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Preface xiii Notation xvii Chapter 1 Pressure transient mechanisms and historical 1 development 1.1 Historical development 2 1.1.1 Foundation of waterhammer theory 3 1.1.2 Development of waterhammer analysis from 1910 5 to 1950 1.1.3 The introduction of digital computing methods, 7 in particular the method of characteristics, 1950-1970 1.1.4 Development of the method of characteristics 8 model, 1970-1990 1.2 Mechanism of pressure transient propagation 9 1.2.1 The importance of valve closure time 16 1.2.2 The importance of frictional effects 19 1.2.3 The importance of initial line pressure, gas 21 release and vapour cavity formation 1.3 Transient propagation velocity: the effect of pipe 25 elasticity and free gas 1.3.1 Non circular ducts 32 1.3.2 Thick walled pipes and tunnels 34 1.3.3 Wave speed calculations for a gas-fluid 36 mixture 1.4 Importance of pipe elasticity, comparison to rigid 39 column theory 1.5 Pressure transient propagation in fluid carrying 42 systems Chapter 2 Basic equations defining transient propagation 45 2.1 Momentum equation 46 2.2 Continuity equation 49 2.3 Summary of the equations defining transient 56 propagation 2.4 Review of methods of solution 57 Chapter 3 Transient reflection and transmission within 59 frictionless systems 3.1 Simplification of the wave equations 60 3.1.1 Pressure rise, or fall, on valve operation 62 3.1.2 Transient reflection at a constant pressure 63 boundary 3.1.3 Transient reflection at a closed valve or dead end 64 3.2 Transient transmission and reflection at changes 67 in pipe properties and/or pipe junctions 3.2.1 Application of reflection and transmission 69 coefficients to pipe junctions and changes in pipe properties 3.2.2 Application of transmission and reflection 71 coefficients in the interpretation of pressure transducer records 3.3 Justification for the simplification of the wave 74 equations 3.4 Graphical solution techniques 76 3.5 Summary of the frictionless model 86 Chapter 4 The method of characteristics 87 4.1 Development of the unsteady flow equations of 89 continuity and momentum 4.2 The method of characteristics applied to the 90 solution of the transient propagation equations 4.2.1 Velocity-pressure formulation 91 4.2.2 Velocity-head formulation 92 4.3 Interpolation as a means of defining base 101 conditions 4.4 Implications of interpolation within a transient 108 analysis 4.5 Improved interpolation methods 111 4.6 Setting up the base conditions for a transient 117 analysis 4.7 Frictional representation 118 4.8 Network solutions 119 Chapter 5 Pressure transient network boundary 121 conditions 5.1 Definition of boundary equations within the 122 method of characteristics 5.2 Pipe and network application of the method of 124 characteristics 5.3 Boundary conditions 127 5.3.1 Reservoirs terminating a pipeline 127 5.3.2 Valve terminating a pipeline 129 5.3.3 Boundary formed by a valve at the junction of 134 two pipes 5.3.4 Boundary formed by a closed valve or 138 dead-ended pipe 5.3.5 Orifice between two pipes 140 5.3.6 Boundary formed by the junction of two pipes 141 5.3.7 Junctions of three or more pipes 144 5.3.8 Trapped air at a dead end or closed valve 147 5.3.9 Steady pump discharge 151 5.4 Variable pump discharge and speed 154 5.5 Hydraulic turbine 166 5.5.1 Pump-turbine performance characteristics 168 5.6 Summary of boundary conditions 170 5.7 Column separation and gas release 170 5.7.1 Column separation upstream of a closing valve 175 or at a dead-ended pipe within a complex pipe network 5.7.2 Column separation downstream of a closing valve, 180 including gas release effects 5.7.3 Vapour only column separation downstream of a 181 closing valve 5.7.4 Cavity boundary conditions including released air 186 5.7.5 Column separation within a pipe length 196 5.8 Development of individual boundary conditions 198 Chapter 6 Surge control and suppression 203 6.1 Control of valve operation generated transients 204 6.2 Outwards relief valves as surge controllers 206 6.3 Inwards relief valves to alleviate column 208 separation 6.4 Accumulators as surge alleviators 213 6.5 Air vessels as a means of surge control 218 6.5.1 Representation of air vessel boundary conditions 219 6.5.2 Application of the air vessel boundary conditions 222 6.5.3 By-pass arrangement for inflow to an air vessel 226 6.6 Case study - Worsham to Burford pumping main 230 6.6.1 Simulations - no pressure surge protection 230 6.6.2 Simulations - existing air vessel at node 2 231 6.7 Air inlet/outlet pressure surge alleviation valve 237 6.7.1 Representation of air inlet/outlet valves 237 6.7.2 Calculation of CON and COU 242 6.7.3 Critical mass rate of inflow of air 242 6.7.4 Critical mass rate multiplier for outflow of air 243 (COU) 6.8 Typical pumping main 243 6.9 Flow continuation as a form of surge suppression 244 6.10 Surge pressure simulations 250 Chapter 7 Free surface wave and air pressure transient 251 propagation 7.1 Open channel unsteady flow 252 7.2 Gas flow low amplitude pressure transient 260 propagation 7.3 Open channel and partially filled pipe unsteady 270 flow analysis 7.4 Boundary condition modelling within free 275 surface flow networks 7.4.1 Entry conditions 275 7.4.2 Exit boundary conditions 278 7.4.3 Junction boundary conditions 279 7.5 Applications of the method of characteristics 285 free surface unsteady flow model 7.6 Application of the method of characteristics 287 to the modelling of small amplitude air pressure transients 7.6.1 Entry boundary conditions 290 7.6.2 Exit and junction boundary conditions 290 7.6.3 Moving boundary conditions - trap seal response 292 7.6.4 Sign conventions 294 7.6.5 Air admittance valves 294 7.6.6 Water curtain effects at a branch to stack 296 junction 7.6.7 Positive pressure tranient propagation on air 297 path closure 7.7 Network modelling 299 7.7.1 Air flow stoppages - the generation of positive 303 transients 7.8 Application of the method of characteristics to 308 free surface wave and low amplitude air pressure transient propagation Afterword The computer based simulation of transient 309 propagation Appendix I Wave speed calculations 313 Appendix II Three pipe system computer model 317 Appendix l Bisection technique 335 References 339 Index 349