Essentials of Heat and Fluid Flow in Porous Media 3030998649, 9783030998646

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Arunn Narasimhan

Essentials of Heat and Fluid Flow in Porous Media

Essentials of Heat and Fluid Flow in Porous Media

Arunn Narasimhan

Essentials of Heat and Fluid Flow in Porous Media

Arunn Narasimhan Department of Mechanical Engineering Indian Institute of Technology Madras Chennai, India

ISBN 978-3-030-99864-6 ISBN 978-3-030-99865-3 (eBook) https://doi.org/10.1007/978-3-030-99865-3 Jointly published with ANE Books Pvt. Ltd. In addition to this printed edition, there is a local printed edition of this work available via Ane Books in South Asia (India, Pakistan, Sri Lanka, Bangladesh, Nepal and Bhutan) and Africa (all countries in the African subcontinent). ISBN of the Co-Publisher’s edition: 9789382127215 © The Author(s) 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to my guide Prof. José L. Lage Southern Methodist University, Dallas, TX, USA.

Foreword by Prof. D. Andrew S. Rees

Porous media are ubiquitous; they appear everywhere. Examples range from clothing, through which the rain seeps, to filters and catalytic converters in our cars; brain tissue, lungs and other parts of the bodies of animals; soils, aquifers, sands and even the methane-saturated regolith of Titan. Apart from that very last example, the others frequently form an essential part of our daily lives and we depend on them functioning in order both to live and to live comfortably. Yet despite this very wide-ranging list of applications of porous media, there is little taught at undergraduate or postgraduate level outside of the disciplines of civil, chemical and geological engineering. Perhaps one reason for this is that the simplest form of the momentum equation is Darcy’s law, which is of a lower differential order than its clear fluid equivalent, the Navier–Stokes equations. A two-dimensional flow in a porous medium is, therefore, governed by a second-order equation whereas that of a clear fluid is of fourth order. This simplification appears not to have been welcomed in some quarters, but rather it has been derided as being too easy to study. I do not agree with this point of view. It has even been stated that a motivation for the study of convective flow in porous media is that it serves simply as a test-bed for methods for studying convective flows in clear fluids. I do not agree with this motivation even though it is quite true that many aspects of stability theory for the porous Bénard problem (see Chap. 5) may be undertaken using solely analytical techniques, thereby imparting much knowledge of stability theory which may then be used elsewhere. However, while Darcy’s law in its most primitive form yields a simpler system to solve than do the Navier–Stokes equations, this ceases to be true when Brinkman effects are non-negligible (see Chap. 3). Small Darcy numbers then yield thin boundary layers near surfaces, and these need to be resolved numerically, a difficult task to do well. Geological applications frequently have heterogeneous permeability fields where the permeability varies randomly over many orders of magnitude; the governing equations are then very difficult to solve numerically. Thus we are entering an arena where mathematicians and physicists, for example, may be called upon to employ averaging techniques in order to obtain equivalent macroscopic equations, and where numerical analysts are required to solve very detailed models with discontinuous and highly anisotropic coefficients. Therefore the study of porous media vii

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requires a great plethora of techniques and topics to cover a large range of possible applications. There is clearly a need for an interdisciplinary approach to studying porous media. Many books on porous media have appeared in the last twenty years or so. The very well-acclaimed and popular monograph, convection in porous media, by Nield and Bejan is just about to appear in its fourth edition, but it is meant to be a resource base, a one-stop-shop for researchers to find out what the state-of-the-art is in their particular niche. The two series of books, one edited by Vafai (Handbook of Porous Media) and the other by Ingham and Pop (Transport Phenomena in Porous Media), which are also excellent, provide more detailed reviews of chosen topics by experts in the field. A very recent book edited by Vafai (Porous Media: Applications in Biological Systems and Biotechnology) also consists of a set of reviews by specially chosen authors. But none of these books, as important as they have been for the research community at large, may be described as textbooks. The present book fills an important gap in the market because it has been written specifically for students to acquire the basic knowledge of flows, heat conduction, convection and radiation in porous materials. Finally, I would like to say a few words about the author. Arunn is equally at home with analytical, numerical and experimental techniques. He thinks creatively and strategically, is prolific in his research and is a conscientious teacher. In all of these respects he is following the heritage of his Ph.D. supervisor, Prof. José Lage, and his supervisor’s supervisor, Prof. Adrian Bejan. However Prof. Narasimhan is very much his own man—not content with traditional means of communication within the lecture theatre, conferences and research journals, he is also the founder of two prolific blogs, one in English, the other in Tamil, in which he discourses on a wide range of topics which includes but is not at all limited by those of his research. He has been described by Asian Scientist as ‘one of India’s most energetic professors—both in person and in cyberspace’. I certainly wonder how he finds the time to do all of these things! So I welcome onto the scene this textbook which is so clearly infused with the distilled product of its author’s roving mind, and which is rooted in and motivated by the applications of the theory. D. Andrew S. Rees University of Bath Bath, UK

Foreword by Prof. Pradip Dutta

It is an honour and pleasure to write a foreword to this useful and interesting book. Arunn is a very well-known researcher who has worked extensively in transport phenomena on porous media for the past two decades and has pioneered several activities in emerging applications in this field. I have had several opportunities to interface and collaborate with Arunn in connection with some applications of porous media pertinent to my research field of solidification and phase change materials, and I have always found the discussions extremely useful. The subject of porous media is not new, and this is not the first book in literature on porous media heat transfer and fluid flow. Yet, the scientific and engineering community is still in the quest for a basic understanding of flow and heat transfer in porous media. The treatment is never straightforward, as the basic formulation of heat and fluid flow is at a scale which is generally much bigger than the actual porous structure. Together with this challenge, the diversity of applications brings in a load of complexities in modelling the transport phenomena. For instance, for the solidification community where I belong, one of the biggest challenges is to appropriately model the mushy region as a porous medium which is anisotropic and dynamically evolving in porosity and structure. The main strength of this book lies in addressing most of the complex issues that occur in various applications of porous media including ground water flow, phase change processes and biological tissues. Each application comes with its own complexity sometimes requiring special treatment, but the generalized approach and appropriate classification of problems in this book enable the reader to obtain a better grasp on the subject. The author has covered systematically almost all necessary topics in porous media, including local thermal equilibrium concept, local non-equilibrium concept, Darcy model, non-Darcy effects, conduction heat transfer, forced convection, natural convection, radiation, bio heat transfer, multiphase flow, transient effects and so on. As I read the book, I find that it has covered adequately the areas I have some familiarity and expertise and has triggered my interest in other areas. Such is the fluidity and clarity in writing.

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Overall, I think the book will be of interest to undergraduate and graduate students in science and engineering, interdisciplinary researchers such as in materials processing and bioengineering, and to R&D engineers involved in various industrial applications of porous media. I wish this book success and its readers as much enjoyment and benefit as I had when reading. Pradip Dutta Indian Institute of Science Bangalore, India

Preface

What is a porous medium? When I had this question about fifteen years ago, upon receiving information about my graduate admission involving porous medium research, the only answer I could muster was, ‘Pots’. The ensuing years, under the supervision of an excellent research advisor, unravelled almost everything in our perceivable Universe—from sand to pot, aluminium foam to honeycombs, fur to forest, capillaries to cauliflower, humans to the Milky Way—as a porous medium. This textbook introduces essential concepts that govern heat and fluid flow through porous media at a graduate level. It aims to fill the knowledge gap between the available research monographs concerning transport in porous media and the basic thermo-fluids courses one is required to master as a prerequisite for approaching such monographs. Knowledge of heat and fluid flow through porous media finds extensive applications in several engineering devices spanning the major divisions, mechanical, civil and chemical engineering. Recent ramifications include bioengineering and bio-technology. The first five chapters (1–5) of this book are evolved as an extension of our basic engineering understanding of momentum and heat transport in situations without a porous medium, as learnt in undergraduate fluid mechanics and heat transfer courses. Analogous situations of classical problems of fluid flow and heat transfer are discussed in this spirit to motivate the diligent reader equipped with a first course knowledge of these subjects. To support the learning process, a sizeable collection of ‘end of chapter’ problems is provided to these chapters. The last three chapters (6–8) are structured at an advanced level, introducing ongoing research that utilize directly or in a suitably modified form, the applications of the porous medium modelling learnt in the previous chapters. While a graduate student could start at Chap. 1 and proceed to the research possibilities in Chap. 8, a practising porous medium researcher could do the reverse and begin at Chaps. 8 or 6 and refer to the initial five chapters, for a quick check of fundamentals, as and when required. It is conventional in porous medium textbooks to introduce fluid dynamics through porous media before venturing into heat transfer. This has the advantage of presenting xi

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the subject through its historical evolution. We shall deviate from this convention— but align with another, adopted by heat transfer textbooks—to introduce heat conduction in Chap. 2, before fluid flow through porous medium. This treatment allows the graduate student to appreciate the basic nature of the thermal equilibrium concept that prevails even without flow inside a porous medium and its crucial role in the success of the concept of volume averaging. A historical approach is adopted in Chap. 3 while introducing fluid flow concepts—a method favoured by Truesdell (1974)—to dispel some recurring misunderstandings concerning form drag and transition of flow in porous media. In this context, the section, ‘Physics of Flow through Porous Media’ is a long argument marshalling evidence for the nature of the ‘quadratic drag’ originating in the form or shape of the solid matrix of the porous medium. The usage of ‘inertial drag’ to depict form effects is a misnomer as inertial effects are almost absent in such high-speed flows through low permeable porous media. Later in the chapter, these significant developments are placed in a theoretical framework that is based on the volume-averaging procedure and concept of representative elemental volume (REV) introduced in Chap. 1. Chapters 4 and 5 discuss, respectively, forced and natural convection in porous medium. An excellent research monograph by Nield and Bejan (2006) with extensive coverage of material segregated in relevant sub-topics of convection in porous media is already available. Additionally, recent compilations by Ingham et al. (2004), Bejan et al. (2004), Vadàsz (2008) and the two handbooks on porous media by Vafai (2000, 2005) discuss both basic and advanced topics and ongoing research pertaining to convection in porous media. Instead of trying to surpass these texts, the discussions in these two chapters are restricted to the fundamentals in convection. A list of available convection heat transfer solutions and correlations, with relevant references, is presented in the appendix. However, advanced material that is less discussed elsewhere is included. For instance, heat transfer augmentation using porous media is discussed extensively in Chap. 4, which also highlights recent approaches of treating heat exchangers as mono- and bi-disperse porous media. Likewise, local thermal non-equilibrium and heat generation effects in natural convection are discussed in detail in Chap. 5. Chapter 6 titled ‘Porous Medium Aspects of Biological Systems’, discusses the basics of porous medium aspects being applied today in the effective modelling of human biological organs and functions. With the advancements in biology, health and medicine becoming interdisciplinary, physical and mathematical concepts are being increasingly invoked to model biological processes. Porous medium modelling of bio-fluid and heat flows is an interesting and useful approach to simplify and understand biological phenomena. To complete the treatment of heat transfer in porous medium, Chap. 7 briefly discusses radiation heat transfer in porous medium. The content is largely based on the review chapter by Prof. Howell, in the Handbook of Porous Media, 1st Edition, ed. Vafai (2000). The radiative transport equation (RTE) valid for a porous medium and its solution under LTE and LTNE conditions are presented.

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Chapter 8 introduces a few current research topics that involve porous medium modelling discussed in the preceding chapters. Each section is treated as a separate case study, providing details on the fundamental equations and modelling involved. It should enthuse readers who wish to pursue research in transport in porous media. Solidification phenomenon in porous medium is discussed in the context of a latent heat storage device. The modifications required in the momentum equation and the associated heat transfer effects when temperature dependency of viscosity of the fluid flowing through a porous medium is considered next. To introduce beyond the singlephase treatment of convection in Chaps. 4 and 5, two-phase convection results under LTE and LTNE using a mixture model are presented. Later in the chapter, the lattice Boltzmann method of momentum equation formulation for porous medium flows is presented. A brief introduction to combustion in inert porous media completes our discussion of advanced topics. Most of the advanced topics discussed throughout the book are based on my involvement in the associated research and should by no means be considered either exhaustive or superior to other ongoing research. Several ongoing and fresh research topics exist in porous medium research, which are not presented here. From direct measurement of averaged properties to heterogeneity and anisotropy to the implications of local thermal non-equilibrium to underground coal gasification to upscaling in geology to fractal porous media to stochastic approaches to biological modelling, much unexplored and exciting research terrain awaits your time and effort in the field of transport through porous medium. That brings us back to the question: what is a porous medium? As you complete reading this book dear reader, I am sure the journey would make you perceive even the bar code on the back cover as one. Chennai, India July 2012

Arunn Narasimhan

Acknowledgements

In most of mankind, gratitude is merely a secret hope for greater favours —Duc de la Rochefoucauld, Maxims (1665)

As it should be for any human being, I have more people to thank for my life and deeds than it would be proper to include inside a thermal science textbook. Restrict then I must my acknowledgements to only those related to the sphere of action culminating in this book. I dedicate this effort to my Ph.D. advisor Prof. José Lage, who taught me the basics of porous media. Neither I nor he would have anticipated a decade down the path, consequences such as this. My friend, Prof. Vladimir Koulich of the Nanyang Technological University, Singapore, in whose rich cultural sweep (from Archaeopteryx and Etruscans to Fractals and Sanskrit to Xanthipe and Zeno), I could always find my intellectual banter and balance, has been with me since his ‘senior student’ days with Prof. Lage. I was fortunate to have served four years as a teaching assistant to Prof. Jack Holman, whose tutorship is my constant academic guide. Research collaboration with Prof. Donald A. Nield, (Department of Engineering Science, University of Auckland), his rigour in doing science and integrity in appraising it, has been beneficial for this novice. Prof. D. Andrew S. Rees of the University of Bath, UK, has been a senior research collaborator and appraiser of my research. I thank him also for the foreword. Prof. Pradip Dutta from the Indian Institute of Science, Bangalore, who has also provided the foreword, is another individual whose collaboration and association is a growing influence in my career. For two summers, he had invited me for research collaboration to his laboratory in IISc., which is becoming my second academic home. I also thank Prof. K. Muralidhar of IIT Kanpur for his personal encouragement and continued academic support. Above all, I remain awed by Prof. Adrian Bejan, my ‘grand advisor’, whose works and words have been my source of inspiration, to remain in Mechanical Engineering and contribute.

