Computational Techniques for Multiphase Flows

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Computational Techniques For Multi-Phase Flows

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Contents

Preface

Chapter 1. Introduction

1.1 CLASSIFICATION AND PHENOMENOLOGICAL DISCUSSION

1.2 TYPICAL PRACTICAL PROBLEMS INVOLVING MULTI-PHASE FLOWS

1.3 COMPUTATIONAL FLUID DYNAMICS AS A RESEARCH TOOL FOR MULTI-PHASE FLOWS

1.4 COMPUTATIONAL FLUID DYNAMICS AS A DESIGN TOOL FOR MULTI-PHASE FLOWS

1.5 IMPACT OF MULTI-PHASE FLOW STUDY ON COMPUTATIONAL FLUID DYNAMICS

1.6 SCOPE OF THE BOOK

Chapter 2. Governing Equations and Boundary Conditions

2.1 BACKGROUND OF DIFFERENT APPROACHES

2.2 AVERAGING PROCEDURE FOR MULTI-PHASE FLOW

2.3 EQUATIONS OF MOTION FOR CONTINUOUS PHASE

2.3.1 Conservation of Mass

2.3.2 Conservation of Momentum

2.3.3 Conservation of Energy

2.3.4 Interfacial Transport

2.3.5 Effective Conservation Equations

2.4 COMMENTS AND OBSERVATIONS ON THE GOVERNING EQUATIONS FOR THE TWO-FLUID MODELLING APPROACH

2.5 EQUATIONS OF MOTION FOR DISPERSE PHASE

2.6 TURBULENCE IN TRANSPORT PHENOMENA

2.6.1 Reynolds-Averaged Equations

2.6.2 Reynolds-Averaged Closure

2.6.3 Some Comments on the k–ε Model and Implications of Other Turbulence Models

2.6.3.1 Shear Stress Transport (SST) Model

2.6.3.2 Reynolds Stress Model

2.6.3.3 Near-Wall Treatment

2.6.4 Some Comments on Turbulence Modelling of the Disperse Phase

2.7 DIFFERENTIAL AND INTEGRAL FORM OF THE TRANSPORT EQUATIONS

2.7.1 A Comment on Multi-Fluid Model

2.8 BOUNDARY CONDITIONS AND THEIR PHYSICAL INTERPRETATION

2.8.1 Comments on Some Wall Boundary Conditions for Multi-Phase Problems

2.9 SUMMARY

Chapter 3. Solution Methods for Multi-Phase Flows

3.1 INTRODUCTION

3.2 MESH SYSTEMS: CONSIDERATION FOR A RANGE OF MULTI-PHASE FLOW PROBLEMS

3.2.1 Application of Structured Mesh

3.2.2 Application of Body-Fitted Mesh

3.2.3 Application of Unstructured Mesh

3.2.4 Some Comments on Grid Generation

3.3 EULERIAN–EULERIAN FRAMEWORK: NUMERICAL ALGORITHMS

3.3.1 Basic Aspects of Discretization – Finite Difference Method

3.3.2 Basic Aspects of Discretization – Finite-Volume Method

3.3.3 Basic Approximation of the Diffusion Term Based upon the Finite-Volume Method

3.3.4 Basic Approximation of the Advection Term Based upon the Finite-Volume Method

3.3.5 Some Comments on the Need for TVD Schemes

3.3.6 Explicit and Implicit Approaches

3.3.7 Assembly of Discretized Equations

3.3.8 Comments on the Linearization of Source Terms

3.4 SOLUTION ALGORITHMS

3.4.1 The Philosophy Behind the Pressure-Correction Techniques for Multi-Phase Problems

3.4.1.1 SIMPLE Algorithm for Mixture or Homogeneous Flows

3.4.1.2 A Comment on Other Pressure-Correction Methods

3.4.1.3 Evaluation of the Face Velocity in Different Mesh Systems

3.4.1.4 Iterative Procedure Based on the SIMPLE Algorithm

3.4.1.5 IPSA for Multi-Phase Flows

3.4.1.6 IPSA-C for Multi-Phase Flows

3.4.1.7 Comments on the Need for Improved Interpolation Methods of Evaluating the Face Velocity in Multi-Phase Problems

3.4.2 Matrix Solvers for the Segregated Approach in Different Mesh Systems

3.4.3 Coupled Equation System

3.5 EULERIAN–LAGRANGIAN FRAMEWORK: NUMERICAL AND SOLUTION ALGORITHMS

3.5.1 Basic Numerical Techniques

3.5.2 Comments on Sampling Particulates for Turbulent Dispersion

3.5.3 Some Comments on Attaining Proper Statistical Realizations

