EQ.0) GOTO 12 11 WRITE(*,’(”OWTH Err.8 - Error in air valve calculation ” ,

* “ ( data line 4”,14,” ) ” ) ’ ) B IF( ICH.EQ.0) STOP WRITE(*,’(” Hit RETURN to continue”)’) READ(*,’(A)’) CH

12 RETURN END

SUBROUTINE CADEVICE (II,UO,QO,RO,SO,PO,VO,Vl,V2,V3,V4,V5,V6,B) C _ _ _ _ _ _ _ _ _ - _ - _ _ _ _ _ _ - calculates parameters of integrated damping device

COMMON /GOV/ CHM1,CHN,CHP1,CHPU,CHQl,CHQU,DT,G,IS(50,2),IT,ITISKT, * ITT,ITTOT,ITTZ,ITV,ITZ,M(5O),N(5O),NIl,NS,NU,PI,RO, it UBEZIT,UNT,UU,UUST,UVT,UZKR,X(5O),Xl(50),Y(lOO)

336

Appendix A

COMMON /PAR/ IC(2000),RC(2000,6)

INTEGER B,X,Xl,Y LOGICAL UO,UBEZIT,UNT,UU,UUST,UVT,UZKR CHARACTER CH

ICH=O UO=.TRUE. IF ( . NOT. UZKR') GOTO 6 Steady state

vo=o. v1 =o. IF(IC(B).EQ.l) GOTO 2 RO=O. so=o. UO=.FALSE. GOTO 80 EE=RO*G+RC(B,3)tRC(B,4)

EEI= RC( B.l )tRC( Bt10.1) I=IABS( IC( Bt10) ) IF(I.EQ.0) GOTO 4 Il=IABS( IC( I ) ) DO 3 J-1,Il 12=JtI-l EE1= EEltRC(I2,l) IF(EEl.EQ.O..OR.QO.LE.O.) GOTO 5 UO=.FALSE. RO=.5/EEl/QO SO=Q0/2.-EE*RO GOTO 80 PO=EE GOTO 80

Total volume of chamber

ROA=1.22 PA= 101325. EE: RC(B,S) VC= EE*RC(B,3) I= IC(Bt10) IF(I.EQ.0) GOTO 8

DO 7 J=I,Il I2= JtI-1 VC= VC+(RC(B ,3)- RC(12,3))* (RC(I2,5) -EE) EE= RC( I2,5) VG=VC

Initial state of air chamber

IF(II.GE.0) GOT034 v1- 0. V4=0. IF(IC(B).NE.2) GOTO 20 IF(RC(B,S) .LE.O..OR. RC(B410,3) .LE.O..OR.VC.LE.O..OR.

IF(ICH.EQ.-1) GOTO 80

IF(QO.LT.(-CHQU).OR.PO.LT.EE) GOTO 1

11. IC(1)

*(VC-RC(BtlO,3)).LT.O.) ICH= -1

Level of liquid in the basic state

EE= (VC-RC( Bt10,3) ) /RC( B,5) I= IC(Bf10)

337

Appendix A

IF( I. EQ. 0

EEl= RC(B DO 9 J = 1

11. IC(1)

12. ItJ-1

GOTO 10

5) I1

JF(EE.LE.RC(I2,B)) GOTO 10 IF(RC(I2,5).LE.O.) ICH= - 1 IF(ICH.EQ.-1) GOTO 80 EE= (EE-RC(I2,3)) *EEl/RC(12,5)t RC(I2,3)

9 EEl- RC(I2,5) 10 V3:EE

C Volume of liquid

IF(EE.LT.O..OR.EE.GE.RC(B,3).0R.RC(BtlO,5).EQ.O IF(ICH.EQ.-1) GOTO 80 EE1= RC(B,4)-RO*G*EE+ RC(B+l0,4)*(RC(B+lO,3)/VC

IF(PO.GT. (RC(B,4)-RC(BtlO,4) -RO*G*EE)) GOTO 1 JCH: -1 vo= 0. GOTO 80

*-RC( Bt 10,4 )

) ICH= -1

**RC( Bt 10,5 )

11 JF(PO.GT.EE1) GOTO 12 VO= VC-RC(B+10,3)~((PO-RC(B,4)tRO*G*EE+RC(BtlO,4))/RC(BtlO,4))

GOTO 8 0 ***( -1. /RC( B+10,5 ) )

12 Jl.1 EE4= RC(B,3)/2. EE1= EE4

13 EE2- EEl*RC(LI,5) I = IC( BtlO) IF(I.EQ.0) GOTO 15 Il=IC( I) EE3=RC ( B ,5 ) DO 14 J=l,Il I2 = ,I t I ~ 1 IF(RC(I2,3).GE.EEl) GOTO 15 EE2= EE2+( EEl -RC( I2,3) ) * ( HC( I2,5)-EE3)

1 4 EE3=RC(I2,5) 15 EE3=RC(R,4)tRO*G*(EEl-EE) tRC(Bt10,4)*(RC(Btl0,3)/

+(VC-EE2))**RC(BtlO,5)-RC(Btl0,4) IF(ABS(EE3~PO) .LE.CHPl) GOTO 16 EE4= EE4/2. EEl=EEltEE4 IF(EE3.GT.PO) EEl=EElE2.*EE4 J1= J1+ 1 IP(Jl.GE.100) GOTO 17 GOTO 13

16 VO=EE2 V4=EE1 GOTO 80

GOTO 80 17 ICHz-2

C Initial state of overflow and air v a l v e C

20 V3.0.

IF(PO.GE.RC(B,4)) GOTO 22

C Empty chamber

TCH=1 vo.0.

338

Appendix A

IF(IC(B).LE.l) GOTO 80 Vl=VC*ROA GOTO 80

C Chamber partly full of liquid

22 IF(PO.GT.(RC(B,4)+RO*G*RC(B,3))) GOTO 27 EEl=(PO-RC(B,4))/RO/G V4=EE1 EE3 =RC ( B ,5 ) EE2=EEl*EE3 I=IC( B+ 10 ) IF( I.EQ.0) GOTO 24

DO 23 J=l,Il 121JtI - 1 IF(EEl.LE.RC(I2,3)) GOTO 24 EE2=EE2+( EE1- RC( I2,3) )*(RC( 12,5)-EE3)

Il=IC( I)

23 EE3=RC( 12,s) 24 VO=EE2

IF( IC(B).EQ.3)Vl=(VC-VO)*ROA GOTO 80

C Air valve full of liquid

27 VO=VC V4 =RC( B ,3 ) IF(IC(B).LE.l) GOTO 28 RO=O. so=o. UO=.FALSE. QO=O. 0 GOTO 80

C Overflow full of liquid

28

29 30

EEl=RC(B,l)+RC(B+lO,l) I=IC(BtlO) IF( I.EQ.0) GOTO 30

DO 29 J=l,Il

EEl=EEl+RC(I2,1) IF( EE1 .EQ.O. ) ICH-2

Il=IC(I)

12.1 tJ-1

GOTO 80

C New parameters of integrated damping device C C Change in volume of liquid

_ _

34 IF(II.EQ.0) GOTO 35

35 IF(VO.LT.VC)GOTO 50 VO=VOtQO*DT

V4=RC(B,3) I=IC(B) IF(I.EQ.0) 1.1 GOTO (36,42,44).1

c Overflow full of liquid

36 EEl=RC(B,l) tRC(Bt10,l) EE2=RC( B ,2 ) +RC( B+ 10,2 ) EE3.0. EE4=RC(B,5) EEB=RC( B,6 )

339

Appendix A

EE=O. I=IC(Bt10) IF(I.EQ.0) GOTO 38 Il=IC( I) DO 37 J=l,Il 12=ItJ-1 EEl=EEltRC(I2,1) EE2=EE2+RC(I2,2) IF ( EE5 . EQ . 0 . ) GOT040 EE=EE+EE4*(RC(I2,3)-EE3)/EE5**2 EE3=RC(I2,3) EE4=RC(I2,5)

