Analysis of Transport Models and Computation Algorithms for Flow Through Porous Media

ANALYSIS OF TRANSPORT MODELS AND COMPUTATION ALGORITHMS FOR FLOW THROUGH POROUS MEDIA

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of The Ohio State Univeristy

Bader Shabeeb Al-Azmi, M.S.

*****

The Ohio State University 2003

Dissertation Committee: Approved by

Professor Kambiz Vafai, Adviser

Professor Robert Henry Essenhigh ______Professor Sudhir Sastry Adviser Department of Mechanical Engineering

ABSTRACT

Computational investigation of variant models and boundary conditions in the area of fluid flow and heat transfer in porous media is presented in this dissertation.

This study is divided into four major parts. In the first part, a summary of variant models of fluid flow and heat transfer through porous media found in the literature is presented. These variances can be categorized into four primary sections, constant porosity, variable porosity, thermal dispersion and local thermal non-equilibrium.

Models for constant porosity and variable porosity are presented in terms of Darcy,

Brinkman and Forchheimer terms. The second part focuses on the interfacial conditions between a porous layer and a fluid layer. It is found that these interface conditions can be classified as a slip or a no slip. The no slip conditions assume a continuity of the property, velocity and/or temperature, while a discontinuity at the interface is assumed for the slip interface conditions. It is shown that in general, the variances have a more pronounced effect on the velocity field and a substantially smaller effect on the temperature field and even a smaller effect on the Nusselt number distributions. When constant heat flux boundary conditions are present, it is found that researchers use inconsistent wall temperature and heat flux boundary conditions at the solid walls of the porous medium. Therefore, the third topic of this

ii dissertation explores the problem of constant heat flux boundary conditions in porous media under local thermal non-equilibrium conditions. For the second and the third parts, correlations that relate various models to each other are presented. The fourth and last part deals with the additional effects of variable porosity, thermal dispersion and local thermal non-equilibrium to the problem of free surface flows in porous media. Results show that the involvement of these effects can be significant for some cases. The finite difference method is used in generating all numerical results in this study. Throughout this study pertinent parameters such as porosity, Darcy number,

Reynolds number, inertia parameter, particle diameter and solid-to-fluid conductivity ratio are used to demonstrate the results of the analyses.

iii

DEDICATED TO MY PARENTS,

MY WIFE AND MY CHILDREN

ACKNOWLEDGMENTS

First of all, I would like to express my gratitude and indebtedness to my advisor Prof. Vafai for his guidance throughout the course of this study. I really appreciate his encouragement, patience, expertise and unlimited trust. I learned from him how to conduct research but most importantly he taught me other great values that

I can use in my life. Prof. Vafai, it was a great honor to work with you.

I also would like to thank Prof. Essenhigh, Prof. Sastry and Prof. Rich for accepting to participate as members in my graduate committee. Their comments were helpful throughout the period of my study. Particularly I thank Prof. Essenhigh who also served in my M.S. committee.

Special thanks go to my sponsor Kuwait University for providing me with the financial support. Members of the department of mechanical engineering at Kuwait

University who helped me in different ways should not be forgotten specially Dr. Ali

Chamkha, Dr. Mohammed Al-Rifae and Dr. Mohammed Al-Fares. Help from my colleagues Dr. Khalil Khanafer, Ahmad Ali and Abdulraheem Khaled is really

v appreciated. Discussions with Dr. Khalil Khanafer were very helpful in many occasions.

Words can not explain my indebtedness to my father. His continuous love and encouragement gave me a great motivation to pursue my graduate studies. This dissertation is mainly dedicated to my best friend, my father. I know it would not be possible to finish this work without my mother’s prayers. I thank you for raising me and taking care of me especially during my high school exams. The only thing I regret is being away from you and my father for the past six years. Also, special thanks go to my brothers and sisters in Kuwait. I thank my brothers Nawaf, Anwar, Saad and

Shrai’an for their continuous encouragement and support.

Last but not least, I would like to express my deep gratitude to my little family, my beloved wife, my sweetheart Sarah and my son Mohammed. It is for my wife that

I am mostly grateful. I really appreciate your incredible and continuous sacrifices.

Your presence and endless help gave me a huge momentum to continue my study. I thank my daughter Sarah and my son Mohammed, who was born during my study, for being the greatest inspiration in my life.

VITA

February 14, 1974 …………………….…...... Born- Kuwait City, Kuwait

June 1996 ………………………………………...... B.S. Mechanical Engineering, Kuwait University

August 1996 – September 1997 ………………….... Teacher Assistant at Kuwait, University

September 1999 ……………………...……..…...…. M.S. Mechanical Engineering, The Ohio State University

September 1999 – Present …………………………. Ph.D. student at The Ohio State University

PUBLICATIONS

Research Publications

1. Alazmi, B., and Vafai, K., 2000, “Analysis of Variants within the Porous Media Transport Models,” J. Heat Transfer, 122, pp. 303-326.

vii 2. Alazmi, B., and Vafai, K., 2001, “Analysis of Fluid Flow and Heat Transfer Interfacial Conditions Between a Porous Medium and a Fluid Layer,” Int. J. Heat Mass Transfer, 44, pp. 1735-1749.

3. Alazmi, B., and Vafai, K., 2002, “Constant wall heat flux boundary conditions in porous media under local thermal non-equilibrium conditions,” Int. J. Heat Mass Transfer, 45, pp. 3071-3087.

FIELDS OF STUDY

Major Field: Mechanical Engineering

viii

TABLE OF CONTENTS

Page

ABSTRACT …………………………………………………………………...... ii DEDICATION ………………………………………………………………...... iv ACKNOWLEDGMENTS …………………………………………………..….. v VITA …………………………………………………………………………..…. vii LIST OF TABLES ………………………………………………………..…….. xii LIST OF FIGURES …………………………………………………….….…… xiv NOMENCLATURE …………………………………………………….………. xxii

CHAPTERS:

1. INTRODUCTION …………………………………………………………..... 1

2. ANALYSIS OF VARIANTS WITHIN THE POROUS MEDIA TRANSPORT MODELS ……………………………………………..…….. 5 2.1 Introduction ………………………………………….....……….... 5 2.2 Constant porosity ………………………………………….....…… 10 2.3 Variable porosity ………………………………………………..... 13 2.4 Thermal dispersion …………………………………………...…... 18 2.5 Local thermal non-equilibrium ………………………………...... 22

ix 3. ANALYSIS OF FLUID FLOW AND HEAT TRANSFER INTERFACIAL CONDITIONS BETWEEN A POROUS MEDIUM AND A FLUID LAYER ………………...………………………. 26 3.1 Introduction ……………………………………………………………... 27 3.2 Analysis ……………...... 29 3.3 Results and discussion ...... 35 3.3.1 Fluid flow …………………………………………………. 36 3.3.2 Heat transfer ………………………………………………. 58 3.4 Conclusions ……………………………………………………………... 69

4. CONSTANT WALL HEAT FLUX BOUNDARY CONDITIONS IN POROUS MEDIA UNDER LOCAL THERMAL NON-EQUILIBRIUM CONDITIONS ………………………………………71 4.1 Introduction ……………………………………..………………... 72 4.2 Analysis ………………………………..…………………………. 74 4.3 Numerical methodology ………………………………………..… 82 4.4 Results and discussion …………………………………………… 82 4.4.1 Constant porosity with no thermal dispersion …………..….. 86 4.4.2 Variable porosity with thermal dispersion …………….…..…107 4.5 Conclusions ………………………………………………..……… 120

5. ANALYSIS OF VARIABLE POROSITY, THERMAL DISPERSION AND LOCAL THERMAL

NON-EQUILIBRIUM ON FREE SURFACE FLOWS THROUGH POROUS MEDIA ……………………….…...... 121 5.1 Introduction ………………………………………………………. 122 5.2 Analysis ……………………………………………………..……. 123 5.3 Numerical solution ………………………………………..……… 130 5.4 Results and discussion ……………………………………..………133 5.4.1 Constant porosity …………………………………………... 135

x 5.4.2 Variable porosity …………………………………...... …..… 139 5.4.3 Thermal dispersion ………………………..……………..…. 143 5.4.4 Local thermal non-equilibrium …..…………………...... ……160 5.5 Conclusions ………………………………….……………..…..… 171

6. CONCLUSIONS ...... 172

LIST OF REFERENCES ………………………………………………………. 176

LIST OF TABLES

Table Page

2.1 Relationship between various models and the pertinent literature for the constant porosity category …………………….……. 12

2.2 Different models of the constant porosity category ...... 12

2.3 Relationship between various models and the pertinent literature for the variable porosity category ….………………………. 16

2.4 Different models of the variable porosity category …………………... 17

2.5 Relationship between various models and the pertinent literature for the thermal dispersion category ………………………… 20

2.6 Different models of transverse thermal dispersion …………………… 21

2.7 Relationship between various models and the pertinent literature for the local thermal non-equilibrium category ……………. 24

2.8 Different models of the fluid to solid heat transfer coefficient and the fluid to solid specific area ….….…………..…...... 25

3.1 Primary categories of fluid flow interface conditions between a porous medium and a fluid layer …………………….……. 33

3.2 Primary categories of heat transfer interface conditions between a porous medium and a fluid layer …………………….……. 34

4.1 Summary of different models based on the two approaches of constant wall heat flux

boundary conditions …………………………………….………..…... 80

xii Summary of corresponding references to different 4.2 models based on the two approaches of constant

wall heat flux boundary conditions ………………………………....… 81

4.3 Mathematical representation of the wall heat flux and the temperature gradients for some special cases ………………………. 96

xiii

LIST OF FIGURES

Figure Page 2.1 Schematic Diagram of the problem, analysis of variants within the porous media transport models, and the corresponding coordinate systems ……………………………...... 7

3.1 Schematic of the physical system and the coordinate system for the problem of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer …………………………………………………... 32

3.2 Figure 3.2: Comparison between the exact solution of Vafai and Kim (1990a) and the present numerical results ……..… 41

3.3 Effect of changing the effective viscosity on velocity for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Da=0.001, Re=1.0 …………………….. 42

3.4 Effect of changing the effective viscosity on temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Da=10-3, Re=1.0 …………………….... 43

3.5 Velocity field for the interface between a porous medium -3 and a fluid layer, ε=0.7, Λ=1.0, Da=10 , Re=1.0, β1=1.0, β2=1.0, αT=10.0, φ=10 ……………………………..…………….. 44

3.6 Temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer, ε=0.7, -3 Λ=1.0, Da=10 , Re=1.0, β1=1.0, β2=1.0, αT=10.0, φ=10 ………... 45

3.7 Effect of porosity variation on the velocity field for the interface between a porous medium and a fluid layer, Λ=1.0, Da=0.001, Re=1.0, β1=1.0, β2=1.0, ε=0.5 ………………... 46

xiv

3.8 Effect of porosity variation on the velocity field for the interface between a porous medium and a fluid layer, 47 Λ=1.0, Da=0.001, Re=1.0, β1=1.0, β2=1.0, ε=0.9 ………………...

3.9 Effect of inertia parameter variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Da=0.001, Re=1.0, β1=1.0, β2=1.0, Λ=0.1 ………..... 48

3.10 Effect of inertia parameter variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Da=0.001, Re=1.0, β1=1.0, β2=1.0, Λ=10.0 ……………..... 49

3.11 Effect of Darcy number variation on the velocity field for the interface between a porous medium and a fluid layer, -4 ε=0.7, Λ=1.0, Re=1.0, β1=1.0, β2=1.0, Da=10 ………………..... 50

3.12 Effect of Darcy number variation on the velocity field for the interface between a porous medium and a fluid layer, -3 ε=0.7, Λ=1.0, Re=1.0, β1=1.0, β2=1.0, Da=10 ………………..... 51

3.13 Effect of Reynolds number variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Λ=10.0, Da=0.001, β1=1.0, β2=1.0, Re=10.0 ……………... 52

3.14 Effect of Reynolds number variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Λ=10.0, Da=0.001, β1=1.0, β2=1.0, Re=100.0 …………..... 53

3.15 Effect of β1 variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Da=0.001, Re=1.0, β2=1.0, β1=0.1 ……………………………….. 54

3.16 Effect of β1 variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Da=0.001, Re=1.0, β2=1.0, β1=10 ……………...………………… 55

3.17 Effect of β2 variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Da=0.001, Re=1.0, β1=1.0, β2=0.1 ……………………………….. 56

3.18 Effect of β2 variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Da=0.001, Re=1.0, β1=1.0, β2=10 ……………………………...… 57

3.19 Effect of porosity variation on temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer, Λ=1.0, Da=10-3, Re=1.0, β1=1.0, αT=10.0, φ=10.0 ………………………………... 62

3.20 Effect of inertia parameter variation on temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer, ε=0.7, -3 Da=10 , Re=1.0, β1=1.0, αT=10.0, φ=10.0 ……………………… 63

3.21 Effect of Darcy number variation on temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Re=1.0, β1=1.0, αT=10.0, φ=10.0 ………………………... 64

3.22 Effect of Reynolds number variation on temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer, ε=0.7, -3 Λ=1.0, Da=10 , β1=1.0, αT=10.0, φ=10.0 ……………………….. 65

3.23 Effect of β1 variation on the temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, -3 Da=10 , Re=1.0, φ=10.0, αT=10.0 ………………………………. 66

3.24 Effect of αT variation on the temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, -3 Da=10 , Re=1.0, φ=10.0, β1=1.0 ………………………………… 67

3.25 Effect of φ variation on the temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, -3 Da=10 , Re=1.0, β1=1.0, αT=10.0 ………………………………. 68

4.1 Schematic diagram of the problem of constant wall heat flux boundary conditions in porous media under LTNE conditions and the corresponding coordinate system ………….…. 79 xvi

4.2 Validation of the present work. (a) Comparison of the velocity profiles with the analytical solution of Vafai and -3 Kim (1989). Da=10 , Λ=70.372, Rep=200. (b) Local Nusselt numbers of the present study against the results of Amiri and -6 Vafai (1994), αs/αf=25.6, Da=1.36×10 and Rep=100 ………...... 85

4.3 Effect of porosity “ε” on the Nusselt number excluding the effects of variable porosity and thermal dispersion. Rep=100, -5 Da=10 , Λ=10, dp=0.008, κ=23.75. (a) Fluid phase. (b) Solid phase. (c) Total Nusselt number ……………………….. 97

4.4 Effect of the solid-to-fluid conductivity ratio “κ=ks/kf” on the Nusselt number excluding the effects of variable porosity and thermal dispersion. ε=0.9, Rep=100, Λ=10, -5 Da=10 , dp=0.008. (a) Fluid phase. (b) Solid phase. (c) Total Nusselt number …………………………………………. 99

4.5 Effect of particle Reynolds number “Rep” on the total Nusselt number excluding the effects of variable porosity and thermal dispersion. (a) ε=0.9, Da=10-5, -5 Λ=10, dp=0.008, κ=23.75. (b) ε=0.5, Da=10 , Λ=10, dp=0.008, κ=1.0 …………………………………………………... 101

4.6 Effect of Darcy number “Da” on the total Nusselt number excluding the effects of variable porosity and thermal dispersion. (a) ε=0.9, Rep=100, Λ=10, dp=0.008, κ=23.75. (b) ε=0.5, Rep=100, Λ=10, dp=0.008, κ=1.0 ……………………... 102

4.7 Effect of the Inertia parameter “Λ” on the total Nusselt number excluding the effects of variable porosity and thermal dispersion. (a) ε=0.9, Rep=100, -5 Da=10 , dp=0.008, κ=23.75. (b) ε=0.5, Rep=100, -5 103 Da=10 , dp=0.008, κ=1.0 …………………………………………

4.8 Effect of the particle diameter “dp” on the total Nusselt number excluding the effects of variable porosity and thermal dispersion. (a) ε=0.9, Rep=100, -5 Da=10 , Λ=10, κ=23.75. (b) ε=0.5, Rep=100, Da=10-5, Λ=10, κ=1.0 …………………………………………..... 104

xvii 4.9 Effect of the solid-to-fluid conductivity ratio “κ” on the -5 temperature gradients. ε=0.5, Rep=100, Da=10 , Λ=10, dp=0.008. (a) Fluid phase. (b) Solid phase ……………………….. 105

4.10 Effect of the solid-to-fluid conductivity ratio “κ” on the difference between the wall temperature and the mean -5 temperature. ε=0.5, Rep=100, Da=10 , Λ=10, dp=0.008. (a) Fluid phase. (b) Solid phase …………………………………. 106

4.11 Effect of the free stream porosity “ε∞”on the Nusselt number including the effects of variable porosity and thermal dispersion. B=0.98, c=2, Rep=100, dp=0.008, κ=23.75. Including the effects of variable porosity and thermal dispersion. (a) Fluid phase. (b) Solid phase. (c) Total Nusselt number …………………………………………. 114

4.12 Effect of the solid-to-fluid conductivity ratio “κ” on the total Nusselt number including the effects of variable porosity and thermal dispersion. ε∞=0.5, b=0.98, c=2, Rep=100, dp=0.008, (εw=0.99). Including the effects of variable porosity and thermal dispersion. (a) Fluid phase. (b) Solid phase.(c) Total Nusselt number ………………………... 116

4.13 Effect of the particle Reynolds number “Rep” on the total Nusselt number including the effects of variable porosity and thermal dispersion. (a) ε∞=0.5, b=0.98, c=2, dp=0.008, κ=23.75, (εw=0.99). (b) ε∞=0.4, b=0.25, c=2, dp=0.008, κ=1.0, (εw=0.5) ………………………………………... 118

4.14 Effect of the particle diameter “dp” on the total Nusselt number including the effects of variable porosity and thermal dispersion. (a) ε∞=0.5, b=0.98, c=2, Rep=100, κ=23.75, (εw=0.99). (b) ε∞=0.4, b=0.25, c=2, Rep=100, κ=1.0, (εw=0.5) ………………………………………………….... 119

5.1 Schematic diagram of the free surface front and the corresponding coordinate system ………………………………… 129

5.2 Comparison between the present results and the numerical results in Chen and Vafai (1996). (a) Temporal free surface distribution using constant Darcy number. (b) Temperature contours for -4 -6 Rek=5.72×10 , Da=1.0×10 at t=0.5 s ……………...... 132

xviii

5.3 Progress of the interfacial front for (a) Constant porosity category with Da=1.0×10-6, ε=0.8, Λ=10.0 and Re=100. (b) Variable porosity category with ε∞=0.45, b=0.98, c=2.0, Re=100 and dP/H=0.05 …………………………………… 134

5.4 Effect of Darcy number for the constant porosity category using Λ=10.0, Re=100 and ε=0.8 on (a) The temporal free surface front (b) The total time to reach the channel exit (τ max) …………………………………. 136

5.5 Effect of Inertia parameter for the constant porosity category using ε=0.8, Da=1.0×10-6 and Re=100 (a) On the temporal free surface front (b) On the total time to reach the channel exit (τ max) …………………………….. 137

5.6 Effect of Reynolds number for the constant porosity category using ε=0.8, Da=1.0×10-6 and Λ=10.0 (a) The temporal free surface front (b) The total time to reach the channel exit (τ max) ……………………………………. 138

5.7 Effect of Reynolds number for the variable porosity category using ε∞=0. 45, b=0.98, c=2.0 and dP/H=0.05 on (a) The temporal free surface front (b) The total time to reach the channel exit (τ max) …………………………………… 141

5.8 Effect of particle diameter for the variable porosity category using ε∞=0. 45, b=0.98, c=2.0 and Re=100 on (a) The temporal free surface front (b) The total time to reach the channel exit (τ max) ……………………………… 142

5.9 Temporal dimensionless temperature profiles including thermal dispersion effects, ε=0.8, Da=10-6, Re=100, Λ=10, κ=15.0 and dP/H=0.05 (a) τ =0.25 (b) τ =0.5 (c) τ = τ max ……………………………………………………….. 146

5.10 Temporal dimensionless temperature profiles excluding thermal dispersion effects, ε=0.8, Da=10-6, Re=100, Λ=10, κ=15.0 and dP/H=0.05 (a) τ =0.25 (b) τ =0.5 (c) τ = τ max ……………………………………………………… 148

xix

5.11 Effect of porosity for the thermal dispersion category -6 using Da=10 , Re=100, Λ=10, κ=15.0 and dP/H=0.05 on (a) Dimensionless temperature profiles using ε=0.7 (b) Dimensionless temperature profiles using ε=0.9 (c) Total Nusselt number ………………………………………... 150

5.12 Effect of Darcy number for the thermal dispersion category using ε=0.8, Re=100, Λ=10, κ=15.0 and dP/H=0.05 on (a) Dimensionless temperature profiles using Da=10-4 (b) Dimensionless temperature profiles using Da=10-8 (c) Total Nusselt number ………………………... 152

5.13 Effect of Reynolds number for the thermal dispersion category using ε=0.8 Da=10-6, Λ=10, κ=15.0 and dP/H=0.05 on (a) Dimensionless temperature profiles using Re=50 (b) Dimensionless temperature profiles using Re=150 (c) Total Nusselt number …………………………. 154

5.14 Effect of particle diameter for the thermal dispersion category using ε=0.8 Da=10-6, Re=100, Λ=10 and κ=15.0 on (a) Dimensionless temperature profiles using dP/H=0.01 (b) Dimensionless temperature profiles using dP/H=0.1 (c) Total Nusselt number ……………… 156

5.15 Effect of solid to fluid conductivity ratio for the thermal dispersion category using ε=0.8 Da=10-6, Re=100, Λ=10 and dP/H=0.05 on (a) Dimensionless temperature profiles using κ=5.0 (b) Dimensionless temperature profiles using κ=30.0(c) Total Nusselt number ………………………………..... 158

5.16 Dimensionless temperature profiles for the LTNE category using ε=0.8, Da=10-6, Re=100, Λ=10, κ=15.0 and dP/H=0.05 (a) Fluid phase (b) Solid phase …………………... 163

5.17 Dimensionless temperature profiles for the LTNE category using ε=0.9, Da=10-8, Re=200, Λ=100, κ=5.0 and dP/H=0.1 (a) Fluid phase (b) Solid phase ……………………. 164

5.18 Effect of porosity on average Nusselt numbers for the LTNE category, Re=100, Da=10-6, Λ=10, κ=15.0 and dP/H=0.05 (a) Excluding thermal dispersion effects (b) Including thermal dispersion effects ……………………………... 165

5.19 Effect of Darcy number on average Nusselt numbers for the LTNE category, ε=0.8, Re=100, Λ=10, κ=15.0 and dP/H=0.05 (a) Excluding thermal dispersion effects (b) Including thermal dispersion effects …………………………. 166

5.20 Effect of Inertia parameter on average Nusselt numbers for the LTNE category, ε=0.8, Da=10-6, Re=100, κ=15.0 and dP/H=0.05 (a) Excluding thermal dispersion effects (b) Including thermal dispersion effects …………………………. 167

5.21 Effect of Reynolds number on average Nusselt numbers for the LTNE category, ε=0.8, Da=10-6, Λ=10, κ=15.0 and dP/H=0.05 (a) Excluding thermal dispersion effects (b) Including thermal dispersion effects …………………………. 168

5.22 Effect of particle diameter on average Nusselt numbers for the LTNE category, ε=0.8, Da=10-6, Re=100, Λ=10 and κ=15.0 (a) Excluding thermal dispersion effects (b) Including thermal dispersion effects …………………………. 169

5.23 Effect of solid-to-fluid thermal conductivity ratio on average Nusselt numbers for the LTNE category, ε=0.8, Da=10-6, Re=100, Λ=10 and dP/H=0.05 (a) Excluding thermal dispersion effects (b) Including thermal dispersion effects ………………….. 170

xxi

NOMENCLATURE

-1 asf = specific surface area of the packed bed, [m ] b, c = porosity variation parameters -1 -1 cp = specific heat at constant pressure, [J kg K ] dp = particle diameter [m] dV = parameter defined in Table 2.8. 4ε/ asf

Dh = Hydraulic diameter [m] Da = Darcy number, K/H2 F = Geometric function in the momentum equation. h = heat transfer coefficient, [W m-2 K-1] -2 -1 hsf = fluid-to-solid heat transfer coefficient, [W m K ] H = Half the height of the channel, [m] J = Unit vector aligned along the pore velocity k = thermal conductivity, [W m-1 K-1] K = Permeability [m2] L = Length of the packed bed, [m]

Nu = Local Nusselt number, Hdh/keff

____ Nu = Average Nusselt number P = Pressure, [N/m2]

Pr = Prandtl number, µf CPf/kf

Re = Reynolds number, u∞H/νf

Re = Permeability Reynolds number, u∞ Κ /νf

Rep = Particle Reynolds number, ucdp/νf T =Temperature, [K]

Tm = Mean temperature, [K] xxii u =Velocity in x-direction, [m s-1] -1 uint = Interface velocity, [m s ]

U = Non-dimensional velocity, u/uc

Uint = Non-dimensional interface velocity

Um =Mean velocity V = Velocity vector, [m s-1] x0 = Free surface front location, [m] x, y = Cartesian coordinates, [m] X, Y = Non-dimensional coordinates, x/L and y/H.

