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1 Introduction to Multiphase Flow 1

1.1 Introduction 1

1.1.1 Scope 1

1.1.2 Multiphase Flow Models 2

1.1.3 Multiphase Flow Notation 3

1.1.4 Size Distribution Functions 6

1.2 Equations of Motion 8

1.2.1 Averaging 8

1.2.2 Continuum Equations for Conservation of Mass 9

1.2.3 Disperse Phase Number Continuity 10

1.2.4 Fick’s Law 11

1.2.5 Continuum Equations for Conservation of Momentum 12

1.2.6 Disperse Phase Momentum Equation 14

1.2.7 Comments on Disperse Phase Interaction 15

1.2.8 Equations for Conservation of Energy 16

1.2.9 Heat Transfer between Separated Phases 19

1.3 Interaction with Turbulence 21

1.3.1 Particles and Turbulence 21

1.3.2 Effect on Turbulence Stability 24

1.4 Comments on the Equations of Motion 25

1.4.1 Averaging 25

1.4.2 Averaging Contributions to the Mean Motion 26

1.4.3 Averaging in Pipe Flows 27

1.4.4 Modeling with the Combined Phase Equations 28

1.4.5 Mass，Force，and Energy Interaction Terms 28

2 Single-Particle Motion 30

2.1 Introduction 30

2.2 Flows Around a Sphere 31

2.2.1 At High Reynolds Number 31

2.2.2 At Low Reynolds Number 32

2.2.3 Molecular Effects 37

2.3 Unsteady Effects 38

2.3.1 Unsteady Particle Motions 38

2.3.2 Effect of Concentration on Added Mass 41

2.3.3 Unsteady Potential Flow 41

2.3.4 Unsteady Stokes Flow 44

2.4 Particle Equation of Motion 48

2.4.1 Equations of Motion 48

2.4.2 Magnitude of Relative Motion 52

2.4.3 Effect of Concentration on Particle Equation of Motion 53

2.4.4 Effect of Concentration on Particle Drag 55

3 Bubble or Droplet Translation 60

3.1 Introduction 60

3.2 Deformation Due to Translation 60

3.2.1 Dimensional analysis 60

3.2.2 Bubble shapes and terminal velocities 61

3.3 Marangoni Effects 66

3.4 Bjerknes Forces 68

3.5 Growing or Collapsing Bubbles 69

4 Bubble Growth and Collapse 73

4.1 Introduction 73

4.2 Bubble Growth and Collapse 73

4.2.1 Rayleigh-Plesset Equation 73

4.2.2 Bubble Contents 75

4.2.3 In the Absence of Thermal Effects； Bubble Growth 78

4.2.4 In the Absence of Thermal Effects； Bubble Collapse 81

4.2.5 Stability of Vapor/Gas Bubbles 82

4.3 Thermal Effects 84

4.3.1 Thermal Effects on Growth 84

4.3.2 Thermally Controlled Growth 85

4.3.3 Cavitation and Boiling 89

4.3.4 Bubble Growth by Mass Diffusion 89

4.4 Oscillating Bubbles 91

4.4.1 Bubble Natural Frequencies 91

4.4.2 Nonlinear Effects 93

4.4.3 Rectified Mass Diffusion 95

5 Cavitation 97

5.1 Introduction 97

5.2 Key Features of Bubble Cavitation 97

5.2.1 Cavitation Inception 97

5.2.2 Cavitation Bubble Collapse 99

5.2.3 Shape Distortion during Bubble Collapse 101

5.2.4 Cavitation Damage 104

5.3 Cavitation Bubbles 106

5.3.1 Observations of Cavitating Bubbles 106

5.3.2 Cavitation Noise 109

5.3.3 Cavitation Luminescence 115

6 Boiling and Condensation 116

6.1 Introduction 116

6.2 Horizontal Surfaces 117

6.2.1 Pool Boiling 117

6.2.2 Nucleate Boiling 119

6.2.3 Film Boiling 120

6.2.4 Leidenfrost Effect 121

6.3 Vertical Surfaces 122

6.3.1 Film Boiling 122

6.4 Condensation 125

6.4.1 Film Condensation 125

7 Flow Patterns 127

7.1 Introduction 127

7.2 Topologies of Multiphase Flow 127

7.2.1 Multiphase Flow Patterns 127

7.2.2 Examples of Flow Regime Maps 129

7.2.3 Slurry Flow Regimes 131

7.2.4 Vertical Pipe Flow 132

7.2.5 Flow Pattern Classifications 134

7.3 Limits of Disperse Flow Regimes 136

7.3.1 Disperse Phase Separation and Dispersion 136

7.3.2 Example：Horizontal Pipe Flow 138

7.3.3 Particle Size and Particle Fission 140

7.3.4 Examples of Flow-Determined Bubble Size 141

7.3.5 Bubbly or Mist Flow Limits 142

7.3.6 Other Bubbly Flow Limits 143

7.3.7 Other Particle Size Effects 144

7.4 Inhomogeneity Instability 144

7.4.1 Stability of Disperse Mixtures 145

7.4.2 Inhomogeneity Instability in Vertical Flows 148

7.5 Limits on Separated Flow 151

7.5.1 Kelvin-Helmoltz Instability 151

7.5.2 Stratified Flow Instability 153

7.5.3 Annular Flow Instability 154

8 Internal Flow Energy Conversion 155

8.1 Introduction 155

8.2 Frictional Loss in Disperse Flow 155