37 EE5=RC(I2,6) 38 IF(EES.EQ.O.)GOT040

39 IF(QO.LT.O.)EEl=EEB EE=EE+EE4*(RC( B,3)-EE3)/EE5**2

IF(2.*EEl*QO+RO*EE/DT.NE.O.)GOTO 41 vo-vc PO=RC( B ,4 ) tRO*G*RC( B ,3 ) GOTO 80

GOTO 80

IF(QO.LT.O.)EEl=-EEl RO=1./(2.*EEl*QO+RO+EE/DT) SO=Q0-RO*(RC(B,4)+RO*G*RC(B,3)tEEl*QO**2t(QO-V2)*RO

40 ICHZ-1

41 VO=VC

* *EE/DT) UO=.FALSE. GOTO 80

C Air chamber full of liquid

42 IF(QO.GT.CHQ1) ICH=3 RO=O. so.0.

43

C

44

45

46

UO=.FALSE. IF(QO.GT.CHQ1) Vl=VO-QO*DT IF( V1. GE , VC ) GOTO 43 EE=RC(B,4)+RO*G*(RC(B,3)+RC(B+10.4) *(RC(BtlO,3)/(VC-V1)) ***RC( B+ 10,5 ) -RC( B+ 10,4 ) IF(PO.LT.EE.AND.QO.LE.CHQ1) VO=V1 GOTO 80 ICH=-3 GOTO 80

Air valve full of liquid

IF( RC( Bt ln,6 ) . GT . 0 . ) GOTO 46 IF(QO.GT.CHQ1) ICH = 3 RO-0. so=o. UO=.FALSE. IF(QO.GT.CHQl.AND.II.EQ.1) VS=VO-DT*QO IF(V5.GE.VC) GOTO 45 EE=PA*(Vl/ROA/(VC-V5))**RC(BtlO,5)-PAiRO*G*RC(B,3)tRC(B,4) IF(PO.LT.EE.AND.QO.LE.CHQ1) VO=V5 IF(PO.LT.EE.AND.QO.LE.CHQ1) GOTO 49 GOTO 80 ICH=-3 GOTO 80 IF(QO.LE.CHQl.OR.II.LE.0) GOTO 48 V5 =VO -DT*QO IF((VC-V5).LE.O.) GOTO 48 EE=PA+(Vl/ROA/(VC-V5))**RC(Bt10,5) EE1=2*Vl*EE/(VC-V5)*(l.-(PA/EE)**2)

340

Appendix A

IF(EE1.I.T.O.) GOTO 48 EE2=2.*DT*PI*RC(BtlO,6)**2 *G*O IF(EB2.GE.Vl) GOTO 48

47 ICH-4 48 V1-0.

EE=RC(B,4)+RO*G*RC(B,3) IF(PO.LT.EEtCHPl.AND.QO.LT.CHQ1 RO=O. so.0. UO=.FALSE. GOTO 80

UO= .TRUE. GOTO 80

49 PO=EE

10285+SQRT(BE1)

GOTO 49

C Overflow,air chamber and air valve partly f u l l of liquid C Level of liquid

50 IF(VO.LE.0.) GOTO 65 EE3.0. EE2=0. EE1= RC ( B ,5 ) EE=VO/RC( B, 5 ) I =IC( Bt 10 ) IF(I.EQ.0) GOTO 54

DO 51 J=l,Il I2=I tJ-1 IF(EE.LE.RC(I2,3)) GOTO 54 EE2=EE2+EEl*(RC(I2,3)-EE3) EE=RC( I2,3)t( VO-EEZ)/RC( I2,5 ) EE1 =RC( I2,5 )

51 EE3=RC( I2,3)

Il=IC( I)

C Surplus pressure of air

54 I=IC(B) IF(I.EQ.O) 1=1 V4=EE GOTO( 56,57,68 ) , I

GOTO 58 56 EEl=O.

57 EE1~RC(BtlO,4)*(RC(BtlO,3)/(VC-VO))**RC(Bt10,5)-RC(Bt10,4)

C Pressure losses due to friction and effect of inertia

58 EE2=RC( B, 1 )tRC( Bt10.1) EE3=RC( B ,2 ) +RC( 8t 10,2 ) EE4 =RC ( B , 5 ) EE5 =RC ( B ,6 ) EEG=EE4*EE/EE5**2 EE7.0. EE8=0.0 I=IC( Bt10) IF(I.EQ.O)GOTO 60

DO 59 J.1.11 12zJtI - 1 IF(EE.LT.RC( I2,3))GOTO G O EE2=EE2tRC(I2,1) EE3=EE3tRC(I2,2)

Il=IC( I)

EE7=EE7+EE4*(RC(I2,3)-EE8)/EE5 E E 6 = E E 7 t R C ( I 2 , 5 ) * ( E E R c ( 1 2 , 3 ) ) EE8=RC( I2,3) EE4=RC( I2,5)

**2 /RC( I2,6 )**2

34 1

Appendix A

59 EE5=RCf12.61 I - - . - . . . _ _ ~ ...

60 IF(QO.LT.O.)EE2=-EE3 IF(2.*EE2*QOtRO/DT*EE6.NE.O.)GOTO 61 PO=RC(B,4)+EEl+RO*G*(EE-V3) GOTO 80

SO=QO-RO*(RC(B,4)+EEl+RO*G*(EE-V3)+EE2*QO**Z t(QO-V2)*RO/DT 61 RO=l./(2.*EE2*QOtRO/DT*EE6)

**EE6 ) UO=.FALSE. GOTO 80

C Overflow, air chamber and air valve empty C Surplus pressure of a i r

65 I=IC(B) IF(I.EQ.0) 1.1 V4.0. GOTO( 66,67,68 ) , I

GOTO 74

GOTO 74

EE1.O. Vl=ROA*( VC-VO ) GOTO 74

IF(II.EQ.O)GOTO 74 V1= ROA*(VC-VO)t (Vl-ROA*(VC-V0))**2/(Vl-ROA*(VC-VO)tDT*PI*

66 EEl=O.

67 EE1~RC(B+lO,4)*(RC(B+lO,3)/(VC-VO))**RC(BilO,5) -RC(B+10,4)

68 IF(Vl.GT.ROA*(VC-V0))GOTO 69

69 EEl=PA*(Vl /ROA/(VC-VO))**RC(B+lO,5)-PA

*RC(B+10,6)**2 *G*O.l0285*SQRT(2*V1*(EEl+PA)/ (VC-VO)* *(l.-(PA/(EEltPA))**2))) EEl=PA*(Vl/ROA/ (VC-VO))**RC(B+lO,5)PA

74 IF(VO.GT.O.)GOTO 58 75 EE=RC(B,l)

IF(QO.LT.O.)EE=-RC(B.2) IF(EE.NE.O..AND.QO.NE.O.)GOTO 76 PO=RC(B,4)tEEl-RO*G*V3 GOTO 80

SO=QO-RO*(RC(B,4)iEEltEE*QO**2-RO*ti*V3) UO=.FALSE.

IF(ICH.EQ.0) GOTO 8 1

76 RO=l./B./EE/QO

80 IF(II.NE.0) V2-90

C Error

WRITE ( * , ' ( ''OWTI1 Err.9 - Error in integrated damping device"/ * " (Type of error",I4,", first line of data 4",14,")")') * ICH,B IF( ICH.LT.0) STOP WRITE(*,'(" Hit RETURN to continue")') READ( *, ' ( A ) ' ) CH

81 RETURN END

SUBROUTINE PQTIME (UO,RO,SO,PO,B) C _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ calculates parameters of pressure/time and C discharge/time curves

COMMON /GOV/ CHM1,CHN,CHP1,CHPU,C~IQl,CHQU,DT,G,IS(50,2),IT,ITISI~T, * ITT,ITTOT,ITTZ,ITV,ITZ,M(5O),N(5O),NIl,NS,NU,PI,RO, * UBEZIT,UNT,UU,UUST,UVT,UZKR,X(5O),Xl(50),Y(lOO) COMMON /PAR/ IC(200O),RC(2000,6)

342

Appendix A

INTEGER B,X,Xl,Y LOGICAL UO,UBEZIT,UNT,UU,UUST,UVT,UZKR

EE=ITV*DT IF( UZKR) EE=O. EE1 =EE

C Periodical function

IF(EEl.LT.RC(B,2).oR.RC(B,l).GE.RC(B,2)) GOTO 1 EEl=EEl-(l.tAINT((EEl-RC(B,2))/(RC(B,2)-RC(B,l)))) * *( RC( B, 2)-RC( B, 1 ) )

C Value of function without damping

1 J=IC(B) EE3=RC( J, 1 ) IF( EE1. GT. EE3 ) GOT02 EE2=RC( J,2) GOTO 10

I = Jt J1/3 11=2+MOD(J1,3)+1 IF(EE3.GT.RC(I,Il)) GOTO 12 EE3=RC( 1,Il) IF(EEl.LT.RC(1,Il)) GOTO 3 EE2=RC( I, Iltl ) GOTO 10

I3=11

I=J+J1/3 I1=2*MOD(J1,3)t1 IF(EE3.LT.RC(I,Il)) GOTO 12 EE3=RC( I, I1 ) IF(KC(I,Il).GT.EEl) GOTO 3 EE2~RC(I,I1t1)t(RC(I2,I3tl)-RC(I,Iltl))/(RC(I2,I3)-RC(I,Il))