Greek Symbols α = Thermal diffusivity, [m2 s-1] α* = Velocity slip coefficient in Table 3.1.

αT = Temperature slip coefficient in Table 3.2. ε = Porosity φ = Parameter in Table 3.2. γ = parameter in equation (5.23).

κ = Solid-to-fluid thermal conductivity ratio, ks/kf λ = Temperature boundary condition Parameter in Table 3.2 3/2 Λ = Inertia parameter, ε FucH/ν µ = Dynamic Viscosity, [kg m-1 s-1] ν = Kinematics Viscosity, [m2 s-1] θ = Dimensionless temperature, [(Tw-T)/(Tw-Te)] Θ =Dimensionless temperature, [(Tw-T)/(Tw-T∞)] ρ = Density, [kg m-3] τ = dimensionless time, equation (5.22). ξ = Dimensionless axial length scale, x/L.

Subscripts c = Convective component d = dispersion e = inlet xxiii eff = effective property f = fluid int = interfacial m = mean max = maximum s = solid t = total w = wall ∞ = free stream - = Plain medium side + = Porous medium side

Other symbols < > = ‘local volume average’ of a quantity

xxiv

CHAPTER 1

INTRODUCTION

Transport processes through porous media have gained extensive attention in recent years due to their wide range of practical applications in contemporary technology. These applications include areas such as geothermal engineering, building thermal insulation, chemical catalytic reactors, petroleum reservoirs, crude oil extraction, heat exchangers, coal combustors, solar collectors, injection molding, die filling, drying technology, industrial and agricultural water distribution, food processing, electronic cooling, heat pipe technologies, industrial and commercial air filtration, energy storage units, nuclear waste repositories, and numerous other applications. Darcy’s law (1856) was utilized in many early investigations to examine the fluid flow in porous media. Fundamentals and details of Darcy’s law can be found in Scheidegger (1974). Recently, Non-Darcian effects, solid boundary effect and inertia effect, became a more popular tool to investigate the momentum and energy transports through porous media. These effects along with the well known Darcy’s law form the generalized model which is also known as the Brinkman-Forchheimer-

1 extended Darcy model. Governing momentum and energy equations that account for these effects were first established in the pioneering study of Vafai and Tien (1981). In their study, the local volume averaging technique was utilized to derive the macroscopic governing equations. More recently, other effects such as variable porosity, partially porous configurations, thermal dispersion and local thermal non- equilibrium (LTNE) attracted more attention because of their compatibility with applications of the modern advanced technology. A comprehensive overview of modern procedures and techniques of transport processes through porous media can be found in the Handbook of Porous Media edited by Vafai (2000).

Numerical simulation of fluid flow and heat transfer model variants within porous media is presented in this study. Four interrelated primary topics of transport phenomena through porous media are considered in this study. The first topic characterizes in detail various aspects of different physical phenomena affecting transport through porous media. These include variable porosity, local thermal non- equilibrium and thermal dispersion. The second topic is the investigation of the fluid flow and heat transfer at the interface between a porous medium and an adjacent layer of a homogeneous fluid. Analysis of the fluid flow and the heat transfer boundary conditions at the interface is the main issue of this topic. The third topic is the analysis of the constant heat flux boundary condition and its significance on the heat transfer process through porous media. For this part a two-equation model is used to investigate how the heat input from the impermeable wall to the porous medium is

2 distributed between the solid matrix and the fluid phase. The fourth topic is the analysis of the free surface transport in porous media. For this topic, the generalized momentum equation is utilized in the analysis of the free surface transport in porous media. Particularly, boundary and inertia effects as well as the effect of variable porosity are considered. For the thermal transport part, effects of using thermal dispersion and local thermal non-equilibrium are also considered. In addition to the numerical presentation of the results, detailed physical interpretation of the topics under consideration will be presented during the course of this study.

Due to ever increasing use of porous media in various applications, the variants among different complex models require a systematic and detailed investigation so as to create a clearer picture for their effects on different physical phenomena. This leads to an investigation of each of the four mentioned topics. Detailed results of the first topic are presented in an earlier study (B. Al-Azmi 1999) and will be summarized in this work. For the interface region between a porous medium and a homogeneous fluid, a number of different models are found in the literature for the fluid flow boundary conditions as well as the heat transfer boundary conditions. Therefore, more investigation is needed for a better understanding of this problem. The third topic is the division of the input constant heat flux between the solid matrix and the fluid phase of the porous medium. For this problem, two approaches are found in the literature for utilizing problems that involve constant heat flux boundary conditions.

Accordingly, investigation of the applicability of each approach would be very helpful

3 for modeling problems of this type. The last topic is the analysis of the free surface transport in porous media. Earlier studies of this problem in the literature considered specific cases. As a result, it is found desirable to think of solving this problem using the generalized model which accounts for the effects of variable porosity, thermal dispersion and local thermal non-equilibrium (LTNE).

CHAPTER 2

ANALYSIS OF VARIANTS WITHIN THE POROUS MEDIA TRANSPORT MODELS

2.1 INTRODUCTION

Modeling of the non-Darcian transport through porous media has been the subject of various recent studies due to the increasing need for better understanding of the associated transport processes. This interest stems from the numerous practical applications which can be modeled or can be approximated as transport through porous media such as thermal insulation, packed bed heat exchangers, drying technology, catalytic reactors, petroleum industries, geothermal systems and electronic cooling. As mentioned earlier in chapter 1, a summary of the pertinent variants within the porous media transport models will be presented in this chapter. Detailed results and comprehensive comparisons between these models can be found in an earlier study (Al-Azmi 1999) and will not be repeated here for the purpose of brevity. These models can be categorized into four major categories, constant porosity models,

5 variable porosity models, thermal dispersion models and local thermal non- equilibrium (LTNE) models.

To analyze these categories in modeling the transport processes through porous media, a fundamental configuration shown in Fig. 2.1 is selected. This configuration consists of a parallel plate channel with constant heat flux qw or constant wall temperature Tw. The height and the width of the channel are 2H and L respectively.

The velocity of the upstream flow is uc and its temperature is Te. This configuration allows an investigation of all the major aspects described earlier. In what follows, the summarized results for each category are presented separately. The pertinent works within each area resulting in a true variance were selected within each category. The physics of the two fundamental configurations considered here have been analyzed in detail in Vafai and Kim (1989) and Amiri et. Al. (1995) and will not be considered again in this study.

Coordinates for qw q w or Tw

y u c x 2H T e y

x qw or Tw L

Coordinates for Tw

Figure 2.1: Schematic Diagram of the problem, analysis of variants within the porous media transport models, and the corresponding coordinate systems.

7 The governing equations for this part assuming fully developed conditions, can be written as given in Vafai and Tien (1981), Vafai (1984), Vafai and Kim (1989) and

Amiri and Vafai (1994):

Continuity Equation

∇ ⋅ V = 0 (2.1)

Momentum Equation

ρ µ ρ Fε µ f (V.∇)V = − V − f []V . V J + ∇ 2 V − ∇ P (2.2) ε Κ Κ ε

For variable porosity case, the permeability of the porous medium K and the geometric function F can be represented as in Ergun (1952) and Vafai (1984, 1986)

ε 3d 2 Κ = P (2.3) 150(1−ε) 2

1.75 F = (2.4) 150ε 3

According to Benenati and Brosilow (1962) and Vafai (1984, 1986), the porosity distribution throughout the porous medium can be presented by the following equation

 -c y  ε =ε ∞ 1+ bexp( ) (2.5)  d p 

8 Energy Equation

keff  (ρ cP ) f V .∇ T = ∇. .∇ T  (2.6)  ε 

For thermal dispersion (Amiri and Vafai, 1994; Amiri et al., 1995)

keff = ko + kd (2.7)

where ko is the stagnant thermal conductivity and kd is the dispersion conductivity.

For LTNE, two separate energy equations are required (Vafai and Amiri, 1998; Amiri and Vafai, 1994; Amiri et al., 1995)

Fluid phase energy equation

()ρ CP f V .∇ Tf = ∇.{k feff .∇ Tf }+ hsf asf ( Ts − Tf ) (2.8)

Solid phase energy equation

0= ∇.{kseff .∇ Ts }− hsf asf ( Ts − Tf ) (2.9)

Where

k feff = ε k f (2.10) and

kseff =(1−ε) ks (2.11) 9 2.2 CONSTANT POROSITY

Darcy law was a prevalent means to investigate the fluid flow in porous media until the appearance of the pioneering study of Vafai and Tien (1981). They presented and characterized the boundary and inertial effects in forced convective flow through a porous medium. Later, Vafai and Tien (1982) investigated the boundary and inertial effects on convective mass transfer in porous media. Vafai and Kim (1989) used the

Brinkman-Forcheimer-extended Darcy model to obtain a closed-form analytical solution for fully developed flow in a porous channel subject to constant heat flux boundary conditions. Hadim (1994) performed a numerical study to analyze steady forced convection in a channel filled or partially filled with a porous medium and containing discrete heat sources. Kaviany (1985) studied the flow through a constant porosity medium bounded by isothermal parallel plates using the Brinkman-extended flow model and constant matrix porosity. Lauriat and Vafai (1991) presented a comprehensive study of forced convective heat transfer in porous media through a channel or over a flat plate. Other research works consider various problems of the flow and heat transfer through a constant porosity medium (Beckerman and Viskanta,

1987; Kim and Choi, 1996; Kladias and Prasad, 1990; Nield et al., 1996; Sung et al.,

1995; You and Chang, 1997a, 1997b; Neale and Nader, 1974; Poulikakos and

Kazmierczak, 1987; Kim et al., 1994; Chen and Vafai, 1996; Nakayama et al., 1990;

Hong et al., 1985; Kaviany, 1987; Kuznetsov, 1996; Lan and Khodadadi, 1993;

Nakayama et al., 1988; Ould-Amer et al., 1998; Vafai and Kim, 1995a, 1995b).

10 The left hand side term of the momentum equation (2.2) is the convective term which is found to be identical in all the above mentioned references. The last term on the right hand side, the pressure gradient, is also the same in all references. Different forms for the other three terms in the momentum equation are found in the literature.

The first term on the right hand side is known as Darcy term, the second term is called

Forchheimer term while the third term is called Brinkman term. The association between various models and the pertinent literature for the constant porosity category is given in Table 2.1. This comprehensive analysis resulted in three different models for this category as shown in Table 2.2.

11 Model References C1 Vafai and Tien (1981, 1982), Vafai and Kim (1989), Kaviany (1985), Lauriat and Vafai (1991), Hong et al. (1985), Kaviany (1987), Kuznetsov (1996), Lan and Khodadadi (1993), Nakayama et al. (1988), Ould-Amer et al. (1998), Vafai and Kim (1995a, 1995b). C2 Kim and Kang (1994), Chen and Vafai (1996), Nakayama et al. (1990). C3 Hadim (1994), Beckermann and Viskanta (1987), Kim and Choi (1996), Kladias and Prasad (1990), Nield et al. (1996), Sung et al. 1995, You and Chang (1997a, 1997b), Neale and Nader (1974), Poulikakos and Kazmierczak (1987).

Table 2.1: Relationship between various models and the pertinent literature for the constant porosity category.

Model Darcy Forchheimer Brinkman C1 µ Fε µ u ρ u 2 ∇ 2u Κ Κ ε C2 µ F µ u ρ u 2 ∇ 2u Κ Κ ε C3 µ F µ∇ 2u u ρ u 2 Κ Κ

Table 2.2: Different models of the constant porosity category.

12 Pertinent parameters such as porosity, Darcy number, inertia parameter and

Reynolds number are used to investigate the differences between these three models.

Results show some differences in the velocity profiles between these models. Also, it is shown that using different models has a substantially less impact on the temperature field and Nusselt number distribution. A useful set of correlations for conversion of models C2 and C3 into C1 are listed below:

______−1.173 0.2812 0.5357 0.0636 Nu C2 = 0.9998Nu C1 + 0.23ε Λ Da ReH (2.12)

______−1.2582 0.45 0.1788 Nu C3 = 0.0002 Nu C1 + 0.3664ε Da ReH (2.13)

2.3 VARIABLE POROSITY

The second category in modeling of the transport processes through porous media is based on variable porosity media. A number of experimental and theoretical studies have shown that variation of porosity near a solid boundary has a significant effect on the velocity fields in packed beds resulting in an appreciable flow maldistribution, which appears as a sharp peak near the solid boundary and decreases to an almost constant value at the center of the bed. This phenomenon is known as the channeling effect. Vafai (1984, 1986) and Vafai et al. (1985) investigated analytically and experimentally the channeling effect on external forced convective flow and heat

13 transfer. Poulikakos and Renken (1987) presented a numerical study of the variable porosity effects in a channel bounded by two isothermal parallel plates and in a circular pipe. A number of investigations have considered the effect of variable porosity on fluid flow and heat transfer in porous media (Renken and

Poulikakos,1988; Hunt and Tien, 1988b; Hsiao et al., 1992).

When variable porosity effects are considered, expression (Eq. 2.5) is used for the porosity. Consequently, permeability (Κ) and the geometric function (F) that appear in the momentum equation are no longer constant and are expressed as in Eq.

(2.3) and Eq. (2.4) respectively. The association between various models and the pertinent literature for the variable porosity category is given in Table 2.3. Four variant models have been found in literature for variable porosity media category as shown in Table 2.4. It can be seen that models V2 and V3 have the same Darcy and

Forchheimer terms while the only difference between them is the presentation of the

Brinkman terms. Models V1 and V4 have the same Forchheimer term, models V1 and

V3 have the same Brinkman term while models V1, V2, and V3 have the same formation for the Darcy term. The pertinent parameters in this category are chosen to be similar to those used by Vafai (1984). These parameters are the pressure gradient, the particle diameter, the free stream porosity, and the constants b and c in equation

(2.5). Models V2 and V3 are closer to each other due to their similar representations for the Darcy and Forchheimer terms. On the other hand, models V1 and V4 are closer to each other due to similar Forchheimer terms. Again detailed results are given in (B. 14 Al-Azmi 1999) and will not be repeated here again. A useful set of correlations for conversion of models V2 , V3 and V4 into V1 are listed below:

______−0.0041051  dp  0.0092769 0.1534 0.0348 −0.0218 Nu V2 = −3.927 +1.2366 Nu V1  d p (ε ∞ ) b c (2.14)  dx 

______−0.005978  dp  0.0058872 0.1621 0.0363 −0.0228 Nu V3 = −4.6447 +1.2441 Nu V1  d p (ε ∞ ) b c (2.15)  dx 

______0.0125  dp  0.0152 0.0578 0.0253 −0.01 Nu V4 = 4.7358 + 0.9921 Nu V1  d p (ε ∞ ) b c (2.16)  dx 

Model References V1 Vafai (1984), Vafai (1986), Vafai et al. (1985), Vafai and Amiri (1998), Amiri and Vafai (1994), Amiri et al. (1995), Amiri and Vafai (1998). V2 Lauriat and Vafai (1991), Poulikakos and Renken (1987), Renken and Poulikakos (1988). V3 Hunt and Tien (1988b), Hong et al. (1987), Chen (1996, 1997a, 1997b), Cheng et al. (1988), Chen et al. (1996). V4 Hsiao et al. (1992), David et al. (1991), Hsu and Cheng (1990), Fu et al. (1996).

Table 2.3: Relationship between various models and the pertinent literature for the variable porosity category.

Model Darcy Forchheimer Brinkman 2 150(1 − ε ) 1.75(1− ε ) 2 µ V1 µ u ρ u ∇2u ε 3 d 2 ε 2 d p p ε 150(1− ε )2 1.75(1− ε ) V2 ρ u2 2 µ 3 2 u 3 µ∇ u ε d p ε d p 150(1 − ε )2 1.75(1− ε ) V3 ρ u2 µ 2 µ 3 2 u 3 ∇ u ε d p ε d p ε V4 2 150(1− ε) 1.75(1 − ε ) 2 µ u ρ u µ∇2u ε 2 d 2 ε 2 d p p

Table 2.4: Different models of the variable porosity category.

17 2.4 THERMAL DISPERSION

The third category in modeling of the transport processes through porous media relates to thermal dispersion. The effect of thermal dispersion has been studied by a number of researchers in the past few years and has been shown to enhance the heat transfer process. These studies have tried to correlate the experimental data to a formulation for the thermal dispersion conductivity or diffusivity. The association between various models and the pertinent literature for the thermal dispersion category is given in Table 2.5. A detailed analysis of the research works in this area reveals the existence of five pertinent models as displayed in Table 2.6. The present section considers the effects of using these five variant models for the transverse thermal dispersion conductivity on the transport processes in porous media. For this category, a constant porosity assumption was invoked since the variable porosity category was analyzed earlier.

When thermal dispersion effects are considered, the effective conductivity in the energy equation becomes a combination of the stagnant and dispersion conductivities. In the present investigation, γ=0.1 was used for model D1, γ=0.04 was used for model D2, and γ=0.17 and w=1.5 were used for model D3. It is easier to observe the differences between models D1, D2, and D4. However, models D3 and D5 have different structures requiring a more careful set of comparisons. A

18 comprehensive numerical study was performed to analyze the variances between the five cited models.

The effects of porosity, inertia parameter, Darcy number, Reynolds number, and the particle diameter on the variances within the thermal dispersion category are best illustrated in terms of their effects on the temperature and the local Nusselt number profiles. Detailed results can be found in Al-Azmi (1999) and will not be listed here again. However, a useful set of correlations that convert results of models D2, D3, D4 and D5 into results of D1 are listed below:

______0.9983 Nu D2 = 4.8126 + 0.9898(4.8132 + Nu D1 ) (2.17) ____1.0593 0.0634 1.0442 0.7976 1.2389 0.9967 +0.9581ε Λ Da Rep d p Nu D1

____ 0.2447 0.1497 Nu D3 = 29.5369 + 0.3445ε (61.4249 + Λ) (2.18) ____ 0.2314 −0.1152 0.511 −0.2456 ×Da Re p d p Nu D1

______0.9983 Nu D4 = 45.8467 + 0.8801Nu D1 (2.19) ____ 1.0548 -0.3152 0.8653 0.7076 1.3521 1.05 +0.721ε Λ Da Re p d p Nu D1

____ 0.2971 0.4933 Nu D5 = -71.2195 + 0.137ε (389.0253 + Λ) (2.20) ____ 0.2098 −0.0354 0.3744 −0.3873 ×Da Re p d p Nu D1

Model References Hong et al. (1987), Hong and Tien (1987), Vafai and Amiri (1998), Amiri D1 and Vafai (1994), Amiri et al. (1995), Amiri and Vafai (1998) Chen (1996, 1997a, 1997b), David et al. (1991), Hsu and Cheng (1990), D2 Jang and Chen (1992). Hsu and Cheng (1990), Cheng et al. (1988), Fu et al. (1996), Vafai and D3 Amiri (1998), Hwang et al. (1995). D4 Chen et al. (1996). D5 Hunt and Tien (1988a), Hunt and Tien (1988b)

Table 2.5: Relationship between various models and the pertinent literature for the thermal dispersion category.

Model Dispersion Conductivity Notes γ = 0.1 D1 γ ρ C Ud P p γ = 0.2 1− ε γ ρ C Ud γ = 0.04 D2 ε P p γ = 0.02  − y  γρ CP U d p 1− exp( )  wH  γ=0.17, w=1.5 D3 γ=0.12 ,w=1.0 γ=0.3, w=3.5

γ=0.375, w=1.5 1−ε D4 0.01 ρ C Ud ε 2 P p

D5 0.025ρ CP U Κ

Table 2.6: Different models of transverse thermal dispersion.