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At IIT Madras, I should begin my thanks with Prof. Babu Viswanathan, who has been my exemplary mentor, inspirational colleague and honest friend—a combination in academia that is rare and precious, like Dennis the Menace. Years ago, he convinced me about joining IIT Madras as a faculty member, just as he convinced me at this stage of my career to write this book. Prof. T. Sundarrajan has been another mentor throughout my tenure. Secure in his knowledge and sagacious in his association, he has been a source of confidence and calming influence. I also thank Prof. Satya Chakravarthy of the Aerospace Department, who is more my ebullient friend than the enthusiastic collaborator that he is. Prof. T. T. Narendran of the Department of Management Studies, who took a special liking to my taste in Carnatic music, has been my trusted friend. I thank Prof. S. Pushpavanam of the Department of Chemical Engineering and others who always provide their time to make my academic life engaging and pleasant. Finally, I thank all my colleagues in the Heat Transfer and Thermal Power Laboratory for shaping my competitive spirit. I thank my research students for the diligence and sincerity they have shown in unravelling some of our ideas in thermal science. In no particular order, here are their names: Akhilesh Prabhakar, Prasad Bhave, J. Wilson, K. S. Raju, Ankit Somani, Krishna Reddy, Kaushal Kumar Jha, P. V. Ramana, V. G. Ramanathan, C. Sundarraj, M. Manish, S. Jaichander. Apart from these, several students who have taken my courses in the past decade have contributed to my better understanding of the subjects I taught. The Januray–May 2012 batch of students who took my course on Heat and Mass Transport in Porous Media offered me suggestions in shaping the text and problems in this book. I thank all of them for their effort and assistance. Chennai, India July 2012

Arunn Narasimhan

Contents

1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Porous Medium Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Non-traditional Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Bi-disperse Porous Medium (BDPM) . . . . . . . . . . . . . . . . . . . 1.1.3 Porous Bio-materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Volumetric Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Surface Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Percolation and Tortuosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Tortuosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Volume-Averaging Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Homogeneous Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Representative Elemental Volume . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Some Clarifications on REVs . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 5 5 6 6 7 8 8 9 10 10 12 13 15

2 Heat Conduction in Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 First Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Local Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Porous Medium Energy Equation . . . . . . . . . . . . . . . . . . . . . . . 2.3 Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Effective Stagnant Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Limiting Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Experimental Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Description of SGHP Apparatus . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Determination of ke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 19 24 25 26 26 26 27 32 32 34

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2.5 Thermal Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Local Thermal Non-equilibrium Model . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 LTNE Formulation of PM Energy Equation . . . . . . . . . . . . . . 2.6.2 Interface Heat Transfer Coefficient . . . . . . . . . . . . . . . . . . . . . 2.7 Transient Heat Conduction in Porous Medium . . . . . . . . . . . . . . . . . . 2.7.1 1D Transient PM Heat Conduction Under LTE . . . . . . . . . . . 2.7.2 Lumped Capacitance Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 1D Transient PM Heat Conduction Under LTNE . . . . . . . . . 2.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Fluid Flow Through a Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Darcy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Hazen–Dupuit–Darcy Model . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Physics of Flow Through Porous Media . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Inertialess Motion and the HDD Model . . . . . . . . . . . . . . . . . 3.3.2 From Newton’s Principia to the Nature of Form Drag . . . . . 3.4 Extensions of the HDD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Brinkman Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Generalized Momentum Equation . . . . . . . . . . . . . . . . . . . . . . 3.5 Permeability and Form Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Analytical Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Fictitious Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Experimental Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Viscometry and Permeametry . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5 Protocol for Permeability Measurement . . . . . . . . . . . . . . . . . 3.6 Flow Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 What Is Transition? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Characterization of the Transition . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Prediction of Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.4 PM Transition Flow and Turbulent Flow . . . . . . . . . . . . . . . . . 3.7 Miscellaneous Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Hydrodynamic Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Wall Channelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Porosity and Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.4 Secondary Effect in the Determination of K and C . . . . . . . . 3.7.5 Boundary and Interface Conditions . . . . . . . . . . . . . . . . . . . . . 3.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 49 49 51 53 54 56 61 61 64 65 65 66 67 72 73 77 78 78 80 82 82 82 83 84 85 86 89

4 Forced Convection Through Porous Medium . . . . . . . . . . . . . . . . . . . . . . 4.1 Energy Equation with Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Local Thermal Non-equilibrium (LTNE) Situation . . . . . . . . 4.1.2 Local Thermal Equilibrium (LTE) Situation . . . . . . . . . . . . . .

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4.2 Forced Convection in Porous Medium Over a Flat Plate . . . . . . . . . . 4.3 Forced Convection in Porous Medium Channel . . . . . . . . . . . . . . . . . 4.3.1 Bulk Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Nusselt Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Channel Filled with Porous Medium . . . . . . . . . . . . . . . . . . . . 4.3.4 A Measurable Nu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Heat Transfer Enhancement Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Basic Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Heat Exchangers as Porous Media . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Porous Medium Inter-connectors . . . . . . . . . . . . . . . . . . . . . . . 4.5 Other Forced Convection Configurations . . . . . . . . . . . . . . . . . . . . . . . 4.6 Viscous Dissipation Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96 99 99 102 103 104 106 107 110 111 117 118 120

5 Natural Convection Through Porous Medium . . . . . . . . . . . . . . . . . . . . . 5.1 Natural Convection Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Vertical Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Horizontal Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Natural Convection with Vertical Thermal Gradient . . . . . . . . . . . . . 5.2.1 Onset of Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Natural Convection with Horizontal Thermal Gradient . . . . . . . . . . . 5.3.1 Isothermal Side-Wall Heated Configuration . . . . . . . . . . . . . . 5.3.2 Isoflux Side-Wall Heated Configuration . . . . . . . . . . . . . . . . . 5.3.3 High Ra Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Heatline Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Non-Darcy, LTNE and Heat Generation Effects . . . . . . . . . . . . . . . . . 5.5.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Departure from Darcy Flow Effects . . . . . . . . . . . . . . . . . . . . . 5.5.4 Heat Generation Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Prediction of Critical Internal Heat Generation . . . . . . . . . . . 5.5.6 Correlation for Nu in Porous Annulus . . . . . . . . . . . . . . . . . . . 5.6 Viscous Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 123 127 127 127 130 130 130 134 134 135 138 138 141 142 142 144 145 146 147

6 Porous Medium Aspects of Biological Systems . . . . . . . . . . . . . . . . . . . . 6.1 Introduction to Bio-thermofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Porous Medium Modelling in Bio-heat Transport . . . . . . . . . . . . . . . 6.2.1 Chen and Holmes Bio-heat Model . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Porous Medium Bio-heat Model with LTNE . . . . . . . . . . . . . 6.2.3 Other Recent Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Modelling Drug Delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Transscleral Porous Medium Model . . . . . . . . . . . . . . . . . . . . 6.3.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 149 151 151 152 153 155 155 156

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Contents

6.3.3 Determination of Porosity and Permeability of the Sclera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Anecortave Acetate Case Study . . . . . . . . . . . . . . . . . . . . . . . . 6.3.6 Effect of Choroidal Blood Flow on Drug Diffusion . . . . . . . . 6.3.7 Transdermal Drug Delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 LDL Transport Across Arterial Tissues . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Porous Medium Modelling in Bio-mass Transport . . . . . . . . . . . . . . . 6.5.1 Mass Transport in Tissue Regeneration . . . . . . . . . . . . . . . . . . 6.5.2 Porous Medium Model for Lung Diffusion . . . . . . . . . . . . . . 6.5.3 Porous Medium Models of Microbial Transport . . . . . . . . . .

157 158 159 160 161 161 164 164 167 170

7 Radiation Heat Transfer in Porous Medium . . . . . . . . . . . . . . . . . . . . . . 7.1 The Radiative Transfer Equation (RTE) . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Energy Equation with Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Radiative Property Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Solving the RTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Coupling of RTE with Other Heat Transfer Modes . . . . . . . . . . . . . .

173 173 174 176 177 178

8 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Phase Change in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Porous Medium LHTES Device . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 The Enthalpy Model and Governing Equations . . . . . . . . . . . 8.1.3 Effective Properties and Boundary Conditions . . . . . . . . . . . . 8.1.4 Exergy Criterion and Optimum φ . . . . . . . . . . . . . . . . . . . . . . . 8.1.5 Scale Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Variable Viscosity Porous Medium Flows . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Temperature-Dependent Viscosity Effects . . . . . . . . . . . . . . . 8.2.2 Variable Viscosity and HDD Model . . . . . . . . . . . . . . . . . . . . . 8.2.3 Limiting Case of the HDD Model . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Modified HDD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5 M-HDD Model Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.6 Hydrodynamics of μ(T ) Flows . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Flow and Convection in Bi-disperse Porous Media . . . . . . . . . . . . . . 8.3.1 Laminar Flow Through BDPM Channel . . . . . . . . . . . . . . . . . 8.3.2 Governing Equations and Boundary Conditions . . . . . . . . . . 8.3.3 Effects of Internal Permeability (Da I ) . . . . . . . . . . . . . . . . . . 8.3.4 Effects of Macro-pore Volume Fraction (φ E ) . . . . . . . . . . . . . 8.3.5 Natural Convection in BDPM Enclosure . . . . . . . . . . . . . . . . . 8.4 Two-Phase Flow Through Porous Media . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Mixture Model Governing Equations . . . . . . . . . . . . . . . . . . . 8.4.2 Mixture Variables and Coefficients . . . . . . . . . . . . . . . . . . . . . 8.4.3 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Sample Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179 179 179 180 181 183 184 186 186 188 189 190 191 192 194 195 195 198 200 202 204 205 206 207 209

Contents

8.5 LBM Formulation for Porous Medium Flows . . . . . . . . . . . . . . . . . . . 8.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 d2q9 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 d2q9i Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Combustion in Inert Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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209 209 210 212 214

Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Appendix A: Porosity and Permeability Values . . . . . . . . . . . . . . . . . . . . . . . 219 Appendix B: Convection Results and Correlations . . . . . . . . . . . . . . . . . . . . 221 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

About the Author

Dr. Arunn Narasimhan is a Professor in the Department of Mechanical Engineering at the Indian Institute of Technology Madras, India. His major research interests for the past fifteen years are heat and fluid flow in porous media and biological systems. His publications include eight invited book chapters (two in the Handbook of Porous Media), two invited review articles and over hundred peer-reviewed research articles in international journals and conferences. He has also written few novels and popular science books in Tamil language.

xxiii

Chapter 1

Basic Concepts

In this chapter we discuss introductory concepts, starting with an understanding of the definition of a porous medium. We proceed to define and discuss other terminologies like porosity, connectivity, consolidation, percolation, tortuosity, homogeneity, isotropy and anisotropy. The concept of volume averaging is also introduced from first principles. These basic concepts and terminology that characterize a porous medium are to be understood before formulating the governing conservation equations of mass, momentum and heat for analysing transport phenomena in such porous media.

1.1 Porous Medium Definition What is a porous medium? A popular definition is “A porous medium is a solid structure with interconnected voids.” Suppose we take three different substances made of two materials, Material 1 and Material 2. Material 2 is the void in the above definition. Given in Fig. 1.1 is the two-dimensional cut-section view of the three substances. Which of the three substances is a porous medium? According to the above definition, all three substances can be treated as porous medium. However, the configuration in Fig. 1.1a can easily be considered as two separate homogeneous materials for analysing transport phenomena in and between the constituents. A definition for porous medium, suited for analysing transport phenomena is, “It is a region in space comprising at least two homogeneous material constituents, presenting identifiable interfaces between them in a resolution level, with at least one of the constituents remaining fixed or slightly deformable ”. The requirement that one material should be stationary is for convenience of analysing transport phenomena in the porous medium, when the other constituent exhibits relative motion. © The Author(s) 2023 A. Narasimhan, Essentials of Heat and Fluid Flow in Porous Media, https://doi.org/10.1007/978-3-030-99865-3_1

1

2

1 Basic Concepts

material 2 (voids)

material 1

(A)

(B)

(C)

Fig. 1.1 Porous medium definition

The simple definition ‘A porous medium is a solid structure with interconnected voids’ implies a requirement of flow through the porous medium. This requirement of flow is not mandatory for the definition. Both constituents can also remain immobile. For instance, a composite material like Fig. 1.1c where both shaded and white regions represent two fixed solid materials can be treated as a porous medium. A closed region with only internal pores is also a porous medium. But access to these internal pores by physical or indirect means is essential to perform further analysis in transport phenomena. Observe the inclusion of material constituents in our definition to suggest that the voids need not be empty. Furthermore, unlike in Fig. 1.1, there can be more than two material constituents arranged in any disorderly fashion, forming a porous medium according to our definition. One restriction is that, in a given visual resolution level, distinct interfaces must demarcate these constituents. An example is a two-phase flow through a fixed metal mesh, a configuration that is often seen in heat exchanging engineering appliances. A region filled with a seemingly continuous substance at one visual resolution level may resolve into a porous structure when magnified to a finer level of visual resolution. Human skin surface is smooth but reveals the distribution of sweat pores only at a finer resolution level. Clothing, from sweat shirt to baby diaper, is a related example. Effectiveness of perspiration-induced heat exchange to the environment through clothing requires a porous medium analysis and is an important design requirement in the textile industry.

1.1.1 Non-traditional Examples Based on the above discussions, it is obvious that a metal mesh insert or a layer of sand is a porous medium. What about the example pictures shown in Fig. 1.2; do they satisfy the definition of a porous medium? They all do, with certain assumptions.

1.1 Porous Medium Definition

(A)

3

(B)

(C)

Flow

Energy transfer

(D)

(E)

Fig. 1.2 Porous medium examples: a Metal mesh, b Beads, metal and wood balls, c Aluminium metal foam (Courtesy Prof. Josè Lage), d Typical porous medium enhanced convection heat exchanger configuration, e Carbon fibre inside phase change material based composite heat sink (PCM-CHS) cooling space-craft batteries (Courtesy NASA public domain website)

All of these have at least two constituents with an identifiable interface between them even at the level of resolution of the human eye. For cooked rice, the solid constituent does not always remain solid. Before it is cooked, rice is dry and solid and is surrounded by air in the interstices. However, when this air is replaced and saturated with water and cooked, the rice swells and some water percolates into the rice. The final cooked product, despite presumably conserving mass, would occupy perhaps a larger volume. This volume change is because of phase changes in both water and the earlier solid rice. However, if we cook the rice-water mix in a fixed container and allow the water to drain off, the remaining cooked rice would be surrounded by air in the interstices as before. But the rice would swell and so the voids would occupy less volume than before. It can thus be stated that the volumetric porosity (defined rigorously in the next section) of the rice-air porous medium has reduced in time. The picture not only shows these voids but also the bigger pores that were formed when the water vapour evaporated into the surrounding. Bread is a porous medium where the pores are not always connected. The pores on the surface and the internal pores that are seen on slicing do not have a connection between them. If we blow air gently from one side, it need not necessarily emerge at

4

1 Basic Concepts

the other side of the loaf or slice. In this type of porous medium, if the solid portion of the bread is replaced by a metal, local hot spots can result from lack of flow-induced cooling. Unlike cooked rice and bread, the human hair scalp and washing net porous media have mobile solid constituents. This allows the possibility of several geometrical structures from the same porous medium configuration. Usually, the hair or the solid of the washing net is assumed stationary when such configuration are treated as porous media for analysing flow or heat transport. One such heat transfer analysis over surface covered with hair is given in Bejan and Lage (1991). Observe also that in all the examples in Fig. 1.3, the solid constituent is slightly deformable, a phrase we used in the definition. This deformability can be modelled using perturbation methods and, in principle, does not significantly alter the form of the governing conservation equations for studying transport phenomena. On the other hand, a single strand of human hair in Fig. 1.3c is not a porous medium under normal visual resolution; it comprises only one material with no identifiable interfaces. If we refine the resolution by observing the hair strand under a microscope, then a porous structure is seen even before we reach atomic level. It may not be worthwhile to treat the individual hair strand as a porous medium at this finer resolution level in order to study the heat and fluid transport through hair.

(A)

(D)

(B)

(E)

(C)

(F)

Fig. 1.3 Non-traditional examples of porous media: a Cooked rice, b Bread, c Human scalp, d Washing net, e Human femur (thigh bone), f Dog lung alveoli (Courtesy Prof. Josè Lage)

1.1 Porous Medium Definition

5

1.1.2 Bi-disperse Porous Medium (BDPM) An extension of a porous medium where the solid constituent of the parent porous medium itself is a porous medium is identified as a bi-disperse porous medium by Nield and Bejan (2006). The length scales of the pores of the two porous media in principle differ by an order of magnitude. Cooked rice shown in Fig. 1.2 exhibits bidispersivity. The dry rice grain allows water to soak into it during cooking, rendering it a porous medium of pore scale much finer than the water vapour pores formed between the cooked rice grains. Hence, cooked rice is a bi-disperse porous medium, a conventional porous medium whose solid is replaced by another porous medium. One can consider examples ranging from cauliflowers to data centres as bidisperse porous media. While as a whole, a cauliflower can be identified as a porous medium, closer inspection of its solid chinks would reveal a finer porous structure, categorizing it as a bi-disperse porous medium. Similarly, the distribution of racks of servers of a data centre can be modelled as a macro-porous medium formed by the individual server stack considered as the micro-porous media—another case of bi-disperse porous medium. Possibilities of extensions into several such resolution levels yielding finer and finer (micro- and nano-level) porous structures have suggested treatment of porous media geometry as fractals, detailed in Yu and Li (2001).

