Emerging Topics in Heat and Mass Transferin Porous Media

Theory and Applications of Transport in Porous Media

Series Editor:Jacob Bear, Technion – Israel Institute of Technology, Haifa, Isreal

Volume 22

The titles published in this series are listed at the end of the volume.

Emerging Topics in Heatand Mass Transferin Porous Media

From Bioengineering and Microelectronicsto Nanotechnology

Peter VadaszEditor

Department of Mechanical Engineering,Northern Arizona University, Flagstaff, AZ, U.S.A.

EditorPeter VadaszDepartment of Mechanical EngineeringNorthern Arizona UniversityFlagstaff, AZ, U.S.A.peter.vadasz@nau.edu

ISBN: 978-1-4020-8177-4 e-ISBN: 978-1-4020-8178-1

Library of Congress Control Number: 2008921941

c© 2008 Springer Science+Business Media B.V.No part of this work may be reproduced, stored in a retrieval system, or transmittedin any form or by any means, electronic, mechanical, photocopying, microfilming, recordingor otherwise, without written permission from the Publisher, with the exceptionof any material supplied specifically for the purpose of being enteredand executed on a computer system, for exclusive use by the purchaser of the work.

Cover illustration: A fractal resulting from evaluating the projection of the trajectory of differences be-tween numerical solutions of natural convection in porous media within the weak turbulent (chaotic)regime window at different accuracy levels (see pages 128–129). Figure 6d (modified) was published inP. Vadasz and S. Olek, Convergence and Accuracy of Adomian’s Decomposition Method for the Solutionof Lorenz Equations, International Journal of Heat and Mass Transfer, Vol. 43(10), pp. 1715–1734, 2000.

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Preface

This book is a synthesis of emerging topics in heat and mass transfer in porousmedia. It brings together some of the world leaders in research on transport phe-nomena in porous media to present the state of the art of its theory as well as theapplication of the theory in emerging fields such as bioengineering, microelectronicsand nanotechnology. The well renowned scientists presenting their findings in thereview chapters presented are not only among the best world leaders in their field,they also capture the research that is undertaken in all the parts of the globe, from theFar East (Hong-Kong), the Southern Hemisphere (New Zealand and South Africa)to Europe and America.

The book is separated into two parts. The first presents the state of the art ofthe theory of heat and mass transfer in porous media and can be used in both thetraditional (underground flow, filtering and reservoir engineering) as well as in themore recent emerging applications. The second part deals with emerging topics andapplications of the theory to bioengineering, microelectronics, and nanotechnology.

Traditionally, the topic of transport phenomena in porous media was almost exclu-sively reserved to the field of underground flow (water, oil, gas, etc.) and filtering. Withsome singular exceptions on applications to drying processes of fabric, the develop-ment of the theory of transport phenomena in porous media was historically drivenby the needs of technologies linked to reservoir engineering or civil engineering. Aturning point in this development was reached in the early part of the second half in thetwenty century when special attention to heat transfer in porous media yielded an ex-ceeding expansion of interest. This development continued in the twenty first centuryand reached recently such an impressive use in a diverse collection of technologicalapplications that created the motivation behind the preparation of this book.

The book starts with introducing the theoretical aspects of heat transfer in porousmedia by introducing the state of the art on the topic. It begins by introducing thetopic of conduction in porous media subject to Lack of Local Thermal Equilibrium –LaLotheq (or Local Thermal Non-Equilibrium – LTNE) and its link to Dual-Phase-Lagging (DuPhlag), the latter having a wider range of applications. It follows intoheat convection effects in porous media, extending the existing knowledge to gen-eralized heterogeneity effects, instability of unsteady boundary layers, transition toweak turbulence and chaos, gravity-modulated convection and thermal-vibrationalporous media convection.

v

vi Preface

This state of the art theoretical background is followed by chapters dealing withits application to emerging fields such as bioengineering, microelectronics, andnanotechnology. Bio- convection effects in porous media are presented in detailfollowed by the porous media application to macromolecular transport in arterialwalls in particular, and to flow and heat transfer in biological tissues in general. Theemerging application of metal foams as passive thermal control systems in coolingof microelectronics introduces another interesting aspect of heat transfer in porousmedia. The book concludes with the introduction of modeling of heat conduction innanofluid suspensions as a derivative of interface heat transfer modeling in porousmedia.