21 2.5 LOCAL THERMAL NON-EQUILIBRIUM (LTNE)

The fourth category in modeling of the transport processes through porous media is the local thermal non-equilibrium between the fluid and solid phases. The association between various models and the pertinent literature for the thermal dispersion category is given in Table 2.7. Table 2.8 shows three variant models for the fluid to solid heat transfer coefficient hsf and for the specific surface area of the packed bed asf, corresponding to the pertinent investigations in the LTNE area. The effects of porosity, inertia parameter, Darcy number, Reynolds number, particle diameter and ratio of fluid to solid conductivities on temperature and Nusselt number profiles of the models shown in Table 2.8 are analyzed. It should be noted that the thermal conductivities of the solid and fluid appear in the relationship for hsf for model E3 while only the fluid phase conductivity appears in the hsf equation for models E1 and

E2. As such, the solid to fluid thermal conductivity ratio will have a significant effect on the variances among the three models.

Two energy equations are required, one for the fluid phase (Eq. 2.8) and another for the solid phase (Eq. 2.9), instead of a single energy equation when LTNE effects are considered. These two equations are coupled and an iterative solution is usually used to obtain a numerical solution. Pertinent parameters such as porosity, inertia parameter, Darcy number, Reynolds number, particle diameter and solid to fluid conductivity ratio are used to investigate the variances between these models.

22 Detailed results can be found in Al-Azmi (1999) and will not be listed here again.

Correlations in terms of the pertinent parameters that convert results, average fluid and solid Nusselt numbers, of models E2 and E3 into model E1 are listed below:

______0.8152 NuE2f = 2.4262+1.2043(NuE1f ) ____ (2.21) 0.2.2892 0.0485 0.1749 0.1567 −0.0531 ks -0.4402 1.268 +0.7792ε Λ Da Rep dp ( ) (NuE1f ) kf

______1.1645 Nu E3f = -77.5642 + 0.2554(Nu E1f ) (2.22) 0.5343 0.0139 -0.0201 0.1605 0.0328 k s -0.0019348 + 81.2051ε (0.3576 + Λ) Da Rep d p ( ) k f

______0.8054 Nu E2s = -10.5362 +1.7505(Nu E1s ) ____ (2.23) 1.469 0.0187 -0.3453 0.6346 k s 0.3848 2.2807 +0.0515 ε Da Rep d p ( ) (Nu E1s ) k f

______0.9753 Nu E3s = -9.566+ 0.7751(Nu E1s )

____ (2.24) 1.6719 0.0154 -0.011 0.7615 -1.1829 k s -0.6435 -1.5434 +28.018 ε (0.0463+ Λ) Da Re p d p ( ) (Nu E1s ) k f

Model References

Vafai and Amiri (1998), Amiri and Vafai (1994), Amiri E1 et al. (1995), Amiri and Vafai (1998).

E2 Hwang et al. (1995).

E3 Dixon and Cresswell (1979)

Table 2.7: Relationship between various models and the pertinent literature for the local thermal non-equilibrium category.

Model hsf asf Notes

1 k 2 +1.1Pr 3 Re0.6  f   6(1−ε) E1 d p d p

 d  k f  0.33 1.35 2 0.004 V  Pr Re 20.346(1−ε) ε    Re< 75  d p  d p  E2 d p

 k f  0.33 0.59 1.064 Pr Re    d p  Re>350 −1  d ε d   p p  6(1−ε) E3 1 2 + 0.2555Pr 3 Re 3 k 10ks   f  d p

Table 2.8: Different models of the fluid to solid heat transfer coefficient and the fluid to solid specific are

CHAPTER 3

ANALYSIS OF FLUID FLOW AND

HEAT TRANSFER INTERFACIAL CONDITIONS

BETWEEN A POROUS MEDIUM AND A FLUID LAYER

In this chapter Different types of interfacial conditions between a porous medium and a fluid layer are analyzed in detail. Five primary categories of interface conditions were found in the literature for the fluid flow at the interface region.

Likewise, four primary categories of interface conditions were found in the literature for the heat transfer at the interface region. These interface conditions can be classified into two main categories, slip and non-slip interface conditions. The effects of the pertinent parameters such as Darcy number, inertia parameter, Reynolds number, porosity and slip coefficients, on different types of interface conditions are analyzed while fluid flow and heat transfer in the neighborhood of an interface region are properly characterized. A systematic analysis of the variances among different boundary conditions establishes the convergence or divergence among competing

26 models. It is shown that in general, the variances have a more pronounced effect on the velocity field and a substantially smaller effect on the temperature field and even a smaller effect on the Nusselt number distributions. For heat transfer interface conditions, all four categories generate results, which are quite close to each other for most practical applications. However, small discrepancies could appear for applications dealing with large values of Reynolds number and/or large values of

Darcy number. Finally, a set of correlations is given for interchanging the interface velocity and temperature as well as the average Nusselt number among various models.

3.1 INTRODUCTION

Fluid flow and heat transfer characteristics at the interface region in systems which consist of a fluid-saturated porous medium and an adjacent horizontal fluid layer have received considerable attention. This attention stems from the wide range of engineering applications such as electronic cooling, transpiration cooling, drying processes, thermal insulation, porous bearing, solar collectors, heat pipes, nuclear reactors, crude oil extraction and geothermal engineering. The work of Beavers and

Joseph (1967) was one of the first attempts to study the fluid flow boundary conditions at the interface region. They performed experiments and detected a slip in the velocity at the interface. Neale and Nader (1974) presented one of the earlier attempts 27 regarding this type of boundary condition in porous medium. In this study, the authors proposed a continuity in both the velocity and the velocity gradient at the interface by introducing the Brinkman term in the momentum equation for the porous side. Vafai and Kim (1990a) presented an exact solution for the fluid flow at the interface between a porous medium and a fluid layer including the inertia and boundary effects.

In this study, the shear stress in the fluid and the porous medium were taken to be equal at the interface region. Vafai and Thiyagaraja (1987) analytically studied the fluid flow and heat transfer for three types of interfaces namely, the interface between two different porous media, the interface separating a porous medium from a fluid region and the interface between a porous medium and an impermeable medium.

Continuity of shear stress and heat flux were taken into account in their study while employing the Forchheimer-Extended Darcy equation in their analysis. Other studies consider the same set of boundary conditions for the fluid flow and heat transfer used in Vafai and Thiyagaraja (1987) such as Vafai and Kim (1990b), Kim and Choi

(1996), Poulikakos and Kazmierczak (1987) and Ochao-Tapia and Whitaker (1997).

Ochoa-Tapia and Whitaker (1995a, 1995b) have proposed a hybrid interface condition (a hybrid between models 2 and 5 discussed later on) in which a jump in the shear stress at the interface region is assumed. In their study, the shear stress jump is inversely proportional to the permeability of the porous medium. This proposed set of interface conditions was used in other references such as Kuznetsov (1996, 1997, 28 1998a, 1998b, 1999). More recently, Ochoa-Tapia and Whitaker (1998a) have presented another shear stress jump boundary condition where the inertia effects become important. The same investigators, Ochoa-Tapia and Whitaker (1998b), have also presented another hybrid interface condition for the heat transfer part in which they introduce a jump condition to account for a possible excess in the heat flux at the interface. Sahraoui and Kaviany (1994) have proposed yet another hybrid interface condition for the heat transfer part. They used the continuity of the heat flux at the interface along with a slip in the temperature at the interface. The main focus of the present study is to critically examine the differences in the fluid flow and heat transfer characteristics due to different interface conditions, including all the aforementioned models. The current study complements a prior work by Alazmi and Vafai (2000) in which they presented a comprehensive analysis of variants within the transport models in porous media. In their study, four major categories namely, constant porosity, variable porosity, thermal dispersion and local thermal non-equilibrium were considered in great detail.

3.2 ANALYSIS

A comprehensive synthesis of literature revealed five primary categories for interface conditions for the fluid flow and four primary forms of interface conditions for the heat transfer between a porous medium and a fluid layer. Table 3.1 summarizes

29 the models for the fluid flow part while Table 3.2 summarizes the models for the heat transfer part. The fundamental configuration used by Vafai and Kim (1990a), representing the interface region between a porous medium and a fluid layer, is used in the current study. This configuration consists of a fluid layer sandwiched between a porous medium from above and a solid boundary from below. The physical configuration and the coordinate system are shown in Fig. 3.1. For the porous region, the governing equations are, Vafai and Tien (1981):

ρ µ ρ Fε µ f 〈(V.∇)V〉 = − V − f []〈V〉.〈V〉 J + ∇2 〈V〉 − ∇〈P〉 (3.1) ε Κ Κ ε

α 〈V〉.∇〈T〉 = eff ∇2 〈T〉 (3.2) ε

keff (3.3a, b) where αeff = and keff = εk f + (1− ε)k s ρ f cPf

The governing equations for the fluid region can be written as Vafai and Kim (1990a) and Vafai and Thiyagaraja (1987):

µ∇2 〈V〉 = ∇〈P〉 (3.4)

2 (3.5) 〈V〉.∇〈T〉 = α f ∇ 〈T〉

k f (3.6) where α f = ρ f cPf

The pertinent dimensionless parameters for this problem are, Vafai and Kim (1990a):

30 Κ , u∞H , Fε H (3.7a, b, c) Da = 2 Re = Λ = H ν f Κ

Primary categories of interface conditions utilized in the literature for the fluid flow are given in Table 3.1 while those for the heat transfer are given in Table 3.2.

u∞

y H

x Tw

Figure 3.1: Schematic of the physical system and the coordinate system for the problem of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer.

Model Velocity Velocity gradient References: Neale and Nader (1974), du du Vafai and Kim (1990a, 1 u =u | = | + - dy + dy − 1990b), Jang and Chen (1992) Vafai and Thiyagaraja du du (1987), Kim and Choi 2 u =u µ | = µ | + - eff dy + dy − (1996) and Poulikakos and Kazmierczak (1987) Ochao-Tapia and µ du du µ Whitaker (1995a, 3 u+=u- |+ −µ |− = β 1 u 1995b), Kuznetsov ε dy dy Κ (1996, 1997a, 1998, 1999) µ du du | −µ | = ε dy + dy − Ochao-Tapia and 4 u =u µ 2 + - β u + β ρ u Whitaker (1998a) 1 Κ 2

* Beavers and Joseph a du α 5 |− = ()uint − u∞ (1967), Sahraoui and dy Κ Kaviany (1992)

Table 3.1: Primary categories of fluid flow interface conditions between a porous medium and a fluid layer.

a A Forchheimer term is added to the momentum equation in the porous side for the purpose of comparison.

Model Temperature Temperature gradient References: Neale and Nader (1974), Vafai and Thiyagaraja (1987), Vafai and Kim (1990b), Kim and Choi (1996), ∂T− ∂T+ I T+=T- keff = k f Poulikakos and ∂y ∂y Kazmierczak (1987), Ochao-Tapia and Whitaker (1997), Kuznetsov (1998, 1999), Jang and Chen (1992) ∂T ∂T φ + k | = k | Ochao-Tapia and II T =T f − eff + + - ∂y ∂y Whitaker (1998b)

Sahraoui and dT α | = T (T − T ) ∂T ∂T Kaviany (1994) III + + − k − = k + dy λ eff ∂y f ∂y using fluid flow of model 1 Sahraoui and dT αT Kaviany (1994) | = (T − T ) ∂T− ∂T+ IV + + − k = k dy λ eff ∂y f ∂y using fluid flow of model 3

Table 3.2: Primary categories of heat transfer interface conditions between a porous medium and a fluid layer.

3.3 RESULTS AND DISCUSSION

A comprehensive analysis of fluid flow and heat transfer for the interface region between a fluid layer and an adjacent porous layer can be found in Vafai and

Thiyagaraja (1987). Therefore, the current study concentrates on analyzing and synthesizing the effects of different interface conditions. The presentation of the results in the current study is given in terms of the velocity fields for models given in

Table 3.1 and in terms of temperature and Nusselt number distributions for models given in Table 3.2. In presenting the results it is useful to introduce the following dimensionless variables, see Vafai and Kim (1990a),

x y X = , Y = L H

u (T − T ) U = and Θ = w (3.8) u ∞ (T − T∞ )

The local Nusselt number for the lower wall is defined as:

hH Nu = (3.9) k f

Where hx is the local heat transfer coefficient at the wall, which is defined as

k ∂T h = f (3.10) |y=0 (Tw − Tm ) ∂y

3.3.1 FLUID FLOW:

Comparison between the present numerical results and the exact solution of

Vafai and Kim (1990a) is shown in Fig. 3.2. Considering the first two models in Table

3.1, it can be seen that model 1 becomes identical to model 2 when the effective viscosity of the porous medium equals the viscosity of the fluid. For model 2, when there is a significant difference between the viscosity of the fluid and the effective viscosity of the porous medium, the slope of the velocity profile in the porous medium is not the same as the slope on the other side of the interface. On the other hand, the velocity gradients for model 1 are equal on both sides. It should be noted that in the current study, based on the analysis presented in Vafai and Tien (1981), the effective viscosity of the porous medium for model 2 is taken to be µf/ε. Therefore, for a higher porosity medium, the value of the effective viscosity is close to that of the fluid’s.

Models 3 and 4 present a jump in the shear stress while model 5 introduces a slip in the velocity at the interface region.

When the inertia effects are negligible, model 4 becomes identical to model 3.

Therefore, model 3 is a special case of model 4. Neale and Nader (1974) proposed a mathematical representation for the slip coefficient for model 5, they predicted that

α*=√(µeff/µ). Model 5 indicates that the velocity gradient in the fluid side is proportional to the difference between the interfacial velocity and the free stream

36 velocity in the porous medium while inversely proportional to the permeability of the porous medium. The velocity slip coefficient α* for model 5 is determined numerically by matching the velocity gradient of model 5 for each individual case according to the physical parameters for this case as described in Sahraoui and

Kaviany (1992). According to the experimental study of Gilver and Altobelli (1994), the ratio of the effective viscosity, µeff, to the fluid viscosity, µf, is in the range of

5.1<µeff/µf<10.9. A mean value of 7.5 was recommended by this study. Physically this range of values appears to be quite high. Nevertheless, a side study for the effect of choosing the effective viscosity is performed in this study and presented in Figs. 3.3 and 3.4. Figure 3.3 shows that changing the effective viscosity from µf to 7.5µf has a relatively minor effect on the velocity profiles considering such a wide range of variations for the effective viscosity. The smaller velocity peak and interfacial velocity are associated with the highest effective viscosity. The effect of changing the effective viscosity on the temperature and Nusselt number distributions is shown in Fig. 3.4. It is clear that changing the effective viscosity even within such a wide range has an insignificant effect on the thermal field.

The comparative representations of velocity given in Fig. 3.5 as well as temperature and Nusselt number given in Fig. 3.6 will be used as baseline for studying the effects of different pertinent parameters. Comparisons between the five primary fluid interface conditions are given in Figs. 3.7 to 3.18. Figure 3.5 is used as a baseline in which moderate values of the pertinent parameters are used. Figures 3.7-3.8 show

37 the effect of porosity variation on the velocity field. It is clear that models 1 and 5 are unaffected by changing the porosity, this is expected as the boundary conditions in these models do not include a porosity term. Model 2 is the most affected by changing the porosity. An increase in the porosity causes model 2 to approach models 1 and 5 and a decrease in the porosity drives it in the opposite direction, i.e., toward models 3 and 4. In other words, higher porosity creates a higher peak velocity and smaller porosity creates a smaller peak velocity for model 2 in the plain medium.

The effect of variations in the inertia parameter is shown in Figs. 3.9-3.10. It is clear that an increase in the inertia parameter results in a slightly larger peak velocity for model 5 as compared to model 1. At the same time, an increase in the inertia parameter results in a smaller interfacial velocity for the first four models and a slightly larger interfacial velocity for the fifth model. The slight increase in the interfacial velocity for model 5 may be ascribed to the fact that this is the only model that does not account for the boundary effect in the momentum equation within the porous medium. The effect of Darcy number variation on the velocity field is shown in

Figs. 3.11-3.12. It can be seen that for smaller Darcy numbers, the velocity profiles of all the five primary models are very close to each other, while large Darcy numbers induces a discrepancy among the models. This is because higher Darcy numbers translate into higher permeabilities in the porous side of the interface, which in turn results in a smaller ratio between the average velocity in the plain medium and the

Darcian velocity. As a result of this smaller velocity ratio, the discrepancy between

38 these models becomes more pronounced. However, it should be noted that for most practical applications Da > 10-4 and as such the variation among the five models becomes less significant.

The effect of variations in Reynolds number is shown in Fig. 3.13-3.14. For larger Reynolds numbers the deviation between the velocity profiles for models 3 and

4 increases due to the presence of u2 term in the interface condition for model 4.

Reynolds number has a similar effect as the inertia parameter had on models 1 and 5.

A similar result was found in Vafai and Kim (1990a). Figures 3.15-3.16 show the effect of variations in the coefficient β1. It indicates that for smaller values of β1 the velocity profiles for models 2 and 3 collapse on each other. On the other hand, for larger β1, velocity profiles for models 3 and 4 approach each other. This is because the linear term in the interface condition for model 4 becomes more dominant for larger values of β1.

Large values of β1 may cause a sharp change in the velocity gradient on both sides of the interface as shown in Fig. 3.16. The effect of variations in parameter β2 on the velocity field is shown in Figs. 3.17-3.18. Smaller values of β2 causes the velocity profile for model 4 to collapse on that for model 3 while larger values of β2 increases the divergence between these two models and creates a sharper slope for the velocity profile for model 4.

39 In general, models 1 and 5 are closer to each other while models 3 and 4 are closer to each other. Model 2 usually falls in between the results for models 1, 5 and models 3, 4. Interfacial velocities for models 2, 3, 4 and 5 in terms of the interfacial velocity for model 1 and pertinent parameters such as the Darcy number, the Reynolds number, the inertia parameter and the porosity are given below:

U = [−0.057 +1.0194ε 0.8762 + 0.00082544Λ int 2 (3.11) 0.3893 +0.0363(1000 Da) + 0.0010228Re] Uint1

11.7265 Uint3 = [0.7077ε + 0.0202Λ − 54.9355Da − 0.0082205Re−0.053β1]Uint1 (3.12)

U = [−14.9258 + 0.4185ε + 0.0012816 Λ +15.1505(1000Da)0.000643 int 4 (3.13) 2 −0.0117 Re− 0.0848 β1 + 0.0050702 β1 − 0.0129 β2 ]Uint1

Uint5 = []1.0107 − 0.169ε + 0.0077721Λ + 58.9397 Da + 0.005532Re Uint1 (3.14)

In the above correlations, Uint1 refers to the interface velocity based on model 1 and Uint2, Uint3, Uint4 and Uint5 refer to interface velocities based on models 2, 3, 4 and 5 respectively.

Exact (Vafai and Kim) 3.5 Numerical

2.5

Y 2

1.5

0.5

0 0 2 4 6 8 10 12 14 16 18 U

Figure 3.2: Comparison between the exact solution of Vafai and Kim (1990a) and the present numerical results.

1.8

1.6

1.4 µ = µ eff f 1.2 µ = 4µ eff f µ = eff 2µ f

Y 1

0.8

0.6 µ = 7.5µ eff f 0.4

0.2

0 0 20 40 60 80 100 120 140 U

Figure 3.3: Effect of changing the effective viscosity on velocity for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Da=0.001, Re=1.0.

X 12 10 8 6 4 2 0 1 0

0.8 5

µ eff = µ f 0.6 10 µ eff =2 µ f µ = µ

Θ eff f Nu µ eff =4 µ f 0.4 15 µ eff =2 µ f µ eff =7.5 µ f µ eff =4 µ f 0.2 20

µ eff =7.5 µ f

0 25 0 0.5 1 1.5 2 Y

Figure 3.4: Effect of changing the effective viscosity on temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer,

ε=0.7, Λ=1.0, Da=10-3, Re=1.0.