1.1.3 Porous Bio-materials Nature, at all levels—microscopic to large natural objects—can be understood better through porous medium modelling approach. On the large scale, vegetation may be considered porous medium. The dense packing of trees in a tropical rain forest and an orderly packing of crops in a field are examples of heterogeneous porous media, of which, the plants/trees are the solid constituents and permeating air, the consolidated fluid constituent. The leaves and branches of the trees are porous media at a smaller scale, leading to the possibility of viewing the entire land cover as a bi-disperse porous medium. Modelling forest as a porous medium (bi- or mono-disperse) can help in understanding natural events such as seed transport, spread of forest fires, etc. Moving closer to the ground brings us to another set of porous media—the soil surface. Soil, composed of sand, clay and humus, with interspersed water and air, can easily be modelled as a porous medium to know more of its properties. This is especially important for civil engineering constructions on unstable terrain such as on slopes or over clayey/marshy soils. Marine-bed and riverbed are also naturally porous, the solid constituent of which is sand, the fluid being water and modelling studies can help in understanding the stability of land vis-à-vis sand quarrying, water extraction and oil rigging activities. An example is understanding the sinking of Venice due to excess water extraction from artesian wells in the mainland that led to disturbances in the fluid constituent of the sea-bed through underground flow (which itself is a

6

1 Basic Concepts

classic example of flow through porous medium), resulting in compaction of the solid constituent of the porous medium and subsequent subsidence of land. In oil exploration and rigging, the use of acids to erode rocks changes the internal structure of the porous medium and is a good specimen for porous medium modelling. Another geo-environmental phenomenon that can be explained through porous medium approach is the transport of contaminants through water, soil and air leading to pollution. Ground water pollution is caused by the movement of contaminants carried by water, through soil—a case of flow through porous medium, as is air pollution and smog caused by pollutant-laden air movement across buildings and other structures. These phenomena exhibit all aspects of heat and mass transfer through porous medium and solutions can be recommended through analysis of models. Life itself is filled with porous media. The marine sponge, coral reef and loaf of bread are obviously porous to the naked eye, but equally typical are the more subtle components of life—cells, tissues and organs—which are porous at various levels of magnification. The lung with its branched tracheal bifurcations, the complex network of capillaries, and the sweat glands, pores, hair and fur on skin serve functions of heat and/or mass transfer using what may be easily seen as the porous medium architecture. Biomimetic and biomedical engineering developments that attempt to duplicate nature’s materials can benefit from porous medium insights. Medical implants including bone and cartilage prosthetics and scaffolds and drug delivery hinge on understanding mass transfer through the porous matrix. Some engineering applications that mimic nature’s porous structure include hydrophobic surfaces (from the nano-level porosity found in lotus leaves), porous ceramic scaffolds (from the micro-porous nacre) and strong fibres (from the nano-porous spider silk structure) and their development can be assisted by porous medium modelling studies.

1.2 Porosity 1.2.1 Volumetric Porosity Since a porous medium comprises at least two material constituents, it is useful to know the relative volume of each constituent. For instance, knowledge of the relative volumes is necessary to analyse heat transfer in a porous medium composed of materials with very different thermal-physical properties like heat capacity and thermal conductivity. The volumetric porosity, φ, of a porous medium is defined as the ratio of pore volume and the total volume, given by φ=

Vp pore-volume = . total-volume V

(1.1)

1.2 Porosity

7

For a conventional porous medium made only of a solid and fluid constituent, the pores are understood to be filled by the fluid constituent. Hence, the converse of the volumetric porosity, (1 − φ), is called the solidity of the porous medium. Further, the definition in Eq. (1.1) implies 0  φ  1 as the range of the volumetric porosity, with the extreme values rendering the region with only a single constituent. However, following our porous medium definition, the ‘pore’ can be occupied by a solid, when the porous medium is formed by two or more solid constituents. For more than two constituents, the volume fraction of each constituent can be defined similar to Eq. (1.1), as φi=1,2,3,...n where i labels each constituent. While volume fractions of individual constituents may not always exhaust the range 0  φ  1, the sum of all the volume fractions equals the total volume of the region occupied by the porous medium, as given by n 

φi = 1.

(1.2)

i=1

1.2.2 Surface Porosity The volumetric porosity, φ, is representative of the volume fraction in three dimensions. A surface porosity or surface fraction can be defined as the ratio between the total area of one constituent of the porous medium and the total area of the sample considered. φs =

Ai ‘i  th constituent area = . total area A

(1.3)

From Eq. (1.3), it is clear that φs is direction dependent as both Ai and A are directiondependant. In Cartesian coordinates, the equation can be written as: φs (x, y, z) = f (φs,x (x), φs,y (y), φs,z (z)).

(1.4)

For a typical porous medium, the surface and volumetric porosity are not identical. In fact, it is difficult to find a real porous medium with φ = φs (x, y, z). When the porous medium is isotropic, a relationship can be written as 1 φ= X

X

1 φs (x, y, z)dx = Y

0

=

1 Z

Y φs (x, y, z)dy 0

Z φs (x, y, z)dz, 0

(1.5)

8

1 Basic Concepts

which, when the φs is uniform, reduces to φ = φs . In general, since the surface porosities are direction-dependant, a useful parameter is the average surface porosity defined as 1 φs = V

S j φs, j ( j)d j.

(1.6)

0

Different methods of measurement of porosity are practised, viz. Summation of fluids method, using the device Boyle’s law porosimeter, Washburn–Bunting method and wet and dry weights method. Several of these methods are explained in detail in the classic text on flow through porous media, Bear (1988).

1.3 Percolation and Tortuosity A porous medium can be made of two well-connected constituents. A well-connected constituent is one in which a straight line can link any two points by passing only through the region occupied by that constituent. Consider a bowl of cereal rolls soaked in hot milk, the top view of which is shown in Fig. 1.4. The region satisfies our definition for a porous medium. Two points A and B in the milk region on the LHS can be linked without crossing the cereals. The milk constituent is well connected, while the cereal constituent is not. On the RHS figure, however, even the milk constituent is not well connected, as one cannot pass from point C in the milk to D without crossing the cereal. The cereal constituent of the RHS figure, on the other hand, can be said to be locally connected. When a connected constituent provides a seamless interface among its parts (several points of contact between two successive parts), it is consolidated. The milk constituent on LHS and RHS of the figure is consolidated while the cereal constituent is not.

1.3.1 Percolation A constituent is said to percolate through a porous medium, when an uninterrupted line can be traced through that constituent connecting any two boundary points. A percolating constituent is required to be well connected, if not also consolidated. The milk constituent on the LHS figure can percolate while the cereal constituent cannot. Neither milk nor cereal can percolate on the RHS figure of Fig. 1.4.

1.3 Percolation and Tortuosity

9

Milk-Cereal Porous Medium

A B

C D

Milk: Connected Cereal: Disconnected

Milk: Disconnected Cereal: Disconnected (locally connected)

Fig. 1.4 Connectivity and percolation

1.3.2 Tortuosity Tortuosity is a measure of how tortuous the path through which a specific parameter transports through a porous medium that can percolate. Originally introduced as a length ratio by Carmen in 1937, it represents the tortuous path of the well connected pores of a porous medium. It can be defined as a length ratio χ=

lapp , l

(1.7)

where lapp is the apparent length between two boundary points, which are connected by pores over l, the actual pore distance. Percolation and tortuosity are direction dependent. For instance, the apparent length, lapp , between two boundary points across the cross section of a porous medium 2 = l x2 + l y2 in Cartesian coordinates. Thus for fixed actual length can be resolved as lapp 2 l, the tortuosity can be written as χapp = χx2 + χ y2 , where χx and χ y are the respective x- and y-direction tortuosities. A porous medium may be composed of several percolation directions, and hence an average tortuosity, a lumped parameter can be proposed to represent the internal geometry. The introduction of the concept of tortuosity is to extend the use of governing transport equations established already for a continuum, to describe transport in porous media, with a small modification. The required modifying parameter would be tortuosity, a possibly distinct property for each porous medium that captures all the effects of internal geometric complexities. Unfortunately, this procedure has disadvantages. The determination of tortuosity, even in the case of porous media with relatively simple internal geometry, is not straightforward. The measurement of lapp and l

10

1 Basic Concepts

requires internal access of porous media, which in a sense renders the solution equally cumbersome as the problem of formulating governing equations valid for complex geometries. Only for porous media with simple, ordered geometries like a packed bed of spheres, the concept of tortuosity provides access to solution methodology with minor modifications to the established governing equations for continuum. The concept of volume averaging and homogenization of the porous media, viewed from this perspective, is a viable alternative. We shall introduce this idea in the ensuing sections of this chapter.

1.4 Volume-Averaging Procedure Two major methods of analysing heat and fluid transport through porous media are the homogenization approach and statistical ensemble approach. We shall subscribe to the homogenization approach in this book for its similarity to existing continuum concepts. Central to the homogenization method of analysis is the idea of volume averaging. It provides a framework to treat any porous region made of at least two constituents into an equivalent homogeneous space, suitable for continuum analysis. To introduce this, we shall begin with understanding what a homogeneous porous medium is. For rigorous and extensive treatment of this method, readers are directed to the excellent monograph by Whitaker (1999).

1.4.1 Homogeneous Porous Medium We are aware of the concept of homogeneity from undergraduate engineering sciences. A homogeneous material has its properties invariant with location and size of the material portion from which those properties are measured or determined. According to this definition, interestingly, even heterogeneous material are often classified as homogeneous because of the difficulty in sizing (or seeing) their individual heterogeneity in a particular visual resolution level. The location or size of the individual heterogeneity could be small and hence negligible when compared with the size of the material sample. On the other hand, uniformly spread heterogeneity in a particular material sample may allow the heterogeneous material to be treated as a homogeneous material. A porous medium comprising at least two homogeneous material constituents, presenting identifiable interfaces between them in a resolution level, is considered homogeneous in this sense. Figure 1.5 explains the concept of homogeneity and the associated volume-averaging procedure for a porous medium. Observe that Fig. 1.5a is a two-dimensional geometric structure comprising a solid (shaded squares) and fluid (pores) constituents that fits our definition of a porous

1.4 Volume-Averaging Procedure

11

L

solid

fluid (pores)

(B)

L

REV size

1 (A)

1

L

0

2

2

4

5

REV size

±2%

1

3

3

4

5

∞

Fig. 1.5 Volume averaging and homogeneous porous medium: a actual porous medium, b equivalent homogeneous porous medium model, c volume averaging and determination of homogeneity for property  of porous medium, d determination of REV size of porous medium

medium. Based on a chosen property that we can measure, this porous medium can be tested for its homogeneity. Let us consider the volumetric porosity, φ, introduced in the previous section, as a suitable property for analysing the homogeneity of the porous medium in Fig. 1.5a. Obviously, in 2D, the volume fraction reduces to a surface area fraction, φs (marked as  in the graph of Fig. 1.5). Figure 1.5c illustrates a method to measure this surface fraction. Considering the area of the sample size enclosed by the square formed by dashed lines and marked as 1 in Fig. 1.5c, we can see that the surface fraction of the white region would be unity. If we now move this square around the region elsewhere, and repeat the measurement of surface fraction, it would obviously oscillate wildly over the range between 0 and 1. The porous medium in Fig. 1.5a cannot be considered homogeneous with respect to surface fraction at this visual resolution level or sample size. If we proceed to increase this sample size and use the region marked 2, the same measurement would result in a surface fraction between one and zero and possibly, neither equal to 1 nor 0. Proceeding in this manner, we can perform subsequent

12

1 Basic Concepts

1

REV size

±2%

0

1

2

3

4

5

Fig. 1.6 Determination of REV size

experiments by increasing the measurement window. The plotted result of surface fraction could resemble the graph of Fig. 1.5. The  symbol on the ordinate represents the surface or volume fraction of the sample and the abscissa represents the increasing size of the measurement window. After initial wiggles for small values of the measurement window, as the size of the sampling area (or volume) increases, the surface (or volume) fraction settles to a certain mean value, within a tolerable percentage of error (taken as ±2% in Fig. 1.6). For example porous medium considered in Fig. 1.5a, this ‘settling down’ or homogenization happens beyond a sample area size represented as 4 (arbitrarily, not to scale). Further increase in the sampling size (to 5 and so on) should result only in identical fixed value for the surface fraction. Beyond the visual resolution of this sampling area, the porous medium exhibits homogeneity, in terms of surface (or volume) fraction.

1.4.2 Representative Elemental Volume The porous medium in Fig. 1.5a is considered identical to the equivalent homogeneous porous medium model of Fig. 1.5b, with each finite point in the mathematical space of Fig. 1.5b representing a finite surface area (or volume, in 3D) equal at least to the sample size marked 4 in Fig. 1.5c. This procedure is called volume averaging, where the porous medium in Fig. 1.5a is, in principle, held afar, such that it looks as Fig. 1.5b, without any distinction among its constituents. The sampling fraction (volume, in 3D) beyond which homogeneity can be claimed for a porous medium based on the volume fraction is called the representative elemental volume (REV) for that porous medium. Representative, because, the structure

1.4 Volume-Averaging Procedure

13

contained inside that volume represents completely the entire porous medium; elemental because, with the help of such volume regions, we can construct the entire porous medium. Observe in our example that once a critical sample size is determined based on the invariance of the surface fraction, the actual porous medium can be constructed by repeating the critical sample size endlessly. This is possible because, by virtue of the procedure we followed, the determined critical sample size and the total porous medium should both yield identical surface fractions. In general, a volume averaged quantity (property) G associated with such REV is defined as    1 G i dV, (1.8) G i = G i  = Vi Vi

where V is the total REV and the subscript i indicates the respective constituent volume. The angle brackets in the above definition indicate, by convention, a volume averaged property variable. It is clear from the above definition for a volume averaged REV variable that, once volume averaging has been performed, the local variations (within the REV) of properties like T, p, v for each constituent have been subsumed in their respective volume averaged variables. Analogous to the continuum concept defined for single constituents, a homogeneous porous medium is defined over a porous continuum. Every point in the porous continuum represents not the individual solid or fluid constituent, but the combined porous medium.

1.4.3 Some Clarifications on REVs Some clarifications on the concept of volume averaging, REVs and homogeneous porous medium models are provided here. First, when we say that the REV can be repeated to reconstruct, we only reconstruct a porous medium model that is equivalent to the actual porous medium. Repeating the area inside sample size 4 of Fig. 1.5c endlessly in x and y directions would not yield a porous medium that exactly matches the actual porous medium in Fig. 1.5a. They both are equivalent in terms of their surface fraction. But when we construct REVs, we lose out on certain associated information from the actual geometric structure. Next, a note on the mathematical space of Fig. 1.5b is laid over a porous continuum. Since each point in this porous continuum is supposed to represent a finite REV of the actual porous medium, it is often misunderstood that this mathematical space is discrete. With the backdrop of the iterative computational methods that are in vogue, it is only tempting to equate the REV to a control volume grid. This is not so.

14

1 Basic Concepts

REVs of the actual porous medium have imaginary boundaries. Two REVs in Fig. 1.5a, representing two successive points in the mathematical porous continuum space of Fig. 1.5b, can overlap. Therefore, the resulting porous continuum housing the homogeneous porous medium model is a continuous mathematical space, unlike a ‘control volume’-type computational grid-space, which discretize space. Of course, the ‘porous continuum’ space can be discretized, just like its counterpart in conventional continuum, and differential conservation equations valid on this porous continuum—we will encounter them in subsequent chapters—can be algebraized for seeking solutions through conventional computational methods, including the control volume approach. We approximate reality by averaging over a sufficiently small elementary volume and onto an exact mathematical space. But for the case of porous media made of recurring simple ordered shapes, the penalty is the loss of information about geometric structure and related effects (like solid–fluid shear stress and heat transfer at the REV scale). This is the signature of volume averaging. This has to be overcome with suitable REV-level modelling, closure and quantification. In this context, microscopic scale refers to the micro level of the individual solid or fluid constituent of the porous medium (within the individual constituents in Fig. 1.5a), while macroscopic scale refers to the coarser combined region of solid and fluid of the porous medium at the representative elemental volume level (Fig. 1.5c). The global scale refers to the larger length scales of the finite volume of the porous medium (L), where experimental measurements result in the determination of porous medium properties like permeability and form coefficient. A biological region formed by tissue irrigated by blood flow is a porous medium at the global scale and the representative elemental construct containing finite small blood and tissue volume is the macroscopic scale, while the region within this REV can be at the microscopic scale. The picture of the porous media in Figs. 1.2 and 1.3, viewed from afar, would blur the interfaces between the solid and fluid (pore) constituents, resulting in a continuous indistinguishable haze. Such a coarser visual resolution (or equivalently, the sample size) is essential for treating a region suitable for volume averaged porous medium analysis. The length scale of the REV, say l, is considered to be much larger than the largest pore (d p ) of the porous medium region within the REV and much smaller than the characteristic length of the actual porous medium under investigation. Volume averaging, the procedure explained in this section, can be performed on any finite region involving at least two constituents, even when, for instance, the region involves a solid block with one pore or a big pore bounded by a solid wall. Obviously, these two extremes can be dealt with satisfactorily, using established continuum concepts and governing equations. A porous medium modelling for such geometries is redundant, if not tedious and redundant. Even when we are able to define a REV for a porous medium, there is no assurance that analysis performed with the equivalent homogeneous porous medium model representing such REVs on a mathematical porous continuum, should yield useful predictive results to compare with reality. In general, a homogeneous porous medium

1.4 Volume-Averaging Procedure

15

approach provides useful results when the representative pore size of the porous medium is much smaller than the smallest length scale of the domain of interest. Sweating from skin, for instance, can be investigated using a porous medium modelling approach, when the individual sweat pores are sufficiently small and many in a much larger human skin surface. Finally, a porous medium region can comprise several pore scales. When such a region can be rigorously segregated into sub-regions of similar pore scale, REVs of differing sizes could be defined over each of the sub-regions. The actual heterogeneous porous medium becomes piecewise homogeneous. It would then be amenable to the equivalent porous medium modelling approach explained in this section. When such a segregation is not possible, volume-averaging procedure becomes tedious when performed on such a heterogeneous porous medium. We restrict our attention to homogeneous porous media in this book.