The book should be of interest to scientists, researchers, engineers and graduatestudents that intend pursuing an application of transport phenomena in porous me-dia or intend working on one of the emerging technologies covered in the book. Inaddition industry leaders that want to engage their teams in a deeper understandingof the concepts underlying the new emerging technologies should also be interestedin the work reported in this book. Finally, since the topics covered are truly inter-disciplinary and cross-disciplinary the disciplines that should find an interest arequite diverse, all engineering fields, biological sciences and physical sciences arejust examples.

September 2007 Peter VadaszFlagstaff, AZ, USA

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Dual-Phase-Lagging and Porous-Medium Heat Conduction Processes . . . . 1Liqiu Wang, Mingtian Xu, and Xiaohao Wei1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Well-Posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Solution Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Thermal Oscillation and Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1 Thermal Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Equivalence Between Dual-Phase-Lagging and Porous-Medium HeatConduction Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Heat Transfer Analysis Under Local Thermal Non-equilibrium Conditions 39A. Haji-Sheikh and W.J. Minkowycz1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.1 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2 Physical Interpretation of Relaxation Times . . . . . . . . . . . . . . . . . . . 43

3 Temperature Field with Stationary Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1 Temperature Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Temperature Field with Moving Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 Remarks and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

vii

viii Contents

General Heterogeneity Effects on the Onset of Convectionin a Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63D.A. Nield1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.1 Thermal Convection in a Square Enclosure . . . . . . . . . . . . . . . . . . . 703.2 Thermal Convection in a Tall Rectangular Enclosure . . . . . . . . . . . 713.3 Double Diffusive Convection in a Square Enclosure . . . . . . . . . . . . 71

4 Non-Uniform Basic Temperature Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . 735 Bidisperse Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756 Enclosure of Variable Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 Strong Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

The Instability of Unsteady Boundary Layers in Porous Media . . . . . . . . . . 85D.A.S. Rees, A. Selim, and J.P. Ennis-King1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874 Linearised Stability Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895 Comparison of the Methods Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.1 Quasi-Static Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.2 Local Rayleigh Number Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3 Energy Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.4 Amplitude Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6 Isolated Small-Amplitude Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 Other Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.1 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.2 Ramped Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.3 Internal Heat Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.4 Local Thermal Nonequilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8 Nonlinear Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Analytical Transition to Weak Turbulence and Chaotic NaturalConvection in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Peter Vadasz1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1112 Problem Formulation and Reduced Set of Equations . . . . . . . . . . . . . . . . . . 1133 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174 Computational and Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Contents ix

5 Compatible Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Natural Convection in Gravity-Modulated Porous Layers . . . . . . . . . . . . . . . 133Saneshan Govender1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1332 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1343 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364 Weak Non-linear Anlaysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405 Pendulum Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Thermal Vibrational Convection in a Porous Medium Saturatedby a Pure or Binary Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Yazdan Pedramrazi, Marie-Catherine Charrier-Mojtabiand Abdelkader Mojtabi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

1.1 What is Thermal Vibration? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1491.2 A Brief History of Thermal Vibration in Porous Media:

Suppression of Motion and Generation of Motion . . . . . . . . . . . . . 1502 The Effect of Vibration in Horizontal Porous Layer Saturated by a Pure

Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1512.1 Infinite Horizontal Porous Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1512.2 Confined Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1632.3 Some Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

3 Influence of Mechanical Vibration on a Porous Media Saturated by aBinary Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1673.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1683.2 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1693.3 Numerical Simulations in a Confined Cavity