2 model 1 1.8 2 3 1.6 4 5 1.4

1.2 Model 2

Y 1 Models 1, 5

0.8

0.6 Models 3, 4 0.4

0.2

0 0 20 40 60 80 100 120 140 U

Figure 3.5: Velocity field for the interface between a porous medium and a fluid layer,

-3 ε=0.7, Λ=1.0, Da=10 , Re=1.0, β1=1.0, β2=1.0, αT=10.0, φ=10

X 12 10 8 6 4 2 0 1 0

0.8 5

0.6 10

Θ Models I, II, Nu 0.4 III, IV 15 Models I, II, III, IV 0.2 20

0 25 0 0.5 1 1.5 2 Y

Figure 3.6: Temperature field and Nusselt number distribution for the interface

-3 between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Da=10 , Re=1.0, β1=1.0,

β2=1.0, αT=10.0, φ=10

2 model 1 1.8 2 3 1.6 4 5 1.4

1.2 Model 2

Y 1 Models 1, 5

0.8

0.6 Models 3, 4 0.4

0.2

0 0 20 40 60 80 100 120 140 U

Figure 3.7: Effect of porosity variation on the velocity field for the interface between a porous medium and a fluid layer, Λ=1.0, Da=0.001, Re=1.0, β1=1.0, β2=1.0, ε=0.5

2 model 1 1.8 2 3 1.6 4 5 1.4

1.2 Models 1, 5

Y 1 Model 3 0.8

0.6 Model 4 0.4

0.2 Model 2 0 0 20 40 60 80 100 120 140 U

Figure 3.8: Effect of porosity variation on the velocity field for the interface between a porous medium and a fluid layer, Λ=1.0, Da=0.001, Re=1.0, β1=1.0, β2=1.0, ε=0.9

2 model 1 1.8 2 3 1.6 4 5 1.4

1.2 Models 1, 5

Y 1 Model 2

0.8

0.6 Models 3, 4 0.4

0.2

0 0 20 40 60 80 100 120 140 U

Figure 3.9: Effect of inertia parameter variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Da=0.001, Re=1.0, β1=1.0, β2=1.0,

Λ=0.1

2 model 1 1.8 2 3 1.6 4 5 1.4

1.2 Models 1, 5

Y 1 Model 2 0.8

0.6 Models 3, 4 0.4

0.2

0 0 20 40 60 80 100 120 140 U

Figure 3.10: Effect of inertia parameter variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Da=0.001, Re=1.0, β1=1.0, β2=1.0,

Λ=10.0

2 model 1 1.8 2 3 1.6 4 5 1.4

1.2 Models 1, 5

Y 1 Model 2 0.8

0.6 Models 3, 4 0.4

0.2

0 0 200 400 600 800 1000 1200 1400 U

Figure 3.11: Effect of Darcy number variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Re=1.0, β1=1.0, β2=1.0,

Da=10-4

2 model 1 1.8 2 3 1.6 4 5 1.4

1.2 Model 2

Y 1 Models 1, 5

0.8

0.6 Models 3, 4 0.4

0.2

0 0 20 40 60 80 100 120 140 U

Figure 3.12: Effect of Darcy number variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Re=1.0, β1=1.0, β2=1.0,

Da=10-3

model 1 1.8 2 3 1.6 4 5 1.4

1.2

Model 2 Models 1, 5

Y 1

0.8

0.6 Model 4 0.4

0.2 Model 3

0 0 20 40 60 80 100 120 140 U

Figure 3.13: Effect of Reynolds number variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Λ=10.0, Da=0.001, β1=1.0, β2=1.0,

Re=10.0

2 model 1 1.8 2 3 1.6 4 5 1.4

1.2 Model 1 Model 2 Model 5

Y 1

0.8

0.6 Model 4 0.4

0.2 Model 3 0 0 50 100 150 U

Figure 3.14: Effect of Reynolds number variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Λ=10.0, Da=0.001, β1=1.0, β2=1.0,

Re=100.0

2 model 1 1.8 2 3 1.6 4 5 1.4

1.2 Models 1, 5

Y 1 Models 2, 3 0.8

0.6 Model 4 0.4

0.2

0 0 20 40 60 80 100 120 140 U

Figure 3.15: Effect of β1 variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Da=0.001, Re=1.0, β2=1.0, β1=0.1

2 model 1 1.8 2 3 1.6 4 5 1.4

1.2 Models 1, 5 Model 2

Y 1

0.8

0.6 Models 3, 4 0.4

0.2

0 0 20 40 60 80 100 120 140 U

Figure 3.16: Effect of β1 variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Da=0.001, Re=1.0, β2=1.0, β1=10

2 model 1 1.8 2 3 1.6 4 5 1.4

1.2 Models 1, 5

Y 1 Model 2 0.8

0.6 Models 3, 4 0.4

0.2

0 0 20 40 60 80 100 120 140 U

Figure 3.17: Effect of β2 variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Da=0.001, Re=1.0, β1=1.0, β2=0.1

2 model 1 1.8 2 3 1.6 4 5 1.4

1.2 Models 1, 5

Y 1 Model 2

0.8

0.6 Model 4 0.4

0.2 Model 3 0 0 20 40 60 80 100 120 140 U

Figure 3.18: Effect of β2 variation on the velocity field for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0, Da=0.001, Re=1.0, β1=1.0, β2=10

3.3.2 HEAT TRANSFER:

Figure 3.4 shows the temperature profiles and the local Nusselt number distributions for different ratios of µeff/µf. In this figure, the temperature and Nusselt number distributions for the entire wide range of variations in µeff/µf are quite close to each other. The first type of heat transfer interface condition, model I is based on a continuity of temperature as well as heat flux at the interface region. The second model includes a jump in the heat flux at the interface region. As a logical extension the fluid flow model 1 is in conjunction with the heat transfer models I and II. The third (III) and the fourth (IV) heat transfer interface condition models have a jump in the temperature at the interface. However, the heat transfer interface model III utilizes the fluid flow model 1 while thermal model IV utilizes the fluid flow model 3.

The effects of the pertinent parameters on the temperature and Nusselt number distributions are shown in Figs. 3.19 to 3.25. Figure 3.6 is used as the datum for temperature and Nusselt number comparisons. Figure 3.19 shows the effect of porosity. Higher porosity results in a better agreement between models I and II while it decreases the discrepancies between the interfacial temperatures for models III and IV.

In other words, higher porosity causes the results for fluid flow models 1 and 3 to be closer to each other, which in return coalesces the results for models III and IV.

58 The effects of inertia parameter variation are shown in Fig. 3.20. It has a relatively insignificant effect on the convergence or divergence of the temperature and

Nusselt number profiles for the four models shown in Table 3.2. However, a careful examination indicates an increase in the inertia parameter produces convergence in the interfacial temperature for all the four models. This is due to the higher values of the inertia parameter, which results in larger convective heat transfer, which in turn decreases the effect of different interface conditions on the heat transfer results.

The effects of Darcy number variation, Da, are shown in Fig. 3.21. It is clear that for more practical values of Darcy number, all the four models displayed in Table

2 converge. For very large Darcy numbers, the discrepancy in the interfacial temperatures for the four models becomes more pronounced. Model IV has the highest interface temperature, model III has the next highest while models I and II have the lowest interfacial temperatures. However, the Nusselt numbers for the first three models are very close to each other with a slight deviation from the fourth model as seen in Fig. 3.21. Figure 3.22 displays the effect of Reynolds number variation on the temperature field and the Nusselt number amongst the four models. As the

Reynolds number increases a better agreement is achieved among the four models.

The effect of variations in β1 is shown on Fig. 3.23. It is found that higher values of β1 create a slightly higher interfacial temperature for model IV while smaller values of β1 result in a closer agreement among the other three models. However,

59 overall all four models are in very close agreement even for significant variations in

β1. Figure 3.24 shows the effect of variations in the temperature slip coefficient on temperature and Nusselt number profiles. It can be seen that the effect of the temperature slip coefficient is insignificant compared to the effects of the Darcy number and the Reynolds number. Finally, the effect of variations in φ is shown in

Fig. 3.25. As can be seen for larger φ, the temperature distribution for model II in the non-porous region starts to deviates from the other three models. However, the effect of parameter φ on the Nusselt number is insignificant. This is because the temperature gradient at the lower surface is not significantly affected by changing φ. In general, for most cases the results obtained from all the four models are quite close to each other.

However, for few cases described earlier, there are some minor deviations. However, even those deviations still can be considered insignificant. Interfacial temperatures for models II, III and IV in terms of the pertinent parameters and the non-dimensional interfacial temperature of the first model are given below:

Θ = [1.0132 − 0.0077072ε − 0.0010277Λ +1.0272 Da int 2 (3.15) −0.0063711Re− 0.0024633φ]Θint1

Θ = [0.3765 + 0.6459ε − 0.0194Λ + 48.2034 Da int 3 (3.16) −0.0444Re− 0.0023708αT ]Θint1

Θ = [0.5576 + 0.4869ε − 0.0181Λ + 60.4243Da − 0.0349Re int 4 (3.17) −0.0045598β1 − 0.0011282αT ]Θint1 60 Also, average Nusselt numbers for models II, III and IV in terms of the pertinent parameters and the average Nusselt number for the first model are given below:

____ Nu = [1.0 − 0.0487 Da + (12.586ε + 0.27205Λ 2 (3.18) ____ −6 +6.8477Re− 5.0257φ )10 ] Nu1

____ Nu = [0.9999 + 0.1952 Da + ( 60.873ε −10.0992Λ 3 (3.19) ____ −6 −14.895Re− 0.91682αT )10 ] Nu1

____ Nu = [1.5686ε + 0.0069482Λ + 6.0586 Da + 0.003314Re 4 (3.20) ____

− 0.0065775β1 − 0.0015485αT ] Nu1

In the above correlations, Θint1, Θint2, Θint3 and Θint4 are the non-dimensional

______interface temperatures and Nu1, Nu2 , Nu3 and Nu4 are the average Nusselt numbers for models I, II, III and IV respectively.

X 12 10 8 6 4 2 0 1 0

0.8 5 Models I, II, III, IV (ε =0.5 and ε =0.9) 0.6 10 Θ Models I, II, Nu 0.4 III, IV (ε =0.9) 15 Models I, II, 0.2 III, IV (ε =0.5) 20

0 25 0 0.5 1 1.5 2 Y

Figure 3.19: Effect of porosity variation on temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer, Λ=1.0,

-3 Da=10 , Re=1.0, β1=1.0, αT=10.0, φ=10.0.

X 12 10 8 6 4 2 0 1 0

0.8 5 10 0.6

Θ Models I, II, III, IV 15 Models I, II, III, IV Nu 0.4 Λ=0.1 and Λ=10 Λ=0.1 and Λ=10 20

0.2 25

0 30 00.511.52 Y

Figure 3.20: Effect of inertia parameter variation on temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer,

-3 ε=0.7, Da=10 , Re=1.0, β1=1.0, αT=10.0, φ=10.0.

X 12 10 8 6 4 2 0 1 0

10 0.8 Models I, II, III, IV -3 20 0.6 Da=10

Θ Models I, II, III, IV 30 Nu -4 Models I, II, III, IV 0.4 Da=10 -4 Da=10 40

0.2 Models I, II, III, IV 50 Da=10-3 0 60 00.511.52 Y

Figure 3.21: Effect of Darcy number variation on temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer,

ε=0.7, Λ=1.0, Re=1.0, β1=1.0, αT=10.0, φ=10.0.

X 12 10 8 6 4 2 0 1 0

0.8 30 Models I, II, III, IV 0.6 Models I, II, III, IV Re=10

Θ Re=100 60 Nu 0.4 Models I, II, III, IV Re=100 90 0.2 Models I, II, III, IV Re=10 0 120 00.511.52 Y

Figure 3.22: Effect of Reynolds number variation on temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer,

-3 ε=0.7, Λ=1.0, Da=10 , β1=1.0, αT=10.0, φ=10.0.

X 12 10 8 6 4 2 0 1 0

0.8 5

0.6 Models I, II, III, IV 10 β =0.1 and β =10

Θ 1 1 Nu 0.4 Models I, II, III, IV 15 β1=0.1 and β1=10 0.2 20

0 25 00.511.522.5Y

Figure 3.23: Effect of β1 variation on the temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0,

-3 Da=10 , Re=1.0, φ=10.0, αT=10.0

X 12 10 8 6 4 2 0 1 0

0.8 5

0.6 10 Models I, II, III, IV Θ Nu 0.4 αT=1 and αT=100 Models I, II, III, IV 15 αT=1 and αT=100 0.2 20

0 25 0 0.5 1Y 1.5 2 2.5

Figure 3.24: Effect of αT variation on the temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0,

-3 Da=10 , Re=1.0, φ=10.0, β1=1.0

X 12 10 8 6 4 2 0 1 0

0.8 5

Models I, II, III, IV 0.6 10 φ=1 and φ=100 Θ

Model II Nu 0.4 φ=100 15 Models I, III, IV (φ=1 and φ=100) 0.2 20 and model II (φ=1)

0 25 0 0.5 1 1.5 2 Y

Figure 3.25: Effect of φ variation on the temperature field and Nusselt number distribution for the interface between a porous medium and a fluid layer, ε=0.7, Λ=1.0,

-3 Da=10 , Re=1.0, β1=1.0, αT=10.0

3.4 CONCLUSIONS:

A comprehensive comparative analysis of the hydrodynamic and thermal interfacial conditions between a porous medium and a fluid layer is presented in this work. Five primary categories for the hydrodynamic interface conditions and four primary forms for the thermal interface conditions were analyzed in detail. The main objective of the present study was to analyze the variances among these models and to attest the effects of using them on the characteristics of heat and fluid flow at the interface region. The results of this investigation systematically quantify and characterize the effect of the pertinent controlling parameters on the variances among different interface conditions. It is shown that for most cases the variances within different models, for most practical applications, have a negligible effect on the results while for few cases the variations can become significant. In general, the variances have a more pronounced effect on the velocity field and a substantially smaller effect on the temperature field and yet even smaller effect on the Nusselt number distribution. For hydrodynamic categories, results from models 1 and 5 of Table 3.1 generate very close results which tend to cluster quite closely and models 3 and 4 generate results which are relatively close to each other, while model 2 generally falls in between these two sets. For heat transfer interface conditions displayed in Table 2, all four categories generate results which are quite close to each other for most practical applications. However, small discrepancies could appear for applications

69 dealing with small values of Re and/or larger values of Da. The effect of choosing the effective viscosity was found to have a relatively small influence on the velocity field and an insignificant effect on the temperature and local Nusselt number distributions.

Finally, a set of correlations were provided for interchanging the interfacial velocity, the interfacial temperature and the average Nusselt number among different models.

CHAPTER 4

CONSTANT WALL HEAT FLUX BOUNDARY CONDITIONS IN POROUS MEDIA UNDER LOCAL THERMAL NON-EQUILIBRIUM CONDITIONS

Boundary conditions for constant wall heat flux in the absence of local thermal equilibrium conditions are analyzed in this work. Effects of variable porosity and thermal dispersion are also analyzed. Different forms of constant heat flux boundary conditions found in the literature were investigated in this work. The effects of pertinent parameters such as porosity, Darcy number, Reynolds number, inertia parameter, particle diameter and solid-to-fluid conductivity ratio were analyzed.

Quantitative and qualitative interpretations of the results are utilized to investigate the prominent characteristics of the models under consideration. Limiting cases resulting convergence or divergence of the models are also considered. Results are presented in terms of the fluid, solid and total Nusselt numbers.

71 4.1 INTRODUCTION

In recent years, the problem of local thermal non-equilibrium (LTNE) has received considerable attention due to its relevance in a wide variety of engineering applications such as electronic cooling, heat pipes, nuclear reactors, drying technology, multiphase catalytic reactors and others. The use of two-equation model is required for these types of problems. Each energy equation, one for the fluid phase and the other for the solid phase, requires a boundary condition at the solid boundary.

Hitherto, it is not clear what two boundary conditions might be used for the case of constant wall heat flux. In contrast, boundary conditions for constant wall temperature are clear, both phases should have a temperature that equals a prescribed wall temperature.

The study of Amiri et al. (1995) was one of the first attempts to highlight this problem and present two different approaches for boundary conditions for constant wall heat flux. The first approach presented in their work was based on assuming the total heat flux qw being divided between the two phases depending on the physical values of their effective conductivities and their corresponding temperature gradients at the wall. The second approach also presented in Amiri et al. (1995) assumes that each of the individual phases at the wall receives an equal amount of the total heat flux qw. In their study, good agreements were found between the numerical results using the second approach and the available experimental results. On the other hand, Hwang

72 et al. (1995) used the first approach and found good agreement between their numerical and experimental results. Lee and Vafai (1999) used the first approach to obtain analytical solutions for the temperature profiles, the temperature difference between the two phases, and the Nusselt number. Several other studies such as Kim and Kim (2000), Hwang and Chao (1994), Nield (1998), Nield and Kuznetsov (1999),

Kuznetsov (1997b) and Calmidi and Mahajan (1998) considered the first approach to investigate the problem of LTNE in porous media. Martin et al. (1998) used the first approach with the assumption of local thermal equilibrium at the wall. That is the temperature gradients for both phases were taken to be the same in their work. Jiang and Ren (2001) considered four boundary conditions for the case of constant heat flux.

These were composed of the two approaches presented by Amiri et al (1995), the model presented by Martin et al. (1998), and a fourth model which can be considered as a very special case of the first approach. They reported that using the second approach corresponds well to the experimental results of Jiang et al. (1999a).

Correspondingly, the second approach was employed in other studies such as Jiang et al. (1999b), Jiang et al. (1996), Ichimiya and Matsuda (1997), Mahmid et al. (2000) to study the problem of thermal non-equilibrium.

The main objective of the present study is to analyze the effect of using different boundary conditions for the case of constant wall heat flux under LTNE conditions. In addition to the above-mentioned boundary conditions, five pertinent new boundary conditions are introduced in the present study. Pertinent parameters

73 such as porosity, Reynolds number, Darcy number, inertia parameter, particle diameter, and solid-to-fluid conductivity ratio are considered to assess and compare the physical features of different boundary conditions under investigation.

4.2 ANALYSIS

A fundamental unit for analyzing this problem is selected. As such a parallel plate channel subject to an imposed constant heat flux is considered to analyze the proper set of boundary conditions in porous media under LTNE conditions. Figure 4.1 shows the schematic diagram of the problem under consideration. The steady state volume averaged governing equations are, Amiri et al. (1995) and Amiri and Vafai

(1994),

Continuity

∇ V = 0 (4.1)

Momentum

ρ µ ρ Fε µ f ()V∇ V = − f V − f ()V V J + f ∇ 2 V − ∇ P f (4.2) ε Κ Κ ε

74 Fluid phase energy

f f f s f ρ C V ∇ T = ∇k ∇ T  + h a  T − T  (4.3) f p f f  feff f  sf sf  s f 

Solid phase energy

s  s f  0= ∇()kseff ∇ Ts − hsf asf  Ts − T f  (4.4)  

The fluid-to-solid heat transfer coefficient is expressed as, Amiri et al. (1995) and

Amiri and Vafai (1994),

0.6 k  1  ρ ud   h = f 2 +1.1Pr 3  f p   (4.5) sf d   µ   p   f  

While the specific surface area of the bed can be expressed as

6(1− ε) asf = (4.6) d p

Effective conductivities of both phases are defined as

k feff = ε k f (4.7)

kseff = (1− ε ) ks (4.8)

75 In some cases, it is important to account for the effects of variable porosity and thermal dispersion. Effective conductivities of the fluid phase can be represented as

Amiri et al. (1995),

  ρ u d  k = ε + 0.5Pr f p  k (4.9) ()feff x    f   µ 

  ρ u d  k = ε + 0.1 Pr f p  k (4.10) ()feff y    f   µ 

and variable porosity can be expressed as Vafai (1984),

  − c y  ε = ε 1+ b exp  (4.11) ∞      d p 

Furthermore, permeability and the geometric function F are expressed as Vafai (1984),

ε 3 d 2 Κ = p (4.12) 150(1− ε)

1.75 F = (4.13) 150ε 3

Local Nusselt numbers for both phases are defined according to Amiri and Vafai

(1994) as

 f  2H  ∂ T f  Nu = − (4.14) f f f  ∂y  T f − T f   w m   y=0

76  s  2H  ∂ Ts  Nus = − (4.15) T s − T s  ∂y  s w s m   y=0

f Where the mean temperatures T and T s are defined as f m s m

f f 1 H T = u T dy (4.16) f m ∫ f U m H 0

H s s 1 T = T dy (4.17) s m ∫ s H 0

Where

1 H U m = ∫ u dy (4.18) H 0

______The average Nusselt numbers ( Nu f and Nu s ) over the length of the bed are expressed as

___ 1 L Nu f = Nu dx (4.19) ∫ f L 0

___ 1 L Nu s = Nu dx (4.20) ∫ s L 0

____ Consequently, the total Nusselt number ( Nu t ) is defined as the summation of

______Nu f and Nu s :

______Nu t = Nu f + Nu s (4.21)

77 Mathematical representations of the pertinent boundary conditions for the case of constant heat flux are shown in Table 4.1. Corresponding references to each model are listed in Table 4.2. Six models (1A, 1B, 1C, 1D, 1E and 1F) are based on the first approach and two models (2A and 2B) are based on the second approach presented in

Amiri et al. (1995).

uc H y Te

x qw L

Figure 4.1: Schematic diagram of the problem of constant wall heat flux boundary conditions in porous media under LTNE conditions and the corresponding coordinate system.

Approach Model Mathematical Representation

∂Tf ∂Ts qw = −k f − ks T f = Ts = Tw 1A eff ∂y |wall eff ∂y |wall AND w w

∂Tf qw = −[]εk f + (1−ε)ks T f = Ts = Tw 1B ∂y |wall AND w w

∂Ts qw = −[εk f + (1−ε)ks ] T f = Ts = Tw 1C ∂y |wall AND w w

q ∂Tf ∂Ts f ε qw = −k f − ks = a eff | eff | AND 1D ∂y wall ∂y wall qs 1− ε

∂Tf ∂Ts q f k f qw = −k f − ks = eff | eff | AND First 1E ∂y wall ∂y wall qs ks

∂Tf ∂Ts q f ε k f qw = −k f − ks = a eff | eff | AND 1F ∂y wall ∂y wall qs (1− ε)ks

∂Tf ∂Ts qw = −k f = −ks 2A eff ∂y |wall eff ∂y |wall

∂T f ∂Ts Second qw = −k f = −k s 2B ∂y |wall ∂y |wall

Table 4.1: Summary of different models based on the two approaches of constant wall heat flux boundary conditions.

a Free stream porosity ‘ε∞’ is used when the effect of variable porosity is considered.

Approach Model References Amiri et al. (1995), Hwang et al. (1995), Lee and Vafai (1999), Kim and Kim (2000), Hwang and Chao (1994), 1A Nield (1998), Nield and Kuznetsov (1999), Kuznetsov (1997),

1B Calmidi and Mahajan (1998), Martin et al. (1998) First 1C Present study 1D Present study

1E Present study

1F Present study

2A Amiri et al. (1995) Second Martin et al. (1998), Jiang and Ren (2001), Jiang et al. 2B (1999)

Table 4.2: Summary of corresponding references to different models based on the two approaches of constant wall heat flux boundary conditions.

4.3 NUMERICAL METHODOLOGY

The governing equations were solved numerically using the finite difference method. No-slip boundary conditions at the wall were employed for the momentum equation while appropriate boundary conditions from Table 4.1 were used for the energy equations. The momentum equation was solved by a tridiagonal scheme after linearizing the Forchheimer term. Since the energy equations were coupled, an implicit iterative method was used to solve the temperature fields. Central differencing was used for the diffusion terms while upwind differencing was used for the convective terms. The convergence criterion was satisfied when the absolute difference between two consecutive iterations was less than 10-6. Variable grids in the y-direction and uniform grids in the x-direction were employed in the present study.

4.4 RESULTS AND DISCUSSION

Comparison between the numerical result for the velocity and the exact solution given in Vafai and Kim (1989) is shown in Fig. 4.2a. Since there is no analytical solution of the LTNE that counts for the effects of axial conduction, variable porosity, and thermal dispersion, the current results are compared with the numerical results of Amiri and Vafai (1994). Excellent agreement was found between

82 the present results and the results presented in Amiri and Vafai (1994) as shown in

Fig. 4.2b.