3.5.4 Evaluation of Source Terms for the Continuous Phase

3.6 INTERFACE-TRACKING/CAPTURING ALGORITHMS: BASIC CONSIDERATIONS OF INTERFACE-TRACKING/CAPTURING METHODS

3.6.1 Algorithms Based on Surface Methods: with Comments

3.6.1.1 Markers on Interface (Surface Marker Techniques)

3.6.1.2 Surface-Fitted Method

3.6.2 Algorithms Based on Volume Methods: with Comments

3.6.2.1 Markers in Fluid (MAC Formulation)

3.6.2.2 Volume of Fluid (VOF)

3.6.2.3 Level Set Method

3.6.2.4 Hybrid Methods

3.6.3 Computing Surface Tension and Wall Adhesion

3.7 SUMMARY

Chapter 4. Gas–Particle Flows

4.1 INTRODUCTION

4.1.1 Background

4.1.2 Classification of Gas–Particle Flows

4.1.3 Particle Loading and Stokes Number

4.1.4 Particle Dispersion Due to Turbulence

4.2 MULTI-PHASE MODELS FOR GAS–PARTICLE FLOWS

4.2.1 Eulerian–Lagrangian Framework

4.2.2 Eulerian–Eulerian Framework

4.2.3 Turbulence Modelling

4.2.3.1 Gas Phase

4.2.3.2 Particle Phase in Lagrangian Reference Frame

4.2.3.3 Particle Phase in Eulerian Reference Frame

4.2.4 Particle–Wall Collision Model

4.2.4.1 Lagrangian Reference Frame

4.2.4.2 Eulerian Reference Frame

4.3 WORKED EXAMPLES

4.3.1 Dilute Gas–Particle Flow over a Two-Dimensional Backwards Facing Step

4.3.1.1 Numerical Features

4.3.1.2 Numerical Results

4.3.1.3 Conclusion

4.3.2 Dilute Gas–Particle Flow in a Three-Dimensional 90° Bend

4.3.2.1 Numerical Features

4.3.2.2 Numerical Results

4.3.2.3 Conclusion

4.3.3 Dilute Gas–Particle Flow Over an In-Line Tube Bank

4.3.3.1 Numerical Features

4.3.3.2 Numerical Results

4.3.3.3 Conclusion

4.4 SUMMARY

Chapter 5. Liquid–Particle Flows

5.1 INTRODUCTION

5.1.1 Background

5.1.2 Some Physical Characteristics of Flow in Sedimentation Tank

5.1.3 Some Physical Characteristics of Slurry Transport

5.2 MULTI-PHASE MODELS FOR LIQUID–PARTICLE FLOWS

5.2.1 Mixture Model

5.2.1.1 Modelling Source or Sink Terms for Flow in Sedimentation Tank

5.2.1.2 Modelling Source or Sink Terms for Flow in Slurry Transportation

5.2.2 Turbulence Modelling

5.3 WORKED EXAMPLES

5.3.1 Liquid–Particle Flows in Sedimentation Tank

5.3.1.1 Numerical Features

5.3.1.2 Numerical Results

5.3.1.3 Conclusion

5.3.2 Sand–Water Slurry Flow in a Horizontal Straight Pipe

5.3.2.1 Numerical Features

5.3.2.2 Numerical Results

5.3.2.3 Conclusion

5.4 SUMMARY

Chapter 6. Gas–Liquid Flows

6.1 INTRODUCTION

6.1.1 Background

6.1.2 Categorization of Different Flow Regimes

6.1.3 Some Physical Characteristics of Boiling Flow

6.2 MULTI-PHASE MODELS FOR GAS–LIQUID FLOWS

6.2.1 Multi-Fluid Model

6.2.1.1 Inter-Phase Mass Transfer

6.2.1.2 Inter-Phase Momentum Transfer

6.2.1.3 Inter-Phase Heat Transfer

6.2.2 Turbulence Modelling

6.3 POPULATION BALANCE APPROACH

6.3.1 Need for Population Balance in Gas–Liquid Flows

6.3.2 Population Balance Equation (PBE)

6.3.3 Method of Moments (MOM)

6.3.3.1 Quadrature Method of Moments (QMOM)

6.3.3.2 Direct Quadrature Method of Moments (DQMOM)

6.3.4 Class Methods (CM)

6.3.4.1 Average Quantities Approach

6.3.4.2 Multiple Size Group Model

6.4 BUBBLE INTERACTION MECHANISMS

6.4.1 Single Average Scalar Approach for Bubbly Flows

6.4.1.1 Wu et al. (1998) Model

6.4.1.2 Hibiki and Ishii (2002) Model

6.4.1.3 Yao and Morel (2004) Model

6.4.2 Multiple Bubble Size Approach for Bubbly Flows

6.4.2.1 DQMOM Model

6.4.2.2 MUSIG Model

6.4.3 Comments of Other Coalescence and Break-Up Kernels