8.2.1 Horizontal Flow 155

8.2.2 Homogeneous Flow Friction 157

8.2.3 Heterogeneous Flow Friction 159

8.2.4 Vertical Flow 161

8.3 Frictional Loss in Separated Flow 163

8.3.1 Two-Component Flow 163

8.3.2 Flow with Phase Change 168

8.4 Energy Conversion in Pumps and Turbines 172

8.4.1 Multiphase Flows in Pumps 172

9 Homogeneous Flows 176

9.1 Introduction 176

9.2 Equations of Homogeneous Flow 176

9.3 Sonic Speed 177

9.3.1 Basic Analysis 177

9.3.2 Sonic Speeds at Higher Frequencies 181

9.3.3 Sonic Speed with Change of Phase 182

9.4 Barotropic Relations 186

9.5 Nozzle Flows 187

9.5.1 One-Dimensional Analysis 187

9.5.2 Vapor/Liquid Nozzle Flow 192

9.5.3 Condensation Shocks 195

10 Flows with Bubble Dynamics 199

10.1 Introduction 199

10.2 Basic Equations 200

10.3 Acoustics of Bubbly Mixtures 200

10.3.1 Analysis 200

10.3.2 Comparison with Experiments 203

10.4 Shock Waves in Bubbly Flows 205

10.4.1 Shock-wave Analysis 205

10.4.2 Shock-wave Structure 208

10.5 Finite Bubble Clouds 210

10.5.1 Natural Modes of a Spherical Cloud of Bubbles 210

10.5.2 Response of a Spherical Bubble Cloud 214

11 Flows with Gas Dynamics 217

11.1 Introduction 217

11.2 Equations for a Dusty Gas 217

11.2.1 Basic Equations 217

11.2.2 Homogeneous Flow with Gas Dynamics 219

11.2.3 Velocity and Temperature Relaxation 220

11.3 Normal Shock Wave 221

11.4 Acoustic Damping 224

11.5 Other Linear Perturbation Analyses 227

11.5.1 Stability of Laminar Flow 227

11.5.2 Flow over a Wavy Wall 228

11.6 Small Slip Perturbation 229

12 Sprays 232

12.1 Introduction 232

12.2 Types of Spray Formation 232

12.3 Ocean Spray 233

12.4 Spray Formation 234

12.4.1 Spray Formation by Bubbling 234

12.4.2 Spray Formation by Wind Shear 235

12.4.3 Spray Formation by Initially Laminar Jets 237

12.4.4 Spray Formation by Turbulent Jets 239

12.5 Single-Droplet Mechanics 243

12.5.1 Single-Droplet Evaporation 243

12.5.2 Single-Droplet Combustion 245

12.6 Spray Combustion 249

13 Granular Flows 252

13.1 Introduction 252

13.2 Particle Interaction Models 253

13.2.1 Computer Simulations 255

13.3 Flow Regimes 255

13.3.1 Dimensional Analysis 255

13.3.2 Flow Regime Rheologies 256

13.3.3 Flow Regime Boundaries 259

13.4 Slow Granular Flow 259

13.4.1 Equations of Motion 259

13.4.2 Mohr-Coulomb Models 260

13.4.3 Hopper Flows 261

13.5 Rapid Granular Flow 263

13.5.1 Introduction 263

13.5.2 Example of Rapid Flow Equations 264

13.5.3 Boundary Conditions 267

13.5.4 Computer Simulations 267

13.6 Effect of Interstitial Fluid 268

13.6.1 Introduction 268

13.6.2 Particle Collisions 268

13.6.3 Classes of Interstitial Fluid Effects 270

14 Drift Flux Models 272

14.1 Introduction 272

14.2 Drift Flux Method 273

14.3 Examples of Drift Flux Analyses 274

14.3.1 Vertical Pipe Flow 274

14.3.2 Fluidized Bed 276

14.3.3 Pool Boiling Crisis 278

14.4 Corrections for Pipe Flows 282

15 System Instabilities 284

15.1 Introduction 284

15.2 System Structure 284

15.3 Quasistatic Stability 286

15.4 Quasistatic Instability Examples 288

15.4.1 Turbomachine Surge 288

15.4.2 Ledinegg Instability 288

15.4.3 Geyser Instability 289

15.5 Concentration Waves 290

15.6 Dynamic Multiphase Flow Instabilities 292

15.6.1 Dynamic Instabilities 292

15.6.2 Cavitation Surge in Cavitating Pumps 292

15.6.3 Chugging and Condensation Oscillations 293

15.7 Transfer Functions 297

15.7.1 Unsteady Internal Flow Methods 297

15.7.2 Transfer Functions 298

15.7.3 Uniform Homogeneous Flow 300

16 Kinematic Waves 302

16.1 Introduction 302

16.2 Two-Component Kinematic Waves 303

16.2.1 Basic Analysis 303

16.2.2 Kinematic Wave Speed at Flooding 304

16.2.3 Kinematic Waves in Steady Flows 305

16.3 Two-Component Kinematic Shocks 306

16.3.1 Kinematic Shock Relations 306

16.3.2 Kinematic Shock Stability 308

16.3.3 Compressibility and Phase-Change Effects 309

16.4 Examples of Kinematic Wave Analyses 311

16.4.1 Batch Sedimentation 311

16.4.2 Dynamics of Cavitating Pumps 313

16.5 Two-Dimensional Kinematic Waves 318

Bibliography 321

Index 341