Table 4.1 displays the detailed mathematical representation of the boundary conditions for the case of constant wall heat flux. Models 1A and 2A are the conventional boundary conditions presented by Amiri et al. (1995). Model 1A indicates that qw is divided between the two phases on the basis of their effective thermal conductivities and their corresponding temperature gradients. Model 2A suggests that each individual phase receive exactly the same amount of the heat flux qw. These two approaches are broad and other models can be considered as extensions of them. Models 1B and 2B were used in previous studies of Calmidi and Mahajan

(1998), Martin et al. (1998), Jiang and Ren (2001) and Jiang et al. (1999). In model

1B, the total heat flux qw is assumed to have the same representation as the case of local thermal equilibrium. In other words, it is assumed that both phases have the same temperature and temperature gradient at the wall. As mentioned above, the present problem requires only two boundary conditions at the wall. However, having the same temperature and temperature gradient at the wall in addition to the constant heat flux boundary condition provides three boundary conditions, this leads to an over specified problem. As such, the fluid phase temperature gradient of model 1B is chosen to represent the heat flux at the solid boundary and model 1C is introduced to investigate the effect of using the solid phase temperature gradient instead of the fluid phase temperature gradient, as in model 1B, in the expression of the wall heat flux.

The generic models (1D, 1E and 1F) are extensions of the first approach, they are introduced for the first time in the present study. Model 1D suggests that the ratio of the fluid phase heat flux to the solid phase heat flux (qf/qs) depend on the porosity of the porous medium. More precisely, it is assumed that the ratio qf/qs is proportional to the ratio ε/(1-ε). Model 1E states that the ratio qf/qs is proportional to the fluid-to- solid conductivity ratio κ. As shown in Table 1, model 1F is a combination of the previous two models. Finally, model 2B advocates the second approach and states that the heat flux of each phase is equal to the thermal conductivity of each phase multiplied by the temperature gradient. When the thermal conductivities of both phases are constants (thermal dispersion effects are excluded), model 2B becomes identical to model 1D. This example explains the probable equivalence between the models listed in Table 4.1. Therefore, an extensive investigation is performed in the present study to clarify the differences and/or the similarities between the eight models. Effects of porosity, Darcy number, Reynolds number, inertia parameter, particle diameter, and the solid-to-fluid conductivity ratio are presented. Physical properties and configuration are chosen to be similar to the ones in Amiri et al.

(1995). Results for the simplified case where the porosity is considered constant and effects of thermal dispersion are excluded are presented in Figs. (4.3-4.10). Results of the generalized model that accounts for the effects of variable porosity and thermal dispersion are demonstrated in Figs. (4.11-4.14).

(a) 1.2 Exact solution of Vafai and Kim (1989) Present results 1

0.8

U 0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 Y

(b) 700 Fluid phase Nusselt number [Amiri and Vafai (1994)] Fluid phase Nusselt number (Present) 600 Solid phase Nusselt number [Amiri and Vafai (1994)] Solid phase Nusselt number (Present) 500

400 NU 300 200

100

0 0 0.2 0.4 0.6 0.8 1 X

Figure 4.2: Validation of the present work. (a) Comparison of the velocity profiles -3 with the analytical solution of Vafai and Kim (1989). Da=10 , Λ=70.372, Rep=200. (b) Local Nusselt numbers of the present study against the results of Amiri and Vafai -6 (1994), αs/αf=25.6, Da=1.36×10 and Rep=100. 85 4.4.1 CONSTANT POROSITY, NO THERMAL DISPERSION

It is clear that all the boundary conditions shown in Table 4.1 depend on the porosity, the thermal conductivities, and the temperature gradients. An important aspect of the present work is the comparison of these different models when the porosity is constant and the thermal dispersion effect is excluded. Table 4.3 shows the mathematical representation for qw for each model for the special cases of ε=0.5 or

κ=1, and the combination of ε=0.5 and κ=1. As shown in Table 4.3, it is found that models 1D, 1F, and 2B are the same when the solid-to-fluid conductivity ratio has a value of unity. When the porosity has a value of 0.5, models 1E and 1F become identical. When the two mentioned effects are combined (ε=0.5 and κ=1), models 1D,

1E, 1F, and 2B become identical. Models 1A, 1B, and 1C become identical when the fluid and solid temperature gradients are the same, which happens in case of thermal equilibrium at the wall. In general, all the models converge to the same limit when the case (ε=0.5 and κ=1) is satisfied. It is clear from Tables 4.1 and 4.3 that model 1A always falls between models 1B and 1C. In fact, when models 1B and 1C give the same results, it means that there is a thermal equilibrium at the wall. In other words, models 1B and 1C do not become identical unless the temperature gradients of both phases are the same at the solid boundary. It is important to note that temperature gradients of model 2A for the special case of (ε=0.5 and κ=1) are almost twice the ones for the other models.

For simplicity, model 1D will not be mentioned in this discussion since it becomes identical to model 2B. Effect of the porosity is illustrated in Fig. 4.3. Figure

4.3a shows the effect of porosity on the fluid phase Nusselt number. For relatively small porosities, models 1A, 1B, 1C, 1E and 1F compose a regime of lower Nusselt number values while models 2A and 2B form another regime of higher Nusselt number values. Models 1A, 1B, 1F and 2B converge to the same limit for very high porosities. In addition, models 2A and 1C are close to these models although they do not approach the same limit. On the other hand, model 1E diverges from all the other models for very high porosities. Figure 4.3 and Table 4.3 show that models 1E and 1F are exactly alike for the special case of ε=0.5. The same is found in Fig. 4.3 for models

2A and 2B. However, results of Table 4.3 suggest that temperature gradients of model

2A are twice those of model 2B for the special case of ε=0.5. It is clear that models 1A and 1B approach the same limit as the porosity approaches unity. Theoretically, model

1A is the same as model 1C when the porosity approaches zero while it is the same as model 1B when the porosity approaches unity. So, model 1A would fall between models 1B and 1C for moderate porosity values. An increase in the porosity causes the

____ fluid phase Nusselt numbers ( Nu f ) of models 1A, 1B, 1C and 1F to increase while an increase in the porosity results in the opposite effect for models 2A, 2B and 1E. As for the solid phase Nusselt number, Fig. 4.3b shows that small porosities cause models

1A, 1C, 1E, and 1F to converge while large porosities cause the models to diverge except models 1A and 1B where they approach the same limit when the porosity 87 approaches unity. An increase in the porosity causes the solid phase Nusselt numbers

______( Nus ) of all the models to increase except model 2B where Nu s decreases gradually as the porosity increases. Figure 4.3c demonstrates the effect of porosity on the total

Nusselt number. As mentioned earlier, models 2A and 2B still have the intersection when the porosity is 0.5. Again, the same happens for models 1E and 1F. It is clear that models 1A, 1B, 1F, and 2A converge to the same value when the porosity approaches unity. Models 1C and 2B are still relatively close to these models while model 1E is quite apart from all the models. For relatively small porosities, models

1A, 1B, 1C, 1E and 1F compose a regime of lower Nusselt number values while models 2A and 2B form another regime of higher Nusselt number values. Model 1A intersects with model 2A for a porosity value around 0.93. This affirms that the two conventional models may give the same results under some particular circumstances.

Generally, the fluid phase Nusselt numbers have higher values than the solid phase

Nusselt numbers which makes the total Nusselt numbers more affected by the fluid phase Nusselt numbers. This result can be observed by comparing Figs. 4.3a and 4.3c.

Figure 4.4 portrays the effect of the solid-to-fluid conductivity ratio (κ). This ratio κ has a significant impact on the behavior of the represented models. As seen in

Fig. 4.4a when the solid-to-fluid conductivity ratio approaches unity, models 1E and

2A converge to the same limit while the other models approach a different limit. On

____ the other hand, Nu s of models 1A, 1B, 1C, 1E, and 1F converge to the same limit for

88 the case of very large values of κ as shown in Fig. 4.4b. Some intersections between the models occur in the intermediate range. For example, model 2A intersects with models 1A, 1B, 1C and 1F while there is common point between model 2B and models 1A, 1B and 1C. It is interesting to note that models 2B and 1F are the same when κ=1. Temperature gradients of models 1A, 1B, 1C, 1D, 1F and 2B are of the same order of magnitude when κ=1 as seen in Table 4.3.

It is found that all the Nusselt number profiles decrease as the conductivity ratio increases except the fluid phase Nusselt number for model 2A. However, the total Nusselt number of model 2A decreases as κ increases as shown in Fig. 4.4c. The numerous intersections of total Nusselt number profiles for different models demonstrate the fact that κ has a substantial impact on these models. Generally, models 1D and 2B form the upper bound for the fluid phase Nusselt number profiles while model 1E forms the lower bound. On the other hand, the opposite is true for the solid phase Nusselt number profiles where model 1E is the upper bound while models

____ 1D and 2B form the lower bound. The total Nusselt number, the summation of Nu f

____ and Nu s , provides further insight on the general variances of these models.

Figure 4.5 exemplifies the effect of the particle Reynolds number. It is evident

____ from Fig. 4.5 that an increase in Rep results in an increase in Nu t for all models. For small values of Rep, all the models except model 1E are considerably closer to each 89 ____ other as shown in Fig. 4.5a. Model 1E has the lowest Nu t while model 1B has the

____ highest Nu t . Also, there are no intersections between the models as the value of Rep changes. Again, the profiles of model 1A falls in between the ones for models 1B and

1C. As shown in Fig. 4.5b, models 1A, 1D, 1E, 1F, 2A and 2B coincide when ε=0.5 and κ=1. Models 1B and 1C approach the other models for small values of Rep where model 1B forms the upper bound and model 1C forms the lower bound. On the other hand, they diverge from the other models as the value of Rep increases. Table 4.3 shows that temperature gradients of model 2A are twice those of models 1D, 1E, 1F and 2B for the case (ε=0.5 and κ=1). At the same time, Fig. 4.5b shows that model 2A is identical to models 1A, 1D, 1E, 1F and 2B. This interesting result regarding model

2A will be discussed later in this section.

Effects of Darcy number on the variant models are depicted in Fig. 4.6. It is

____ clear that an increase in Da leads to a gradual decrease in the Nu t for all models under consideration. Moreover, models 1B and 2A tend to be closer to each other under higher values of Da. In fact, they intersect when Da is around 4×10-4. Figure 6a shows that once again model 1E is apart from all the other models. However, when the values of the input parameters change as in Fig. 4.6b model 1E coincides with all the other models except models 1B and 1C. Figure 4.6b displays similar results to the ones in Fig. 4.5b although model 1A diverges from models 1D, 1E, 1F, 2A and 2B for small values of Da. The reason for this behavior is that smaller values of Da cause the

90 difference between the fluid and solid temperature gradients of model 1A to increase.

This conclusion can be verified from results of models 1B and 1C, which diverge as

Da decreases. Figures 4.6a and 4.6b show that higher values of Da decrease the disparity between the variant models under consideration.

Effects of the inertia parameter Λ on all models are shown in Fig. 4.7.

Although the Nusselt numbers of all models exhibit a slight increase as Λ increases, it

____ is found that the effect of Λ is insignificant on Nu t for all the listed models in Table

4.1. The total Nusselt number profiles are found to have an almost linear relationship with the inertia parameter Λ. Once again, model 1E is apart from other models and model 1A is sandwiched between models 1B and 1C.

____ Figure 4.8 shows the effect of the particle diameter on Nu t . In general, an

____ increase in the particle diameter dp results in a decrease in Nu t for all the models and an increase in dp attains a convergence between models 1B, 2A and 2B. Again, model

______1E has the lowest Nu t while model 2B has the highest Nu t . When the porosity is constant and the thermal dispersion effects are omitted, the particle diameter only affects hsf and asf. For the case of Fig. 4.8a, higher values of dp cause all the models except 1E to converge while for the case of Fig. 4.8b, smaller values result in convergence among all models.

91 It is worth noting that results obtained based on model 2A are of the same order as other models. Although the temperature gradients of model 2A are twice the values of other models for the case of (ε=0.5 and κ=1) as shown in Table 4.3, its

Nusselt number is exactly the same as others when (ε=0.5 and κ=1) as shown in Figs.

4.5-4.8. Reviewing the definition of the Nusselt number (eqs. 4.14-4.15) explains this phenomenon. In fact, it is evident that the difference between the wall temperature and the mean temperature for model 2A is twice that of the other models. Figure 4.9 demonstrates the effect of κ on the temperature gradients while Fig. 4.10 displays the effect of κ on the difference between the wall temperature and the mean temperature.

____ Although Nu t of model 2A is of the same order of magnitude as the total Nusselt number of other models, its wall temperature and its temperature gradients reach much greater values than other models. It is clear that the order of magnitude of the temperature gradients of model 2A is nearly twice that obtained from other models for the case (κ=1). The same is true for the temperature difference between the wall temperature and the mean temperature. As κ varies, results of the ratio of the temperature gradient to the temperature difference of model 2A and other models change accordingly. Therefore, the Nusselt number of model 2A is of the same order of magnitude as the other models.

At this point, it is found that relating other models to a reference model is desirable. However, designating one model over the others is not an easy issue since some previous studies validated each of the two primary models. In addition to that, 92 the mechanics of splitting the heat flux between the two phases is not yet resolved.

Moreover, it is expected that various effects might cause a set of experimental results to agree with one model over the others. These effects include the variable porosity, the thermal dispersion and the wall thickness. It is known that when the wall boundary has a finite thickness composed of a high conductivity material, the two phases should have the same wall temperature. Therefore, for this class of applications, model 1A is preferable. On the other hand, model 2A is anticipated to be a good representative boundary condition for applications with high wall temperatures and high temperature gradients. Figure 4.9 shows that fluid phase and solid phase temperature gradients of model 2A are the highest among all models under consideration. In addition, results of

Fig. 4.10 show that solid and fluid wall temperatures of model 2A are the highest among all models under consideration. The results in this work indicate that depending on the application area either model 1A or model 2A will be a better representative boundary condition. Following is a set of correlations that relate other models to model 2A:

0.1  ____  3.67 + 2.45 ε 2.55 10.56 + 0.35 (κ / 23.75)  Re   Nu  =[ − + 4.04 p  −  t  3.55 −5.7 −9    1A 0.446 + 0.1 ε 1.48 + 9.2 (κ / 23.75) ×10  100  0.1 0.02 ____ 0.75 0.18  d p   Λ    5   0.44()Da ×10 − 3.94  + 0.51  ] Nut  (4.22)  0.008  100   2 A

93 0.12  ____  3.87 +1.782 ε 1.55 21.39 + 0.38 (κ / 23.75)1.32  Re   Nu  =[ − + 5.18 p  −  t  46.4 −5.79 −8    1B 0.435 + 0.021 ε 2.886 +1.7 (κ / 23.75) ×10  100  0.13 0.062 ____ 0.71 0.16  d p   Λ    5   0.87()Da ×10 − 4.72  + 0.249  ] Nut  (4.23)  0.008  10   2 A

 ____  3.4 + 0.205 ε 1.83 27.89 + 7.15 (κ / 23.75) −4.69 ×10−5  Nut  =[ 27.43 − −4.81 −6 +  1C 0.455 + 0.0056 ε 3.73 + 6.79 (κ / 23.75) ×10 −0.14 −0.076  Re p  5 −0.0018  d p  0.164  − 3.02()Da ×10 − 0.275  +  100   0.008  (4.24) −1.94×10−4 1.13  Λ   ____  3.196  ] Nut  10    2 A

____ −0.019   5.37  κ   Nut  =[9.253 − 4.309ε + 2.204 EXP()1.829 + 0.09ε − 8.537   +  1D,2B  23.75  0.21  κ   Re    −3  p  2.102 EXP2.197 + 3.64   ×10  + 6.937  + (4.25)  23.75   100  0.04 1.26 ____ 0.59 −0.01  d p   Λ    5   26.48()Da ×10 − 57.06  + 0.016  ] Nut   0.008  10   2 A

____ 0.25   0.004  κ   Nut  =[ − 6.915 − 4.096ε + 0.052 EXP()−1.891+ 4.876ε − 5.37  +  1E  23.75  0.1   κ   Re p  5 0.13 2.688 EXP2.292 + 0.017   + 2.688  − 0.355()Da ×10 −  23.75  100  (4.26) 0.033 0.76 1.19 ____  d p   Λ      11.83  + 0.0027  ] Nut   0.008  10   2 A

 ____   9.68×10−3 + 0.433 ε 2.9   6119.63  Nu =[  ×  +  t   −4 −11.39   1.27   1F  27.116 +1.82×10 ε   27.69 + 6.984(κ / 23.75)  0.023 0.014  Re p  5 0.24  d p  12.866  − 0.225()Da ×10 − 23.512  +  100   0.008  0.0005 0.77  Λ   ____  11.651  ] Nut  (4.27) 10   2 A

Limiting cases where models under consideration converge or diverge can be obtained by combining the results of Figs. 4.3-4.8. As mentioned earlier, the inertia parameter effect is insignificant on the convergence or divergence of the models. In addition, effect of the particle diameter is found to be vacillating since it depends on the choice of other parameters as shown in Fig. 4.8. Generally, a porosity value of 0.5, a solid-to-fluid conductivity ratio of unity, low particle Reynolds number values and high Darcy number values cause the models to converge. On the other hand, low porosity values, high solid-to-fluid conductivity ratios, high particle Reynolds number values and small Darcy number values cause the models to diverge.

95 Model κ = 1 ε=1/2 κ = 1 and ε=1/2 (ks=kf=k)  ∂T   ∂T  1A  ∂Tf ∂Ts  1 f ∂Ts k f ∂Ts q = −kε + ()1− ε  qw = − k f + ks  qw = −  +  w   2  ∂y ∂y  2  ∂y ∂y   ∂y ∂y  y=0   y=0   y=0

1B  ∂T  k + k  ∂T   ∂T  q = −k f  q = − f s  f  q = −k f  w  ∂y  w 2  ∂y  w     y=0   y=0  ∂y  y=0 1C  ∂T  k + k  ∂T   ∂T  q = −k s  q = − f s  s  q = −k s  w  ∂y  w 2  ∂y  w     y=0   y=0  ∂y  y=0 1D  ∂T   ∂T   ∂T  q = −k f  q = −k  f  q = −k f  w  ∂y  w f  ∂y  w  ∂y    y=0   y=0   y=0  ∂T   ∂T   ∂T  = −k s  = −k  s  = −k s   ∂y  s       y=0  ∂y  y=0  ∂y  y=0 1E  ∂T  k + k  ∂T   ∂T  q = −2ε k f  q = − f s  f  q = −k f  w  ∂y  w 2  ∂y  w  ∂y    y=0   y=0   y=0  ∂T  k + k  ∂T   ∂T  = −2(1− ε)k s  = − f s  s  = −k s       ∂y   ∂y  y=0 2  ∂y  y=0   y=0 1F  ∂T  k + k  ∂T   ∂T  q = −k f  q = − f s  f  q = −k f  w  ∂y  w 2  ∂y  w  ∂y    y=0   y=0   y=0

 ∂T  k f + ks  ∂T   ∂T  = −k s  = −  s  = −k s   ∂y  2  ∂y      y=0   y=0  ∂y  y=0

 ∂Tf  k f  ∂Tf  k  ∂T f  qw = −ε k  q = −   q = −   2A  ∂y  w   w     y=0 2 ∂y 2 ∂y   y=0   y=0  ∂Ts  = −(1− ε)k  ks  ∂Ts  k  ∂Ts    = −   = −    ∂y  y=0   2 ∂y 2  ∂y  y=0   y=0  ∂T   ∂T   ∂T  q = −k f  f q = −k f  w   qw = −k f   w   2B  ∂y  ∂y ∂y y=0   y=0   y=0

 ∂Ts   ∂T   ∂Ts  = −k  = −k  s  = −k   ∂y  s   ∂y y=0  ∂y  y=0   y=0

Table 4.3: Mathematical representation of the wall heat flux and the temperature gradients for some special cases

(a) 450 2A 400 1D, 2B 350

300 1A 1C 250 Nu f 200 1F 1E 1B 150 100 50

0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ε

(b) 120 2A

100 1F 1E

80 1B Nu s 60

40 1A 1C

20 1D, 2B 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ε

continued

Figure 4.3: Effect of porosity “ε” on the Nusselt number excluding the effects of -5 variable porosity and thermal dispersion. Rep=100, Da=10 , Λ=10, dp=0.008, κ=23.75. (a) Fluid phase. (b) Solid phase (c) Total Nusselt number.

