Contents - Fundamentals of Multiphase Flow

FUNDAMENTALS OF MULTIPHASE FLOW

Preface 2
Nomenclature 11

CHAPTER 1. INTRODUCTION TO MULTIPHASE FLOW
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
CHAPTER 2. SINGLE PARTICLE MOTION
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
CHAPTER 3. BUBBLE OR DROPLET TRANSLATION
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
CHAPTER 4. BUBBLE GROWTH AND COLLAPSE
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 121 120
4.4.1 Bubble natural frequencies 120
4.4.2 Nonlinear effects 124
4.4.3 Rectified mass diffusion 126
CHAPTER 5. CAVITATION
5.1 INTRODUCTION 128
5.1.1 Cavitation inception 128
5.1.2 Cavitation bubble collapse 131
5.1.3 Shape distortion during bubble collapse 133
5.1.4 Cavitation damage 136
5.2 CAVITATION BUBBLES 139
5.2.1 Observations of cavitating bubbles 139
5.2.2 Cavitation noise 142
5.2.3 Cavitation luminescence 149
CHAPTER 6. BOILING AND CONDENSATION
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
CHAPTER 7. FLOW PATTERNS
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
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
CHAPTER 8. INTERNAL FLOW ENERGY CONVERSION
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
CHAPTER 9. HOMOGENEOUS FLOWS
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
CHAPTER 10. FLOWS WITH BUBBLE DYNAMICS
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
10.4.1 Normal shock was 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
CHAPTER 11. FLOWS WITH GAS DYNAMICS
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
CHAPTER 12. SPRAYS
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
CHAPTER 13. GRANULAR FLOWS
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
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
CHAPTER 14. DRIFT FLUX MODELS
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
CHAPTER 15. SYSTEM INSTABILITIES
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
CHAPTER 16. KINEMATIC WAVES
16.1 INTRODUCTION 365
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

	Preface		2
	Nomenclature		11

CHAPTER 1.	INTRODUCTION TO MULTIPHASE FLOW
	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
CHAPTER 2.	SINGLE PARTICLE MOTION
	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
CHAPTER 3.	BUBBLE OR DROPLET TRANSLATION
	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
CHAPTER 4.	BUBBLE GROWTH AND COLLAPSE
	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 121		120
		4.4.1 Bubble natural frequencies	120
		4.4.2 Nonlinear effects	124
		4.4.3 Rectified mass diffusion	126
CHAPTER 5.	CAVITATION
	5.1 INTRODUCTION		128
		5.1.1 Cavitation inception	128
		5.1.2 Cavitation bubble collapse	131
		5.1.3 Shape distortion during bubble collapse	133
		5.1.4 Cavitation damage	136
	5.2 CAVITATION BUBBLES		139
		5.2.1 Observations of cavitating bubbles	139
		5.2.2 Cavitation noise	142
		5.2.3 Cavitation luminescence	149
CHAPTER 6.	BOILING AND CONDENSATION
	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
CHAPTER 7.	FLOW PATTERNS
	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
		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
CHAPTER 8.	INTERNAL FLOW ENERGY CONVERSION
	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
CHAPTER 9.	HOMOGENEOUS FLOWS
	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
CHAPTER 10.	FLOWS WITH BUBBLE DYNAMICS
	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
		10.4.1 Normal shock was 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
CHAPTER 11.	FLOWS WITH GAS DYNAMICS
	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
CHAPTER 12.	SPRAYS
	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
CHAPTER 13.	GRANULAR FLOWS
	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
	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
CHAPTER 14.	DRIFT FLUX MODELS
	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
CHAPTER 15.	SYSTEM INSTABILITIES
	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
CHAPTER 16.	KINEMATIC WAVES
	16.1 INTRODUCTION		365
	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