208 Hydraulic analysis of unsteady flow in pipe networks

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1 References 1. Allievi, L. Teoria generale del moto perturbato dell' acqua nei tubi in pressione, Milan Translated into English by E. E. Halmos, The Theory of waterhammer. Anz. Soc. Civil Eng., Schnyder, 0. Druckstosse in Pumpensteigleitungen. Schweiz Bauztg., 94, Nos. 22 and 23, Bergeron, L. Etudes des variations de regime dans les conduites d'eau. Rev. gen. Hydraulique, Nos. 1 and 2 (1935). 4. Zienkiewicz, 0. C. and Hawkins, P. Transmission of waterhammer pressures through surge tanks. Proc. Inst. Mech. Eng., 68, No. 25 (1954). 5. Angus, R. W. Waterhammer in pipes, including those supplied by centrifugal pumps; graphical treatment. Proc. Inst. Mech. Eng., pp. 136 and 245 (1937). 6. Angus, R. W. Waterhammer pressures in compound and branched pipes. Trans. Anz. Soc. Civ. Eng., pp. 104 and 340 (1939). 7. Lax, P. and Wendroff, B. Systems of conservation laws, Comm. Pure Appl. Maths, XII, (1960). 8. Lax, P. Weak solutions of nonlinear hyperbolic equations and their numerical computations, Comnz. Pure Appl. Maths, VII, (1954). 9. Lister, M. The numerical solution of hyperbolic partial differential equations by the method of characteristics, in Mathematical Methods for Digital Computers (ed. Wilf, A and Ralston, H. S.) Wiley, New York (1960). 10. Courant, R., Friedrichs, K. and Lewy, H. On the Partial Differential Equations of Mathematical Physics, New York University Institute of Mathematics, translated by P. Fox (1956). 11. Price, R. K. Comparison of four numerical methods for flood routing, J. Hydr. Div., Anz. Soc. Civ. Eng., July (1974). 12. Pearsall, I. The velocity of waterhammer waves in Symposium on Surges in Pipelines, Inst. Mech. Eng., 180 ( ). 13. Karplus, M. B. The velocity of sound in a liquid containing gas bubbles, Armour Research Earth Foundation Report, June (1958). 14. Fox, J. A. An Introduction to Engineering Fluid Mechanics, Macmillan, London (1974). 207

2 208 Hydraulic analysis of unsteady flow in pipe networks 15. Suter, P. Representation of pump characteristics for calculation of waterhammer, Sulzer Review, No Edgell, G. Pressure Transients in Tunnels; Extension of Theory to!"eversible, Nonadiabatic Flow. Limited publication by Leeds University, Department of Civil Engineering, April (1974). 17. Stoker, J. J. Water Waves, Pure and Applied Mathematics, vol. 4, The Institute of Mathematical Sciences, New York University (1957) 18. Swaffield, J. A. A Study of column separation following valve closure in a pipeline carrying aviation kerosine, Proc. lnst. Mech Eng., No. 23 (1969). 19. Marsden, N. and Fox, J. A. An alternative approach to the problem of column separation in an elevated section of pipeline, Proceedings of 2nd International Symposium on Pressure Surges, September 1976.

3 Bibliography I. Angus, R. W. Simple graphical solution for pressure rise in pipes and pump discharge lines, J. Inst. Ca1Ulda, February, (1935). 2. Donsky, B. Complete pump characteristics and the effects of specific speeds on hydraulic transients, J. Basic Eng., December, (1961). 3. Fox, J. A. The use of the digital computer in the solution of waterhammerproblems,proc. Inst. Civ. Eng., 39, (1968) 4. Fox, J. A. and Henson, D. A. The prediction of the magnitudes of pressure transients generated by a train entering a single tunnel, Proc. /nst. Civ. Eng., 49, (1971). 5. Henson, D. A. and Fox, J. A. Transient flows in tunnel complexes of the type proposed for the channel tunnel, Proc. Inst. Mech. Eng., 188, No. I 5, I (1974). (Two papers). 6. Jaeger, C. Engineering Fluid Mechanics, Blackie, London (1956). 7. Knapp, R. T. Complete characteristics of centrifugal pumps and their use in prediction of transient behaviour, Trans, Am Soc. Civ. Eng., 59, (I 939). 8. Lax, P. D. and Richtmyer, R. D. Survey of the stability of finite difference equations, Comm Pure. Appl. Maths, ix, (I 956). 9. Parmakian, J. Waterhammer A1Ullysis, Dover, New York (1963). I 0. Pickford, J. A1Ullysis of Surge, Macmillan, London ( 1969). 11. Rich, G. Hydraulic Transients, Dover, New York (1963) 12. Stepanoff, A. J. Centrifugal and Axial Flow Pumps, Wiley, New York (1948). 13. Stoker, J. 1. Water Waves, Pure and Applied Mathematics, vol. 4, The Institute of Mathematical Sciences, New York University (1957). 14. Streeter, V. and Wylie, E., Hydraulic Transients, McGraw-Hill, New York (1967). 15. Symposium on surges in pipelines, Proc. Inst. Mech. Eng., 180, part E ( ). 16. Wilf, H. S. and Ralston, A. Mathematical Methods for Digital Computers, Wiley, New York (1960). 209