1.5 Problems (1) Discussion Questions: (a) Can two perfectly miscible materials form a porous medium? Justify. (b) Can a homogeneous material satisfy the porous medium definition? Give two reasons. (c) State the condition when a porous medium constituent percolates. (d) Why is visual resolution important for characterizing a porous medium as homogeneous? (e) The region delimited by the human hair, saturated with air, can fulfil our definition of a porous medium. State reasons whether the hair-strand constituent connected in this case (f) Provide a situation in which the same substance can be present as two distinct constituents forming a porous medium. (2) (A) State the relationship between surface (φs ) and volumetric (φ) porosities in a 3D porous medium (B) Provide two distinct engineering examples when φs = φ (3) Find the expressions of surface and volumetric porosity of the given face-centred cubic (FCC) unit cell in terms of sphere diameter and distance from origin. (4) (A) Define tortuosity (B) What is the tortuosity of a, straight-line path (C) halfcircle path? (5) Find tortuosity of a curve (representing a void geometry in a porous medium) y 2 = x over the range [0, x] as a function of x. (6) Consider a fully developed laminar fluid flow through a single flexible hose of length S and cross-sectional area A, from point 1 to point 2, as shown in the figure. These points are aligned with the horizontal direction and apart by a distance Sapp < S. The cross-sectional averaged speed v of the fluid in the hose is

16

1 Basic Concepts

constant and uniform along the entire hose. The hydrostatic pressure difference between point 1 and point 2 is also constant and equal to p = p2 − p1 < 0. Unit vector along x direction 1

nx

Hose of length S

2

Fluid flow

Sapp

(a) Beginning from its vector form, write the Hagen–Poiseuille equation for fluid flow through the hose, relating the cross-sectional averaged fluid speed v to the pressure-drop from point 1 to point 2, i.e. p. (b) Derive an expression for the Hagen–Poiseuille equation in terms of the distance Sapp and write the result in terms of the hose tortuosity defined as ξ = Sapp /S. (c) c) Find an algebraic expression for an effective constant κeff , in terms of ξ and κ = v/(p/S), where κ is the hydraulic conductivity of the hose, so that v = −κeff p/Sapp . (d) d) If the hose curvature deforms the cross-sectional area along the direction of flow, how will the result for effective constant κeff found in (c) will vary? (7) A tube with uniform cross-sectional area and hydraulic conductivity κ is used to link an inlet point to an outlet point following a tortuous path. Under laboratory test, one can measure the apparent length Sapp from inlet to outlet, the applied P from inlet to outlet, and the fluid speed v into the tube. Suggest a method to estimate the tortuosity using these parameters. (8) Derive equivalent expressions in the cylindrical coordinates for volumetric porosity φ and relate it to the corresponding surface porosity, φs , in a two-constituent porous cylinder when, (a) the two constituents form two co-axial cylinders of equal radii r , lying one over the other (b) the two constituents form two co-axial cylinders, one with radius r1 inside the other with inner radius r2,i = r1 and outer radius r2,o . (9) In Cartesian coordinates, we were able to relate the surface porosities in every direction (x, y, z) to the volumetric porosity through an integration along the respective direction. Derive a similar relationship between the volumetric and surface porosities of all three directions (r, θ, z) in the cylindrical coordinates. Which of these relations are valid (i.e. in which of these directions, the integration of the respective surface porosity leads to the volumetric porosity)?

1.5 Problems

17

(10) A solid block is drilled with capillaries to form a porous medium. The straight holes are of d = 2.0 cm and L = 30d. They are arranged in a 4 × 4 distribution with a 4d spacing between each row and/or column. The corner rows and columns are also separated from their centres, by a 4d spacing from the bounding solid walls of the block. (a) Calculate the volumetric porosity φ of the porous medium. (b) Is the surface porosity φs of this porous medium isotropic? (c) Calculate the representative elemental volume (REV) of this porous medium and determine the value of REV beyond which it can be regarded as a homogeneous porous medium.

Chapter 2

Heat Conduction in Porous Medium

2.1 Introduction It is conventional in expositions of transport through porous media, to introduce fluid dynamics through porous media before venturing into heat transfer. This has the advantage of presenting the subject through its historical evolution—a method supported by Truesdell (1974). We shall deviate from this convention—but align with another, adopted by heat transfer textbooks—to introduce heat conduction before flow through porous media. This approach has advantages. For instance, it introduces the important concept of local thermal equilibrium at the REV scale, as an independent concept without requiring fluid dynamics of porous media. Even at the rudimentary level, a porous medium is made of at least two constituents. Investigating energy transfer would require separate governing equations for the solid and fluid constituents with a closure model that accounts for possible interface energy transfer. We shall proceed to formulate this from first principles, marshalling some of the basic concepts learnt, perhaps, in an undergraduate thermalfluids course.

2.2 First Law of Thermodynamics Consider a generic porous medium region as shown in Fig. 2.1. This region can be modelled as a homogeneous porous medium following the volume averaging procedure explained in Chap. 1. The first law of thermodynamics can be applied at the REV level, by considering a control volume in Cartesian coordinates around the REV as shown in Fig. 2.1. Additionally, since the fluid is stagnant and hence the porous medium region immobile, the REV is a thermodynamic closed system capable of energy transfer interactions with the surroundings. The REV is composed of two constituents, solid, occupying surface area As along the y − z plane and fluid (pore) occupying surface © The Author(s) 2023 A. Narasimhan, Essentials of Heat and Fluid Flow in Porous Media, https://doi.org/10.1007/978-3-030-99865-3_2

19

20

2 Heat Conduction in Porous Medium

Fig. 2.1 Schematic for formulating the porous medium energy equation

area A f , with an identifiable interface of arbitrary shape separating them. For simplicity during formulation, let the REV geometry remain invariant along x-direction. This yields the surface porosity φs = A f /A yz−total equal to the volumetric porosity φ of the REV and the porous medium region. The total energy for the REV-system is written as (E = E s + E f ). Consider first, the solid constituent. Applying the first law of thermodynamics on the solid volume of the REV, we could write δ Q s − δWs = dE s .

(2.1)

Equation (2.1) when expressed as a rate equation, (i.e. variables expressed as divided per time) results in q s − ws =

dE s . dt

(2.2)

In Eq. (2.2) each of the terms is expressed in Watts, while in Eq. (2.1) it is in Joules. For the REV in Fig. 2.1, assume two heat transfer interaction across the surface area A(= As + A f ), one at location x and the other one at (x + dx). For the moment, assuming no work interactions at the REV scale, we can nullify the second term on the LHS of Eq. (2.2). Equation (2.2) can be rewritten thus:

2.2 First Law of Thermodynamics

21

qs,x − qs,(x+dx) =

dE s . dt

(2.3)

We must now express the terms in Eq. (2.3) in terms of measurable quantities. The RHS is expressed as a combination of fundamental quantities as E s = ρs (As dx)u s ,

(2.4)

where u s is the specific (per unit mass) internal energy of the solid constituent and the product of the rest of the symbols in RHS should result in a mass of the solid constituent. Observe that we assume the block in Fig. 2.1 to have negligible kinetic and potential energies, at least during the heat transfer process. This results in the total energy E s depending only on the change in the internal energy of the solid constituent. Further, the change in this internal energy, being something that is not accounted for in the kinetic and potential energies, can be related to the change in the temperature of the solid constituent of the REV as du s = cs · dTs ,

(2.5)

where cs is the specific heat of the solid material. Observe that the discussion so far is valid for the fluid constituent of the REV of Fig. 2.1; E f may be expressed by a similar equation for the internal energy change of the fluid constituent du f using the fluid counterpart variables. A note on temperatures is in order here. It is evident that our analysis is valid on a porous continuum as shown in Fig. 2.1. Following the definition for a volume averaged quantity (property) G associated with such REV from Eq. (1.1), the definition for volume averaged temperature, in this sense, for the solid constituent is 1 Ts  = Vs

   Ts dV,

(2.6)

T f dV,

(2.7)

Vs

and for the fluid constituent is 1 T f  = Vf

   Vf

where V is the total REV while the subscripts s and f signify the respective constituent volumes inside the REV—for instance, in the REV of Fig. 2.1, Vs = As · δx = (A − A f ) · δx. The angle brackets in the above definitions indicate, by convention, that these are volume averaged quantities, an indication we refrain from using in subsequent text, unless stated otherwise.

22

2 Heat Conduction in Porous Medium

Using the simplifications in Eqs. (2.4)–(2.7), the RHS of Eq. (2.3) is rewritten as ∂ Ts ∂ Es = ρs cs (As dx) . ∂t ∂t

(2.8)

Equation (2.8) involves in the RHS, the density, specific heat, the cross-sectional area (see Fig. 2.1), the length along which the heat current travels and the change in temperature in time. All of these are measurable, in a global scale for the porous medium, with fair accuracy using our existing technology. We shall proceed to express the LHS of Eq. (2.3), the heat interaction in terms of measurable quantities. Jean Baptiste Joseph Fourier (1768–1830) showed a way to do this by assuming the local heat current q to be proportional to the local spatial temperature difference, which at x would be, qx ∝ − ∂∂Tx . The proportionality was resolved later using experiments so that the relationship is written as qx = −k · A ·

∂T , ∂x

(2.9)

where k is a material property called the thermal conductivity. Equation (2.9) is called the constitutive equation for k because it defines what ‘k’ is, and hence must be evoked to measure k (as done in experiments to measure k). However, Eq. (2.9) enables q to be determined by measuring local temperatures. The negative sign for the RHS term indicates that heat current flows in the direction of the negative gradient of temperature, i.e. from higher temperature to lower temperature. In Fig. 2.1, temperature is assumed to decrease along the positive x, the direction of the heat current vector. Equation (2.9), the Fourier law, is distinct from Eq. (2.2) or (2.3), the first law of thermodynamics written for the block. Applying Eq. (2.9) for the solid constituent of the REV at location x, would result in qs,x = −ks · A ·

∂ Ts . ∂x

(2.10)

Observe the use of ks in Eq. (2.10). Similarly, the qs at location (x + dx) can be written using a Taylor series expansion truncated to the first two terms as qs,(x+dx) = qs,x +

∂qs,x dx. ∂x

(2.11)

Applying the Fourier’ assumption at location (x + dx), Eq. (2.11) could be rewritten as   ∂ Ts ∂ Ts ∂ qs,(x+dx) = −ks · As · −ks · As · dx. (2.12) + ∂x ∂x ∂x

2.2 First Law of Thermodynamics

23

Combining Eqs. (2.8), (2.10), (2.11) and (2.3) and after cancelling a few terms and rearranging, the first law as applied to the solid constituent of the REV in Fig. 2.1, in terms of measurable quantities would be ∂ ∂x

  ∂ Ts ∂ Ts ks · = ρs cs . ∂x ∂t

(2.13)

To generalize this equation further, the work term, kept until now as zero, can be added. For instance, work interaction in a solid block can happen by means of stretching or shrinking of the block or by passing electricity through it. By passing electric current through it, work is done on the block system. The result of this work transfer interaction is to add energy to the solid block, which could either be stored or be released to the surrounding. Hence, this energy (work) inventory has to be accounted for in the first law energy balance, for instance, as written in Eq. (2.2). The work interaction of passing an electric current through the block is modelled usually as a heat generation quantity measured in Watts per unit volume of the block. This heat generation is understood to have its origin in the ohmic losses that the electric current exhibits, resulting in dissipation of losses out of the block as heat transferred to the surroundings. The solid block, in principle, should become hotter because of the passage of electric current, if the ‘generated’ energy is not released to the surroundings. A similar argument can be made for the solid constituent of the REV, if the porous medium region in Fig. 2.1 is a mesh with interconnected pores with electric current passing through it. An example of direct heat generation appropriate in a porous medium context is that of the fissile fuel inside a nuclear reactor. This fissile material, which generates heat internally because of nuclear fission, can be treated as a heat generating porous medium. Including the work interaction term in the form of heat generation (W/m 3 ), the first law of thermodynamics for the closed system of the solid constituent of the REV in Fig. 2.1 reads ∂ (1 − φ) ∂x

  ∂ Ts ∂ Ts ks · + (1 − φ)q˙s  = (1 − φ)ρs cs . ∂x ∂t

(2.14)

Observe the substitution (1 − φ) = (A − As )δx to validate Eq. (2.14) for the entire solid volume region of the REV. The first term in the LHS is the longitudinal heat conduction that is the resultant difference between the q at x and (x + dx) location in Fig. 2.1. Observe the use of (1 − φ) in the second heat generation term. Both these terms bring in or send out energy. The total of these two should be balanced by the energy inventory of the block, represented by the RHS term, also known as thermal inertia. This term determines how hot the block must become for a given energy interaction (on the LHS). Smaller the thermal capacity ((ρc)s ), faster will be the temperature raise.

24

2 Heat Conduction in Porous Medium

Following a similar procedure, the first law of thermodynamics applied to the fluid (pore) region of the REV in figure would result in an energy balance equation of the form   ∂Tf ∂Tf ∂ kf · + φ q˙f  = ρ f c P, f . (2.15) ∂x ∂x ∂t Note the presence of heat generation q˙f  weighted by φ, the measure of relative fluid volume.

2.2.1 Local Thermal Equilibrium Ideally, even within the REV of Fig. 2.1, heat transfer between the solid and fluid constituent across the solid–fluid (pore) interface A f s is possible. This implies that information about the energy exchange at the interface is necessary to solve Eqs. (2.14) and (2.15). Following the volume averaged definitions of temperatures in Eqs. (2.6) and (2.7), at the solid–fluid interface A f −s the following interface conditions are valid 1 V

 

1 T f d s= V

Afs

  Ts d s,

(2.16)

ks ∇Ts · d s,

(2.17)

Afs

and 1 V

  Afs

1 k f ∇T f · d s= V

  Afs

where d s = n f s dA f s , with n f s , the unit vector normal locally along A f s from fluid to solid. The T f and Ts in the above two equations represent the actual local temperatures. To solve the energy equations Eqs. (2.14) and (2.15), the interface flux Q f s defined in Eq. (2.17) must be known. A way to proceed is to ‘assume away’ this problem by invoking the local thermal equilibrium (LTE) to prevail between the lumped volume averaged temperatures of the solid and fluid constituents inside the REV of Fig. 2.1. This explicitly means Ts  = T f  = T, rendering the interface flux, Eq. (2.17), Q f s = 0.

(2.18)

2.2 First Law of Thermodynamics

25

The LTE assumption implies that the temperature T can no longer be distinguished as either of the solid or fluid constituent, but defined on a porous continuum as a porous medium temperature. However, this does not preclude heat transfer (conduction, in this context) in the porous region of Fig. 2.1. Obviously, factors that induce large temperature gradients at the REV scale would hamper the validity of the LTE assumption. A short discussion on local thermal non-equilibrium (LTNE) effects at the end of the chapter highlights these aspects.

2.2.2 Porous Medium Energy Equation Assuming LTE to prevail, Eqs. (2.14) and (2.15) are added term-wise to yield the governing energy equation for porous medium (ρc P )e

∂T = ∇ · ke ∇T + q˙e , ∂t

(2.19)

where (ρc P )e = φ(ρc P ) f + (1 − φ)(ρc P )s ke = φk f + (1 − φ)ks

(2.20) (2.21)

q˙e = φ q˙ f + (1 − φ)q˙s .

(2.22)

The assumptions that are made to obtain the above form of the energy equation are: • Stagnant fluid, saturating all of the pores • Local thermal equilibrium (LTE) between solid and fluid constituent at the REV • Averaged effective thermal–physical properties For steady state, without heat generation in the porous region having constant thermal–physical properties, Eq. (2.19) reduces to the familiar form ke ∇ 2 T = 0.

(2.23)

d2 T = 0. dx 2

(2.24)

The 1D form of this would be ke

The solution of the above equation for two realistic boundary conditions enables the determination of the effective stagnant thermal conductivity of the porous medium. This is briefly discussed later.