2 Jl=IC(J)-l

3 12.1

51.51-1

* *( EEl-RC( I, 11) )

C Effect of damping

10 IF(EE.GT.KC(B,3).AND.RC(B,4).NE.O.) E E ~ Z R C ( B , ~ ) + ( E E ~ - R C ( B , ~ ) ) * * EXP(-RC(B,4)*(EE-RC(B,3)))

C Coefficients of linear equations

IF( RC( B, 5 ) . LT. 0.5 ) GOTO 11 SO=EE2 RO=O. UO=.FALSE. GOTO 13

UO- .TRUE. GOTO 13

11 PO=EEZ

<- Error

1 2 WRITE(*,’(”OWTH Err.14 Error in data of function”, * ” ( beginning 4”,14,” ) ’ I ) ’ ) IC(B) STOP

1 3 RETURN END

343

Appendix A

SUBROUTINE LOSSES(U,Q,R,S,D,B) C - _ _ - - _ - _ - _ _ _ _ - _ _ _ calculates parameters of local losses

COMMON /GOV/ CHM1,CHN,CHP1,CHPU,CHQl,CHQU,DT,G,IS(50,2),IT,ITISKT, * ITT,ITTOT,ITTZ,ITV,ITZ,M(5O),N(5O),NIl~NS,NU,PI,RO, * UBEZIT,UNT,UU,UUST,UVT,UZKR,X(50),X1(5O),Y(lOO) COMMON /PAR/ IC(2000),RC(2000,6)

INTEGER B,X,Xl,Y LOGICAL U,UBEZIT,UNT,UU,UUST,UVT,UZKR

C Coefficient of linear equations

U= . FALSE. EE=8.*RO/PI/PI/D**4*Q IF(Q.LT.0) GOTO 1 R=E.*RC(B,l)*EE S=-RC(B,l)*Q*EE GOTO 2

S=RC( B ,2 ) *Q*EE 1 R=-E.*RC(B,B)*EE

2 RETURN END

SUBROUTINE VALVE(U,Q,R,S,D,VN,B) C _ _ _ _ _ _ _ - - - - - - - - - calculates parameters of valve

COMMON /GOV/ CHM1,CHN,CHP1,CHPU,CHQl,CHQU,DT,G,IS(50,2),IT,ITISKT, * ITT,ITT6T,ITTZ,ITV,ITZ,M(50),N(50),NIl,NS,NU,PI,RO, * UBEZIT, UNT, UU, UUST, UVT,UZKR,X( 50 ) ,X1( 50 ) ,Y ( 100 ) COMMON /PAR/ IC(2000),RC(2000,6)

INTEGER B,Bl,X,Xl,Y LOGICAL U,UBEZIT,UNT,UU,UUST,UVT,UZKR

ET=ITV*DT Bl=IC( 9 ) IF (B1.LE.O) GOTO 1

C Parameters of linear equations

E2=FCE(ET,O.O,O.O,O.O,El,B) IF(IC(BtlO).GT.O) E2=FCE(E2,O.O,O.O,O.O,El,-IC(B+~O)) IF(E2.LE.0.00001) GOTO 1 U= . FALSE. R: 1 6. *RO*ABS ( Q ) / ( D*D*PI ) **2* ( 1 . / ( E2*E2 ) - 1 . ) S=-R*Q/2. VN=E2 GOTO 10

1 U=.TRUE. VN=O. 0

10 RETURN END

SUAROUTINE NRFV (U,Q,R,S,P,PO,J) c ~ ~ _ _ _ _ _ _ - - - - - - - _ _ _ calculates parameters of non-return flap valve

LOGICAL U

IJ=.FALSE. R.0. S=Q.

344

Appendix A

JF(Q.LT.O..OR.Q.~Q.O..AND.(J.EQ.l.AND.PO.LT.P.OR. * J.EQ.2.AND.P.LT.PO)) U=.TRUE.

RETURN END

SUBROUTINE PUMP(II,U,Q,R,S,VN,VNl,VN2,VN3,VN4,VN5,VNU,VNUl, * VNU2,VNU3,P,PO,J,B,AM,BM) C calculates parameters of pump and turbine

COMMON /GOV/ CHM1,CHN,CHP1,CHPU,CHQl,CHQU,DT,G,IS(50,2),IT,ITISKT, * ITT,ITTOT,ITTZ,ITV,ITZ,M(5O),N(5O),NIl,NS,NU,PI,RO, * UBEZIT,UNT,UU,UUST,UVT,UZKR,X(5O),Xl(50),Y(lOO) COMMON /PAR/ IC(2000),RC(2000,6)

INTEGER B,X,Xl,Y LOGICAL U.UBEZIT,UNT,UPOM,URVT,USETRO,UTURB,UU,UUST,UVNK,UVT,

* UZAP,UZKR

C Electric motor or generator switch on and of f

UZAP=.TRUE.

I=IC(B)

IF(ET.GT.RC(B,I)) UZAP=.NOT.IIZAP I=I-1 GOTO 10

ET~ITV*DT-0.000001

10 IF(I.LE.0) GOTO 11

C Position of blades

11 UTURB= IC( Bt 10 ) . GT . 0 URVT=IC(Bt20).GT.O.AND.UTURB UVNK=IC(Bt190).EQ.O.AND.UZAP.AND..NOT.URVT POMO=RC(B,4) IF(UZAP.AND.URVT) POMO=PONOtFCE(ET,O.,O.,O.,DER,Bt30) USETRO=POMO.LE.O. IF(.NOT.UTURB) GOTO 17 Il=IC( Bt10) IF(RC(BtlO,l).EQ.O..OR.RC(Bt10,2).EQ.O..OR.RC(Il,l).EQ.O.

* .OR.RC(Il,2).EQ.O..OR.RC(Il,3).EQ.O.) GOTO 70 POMD=ABS(RC(BilO,l)/RC( I1,l)) POMI=ABS( RC( Bt 10,2 )/RC( I1,2 ) ) IF(URVT) GOTO 15 AM=FCE(ET,O.,O.,O.,DER,Bt30) BM=FCE(ET,O.,O.,O.,DER,B+$O) GOTO 16

C Turbine governor

15 IF(II.LT.0.) VN.RC(B.6) IF(II.LT.O.AND.UUST) VN=VNU3 EZ=FCE(ET,O.,O.,O.,DER,Bt90) EW=FCE(ET,O.,O.,O.,DER,Bt130) EM=FCE(ET,O.,O.,O.,DER,Btl70) DPyPO-P IF( J . EQ .2 )DP= -DP IF(II.EQ.-1) DP=RC(B+20,6) IF(JI.EQ.-1.AND.UUST) DPzVNU2 VN3=DP CALL GOVERN (II,VN,VN4,VN5,VNU,VNUl,EZ,EW,EM,RC(Bt20,1),

* RC(B+20,2),RC(B+20,3),RC(Bt20,4),DP,AM,BM,RC(Bt220,1), * RC(B+220,2),RC(B+220,3),RC(Bt220,4),RC(Bt220,5),RC(Bt220,6), * RC(B+230,1),RC(B+230,2),IC(B+2O),B) IF(II.EQ.0) GOTO 17

345

Appendix A

16 IF(RC(B+lO,3).GT.O.) AM=AM*RC(Btl0,3) IF(RC(BtlO,Q).GT.O.) BM=BM*RC(Bt10,4)

C Initial speed

17 IF(II.LT.0) VN=RC(B,G) lF(II.LT.O.AND.UUST) VN=VNU3 IF(USETR0) VN.O.0 IF(UVNK) VN=RC(B+lO,G)

20 IF (.NOT.USETRO) GOTO 21

C Speed determined from equality of momentum of pump C or turbine and electric motor

NITER=O OPR=RC( B,5) IF(OPR.EQ.0.) OPR=l. UPOM=.TRUE.

