Multi phase Contractors Contents Preface page 2 Contents 10 Nomenclature 11 1 INTRODUCTIONTOMULTIPHASEFLOW 19 1.1 INTRODUCTION 19 1.1.1 Scope 19 1.1.2 Multiphase flow models 20 1.1.3 Multiphase flow notation 22 1.1.4 Size distribution functions 25 1.2 EQUATIONS OF MOTION 27 1.2.1 Averaging 27 1.2.2 Conservation of mass 28 1.2.3 Number continuity equation 30 1.2.4 Fick’s law 31 1.2.5 Equation of motion 31 1.2.6 Disperse phase momentum equation 35 1.2.7 Comments on disperse phase interaction 36 1.2.8 Equations for conservation of energy 37 1.2.9 Heat transfer between separated phases 41 1.3 INTERACTION WITH TURBULENCE 42 1.3.1 Particles and turbulence 42 1.3.2 Effect on turbulence stability 46 1.4 COMMENTS ON THE EQUATIONS OF MOTION 47 1.4.1 Averaging 47 1.4.2 Averaging contributions to the mean motion 48 1.4.3 Averaging in pipe flows 50 1.4.4 Modeling with the combined phase equations 50 1.4.5 Mass, force and energy interaction terms 51 4 2 SINGLE PARTICLE MOTION 52 2.1 INTRODUCTION 52 2.2 FLOWS AROUND A SPHERE 53 2.2.1 At high Reynolds number 53 2.2.2 At low Reynolds number 56 2.2.3 Molecular effects 61 2.3 UNSTEADY EFFECTS 62 2.3.1 Unsteady particle motions 62 2.3.2 Effect of concentration on added mass 65 2.3.3 Unsteady potential flow 65 2.3.4 Unsteady Stokes flow 69 2.4 PARTICLE EQUATION OF MOTION 73 2.4.1 Equations of motion 73 2.4.2 Magnitude of relative motion 78 2.4.3 Effect of concentration on particle equation of motion 80 2.4.4 Effect of concentration on particle drag 81 3 BUBBLE OR DROPLET TRANSLATION 86 3.1 INTRODUCTION 86 3.2 DEFORMATION DUE TO TRANSLATION 86 3.2.1 Dimensional analysis 86 3.2.2 Bubble shapes and terminal velocities 88 3.3 MARANGONI EFFECTS 91 3.4 BJERKNES FORCES 95 3.5 GROWING BUBBLES 97 4 BUBBLE GROWTH AND COLLAPSE 100 4.1 INTRODUCTION 100 4.2 BUBBLE GROWTH AND COLLAPSE 100 4.2.1 Rayleigh-Plesset equation 100 4.2.2 Bubble contents 103 4.2.3 In the absence of thermal effects; bubble growth 106 4.2.4 In the absence of thermal effects; bubble collapse 109 4.2.5 Stability of vapor/gas bubbles 110 4.3 THERMAL EFFECTS 113 4.3.1 Thermal effects on growth 113 4.3.2 Thermally controlled growth 115 4.3.3 Cavitation and boiling 118 4.3.4 Bubble growth by mass diffusion 118 4.4 OSCILLATING BUBBLES 120 4.4.1 Bubble natural frequencies 120 5 4.4.2 Nonlinear effects 124 4.4.3 Rectified mass diffusion 126 5 CAVITATION 128 5.1 INTRODUCTION 128 5.2 KEY FEATURES OF BUBBLE CAVITATION 128 5.2.1 Cavitation inception 128 5.2.2 Cavitation bubble collapse 131 5.2.3 Shape distortion during bubble collapse 133 5.2.4 Cavitation damage 136 5.3 CAVITATION BUBBLES 139 5.3.1 Observations of cavitating