26

2 Heat Conduction in Porous Medium

2.3 Second Law of Thermodynamics A second law of thermodynamics statement for the porous medium region in Fig. 2.1 can be formulated following the exposition in Sects. 1.4 and 12.4 of Bejan (2004). The second law statement for the porous medium region in Fig. 2.1 in terms of the entropy generated Sgen is Sgen =

 qi   ∂ SC V − − ms ˙ + ms ˙ ≥ 0. ∂t Ti inlet outlet

(2.25)

For the porous medium region in Fig. 2.1 with q˙ = 0, and heat conduction as the predominant irreversibility, the second law statement, with T representing absolute temperature, would be  = Sgen

ke (∇T )2 ≥ 0. T2

(2.26)

Since the fluid constituent is stationary, viscous dissipation is absent in Eq. (2.27) as a source of irreversibility. This will be included while discussing convection through porous media in subsequent chapters.

2.4 Effective Stagnant Thermal Conductivity The solution of the porous medium energy equation, Eq. (2.19), requires the effective thermal–physical properties associated with the porous medium to be known ahead. The effective stagnant thermal conductivity, ke , Eq. (2.22), is an offshoot of the presumed parallel arrangement of the solid and fluid constituents of the porous medium in Fig. 2.1, when volume averaging is performed. Other arrangements are possible, while the measured property, ke , could be different due to the complexity of internal structure in actual porous media.

2.4.1 Limiting Models Analytical determination of effective properties when the region of study is composed of more than one material constituent can perhaps be traced back to James Clarke Maxwell who in his treatise, Maxwell (1873), proposed models for electrical resistance of composites. Analogously, analytical models for estimating the effective stagnant thermal conductivity of a porous medium composed of basic arrangements of two constituents can be considered.

2.4 Effective Stagnant Thermal Conductivity

27

The first one, obviously, is the parallel model where the heat current traverses through the porous medium alternately between the solid and fluid constituents. The effective stagnant thermal conductivity in this geometric situation would be the arithmetic mean of ks and k f , that has already been introduced during the energy equation formulation, ke = φk f + (1 − φ)ks ,

(2.27)

where φ, the volumetric porosity, ks , the thermal conductivity of the solid constituent and k f , the thermal conductivity of the fluid constituent are taken to be known quantities. The counterpart to the above is the series model, where the heat current traverses through the porous medium simultaneously through the solid and fluid constituents. In this case, ke would be the harmonic mean of ks and k f , given by 1 φ (1 − φ) = + . ke kf ks

(2.28)

The above two models provide the upper and lower bound for ke . In line with the nature of above two means, a third model was introduced in Nield (1991) (also suggested earlier by Combarnous and Bories (1975), which proposed the geometric mean of the constituent conductivities, as φ

ke = k f + ks(1−φ) .

(2.29)

Obviously, the ke calculated using Eq. 2.29 would fall between the other two means, the upper and lower bounds and hence could be considered as a practical alternative, when ks /k f is not too different from 1. The nature of the predicted ke using the above three models is shown in Fig. 2.2. Nozad et al. (1985) and Prasad et al. (1989) conducted experiments to determine ke for saturated porous media arranged as packed bed of spheres of several solid– fluid material combinations. The range of volumetric porosities of the porous media considered in these experiments was 0.3 ≤ φ ≤ 0.45 and hence the prediction curves by the three models for φ = 0.4 can be compared with the experimental data in Fig. 2.2. As expected, the nature of ke is markedly different between ks /k f ∼ 1 and ks /k f 1. The models would yield the limiting results of ke = k f when φ = 1 and ke = ks when φ = 0, which are shown as the bounding curves in Fig. 2.2.

2.4.2 Other Models In actual porous media, contact resistance between adjacent portions of the solid constituent, local movement of the fluid constituent, and non-contact with the heat

28

2 Heat Conduction in Porous Medium

106 ke/kf 105

upper bound (ke = ks)

AM (parallel layers)

0 0.4 0.8

GM

104

HM (series layers) 0.4

Prasad et. al (1989)

103

Nozad et. al (1985)

φ

102 101

0.8

100

0.4 0.8 1

lower bound (ke = kf)

10-1 10-1

100

101

102

103

104

105 106 ks/kf

Fig. 2.2 Effective stagnant thermal conductivity prediction using three analytical models

transfer boundaries are some of the factors that affect the value and determination of stagnant effective thermal conductivity. Over several decades, experimental efforts have attempted to propose ke models that are more general and predictive than the limiting models discussed in the previous section. Three review chapters, Cheng and Hsu (1998), Hsu (2000), Hsu (2005), discuss these experiments and the associated ke models in detail. We shall introduce and compare the results of some of these ke models for exposition. As mentioned earlier, the limiting models in the previous section work better for ks /k f ∼ 1. Krupiczka (1966 Polish, 1967 English) proposed a simple model for predicting ke that worked better when ks /k f > 1 (mentioned as λ = k f /ks < 1 in the original paper) ke = ks(1−n) · k nf ,

(2.30)

where n is related to porosity φ by the relation n = 0.280 − 0.757log10 φ + 0.057log10

kf . ks

(2.31)

In Fig. 2.3, predictions by Eq. 2.31 of the experimental data are shown to be in better agreement than the other limiting models discussed in the previous subsection. Kunii and Smith (1960) proposed an unit cell concept applicable for a porous medium formed by packed bed of spheres with or without point contact among their

2.4 Effective Stagnant Thermal Conductivity

29

104 ke/kf 103

0 0.4

upper bound (ke = ks)

AM (parallel layers) GM

0.8

HM (series layers) Krupiczka (1967) 0.4

Nozad et. al (1985)

102

Prasad et al (1989)

φ 0.4

101

0.8 0.4 0.8 1

100

lower bound (ke = kf)

101

102

103

104 ks/kf

Fig. 2.3 Effective stagnant thermal conductivity prediction using Krupiczka (1967) model, Eq. (2.31)

solid constituent. They modelled this arrangement as a fluid layer in parallel with a second layer consisting of fluid and solid layers in series. The effective thermal conductivity is given by ke /k f = φ + (1 − φ)k ∗ /k f ,

(2.32)

where k ∗ is the effective thermal conductivity of the second layer, given as k ∗ /k f = (ks /k f )/(α K (ks /k f ) + 2/3),

(2.33)

with α K being an empirical parameter depending on the porosity. The final expression for ke /k f is ke /k f = φ +

(1 − φ) (ks /k f ) . (α K (ks /k f ) + 2/3)

(2.34)

Although the ke in Eq. (2.34) is a more advanced model than those discussed in the previous subsection, the applicability is limited by the large associated uncertainty incurred while determining α K , an empirical parameter. Hsu et al. (1995) applied the previous unit cell concept to periodic in-line arrays of square cylinders, used by Nozad et al. (1985) for 2D direct numerical simulation, to obtain a model ke /k f as

30

2 Heat Conduction in Porous Medium

γa (1 − γc )(ks /k f ) (1 − γa )(ks /k f ) + γa (1 − γa )(ks /k f ) + . (1 − γa γc )(ks /k f ) + γa γc

ke /k f = γa γc (ks /k f ) +

(2.35)

In Eq. (2.35), γa = (a/le ) and γc = (c/a) are the dimensionless length ratios of the unit cell, related to the volumetric porosity φ by 1 − φ = γa2 + 2γc γa (1 − γa ).

(2.36)

The γ terms attempt to capture the degree of non-consolidation—and hence, to an extent, the contact resistance for conduction heat transfer—in the solid constituent of a simple periodic porous medium. When the square cylinders are non-touching, i.e. disconnected and non-consolidated, the γc = 0 since c = 0. From Eq. (2.36) it follows that γa = (1 − φ)1/2 and Eq. (2.35) reduces to ke /k f = [1 − (1 − φ)1/2 ] +

(1 − φ)1/2 (ks /k f ) . (2.37) [1 − (1 − φ)1/2 ](ks /k f ) + (1 − φ)1/2

In this situation, the configuration considered by Hsu et al. (1995) is identical to that of Kunii and Smith (1960) and hence expected to yield similar predictions. The major difference is that, while the α K of Eqs. (2.34)–(2.36) is an empirical parameter, the γ s of Eqs. (2.35)–(2.37) are deterministic and related to porosities and internal geometry of the porous medium. However, the accurate measurement of these length scales at the pore level remains intractable. The limiting model for the non-touching case, Eq. (2.37) is the useful outcome. As an extension for the 2D unit cell configuration of periodic square cylinders, a 3D model was also proposed by Hsu (1998), which considered interconnected periodic cubes as the unit cell configuration. The model proposed from this study reads ke /k f = 1 − γa2 − 2γc γa + 2γc γa2 + (ks /k f )γc2 γa2 (ks /k f )γa2 (1 − γc2 ) (ks /k f ) + γa (1 − (ks /k f )) 2(ks /k f )γa γc (1 − γa ) , + (ks /k f ) + γc γa (1 − (ks /k f ))

+

(2.38)

where γa and γc are related to φ through 1 − φ = (1 − 3γc2 )γa3 + 3γc2 γa2 .

(2.39)

As with the 2D case, for non-touching cubes, γc = (1 − φ)1/3 and Eq. (2.38) reduces to

2.4 Effective Stagnant Thermal Conductivity

ke/kf 103

31

3D model 2D model Curve fit

A.P

Prasad et al (1989) Nozad et. al (1985)

G.P

102

0.02 0.13

c/a

101 H.P

101

102

c/a = 0

103 ks/kf

Fig. 2.4 Effective stagnant thermal conductivity prediction using several ke models including the 2D and 3D models of Hsu et al. (1995). The curve-fit equation is provided in Eq. (2.41)

ke /k f = [1 − (1 − φ)2/3 ] +

(1 − φ)2/3 (ks /k f ) . (2.40) [1 − (1 − φ)1/3 ](ks /k f ) + (1 − φ)1/3

The experimental data for ke along with the predictions from the Kunii and Smith (1960) and Hsu et al. (1995) models discussed so far are shown in Fig. 2.4. The curve fit shown in Fig. 2.4 yields a power-law relation of the form (ke /k f ) = 1.265 · (ks /k f )φ ,

(2.41)

where φ is the average volumetric porosity; φ ∼ 0.36 for all the reported experimental data. The root mean square (RMS) error for the predictions of the experimental data by the models can be calculated using  RMS error =

(y − yexp )2 , n

(2.42)

where y is the (ke /k f ) predicted by the model and yexp is the corresponding experimental data in Figs. 2.3 and 2.4; n = 42 is the total data points available. The RMS error values for parallel, series model, geometric mean prediction model, Hsu 2D model with c/a = 0, Hsu 2D with c/a = 0.02, Hsu 3D model with c/a = 0, Hsu 3D model with c/a = 0.13, Krupickza’s formula, Eq. (2.30) and the

32

2 Heat Conduction in Porous Medium

curve fit, Eq. (2.41) are given, respectively, as 660.33, 54.07, 10.11, 9.22, 7.51, 8.39, 4.98, 3.47 and 2.57. Obviously, 2D and 3D models proposed by Hsu et al. (1995) predict the available experimental data for ke better than the three simpler analytical models, which offer their limiting trends. Krupickza’s formula, Eq. (2.30), although requiring an empirical relationship for determining n, due to its accuracy for ks /k f > 1, yields predictions with less error. Interestingly, the simpler power-law-type curve-fit Eq. (2.41) results in the least RMS error for the range 0.1 ≤ (ks /k f ) ≤ 104 , which covers most of the engineering solid–fluid material combinations. This curve-fit function would suffice as a quick predictive alternative. The predictive models discussed are for homogeneous and isotropic porous media with simple ordered geometric arrangement of two constituents. There is scope for further research for determining ke of porous media that are heterogeneous, anisotropic and unsaturated. The effect of contact thermal resistance must also be specifically addressed.

2.4.3 Experimental Determination Effective stagnant thermal conductivity is measured by either steady-state or transient methods. The absolute steady-state thermal conductivity measurement of two-phase materials using square guarded hot plate (SGHP) apparatus is described here. This method is applicable for homogeneous materials and can be used effectively to estimate ke for all types of low thermal conductivity materials. The principle of guarded hot plate for measuring thermal conductivity is based on one-dimensional Fourier heat conduction through a porous medium. In essence, it is the steady-state solution of Eq. (2.24) in the absence of heat generation, which upon rearranging can be written as ke =

Q × x . A × ( T )

(2.43)

The heat power Q across the cross-sectional area A of the porous medium sample, at steady state, results in a T over a distance x of the cross section of the sample. The measurement of ke would then involve the measurement of heat power Q and temperature difference T .

2.4.4 Description of SGHP Apparatus The schematic of a typical square guarded hot plate apparatus is shown in Fig. 2.5. It consists of a hot plate stack, cooling system, power supply, data acquisition and

2.4 Effective Stagnant Thermal Conductivity

33

Cold Plate

Hot Plate Specimen

Edge guard

Fig. 2.5 Schematic of SGHP apparatus for the determination of ke (Courtesy Prof. Srinivasa Reddy, IIT Madras)

control systems. At steady-state the hot plate (middle) provides one-dimensional heat transfer through the porous medium test specimens. In recent experiments reported in Reddy et al. (2008), the hot plate is made of 8 mm thick copper plate with 300 mm × 300 mm square cross section and a heating element is embedded in it. The hot plate comprises a metre plate (150 mm × 150 mm) section at the centre, surrounded by a guard section. The metre plate and guard plate are separated by a 2 mm guard gap. The guard is maintained at same temperature of the metre plate to obtain one-dimensional heat flow. The porous medium specimen is kept between hot and cold plates. In order to improve the contact between the heated surface and the sample surface a film of high conductivity material may be applied between the two. An axial force is also applied using a suitable arrangement, so that contact between the surfaces is thermally good. The SGHP apparatus can accommodate two specimens at a time, with a mirror image across the plane of symmetry. The edge guard and edge insulation are placed around the specimen to reduce the lateral heat loss from the sample to the environment. The picture in Fig. 2.5 shows the inside view near the test section arrangement. The guard plate and the metre plate are maintained at same temperature to obtain one-dimensional heat flow from hot plate to cold plate through the specimen. The cold

34

2 Heat Conduction in Porous Medium

plate is maintained at constant temperature during measurements by a combination of fluid flow through the coolant blocks and supply of heat. The temperature data are collected from various nodal points of hot plate, cold plate, edge guard heater and coolant circuit using temperature sensors (K-type thermocouples). Based on these data, a temperature balance is attained between metre and guard heater, cold and hot plate and hot plate and edge guard heaters. The proportional integral derivative (PID) temperature controllers and voltage programmable power supply units are used to maintain the various heater plates at specified temperatures.

2.4.5 Determination of ke The thermal conductivity of two-phase materials depend on various factors such as density, moisture content, structure and composition and operating temperature of these materials. The SGHP was used to measure thermal conductivity of low density (< 200kg/m3 ) fibre insulation materials and high density (2000 kg/m3 ) insulation plate. The thermal conductivity of porous materials involving one solid and one fluid has been evaluated at various temperatures ranging from 323 to 623 K and thermal conductivity range up to 2 W/mK.