C Momentum of pump or turbine

21 IF(ABS(VN).GT.CHN) GOTO 23

C Speed equals zero

IF(UTURB) GOTO 22 POM=RC( B+ 10,3 ) IF(Q.LT.O.0) POM=RC(Btl0,4) VNl=POM*Q*Q GOTO 30

IF(Q.LT.0.) 1.4 VN1=Q*Q/POMD+POMI*FCEl(O.,A~,BM,DER,I,l,Il,U) GOTO 30

22 1=3

C Speed does not equal zero

23 IF(UTURB) GOTO 27 IF(RC(B,S).EQ.O.) GOTO 70 POMl=(VN/RC(B,5))**2 POM2=Q/ABS( VN )*RC( B, 5 ) NC=B+100 IF(IC(NCt40).NE.O .AND.(VN.LT.O.O.OR.IC(NC).EQ.O)) NCzNCi40 VNl=POM1itFCE(POM2,RC(B~lO,4),RC(B~lO,3),~.O,~ER,NC) GOTO 30

IF(VN.LT.O.)I-20 POMN=ABS(VN/RC(I1,3)) VN1=POMN+*2+POMD**5itPOMI*FCEl(Q/PO~N/~OMD**3,AM,~M,D~R,

27 I=14

* O,I,Il,LI)

C Momentum of electric motor or generator

30 IF(UVNK) GOTO 39 VN2-0.0 IF( .NOT.UZAP) GOTO 40 IF(URVT) GOTO 38 POM=RC( B+10,6 ) VN2=FCE(VN,RC(B+lO,5),-RC(BtlO,5),PO~,DER,Etl80) IF(IC(B+lEO).GE.O.AND.VN.GT.POM)

GOTO 40 it VN2~-FCE(2*POM-VN,RC(B~lO,5),-RC(BilO,5),POM,DER,Btl8O)

38 VN2=FCE(ET,O.,O.,O.,DER,Bt70)tFCE(ET,O..O.,O.,DER,B+50)~ *(VN-RC(B+lO,G)) GOTO 40

346

Appendix A

39 VN2=-VNI GOTO 50

C Change of speed

40 IF(USETR0) GOTO 41 IF(II.LE.0) GOTO 50 VN=VN+(VNltVN2)*DT/(2.*PI*POMO) GOTO 50

41 DM=VNl+VN2 IF(ABS(DM).LE.CHMl) GOTO 50 IF(NITER.GT.O.AND.DM*OPR.LT.O.) UPOM=.FALSE. OPR=SIGN(OPR,DM) IF(UP0M) GOTO 42 OPR=O.S*OPR

NITER=NITER+l IF(NITER.LT.40) GOT0.21 CHMl =ABS ( DM )

42 VN=VN+OPR

C Coefficients of linear equations

50 IF (UTURB) GOTO 60 IF(IC(NC).LT.O) GOTO 52 I F ( ABS(VN) .GT.CHN) GOTO 51 POMl =RC( Bt 10,l) IF(Q.LT.O.0) POMl=RC(B+lO,2) R=2*POMl*Q S=-R*Q/2.0 U = . FALSE. GOTO 80

51 IF(RC(B,S).EQ.O.) GOTO 70 POMl=ABS(VN)/RC(B,S) POMZ=Q*RC( B,5)/ABS( VN) NC=B+20 IF(IC(NCt40).GT.O .AND.(VN.LT.O.O.OR.IC(NC).EQ.O)) NC=NC+IO S=POM1+POMl*FCE(POM2,RC(BtlO,2),RC(BtlO,l),O.O,DER,NC) R=DER*POMl S=S-Q*DER*POMl U= . FALSE. GOTO 80

52 U=.TRUE. GOTO 80

60 IF(ABS(VN 1.1 IF( Q.LT. 0 R.2. *FCE S = -R*Q/Z GOT0 80

6 1 1.2

.GT.CHN) GOTO 61

) 1=2 (O.,AM,BM,DER,I,1,I1,U)/POMD**4*POMI*Q

IF(VN.LT.0.) 1=8 POMN=ABS(VN/RC(I1,3)) POM=FCE1(Q/POMN/POMD**3,AM,BM,DER,O,I,I1,U) R=DER*POMN/POMD*POMI S=POMN**2*POMD**2*POMI*POM-Q*R GOTO 80

C Error

70 WRITE(+,’(”OWTH Err.10 - Error in data for turbine ‘ I ,

* ” ( First line of data 4”,14,” ) ” ) ’ ) B STOP

80 RETURN END

347

Appendix A

SUBROUTINE COND (II,U,Q,R,S,D,B) calculates parameters of condenser

COMMON /GOV/ CHM1,CHN,CHP1,CHPU,CHQl,CHQU,DT,G,IS(50,2),IT,ITISKT,

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ C

* ITT,ITTOT,ITTZ,ITV,ITZ,M(5O),N(5O),NIl,NS,NU,PI,RO, * UBEZIT,UNT,UU,UUST,UVT,UZKR,X(50),Xl(50) Y(100)

* UO( 50),UD( 50)

* V6( 50)

COMMON /JUN/ C~(50),KS(5O),PO(50),PS(5O),QO(5O),RO(50 ,S0(50),

COMMON /PAR/ IC(200O),RC(2000,6) COMMON /DAM/ UV(5O),VO(5O),V1(5O),V2(50),V3(5O),V4(50 ,V5(50),

INTEGER B,CS,X,Xl,Y LOGICAL U,UO,UBEZIT,UD,UNT,UU,UUST,UV,UVT,UZKR

ROA=1 .22 1=1 IF(UZKR.OR.II.LT.0) GOTO 1

C Effect of the levels of the liquid in the chambers of the connected C integrated damping devices

TSZ=IFIX( RC( B, 5)t. 1 ) ISK=IFIX( RC( B,6)t. 1 ) IF(ISZ.LT.l.OR.ISZ.GT.NS) GOTO 25 IF(ISK.LT.l.OR.1SK.GT.NS) GOTO 25 HZ=V4( ISZ) HK=V4(ISK) HH=RC( B ,3 ) HD=RC( B, 4 ) IF(HH.LE.HD) GOTO 25 IF(H2.GE.HH.AND.HK.GE.HH) 1=1 IF(HZ.GE.HH.AND.HK.LT.HH.AND.HK.GT.HD) I=2 IF(H2.GE.HH.AND.HK.LE.HD) 1=3 IF(HZ.LT.HH.AND.HZ.GE.HK.AND.HK.GT.HD) I=4 IF(HZ.LT.HH.AND.HZ.GT.HD.AND.HK.LE.HD) 1=5 IF(HZ.LT.HH.AND.HZ.GT.HD.AND.HK.GE.HH) 1=6 IF(H2.LE.HD.AND.HK.GE.HH) 1.7 IF(HZ.GT.HD.AND.HZ.LT.HK.AND.HK.LT.HH) I=8 IF(HZ.LE.HD.AND.HK.GT.HD.AND.HK.LT.HH) 1.9 lF(H2.LE.HD.AND.HK.LE.HD) 1.10

1 U=.FALSE. El -8. *RC( B, 1 )*RO/ ( D*D*PI )**2. E2=RO*G It- 2. *El*ABS ( Q ) S=-R/2.*Q GOT0(11,12,13.14,15,16,17,18,19,20),1

11 GOTO 30 12 S-StEE*( (IIH+HD)/2.-(HK+HD)/2.)

IF( (PO(ISZ)-E2*HH),LT.(PO(ISK)-E2*HK)) GOTO 21 GOTO 30

IF((PO(ISZ)-E2*HH).LT.(PO(ISK)-B2+HK)) GOTO 21 GOTO 30

S=-R/2.*QtE2+( (HZtHD)/Z. - ( HKtHD)/2. ) GOTO 21

S z R/2.*QtEz*((HZ+HD)/2.-HK)

1 3 S=S+EZ*((HHtHD)/Z.-HK)

1 4 R=R*( (HH-HD)/(HZ-€ID) )**2

1 5 R=R*( (HH-HD)/(HZ-€ID) )**2

GOTO 21 S = S 1 E2* ( ( HZt HD ) /2. - ( HHtHD ) /2 . ) IF((PO(ISZ)-E2*HZ).GT.(PO(ISK)-E2*HH)) GOTO 21

1 6

GOTO 30

348

Appendix A

17 S=StE2*(HZ (HHtHD)/2.) IF( (PO( ISZ)-E2*HZ).GT.(PO(ISK)-E2*HH)) GOTO 21 GOTO 30

18 R=R*((HH-HD)/(HK-HD))**2 S=-R/2. *Q+E2*( (HZtHD)/2. - ( HK+HD)/2. ) GOTO 21

S=-R/2.*Q+E2*(HZ-(HK+HD)/2.) GOTO 21

1 9 R=R*( (HH-HD)/(IIK-HD) )**2

20 U=.TRUE.

C Equalizing of air pressure in chambers

21 El=V6( ISZ)-VO(ISZ) EB=VG(ISK)-VO(ISK) IF(El.LE.O..OR.EZ.LE.O.) GOTO 30 E3=Vl( ISZ)+Vl( ISK) Vl(ISZ)-EltE3/(El+E2) V1( ISK)=E2+E3/( EltE2) GOTO 30

c Error

25 WRITE(*,’(”OWTH Err.16 - Error in data of condenser”, it ” ( First line of data 4 ” , 1 4 , ” ) ” ) I ) B STOP

30 RETURN END

SUBROUTlNE BVALVE (II,i[,Q,R.S,P,PO.ALU,AL,DAL,MQ,MH,B,J) _ _ _ _ _ _ _ _ _ _ - - - - - - - calculates parameters of controlled butterfly valve