bubbles 139 5.3.2 Cavitation noise 142 5.3.3 Cavitation luminescence 149 6 BOILING AND CONDENSATION 150 6.1 INTRODUCTION 150 6.2 HORIZONTAL SURFACES 151 6.2.1 Pool boiling 151 6.2.2 Nucleate boiling 153 6.2.3 Film boiling 154 6.2.4 Leidenfrost effect 155 6.3 VERTICAL SURFACES 157 6.3.1 Film boiling 158 6.4 CONDENSATION 160 6.4.1 Film condensation 160 7 FLOW PATTERNS 163 7.1 INTRODUCTION 163 7.2 TOPOLOGIES OF MULTIPHASE FLOW 163 7.2.1 Multiphase flow patterns 163 7.2.2 Examples of flow regime maps 165 7.2.3 Slurry flow regimes 168 7.2.4 Vertical pipe flow 169 7.2.5 Flow pattern classifications 173 7.3 LIMITS OF DISPERSE FLOW REGIMES 174 7.3.1 Disperse phase separation and dispersion 174 7.3.2 Example: horizontal pipe flow 176 7.3.3 Particle size and particle fission 178 7.3.4 Examples of flow-determined bubble size 179 7.3.5 Bubbly or mist flow limits 181 7.3.6 Other bubbly flow limits 182 6 7.3.7 Other particle size effects 183 7.4 INHOMOGENEITY INSTABILITY 184 7.4.1 Stability of disperse mixtures 184 7.4.2 Inhomogeneity instability in vertical flows 187 7.5 LIMITS ON SEPARATED FLOW 191 7.5.1 Kelvin-Helmoltz instability 192 7.5.2 Stratified flow instability 194 7.5.3 Annular flow instability 194 8 INTERNAL FLOW ENERGY CONVERSION 196 8.1 INTRODUCTION 196 8.2 FRICTIONAL LOSS IN DISPERSE FLOW 196 8.2.1 Horizontal Flow 196 8.2.2 Homogeneous flow friction 199 8.2.3 Heterogeneous flow friction 201 8.2.4 Vertical flow 203 8.3 FRICTIONAL LOSS IN SEPARATED FLOW 205 8.3.1 Two component flow 205 8.3.2 Flow with phase change 211 8.4 ENERGY CONVERSION IN PUMPS AND TURBINES 215 8.4.1 Multiphase flows in pumps 215 9 HOMOGENEOUS FLOWS 220 9.1 INTRODUCTION 220 9.2 EQUATIONS OF HOMOGENEOUS FLOW 220 9.3 SONIC SPEED 221 9.3.1 Basic analysis 221 9.3.2 Sonic speeds at higher frequencies 225 9.3.3 Sonic speed with change of phase 227 9.4 BAROTROPIC RELATIONS 231 9.5 NOZZLE FLOWS 233 9.5.1 One dimensional analysis 233 9.5.2 Vapor/liquid nozzle flow 238 9.5.3 Condensation shocks 242 10 FLOWS WITH BUBBLE DYNAMICS 246 10.1 INTRODUCTION 246 10.2 BASIC EQUATIONS 247 10.3 ACOUSTICS OF BUBBLY MIXTURES 248 10.3.1 Analysis 248 10.3.2 Comparison with experiments 250 10.4 SHOCK WAVES IN BUBBLY FLOWS 253 7 10.4.1 Normal shock wave analysis 253 10.4.2 Shock wave structure 256 10.4.3 Oblique shock waves 259 10.5 FINITE BUBBLE CLOUDS 259 10.5.1 Natural modes of a spherical cloud of bubbles 259 10.5.2 Response of a spherical bubble cloud 264 11 FLOWS WITH GAS DYNAMICS 267 11.1 INTRODUCTION 267 11.2 EQUATIONS FOR A DUSTY GAS 268 11.2.1 Basic equations 268 11.2.2 Homogeneous flow with gas dynamics 269 11.2.3 Velocity and temperature relaxation 271 