Buoyancy-Thermocapillary Convection of Volatile Fluids in Confined and Sealed Geometries

by Tongran Qin

Overview

This thesis represents the first systematic description of the two-phase flow problem. Two-phase flows of volatile fluids in confined geometries driven by an applied temperature gradient play an important role in a range of applications, including thermal management, such as heat pipes, thermosyphons, capillary pumped loops and other evaporative cooling devices.  Previously, this problem has been addressed using a piecemeal approach that relied heavily on correlations and unproven assumptions, and the science and technology behind heat pipes have barely evolved in recent decades. The model introduced in this thesis, however, presents a comprehensive physically based description of both the liquid and the gas phase.

The model has been implemented numerically and successfully validated against the available experimental data, and the numerical results are used to determine the key physical processes that control the heat and mass flow and describe the flow stability. One of the key contributions of this thesis work is the description of the role of noncondensables, such as air, on transport. In particular, it is shown that many of the assumptions used by current engineering models of evaporative cooling devices are based on experiments conducted at atmospheric pressures, and these assumptions break down partially or completely when most of the noncondensables are removed, requiring a new modeling approach presented in the thesis.

Moreover, Numerical solutions are used to motivate and justify a simplified analytical description of transport in both the liquid and the gas layer, which can be used to describe flow stability and determine the critical Marangoni number and wavelength describing the onset of the convective pattern. As a result, the results presented in the thesis should be of interest both to engineers working in heat transfer and researchers interested in fluid dynamics and pattern formation.

Product Details

ISBN-13:	9783319613314
Publisher:	Springer-Verlag New York, LLC
Publication date:	07/25/2017
Series:	Springer Theses
Sold by:	Barnes & Noble
Format:	eBook
Pages:	209
File size:	4 MB

About the Author

Dr Tongran Qin was awarded a PhD degree by Georgia Institute of Technology in 2015.

Table of Contents

CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Previous Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Buoyancy-Thermocapillary Convection . . . . . . . . . . . . . . 3

1.2.2 Modeling of Two-Phase Cooling Devices . . . . . . . . . . . . . 7

1.2.3 Effect of Noncondensable Gases . . . . . . . . . . . . . . . . . . 14

1.2.4 Direct Numerical Simulation of Two-Phase Flows . . . . . . . . . 18

1.3 Objectives of This Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

CHAPTER 2 MATHEMATICAL MODEL . . . . . . . . . . . . . . . . . . . . 24

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Fundamental Equations for Simple Fluid Flow . . . . . . . . . . . . . . . 25

2.3 Fundamental Equations for Binary Fluid Flow . . . . . . . . . . . . . . . 26

2.4 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.1 Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.2 Molecular Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.3 Material Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Simplified Form of the Governing Equations . . . . . . . . . . . . . . . . 33

2.6 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.6.1 Liquid-Gas Interface . . . . . . . . . . . . . . . . . . . . . . . . 35

2.6.2 Inner Surface of the Cavity . . . . . . . . . . . . . . . . . . . . . 42

2.6.3 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 42

CHAPTER 3 CONVECTION AT ATMOSPHERIC CONDITIONS . . . . . . 44

3.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1.1 Steady Unicellular and Multicellular Flow . . . . . . . . . . . . . 48

3.1.2 Oscillatory Multicellular Flow . . . . . . . . . . . . . . . . . . . 52

3.2 Comparison with Experiments . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 The Effect of the Interfacial Curvature . . . . . . . . . . . . . . . . . . . 59

3.4 Three-Dimensional Effects . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.5 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5.1 Fluid Flow and Temperature in the Liquid Layer . . . . . . . . . . 71

3.5.2 Fluid Flow, Temperature, and Composition in the Gas Layer . . . 73

3.6 End Wall Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.7 Validity of the One-Sided Models . . . . . . . . . . . . . . . . . . . . . . 79

3.7.1 Interface Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.7.2 Phase Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.7.3 Newton’s Law of Cooling . . . . . . . . . . . . . . . . . . . . . . 82