COMMON /GOV/ CHM1,CHN,CHP1,CHPU,CHQ1,CI1QU,DT,G,IS(50,2),IT,ITlSKT,

C

* ITT,ITTOT,ITTZ,ITV,ITZ,M(50),N(50),NI1,NS,NU,PI,RO, * UBEZ~T,UNT,UU,UUST.UVT,UZKR.X(5O),XI(5O),Y(lOO) COMMON /PAR/ IC(200O),RC(2000,6)

REAL MG,MH,MI,MQ INTEGER B,X,Xl,Y LOGICAL U,UBEZIT,UC,UNT,UU,UUST,UVT,UZKR

IF(I1) 1,50,10

C Preparation

1 AL=RC(B,2) IF( UUST ) AL-ALU DAL.0. MH-0. MQ=O. GOTO 50

C Cavitation coefficient

10 EI=AMIN~(PO,P) E2=ABS ( PO-P ) IF(E2.EQ.O.) E3.1000. IF( E2 .NE. 0. ) E3=( E1-RC( B, 5 ) )/E2

C State of equilibrium

E2=RC( B, 4 ) E1=0.

349

Appendix A

11.1

IF(IL.EQ.1) ALR-0. IF(I1 .EQ.2) ALR=RC(B, 4 ) I=IC(B)

1 5 AI,R=(EltE2)/2.

MQ~RC(B,1)**3.*(PO-P)*FCE2(ALR,E3,Q,It20,10,4,DER) IF( J.EQ.2) MQz-MQ MG=FCE(ALR,O.,O.,O.,DER,-(It8)) MH-0. IF(ALR.LT.AL) MH=(ALR-AL)+fALR-AL)/DT/DT*FCE(ALR,O.,O.,O.,DER,

M=-RC(B,6)*((ALR-AL)/DT-DAL)/DT E=MQtMGtMHtMI IF(ALR.LE.O..AND.E.LE.O..OR.ALR.GE.RC(B,4).AND.E.GE.

I1=11+1 IF(Il.LT.3) GOTO 15 IF(E.LT.0.) EI=ALR IF(E.GT.0.) El-ALR IF(Il.GT.30) GOTO 1 6 GOTO 15

* -(It12))

* O..OR.ABS(E).LE.CHMl) GOTO 20

1 6 IF( .NOT.UZKR.OR.IT.GT.ITTOT.OR.UU) CHMl=ABS(E)

C Time

20 ET=ITV*DT

C

21

22

C

25

C

30

C

4 0

C

IF( UZKR) ET=O . Switching on and off oil pump

UC= . TRUE. I=IC( Bt10) IF(I.LE.0) GOTO 25 IF( ET. GE. RC( B+ 10, I ) )UC= .NOT. UC I=I-1 GOTO 22

Tilt angle of flap in the state of equilibrium

IF(.NOT.UZKR) GOTO 30 AL= ALR IF(UC) ALzAMAXl(AL,RC(B,J)) DAL.0. GOTO 50

Rate of rotation of flap - oil pump switched off

IF(UC) GOTO 4 0 DAL=(ALR-AL)/DT AL=ALR GOTO 50

Rate of rotation of flap - oil pump switched on

I=IC( B ) + 1 6 El =AL

IF( AL.GT.RC( B, 3 ) ) AL=RC( B,3) AL:ALtDT*FCE(AL,O.,O.,O.,DER,-I)

AL=AMAXl(ALR,AL)

Coefficients of linear equations

DAL=( AL-El )/DT

50 IF(,AL.LT.O) AL-0

350

Appendix A

IF(AL.GT.RC(B,4)) AL=RC(B,Q) I=IC(B) IF(Q.LT.O.AND.IC(It4).GT.O.OR.IC(I).LE.O) I=It4 E=FCE(AL,O.,O.,O.,DER,-I) IF(E.LE.O..OR.RC(B,l).LE.O.) GOTO 60 U=.FALSE. R=l6.+RO*ABS(Q)/(PI*RC(B.l)**2.)~*2.+(l./(E*E)-l.) S=-R/2 .+Q GOTO 70

60 U=.TRUE.

70 RETURN END

35 1

Appendix B

List of files on WTHM diskette

The WTHM diskette contains the following files: WTHM. EXE - WTHM program for calculating water hammer in a pipe-line

systems (in translated form); WTHD. EXE - WTHD program for creating and checking the input data files

for the WTHM program (in translated form); WTHG.EXE - WTHG program for plotting the results of the calculations

obtained with the aid of the WTHM program (in translated form);

D1. WTH; D2. WTH; D3. WTH; D4. WTH; D5A. WTH; D5B. WTH; D6. WTH; D7. WTH; D8. WTH; D9. WTH; D 10. WTH; Dl 1. WTH; D12. WTH; D13. WTH; D14. WTH

- Input data files for the examples presented in Sections 14.1 to 14.14

352

References

ABE, T., FUJIKAWA, T., ITO, s.: A calculating method of pulsation in a piping system. Bull.

A L L I ~ V I , L. R.: Teorie uber die veranderliche Bewegung des Wassers in Leitungen. Springer, Berlin 1909. ANGUS, R. W.: Air chambers and valves in relation to water hammer. Trans. ASME 59, p. 661, 1937. BAKES, F.: Solution of water hammer in a pressure system - algorithm of the solution (Rieienie hydraulickeho razu v tlakovom systtme - algoritmus rieienia). Vodohospodaisky Easopis SAV 3, p. 296, 1967. BERGERON, L.: Waterhammer in hydraulics and wave surges in electricity. John Wiley & Sons, Inc., New York, 1961. CHARNYJ, I. A,: Unsteady motion of a real liquid in a pipe-line (qAPHbIft, A. A.: Heyc-

1951. ChAUDHRY, M. H., HUSSAINI, M. Y.: Second-order accurate explicit finite-difference schemes for waterhammer analysis. Trans. ASME 107, 4, pp. 523-529, 1985. CHAUDHRY, M. H., Ruus, E.: Surge tank stability by phase plane method. Proc. ASCE 97,

DE BERNARDINIS, B., FEDERICI, G., SICCARDI, F.: Transient with liquid column separation: Numerical evaluation and comparison with experimental results. L’Energia Elettrica 52,9, pp.

DONSKY, B.: Complete pump characteristics and the effects of specific speeds on hydraulic transients. Trans. ASME 83, D, 4, pp. 685-696, 1961. DONSKY. B., BYRNE, R. M., BARTLETT, P. E.: Upsurge and speed-rise charts due to pump shutdown. Proc. ASCE 105, HY6, pp. 661-674, 1979. DRIELS, M. R.: Design of pressure transient control system. Proc. ASCE 101, HY5, pp.

DRIELS. M. R.: The reduction pressure surge in a simple pipe Row system. Journal of Mechanical Engineering Science 18, 2, pp. 66-72, 1976. EVANS, W. E., CRAWFORD, C. C.: Design charts for air chambers on pump lines. Trans. ASCE

FOK, A. T. K.: Design charts for air chamber on pump pipe lines. Proc. ASCE 104, HY9, pp.

FRANCE, I? W.: Mathematical models for surge analysis. Eng. Analysis 1,2, pp. 107-109, 1984. GABRIEL, I?, KRATOCHV~L, J., SEREK, M.: Computer applications in water conservation and hydraulic projects (Vypotetni technika pro obor vodni hospodafstvi a vodni stavby). SNTL, Praha 1982. GILL, M. A.: Oscillations in surge tanks. Proc. ASCE 100, HYlO, pp. 1369-1381, 1974.

JSME 13, 59, pp. 678-687, 1970.

TaHOBHBuleeCR nBHXeHHe PeaJIbHOfi XKJWOCTH B ~py6ax). rOCTeXTeOpeTM3naT, MocKea

HY4, pp. 489-503, 1971.

471-477, 1975.

437-448, 1975.

119, 2710, pp. 1025-1036, 1954.

1289-1303, 1978.