11.3 NORMAL SHOCK WAVE 272 11.4 ACOUSTIC DAMPING 275 11.5 LINEAR PERTURBATION ANALYSES 279 11.5.1 Stability of laminar flow 279 11.5.2 Flow over a wavy wall 280 11.6 SMALL SLIP PERTURBATION 282 12 SPRAYS 285 12.1 INTRODUCTION 285 12.2 TYPES OF SPRAY FORMATION 285 12.3 OCEAN SPRAY 286 12.4 SPRAY FORMATION 288 12.4.1 Spray formation by bubbling 288 12.4.2 Spray formation by wind shear 289 12.4.3 Spray formation by initially laminar jets 292 12.4.4 Spray formation by turbulent jets 293 12.5 SINGLE DROPLET MECHANICS 299 12.5.1 Single droplet evaporation 299 12.5.2 Single droplet combustion 301 12.6 SPRAY COMBUSTION 305 13 GRANULAR FLOWS 308 13.1 INTRODUCTION 308 13.2 PARTICLE INTERACTION MODELS 309 13.2.1 Computer simulations 311 13.3 FLOW REGIMES 312 13.3.1 Dimensional Analysis 312 13.3.2 Flow regime rheologies 313 13.3.3 Flow regime boundaries 316 8 13.4 SLOW GRANULAR FLOW 317 13.4.1 Equations of motion 317 13.4.2 Mohr-Coulomb models 317 13.4.3 Hopper flows 318 13.5 RAPID GRANULAR FLOW 320 13.5.1 Introduction 320 13.5.2 Example of rapid flow equations 322 13.5.3 Boundary conditions 325 13.5.4 Computer simulations 326 13.6 EFFECT OF INTERSTITIAL FLUID 326 13.6.1 Introduction 326 13.6.2 Particle collisions 327 13.6.3 Classes of interstitial fluid effects 329 14 DRIFT FLUX MODELS 331 14.1 INTRODUCTION 331 14.2 DRIFT FLUX METHOD 332 14.3 EXAMPLES OF DRIFT FLUX ANALYSES 333 14.3.1 Vertical pipe flow 333 14.3.2 Fluidized bed 336 14.3.3 Pool boiling crisis 338 14.4 CORRECTIONS FOR PIPE FLOWS 343 15 SYSTEM INSTABILITIES 344 15.1 INTRODUCTION 344 15.2 SYSTEM STRUCTURE 344 15.3 QUASISTATIC STABILITY 347 15.4 QUASISTATIC INSTABILITY EXAMPLES 349 15.4.1 Turbomachine surge 349 15.4.2 Ledinegg instability 349 15.4.3 Geyser instability 350 15.5 CONCENTRATION WAVES 351 15.6 DYNAMIC MULTIPHASE FLOW INSTABILITIES 353 15.6.1 Dynamic instabilities 353 15.6.2 Cavitation surge in cavitating pumps 354 15.6.3 Chugging and condensation oscillations 356 15.7 TRANSFER FUNCTIONS 359 15.7.1 Unsteady internal flow methods 359 15.7.2 Transfer functions 360 15.7.3 Uniform homogeneous flow 362 16 KINEMATIC WAVES 365 16.1 INTRODUCTION 365 9 16.2 TWO-COMPONENT KINEMATIC WAVES 366 16.2.1 Basic analysis 366 16.2.2 Kinematic wave speed at flooding 368 16.2.3 Kinematic waves in steady flows 369 16.3 TWO-COMPONENT KINEMATIC SHOCKS 370 16.3.1 Kinematic shock relations 370 16.3.2 Kinematic shock stability 372 16.3.3 Compressibility and phase change effects 374 16.4 EXAMPLES OF KINEMATIC WAVE ANALYSES 375 16.4.1 Batch sedimentation 375 16.4.2 Dynamics of cavitating pumps 378 16.5 TWO-DIMENSIONAL SHOCKS 383 Bibliography 385 Index 407 10
Contents
