Hydraulic Transients and Computations (eBook)

Preface 5
Contents 7
1 Introduction 13
1.1 History of Development of Computational Methods 15
1.1.1 Main Contributions 15
1.1.2 Confusion in Computations and Computational Methods 19
1.2 Rigid and Elastic Water Column Theories 20
1.3 Wave Tracking Method (WTM) 22
1.4 Method of Characteristics (MOC) 25
1.5 CFD and Its Restrictions 25
1.6 Design Aspects of Hydraulic Systems 26
1.7 Objectives and Main Content of This Reference Book 27
References 28
2 Stationary Flows and Flow Regulations 31
2.1 Laws of Flow Friction and Pressure Drop 32
2.1.1 Laminar Flows 33
2.1.2 Turbulent Flows in Hydraulically Smooth Pipelines 33
2.1.3 Turbulent Flows in Hydraulically Rough Pipelines 34
2.1.4 Resistance Constants 34
2.2 Flow Resistance Constants in Pipeline Systems 35
2.2.1 Pipeline Network of Pipes in Series Connection 35
2.2.2 Pipeline Network of Pipes in Parallel Connection 35
2.2.3 Shock Losses and Borda-Carnot Formula 37
2.3 Hydraulic Characteristics of Regulation Organs 40
2.3.1 Characteristic of the Injector Nozzle of the Pelton Turbine 41
2.3.2 Characteristic of the Spherical Valve 43
2.3.3 Characteristic of the Butterfly Valve 44
2.3.4 Characteristic of the Gate Valve 46
2.4 Flow Regulations in Pipeline Systems 48
2.4.1 Simple Pipeline Systems with One Regulation Valve 49
2.4.2 Pipeline Systems with Two or More Regulation Valves 49
References 50
3 Transient Flows and Computational Methods 52
3.1 Occurrence of Hydraulic Transients in Hydropower Stations 53
3.2 Method of Rigid Water Column Theory 55
3.2.1 Restrictions of Application 55
3.2.2 Flows in Pipelines of Constant Cross-Sectional Area 58
3.2.3 Flows in Specially Curved Pipelines of Constant Cross-Sectional Area 60
3.2.4 Flows in Stepped Pipes 62
3.3 Method of Elastic Water Column Theory 63
3.3.1 Joukowsky’s Equation 64
3.3.2 Primary and Reflected Shock Waves 66
3.3.3 Influence of Wave Speed on Accuracies of Transient Computations 69
3.3.4 Self-stabilization of Transient Flows by Regulation Valves 72
3.3.5 Pipe Elasticity and Size Response to the Pressure Shock 76
3.3.6 Momentum Equations in Pipe Flows 78
3.3.7 Continuity Equation in Pipe Flows 81
3.3.8 Wave Propagation Speed 83
3.4 Transverse Relation Between Rigid and Elastic Water Column Theories 84
References 86
4 Rigid Water Column Theory and Applications 87
4.1 Flow Oscillations in Open Pipeline Systems 88
4.1.1 Damped Flow Oscillation in a U-Tube 88
4.1.2 Damped Flow Oscillation Between a Lake and a Surge Tank 95
4.2 Flow Regulation and Computations of Shock Pressures 101
4.2.1 Flow Regulations 101
4.2.2 Computational Algorithms and Simplifications 102
4.2.3 Pressure Response by Closing the Injector Nozzle 107
4.2.4 Pressure Response by Opening the Injector Nozzle 110
4.2.5 Pressure Response by Stepped Pipelines 111
5 Surge Tank Functionality and System Stability 113
5.1 Functionalities of the Surge Tank 114
5.2 Momentum Equations and Numerical Solutions 115
5.2.1 Basic Computations 115
5.2.2 Simplifications of Computations 120
5.2.3 Reaction of the Surge Tank on the Turbine Start 121
5.2.4 Reaction of the Surge Tank on the Turbine Shutdown 126
5.2.5 Damping Effect of the Surge Tank Throttle Area 128
5.3 System Stability Performance and the Thoma Criterion 129
5.3.1 System Instability Owing to External Stimulations 129
5.3.2 Thoma Criterion 130
References 134
6 Elastic Water Column Theory and Fundamentals 135
6.1 Transient Flow Mechanics and Differential Equations 135
6.2 Wave Equation and Wave Parameters 138
References 139
7 Wave Tracking Method 140
7.1 Fundamental Equations of the Wave Tracking Method 141
7.1.1 Joukowsky’s Equation in Upstream Flows 143
7.1.2 Joukowsky’s Equation in Downstream Flows 144
7.1.3 Joukowsky’s Equation in Spatial Scale 145
7.2 Multiple Initial and Boundary Conditions 146
7.3 Generation of Primary Shock Waves 146
7.3.1 Regulation Mechanism 147
7.3.2 Upstream Shock Waves F of Primary Order 148
7.3.3 Downstream Shock Waves f of Primary Order 149
7.3.4 Shock Waves by Predefined Flow-Rate Regulation 151
7.3.5 Connection of Shock Waves on Both Sides of a Hydraulic Machine 151
7.3.6 Application Cases with a Pump and a Francis Turbine 153
7.4 Conservation and Traveling Laws of Shock Waves at Series Junctions of Pipes 158
7.4.1 Conservation Laws of Shock Waves 158
7.4.2 Traveling Laws of Shock Waves 159
7.4.3 Traveling Laws of Using Volume Flow Rate 160
7.5 Conservation and Traveling Laws of Shock Waves at T-Junctions 161
7.5.1 T-Junction of Diverging Flows 161
7.5.2 T-Junction of Converging Flows 164