3.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

CHAPTER 4 CONVECTION UNDER PURE VAPOR . . . . . . . . . . . . . 88

4.1 Fluid Flow and Temperature Fields . . . . . . . . . . . . . . . . . . . . . 89

4.2 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.2.1 Interfacial Temperature . . . . . . . . . . . . . . . . . . . . . . . 92

4.2.2 Thermal Boundary Layer Thickness . . . . . . . . . . . . . . . . 97

4.2.3 Interfacial Flow Speed . . . . . . . . . . . . . . . . . . . . . . . 98

4.2.4 Newton’s Law of Cooling . . . . . . . . . . . . . . . . . . . . . . 99

4.3 Comparison of Different Phase Change Models . . . . . . . . . . . . . . 101

4.4 Dependence on the Accommodation Coeffcient . . . . . . . . . . . . . . 105

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

CHAPTER 5 CONVECTION AT REDUCED PRESSURES . . . . . . . . . . 110

5.1 Solutions in the Bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.1.1 Flow Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.1.2 Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.1.3 Concentration Field . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.1.4 Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2 Solutions at the Interface . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.2.1 The Temperature and Velocity Profiles . . . . . . . . . . . . . . . 120

5.2.2 Mass Flux Due to Phase Change . . . . . . . . . . . . . . . . . . 123

5.2.3 The Concentration Profile . . . . . . . . . . . . . . . . . . . . . . 126

5.3 Mass and Heat Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.3.1 Mass Flux of Vapor . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.3.2 Thick Liquid Layers . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.3.3 Thin Liquid Layers . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

CHAPTER 6 LINEAR STABILITY ANALYSIS . . . . . . . . . . . . . . . . . 145

6.1 Governing Equations and the Base Solution . . . . . . . . . . . . . . . . 145

6.2 Boundary Conditions for the Perturbations . . . . . . . . . . . . . . . . . 147

6.3 Evolution Equations for the Perturbations . . . . . . . . . . . . . . . . . . 150

6.4 Comparison with Experimental and Numerical Results . . . . . . . . . . . 152

6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . 165

7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.2 Main Contributions of This Work . . . . . . . . . . . . . . . . . . . . . . 170

7.3 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

APPENDIX A DIMENSIONLESS PARAMETERS . . . . . . . . . . . . . . . . 175

APPENDIX B NUMERICAL IMPLEMENTATION . . . . . . . . . . . . . . . 177

B.1 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

B.1.1 Discretization of the Solution Domain . . . . . . . . . . . . . . . 179

B.1.2 Discretization of the Equations . . . . . . . . . . . . . . . . . . . 180

B.1.3 Discretization in Time . . . . . . . . . . . . . . . . . . . . . . . . 183

B.1.4 Linearization of the Transport Equation . . . . . . . . . . . . . . 184

B.1.5 Discretization of the Boundary Conditions . . . . . . . . . . . . . 185

B.2 Moving Mesh Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

B.2.1 Finite Volume Method on the Moving Mesh . . . . . . . . . . . . 188

B.2.2 Mesh Motion Solver . . . . . . . . . . . . . . . . . . . . . . . . 189

B.2.3 Boundary Conditions at the Interface . . . . . . . . . . . . . . . . 193

B.3 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

B.3.1 Pressure-Velocity Coupling . . . . . . . . . . . . . . . . . . . . . 200

B.3.2 Temperature and Density Field . . . . . . . . . . . . . . . . . . . 202

B.3.3 Sequence of Solution . . . . . . . . . . . . . . . . . . . . . . . . 203

B.3.4 Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . 205

APPENDIX C PROCEDURE FOR NUMERICAL SIMULATIONS . . . . . . 207

C.1 Problem Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

C.2 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

C.2.1 Volume Mesh Generation . . . . . . . . . . . . . . . . . . . . . . 209

C.2.2 Surface Mesh Generation . . . . . . . . . . . . . . . . . . . . . . 211

C.3 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . 212

C.4 Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

C.5 Control Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

C.6 Discretization Schemes and the Linear Solver . . . . . . . . . . . . . . . 219

C.7 Mesh Motion Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

C.8 Running the Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

C.9 Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230