353

References

HAINDL. K.: Water hammer in pipelines (Raz v potrubi). Vodni hospodaistvi 12, p. 357, 1953. HAINDL, K.: Water hammer in water-supply and industrial pipe-lines (Hydraulicky raz ve vodovodnich a prbmyslovych potrubich). SNTL, Praha 1963. HAINDL, K.: The protection of low head irrigation pipe-networks against water hammer effects. L'Energia Elettrica 46, 6, pp. 410-415, 1969. HALLIWELL. A. R.: Velocity of a water-hammer wave in an elastic pipe. Proc. ASCE 89, HY4,

HEPWORTH. H. K., RICE, W.: Laminar two-dimensional flow in conduits with arbitrary time-varying pressure gradient. Trans. ASME 37, E, 3, pp. 861-864, 1970. HOLLOWAY. M. B., CHAUDHRY, M. H.: Stability and accuracy of water hammer analysis. Adv. Water. Resour. 8, 3, pp. 121-128, 1985. HoRAK. Z . , KRUPKA, F., S I N D E a R . V.: Physics (Fyzika). Price, Praha 1954. JAEGER. C.: Contribution to the stability theory of systems of surge tanks. Trans. ASME 80,

JAEGER. C.: A review of surge tank stability criteria. J. Basic Eng. pp. 756-783, 1960. JURECKA. J., BiNA. J., ZARUBA. J.: Some results of theoretical solutions of water turbine speed governing systems (NtkterC. vysledky teoretickkho ieieni regulace vodnich turbin). Sbornik HYDROTURBO 87, TrenEin, pp. 409-417, 1987. KALCiK. J.: Technical thermodynamics (Technicka termodynamika). Nakladatelstvi CSAV, Praha 1963. KAPLAN, M., STREETER, V. L., WYLIE, E. B.: Computation of oil pipeline transients. Proc.

KARTVELISHVILY. N. A.: Internal pressure fluctuations in pressure pipes. Hydrotechnical Construction 17, 8, pp. 369-400, 1984. KINNO. H., KENNEDY. J. F.: Water-hammer charts for centrifugal pump systems. Proc. ASCE

KITAGAWA. A.: A method of absorption for surge pressure in conduits. Bull. JSME 22, 165, pp.

KRANENBURG. C.: Gas release during transient cavitation in pipes. Proc. ASCE 100, HY 10,

KROUZA, V.: Centrifugal and kindred pumps (Cerpadla odstiediva a jim piibuzni). Na- kladatelstvi CSAV, Praha 1956. LAMAC', J.: Electrohydraulic control of water turbines (Elektrohydraulickii regulace vodnich turbin). SNTL, Praha 1981. LEAF, G. K., CHAWLA, T. C., MINKowYcz. W. J.: Numerical methods for hydraulic transients. Numerical Heat Transfer 2, I , pp. 1-34, 1979. MOHRI, Y., tiAYAMA. s.: Resonant amplitudes of pressure pulsation in pipclincs. (2nd Report, A calculation by method of transfer matrix). Bulletin JSME 26, 221, pp. 1977-1984, 1983. NECHLEBA. M.: Writer turbines. their design and accessories (Vodni turbiny, jejich konstrukce a piisluienstvi). SNTL, Praha 1962. PARMAKIAN. J.: Pressure surges in pump installations. Trans. ASCE 120, pp. 697-720, 1955. PARMAKIAN. J.: Water hammer analysis. Prentice-Ihil Inc.. New York 1955. PARMAKIAN, J.: Waterhammer design criteria. Proc. ASCE 83, P02, 1216, 1957. PARMAKIAN, J.: Unusual aspects of hydraulic transients in pumping plants. Journal of the Boston Society of Civil Engineers 5 5 , 1, pp. 30-47, 1968. PAVLUCH. L.: Waler hammer following failure of power to a centrifugal pump with a large moment of inertia. National Research Institute of Hcat Engineering, Monographs and memoranda 3, Praha 1963. PAvLucH. L.: Determination of non-stationary phenomena in pumping installations neglecting the cornprcssibility of thc liquid (Stanoveni nestacionrirnich jevd v Eerpacich zaiizenich pfi

pp. 1-21, 1963.

pp. 1574-1584, 1958.

ASCE 93, PL3, pp. 59-72, 1967.

91, HY3, pp. 247-270, 1965.

348-355, 1979.

pp. 1383-1 398, 1974.

3 54

References

r631

zanedbani stlaEitelnosti kapaliny). SVUSS Bkhovice, Monografie a memoranda 22, Praha 1977. PETROVSKIJ. 1. G.: Partial differential equations (Parcialni diferencialni rovnice). PiirodovEdecke nakladatelstvi, Praha 1952. RAITERI. E., SICCARDI, F.: Transients in conduits conveying a two-phase bubbly flow: Experimental measurements of celerity. L’Energia Elettrica 52, 5, pp. 256-261, 1975. REKTORYS. K.: Summary of applied mathematics (Piehled uiite matematiky). SNTL, Praha 1963. Ruus. E.: Charts for water hammer in pipelines with air chambers. Canadian Journal of Civil Engineering 4, 1977. SCHNYDER. 0.: Comparisons between calculated and test results on water hammer in pumping plants. Trans. ASME 59, p. 661, 1937. SHIMADA, M., OKUSHIMA, S.: New numerical model and technique for waterhammer. Journal of Hydraulic Engineering ASCE 110, 6, pp. 736-748, 1984. SHIN. Y. W., CHEN, W. L.: Numerical fluid-hammer analysis by the method of characteristics in complex piping networks. Nuclear Engineering and Design 33, 3, pp. 357-369, 1975. SHIN, Y. W., VALENTIN, R. A.: Numerical analysis of fluid-hammer waves by the method of characteristics. Journal of Computational Physics 20, 2, pp. 220-237, 1976. SMETANA, J.: Hydraulics (Hydraulika). Nakladatelstvi CSAV, Praha 1957. STREETER. V. L.: Valve stroking to control water hammer. Proc. ASCE 89, HY2, pp. 39-66, 1963. STREETER. V. L.: Water hammer analysis of pipelines. Proc. ASCE 90, HY4, pp. 151-172, 1964. STREETER. V. L.: Water hammer analysis. Proc. ASCE 95, HY6, pp. 1959-1972, 1969. STREETER. V. L.: Transients in pipelines carrying liquids or gases. Proc. ASCE 97, TEI, pp.

STREETER. V. L., LAI. Ch.: Waterhammer analysis including fluid friction. Proc. ASCE 88,

STREETER. V. L., WYLIE. E. B.: Hydraulic transients. McGraw-Hill Book Co., New York 1967. TAKAHASHI. K., IKEO, S., TAKAHASHI, Y.: Transient phenomena caused by directional control valve in a hydraulic pipeline. Bull. JSME 16, 102, pp. 1911-1917, 1973. TANAHASHI. T., KASAHARA, E.: Analysis of the hydraulic transient in pipe equipped with an air-chamber. Bull. JSME 12, 54, pp. 1380-1387, 1969. TANAHASHI. T., KASAHARA, E.: Analysis of the waterhammer with water column separation.

TANAHASHI. T., KASAHARA, E.: Comparisons between experimental and theoretical results of the waterhammer with water column separations. Bull. JSME 13, 61, pp. 914-925, 1970. THORLEY, A. R. D., GUYMER, C.: Pressure surge propagation in thick-waled conduits of rectangular cross section. Trans. ASME 98, 3, pp. 455-461, 1976. TRIKHA. A. K.: Variable time steps for simulating transient liquid flow by method of characteristics. Trans. ASME 99, 1, pp. 259-261, 1977. VALOUCH, M.: Five-place logarithmic tables and tables of constants (Pttimistne logaritmickk tabulky a tabulky konstant). SNTL, Praha 1967. WAITERS. G. Z., JEPPSON. R. W., FIAMMER, G. H.: Water hammer in PVC and reinforced plastic pipe. Proc. ASCE 102, HY7, pp. 831-843, 1976. WICHOWSKI, R.: One-dimensional theory of the phenomenon of water hammer in water mains. Part I (Jednowymiarowa teoria zjawiska uderzenia hadraulicznego w przewodach wodociqgowych. Cz&c I). Archiwum Hydrotechniky 30, 4, pp. 333-360, 1983. WICHOWSKI. R.: One-dimensional theory of the phenomenon of water hammer in water

15-29, 1971.

HY3, pp. 79-1 12, 1962.

Bull. JSME 12, 50, pp. 206-214, 1969.

355

References

mains. Part I1 (Jednowymiarowa teoria zjawiska uderzenia hydraulicznego w przewodach wodociqgowych. Czqk 11). Archiwum Hydrotechniky 31, 1-2, pp. 39-60, 1984. .

WIGGERT, D. C., SUNDQUIST, M. J.: Fixed-grid characteristics for pipeline transients. Proc.

WIGGERT. D. C., SUNDQUIST, M. J.: The effect of gaseous cavitation on fluid transients. Trans.

