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

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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 By 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 iv 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. vi 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 xi 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
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