Preface page 2
Contents 10
Nomenclature 11
1 INTRODUCTIONTOMULTIPHASEFLOW 19
1.1 INTRODUCTION 19
1.1.1 Scope 19
1.1.2 Multiphase flow models 20
1.1.3 Multiphase flow notation 22
1.1.4 Size distribution functions 25
1.2 EQUATIONS OF MOTION 27
1.2.1 Averaging 27
1.2.2 Conservation of mass 28
1.2.3 Number continuity equation 30
1.2.4 Fick’s law 31
1.2.5 Equation of motion 31
1.2.6 Disperse phase momentum equation 35
1.2.7 Comments on disperse phase interaction 36
1.2.8 Equations for conservation of energy 37
1.2.9 Heat transfer between separated phases 41
1.3 INTERACTION WITH TURBULENCE 42
1.3.1 Particles and turbulence 42
1.3.2 Effect on turbulence stability 46
1.4 COMMENTS ON THE EQUATIONS OF MOTION 47
1.4.1 Averaging 47
1.4.2 Averaging contributions to the mean motion 48
1.4.3 Averaging in pipe flows 50
1.4.4 Modeling with the combined phase equations 50
1.4.5 Mass, force and energy interaction terms 51
4
2 SINGLE PARTICLE MOTION 52
2.1 INTRODUCTION 52
2.2 FLOWS AROUND A SPHERE 53
2.2.1 At high Reynolds number 53
2.2.2 At low Reynolds number 56
2.2.3 Molecular effects 61
2.3 UNSTEADY EFFECTS 62
2.3.1 Unsteady particle motions 62
2.3.2 Effect of concentration on added mass 65
2.3.3 Unsteady potential flow 65
2.3.4 Unsteady Stokes flow 69
2.4 PARTICLE EQUATION OF MOTION 73
2.4.1 Equations of motion 73
2.4.2 Magnitude of relative motion 78
2.4.3 Effect of concentration on particle equation of motion 80
2.4.4 Effect of concentration on particle drag 81
3 BUBBLE OR DROPLET TRANSLATION 86
3.1 INTRODUCTION 86
3.2 DEFORMATION DUE TO TRANSLATION 86
3.2.1 Dimensional analysis 86
3.2.2 Bubble shapes and terminal velocities 88
3.3 MARANGONI EFFECTS 91
3.4 BJERKNES FORCES 95
3.5 GROWING BUBBLES 97
4 BUBBLE GROWTH AND COLLAPSE 100
4.1 INTRODUCTION 100
4.2 BUBBLE GROWTH AND COLLAPSE 100
4.2.1 Rayleigh-Plesset equation 100
4.2.2 Bubble contents 103
4.2.3 In the absence of thermal effects; bubble growth 106
4.2.4 In the absence of thermal effects; bubble collapse 109
4.2.5 Stability of vapor/gas bubbles 110
4.3 THERMAL EFFECTS 113
4.3.1 Thermal effects on growth 113
4.3.2 Thermally controlled growth 115
4.3.3 Cavitation and boiling 118
4.3.4 Bubble growth by mass diffusion 118
4.4 OSCILLATING BUBBLES 120
4.4.1 Bubble natural frequencies 120
5
4.4.2 Nonlinear effects 124
4.4.3 Rectified mass diffusion 126
5 CAVITATION 128
5.1 INTRODUCTION 128
5.2 KEY FEATURES OF BUBBLE CAVITATION 128
5.2.1 Cavitation inception 128
5.2.2 Cavitation bubble collapse 131
5.2.3 Shape distortion during bubble collapse 133
5.2.4 Cavitation damage 136
5.3 CAVITATION BUBBLES 139
5.3.1 Observations of cavitating bubbles 139
5.3.2 Cavitation noise 142