WOOD, E M.: The application of Heaviside’s operational calculus to the solutions of problems in waterhammer. Trans ASME 59, HYD-59-15, pp. 707-713, 1937. WOOD, D. J., CHAO, S. F?: Effect of pipeline junctions on water hammer surges. Proc. ASCE

WOOD, D. J., JONES, S. E.: Water-hammer charts for various types of valves. Proc. ASCE 99,

WRIGHT, S. J., WYLIE, E. B., TALPIN, L. B.: Matched impedance to control fluid transients. Trans. ASME 105, 2, pp. 219-224, 1983. WYLIE, E. B.: Fundamental equations of waterhammer. Journal of Hydraulic Engineering

WYLIE. E. B., STREETER, V. L.: Fluid transients. McGraw-Hill Book Co., New York 1978. YAMAMOTO, K.: Optimum valve operation in waterhammer. Bull. JSME 27, 228, pp.

ZARUBA, J.: Reduction of the drop in stress at the closing of a pipeline (ZmenSeni podtlakd pii uzavirani potrubi). Vodohospodaisky Easopis SAV XIII, 1, pp. 105-1 15, 1965. ZARUBA, J.: Optimum course of thecontrol ofliquid discharge through a pipe-line (Optimalni prbEh regulace pdtoku kapaliny potrubim). Vodohospodaisky hsopis SAV XV, 3, pp.

ZARIJBA, J.: The effect on water hammer of solid particles in pipelines. Acta technika CSAV

ZARUBA, J.: Calculation of non-stationary flow in complex pipe-line systems on a digital computer (VypoCet nestacionarniho proudgni ve sloiitych systemech potrubi na Eislicovem pofitati). Institute of Hydrodynamics of the Czechoslovak Academy of Sciences, Zprava 229, Praha 1969. ZARUBA, J.: The effect on water hammer of solid particles in pipelines. Romanian Journal of Technical Sciences and Applied Mechanics 16, 2, pp. 419-425, 1971. ZARUBA, J.: Water hammer in pipeline systems (Hydraulicky raz v soustavach potrubi). Academia, Praha 1984. ZHUKOVSKIJ. N. E.: Water hammer in water mains (XYKOBCKW~, H. E.: 0 runpasnsrecKoM ynape B BononpoBonHMx ~py6ax). r o c ~ e x ~ e o p e ~ a s ~ ~ a ~ , MOCKBa 1949.

ASCE 103, HY12, pp. 1403-1416, 1977.

ASME 101, 1, pp. 79-86, 1979.

97, TE3, pp. 441-457, 1971.

HYl, pp. 167-178, 1973.

ASCE 110,4, pp. 539-542, 1984.

1 188-1 193, 1984.

269-282, 1967.

3, pp. 339-350, 1967.

Appendix of References

[87] ANDO, Y., KONW, S.: Experiment and analysis on pressure pulse propagation in a plastically deforming pipe. Nuclear Engineering and Design, 55, 2. pp. 249-259, 1979.

[ 8 8 ] AZOURY, P. H., BAASIRI, M.: Effect of valve-closure schedule on water hammer. Journal of Hydraulic Engineering ASCE, 112, 10, pp. 890-903, 1986.

[89] BJORGE, R. W.: Initiation of waterhammer in horizontal and nearly horizontal pipes containing steam and subcooled water. Journal of Heat Transfer, 106, 4, pp. 835-840, 1984.

[90] CAZACU. M. C.: Influence of viscosity on the waterhammer phenomennon. Mecanique Appliquee, 33, 6, pp. 567-574, 1988.

[91] DOMACHOWSKI, Z., ORLIKOWSKI, C., et al.: Laplace operator analysis of water hammers (in Polish). Prace Instytutu Maszyn Przeplywowych, 83/84, pp. 21-28, 1983.

356

References

FUZY, O., KULLMANN, L.: Simplified models of large pipe networks for simulation of pressure transients, (in German). Prace Instytutu Maszyn Pneplywowych, 83/84, pp. 41-48,

GRUEL, R. L., HUBER, P. W., et al.: Piping response to steamgenerated water hammer. Journal of Pressure Vessel Technology, Transaction of the ASME, 103.3, pp. 219-225, 1981. GRYBOS, R.: Water hammer in water outlet system in a deep coal mine (in Polish). Archiwum Gornictwa, 26, 2, pp. 225-237, 1981. GYORGY, D., KASZLO. G.: Generalization of the theory about pressure wave velocity. Energia es Atomtechnika, 34, 7, pp. 321-328, 1981. JELEV. I.: Damping of flow and pressure oscillations in water hammer analysis. Journal of Hydraulic Research, 27, 1, pp. 91-1 14, 1989. JvARsHEIsHviLi, A. G.: Behavior and presention of hydraulic surges in hydrotransport systems. Journal of Pipelines, 2, I , pp. 35-41, 1982. Li, X., BREKKE. H.: Large amplitude water level oscillations in throttled surge tanks. Journal of Hydraulic Research 27, 4, pp. 537-551, 1989. MAGDALINSKAYA, 1. V., ROZENBERG, G. D.: Experimental study of the attenuation of water-hammer pressure wave. Soviet Physics Doklady, 25, 12, pp. 973-974, 1980. MARTIN, C. S.: Waterhammer highlights of 1977-1980. Journal of Pipelines, I , I , pp.

MIYAKE. Y., INABA, T., BANDO, K., et al.: Panel method analysis of frequency response of fluid in pipe elements (in Japanese). Transaction of the JSME B, 53, 485, pp. 20-26, 1987. MUTO, T., KAYUKAWA, H.: Transient responses fluid lines (2nd report, oil-hammer caused by finite-time valve stroking) (in Japanese). Transaction of the JSME B, 52,481, pp. 3273-3277, 1986. NATHAN, G. K., TAN, J. K., NG, K. C.: Two-dimensional analysis of pressure transients in pipelines. International Journal for numerical Methods in Fluids, 8, 3, pp, 339-349, 1988. OLAJOSSY. A., PUCHALA, R.: The phenomena of hydraulic hammer and pulsation in liquid storing system (in Polish). Archiwum Budovy Maszin, 29, 2, pp. 187-200, 1982. PARADAKIS. C., SCARTON, H., edited by: Fluid transients and acustics in the power industry, presented at the Winter Annual Meeting of the American Society of Mechanical Engineers, San Francisco, California, December 10-15, 1978. New York, The American Society of Mechanical Engineers, 1978. ROTHE. P. H., WIGGERT, D. C., edited by: Fluid transients and structural interactions in piping systems, presented at The Fluids Engineering Conference, Boulder, CO, June 22-24, 1981. New York, The American Society of Mechanical Engineers, 1981. SEN, S., CONTRACTOR, D. N.: Reduction of pressure surges by minimax optimization. Applied Mathematical Modelling, 10, 4, pp. 271-277, 1986. SUZUKI. K.: New hydraulic pressure intensifier using oil hammer. Journal of Fluids En- gineering, 112, I , pp. 56-60, 1990. THIELEN. H., BURMA". W.: Calculation and protection of pipelines laid in the open against undue internal pressure and reactive forces resulting from water hammer (in German). 3R International, 19, 1 I , pp. 622-628, 1980. URATA, Y.: Coupled wave motions of elastic pipes and fluids (9th report, analysis of water hammer) (in Japanese). Transaction of the JSME C, 53,487, pp. 527-533, 1987. WASHIO. S., KONISHI. T., VETA. T.: Research on wave phenomena in hydraulic lines ( I 5 Report, surge calculus in place of method of characteristics) (in Japanese). Transaction of the JSME B, 53, 488, pp. 1247-1253, 1987. WEBB, S. W., CAVES. J. L.: Fluid transient analysis in pipelines with nonuniform liquid density. Journal of Fluids Engineering ASME, 105. 4, pp. 423-428, 1983.

1983-1984.

105-107, 1981.

357

[ I 131 WICHOWSKI, R.: Water-hammer analysis in the complex water supply systems (in Polish). Archiwum Hydrotechniky, 32, I , pp. 73-98, 1984.

[ I 141 WYLIE. E. B., STREETER, V. L.: Multidimensional fluid transients by laticework. Journal of Fluids Engineering ASME, 102, 2, pp. 203-210, 1980.