2.5 Thermal Dispersion When considering a porous medium, vortices and confluence of fluid streams are common. It has been observed and well documented in fluid mechanics literature that a fluid flowing around a solid obstacle could be locally mixed by the vortices created at the trailing edge of the obstacle. Mixing is also induced when two flows, originally following distinct paths due to the presence of neighbouring solid obstacles, meet at a point. This is called hydrodynamic dispersion and has been discussed in detail for porous medium flows in Bear (1988). The strength of the trailing vortices, hence the local mixing, depends on the fluid properties, flow characteristic (speed, velocity variation) and the shape of the obstacle. A related phenomenon is thermal dispersion, modelled as an extended thermal conduction effect, the reason for its inclusion in this chapter. The local fluid mixing caused by the tortuous fluid path imposed by a solid obstacle in a porous medium also influences the local solid–fluid heat transfer. This extra mixing, identified as thermal dispersion, is not specifically accounted for by the effective thermal conductivity models, say, Eq. (2.27). The thermal consequences of this dispersion are similar to the extra mixing caused by turbulence in regular forced convection situations. This led researchers to include the dispersion effect into the effective thermal conductivity ke of the porous medium, similar to the eddy diffusivity concept introduced while modelling turbulent heat transfer. This analysis is not limited to typical porous media (small porosity, packed particles), but also to flows

2.5 Thermal Dispersion

35

through tube bundles, micro-channels, fins and many other configurations found in enhanced convection heat transfer—all of them being porous media, according to our definition in Chap. 1. The modelling of the effect of dispersion on the diffusivity, like the turbulence effect, is complex. Available simple models introduce an additional term in the generic ke model equation. One such extension is to write Eq. (2.27) as a model for the transverse effective conductivity keT ,

keT = φk f + (1 − φ) ks + k f − ks G + k f PeD,

(2.44)

where Pe is a local Pèclet number, Pe = νd/α f , with d, a characteristic length scale and α f , the thermal diffusivity of the fluid equal to k f /(ρc p ). The D is a positive parameter and a function of the internal structure, porosity, contact resistance, etc. It has to be determined from experiments. Majority of thermal dispersion experiments were performed with packed bed of spheres, where the length scale d is usually set equal to the sphere diameter (or particle diameter) d p . The G in the second last term of Eq. (2.44) represents the structural effect (heat flow path, or tortuosity of the matrix) on the effective thermal conductivity of the porous medium. At low flow speeds, when the Pèclet number is small, the dispersion effect can be neglected. The effect of dispersion is always to increase ke . When dispersion is present in the configuration and unaccounted, while modelling, the resulting heat transfer enhancement, obtained by solving the conservation equations, will be conservative. The mixing effect of vortices and/or confluence of fluid streams along the main flow direction could be different from that in the transverse direction. Hence, thermal dispersion is direction dependent and keT exhibits anisotropy. Available experimental results in the literature indicate a predominance of transverse dispersion. References to experimental measurement of dispersion conductivity are available in Nield and Bejan (2006).

2.6 Local Thermal Non-equilibrium Model 2.6.1 LTNE Formulation of PM Energy Equation When a nonzero solid–fluid interface heat flux (Q f s = 0) exists, the equality in Eq. (2.16) is no longer valid. In other words, local thermal equilibrium (LTE) no longer exist between the solid and fluid phase in the REV of the considered porous medium. For this local thermal non-equilibrium (LTNE) situation, Eqs. (2.14) and (2.15) are recast in this situation as follows: φ(ρc p ) f

∂Tf = ∇.(φk f ∇T f ) + φ Q f + h s f av (Ts − T f ), ∂t

(2.45)

36

(1 − φ)(ρc)s

2 Heat Conduction in Porous Medium



∂ Ts = ∇. (1 − φ)ks ∇Ts + (1 − φ)Q  s + h s f av (T f − Ts ), (2.46) ∂t

where h s f is a volumetric heat transfer coefficient and av is the specific surface of the porous medium (surface per unit volume). In the two-energy model, it is crucial to determine the appropriate value of h s f . Correlations exist in the literature to predict values for h s f and av . Local thermal non-equilibrium (LTNE) (i.e. Ts = T f ) effects become important in many applications involving high-speed flows and during high heat flux and step change situations, such as a hot highly conducting fluid flowing through cold and poorly conducting porous matrices. The LTNE effect is not necessarily an unsteady phenomenon, but can also arise in steady flows, for example, when a cold poorly conducting fluid is drawn by suction towards a hot permeable surface. In this case the resulting steady thermal boundary layer within the solid phase extends further from the hot surface than it does in the fluid phase. Further, LTE assumption starts to deviate when the particles or pores are not small enough, when the thermal properties differ extensively, when there is significant heat generation in any of the phases such as in combustion in porous media and in situations involving rapid change of phase such as rapid evaporation and condensation. In studies of nuclear reactor cores and hypothetical nuclear reactor accidents, there is substantial temperature difference between fluid and solid phases. In such situations, the two-energy model plays an indispensable role in the accurate prediction of phase temperatures.

2.6.2 Interface Heat Transfer Coefficient To solve the LTNE formulation of the porous medium energy equation, h f s , the interfacial heat transfer coefficient is to be known, for providing closure. The determination of the interfacial heat transfer coefficient h f s has been the subject of interest in the recent decades of porous medium research. Early studies considered h f s to be heavily influenced by fluid flow through the pores of the porous medium, i.e. local forced convection effect is predominant. Under this situation, Wakao et al. (1979) suggested the correlation: Nˆu = hˆ f s d p /k f = 2 + 1.1Pr 1/3 Re p 0.6 ,

(2.47)

where Nˆu is the Nusselt number for the convective interfacial heat transfer coefficient hˆ f s , Pr , the Prandtl number and Re p the Reynolds number based on the diameter of the solid particle. However, when Re p → 0, the above correlation would suggest hˆ f s d p /k f → h f s d p /k f = 2. In other words, it leads to N u = h f s d p /k f = 2 as the the interfacial heat transfer coefficient for stagnant heat conduction. Obviously, this has limited

2.6 Local Thermal Non-equilibrium Model

37

physical meaning, as the stagnant interfacial heat transfer coefficient N u should depend on the thermo-physical properties of both solid and fluid phase of the porous medium. The review by Hsu (2000) details the research advancements in the determination of h f s , from which we shall recount here, material concerning the basic understanding of the investigations. To estimate a meaningful interfacial heat transfer in the form of h f s (or its non-dimensional form N u) even when the fluid is stagnant, Hsu (1999) proposed a quasi-steady-state composite heat conduction model involving the conductive resistances of both the solid and fluid phases, valid at the pore level of a porous medium. The model essentially consists of two heat conduction layers on the two sides of the interface. For the quasi-steady assumption to be valid, the time scale of microscopic heat conduction should be smaller than the time scale or macroscopic heat conduction. The steady-state heat conduction equation in spherical coordinate system was solved by Hsu (1999) with the boundary conditions T f = T¯ f at r = r o and Ts = T¯s at r = ri . An interfacial heat flux from solid to fluid for each particle was obtained as q=

4πr p k f (T¯s − T¯ f ), B + σ (1−B)

A 1+A

(2.48)

where A = (ro r p )/r p and B = (r p ri )r p are the non-dimensional thickness of the conduction layers. For n particles in a REV, the above equation can be summed to yield an interfacial heat transfer per unit volume as Q f s = nq/v. Using a f s = 6(1 − φ)/d p the final form would become Nu =

h f sdp 2σ = , kf α Aσ + αB

(2.49)

where α A = A(1 + A) and α B = B/(1 − B). The parameters A and B are related by β A2

3+ A 3− B = B2 , 1+ A 1− B

(2.50)

as a result of thermal energy conservation within the two phases. From this the stagnant interfacial heat transfer coefficient N u depends on three parameters, A , σ and β. Here σ and β represent material thermo-physical properties. A depends also on the packing density of the medium, i.e. on its volumetric porosity. Taking the equivalent radius as re /r p = (1 − φ)−1/3 and assuming the fluid wetting the solid phase to be in a quasi-steady state for ro = re results in 1 + A = (1 − φ)−1/3 . From this, A = 0.16 can be obtained when φ = 0.36, for a porous medium of packed bed of spherical particles. Additional results are summarized in the review by Hsu (2000).

38

2 Heat Conduction in Porous Medium

2.7 Transient Heat Conduction in Porous Medium Transient heat conduction studies in porous media can be performed for both LTE and LTNE situations. When LTE prevails, REV-level solid–fluid heat exchange is considered to be absent and one investigates essentially the macroscopic transient heat conduction in the porous medium considered. Once a suitable set of effective thermo-physical properties are known for the porous medium considered, the analysis is equivalent to that performed in continuous media, well documented in standard undergraduate heat transfer textbooks. When LTNE is assumed, a more involved analysis is required, as both Eqs. (2.45) and (2.46) must be solved along with a suitable h f s , which could also be time dependent. However, in this situation, both the microscopic, REV level heat transfer and the macroscopic, spatial heat transfer across the porous medium are considered in the analysis.

2.7.1 1D Transient PM Heat Conduction Under LTE Apart from the ‘early regime’ where the initial transients of the individual phases could result in a nonzero interfacial heat transfer, several engineering situations can be approximated using a mid or ‘late regime’ transient heat conduction model, wherein LTE is assumed. We shall analyse a simple situation of transient heat conduction in a porous medium slab kept in a cooling fluid, when local thermal equilibrium prevails at the REV level. A porous medium of transverse width D and extending endlessly in the transverse direction (see Fig. 2.6) is immersed in a hotter fluid at time t = 0. At what time t, the centre of the plate x = D/2 would ‘feel the heat’ and the temperature of that location rises from its initial temperature, T0 ? To answer this question when heat generation is absent and assuming LTE is valid, i.e. Eq. (2.18) holds, the governing 1D (in space) equation of the situation can be formulated by simplifying Eq. (2.19) as (ρc P )e

∂T ∂2T = ke 2 . ∂t ∂x

(2.51)

Additionally, in writing the above equation, the effective properties, Eqs. (2.21) and (2.22) are assumed to be uniform over the entire porous medium. Owing to the symmetry of the situation we can concentrate on one half of the plate (D/2) for making an order of magnitude estimate of each term in Eq. (2.51) above. The LHS of Eq. (2.51) for instance can be scaled as (ρc P )e

∂T

T ∼ (ρc P )e . ∂t t

(2.52)

2.7 Transient Heat Conduction in Porous Medium

39

Fig. 2.6 Time response of temperature in a porous slab immersed in a cooling fluid

Observe the tilda instead of the equality symbol in Eq. (2.52). Further, the T in Eq. (2.52) is the temperature difference possible for the system (Fig. 2.6) in the time t (unknown) and is as such, unknown. One can make a guess once the scale for the RHS of Eq. (2.51) is written. Proceeding to do so, the RHS of Eq. (2.51) can be scaled as ke

T ∂2T ∂ ∂T 1

T ∼ ke = ke = k . e 2 ∂x ∂x ∂x (D/2) (D/2) (D/2)2

(2.53)

Observe in Eq. (2.53) as well, there appears a T whose exact value is unknown. Using Eqs. (2.52) and (2.53) in Eq. (2.51), a time scale can be found as t∼

(D/2)2 αe

(2.54)

where αe = ke /(ρc P )e is the effective thermal diffusivity of the porous medium within region D. Importantly, it is assumed that both ke and (ρc P )e are known or can be modelled through equations similar to Eqs. (2.21) and (2.22). Equation. (2.54) provides the time scale for when T (x = D/2) > T0 . At this juncture, the two unknown T values in Eqs. (2.52) and (2.53) must be equal. It is the only instance when the T across the spatial distance of D/2 would match

40

2 Heat Conduction in Porous Medium

exactly the T for the time duration t (measured from the initial time of t = 0). Hence our assumption about T while finding the answer in Eq. (2.54) is correct. The result in Eq. (2.54) for determining the heat penetration time compares well with the results from the exact analysis to the problem.

2.7.2 Lumped Capacitance Model For a transient heat conduction situation in porous medium of time scale well beyond the critical time scale in Eq. (2.54), i.e. for t >> tc (∼ (D/2)2 /αe ), the ‘late regime’ transient solution or the lumped capacitance heat conduction model is valid. Observe that the LTE assumption is independent of this criterion and more basic (provides the validity of using a single energy equation to model the transient situation). However, since under ‘late regime’ the local spatial variations of temperature are expected to be negligible, LTE is also expected to prevail. Neglecting spatial variation, when the temperature of the porous medium in Fig. 2.6 varies only in time, i.e. T = T (t), a lumped capacitance (thermodynamic) model solution can be provided, following the analysis performed for continuous media Bejan (1993). The first law of thermodynamics for this ‘closed system’ of a porous medium slab of characteristic size D cooled by a heat transfer process of magnitude h e A(T − T∞ ) at the boundary, would be q = dE P M /dt. The 1D transient heat conduction equation, Eq. (2.51), can be recast as he A dT =− (T − T∞ ) , dt (ρc)e V

(2.55)

subject to the initial condition T = Tc

at t = tc .

(2.56)

The solution to this formulation is

 he A T − T∞ (t − tc ) . = exp − Tc − T∞ (ρc)e V

(2.57)

In the above equation, A and V are the porous medium surface area wetted by the cooling fluid and volume of the porous medium, respectively. (ρc)e is given by Eq. (2.21) noting that c P = c for the solid phase. The h e is the effective convection heat transfer coefficient between the porous medium and the surrounding cooling fluid and has to be separately evaluated. Observe that h e is not that same as h f s , the interfacial heat transfer coefficient between the solid and fluid phase of the porous medium. At the limit φ → 0, the above analysis should reclaim the transient heat conduction results for continuous medium.

2.7 Transient Heat Conduction in Porous Medium

41

Following the analysis in Bejan (1993) for continuous media, one could ask when the above lumped capacitance model solution alone is sufficient to analyse the entire (LTE assumed) transient heat conduction regime of T evolution of the porous medium. With reference to Fig. 2.6, when T = T0 = Tc at t = ti = tc = 0, the conduction heat flux leaving the porous medium would be q  ∼ ke (T0 − Ts )/(D/2), where Ts is the boundary temperature of the porous medium that could be different from both T0 and T∞ . This would be balanced by the convection to the surrounding cooling fluid, h e A(Ts − T∞ ). Taking D/2 = r , the balance would result in T0 − Ts ∼

Bi e (T0 − T∞ ), 1 + Bi e

(2.58)

where Bi e = h e r/ke is the effective Biot number of the system. Since we require T0 − Ts  0 are θ f = θs = 0, at x = 0,

(2.62)

∂θ f ∂θs = = 0, at x = 1. ∂x ∂x

(2.63)

The above system of equations subject to the initial and boundary conditions was solved by Hsu (2000) using a finite difference numerical procedure. The computations were performed for different values of , λ, σ and N u∗. Here, λ takes an equivalent role to β. Note that G is not an independent parameter and depends on φ and σ . Figure 2.7 shows the results of the transient response of the solid and fluid temperatures at different locations in the slab when φ = 0.4, λ = 0.4, σ = 4.0 and N u∗ = 108. It is observed at x = 0.25, T f lags Ts significantly. This temperature lag, initially zero, increases with time and decays to zero when t ∼ 1. Since t is scaled with lm 2 /α f , Ts = T f ∼ T∞ in the porous medium, when t > 1. From Fig. 2.8 it is observed that the difference between Ts and T f is large at shorter time and reduces with increase in time. Further analysis of this problem is available in the review chapter (Hsu 2000).

2.8 Problems (1) Derive an expression to calculate effective thermal conductivity, for a porous medium where temperatures are elevated, solid phase is a poor conductor and pores are large. (Hint: Read Sect. 7.4 to formulate the governing equation including radiation heat flux).

2.8 Problems

43

Dimensionless Temperature

1 fluid

0.6

solid

x=1.0

0.8

x=0.25

0.4

0.2

0 0.001

0.01

0.1

1

Dimensionless Time Fig. 2.7 Time response of solid and fluid temperatures at different locations of an infinitely extended porous slab to a sudden drop of temperature at one wall, the other wall being insulated. Adapted from Hsu (2000)

(2) Consider a porous rectangular slab of dimension 10 × 20 cm2 . The longer sides are insulated and the other two opposite sides are maintained at temperatures of 0 and 100 ◦ C. Find the steady-state temperature distribution in the slab and the heat flux through the slab. (3) Thermal conductivity of sand is found to be 2.2 W/mK in an experiment. Find the porosity of sand, taking thermal conductivities of air and sand particle are 0.03 and 3 W/mK, respectively. (4) A bunch of mangoes are kept on the ground, under a stack of hay for ripening. Assume the ground to be insulated and the heat loss through the sides is negligible compared to the heat loss through the top surface. The thickness and porosity of the mango layer are 6 and 0.4 cm. Corresponding values for the hay layer are 3 and 0.6 cm. Mangoes generate heat at the rate of 15 kW/m3 . Make suitable assumptions on the side and top surface heat transfer coefficients and ambient temperature. Find the (a) heat loss per unit length for the mango distribution. (b) temperature at the top of the mango layer assuming that the thermal conductivity of mangoes is very large. (c) the temperature of the mangoes.