Figure 4.3: (continued)

350 1A 1C 300 Nu t 250 1F 1E 1B 200

150

100

50 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ε

(a) 200 1D, 2B 1B

150

Nu 100 f 2A 1A 1C 50 1F 1E

0 0 1 2 10 10 10 κ

(b) 350 1E 300 2A 250

200 1F 1C 1A Nu s 150 1B 100

50 1D, 2B 0 0 1 2 10 10 10 κ continued

Figure 4.4: Effect of the solid-to-fluid conductivity ratio “κ=ks/kf” on the Nusselt number excluding the effects of variable porosity and thermal dispersion. ε=0.9, -5 Rep=100, Λ=10, Da=10 , dp=0.008. (a) Fluid phase. (b) Solid phase (c) Total Nusselt number 99

Figure 4.4: (continued)

400 1F 350 2A 300 1A 1B Nu t 250 200 1D, 2B 1C 150

100 1E 50 0 1 2 10 10 10 κ

100

(a) 300 1B 2A 250

200 1D, 2B 1A Nu t 150 1C 1F 100

50 1E

0 0 50 100 150 200 Rep

(b) 400 1B

300

1A,1D, 1E, Nu 200 1F, 2A, 2B t

1C 100

0 0 50 100 150 200 Re p

Figure 4.5: Effect of particle Reynolds number “Rep” on the total Nusselt number excluding the effects of variable porosity and thermal dispersion. (a) ε=0.9, Da=10-5, -5 Λ=10, dp=0.008, κ=23.75. (b) ε=0.5, Da=10 , Λ=10, dp=0.008, κ=1.0

101

(a) 2A 240 1B

1D, 2B 200

Nu t 1A 160 1E 1C

120 1F

80 -7 -6 -5 -4 10 10 10 10 Da

(b) 400 1A

350 1B

Nu 300 t 1D, 1E, 1F, 2A, 2B 1C 250

200 -7 -6 -5 -4 10 10 10 10 Da

Figure 4.6: Effect of Darcy number “Da” on the total Nusselt number excluding the effects of variable porosity and thermal dispersion. (a) ε=0.9, Rep=100, Λ=10, dp=0.008,

κ=23.75. (b) ε=0.5, Rep=100, Λ=10, dp=0.008, κ=1.0

102

(a) 220

180

Nu 1D, 2B 2A 1B t 1C 1A 1F 140 1E

100 0 20 40 60 80 100 Λ

(b) 320

1B 300

Nu 280 t 1A, 1D, 1E 1F, 2A, 2B 1C 260

240 0 20 40 60 80 100 Λ

Figure 4.7: Effect of the Inertia parameter “Λ” on the total Nusselt number excluding -5 the effects of variable porosity and thermal dispersion. (a) ε=0.9, Rep=100, Da=10 , -5 dp=0.008, κ=23.75. (b) ε=0.5, Rep=100, Da=10 , dp=0.008, κ=1.0

103

(a) 600 1D, 2B 2A 500 1B 1A 400 1C 1F Nu 300 t

200 1E 100

0 -3 -2 10 d 10 p

(b) 650

600 1C

550 1A, 1D, 1E, 1F, 2A, 2B 500

Nu 450 t 400 1B 350

300

250 -3 -2 10 10 dp

Figure 4.8: Effect of the particle diameter “dp” on the total Nusselt number excluding -5 the effects of variable porosity and thermal dispersion. (a) ε=0.9, Rep=100, Da=10 , -5 Λ=10, κ=23.75. (b) ε=0.5, Rep=100, Da=10 , Λ=10, κ=1.0

104

(a) 5 10 2A 1C

4 10

∂T 1D, 2B f | ∂y w 1A, 1B 3 10

1E, 1F 2 10 0 1 2 10 10 10 κ

(b) 5 10

1C, 1E, 4 10 1F

∂Ts 1B |w ∂y 1A 3 10

1D, 2B 2 10 0 1 2 10 10 10 κ

Figure 4.9: Effect of the solid-to-fluid conductivity ratio “κ” on the temperature -5 gradients. ε=0.5, Rep=100, Da=10 , Λ=10, dp=0.008. (a) Fluid phase. (b) Solid phase.

105

(a) 120 data1 data2 100 data3 2A data4 data5 80 data6 T T 1C data7 mf wf data8 60 1D, 2B

20 1A, 1E, 1F 1B 0 0 1 2 10 10 10 κ

(b) 120 2A data1 data2 100 data3 data4 data5 80 data6 1C 1A, 1E, data7 T Tms ws 1F data8 60

1D, 2B 20 1B 0 0 1 2 10 10 10 κ

Figure 4.10: Effect of the solid-to-fluid conductivity ratio “κ” on the difference -5 between the wall temperature and the mean temperature. ε=0.5, Rep=100, Da=10 ,

Λ=10, dp=0.008. (a) Fluid phase. (b) Solid phase

106 4.4.2 VARIABLE POROSITY, THERMAL DISPERSION

Introducing only the effect of thermal dispersion would affect the problem under consideration implicitly since the effective conductivities remain constant at the wall. Therefore, expressions of Tables 4.1 and 4.3 are still valid and the results would be analogous to the ones from the previous section. On the other hand, introducing the effect of variable porosity would affect the models in Tables 4.1 and 4.3 implicitly as well as explicitly. Models 1D and 1F are greatly affected by the porosity variation since they have a direct dependence on it as shown in Table 4.1. As a result, model 1D is no longer identical to model 2B when the porosity variations are taken into account.

Also, characteristics of model 1F are totally different when effects of variable porosity are included. Consequently, results of Table 4.3 are valid for all the models except models 1D and 1F. When variable porosity is considered, the three parameters Darcy number, particle diameter and free stream porosity (ε∞) are linked through equation

(4.12). Therefore, analyzing two of these three parameters is sufficient. As a result, effects of the free stream porosity and the particle diameter are considered in this section. Effect of the free stream porosity is shown in Fig. 4.11. It is evident that the

Nusselt numbers for Models 1E and 1F are relatively apart from other models.

Although models 1D and 1E approach the same limit as the free stream porosity approaches 0.5, they are quite different as porosity decreases. Likewise, models 1A,

1B and 1C are close to each other when the free stream porosity is 0.5 however, they

107 produce values which are also not far from each other for smaller values of the free stream porosity. Moreover, models 1D and 2A approach the same limit when ε∞ approaches 0.5.

The presented figures are indicative of the general qualitative trend of the results. Effect of the free stream porosity on the solid phase Nusselt numbers is shown in Fig. 4.11b. It can be observed that the presented eight models can be categorized into four groups, the first is models 1A, 1B and 1C, the second is 1D and 2A, the third is 1E and 1F, and the forth is model 2B. Remarkable results are projected in Fig.

4.11c, it is found that models 1A, 1B, 1C, 1D, 2A, and 2B are very close to each other when low free stream porosity values are considered. Also, models 1D, 1E, 1F and 2A approach the same limit as ε∞ approaches 0.5. It is important to notice that model 1D becomes identical to model 2A when ε∞ has a value of 0.5.

Effects of the solid-to-fluid conductivity ratio are shown in Fig. 4.12. It is found in Fig. 4.12a that when κ=1, fluid phase Nusselt numbers for models 1A, 1B,

1C and 2B are almost the same. The same is true for models 1D, 1E, 1F and 2A. For high values of κ, models 1E and 1F diverge from the other models. In Fig. 4.12, models 1D and 2A are identical because a free stream porosity value of 0.5 is used in this figure. Models 1D and 2A diverge from models 1E and 1F at low κ values and approach other models as κ increases. As a result, they intersect with models 1A and

1C. Figure 4.12b shows the results of the solid phase Nusselt numbers. It is found that 108 ______Nu s values are relatively higher than Nu f . Solid phase Nusselt numbers of models

1A, 1B and 1C are found to be close to each other especially for lower κ values.

Again, models 1D and 2A separate from models 1E and 1F at small κ values and approach the other models as the value of κ increases. Model 2B essentially provides

____ the lower bound for small Nu s especially for larger values of κ. Generally, results of

Fig. 4.12c are equivalent to the ones of Fig. 4.12b because of the higher values of

____ Nu s . However, all the models except model 2B converge to the same limit for high values of κ.

In this section, the free stream porosity (ε∞) as well as the porosity at the wall

(εw) are found to affect convergence or divergence of the models under consideration.

Therefore, choosing appropriate values of the free stream porosity and the parameter b would give a chance to generate a value of 0.5 for the porosity at the wall. This might be achieved by choosing a value of 0.25 for the parameter b along with a free stream porosity value of 0.5. It is observed that choosing a value of unity for κ besides a wall porosity value of 0.5 makes models 1D and 1F identical. The same is true for models

1E, 2A and 2B. Effects of the particle Reynolds number on the total Nusselt numbers

____ Nu t are displayed in Fig. 4.13. Figure 4.13a shows that models 1D and 2A are exactly the same when free stream porosity equals 0.5, which is consistent with the results of

Fig. 4.11. The same results are observed for models 1E and 1F. When the particle

109 Reynolds number has a value around 45, the curve of models 1D and 2A intersects with the curve of models 1E and 1F. Models 1D, 2A, 1E and 1F yield the highest values of the total Nusselt number as Rep varies. It is also found that models 1A and

1B are very close for all values of Rep. Model 1C is still close to models 1A and 1B.

Figure 4.13b shows the effect Rep of for the special case of a wall porosity value of

____ 0.5. Unlike results of Fig. 4.13a, all the models experience an increase in their Nu t values as the value of Rep increases. It is observed that models 1D and 1F form the upper bound while models 1A, 1B and 1C form the lower limit. Models 1E, 2A and

2B come in the middle between the previous two sets.

Figure 4.14 shows the effects of the particle diameter on the total Nusselt. It is clear that models 1E and 1F produce results that are quite different from the other models. According to Fig. 14a, the total Nusselt numbers of models 1E and 1F decrease as the particle diameter dp increases while all the other models experience a slight increase as dp increases. Model 1C is quite close to models 1A and 1B, which are very close to each other. Results of models 1E and 1F approach the ones of 1D and

2A as dp increases. Generally, model 2B forms the lower bound while models 1E and

1F form the upper bound for the total Nusselt number profiles in Fig. 4.14a. Figure

4.14b shows the results for the special case of a wall porosity value of 0.5. Three regimes similar to the ones of Fig. 4.13b are found in Fig. 4.14b. However, the behavior of the three regimes in Fig. 4.14b is different and more involved. In fact, the

110 ____ behavior of the models in Fig. 4.14b is quite complex where the Nu t profiles of all the three regimes have local minimum values, which are different from one regime to another. The reason behind this behavior is the dependency of several parameters on dp such as the hsf, asf, the effective conductivities, the porosity and the permeability.

Useful correlations that relate other models to model 2A and the pertinent parameters of this section are listed below:

____   13.89   κ   Nut  = [8.56 − 20.205 ()ε∞ []1+ b − 0.494 EXP 2.541+ 0.065  −  1A  23.75 0.16 0.56  Re p    κ   1.218  + 1.22(ε + 0.557)27.97 + 0.657 ×    ∞     100   23.75  (4.28) 0.81   d   d   ____  EXP1.078 + 0.047 p  − 0.064 p ]  Nu        t   0.008  0.008   2 A

____   14.03   κ   Nut  = [8.01−19.44 ()ε∞ []1+ b − 0.22 EXP 3.214 + 0.068  −  1B  23.75 0.15 0.56  Re p    κ   1.298  + 1.25(ε + 0.553)28.18 + 0.639 ×    ∞     100   23.75  (4.29) 0.81   d   d   ____  EXP1.093 + 0.047 p  − 0.061 p ]  Nu        t   0.008  0.008   2 A

111 ____   13.81   κ   Nut  = [8.79 − 21.35 ()ε∞ []1+ b −1.52 EXP 1.453 + 0.056   −  1C  23.75 0.19 0.53  Re p    κ   1.116  + 1.27(ε + 0.557)27.89 + 0.65 ×    ∞     100   23.75  (4.30) 0.8   d   d   ____  EXP1.098 + 0.055 p  − 0.089 p ]  Nu        t   0.008  0.008   2 A

−0.48 ____  Re     p   Nut  = [1.535 − ()0.464 + 0.11[]1+ b ε ∞ + +0.048  +  1D  100 

  κ  −3 6.51EXP − 4.297 + 0.132   ×10 +  23.75 −0.019   κ   4.31ε 0.022 − 0.725  × (4.31)  ∞     23.75     d   d  ____  p  p   EXP 0.474 −1.573  + 0.856  ]  Nut   0.008   0.008          2 A

0.58 ____  Re    −765.04  p   Nut  = [−1.322 − 0.108()3.759 + 5.176[]1+ b ε ∞ − 0.345  +  1E  100 

  κ  12.28 0.801EXP 1.19 − 0.171  + ()−1.106 + 2.339[]1+ b ε ∞ ×  23.75 (4.32) 0.58 −3.53 ____ 0.92   κ    d p    0.128 +  1.25 +  ] Nu        t   23.75  0.008   2 A

112 ____   0.042   κ   Nut  = [−15.71− 21.23()[]1+ b ε∞ +1.752 EXP 3.05 − 0.039  +  1F  23.75 0.71 0.91  Re p    κ   0.131  + 2.89ε 6.94 + 0.056 ×    ∞     100   23.75  (4.33) −0.11 1.036    d   d    ____   p  p EXP 2.896 − 2.908  + 26.555   ]  Nut   0.008   0.008            2 A

____   21.15   κ   Nut  = [0.767 − 8.627()[]1+ b ε∞ + 0.206 EXP − 0.741+ 0.587   −  2B  23.75 0.037 2.87  Re p    κ   0.481  + 5092.94 ε 21.09 − 5.05×10−6 ×    ∞     100   23.75  (4.34) 1.13    d   d   ____   p  p EXP 7.972 + 0.042   − 92.767  ]  Nut   0.008   0.008           2 A

There is no universal situation where the eight models converge or diverge from each other when variable porosity effects are taken into account. However, it is found that low free stream porosity, high solid-to-fluid conductivity ratio and low particle Reynolds number cause models 1A, 1B, 1C, 1D and 2A to converge. Since many parameters depend on the particle diameter, its effect on the convergence and divergence of the models is significantly dependent on the choice of other parameters.

113

(a) 350

300 2A 2B 250 1D

200 1B Nu 1A f 150 1C 1E 100

50 1F 0 0.3 0.35 0.4 0.45 0.5 ε ∞

(b) 3 10

1F 1E

2 Nu 10 1D s

1A 1B 1C 2A 1 2B 10 0.35 0.4 0.45 0.5 ε ∞

continued

Figure 4.11: Effect of the free stream porosity “ε∞”on the Nusselt number including the effects of variable porosity and thermal dispersion. b=0.98, c=2, Rep=100, dp=0.008, κ=23.75. Including the effects of variable porosity and thermal dispersion. (a) Fluid phase. (b) Solid phase. (c) Total Nusselt number.

114

Figure 4.11: (continued)

2A 1D 1B

Nu t 1C 2B 1A

1E 1F

2 10 0.3 0.35 0.4 0.45 0.5 ε∞

115

(a) 400 2B 350

300 1B 250 1C 1A 1D, 2A

Nu f 200 150 1E, 1F 100

0 0 1 2 10 10 10 κ

(b) 1800 1E, 1F

1500 1D, 2A 1200 1A, 1C 1B Nu s 900

600 2B

300

0 0 1 2 10 10 10 κ

continued

Figure 4.12: Effect of the solid-to-fluid conductivity ratio “κ” on the total Nusselt number including the effects of variable porosity and thermal dispersion. ε∞=0.5, b=0.98, c=2,

Rep=100, dp=0.008, (εw=0.99). (a) Fluid phase. (b) Solid phase. (c) Total Nusselt number.

116

Figure 4.12: (continued)

1D, 2A 1600 1C 1A

1200

Nu t 2B 800

400

1B 0 0 1 2 10 10 10 κ

117 (a) 1100 1E, 1F

900 1D, 2A

700 Nu t

500 1B 1A 1C 2B

300 1 2 10 10 Re p

(b) 520 1D, 1F

420 1E, 2A, 2B

Nu t

320 1C

1A, 1B

220 1 2 10 Re p 10

Figure 4.13: Effect of the particle Reynolds number “Rep” on the total Nusselt number including the effects of variable porosity and thermal dispersion. (a) ε∞=0.5, b=0.98, c=2, dp=0.008, κ=23.75, (εw=0.99). (b) ε∞=0.4, b=0.25, c=2, dp=0.008, κ=1.0, (εw=0.5).

118 (a) 1800

1500 1E, 1F 1200

1D, 2A 1A, 1B 1C Nu 900 t

600

300

2B 0 -3 -2 10 d 10 p

(b) 500

1D, 1F 450

400 1A, 1B

Nu t 350 1E, 2A, 2B 1C 300

250 -3 -2 10 dp 10

Figure 4.14: Effect of the particle diameter “dp” on the total Nusselt number including the effects of variable porosity and thermal dispersion. (a) ε∞=0.5, b=0.98, c=2, Rep=100,

κ=23.75, (εw=0.99). (b) ε∞=0.4, b=0.25, c=2, Rep=100, κ=1.0, (εw=0.5). 119

4.5 CONCLUSION

A comprehensive investigation of variant boundary conditions for constant wall heat flux in porous media under LTNE conditions has been presented in this work. It is evident that using different boundary conditions may lead to substantially different results. Physical properties are found to have a significant impact on the heat transfer predictions using the models listed in Table 4.1. Generally, models 1A and 1B are relatively close to each other while models 1C and 2B are exactly the same when thermal dispersion effects are excluded. Effects of porosity, particle Reynolds number, Darcy number, particle diameter and solid-to-fluid conductivity ratio are found to be significant when variable porosity and thermal dispersion effects are excluded. Effects of inertia

______parameter are found to be relatively insignificant on Nu f , Nu s and Nu t . In general,

____ model 1E forms the lower bound while model 2B forms the upper bound for Nu f . On the other hand, model 1E form the upper bound while model 2B forms the lower bound

____ for Nu s . Including the effects of variable porosity and thermal dispersion altered the

______sequence of the models in terms of their Nu f , Nu s and Nu t predictions. Including these effects causes intermediacy independence between some of the pertinent parameters. Special cases when variant models under consideration converge are obtained. Comprehensive sets of correlation that relate various models to each other in terms of the total Nusselt number are presented. 120

CHAPTER 5

ANALYSIS OF VARIABLE POROSITY, THERMAL DISPERSION AND

LOCAL THERMAL NON-EQUILIBRIUM ON FREE SURFACE

FLOWS THROUGH POROUS MEDIA

Characteristics of momentum and energy transport for free surface flows through porous media are explored in this study. Effects of variable porosity and an impermeable boundary on the free surface front are analyzed. In addition, effects of thermal dispersion and local thermal non-equilibrium (LTNE) are also analyzed.

Pertinent parameters such as porosity, Darcy number, inertia parameter, Reynolds number, particle diameter, and solid-to-fluid conductivity ratio are used to investigate the significance of the above mentioned effects. Results show that considering the effect of variable porosity is significant only in the neighborhood of the solid boundary. Including thermal dispersion effect is found to be quite essential for certain ranges of the pertinent parameters. Similarly, some ranges of the relevant parameters enhance the LTNE between the two phases. Finally, it is shown that adding the effect of thermal dispersion to LTNE increases the sensitivity of LTNE between the two phases.

121

5.1 INTRODUCTION

Incompressible free surface fluid flow in porous media has been the subject of many studies in the last few decades because of its importance in many applications such as geophysics, die filling, metal processing, agricultural and industrial water distribution, oil recovery techniques and injection molding. One of the earliest studies in this field was performed by Muskat (1937) who considered a one-dimensional

Darcy’s flow model to analyze the linear encroachment of two fluids in a narrow channel. Recently, an analytical solution for linear encroachment in two immiscible fluids in a porous medium was presented in Srinivasan and Vafai (1994). They obtained a closed form solution, for the temporal free surface fluid front, that accounts for boundary and inertia effects. Their results show that for higher permeabilities

Muskat’s model underestimates the total time needed for the encroaching fluid to reach the end of the channel. Furthermore, they show that implementing their analytical solution is essential for cases of low mobility ratios.

Later on, Chen and Vafai (1996) investigated the free surface transport through porous media numerically using the Marker and Cell method. They extended the study of Srinivasan and Vafai (1994) to include free surface energy transport in their investigation. Another study performed by Chen and Vafai (1997) considered interfacial tension effects on the free surface transport in porous media. Their results show that surface tension can be neglected for high Reynolds number flows. 122

The main objective of this study is to investigate effects of variable porosity, thermal dispersion and LTNE on the free surface fluid flow and heat transfer through porous media. These effects have been shown to be quite significant, Vafai (1984),

Vafai (1986) and Amiri and Vafai (1994), for a number of practical situations and were not studied in the earlier works, Muskat (1937), Srinivasan and Vafai (1994),

Chen and Vafai (1996) and Chen and Vafai (1997).

5.2 ANALYSIS

Geometry and Physical properties are chosen to be similar to those given in previous related studies, Srinivasan and Vafai (1994) and Chen and Vafai (1996), for the purpose of comparison. Description of the system under consideration is shown in

Fig. 5.1. The volume-averaged governing equations are given as Vafai and Tien

(1981)

Continuity equation

∇ ⋅ V = 0 (5.1)

123 Momentum equation

ρ µ ρ Fε µ f ()V ∇ V = − f V − f []V V J + f ∇2 V − ∇ P f (5.2) ε Κ Κ ε

and fluid phase and solid phase energy equations are, Amiri and Vafai (1994),

f f f s f ρ f c f V ∇ Tf = ∇ (k feff ∇ Tf ) + hsf asf ( Ts − Tf ) (5.3)

s s f 0= ∇(kseff ∇ Ts ) − hsf asf ( Ts − Tf ) (5.4)

The fluid-to-solid heat transfer coefficient and the specific surface area are expressed as Amiri and Vafai (1994)

 0.6  k 1  ρ ud  h = f 2 +1.1Pr 3  f p   (5.5) sf d   µ   p   f  

6(1− ε) asf = (5.6) d p

Effective conductivities of both phases are defined as

k feff = ε k f (5.7)

kseff = (1− ε ) ks (5.8)

124 When effects of thermal dispersion are present, axial and lateral effective conductivities of the fluid phase can be represented respectively as Amiri and Vafai

(1994)

  ρ u d  k = ε + 0.5Pr f p  k (5.9) ()feff x    f   µ 

  ρ u d  k = ε + 0.1 Pr f p  k (5.10) ()feff y    f   µ 

Furthermore, when variation of porosity near the impermeable boundaries is present, porosity, permeability and the geometric function F may be expressed as Vafai (1984) and Vafai (1986)

  − c y  ε = ε 1+ b exp  (5.11) ∞      d p 

ε 3 d 2 Κ = p (5.12) 150(1− ε)

1.75 F = (5.13) 150ε 3

______Results are presented in terms of the average Nusselt numbers ( Nu f and Nu s ). Local

Nusselt numbers for both phases are, Amiri and Vafai (1994),

125

 f  4H  ∂ T f  Nu f = − (5.14) f f  ∂y  T f − T f w m   y=0

 ∂ T s  4H  s  Nus = − (5.15) T s − T s  ∂y  s w s m   y=0

f Where T and T s are the volume averaged mean fluid and solid temperatures f m s m respectively.

As mentioned earlier, boundary and initial conditions are taken exactly similar to previous studies, Srinivasan and Vafai (1994) and Chen and Vafai (1996)

Initial condition

At t = 0, u = v = 0, and T=T∞ (5.16)

Boundary conditions

At x = 0, p = pe , v = 0, T=Te (5.17)

∂ u ∂T At x = x0 , p = p∞ , = 0, − k = h(T − T ) (5.18) ∂ x eff ∂ x ∞

At y = 0, 2H, u = v = 0, T=Tw (5.19)

126

The driven fluid is assumed to have a smaller viscosity compared to the encroaching fluid. Based on this assumption, it is possible to compare the present numerical results to the modified analytical solution given in Chen and Vafai (1996).