4 Index Abbot and Ionescu method 200 Acceleration angular 56 convective 81, 8 5 Actuators 136, 140 Adiabatic 148 Air vessels 39, 64, 68, 124, I 25, 126, 127,134, 135, 136, 201, 202 Allievi expression 17 Allievi interlocking equation 21, 25, 29, 35, 36 Allievi pipe characteristic 35 Amein's four-point method 86, 200 Analytical techniques I, 23 Anchor block 146 Arrays 118,119,120,140,187, 202, 203 for global programming 203 Attenuation 13, 15, 16, 63, 64, 127, 128,134,161 Bends 146 Bernoulli equation 6, 25 Bi-Characteristic method 177 Boiling I 7, 94 Bore 178 Boundary conditions 36, 39, 55, 73, 81, 108, 130, 154, 167, 186 Boundary layer 96 Bubble 17, 72, 87, 88, 89, 90, 91, 93, 94, 95, 99,110, 112 evolution of 88 Buckle, running 50, Buckling mode I 25 Bursts 112 Capacitance 157, 160 Catchments 186, 187 Cavity 50 Celerity 78, 181, 184, 186, 191, 194, 195, 198 Channels, open I 77 equations for unsteady flow in 178 nonprismoidal 180, 182 nonrectangular I 77 prismoidal 180, 182 rectangular I 77 supercf"itical flow in I 85 Characteristic 79, 80, 81 Characteristic equations 73, 74, 80, 84,129,149,181 in gas flow 148 Characteristic line 7 8, 131, 13 2 Choke, see throttle Coefficient of discharge 6, 24 Coefficient of heat transfer!51,!52, 153 Colebrook-White equation 95, 96 Column separation 16, 17, 50, 95 Compressor 65, 125, 154 Computerised methods 85, 201 Continuity equation 26, 29, 30, 46, 66, 67, 68, 70, 73, 85, 147, 178, 179, 193, 198 Corrector-predictor method 71 Courant and Lewy stability criterion 82

5 212 Index Crack, surface 89 Crack, micro- 89, 90 Current 157 Darcy-Weisbach equation 4, 29, 95, 160 Differential equation, hyperbolic partial quasi-linear, 29, 73, 74 Dimensional analysis of rotodynamic machines 57 Dispersion of wave 83 Distensible pipe-elastic fluid theory 2, 9 Domain of dependency 78, 183, 185 Dump tank 39, 73 Dynamic equation 2, 3, 26, 28, 30, 67' 68, 69, 73, 85, 148, 179, 193 Dynamometer mode of pump operation 62, 100, 102, 103, 113 Eagre lines 36, 38, 39, 40, 42, 45, 46, 47, 48,49, 53 plotting of 43 Edgell, G., 150 Efficiency, pump 56, 57, 60 Elasticity effects 9 Elasticity of fluid 6, 9 Elasticity of pipe wall material 6 Elliptic partial differential equations 75 Ends, blank 167 receiving sending 165, 173 Energy, conservation 13 kinetic 14, 46, 55, 191 strain 6, 13 velocity 13 Equation of state 148 Estuaries 186, 187 Euler equation 3 Expansion joints in pipes 18 Fanning equation, see Darcy Weisbach equation Finite difference methods 7, 9, 68, 85, 107, 199 Flood plain 177 Flow, two phase 50, 87, 92 Flywheels 124, 1 25, 126 Fourier analysis 173 Francis turbine 156 Frequencies, dangerous 172 Frequency 173 Friction 4, 13, 23, 28, 29, 36, 49, 66, 67, 68, 96, 99, 108, 121, 128,148 variable 96 Frictional damping 156 Froude number 185, 191, 197 relative 196 Fundamental 156, 173 Galleries 65 Gas 81, 89, 91,92 bubbles 72, 87, 88 calculation of free content 98 compressed 112 evolution 88 release 90 release head 93, 95, 96, 98, 99, 112 Global programming 201 Governor and mechanism 63 Graphical method 36, 50 Graphical techniques 36, 55, 72 Grid, regular rectangular 82, 86 Hagen Poiseuille formula 160 Harmonic 155, 17 3 Harmonic analysis 172, 173 Hartree method 82 Head frictional 2 9, Ill inertia 1 0, 1 5 maximum 56 no flow 55 pump 56 Heat flow 150 transfer 1 52 Henry's Law 88 Hydraulic controls 1, 36, 37, 62, 112,201 Hydraulic radius 28 Hydroelectric installations 156 Hyperbolic partial differential Hypersonic 7 6 equations 7 5