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2 Heat Conduction in Porous Medium

Dimensionless Temperature

1

0.8

t = 0.01

0.6

0.4

0.1

0.2 0.3

0 0

0.2

0.4

0.6

0.8

X

1

Fig. 2.8 Profiles of solid and fluid temperatures in an infinitely extended porous slab at different times after a sudden drop of temperature at one wall, the other wall being insulated. Adapted from Hsu (2000)

(5) The heat loss from a room to the ambient varies with the porosity of the window and door material. Cost of the window and door material increases as C = Aeφ where φ is the porosity of the material and A is a constant. The heat generated in the room is Q, W/m3 and the cost of the air conditioning varies linearly with the load as C = B Q  where Q  is the load on the air conditioner. Determine the optimum porosity of the wall. Make suitable assumptions for the missing data. (6) A spherical particle of radius r1 experiences uniform thermal generation at a rate of q  (W/m3 ). The particle is encapsulated with good thermal contact, by a spherical porous shell of outside radius r2 that is cooled by ambient air. The thermal conductivity of the particle is k1 . The porosity of the shell is 0.2 and the solid phase thermal conductivity ks = k2 and that of the pore fluid is k f = k3 , such that the relation k1 = 2k2 = 4k3 holds. Find the steady-state temperature distribution in the porous layer and also the temperature at the surface. Make suitable assumptions about the ambient convection. (7) Consider two concentric tubes of radius r1 and r2 , with water flowing inside the inner tube. The temperature at the wall-water interface is T . To minimize heat loss, the region r1 < r < r2 is filled with a porous medium of porosity φ and thermal conductivity ks of the solid material and k f of the pore fluid. Both the inner and outer walls have a thermal conductivity of k. Air flows at room conditions and convection heat transfer coefficient h around the outer surface.

2.8 Problems

45

(a) Determine the heat loss and temperatures at r1 and r2 . (b) Derive a criterion to predict at what parameter values (for instance, what value of ke , effective thermal conductivity) of the configuration, the heat loss to the surrounding is less than that of the case without the porous medium in the annulus. (8) Cooked rice is kept in a cylindrical steel box of thickness 5 mm, bore of 10 cm and a height of 15 cm at a temperature of 90 ◦ C. The rice loses its taste when the temperature drops below 70 ◦ C. Assume rice to be a porous medium of porosity 0.3; specific heat capacity of rice particles is 1.4 K J/kg-K and that of air is 1.005 KJ/kg-K. Convection heat transfer coefficient of air is 20 W/m2 -K; thermal conductivity of steel is 50 W/mK. (a) Determine the time before which the rice has to be consumed. (b) Determine the time by considering the rice as a continuous medium (minus the air) and compare the result with the previous value. (9) A person is wearing a woollen sweater. Assume her hand to be a cylinder of radius of 4 cm and length of 0.5 m and the body temperature is 360 ◦ C. The heat transfer coefficient on the outer surface of the sweater is h = 25 W/m2 -K. Assume the sweater to be 5 mm thick and a porous medium of porosity 0.1. Thermal conductivity of wool is 0.04 W/mK. The ambient temperature is 100 ◦ C. Find the heat loss from the hand. (10) Using the approximation formula, calculate the ke of a porous medium with dispersed spherical pores and φ = 0.3, including the radiation heat transfer effects, as derived in the previous problem. Given: ks = 234 W/mK and k f = 0.024 W/mK. (Hint: Consult, A. Helte (1993), Radiative and conductive heat transfer in porous media: Estimation of the effective thermal conductivity, J. Appl. Phys. 73, 7167).

Chapter 3

Fluid Flow Through a Porous Medium

In this chapter, we take a historical approach to introduce some key ideas in flow through porous media. Later in the chapter, we will place these historically significant developments into a theoretical framework that is based on the volume averaging procedure and concept of representative elemental volume (REV) introduced in Chap. 1. For this, we shall begin with conventional definitions of flow variables prevalent in porous medium literature.

3.1 Introduction Any flow through a porous medium happens through its interconnected pores whose solid constituent, by definition, is assumed to be stationary. Employing the volume averaging procedure, the actual porous medium is modelled as a homogeneous porous medium defined over a porous continuum amenable to mathematical analysis. An REV, by definition, can be made of several interconnected pores. For simplicity of explanation consider a fissure-type porous medium. Fissuretype porous medium in 2D longitudinal (length-wise) cross section would appear as evenly spaced parallel plates stacked in y direction. When flow persists through  this porous medium, a velocity profile can develop inside each pore. Let u be the x direction velocity at a local point inside one of the pore. At the same location x, the pore comprise several such velocities along the y cross section (pore scale). At the REV scale then, by taking the ‘pore scale’ to be uniform, for a homogeneous porous medium, the pore averaged velocity could be defined as  up =

u  (y)d(pore-scale).

(3.1)

pore scale

Here, only u  (y) is the measurable quantity that can be measured using perhaps a Pitot probe small enough to access points inside a pore (of an REV) of a porous © The Author(s) 2023 A. Narasimhan, Essentials of Heat and Fluid Flow in Porous Media, https://doi.org/10.1007/978-3-030-99865-3_3

47

48

3 Fluid Flow Through a Porous Medium

medium. The integral is carried only over the fluid cross-sectional area or the pore → scale. In 3D, one may write, − u p as the pore averaged velocity. Since the REV is composed of both solid and fluid constituent, the REV averaged quantity should involve both the solid and fluid areas. When applied to the velocity, this results in the Darcy velocity (3.2) u = φu p , where φ is the surface porosity (pore cross-sectional area/total REV cross-sectional area), which is often equal to the volumetric porosity of a porous medium considered → homogeneous. In 3D, u is written as − u and is called by various names—Darcy velocity, seepage speed and superficial velocity. We shall use Darcy velocity in this text. Darcy velocity, by definition, is not directly measurable using an instrument. In its vectorial form, it is nevertheless a point function defined on a porous continuum. In the simplest case of a homogeneous porous medium channel with uniform cross section, the volumetric flow rate can be measured and used along with known porosity, φ, cross-sectional area, As , and density of the fluid ρ to determine the Darcy velocity. A similar discussion holds for pressure, p  , which is defined conventionally as a scalar point function inside the pores. For a REV then, P is the volume-averaged pressure, which is related to the pore averaged (average of p  ) pressure pp by the relation (3.3) P = φpp . Again, in a homogeneous porous medium channel, the pressure gauges fitted on the walls just before and after the porous medium, measure the average static pressure P defined in Eq. (3.3). Darcy, in his original experiments, measured this pressure using manometers fitted to inlet and outlet headers sandwiching a sand filter. To measure pore pressure, slender and sophisticated manometers that reach into the pores without significantly disrupting the adjacent flow are required. At this stage, using the above definitions, the conventional mass conservation statement can be written for flow through porous media. For a control volume around a unit volume of REV of a homogeneous porous medium, the mass conservation statement in 1D can be written as follows φ

∂ρf + (m˙ out − m˙ in ) = 0. ∂t

(3.4)

Here, ρf is the fluid density, and the partial derivative in time indicates the rate of change of fluid mass within the control volume, while the terms involving m˙ are the → → u , where − u respective flow rates into and out of the control volume. Since m˙ = ρf − is given by Eq. (3.2), the above equation translates in 3D as φ

∂ρf → + ∇ · (ρf − u ) = 0. ∂t

(3.5)

3.1 Introduction

49

Observe that the porosity φ is set independent of time. This is not applicable when swelling and erosion are involved in the porous medium considered. The process of cooking rice discussed in Chap. 1 is an example of swelling where porosity depends on time. This requires φ on the LHS of the above equation to be a function of time. During erosion (accompanied or not by chemical reaction), parts of the solid constituent are subsumed in the fluid flow. This situation requires a revised mass conservation statement, which includes the mass change of both the constituents inside the time derivative.

3.2 Historical Overview 3.2.1 Darcy Model Around the early nineteenth century, the no-slip condition at the wall for a flowing fluid was recognized first by Pierre-Louis-Georges Du Buat (1734–1809). The noslip condition was being appreciated among others, through the works of Prony, who tried to explain free surface flow over rigid inclined surfaces wherein he used a velocity profile taking zero value at the inclined surface. It was a time to follow and fill the relevant gaps of the blazing trail of Bernoulli, Lagrange, Newton and Euler, who a century earlier had merged mathematics and fluid mechanics and created the analytical foundations of fluid mechanics, most of which stand unaltered today, after three hundred years. Hagen (1839) and Poiseuille (1840) through experiments and theory understood fully developed laminar flow inside circular cross-sectioned pipes. The understanding corroborated and provided practical use to the shear flow theory derived from the contributions of Navier (1822) and Stokes (1845). Henry Philibert Gaspert Darcy was the ‘Dean of the School of Bridges and Roadways’ and was involved in planning the fountains of the city of Dijon. While working on filtering hospital waste water, Darcy described his empirical studies on steadystate filtration in detail in a paper, Les Fontaines Publiques de la Ville de Dijon, Victor Dalmont, Paris, in 1856. He demonstrated that the water volume passing through a sand layer is proportional to the pressure-drop across the length of the layer, leading to a relation for α, identified by Darcy as the hydraulic conductivity of the porous medium, as α = U/(P/L). (3.6) In Eq. (3.6), U is the channel cross-sectional averaged fluid speed, also known as the seepage speed, and P is the hydrostatic pressure difference (i.e. inlet minus outlet) across the porous medium layer of length L. For historical accuracy, we record that Darcy performed experiments in vertical sand filters and measured flow rate and the pressure-drop as head difference, as represented in Fig. 3.1. Three observations regarding Eq. (3.6) are fundamentally important. The first is that liquid water, the fluid used by Darcy, was not heated/cooled along the sand

50

3 Fluid Flow Through a Porous Medium

Fig. 3.1 Summary of historical evolution of momentum conservation in flow through porous media

layer. Therefore, we can consider his experiment isothermal. The second is that, in the proposed form, Eq. (3.6) involves no fluid material property, a direct consequence of Darcy using only liquid water and not heating/cooling it throughout his experiments. Hence, Eq. (3.1) is valid essentially for incompressible fluid in isothermal flow1 through a porous medium. The third observation is that Eq. (3.6) is an algebraic equation involving crosssectional average quantities, which we call global quantities, and can be measured— as Darcy did—experimentally. It is not the differential momentum conservation equation but a solution to one with prescribed boundary conditions. This point shall be elaborated in a later section discussing measurement of porous medium hydraulic properties. Also, the influence of local (pore level) changes inside either by the porous medium or the fluid upon the hydraulic conductivity is not apparent from the variables involved in the construct of Eq. (3.6). Further historic details are available in an excellent review by Lage (1998). Hazen (1892) essentially repeated Darcy’s experiments, however, by altering the inlet temperatures of water flowing through sand columns. These experiments allowed him to distil the α in Eq. (3.6) as α = α50 · (60/(T + 10)), where α50 measured at T = 50 ◦ F is the specific hydraulic conductivity and (60/(T + 10)) is the Fahrenheit factor. 1

Although Darcy didn’t explicitly state this, it is highly probable that the experiments were performed at ambient temperature (which, on a winter day, is about 17 ◦ C, in Dijon, France).

3.2 Historical Overview

51

Similar to Helmholtz reducing the data of Poiseuille’s earlier, Forchheimer (1901) realized subsequently that the group (60/(T + 10)) in Hazen’s results resembled dependence of the viscosity of water on temperature, namely, μ(T )/μ(50 ◦ F), to less than one per cent for 40 ◦ F ≤ T ≤ 90 ◦ F. This led to the resolution of α as K /μ, where K is called the specific permeability, a property of the porous medium and μ the dynamic viscosity of the fluid. It is worth mentioning that the specific permeability K = [μ(50 ◦ F)α50 ], does not appear as such in Hazen’s original equation (see Hazen 1892), which nevertheless unveiled the influence of temperature on the hydraulic conductivity of Eq. (3.6), by introducing the Fahrenheit-factor, 60/(T + 10). With this development, Eq. (3.6) took its present form as μ P = U. L K

(3.7)

This equation can be identified as Darcy’s law, a particular case of the momentum balance statement for flow through porous media. Importantly, it is the constitutive relation that defines permeability, K , m2 , a hydraulic property of a porous medium. Hence, the important contribution of Darcy’s (1856) paper is perhaps the recognition of permeability, K , m2 (in its primitive form α in Eq. (3.6)) as a porous medium property.

3.2.2 Hazen–Dupuit–Darcy Model Kozeny (1927) provided a physical explanation for the viscous dependency of Hazen– Darcy equation. By associating Hagen’s (1839) and Poiseuille’s (1840) theories of flow through capillaries with Darcy’s empirical equation of filtration, Eq. (3.1), Kozeny demonstrated that, by assuming uniform pressure-drop and integrating Poiseuille’s partial differential equation along a certain capillary length, an equation similar to Darcy’s equation can be obtained. The analogy with Hagen–Poiseuille flow allows the interpretation of the Darcy’s equation, Eq. (3.7), as a balance between hydrostatic pressure-drop and global viscous drag, resulting from the viscous stresses along the fluid–solid interface of the pores. Dupuit (1863) was the first scientist to use theoretical principles to derive Darcy’s equation, Eq. (3.6), by considering the resulting contribution of each pore (microlevel) to the flow characteristic of the entire porous layer (global, or cross-sectional average, level). A consequence of Dupuit’s study is the inclusion of a quadratic term in Eq. (3.7) modelling the resistive form drag imposed to the fluid by the solid permeable medium. This resistive force due to the form or shape of the solid was proposed by Newton— from his experiments with falling objects in air—to be proportional to the square of the average fluid velocity and to the fluid density ρ. The resulting equation is

52

3 Fluid Flow Through a Porous Medium

μ P = U + CρU 2 . L K

(3.8)

Equation (3.8) is the global Hazen–Dupuit–Darcy (HDD) model for porous medium flows. It is also known as the Forchcheimer–Darcy model possibly due to the reason that Forchheimer (1901) supported the presence of the quadratic velocity term in the momentum balance equation. More on this, historical oversight is discussed in Lage (1998). Ironically, the experimental data that Darcy used to derive his linear relation, Eq. (3.6), can be used to deduce the quadratic relation, Eq. (3.3). This aspect is discussed in Lage and Antohe (2000). Similar to the use of Hagen–Poiseuille equation in the context of clear fluid flow, Eq. (3.8) is used to predict the global pressure-drop across a channel if it were to be filled with a porous medium. A step towards the generalization of Darcy’s equation valid at a global scale for the porous medium is the ad hoc extension of the global Hazen–Dupuit–Darcy model, Eq. (3.8) by Stanek and Szekely (1974), into the differential form μ→ → → u + ρC|− u |− u, (3.9) −∇p = − K → where − u is the Darcy velocity, also known as the local (macroscopic, or representative elementary volume) seepage fluid velocity. It is related to the intrinsic pore  → average velocity as − u = φu p , where u p = u(x, y)d A, with the integral taken over the local pore cross section in an REV of a porous medium. The ∇ p in Eq. (3.9) is the local (macroscopic) pressure gradient, related to the pore cross-sectional averaged static pressure by p = φpp . The two terms in the RHS of Eq. (3.9) represent the lumped viscous and lumped form effects (forces) within the macroscopic permeable medium and are usually designated as viscous drag and form drag, respectively, as they impede the flow. In this sense, Eq. (3.9) represents the balance between these two drag forces and the pressure force required to overcome them, in order for the flow to maintain average velocity U . The justification for using the same coefficients K and C in the algebraic and → differential forms is quite simple: when the fluid velocity − u and properties μ and ρ are uniform, the cross-sectional average of the unidirectional flow version of Eq. (3.9) leads to Eq. (3.8). Hence, the parameters K and C used in the differential form, Eq. (3.9), must be the same as the ones used in the global form, Eq. (3.8). In other words, a volume integration of the differential momentum equation, Eq. (3.8), should lead to a result similar to that of the experimentally verifiable, global momentum conservation equation, Eq. (3.9). → When the fluid properties μ and ρ are uniform, the fluid velocity − u along a porous medium filled channel with uniform cross section will also be uniform. This → uniformity in − u is because Eq. (3.9) (being a first order PDE) allows only one velocity boundary condition, i.e. the impermeable condition at the channel surfaces. As Eq. (3.9) does not allow the imposition of the non-slip velocity boundary condition, the resulting velocity profile is flat, such as the velocity profile for slug flow (i.e. − → ˆ u = U i).

3.2 Historical Overview

53

Fig. 3.2 Reconciliation of global (experimentally observed) and macro (REV-concept based) momentum conservation statements

Before concluding this section, it must be noted that the above historical overview addresses only the evolution of the global momentum conservation statement for a particular restricted class of porous media, the low permeable media. This is partly due to the nature of the earlier experiments performed (in the nineteenth and early twentieth centuries) to understand the flow through porous media, which involved mostly low permeable media (K < 10−7 m2 ) like sand filters (Fig. 3.1). Noteworthy also, is the almost universal use of constant fluid properties through a porous medium in modern experimental, theoretical and numerical convection heat transfer studies. The readers are directed, for instance, to the textbook Kaviany (1991), the monograph Nield and Bejan (2006) for general reviews of the subject and Lauriat and Vafai (1991), Lage and Narasimhan (2000) for reviews specific to forced convection.