(A = 1 and A = 10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1723.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

New Developments in Bioconvection in Porous Media: BioconvectionPlumes, Bio-Thermal Convection, and Effects of Vertical Vibration . . . . . . 181A.V. Kuznetsov1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1812 Numerical Modeling of a Falling Plume in a Suspension of Oxytactic

Microorganisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1832.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1832.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1842.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

x Contents

3 The Onset of Bio-thermal Convection in a Porous Medium . . . . . . . . . . . . . 1863.1 The Onset of Bio-thermal Convection in a Suspension

of Gyrotactic Microorganisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1893.2 The Onset of Bio-thermal Convection in a Suspension

of Oxytactic Microorganisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1974 Effect of Vertical Vibration on the Onset of Bioconvection

in a Horizontal Porous Layer of Finite Depth . . . . . . . . . . . . . . . . . . . . . . . 2064.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2064.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2064.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2084.4 Basic State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2094.5 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2094.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Macromolecular Transport in Arterial Walls: Currentand Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219K. Khanafer and K. Vafai1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2192 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

2.1 Wall-Free Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2202.2 Fluid-Wall Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2212.3 Multi-Layers Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2232.4 Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

3 Physiological Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2253.1 Endothelium and Internal Elastic Lamina . . . . . . . . . . . . . . . . . . . . . 2263.2 Intima and Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

4 Mathematical Model of Macromolecule Transport within theArterial Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2274.1 Lumen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2274.2 Endothelium and Internal Elastic Lamina . . . . . . . . . . . . . . . . . . . . . 2284.3 Intima and Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

5 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

Flow and Heat Transfer in Biological Tissues: Application of PorousMedia Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237Khalil Khanafer, Abdalla AlAmiri, Ioan Pop, and Joseph L. Bull1 Brain Aneurysm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2371.2 Clinical and Experimental Studies Associated

with the Treatment of Aneurysms Using StentImplantation and Coil Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

1.3 Computational Studies Associated with Combined Use ofStents and Coils for the Treatment of Cerebral Aneurysms . . . . . . 239

1.4 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Contents xi

2 Flow and Heat Transfer in Biological Tissues . . . . . . . . . . . . . . . . . . . . . . . . 2422.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2422.2 Thermal Models for Blood Perfused Tissues . . . . . . . . . . . . . . . . . . 2442.3 Mathematical Modeling of Bioheat Equation Using Porous

Media Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2493 Tissue Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2513.2 Porous Scaffolds for Tissue Engineering . . . . . . . . . . . . . . . . . . . . . 251

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

Metal Foams as Passive Thermal Control Systems . . . . . . . . . . . . . . . . . . . . . 261Shankar Krishnan, Jayathi Y. Murthy, and Suresh V. Garimella1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2612 Mathematical Formulation and Numerical Modeling . . . . . . . . . . . . . . . . . . 2633 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

3.1 Melt Volume Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2723.2 Wall Nusselt Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Nanofluid Suspensions and Bi-composite Media as Derivativesof Interface Heat Transfer Modeling in Porous Media . . . . . . . . . . . . . . . . . . 283Peter Vadasz1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2832 Problem Formulation and the Apparent Paradox . . . . . . . . . . . . . . . . . . . . . . 2853 Solution by the Eigenvectors Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2884 Solution by the Elimination Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2925 Resolution of the Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2956 Experimental Measurement of the Effective Thermal Conductivity of a

Porous Medium via the Transient Hot Wire (THW) Method . . . . . . . . . . . 3016.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3016.2 Concepts and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

7 Application of the Heat Conduction in Porous Mediato Nanofluid Suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3147.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3167.2 Solution and Correction of the THW Results . . . . . . . . . . . . . . . . . . 3187.3 Results, Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 319

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Contributors

Abdalla AlAmiriMechanical Engineering Department, United Arab Emirates University,P.O. Box 17555 Al-Ain, UAE, alamiri@uaeu.ac.ae