6 Impedance 172, 173, 176 characteristic 164, 16 8 concept 164 equation 165 hydraulic 164 loops 169 methods!55 networks 170 Impeller, pump 55, 56, 57, 100, 101, 103, 104, 117 roughness 57 Inductance!57, 160 Inertia, moment of 56, 60, I 04, 122, 123, 124, 125 Interpolation 83, 97, 99, 115, ,140,184,187,198 Isothermal 148 Iterative methods 70, 84, 96 Joints 132, 202 impedance theory of 168 method of dealing with 45 Joukowsky 17 Junctions 14, 39, 46, 48, 65, 73, 167 impedance theory of 167, 169, 170, 173 in open channels 187, 191, 192 n way, 130, 202 Korteweg 17 Lax- Wendroff method 86, 199 Leap frog method 86, 199 Ligget and Woolhiser method 86, 200 Line pack 127, 128, 129 Link method see Route method Lister method 7 4, 18 2 Local losses 5, 46, 68, 130, 133, 154, 191 Lock in 128, 129 Loops in networks 169, 170, 176 Method of characteristics 74, 77, 85, 107 Micronuclei 89, 90 Index 213 Modulus 172 bulk 18, 20, 72, 87, 90, 91 Young's 18, 20, 87 Moens 17 Momentum II Motor I 03, II 8, I 20 Motor armature 56 Network 36, 73, 94, I 00, I 03, 112, 124, 130, 136, 145, 168, 169,172,173,176,178, 201, 206 gas 147 open channel 178 resonant I 6 7 Newtonian theory I Newton-Raphson 154, 188 Newton's 2nd law of motion 2, 28, 148 Nodal method 20 I Nodes 65 Nonprismoidal channel 180, 181, 182 Open channels 177 nonrectangular I 77 prismoidal 177 rectangular 177 Organ pipes!55 Organ piping 65 Orifice 49, 63, 68, 176, see throttle Oscillating component of flow 158, 165 of head 158, 165 Oscillating valve 155, 174 Oscillation forcing 166,172,173,174,176 frequency of 161 mass 62, 65, 66, 134 sinusoidal 162 Oscillatory wave 161 Parabolic partial differential equations 7 5 Pelton wheel!56 Perimeter, wetted 28 Period, pipe 22, 23, 25, 41, 60, 125 Pipe bends 146 blank end of 167

7 214 Index Pipe (contd.) by-pass 126 description for global programming 203, 205 joints 45, 13 2, 168 junctions 46, 48, 73, 130, 154, 167 lengths 73 loops 169 period 22, 23, 25, 41, 60, 125 Poisson's ratio 20 Polytropic efficiency!54 Polytropic index 69, 127 Polytropic process 70, I 27, 148 Potential head equation 84, 99 Power 55 Pressure head amplitude 173 Pressure head, calculation at t:.t time intervals 41 Pressure rise, after instantaneous valve closure 21 Pressure transients, see Transients and waterhammer Price 200 Prismoidal channels 180, 181, 182 Procedure calls, in global programming 205 Propagation constant 161 evaluation of 162 Pump by-pass 120, 1 26 characteristic equation 55, 56, 58, 59, 60, 62, 100, , compound arrangement 123 description for global programming 204 dynamometer behaviour , efficiency 56, 58, 59, 60, 105, 122, 123, 124, 202 four quadrant operation 60, 111, 113, 120 impeller 55, 56, 57 in-line 108, 176 in parallel 1 22 in series 122 power 104, I 05, 106, 107 run-down 107 run-up 120 start-up 106 stations 1 20, 1 21 suction-well 11 0 torque 56 trip 55, 56, 73, 103, 110, 111, 112, 121,124, 126 turbine behaviour 1 03, 111, 113 Pumps 55, 56, 57, 100, 201, 202, 204, 205 Reflection I 0, 13, 3 2, 3 6, 55, 1 00, 128, 173 negative 14, 22, 23, 25, 146 partial 14 Regular rectangular grid method 82, 184 Reimann equations 30, 32, 37 Reservoir characteristic ,,, Reservoirs 39, 40, 50, 55, 65, 144, 145, 146, 154, 167, 173, 187, 190, 191, 201,204, 205 static water level 62, 63, 66, 68 Resistance, electrical 157,!59, 160 Resonance 65, 15 5, 156, 164, 172, 173, 176 Reynolds number 95, 96 Rigid pipe-incompressible fluid theory 2, 66 Riser 63, 64, 68 Route method, in global programming 201, 202 Schnyder-Bergeron graphical 129 Separation 16 Shock wave 186 method 36, 49, 72, 85, Simple harmonic motion 67 Sluice gates 186, 189, 190 Solenoid 126 Sound 1 Specific energy 188, 189 Specific speed 115, 117, 118 Speed, rotational 56 Steady flow component 158 Steady head component 158 Stoker, J. 199 Strain circumferential 18 diametral 18 volumetric 91