The modified analytical solution in Chen and Vafai (1996) is presented in terms of the driven fluid physical properties. However, it is more appropriate to express the solution in terms of the encroaching fluid physical properties. Also, in order to compare the numerical results to the analytical results it is assumed that the viscosity of the driven fluid is much smaller than the viscosity of the encroaching fluid. As such the modified analytical solution is re-written in a simpler form

2   xi  2Κ 2 ∆p − ω + ω + ()1− ω   + ()1− ω 2 t x   L  ε µ2 L 0 = (5.20) L ()1− ω

Again, the above equation (Eq. 5.20) is only a rearranged format of the solution given in Chen and Vafai (1996). The new mobility ratio (ω) is the inverse of the one defined in Chen and Vafai (1996). The present mobility ratio, which is assumed to have a very small value, is defined as

µ ω = 1 (5.21) µ 2

127 Based on the appearance of Eq. (5.20), we also define a dimensionless time (τ) as

τ = γ t (5.22) where

2Κ 2 ∆p γ = 2 (5.23) ε µ2 L

In the above equation, the pressure difference is calculated according to the relation given in Vafai and Kim (1989). Permeability is related to the Darcy number for the constant porosity category while it can be calculated using Eq. (5.12) for the variable porosity category. Water is considered as the encroaching fluid in the present study.

128

Interfacial front ( x 0 )

‘2’ ‘1’ 2H Encroaching Residing y fluid fluid x L

Figure 5.1: Schematic diagram of the free surface front and the corresponding coordinate system

129

5.3 NUMERICAL SOLUTION

The governing equations were discretized using the finite difference method.

First, variable grids in the y-direction and constant grids in the x-direction were implemented in the prediction of the flow field. Then, mesh refinements in the vicinity of the free surface front were applied which cause the grids to be variable in both directions. This procedure is called interface capturing technique (ICT). It is similar to the one given in Sharif and Wiberg (2002). However, the finite difference method is used in the present study instead of the finite element method used in Sharif and

Wiberg (2002). ICT eliminates the need for interpolations and extrapolations in the process of predicting the velocity and temperature fields. Since all the nodes coincide with the free surface, the two coupled energy equations are solved using the alternating direction implicit (ADI) method. An iterative solution is required since the two energy equations are coupled.

The requirement that the variation of velocity and temperature distributions is less than 10-6 between any two consecutive iterations is employed as the criterion for convergence. Numerical experiments were performed to assure the independence of the results from the choice of the grid size. A grid size of 201×501 was found to provide grid independent results. Due to the presence of symmetry and in order to save a considerable amount of CPU time, numerical computations were performed for the lower half of the physical domain.

130

In order to verify the accuracy of the present numerical results, comparisons with previous analytical and numerical results are presented in Fig. 5.2. Figure 5.2a shows comparison between the present numerical results and the modified analytical solution, using a mobility ratio value of zero (ω=0), for the free surface front position.

Inputs used to generate Fig. 5.2a are taken the same as the ones given in (Figure 4 of

Chen and Vafai (1996)). The same is done in Fig. 5.2b which displays the comparison between the current numerical temperature distributions and previous corresponding results (Figure 5a in Chen and Vafai (1996)). Excellent agreement is found between the present numerical results and results given in Chen and Vafai (1996).

Numerical accuracy for results of velocity, temperature and the average

____ Nusselt number ( Nu ) was assessed by varying the convergence criteria and the mesh size of the computational domain. Changing the convergence criterion from the utilized value of 10-6 to 10-8 always results in a deviation less than 0.5 percent, 0.2

____ percent and 0.1 percent in velocity, temperature and Nu results respectively. On the other hand, mesh refinements of 401×1001 are found to cause the results of velocity,

____ temperature and Nu to deviate from those of the utilized mesh size of 201×501 by less than 0.8 percent, 0.3 percent, and 0.2 percent respectively.

131

(a)

(b)

Figure 5.2: Comparison between the present results and the numerical results in Chen and Vafai (1996). (a) Temporal free surface distribution using constant Darcy number. (b) -4 -6 Temperature contours for Rek=5.72×10 , Da=1.0×10 at t=0.5 s.

132

5.4 RESULTS AND DISCUSSION

To show the effect of utilizing the constant porosity assumption, results for both constant and variable porosity will be discussed here. Figure 5.3a shows the progress of the interfacial front using constant porosity as a function of time while Fig.

5.3b shows the same using variable porosity. For both cases the flow is assumed to have the same initial position. However, the development of the variable porosity flow is different than the one using constant porosity. It is evident that the effect of variable porosity, which is the case for a number of engineering applications, is essential in the neighborhood of the solid boundary. This phenomenon is called the channeling effect which was discussed in detail in Vafai (1984) and Vafai (1986) and will not be discussed here. Our goal here is to analyze the effects of pertinent parameters such as porosity, Darcy number and Reynolds number on the residence time for the encroaching fluid τmax for both constant and variable porosity categories.

133

(a)

(b)

Figure 5.3: Progress of the interfacial front for (a) Constant porosity category with -6 Da=1.0×10 , ε=0.8, Λ=10.0 and Re=100. (b) Variable porosity category with ε∞=0.45, b=0.98, c=2.0, Re=100 and dP/H=0.05

134

5.4.1 CONSTANT POROSITY

For the constant porosity category, the reference value of dimensionless time τ is based on the choice of intermediate values for the pertinent parameters. These reference values are 0.8 for the porosity, 10-6 for the Darcy number, 100 for the

Reynolds number and 10 for the inertia parameter. For a wide range of Darcy number

-10 -6 values, 10 to 10 in the present study, it is found that the required τmax to reach the end of the channel is almost the same. However, higher Darcy numbers allow the fluid to reach the end of the channel in a shorter time as shown in Fig. 5.4. Effect of inertia parameter (Λ) on the temporal free surface front location is displayed in Fig. 5.5. It is found that the inertia parameter has less influence on the results when other parameters are fixed. Although, it is worth noting that τmax and the inertia parameter show an inversely linear proportional relation as shown in Fig. 5.5b. As expected,

Reynolds number has the most significant effect on the progress of the free surface front. Its effect is shown in Fig. 5.6 where higher Reynolds numbers require significantly shorter time for the fluid to reach the end of the channel.

135

(a)

(b)

Figure 5.4: Effect of Darcy number for the constant porosity category using Λ=10.0, Re=100 and ε=0.8 on (a) The temporal free surface front (b) The total time to reach the channel exit (τ max)

136

(a)

(b)

Figure 5.5: Effect of Inertia parameter for the constant porosity category using ε=0.8, Da=1.0×10-6 and Re=100 (a) On the temporal free surface front (b) On the total time to reach the channel exit (τ max)

137

(a)

(b)

Figure 5.6: Effect of Reynolds number for the constant porosity category using ε=0.8, Da=1.0×10-6 and Λ=10.0 (a) The temporal free surface front (b) The total time to reach the channel exit (τ max)

138

5.4.2 VARIABLE POROSITY

For the variable porosity category, the reference value of dimensionless time τ is based on the choice of intermediate values for the pertinent parameters. These reference values are 0.45 for the free stream porosity, 100 for the Reynolds number and 0.05 for ratio of particle diameter to channel height. As mentioned earlier, permeability, Darcy number, the geometric function (F) and pressure gradient depend on the choice of the pertinent parameters for the variable porosity category. It is found that parameters (b) and

(c) in Eq. (5.11) affect the velocity profile but have insignificant effect on the location of the free surface front. Typical values for parameters (b) and (c) are assumed based on a previous study, Vafai (1984).

Results of the effect of the Reynolds number in Fig. 5.7 for the variable porosity category and for the constant porosity category given in Fig. 5.6 reinstate the same conclusions. That is, the general behavior of the relation between the free surface position and Reynolds number are not affected by introducing the effect of variable porosity.

Particle diameter appears in the expressions of the variable porosity (Eq. 5.11) and the permeability expression (Eq. 5.12). It is found that changing the particle diameter have an insignificant effect on the results. However, the influence of changing the particle diameter on the permeability is of more significance. Particle diameter affects the value of Darcy number through the definition of permeability. Therefore, it is reasonable to

139 compare the results of Fig. 5.8 for the particle diameter and the ones in Fig. 5.4 for the effects of variations in the Darcy number. The effect of Darcy number in Fig. 5.4 for the constant porosity category is more pronounced than the effects of variations in the particle diameter given in Fig. 5.8 for the variable porosity category.

140

(a)

(b)

Figure 5.7: Effect of Reynolds number for the variable porosity category using ε∞=0. 45, b=0.98, c=2.0 and dP/H=0.05 on (a) The temporal free surface front (b) The total time to reach the channel exit (τ max)

141

(a)

(b)

Figure 5.8: Effect of particle diameter for the variable porosity category using ε∞=0. 45, b=0.98, c=2.0 and Re=100 on (a) The temporal free surface front (b) The total time to reach the channel exit (τ max)

142 5.4.3 THERMAL DISPERSION

The effects of thermal dispersion on the thermal characteristics of the free surface are analyzed in this section. Thermal equilibrium between the two phases is assumed. Introducing the effect of thermal dispersion in the energy equation in general favors conduction over convection. In other words, supplementing dispersion effects to the energy equation gives thermal conduction more dominance. This can be seen by comparing the temporal dimensionless temperature profiles in Figs. 5.9-5.10. Figure

5.9 illustrates the development of the temperature field with time with thermal dispersion effects included while Fig. 5.10 shows the same with thermal dispersion effects excluded while all other input parameters are kept the same. It is found that the total time taken to reach the end of the channel is the same for both cases.

Temperature contours and average Nusselt number curves for different values of porosity are shown in Fig. 5.11. It can be seen that higher porosity allows further thermal penetration of the encroaching fluid into the channel. Also, higher porosities result in higher Nusselt numbers as shown in Fig. 5.11c. In addition, the difference between the two Nusselt numbers (with and without dispersion) widens as porosity increases. Therefore, effect of thermal dispersion becomes more pronounced at higher porosities.

143 Temperature contours for relatively smaller and larger Darcy numbers are quite close as seen in Figs. 5.12a and 5.12b. Overall heat transfer characteristics are almost unaffected. The two Nusselt numbers, with and without the effect of dispersion, are almost the same for different Darcy numbers as shown in Fig. 5.12c.

However, both curves show a slight decrease as Da increases while keeping the same difference. Therefore, effect of Darcy number with the presence of thermal dispersion is relatively insignificant. Inertial parameter variation were also found to have an insignificant effect on the temperature distributions.

Effect of thermal dispersion on temperature contours are shown in Figs. 5.13a and 5.13b. As expected, heat transfer by convection is more dominant at higher

Reynolds numbers. Reynolds number influence, with and without the presence of thermal dispersion, on the Nusselt number is depicted in Fig. 5.13c. As can be seen the effect of thermal dispersion is relatively insignificant at very small Reynolds numbers.

However, as Reynolds number increases, the effect of thermal dispersion becomes more pronounced. Results of particle diameter in Fig. 5.14 show that it is only important to account for the thermal dispersion effects for larger values of dp.

Figures 5.15a and 5.15b present effect of solid-to-fluid conductivity ratio on temperature profiles when thermal dispersion effect is included. This pertinent parameter is the only one that is not considered in previous sections since it appears only in the energy equation. As can be seen in Fig. 5.15 temperature contours undergo

144 a drastic change as this ratio changes. A comparative analysis of order of magnitudes between the conduction and convection mechanisms is enough to explain the shown behavior in these temperature contours. In Fig. 5.15a where a relatively smaller conductivity ratio is considered, the convective mode is more dominant since the effective conductivity is relatively smaller. As can be seen in Fig. 5.15c, neglecting the thermal dispersion effect is a reasonable assumption only for relatively high conductivity ratios.

145

(a)

(b)

continued

Figure 5.9: Temporal dimensionless temperature profiles including thermal dispersion -6 effects, ε=0.8, Da=10 , Re=100, Λ=10, κ=15.0 and dP/H=0.05 (a) τ =0.25 (b) τ =0.5 (c) τ = τ max

146

Figure 5.9: (continued)

(c)

147

(a)

(b)

continued

Figure 5.10: Temporal dimensionless temperature profiles excluding thermal -6 dispersion effects, ε=0.8, Da=10 , Re=100, Λ=10, κ=15.0 and dP/H=0.05 (a) τ =0.25 (b) τ =0.5 (c) τ = τ max 148

Figure 5.10: (continued)

(c)

149

(a)

(b)

continued

Figure 5.11: Effect of porosity for the thermal dispersion category using Da=10-6, Re=100, Λ=10, κ=15.0 and dP/H=0.05 on (a) Dimensionless temperature profiles using ε=0.7 (b) Dimensionless temperature profiles using ε=0.9 (c) Total Nusselt number

150

Figure 5.11: (continued)

(c)

151

(a)

(b)

continued

Figure 5.12: Effect of Darcy number for the thermal dispersion category using ε=0.8, Re=100, Λ=10, κ=15.0 and dP/H=0.05 on (a) Dimensionless temperature profiles using Da=10-4 (b) Dimensionless temperature profiles using Da=10-8 (c) Total Nusselt number 152

Figure 5.12: (continued)

(c)

153

(a)

(b)

continued

Figure 5.13: Effect of Reynolds number for the thermal dispersion category using -6 ε=0.8 Da=10 , Λ=10, κ=15.0 and dP/H=0.05 on (a) Dimensionless temperature profiles using Re=50 (b) Dimensionless temperature profiles using Re=150 (c) Total Nusselt number

154

Figure 5.13: (continued)

(c)

155

(a)

(b)

continued

Figure 5.14: Effect of particle diameter for the thermal dispersion category using ε=0.8 Da=10-6, Re=100, Λ=10 and κ=15.0 on (a) Dimensionless temperature profiles using dP/H=0.01 (b) Dimensionless temperature profiles using dP/H=0.1 (c) Total Nusselt number

156

Figure 5.14: (continued)

(c)

157

(a)

(b)

continued

Figure 5.15: Effect of solid to fluid conductivity ratio for the thermal dispersion -6 category using ε=0.8 Da=10 , Re=100, Λ=10 and dP/H=0.05 on (a) Dimensionless temperature profiles using κ=5.0 (b) Dimensionless temperature profiles using κ=30.0 (c) Total Nusselt number 158 Figure 5.15: (continued)

(c)

159

5.4.4 LOCAL THERMAL NON-EQUILIBRIUM

In previous sections, local thermal equilibrium (LTE) between the solid and fluid phases was assumed. Figure 5.16 displays dimensionless temperature contours, under LTNE conditions while accounts for thermal dispersion, for fluid and solid phases respectively. It is noted that temperatures of both phases are almost indistinguishable when moderate pertinent input parameters are used. When limiting input parameters are used as in Fig. 5.17, temperature difference between the two phases becomes more pronounced.

Effect of porosity variations on the average Nusselt numbers for LTNE category is shown in Fig. 5.18. Figure 5.18a shows this effect while neglecting thermal dispersion. As can be seen the relatively small difference between the two phases

Nusselt numbers remains constant as porosity changes. This is not the case when thermal dispersion effect is implemented in the fluid phase energy equation as demonstrated in Fig. 5.18b. It is worth noting that the solid phase Nusselt number is almost the same with and without the effect of thermal dispersion.

Figure 5.19 shows the effect of Darcy number on the LTNE. When thermal dispersion effect is accounted for, the general trend of the fluid phase Nusselt number is similar to the results when LTE is assumed as in Fig. 5.12. For higher Darcy

160 numbers, the two phases tend to reach thermal equilibrium as their Nusselt numbers intersect. Involvement of thermal dispersion enhances the LTNE between the two phases as shown in Fig. 5.19b. The inertia parameter effect is not significant on the

Nusselt numbers as shown in Fig. 5.20, however, thermal dispersion widens the difference between the two Nusselt numbers as the inertia parameter changes. Solid phase Nusselt numbers are almost the same regardless of the presence or absence of thermal dispersion effects.

Increasing Reynolds number increases the LTNE between the two phases with and without the effect of thermal dispersion as shown in Fig. 5.21. The involvement of thermal dispersion increases the sensitivity of LTNE to Reynolds number. The relation between the Nusselt numbers and Reynolds number remains linear for all cases even with the assumption of LTE as in Fig. 5.13. Effect of particle diameter on LTNE, with and without the presence of thermal dispersion, is displayed in Fig. 5.22. Solid phase

Nusselt numbers are not affected by any change in the particle diameter. However, fluid phase Nusselt numbers increase as the value of particle diameter increases.

Again, including thermal dispersion effects increase the response of the fluid phase

Nusselt number to the changes in the particle diameter.

Finally, the influence of the solid-to-fluid conductivity ratio on the behavior of the LTNE problem is presented in Fig. 5.23. When thermal dispersion is not included,

Fig. 5.23b shows that higher ratios cause a slight increase in the difference between

161 the two Nusselt numbers. On the other hand, the difference between the two Nusselt numbers is almost constant when thermal dispersion effect is considered. It is worth noting that all Nusselt numbers decrease as the solid-to-fluid conductivity ratio increases for the same reasons discussed in the previous section. However, the order of magnitude analysis for the two-equation model is more involved because of the presence of the coupling term which is regarded as an unknown source term in both equations.

162

(a)

(b)

Figure 5.16: Dimensionless temperature profiles for the LTNE category using ε=0.8, -6 Da=10 , Re=100, Λ=10, κ=15.0 and dP/H=0.05 (a) Fluid phase (b) Solid phase

163

(a)

(b)

Figure 5.17: Dimensionless temperature profiles for the LTNE category using ε=0.9, -8 Da=10 , Re=200, Λ=100, κ=5.0 and dP/H=0.1 (a) Fluid phase (b) Solid phase

164

(a)

(b)

Figure 5.18: Effect of porosity on average Nusselt numbers for the LTNE category, -6 Re=100, Da=10 , Λ=10, κ=15.0 and dP/H=0.05 (a) Excluding thermal dispersion effects (b) Including thermal dispersion effects

165

(a)

(b)

Figure 5.19: Effect of Darcy number on average Nusselt numbers for the LTNE category, ε=0.8, Re=100, Λ=10, κ=15.0 and dP/H=0.05 (a) Excluding thermal dispersion effects (b) Including thermal dispersion effects

166

(a)

(b)

Figure 5.20: Effect of Inertia parameter on average Nusselt numbers for the LTNE -6 category, ε=0.8, Da=10 , Re=100, κ=15.0 and dP/H=0.05 (a) Excluding thermal dispersion effects (b) Including thermal dispersion effects

167

(a)

(b)

Figure 5.21: Effect of Reynolds number on average Nusselt numbers for the LTNE -6 category, ε=0.8, Da=10 , Λ=10, κ=15.0 and dP/H=0.05 (a) Excluding thermal dispersion effects (b) Including thermal dispersion effects

168

(a)

(b)

Figure 5.22: Effect of particle diameter on average Nusselt numbers for the LTNE category, ε=0.8, Da=10-6, Re=100, Λ=10 and κ=15.0 (a) Excluding thermal dispersion effects (b) Including thermal dispersion effects

169

(a)

(b)

Figure 5.23: Effect of solid-to-fluid thermal conductivity ratio on average Nusselt -6 numbers for the LTNE category, ε=0.8, Da=10 , Re=100, Λ=10 and dP/H=0.05 (a) Excluding thermal dispersion effects (b) Including thermal dispersion effects

170 5.5 CONCLUSIONS

A comprehensive analysis of variable porosity, thermal dispersion and local thermal non-equilibrium on free surface transport through porous media is presented in this study. Effects of pertinent parameters, porosity of the porous medium, Darcy number, Inertia parameter, Reynolds number, particle diameter and solid-to-fluid conductivity ratio, on the momentum and thermal transport for each category are analyzed and discussed. It is found that introducing the effect of variable porosity is essential in the neighborhood of the solid boundaries. Results show that the thermal dispersion has a substantial effect on the thermal transport process. LTNE between the two phases is found to be very sensitive when thermal dispersion effect is included.

Changes in porosity, Darcy number, Reynolds number and particle diameter are found to significantly affect the involvement of thermal dispersion on the LTNE. Therefore, ignoring these effects can lead to inaccurate estimations of the free surface problem.

171

CHAPTER 6

CONCLUSIONS

Four interrelated primary topics of transport phenomena through porous media were considered in this study. The first topic dealt with various aspects of different physical phenomena affecting transport through porous media. These included variable porosity, local thermal non-equilibrium and thermal dispersion. Summary of pertinent models within each category of the first topic is presented in a tabulated format. Table 2.1 summarizes models for the constant porosity category, Table 2.3 summarizes models for the variable porosity category, Table 2.5 summarizes models for the thermal dispersion category and Table 2.7 summarizes models for the LTNE category. It is shown that for some cases the variances within different models have a negligible effect on the results while for some cases the variances can become significant. In general the variances have a more pronounced effect on the velocity field and a substantially smaller effect on the temperature field and Nusselt number distribution. Therefore, results of this part are essential in resolving any confusion in utilizing either of these models within each category.

172

The second topic considered hydrodynamic and thermal interfacial boundary conditions between a porous medium and an adjacent fluid layer. Hydrodynamic as well as thermal boundary conditions at the interface can be classified as slip or non- slip boundary conditions. Results for the pertinent interfacial models are also presented in Tables 3.1 and 3.2. It is shown that for most cases the variances within different models, for most practical applications, have a negligible effect on the results while for few cases the variations can become significant. In general, the variances have a more pronounced effect on the velocity field and a substantially smaller effect on the temperature field and yet even smaller effect on the Nusselt number distribution. The effect of choosing the effective viscosity was found to have a relatively small influence on the velocity field and an insignificant effect on the temperature and local Nusselt number distributions. Results of this part are believed to be helpful in resolving any ambiguity in choosing the appropriate hydrodynamic and thermal boundary conditions at the interface between a porous medium and an adjacent fluid layer.

The analysis of the constant heat flux boundary condition and its significance on the heat transfer process through porous media was presented next. For this part, two-equation model was utilized to investigate how the heat input from the impermeable wall to the porous medium is divided between the solid matrix and the

173 fluid phase. Eight different boundary conditions for the heat flux at the solid boundary are analyzed in this part. These eight models are based on two primary approaches.