Joseph L. BullVascular Mechanics Lab, Biomedical Engineering Department, University ofMichigan, Ann Arbor, MI 48109, USA, joebull@umich.edu

Marie-Catherine Charrier-MojtabiLaboratoire P.H.A.SE., UFR PCA, Universite Paul Sabatier,118 route de Narbonne,31062 Toulouse cedex, France, cmojtabi@cict.fr

J.P. Ennis-KingCooperative Research Centre for Greenhouse Gas Technologies, CSIRO Petroleum,Private Bag 10, Clayton South, VIC 3169, Australia

Suresh V. GarimellaCooling Technologies Research Center, School of Mechanical Engineering, PurdueUniversity, West Lafayette, IN 47907-2088 USA, sureshg@ecn.purdue.edu

Saneshan GovenderUniversity of Kwa-Zulu Natal, School of Mechanical Engineering, King George VAvenue, Durban 4001 South Africa, govenders65@ukzn.ac.za

A. Haji-SheikhDepartment of Mechanical and Aerospace Engineering, The University of Texas atArlington, Arlington, TX 76019-0023, USA, haji@mae.uta.edu

Khalil KhanaferVascular Mechanics Lab, Biomedical Engineering Department, University ofMichigan, Ann Arbor, MI 48109, USA, khanafer@umich.edu

Shankar KrishnanBell Laboratories, Alcatel-Lucent, Blanchardstown Industrial Park, Dublin-15,Ireland, skrishnan@alcatel-lucent.com

A.V. KuznetsovNorth Carolina State University, Mechanical & Aerospace Engineering, Box 7910,Raleigh, NC 27695, USA, avkuznet@eos.ncsu.edu

xiii

xiv Contributors

W.J. MinkowyczDepartment of Mechanical and Industrial Engineering, University of Illinois atChicago, Chicago, IL 60607-7022, USA, WJM@uic.edu

Abdelkader MojtabiInstitut de mecanique des Fluides, UMR CNRS-INP-UPS N◦5502, Universite PaulSabatier, 118 route de Narbonne, 31062 Toulouse cedex, France, mojtabi@imft.fr

Jayathi Y. MurthyCooling Technologies Research Center, School of Mechanical Engineering, PurdueUniversity, West Lafayette, IN 47907-2088 USA, jmurthy@ecn.purdue.edu

D.A. NieldDepartment of Engineering Science, University of Auckland, Private Bag 92019,Auckland 1142, New Zealand, d.nield@auckland.ac.nz

Yazdan PedramraziReservoir Engineering Research Institute(RERI), 385 Sherman Ave, Suite 5, PaloAlto, CA 94036, USA, yazdan@rerinst.org

Ioan PopFaculty of Mathematics, University of Cluj, R-3400 Cluj, Romania,pop-ioan@yahoo.co.uk; popi@math.ubbcluj.ro

D.A.S. RessDepartment of Mechanical Engineering, University of Bath, Claverton Down, BathBA2 7AY, UK, D.A.S.Ress@bath.ac.uk

A. SelimDepartment of Mechanical Engineering, University of Bath, Claverton Down, BathBA2 7AY, UK

Peter VadaszDepartment of Mechanical Engineering, Northern Arizona University,P.O. Box 15600, Flagstaff, AZ 86011-5600, USA, peter.vadasz@nau.edu

K. VafaiMechanical Engineering Department, University of California, Riverside, CA92505, USA, vafai@engr.ucr.edu

Liqiu WangDepartment of Mechanical Engineering, The University of Hong Kong, PokfulamRoad, Hong Kong, lqwang@hku.hk

Xiaohao WeiDepartment of Mechanical Engineering, The University of Hong Kong, PokfulamRoad, Hong Kong

Mingtian XuInstitute of Thermal Science and Technology, Shandong University, Jinan 250061,China