7.6 Traveling of Shock Waves Through an Orifice 165
7.7 Full Reflection of Shock Waves at the Reservoir 167
7.7.1 Entrance Velocity Effect 167
7.7.2 Quantification of the Entrance Velocity Effect 168
7.8 Total Reflection of Shock Waves at Closed Valve 168
7.8.1 Valve at the Downstream End of a Pipe 168
7.8.2 Valve at Upstream of a Pipe 169
7.9 Reflection of Shock Waves on the Moving Surface of Water in the Surge Tank 170
7.10 Friction Effect on the Propagation of Shock Waves 172
7.10.1 General Friction Effect and Computations 172
7.10.2 Overall Friction Effect in a Round Trip of a Shock Wave 173
7.11 Local Resistance Effect on the Propagation of Shock Waves 175
7.12 Throttle Resistance at the Entrance of the Surge Tank 176
7.13 Two Regulation Organs and Origins of Wave Generations 178
7.14 Shapes of Shock Waves 180
7.15 Computational Examples and Algorithms 181
7.15.1 Closing of the Injector Nozzle 182
7.15.2 Opening of the Injector Nozzle 184
7.15.3 Stepped Pipeline 187
7.15.4 Flow Oscillation in Surge Tanks 189
7.15.5 Flow Regulation Between Two Reservoirs 193
7.16 Evaporation of Flows and Restrictions of Computations 201
References 202
8 Method of Characteristics 203
8.1 Characteristic Lines 203
8.2 Computational Algorithms 205
8.3 Generation of Shock Waves and CB at an Injector Nozzle 207
8.4 Flow State at a Reservoir of Constant Height 209
8.5 Pressure Head at the Closed Valve 210
8.6 Traveling Laws of Shock Waves at Series Junctions of Pipes 211
8.7 Traveling Laws of Shock Waves at T-Junction 213
References 214
9 Method of Direct Computations and Transient Conformity 215
9.1 Method of Direct Computations (MDC) 216
9.1.1 Remarks on the Method of Direct Computations 220
9.1.2 Numerical Computational Algorithms 220
9.2 Validation of the Direct Method and Computation Examples 221
9.2.1 Closing of the Injector Nozzle 222
9.2.2 Opening of the Injector Nozzle 224
9.2.3 Stepped Pipeline 225
9.3 Pressure Jumps at the Beginning and End of Each Flow Regulation 226
9.3.1 Pressure Jumps at the Beginning of Flow Regulations 227
9.3.2 Pressure Jumps at the End of Flow Regulations 228
9.4 Pressure Fluctuations 230
9.4.1 Pressure Fluctuations After Flow Regulations 230
9.4.2 Pressure Fluctuations During Flow Regulations 231
9.5 Transient Conformity of Rigid and Elastic Fluid Flows 233
9.6 Simplifications of Computations 233
9.7 Application to the Two-Step Closing of an Injector Nozzle 236
9.7.1 Comparison Between MDC and WTM 237
9.7.2 Time Increment Effect and Computational Bias 238
9.7.3 Explicit Explanation of the Viscous Friction Effect 239
9.7.4 Explicit Explanation of Overlapping Pressure Shock Waves 240
9.8 Remarks to the Method of Direct Computations 241
10 Hydraulic Characteristics of Pumps and Turbines 243
10.1 Hydraulic Characteristics of the Pump 243
10.1.1 Characteristics in Terms of Coefficients of Discharge and Head 244
10.1.2 Four-Quadrant Diagrams and Operation Map 249
10.1.3 Unification of the Pump and the Valve Characteristics 252
10.2 Hydraulic Characteristics of the Pelton Turbine 258
10.2.1 Characteristic of the Injector 259
10.2.2 Power Output and Flow Regulations 261
10.2.3 Linear Closing Law of Injectors 262
10.2.4 Parabolic Closing Law of Injectors 264
10.2.5 Unification of Characteristics of the Injector and the Spherical Valve 268
10.3 Hydraulic Characteristics of the Francis Turbine 269
10.3.1 Characteristics in Terms of Unit Parameters 271
10.3.2 Master Equation of the Francis Turbine 273
10.3.3 Reconstruction of the Master Equation of the Francis Turbine 275
10.3.4 Unification of the Francis Turbine and a Spherical Valve 278
References 279
11 Application Examples of Complex Transient Computations 281
11.1 Shut-Down of a Pelton Turbine 282
11.2 Pump Emergency Stop with Simultaneous Closing of a Spherical Valve 288
11.2.1 Unified Characteristics and Rotor Dynamics 290
11.2.2 Connection of Shock Pressures on Both Sides of the Pump Unit 291
11.2.3 Determination Equation and Primary Shock Waves 292
11.2.4 Tracking the Shock Waves 293
11.2.5 Numerical Computations 295
11.2.6 Computational Results 296
11.3 Pump Start 298
11.3.1 Computational Specifications and Algorithms 299
11.3.2 Computational Results 300
References 303
Appendix A Nomenclature 304
Appendix B Characteristics of Regulation Organs 307
B.1 Characteristics of Spherical Valves 307
B.2 Characteristics of Butterfly Valves 310
B.3 Characteristics of Gate Valves 311
References 313
Appendix C Computation of the Pump Start by Neglecting the Water Hammer 314
C.1 Quasi-Stationary Flows 315
C.2 Dynamic Start-Up of the Pump 317
Index 321