358

Index

Assumptions 22, 23, 26, 28, 37, 40, 47, 127. 129 Device 47

Blades action 119, 148 guide 115, 148, 327

Bubbles 62

Cavitation 40, 66, 77, 96, 139, 306 Characteristic

method 45, 127 pump 85, 102, l7Y, 271, 306, 313, 316

moment 103, 144, 181 motor 103, 145. 182. 316 pressure 102, 180, 273

invariable 106, 145. 179, 181 variable 108, 145, 146, 187, 321. 327

turbine 85, 180. 181

actual 109 interpolation 110, 321, 327 model 107, 108, 187, 321, 327

valve 85, 91, 178. 185, 271, 279 Computer application 120, 216, 226, 234 Conditions

initial 127. 133, 149, 250 steady state 43, 133, 151, 154, 258

Data 121, 125, 247 input 121, 125, 163, 216

graphical output 210 junction 165 name 0 1 calculation 163 numerical output 210 parameters 166 section 165 subtitle 215 type of calculation 160, 208

output 125, 223, 250, 258 Density 37, 49, 55, 57, 62, 77, 167

damping 47, 68, 120, 135, 174 air chamber 70, 136, 174, 232, 301 air inlet valve 75, 138, 174, 233, 306 cavitation 77, 139, 174, 306 constant pressure 68, 135, 250 discharge 86, 141, 175, 283 integrated device 79, 139, 174, 316, 321 modification 156 overtlow 73, 137, 174, 306, 3 16 pressure 84, 140, 175, 279, 294, 3 10, 3 I3 reservoir 69, 135, 174, 290 surge tank 72, 137, 174, 231 without damping device 68, 135, 258

pressure 47, 89, 120, 141, 178 butterfly valve 93, 143, 183, 316 closed pipe-line 89, 141 condenser 98, 143, 183, 316 control valve 32, 91, 142, 178, 185, 229, 258, 271, 283

local loss 90, 142, 178, 271, 306 modification 157 non-return flap valve 93, 142, 306 pump 32, 86, 102, 144, 178, 230, 271, 290,

turbine controlled by governor 33, 113, 306, 313, 316

146, 193. 327 action blades 119, 204 guide blades 115, 204, 327

178

145, 187, 321

turbine with fixed characteristics 106, 145,

turbine with variable characteristics 108,

without pressure device 89, 141, 250 Diameter

aperture 75, 77, 139, 305 butterfly valve 94 control valve 91

3 59

Index

pipe-line 37, 54 pump 109 turbine 109

admissible 145, 150, 151, 155, 158, 160, 167 Differences

Direction of section 47, 121, 165 Discharge 68, 86, 127, 141, 165 Diskette 120, 216, 352

Elevation 37, 258 Equations of water hammer 22, 23, 26, 37, 127

derivation 37 continuity 39 motion 37

linearisation 43, 127, 129, 135, 141, 150 methods of solution 43 parameters 37, 49

density of the liquid 49, 55, 57, 62 pipe-line diameter 54 pressure losses due to friction 55, 127, 155 velocity of pressure waves 49, 127, 155

scope of application 40 solution 43, 127

accuracy 48, 150, 151, 160 method of characteristics 45, 127, 149 methods 43

discharge 127 pressure 37, 65 velocity of liquid 37, 57

variables 37

Errors in calculation 240, 249 Examples 32, 247

actual 32, 279, 306, 310, 313, 316, 321, 327 calculations 247

abrupt closing 250, 258 air chamber 301 butterfly valve 316 cavitation 306 characteristics 279, 313, 316, 321, 327 condenser 306, 316 control valves 258, 271, 279 discharge 283, 310 electric motor 316 governor-controlled turbine 327 integrated damping device 316, 321, 327 local losses 271, 294, 306 non-return flap valve 306 overflow 306, 316 periodic variations 294 pipe-line network 290

pressure 279, 283, 294, 310, 313 pump 271, 290, 301, 306, 316 reservoir 290 steady state 258, 271, 290, 327 turbine with variable characteristics 321 valve control regime 271, 283

File 120, 163, 352 input 121, 163, 216, 247, 352

graphical and numerical outputs 210, 251,

junctions 165, 251 name of calculation 163, 250 parameters 166

255

accuracy of calculation 167, 321 control valve - alternative method 185,

damping devices 174 governor-controlled turbine 193, 327 pressure devices 178 time interval of printing 167, 251, 295 turbine with variable characteristic 187,

27 1

32 1 sections 165, 251 subtitle 215, 259 type of calculation 208, 258

graphical 210, 225, 255, 262, 268, 277 main 210, 223, 252, 264, 274, 293, 296 numerical 210, 226, 256, 286

output 223

Gas 62 Governor 113, 146, 193, 327 Grid of characteristics 127

Interval calculation 123, 127, 149, 155, 162, 210, 251 printing 124, 167, 208, 210

Iteration 130, 150, 208

Junction 47, 165, 210 number 47, 121 solution 129

iterative 130, 150

Liquid 40, 55, 57, 62, 66 compressible 23, 26, 37 incompressible 22

Modulus of elasticity 50 bulk 49, 58, 62

360

Index

Moment inertial

butterfly valvc 96, 316 motor 105, 306 pump 105, 144 turbine 113, 146, 193, 327

butterfly valve 95, 316 generator 114, 144, 146, 182, 193,321, 327 motor 103, 144, 182, 316 pump 103, 144, 181, 306, 316

torsional

Momentum 22

Particles 57 concentration 58 settled 60

Period 26, 32, 176, 258, 294 Pipe-linc 40, 47

closing 32. 86, 271, 283, 310 abrupt 23, 86, 250. 258 linear 22, 26, 86

diameter 37. 54 expansion bends 51 length 47. 128, 155, 165, 251, 252 modulus of elasticity 50 non-circular 53, 54 scction modification 127, 155 support conditions 51 system 47

thick-walled 52

reference 37, 65, 121, 250, 258

schematisation 47, 120, 247

Plane

Poisson’s ratio 51 Pressure 37, 65, 84

absolute 65. 70, 75, 77, 82, 121, 136, 138, 303 actual 37, 65, 121, 263 atmospheric 65, 76, 77, 138 converted 37, 68, 84, 127, 129, 140 initial 133, 149, 251 losses 37, 55, 90, 91, 128, 142 slow variations 26. 43 wave 23, 26, 40, 49. 155 zero 121, 250

deleting data lines 218 inserting data lines 217 listing data files 219 modifying data lines 218 plotting of functions 220 starting work 216 terminating the WTHD program 222

put 120,226, 352 WTHG for plotting the numerical out-

WTHM for the calculation of water hammer 120, 125, 153, 234, 352

basic layout 234 denotation of variables 243 main program 234 subprograms 238, 333 errors 240

Reduction of water hammer 229 adjustment regime of a valve 229, 283 air chamber 232, 301 other methods 232 pump 230 surge tank 231

References 353, 356 Roughness 55

Section 47, 165, 210 direction 47, 121, 165 length 47, 128, 155 modification 47, 128, 155 number 47, 121 parameters 47, 165 solution 127

Sediment 60 State

initial 133, 135, 141, 149, 250 steady 43, 133, 151, 154

calculation 159, 208 checking 151, 155, 156, 157

abridged calculation 159, 208 several calculations 152, 208

Submission

Subprograms 238, 333 Symbols 13, 243

Thickness of pipe-line wall 50 Time 37, 127, 149, 161, 210, 260 Type of calculation 160, 208

Program 120, 333, 352 WTHD for creating the input data file 120,

163, 216, 352 adding data files 219 adding data lines 217 Values checking data files 219 maximum 151, 254, 256, 257, 300

361

Index

minimum 151, 254, 256, 257, 300

air inlet 75, 138, 174, 306 butterfly 93, 143, 183, 316 control 32, 91, 142, 178, 250, 258, 271, 283

liquid 37, 57 mean cross-sectional 38 pressure wave 24,37,40,49,62, 127,155, 160,

165, 2 10, 260, 290 profile 57

Valve

Velocity

Viscosity 37, 47, 55, 127, 155, 165, 258, 312

Water hammer 21, 28, 247 actual cases 28, 32, 279, 306, 310, 312, 313,

basic equations 37, 127 calculation 22, 23, 26, 43, 120, 125, 149, 154

abridged 154. 258 submission 159 time interval of calculation 161 velocity of pressure waves 160

316, 327

computer application 120 damping devices 68, 135, 156, 174 examples 247 initial state 133, 135, 141, 149, 154 input 121, 125, 159, 163, 216 junctions 47, 129, 150, 165 output 125, 208, 210, 223 preparation 120 pressure devices 89, 141, 150, 157, 178 procedure 149 sections 47, 127, 149, 165 several 152, 208

causes 30 estimating 21, 22, 23, 26 examples 32, 247 measurement 32 methods of solution 43

origin 21, 30 physical principles 22 reduction 229

characteristics 45, 127, 149, 150, 154

362

Documents

Water Hammer in Pipe-Line Systems, J. Zaruba