5.3.3 Cavitation luminescence 149
6 BOILING AND CONDENSATION 150
6.1 INTRODUCTION 150
6.2 HORIZONTAL SURFACES 151
6.2.1 Pool boiling 151
6.2.2 Nucleate boiling 153
6.2.3 Film boiling 154
6.2.4 Leidenfrost effect 155
6.3 VERTICAL SURFACES 157
6.3.1 Film boiling 158
6.4 CONDENSATION 160
6.4.1 Film condensation 160
7 FLOW PATTERNS 163
7.1 INTRODUCTION 163
7.2 TOPOLOGIES OF MULTIPHASE FLOW 163
7.2.1 Multiphase flow patterns 163
7.2.2 Examples of flow regime maps 165
7.2.3 Slurry flow regimes 168
7.2.4 Vertical pipe flow 169
7.2.5 Flow pattern classifications 173
7.3 LIMITS OF DISPERSE FLOW REGIMES 174
7.3.1 Disperse phase separation and dispersion 174
7.3.2 Example: horizontal pipe flow 176
7.3.3 Particle size and particle fission 178
7.3.4 Examples of flow-determined bubble size 179
7.3.5 Bubbly or mist flow limits 181
7.3.6 Other bubbly flow limits 182
6
7.3.7 Other particle size effects 183
7.4 INHOMOGENEITY INSTABILITY 184
7.4.1 Stability of disperse mixtures 184
7.4.2 Inhomogeneity instability in vertical flows 187
7.5 LIMITS ON SEPARATED FLOW 191
7.5.1 Kelvin-Helmoltz instability 192
7.5.2 Stratified flow instability 194
7.5.3 Annular flow instability 194
8 INTERNAL FLOW ENERGY CONVERSION 196
8.1 INTRODUCTION 196
8.2 FRICTIONAL LOSS IN DISPERSE FLOW 196
8.2.1 Horizontal Flow 196
8.2.2 Homogeneous flow friction 199
8.2.3 Heterogeneous flow friction 201
8.2.4 Vertical flow 203
8.3 FRICTIONAL LOSS IN SEPARATED FLOW 205
8.3.1 Two component flow 205
8.3.2 Flow with phase change 211
8.4 ENERGY CONVERSION IN PUMPS AND TURBINES 215
8.4.1 Multiphase flows in pumps 215
9 HOMOGENEOUS FLOWS 220
9.1 INTRODUCTION 220
9.2 EQUATIONS OF HOMOGENEOUS FLOW 220
9.3 SONIC SPEED 221
9.3.1 Basic analysis 221
9.3.2 Sonic speeds at higher frequencies 225
9.3.3 Sonic speed with change of phase 227
9.4 BAROTROPIC RELATIONS 231
9.5 NOZZLE FLOWS 233
9.5.1 One dimensional analysis 233
9.5.2 Vapor/liquid nozzle flow 238
9.5.3 Condensation shocks 242
10 FLOWS WITH BUBBLE DYNAMICS 246
10.1 INTRODUCTION 246
10.2 BASIC EQUATIONS 247
10.3 ACOUSTICS OF BUBBLY MIXTURES 248
10.3.1 Analysis 248
10.3.2 Comparison with experiments 250
10.4 SHOCK WAVES IN BUBBLY FLOWS 253
7
10.4.1 Normal shock wave analysis 253
10.4.2 Shock wave structure 256
10.4.3 Oblique shock waves 259
10.5 FINITE BUBBLE CLOUDS 259
10.5.1 Natural modes of a spherical cloud of bubbles 259
10.5.2 Response of a spherical bubble cloud 264
11 FLOWS WITH GAS DYNAMICS 267
11.1 INTRODUCTION 267
11.2 EQUATIONS FOR A DUSTY GAS 268
11.2.1 Basic equations 268
11.2.2 Homogeneous flow with gas dynamics 269
11.2.3 Velocity and temperature relaxation 271
11.3 NORMAL SHOCK WAVE 272