6.4.4 Modelling Beyond Bubbly Flows – A Phenomenological Consideration

6.5 MODELLING SUB-COOLED BOILING FLOWS

6.5.1 Review of Current Model Applications

6.5.2 Phenomenological Description

6.5.3 Nucleation of Bubbles at Heated Walls

6.5.4 Condensation of Bubbles in Sub-Cooled Liquid

6.6 WORKED EXAMPLES

6.6.1 Dispersed Bubbly Flow in a Rectangular Column

6.6.1.1 Numerical Features

6.6.1.2 Numerical Results

6.6.1.3 Conclusion

6.6.2 Bubbly Flow in a Vertical Pipe

6.6.2.1 Numerical Features

6.6.2.2 Numerical Results

6.6.2.3 Conclusion

6.6.3 Sub-Cooled Boiling Flow in a Vertical Annulus

6.6.3.1 Application of MUSIG Boiling Model

6.6.3.2 Application of Improved Wall Heat Partition Model

6.7 SUMMARY

Chapter 7. Free Surface Flows

7.1 INTRODUCTION

7.2 MULTI-PHASE MODELS FOR FREE SURFACE FLOWS

7.3 RELEVANT WORKED EXAMPLES

7.3.1 Bubble Rising in a Viscous Liquid

7.3.1.1 Numerical Features

7.3.1.2 Numerical Results

7.3.1.3 Conclusion

7.3.2 Single Taylor Bubble

7.3.2.1 Numerical Features

7.3.2.2 Numerical Results

7.3.2.3 Conclusion

7.3.3 Collapse of a Liquid Column (Breaking Dam Problem)

7.3.3.1 Numerical Features

7.3.3.2 Numerical Results

7.3.3.3 Conclusion

7.3.4 Sloshing of Liquid

7.3.4.1 Numerical Features

7.3.4.2 Numerical Results

7.3.4.3 Similar Comparison for the Roll Motion Cases

7.3.4.4 Conclusion

7.4 SUMMARY

Chapter 8. Freezing/Solidification

8.1 INTRODUCTION

8.2 MATHEMATICAL FORMULATION

8.2.1 Governing Equations

8.2.2 Solid–Liquid Interface

8.2.3 Other Boundary Conditions

8.3 NUMERICAL PROCEDURE

8.3.1 Internal Grid Generation

8.3.2 Surface Grid Generation

8.3.3 Optimizing Computational Meshes

8.3.3.1 Objective Function

8.3.3.2 Optimization Algorithm

8.3.4 Transformation of Governing Equations and Boundary Conditions

8.4 WORKED EXAMPLES

8.4.1 Freezing of Water on a Vertical Wall in an Enclosed Cubical Cavity

8.4.1.1 Numerical Features

8.4.1.2 Numerical Results

8.4.1.3 Conclusion

8.4.2 Freezing of Water in an Open Cubical Cavity

8.4.2.1 Numerical Features

8.4.2.2 Numerical Results

8.4.2.3 Conclusion

8.5 SUMMARY

Chapter 9. Three-Phase Flows

9.1 INTRODUCTION

9.2 DESCRIPTION OF PROBLEM IN THE CONTEXT OF COMPUTATIONAL FLUID DYNAMICS

9.3 MODELLING APPROACHES FOR GAS–LIQUID–SOLID FLOWS

9.3.1 Three-Fluid Model

9.3.2 Turbulence Modelling

9.4 EVALUATION OF MULTI-PHASE MODELS FOR GAS–LIQUID–SOLID FLOWS

9.4.1 Three-Phase Modelling of the Air-Lift Pump

9.4.1.1 Numerical Features

9.4.1.2 Numerical Results

9.4.1.3 Conclusion

9.4.2 Modelling of Three-Phase Mechanically Agitated Reactor

9.4.2.1 Numerical Features

9.4.2.2 Numerical Results

9.4.2.3 Conclusion

9.5 SUMMARY

Chapter 10. Future Trends in Handling Turbulent Multi-Phase Flows

10.1 INTRODUCTION

10.2 DIRECT NUMERICAL SIMULATION OF MULTI-PHASE FLOWS

10.2.1 Model Description

10.3 LARGE EDDY SIMULATION OF MULTI-PHASE FLOWS

10.3.1 Model Description

10.3.1.1 Basic SGS Model

10.3.1.2 Dynamics SGS Model

10.4 ON MODELLING GAS–LIQUID–SOLID FLUIDIZATION

10.4.1 Governing Equations

10.4.2 Interface Tracking/Capturing Methods: with Comments

10.4.3 Discrete Particle Model

10.4.4 Particle–Particle Collision

10.4.5 Inter-Phase Couplings

10.4.6 Simulation Results

10.5 SOME CONCLUDING REMARKS

Appendix A. Full Derivation of Conservation Equations

References

Index

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

Y