8 Strain energy 6 Streeter, V. 176 Stress, hoop 18 St. Venant equations 180, 199 Subcritical free surface flow 197, 198 Supercritical free surface flow 76, 185, 186, 197, 198 Superposition Surface tension 88 Surge 197 demand 196 ebb 196 floodorstorm 186,187,189, 196 rejection 196 suppression 124 travelling 1 7 8, 1 92, 1 93, 19 5 Surge tanks 47, 62, 68, 124, 125, 126, 127,201,202 choke ring 63, 68 equation integration 70 Johnson differential 63, 68 mass oscillation 66 pressurised 64, 65, 68, 125, 134 simple 63 transient analysis 65 Suter diagrams 116, 117, 118 Thermodynamics, first law of 149 Throttles 85 causing friction 49, 50, 52, 53, 54 multiple 52 Time level scanning, in global programming 206 Torque, pump 56,115,117,118, 120 Transducer, pressure 143 Transients I, 10, 16, 17, 22, 29, 36, 42, 50, 55, 60,63, 65, 72, 73, 87, 88, 90, 1 00, 1 03, 1 06, 111, 112, 125, 146, 156 Transmission line equation 157 Transmission line theory 161 Turbine gate 62, 63 Turbine mode of operation of a pump 62, 100, 102, 103, 1 07, 111, 113 Turbo-blowers 154 Index 215 Unidimensional flow in channels 177 Vacuum 94 Valve 103, 129, 201 ball 155 butterfly 112, 156 by-pass reflux 1 20, 1 26 characteristic 3 9, 40, 41, 4 2, 52 control 144 downstream closing 112 opening 4 effective area 17 5 fractional opening 35, 39 instantaneous closure 10, 21 linear closure 8 motorised 136, 137, 138, 139, 140, 141, 205 nonreturn or reflux 55, 100, 111, 112,120,121,128,129, 132, 134, 144, 145 part closed 144 reflux, see nonreturn servocontrolled 72, 142, 156, 175 slow closure 6, 17, 22, 23, 45, 125 slow opening 42 slow partial closure 43 stepwise closure 23 sudden closure 21, 125 Vapourous cavitation 94, 95 Vapour pressure 16, 50, 94, 110 Venturi-flume 190 Voltage 157 Volute 100 Waterhammer 1, 9, 28, 36, 60, 73, 77, 78, 85, 86, 93, 121, 134, 156, 157, 160, 181, 183, 184, 186 Waterhammer equations 73, 74, 77 integration of 79 linearisation of 157, 160 Waterhammer theory 1 history of 2 Waves 10, 11, 12, 13, 14, 15, 16, 21, 173 compressional 2

9 216 Waves (contd.) [type 24, 25, 31, 32, 33 F type 24, 25, 31, 32, 33 formation I 0 propagation I 0 shape 15 Wave form 64, 173 Waverider 38, 40, 41, 42, 45, 46, 47, 53 Wavespeed 18, 42, 72, 85, 97-99, Index 151,161,177,185,198, 199 equation 84, 99, 161 magnitude of variability 90 minimum 93 variation in 87,94 Weirs 64, 145, 146, 186, 188, 190 Zone, of influence 78, 183 of quiet 79, 183

ENGINEERING FLUID MECHANICS. CHAPTER 1 Properties of Fluids

ENGINEERING FLUID MECHANICS. CHAPTER 1 Properties of Fluids CHAPTER 1 Properties of Fluids ENGINEERING FLUID MECHANICS 1.1 Introduction 1.2 Development of Fluid Mechanics 1.3 Units of Measurement (SI units) 1.4 Mass, Density, Specific Weight, Specific Volume, Specific