The first approach assumes that the total heat flux is divided between the two phases depending on the physical values of their effective conductivities and their corresponding temperature gradients at the wall. The second approach suggests that each of the individual phases at the wall receives an equal amount of the total heat flux. Results show that using different boundary conditions may lead to substantially different results. Physical properties are found to have a significant impact on the heat transfer predictions using the models listed in Table 4.1. Effects of porosity, particle

Reynolds number, Darcy number, particle diameter and solid-to-fluid conductivity ratio are found to be significant when variable porosity and thermal dispersion effects are excluded. Special cases when variant models under consideration converge are obtained.

The fourth topic dealt with the analysis of the free surface transport in porous media. For this topic, the generalized momentum equation was utilized in the analysis of the free surface transport in porous media. Particularly, boundary and inertia effects as well as the effect of variable porosity were considered. For the thermal transport part, effects of using thermal dispersion and local thermal non-equilibrium were also considered. Variable porosity effect is found to be quite substantial in the neighborhood of the solid boundaries. Also, including thermal dispersion effects are

174 found to enhance the LTNE between the two phases. Results show that ignoring effects of variable porosity, thermal dispersion and LTNE might lead to inaccurate estimations of the free surface transport. The interface capturing technique (ICT) was utilized in predicting the free surface front. Using ICT eliminates the need of interpolation and extrapolation to predict the free surface front position as it progresses toward the channel exit.

Results of this study implicitly suggest the need for future experimental work which will lead to a better understanding in using the proper models and the corresponding boundary conditions. Comprehensive sets of correlation that relate various models to each other were presented throughout the study whenever applicable.

175

LIST OF REFERENCES

Al-Azmi, B., 1999, “Analysis of Variants within the Porous Media Transport Models,” Ph.D. Thesis, The Ohio State University, Columbus, OH

Alazmi, B. and Vafai, K., 2000, “Analysis of Variants within the Porous Media Transport Models,” Journal of Heat Transfer, 122, pp. 303-326.

Amiri, A. and Vafai, K., 1994, “Analysis of Dispersion Effects and Non-Thermal Equilibrium, Non-Darcian, Variable Porosity, Incompressible Flow through Porous Media,” Int. J. Heat Mass Transfer, 37, pp. 939-954.

Amiri, A. and Vafai, K., 1998, “Transient Analysis of Incompressible Flow through a Packed Bed,” Int. J. Heat Mass Transfer, 41, pp. 4259-4279.

Amiri, A., Vafai K. and Kuzay, T. M., 1995, “Effects of Boundary Conditions on Non-Darcian Heat Transfer through Porous Media and Experimental Comparisons,” Num. Heat Transfer, Part A, 27, pp. 651-664.

Beavers, G. and Joseph, D. D., 1967, “Boundary Conditions at a Naturally Permeable Wall,” Journal of Fluid Mechanics, 1967, 30, pp. 197-207.

Beckermann, C. and Viskanta, R., 1987, “Forced Convection Boundary Layer Flow and Heat Transfer Along a Flat Plate Embedded in a Porous Medium,” Int. J. Heat Mass Transfer, 30, pp. 1547-1551.

Benenati, R. F. and Brosilow, C. B., 1962, “Void Fraction Distribution in Beds of Sphere,” AIChE, 8, pp. 359-361.

Chen, H., 1997, “Analysis of Non-Darcian Mixed Convection from Impermeable Horizontal Surface in Porous Media: The Entire Regime,” Int. J. Heat Mass Transfer, 40, pp. 2993-2997.

Chen, H., Chen, T. S. and Chen, C. K., 1996, “Non-Darcy Mixed Convection along Non-Isothermal Vertical Surface in Porous Media,” Int. J. Heat Mass Transfer, 39, pp. 1157-1164.

176

Chen, H., 1997, “Non-Darcy mixed convection over a vertical flat plate in porous media with variable wall heat flux,” Int. Comm. Heat Mass Transfer, 24, pp. 427-437.

Chen, H., 1996, “Technical note: non-Darcy mixed convection from a horizontal surface with variable surface heat flux in a porous medium,” Num. Heat Transfer, Part A, 30, pp. 859-869.

Chen, S. C. and Vafai, K., 1996, “Analysis of Free Surface Momentum and Energy Transport in Porous Media,” Num. Heat Transfer, Part A., 29, pp. 281-296.

Chen, S. C. and Vafai, K., 1997, “Non-Darcian Surface Tension Effects on Free Surface Transport in Porous Media,” Num. Heat Transfer, Part A., 31, pp. 235-254.

Cheng, P. and Hsu, C. T., 1986, “Fully-Developed, Forced Convective Flow through an Annular Packed-Sphere Bed with Wall Effects,” Int. J. Heat Mass Transfer, 29, pp. 1843-1853.

Cheng, P. and Vortmeyer, D., 1988, “Transverse Thermal Dispersion and Wall Channeling in a Packed Bed with Forced Convective Flow,” Chem. Eng. Sci., 43, pp. 2523-2532.

Cheng, P. and Zhu, C. T., 1987, “Effect of Radial Thermal Dispersion on Fully Developed Forced Convection in Cylindrical Packed Tubes,” Int. J. Heat Mass Transfer, 30, pp. 2373-2383.

Cheng, P., Hsu, C. T. and Chowdary, A., 1988, “Forced Convection in the Entrance Region of a Packed Channel with Asymmetric Heating,” J. Heat Transfer, 110, pp. 946-954.

Calmidi, V.V. and Mahajan, R. L., 2000, “Forced Convection in High Porosity Metal Foams,” J. Heat Transfer, 122, pp. 557-565.

Darcy H., 1856, “Les Fontaines Publiques de la Ville de Dijon”, Paris, Victor Dalmont. (in French).

David, E., Lauriat, G. and Cheng, P., 1991, “A Numerical Solution of Variable Porosity Effects on Natural Convection in a Packed-Sphere Cavity,” J. Heat Transfer, 113, pp. 391-399.

Dixon, A. G. and Cresswell, D. L., 1979, “Theoretical Prediction of Effective Heat Transfer Parameters in Packed Beds,” AIChE, 25, pp. 663-676.

177 Ergun, S., 1952, “Fluid Flow through Packed Columns,” Chem. Eng. Pro., 48, pp. 89- 94.

Fu, W., Huang, H. C. and Liou, W. Y., 1996, “Thermal Enhancement in Laminar Channel Flow with a Porous Block,” Int. J. Heat Mass Transfer, 39, pp. 2165-2175.

Gilver, R. C. and Altobelli, S. A., 1994, “A Determination of the Effective Viscosity for the Brinkman-Forchheimer Flow Model,” J. Fluid Mechanics, 258, pp. 355-370.

Hadim, A., 1994, “Forced Convection in a Porous Channel with Localized Heat Sources,” J. Heat Transfer, 116, pp. 465-472.

Hong, J. T. and Tien, C. L., 1987, “Analysis of Thermal Dispersion Effect on Vertical- Plate Natural Convection in Porous Media,” Int. J. Heat Mass Transfer, 30, pp. 43- 150.

Hong, J. T., Tien, C. L. and Kaviany, M., 1985, “Non-Darcain Effects on Vertical- Plate Natural Convection in Porous Media with High Porosities,’ Int. J. Heat Mass Transfer, 28, pp. 2149-2157.

Hong, J. T., Yamada, Y. and Tien, C. L., 1987, “Effect of Non-Darcian Non-Uniform Porosity on Vertical-Plate Natural Convection in Porous Media,” J. Heat Transfer, 109, pp. 356-362.

Hsiao, S. W., Cheng, P. and Chen, C. K., 1992, “Non-Uniform Porosity and Thermal Dispersion Effects on Natural Convection about a Heated Horizontal Cylinder in an Enclosed Porous Medium,” Int. J. Heat Mass Transfer, 35, pp. 3407-3418.

Hsu, T. and Cheng, P., 1990, “Thermal Dispersion in a Porous Medium,” Int. J. Heat Mass Transfer, 33, pp. 1587-1597.

Hunt, M. L. and Tien, C. L., 1988a, “Effects of Thermal Dispersion on Forced Convection in Fibrous Media,” Int. J. Heat Mass Transfer, 31, pp. 301-309.

Hunt, M. L. and Tien, C. L., 1988b, “Non-Darcian Convection in Cylindrical Packed Beds,” J. Heat Transfer, 110, pp. 378-384.

Hwang, G. J., Wu, C. C. and Chao, C. H., 1995, “Investigation of Non-Darcian Forced Convection in an Asymmetrically Heated Sintered Porous Channel,” J. Heat Transfer, 117, pp. 725-732.

Hwang, G. J., and Chao, C. H., 1994, “Heat Transfer Measurement and Analysis for Sintered Porous Channels,” J. Heat Transfer, 116, pp. 456-464.

178 Ichimiya, K., Matsuda, T., and Kawai, Y., 1997, “Effects of a Porous Medium on Local Heat Transfer and Fluid Flow in a Forced Convection Field,” Int. J. Heat Mass Transfer, 40 pp. 1567-1576.

Jang, J. Y. and Chen, J. L., 1992, “Forced Convection in a Parallel Plate Channel Partially Filled with a High Porosity Medium,” Int. Comm. Heat Mass Transfer, 19, pp. 263-273.

Jiang, P. X., and Ren, Z. P., 2001, “Numerical Investigation of Forced Convection Heat Transfer in Porous Media Using a Thermal Non-Equilibrium Model,” Int. J. Heat Fluid Flow, 22, pp. 102-110.

Jiang, P. X., and Ren, Z. P., and Wang, B. X., 1999a, “Experimental Research of Fluid Flow and Convection Heat Transfer in Plate Channels Filled with Glass or Metallic Particles,” Experimental Thermal and Fluid Science, 20, pp. 45-54.

Jiang, P. X., and Ren, Z. P., and Wang, B. X., 1999b, “Numerical Simulation of Forced Convection Heat Transfer in Porous Plate Channels Using Thermal Equilibrium and Nonthermal Equilibrium Models,” Num. Heat Transfer, Part A, 35, pp. 99-113.

Jiang, P. X., and Ren, Z. P., Wang, B. X., and Wang, Z., 1996, “Forced Convective Heat Transfer in a Plate Channel Filled with Solid Particles,” J. Thermal Science, 5, pp. 43-53.

Kaviany, M., 1987, “Boundary Layer Treatment of Forced Convection Heat Transfer from a Semi-Infinite Flat Plate Embedded in Porous Media,” J. Heat Transfer, 109, pp. 345-349.

Kaviany, M., 1985, “Laminar Flow through a Porous Channel Bounded by Isothermal Parallel Plates,” Int. J. Heat Mass Transfer, 28, pp. 851-858.

Kim, D., and Kim, S. J., 2000, “Thermal Interaction at the Interface between a Porous Medium and an Impermeable Wall,” 4th KSME-JSME Thermal Engineering Conference, Kobe, Japan, October 1-6.

Kim, S. J. and Choi, C. Y., 1996, “Convection Heat Transfer in Porous and Overlying Layers Heated from Below,” Int. J. Heat Mass Transfer, 39, pp. 319-329.

Kim, S. J., Kim, D., and Lee, D. Y., 2000, “On the Local Thermal Equilibrium in Microchannel Heat Sinks,” Int. J. Heat Mass Transfer, 43, pp. 1735-1748.

179 Kim, S.Y., Kang, B. H. and Hyun, J. M., 1994, “Heat Transfer from Pulsating Flow in a Channel Filled with Porous Media,” Int. J. Heat Mass Transfer, 37, pp. 2025-2033.

Kladias, N. and Prasad, V., 1990, “Flow Transitions in Buoyancy-Induced Non-Darcy Convection in a Porous Medium Heated from Below,” J. Heat Transfer, 112, pp. 675- 684.

Koh, J.C., Dutton, J.C., Benson, B.A. and Fortini, A., 1977, “Friction Factor for Isothermal and Non-isothermal Flow Through Fibrous Porous Media,” J. Heat Transfer, 99C, pp. 367-373.

Kuznetsov, A. V., 1996, “Analytical Investigation of the Fluid Flow in the Interface Region between a Porous Medium and a Clear Fluid in Channels Partially Filled with a Porous Medium,” Applied Scientific Research, 56, pp. 53-67.

Kuznetsov, A. V., 1997a, “Influence of the Stress Jump Condition at the Porous- Medium/Clear-Fluid Interface on a Flow at a Porous Wall,” Int. Comm. Heat Mass Transfer, 24, pp. 401-410.

Kuznetsov, AV., 1997b, “Thermal Non-Equilibrium, Non-Darcian Forced Convection in a Channel Filled with a Fluid Saturated Porous Medium –a Perturbation Solution,” Applied Scientific Research, 1997b, 57, pp. 119-131.

Kuznetsov, A. V., 1998a, “Analytical Investigation of Coette Flow in a Composite Channel Partially Filled with a Porous Medium and Partially Filled with a Clear Fluid,” Int. J. Heat Mass Transfer, 41, pp. 2556-2560.

Kuznetsov, A. V., 1998b, “Analytical Study of Fluid Flow and Heat Transfer During Forced Convection in a Composite Channel Partly Filled with a Brinkman- Forchheimer Porous Medium,” Flow, Turbulence and Combustion, 60, pp. 173-192.

Kuznetsov, A. V., 1999, “Fluid Mechanics and Heat Transfer in the Interface Region between a Porous Medium and a Fluid Layer: A Boundary Layer Solution,” J. Porous Media, 2, pp. 309-321.

Lan, X. K. and Khodadadi, J. M., 1993, “Fluid Flow and Heat Transfer through a Porous Medium Channel with Permeable Walls,” Int. J. Heat Mass Transfer, 36, pp. 2242-2245.

Lauriat, G. and Vafai, K., 1991, “Forced Convective Flow and Heat Transfer through a Porous Medium Exposed to a Flat Plate or a Channel,” S. Kacac, B. Kilikis, F. A. Kulacki, F. Arnic, eds. Convective Heat and Mass Transfer in Porous Media. Dordrecht: Kluwer Academic, pp. 289-328.

180 Lee, D. Y. and Vafai, K., 1999, “Analytical Characterization and Conceptual Assessment of Solid and Fluid Temperature Differentials in Porous Media,” Int. J. Heat Transfer, 42, pp. 423-435.

Mahmid, A., Ben Nasrallah, S., and Fohr, J. P., 2000, “Heat and Mass Transfer During Drying of Granular Products Simulation with Convective and Conductive Boundary Conditions,” Int. J. Heat Mass Transfer, 43, pp. 2779-2791.

Martin, A. R., Saltiel, C., and Shyy, W., 1998, “Heat Transfer Enhancement with Porous Inserts in Recirculating Flows,” J. Heat Transfer, 120, pp. 458-467.

Muskat, M., 1937, “The Flow of Homogeneous Fluids through Porous Media,” 1st Edition, Edwards, Ann Arbor, MI

Nakayama, A., Kokudai, T. and Koyama, H., 1990, “Non-Darcian Boundary Layer Flow and Forced Convection Heat Transfer over a Flat Plate in a Fluid-Saturated Porous Medium,” J. Heat Transfer, 112, pp. 157-162.

Nakayama, A., Koyama, H. and Kuwahara, F., 1988, “An Analysis on Forced Convection in a Channel Filled with a Brinkman-Darcy Porous Medium: Exact and Approximate Solutions,” Wearme-und Stoffeubertragung, 23, pp. 291-295.

Neale, G. and Nader, W., 1974, “Practical Significance of Brinkman’s Extension of Darcy’s Law: Coupled Parallel Flows within a Channel and a Bounding Porous Medium,” The Canadian Journal of Chemical Engineering, 52, pp. 475-478.

Nield, D. A., Junqueira, S. L. M. and Lage, J. L., 1996, “Forced Convection in a Fluid-Saturated Porous-Medium Channel with Isothermal or Isoflux Boundaries,” J Fluid Mechanics, 322, pp. 201-214.

Nield, D. A., 1998, “Effects of Local Thermal Nonequilibrium in Steady Convective Processes in a Saturated Porous Medium: Forced Convection in a Channel,” J. Porous Media, 1, pp. 181-186.

Nield, D. A., and Kuznetsov, A.V., 1999, “Local Thermal Nonequilibrium Effects in Forced Convection in a Porous Medium Channel: a Conjugate Problem,” Int. J. Heat Mass Transfer, 42, pp. 3245-3252.

Ochoa-Tapia, J. A. and Whitaker, S., 1995a, “Momentum Transfer at the Boundary between a Porous Medium and a Homogeneous Fluid I: Theoretical Development,” Int. J. Heat Mass Transfer, 38, pp. 2635-2646.

181 Ochoa-Tapia, J. A. and Whitaker, S., 1995b, “Momentum Transfer at the Boundary between a Porous Medium and a Homogeneous Fluid II: Comparison with Experiment,” Int. J. Heat Mass Transfer, 38, pp. 2647-2655.

Ochoa-Tapia, J. A. and Whitaker, S., 1997, “Heat Transfer at the Boundary between a Porous Medium and a Homogeneous Fluid,” Int. J. Heat Mass Transfer, 40, pp. 2691- 2707.

Ochoa-Tapia, J. A. and Whitaker, S., 1998a, “Momentum Jump Condition at the Boundary between a Porous Medium and a Homogeneous Fluid: Inertial Effects,” J. Porous Media, 1, pp. 201-217.

Ochoa-Tapia, J. A. and Whitaker, S., 1998b, “Heat Transfer at the Boundary between a Porous Medium and a Homogeneous Fluid: the One-Equation Model,” J Porous Media, 1, pp. 31-46.

Ould-Amer, Y., Chikh, S., Bouhadef, K. and Lauriat, G., 1998, “Forced Convection Cooling Enhancement by Use of Porous Materials,” Int. J. Heat Fluid Flow, 19, pp. 251-258.

Poulikakos, D. and Kazmierczak, M., 1987, “Forced Convection in a Duct Partially Filled with a Porous Material,” J. Heat Transfer, 109, pp. 653-662.

Poulikakos, D. and Renken, K., 1987, “Forced Convection in a Channel Filled with Porous Medium, Including the Effects of Flow Inertia, Variable Porosity, and Brinkman Friction,” J. Heat Transfer, 109, pp. 880-888.

Renken, K. and Poulikakos, D., 1988, “Experiment and Analysis of Forced Convective Heat Transport in a Packed Bed of Spheres,” Int. J. Heat Mass Transfer, 25, pp. 1399-1408.

Sahraoui, M. and Kaviany, M., 1992, “Slip and No-Slip Velocity Boundary Conditions at the Interface of Porous, Plain Media,” Int. J. Heat Mass Transfer, 35, pp. 927-943.

Sahraoui, M. and Kaviany, M., 1994, “Slip and No-Slip Temperature Boundary Conditions at the Interface of Porous, Plain Media: Convection,” Int. J. Heat Mass Transfer, 37, pp. 1029-1044.

Scheidegger, A. E., 1974, “The Physics of Flow through Porous Media”, University of Toronto Press, Toronto

182 Sharif, N. H. and Wiberg, N. E., 2002, “Adaptive ICT Procedure for Non-Linear Seepage Flows with Free Surface in Porous Media”, Communications in Numerical Methods in Engineering, 18, pp. 161-176.

Srinivasan, V. and Vafai K., 1994, “Analysis of Linear Encroachment in Two- Immiscible Fluid Systems in a Porous Medium,” J. Fluids Eng., 116, pp. 135-139.

Sung, H. J., Kim, S.Y. and Hyun, J. M., 1995, “Forced Convection from an Isolated Heat Source in a Channel with Porous Medium,” Int. J. Heat Fluid Flow, 16, pp. 527- 535.

Vafai, K., 1984, “Convective Flow and Heat Transfer in Variable-Porosity Media”, J. Fluid Mechanics, 147, pp. 233-259.

Vafai, K., 1986, “Analysis of the Channeling Effect in Variable Porosity Media,” ASME J. Energy Resources Technology, 108, pp. 131-139.

Vafai, K., 2000, “Handbook of Porous Media”, Marcel Dekker, Inc, New York

Vafai, K. and Amiri, A., 1998, “Non-Darcian Effects in Confined Forced Convective Flows,” In: DB Ingham, I Pop, eds. Transport Phenomena in Porous Media, Pergamon, England: Elsevier Science, pp. 313-329.

Vafai, K., Alkire, R. L. and Tien, C. L., 1985, “An Experimental Investigation of Heat Transfer in Variable Porosity Media,” J. Heat Transfer, 107, pp. 642-647.

Vafai, K. and Kim, S. J., 1989, “Forced Convection in a Channel Filled with a Porous Medium: an Exact Solution,” J. Heat Transfer, 111, pp. 1103-1106.

Vafai, K. and Kim, S. J., 1990a, “Fluid Mechanics of the Interface Region between a Porous Medium and a Fluid Layer-an Exact Solution,” Int. J. Heat Fluid Flow, 11, pp. 254-256.

Vafai, K. and Kim, S. J., 1990b, “Analysis of Surface Enhancement by a Porous Substrate,” J. Heat Transfer, 112, pp. 700-706.

Vafai, K. and Kim, S. J., 1995a, “Discussion of Forced Convection in a Porous Channel with Localized Heat Sources,” J. Heat Transfer, 117, pp. 1097-1098.

Vafai, K. and Kim, S. J., 1995b, “On the Limitations of the Brinkman-Forchheimer- Extended Darcy Equation,” Int. J. Heat Fluid Flow, 16, pp. 11-15.

Vafai, K. and Thiyagaraja, R., 1987, “Analysis of Flow and Heat Transfer at the Interface Region of a Porous Medium,” Int. J. Heat Mass Transfer, 30, pp. 1391-1405.

183

Vafai, K. and Tien, C. L., 1981, “Boundary and Inertia Effects on Flow and Heat Transfer in Porous Media,” Int. J. Heat Mass Transfer, 24, pp. 195-203.

Vafai, K. and Tien, C. L., 1982, “Boundary and Inertia Effects on Convective Mass Transfer in Porous Media,” Int. J. Heat Mass Transfer, 25, pp. 1183-1190.

You, H. I. and Chang, C. H., 1997a, “Determination of Flow Properties in Non- Darcain Flow,” J. Heat Transfer, 119, pp. 190-192.

You, H. I. and Chang, C. H., 1997b, “Numerical Prediction of Heat Transfer Coefficient for a Pin-Fin Channel Flow,” J. Heat Transfer, 119, pp. 840-843.

184