11.4 ACOUSTIC DAMPING 275
11.5 LINEAR PERTURBATION ANALYSES 279
11.5.1 Stability of laminar flow 279
11.5.2 Flow over a wavy wall 280
11.6 SMALL SLIP PERTURBATION 282
12 SPRAYS 285
12.1 INTRODUCTION 285
12.2 TYPES OF SPRAY FORMATION 285
12.3 OCEAN SPRAY 286
12.4 SPRAY FORMATION 288
12.4.1 Spray formation by bubbling 288
12.4.2 Spray formation by wind shear 289
12.4.3 Spray formation by initially laminar jets 292
12.4.4 Spray formation by turbulent jets 293
12.5 SINGLE DROPLET MECHANICS 299
12.5.1 Single droplet evaporation 299
12.5.2 Single droplet combustion 301
12.6 SPRAY COMBUSTION 305
13 GRANULAR FLOWS 308
13.1 INTRODUCTION 308
13.2 PARTICLE INTERACTION MODELS 309
13.2.1 Computer simulations 311
13.3 FLOW REGIMES 312
13.3.1 Dimensional Analysis 312
13.3.2 Flow regime rheologies 313
13.3.3 Flow regime boundaries 316
8
13.4 SLOW GRANULAR FLOW 317
13.4.1 Equations of motion 317
13.4.2 Mohr-Coulomb models 317
13.4.3 Hopper flows 318
13.5 RAPID GRANULAR FLOW 320
13.5.1 Introduction 320
13.5.2 Example of rapid flow equations 322
13.5.3 Boundary conditions 325
13.5.4 Computer simulations 326
13.6 EFFECT OF INTERSTITIAL FLUID 326
13.6.1 Introduction 326
13.6.2 Particle collisions 327
13.6.3 Classes of interstitial fluid effects 329
14 DRIFT FLUX MODELS 331
14.1 INTRODUCTION 331
14.2 DRIFT FLUX METHOD 332
14.3 EXAMPLES OF DRIFT FLUX ANALYSES 333
14.3.1 Vertical pipe flow 333
14.3.2 Fluidized bed 336
14.3.3 Pool boiling crisis 338
14.4 CORRECTIONS FOR PIPE FLOWS 343
15 SYSTEM INSTABILITIES 344
15.1 INTRODUCTION 344
15.2 SYSTEM STRUCTURE 344
15.3 QUASISTATIC STABILITY 347
15.4 QUASISTATIC INSTABILITY EXAMPLES 349
15.4.1 Turbomachine surge 349
15.4.2 Ledinegg instability 349
15.4.3 Geyser instability 350
15.5 CONCENTRATION WAVES 351
15.6 DYNAMIC MULTIPHASE FLOW INSTABILITIES 353
15.6.1 Dynamic instabilities 353
15.6.2 Cavitation surge in cavitating pumps 354
15.6.3 Chugging and condensation oscillations 356
15.7 TRANSFER FUNCTIONS 359
15.7.1 Unsteady internal flow methods 359
15.7.2 Transfer functions 360
15.7.3 Uniform homogeneous flow 362
16 KINEMATIC WAVES 365
16.1 INTRODUCTION 365
9
16.2 TWO-COMPONENT KINEMATIC WAVES 366
16.2.1 Basic analysis 366
16.2.2 Kinematic wave speed at flooding 368
16.2.3 Kinematic waves in steady flows 369
16.3 TWO-COMPONENT KINEMATIC SHOCKS 370
16.3.1 Kinematic shock relations 370
16.3.2 Kinematic shock stability 372
16.3.3 Compressibility and phase change effects 374
16.4 EXAMPLES OF KINEMATIC WAVE ANALYSES 375
16.4.1 Batch sedimentation 375
16.4.2 Dynamics of cavitating pumps 378
16.5 TWO-DIMENSIONAL SHOCKS 383
Bibliography 385
Index 407
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