3.3 Physics of Flow Through Porous Media In the earlier section, we provided a historical perspective of some of the key ideas that shaped our understanding of flow through porous media. In this section, we shall provide a physical perspective of these ideas, relating to established concepts from

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3 Fluid Flow Through a Porous Medium

mechanics that precede them, if not in their origin, surely in engineering curricula. This allows us a rethink on some of the conventions that occur in the current understanding of porous medium flows. This section may be skipped on a first read without losing continuity in the understanding of porous medium flows.

3.3.1 Inertialess Motion and the HDD Model Assume a rigid body is executing uniform motion, i.e. moving with constant velocity in vacuum. Being at rest is only a special case of zero velocity uniform motion with respect to a particular reference frame. Let an applied constant force F (see Fig. 3.3, case A) disrupt this uniform motion and the body accelerates (i.e. increases its velocity by a constant value for each second). The steady acceleration is sustained as long as the force F, persists. When the force is absent, the body would presently execute uniform motion again. However, it will now be moving with a new (increased) constant velocity. In effect, the force F accelerates the rigid body in vacuum. While executing uniform motion (constant velocity), before and after the action of the force F, the body has no inertia. When a force is impressed on it, changing its uniform motion, it experiences inertial motion. If there were more than one force acting on the body, with the resultant of these forces equalling in magnitude and direction to the same F as before, then the body would accelerate in an identical manner. In this more general case, inertial motion is the result of the resultant of all of the forces that were impressed on a body. Hence, the resultant force is called inertial force. To summarize, if there were no accelerated motion, there is no inertial force experienced by the body and the body is said to execute an inertialess motion. Next, consider case B, of Fig. 3.3. Suppose the same rigid body from case A, while executing uniform motion (moving with a constant velocity, say U ), is sent into a uniform cross sectional corridor/duct as shown. The body experiences the friction at the walls acting as an impressed force disrupting the uniform motion. It will decelerate and eventually come to rest because of the frictional resistance at the top and bottom walls. The body decelerates as the frictional force acts in the opposite direction (along −x) of the body’s original direction of motion. In effect, the body executes a decelerating inertial motion and comes to rest. Naturally, if the body has to maintain (retain) its constant velocity U , a force F, has to push it against the frictional force FR . Applying a force F such that, F = FR (resultant of F and FR is zero), would make the body maintain its original uniform motion with velocity U . The situation remains unchanged, even if there are two forces F1 and F2 acting on either ends of the body (left and right faces) such that their resultant is equal to F (in the positive x-direction) and opposite to FR . We conclude, for this case, the body is executing uniform inertialess motion, with friction.

3.3 Physics of Flow Through Porous Media

F

U=c

55

F

FR FR

Solid, rigid body y

U=c

(A)

(B)

x

F

FR FR

Fluid (C)

U=c

F FR

U=c

PM

(D)

Fig. 3.3 Inertialess motion in solid, fluid and porous medium. a Solid, rigid body motion without friction and b with friction; c fluid flow with friction, d fluid flow through porous medium with friction

In case C, the rigid body is replaced by a fluid bulk executing uniform motion with constant velocity U . In principle, each point (located by an x, y pair) of the (fluid) body itself can have a local velocity u(x, y). In this context, when we say that the body moves with constant velocity u, we mean that the average of all of the local velocities, in the x-direction, is a constant equal to U . This type of fluid motion is known as the fully developed flow (a special case being the Hagen–Poiseuille flow) of a viscous fluid, where the pressure gradient across the length of the duct balances the friction drag force experienced by the fluid in contact with the wall, to sustain a flow with velocity U . The situation again resembles the case of two normal forces F1 and F2 acting on either ends (left and right faces) of the fluid body, their resultant F balancing the wall frictional drag forces FR . Here in case C, the fluid body is executing uniform inertialess motion with friction. If the viscosity of the fluid was to suddenly become zero (i.e. ideal fluid with μ = 0), there would be no drag forces on the fluid and the situation in case C reduces to case A (and not case B—the fluid will not feel the presence of the wall due to the absence of viscosity). Thus, the imposed pressure gradient will accelerate the (bulk fluid) flow into an inertial motion. This new flow situation is known as the classical Bernoulli flow—frictionless, accelerated inertial motion of a fluid.

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All reasoning discussed for case A, apply for this type of flow. To sum this situation alternatively, there is no pressure gradient required to maintain the flow with any, constant velocity. Case D of Fig. 3.3 has a mesh connecting the top and bottom walls of the duct, through which the fluid flows. The situation can be treated as similar to case C, only with additional friction between the mesh surface and the flowing fluid. To discuss the special case first, if the fluid viscosity is zero, the situation is again the classical Bernoulli flow—frictionless, accelerated inertial motion of a fluid, through a porous medium (mesh). If we have to maintain an uniform fluid flow with constant velocity U , the frictional drag forces of the entire solid–fluid contact surface—treated again, for simplicity, as FR —have to be balanced by a pressure-drop across the length of the porous duct. The HDD model, Eq. (3.8), governs this type of flow, which is the fully developed, porous medium flow counterpart of the fully developed flow (special case being the laminar Hagen–Poiseuille flow) of case C. The bulk fluid, under an imposed constant pressure-drop, executes uniform, inertialess motion, (only) with friction, through porous media. We proceed with the notion that the system of flow over solid obstructions (a concept from fluid mechanics) can be considered as elemental building blocks of the porous continuum.

3.3.2 From Newton’s Principia to the Nature of Form Drag In order to place the viscosity-related form-drag force of the porous medium in historical context and derive, if possible, some additional insights, we shall recount here, the nature of form drag as propounded for the first time by by Sir Isaac Newton in 1687. Resistance on solid bodies moving in fluids was systematically investigated and reported for the first time in Book 2 of the Principia by Sir Isaac Newton. Incidentally, equivalence of the problem of a solid moving in a stationary fluid and fluid flow over a stationary solid was stated and used with success for the first time, in Book 2. The following is a brief summary of Book 2, strictly within the engineering context of our discussion. Book 2 begins with Sect. 1 on the analysis of resistance force that varies linearly with the velocity. In Sect. 2, the resistance force varying with the quadratic velocity is analysed. Resistances arising as a combination of these two are discussed in Sect. 3. In effect, Sects. 1 through 4 provide solutions for the equations of motion, when the forces acting on the solid body is a function of the unknown velocity. Extensive experiments and their data are reported in Sects. 5 through 7 to experimentally determine the value of the drag coefficient β for flow over solid spheres. Section 8 deals with wave motion, while Sect. 9, is on vortex motion. In the beginning of Sect. 9, Newton hypothesizes about viscosity. Newton envisaged three types of resistance forces exerted on a solid (sphere) by the fluid (particles) flowing (contacting) around it. In the scholium at the end of

3.3 Physics of Flow Through Porous Media

57

Sect. 3, he states that the resistance arises from tenacity, friction and density of the fluid, of which the resistance arising from friction, he sets as proportional to the fluid velocity while the resistance arising form the density is proportional to the square of the velocity. These can be expressed as FR = γ + αv + βv 2 .

(3.10)

γ in the force balance accounts for the force necessary to overcome the tenacity of the fluid. In various sections (such as 1, 2, 7 or 9) of Book 2, this force was either properly accounted for or not explicitly stated. In modern parlance, when we invoke the no-slip condition at the solid surface, this force γ vanishes. Incidentally, the no-slip condition was experimentally ‘discovered’ and was explicitly stated by Du Buat; in 1786, more than a century after Newton’s work was published. For further details, see Neményi (1962) and Giacomelli and Pistolesi (1963). The resistance force αv, arises from the internal friction (lack of lubricity) of the fluid. The coefficient β, which depends on the density of the fluid, was further refined through the definition of a type of resistance, expressed as Fβ =

1 ACρv 2 . 2

(3.11)

It depends on the density ρ of the medium (fluid), the frontal area of the solid (sphere) and on the square of the velocity. We find a ‘proof’ for Eq. (3.11) only as a physical explanation, in the scholium at the end of Sect. 1, while building an argument for studying (in Sect. 2) resistances that vary with the square of the velocity. Newton created impact models that predict the value for C in Eq. (3.11) and compared it with experimental values. In Sects. 5 through 7, he first writes about experiments on a pendulum bob moving through air, water and mercury. Unsatisfactory results from these (as realized from data in the first two editions of the Principia) made him perform further experiments on spheres falling through air and water. He used the results from all of these experiments to determine C and the force F, Eq. (3.11). The third and final edition (1727) of the Principia, in a much expanded Sect. 7 of Book 2, included all of the experiments he and his colleagues performed on falling spheres. See also Truesdell (1968), for more details. Based on his model, he concluded that for incompressible, rarefied medium, ( fρ A < C < fρ A)/2, where f is a shape factor that depended on the form of the solid body (which could be a sphere, a disc or a cylinder) and A, the projected frontal area. For a continuous medium, the group ( fρ A)/4 replaces the ( fρ A)/2 in the aforementioned relation. He compared the values for C obtained from his experiments with his theoretical predictions (which, he felt were correct) and tried to attribute the small differences between them, to the other coefficient α in Eq. (3.10), as a measure of frictional resistance generated by the fluid’s viscosity (lack of lubricity). Perhaps because he was not totally successful (i.e. he had not directly verified/found through experiments, a value for α similar to that of C), he proposed the concept

58

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of viscosity, which he believed was the cause for the resistance group αv, in the hypothesis in the beginning of Sect. 9, the final section of Book 2. The resistance expressed in Eq. (3.11) was further shown to be dependent on the form of the solid bodies tested, in the middle of the eighteenth century (see Giacomelli and Pistolesi 1963; Tokaty 1971 for further details) primarily by Chevalier de Borda, who showed that the resistance of Eq. (3.11) for a disc—axis aligned with the direction of fluid flow—was more than twice greater than that for the sphere, under similar conditions. Newton’s proposition was that the resistance on the disc and sphere should be identical. Herein begins the reason for designating the resistance in Eq. (3.11), as one due to the form of the body, or the form drag experienced by a solid obstruction in a flowing fluid. The prime assumption of Newton’s impact model is that the interaction between the fluid and the solid is similar in style to rigid body collision. This is probably because of the influence of his (own) three laws in Book 1, primarily meant for, and applied on, the interaction of spherical (heavenly) bodies. The post-Newtonian understanding that resistances in flow over bodies are related to the pressure differences across the body, explained better, the importance of shape or form of the body. For instance, Stokes (1850) investigated the problem of flow over spheres, treating the fluid as continua, streamlining around the surface of the sphere, with a pressure distribution that depended on the fluid viscosity. He split the resistance force on the surface of the sphere, due to the fluid motion, into a normal and a tangential component. Of relevant mention is the fact that both the normal component (when expressed through the viscosity affected pressure distribution) and the tangential component (expressed through the shear stress) were directly affected by the viscosity of the fluid. For a formal derivation, see Sect. 2.6, page 56, of Bird et al. (1960). Then he integrated them over the surface of the sphere leading to the Stokes law Fk = 2π μRv + 4π μRv = 6π μRv = (α1 + α2 )v = αv.

(3.12)

The form drag (second term of Eq. (3.12)) results from the integration of the normal (pressure) force and the viscous drag (third term) from the integration of the tangential (shear) force. The subscript k stands for kinetic and signifies the resistance force arising from the motion (kinetic) of the fluid. Note that if the separate origination of the drag terms is set aside for a moment, Eq. (3.12) in its entirety, is identical to Eq. (3.10), with the βv 2 term omitted from it (the γ as discussed immediately after Eq. (3.10), has already been omitted). Newton inherently assumed the omission of the βv 2 term from Eq. (3.10), when he considered resistances varying linearly with velocity, in Sect. 1 of Book 2. Finally, Eq. (3.12) is valid (i.e. predicts the drag, correctly) for the sphere diameter based Re D ∼ 0.1. For a Re D ∼ 1, Eq. (3.12) predicts a drag 10% in error with experiments.

3.3 Physics of Flow Through Porous Media

59

We began, in Eq. (3.10), with a general expression for the resistances on a solid sphere by a moving fluid and questioned through historical evidence, the nature of Eq. (3.11), the second term of Eq. (3.10). With the new knowledge, we built another (Stokes) law, which in principle, agrees with the first term of Eq. (3.10). It would now seem we have deviated afar, but such is not the case. We have come to the conclusion that only the nature of, and certain predictions with, Eq. (3.11), as done by Newton, is faulty. The equation, nevertheless, holds. So, when is Eq. (3.11), Newton’s law of the drags for flow over spherical bodies, valid? We will answer this question, through the introduction of the concept of friction factor. We first define an equation F = A × (K.E.) × f,

(3.13)

with force F (which could be a combination of normal and tangential force between a solid surface and a moving fluid), expressed using, K. E. the kinetic energy per unit volume (= ρv 2 /2), and A being a characteristic area and f , the dimensionless quantity, called the friction factor. This Eq. (3.13) is the definition for f , provided A and K.E. are well defined. The origin of Eq. (3.13) can be thought of as the adaptability of the classical Bernoulli equation to the subject of hydraulics (see Chap. 6 of Bird et al. 1960; Chap. 10 of Granger 1995 for related discussions). For the channel flow in Fig. 3.3c, Bernoulli equation takes the form P = (PL − PR ) =

ρ 2 (v − vL2 ), 2 R

(3.14)

with P representing the pressure difference across the channel (i.e. difference in the normal force felt on the left and right faces of the channels in Fig. 3.3c, respectively) and the RHS representing the kinetic energy/volume change (increase) across the channel. In light of the earlier discussion (in particular, the case of frictionless motion for case C of Fig. 3.4), Eq. (3.14) can be thought of as a scale P ∼ 1, (ρv 2 /2)

(3.15)

depicting the flow, meaning, Bernoulli-type, pressure-drop driven, accelerated flow. Note that in Eq. (3.13), the resistance force F is the sum of the normal and tangential components of the resistance force, depending on the flow situation. Clearly then, for the flow situation, we are discussing above, Eq. (3.13) can be expressed in the form  (Force/area) P ∼ f, (3.16) = (ρv 2 /2) (Kinetic Energy/vol.) with Eq. (3.15) as a particular case of Eq. (3.16) with f = 1 (no friction).

60

3 Fluid Flow Through a Porous Medium

In other words, Eq. (3.13) is a useful tool for the hydraulics engineer to estimate the deviation of a ‘real’ duct flow pressure-drop from the ideal, Bernoulli-type, frictionless flow pressure-drop. Defining a ‘friction-head loss’ using the f as defined in Eq. (3.13) and estimating it, for instance, by rewriting the Hagen–Poiseuille equation (as explained above, in this page), and plugging the ‘friction-head loss’ group into it, the classical Bernoulli equation, Eq. (3.14) was adapted by the hydraulics engineer into the friction-corrected modified Bernoulli equation for duct flow.2 There are other particular cases of Eq. (3.16) or in general, of Eq. (3.13). For instance, the Hagen–Poiseuille equation can be expressed in a form similar to Eq. (3.16), with f = 24/Re D , with Re D based on the channel spacing D (see Fig. 3.3, case C). This time however the interpretation must be done with care. While it is misleading to use a ρv 2 /2 (Kinetic Energy/volume) group—a frictionless, Bernoulli flow concept (see page 100, Bejan (1993)), even for fully developed Hagen–Poiseuille duct flow, early porous medium flow investigators (such as Ward 1964; Ahmed and Sunada 1969 and others) have used a similar group to ‘scale’ the pressure-drop in, Eq. (3.8), the HDD equation—see for instance, Fig. 1.3, a reproduction from Ward (1964), in Nield and Bejan (1992). The HDD equation is a balance of pressure versus frictional forces in the context of porous medium flows (as seen in previous section while discussing Fig. 3.3 case D). While it is already misleading to use ρv 2 /2 for scaling any part of the HDD equation, attempts to either reconcile the ρv 2 /2 group as a derivative of the formdrag or interpret the resulting (after using the group on the HDD) dimensionless group as a friction factor is a conceptual inadequacy. We shall present a suitable scaling in a later section to avoid such misunderstanding. With the above success for laminar flow, similar interpretation of Eq. (3.13) in the turbulent duct flow has been made, with f determined experimentally, for varying surface roughness of the solid surface. The combined result is the familiar ‘Moody diagram’. Continuing our discussion on flow over spheres, Stokes Law, Eq. (3.12) upon rearrangement, can also be expressed in a form similar to Eq. (3.13) and through Eq. (3.16), will again lead to f = 24/Re D —with Re based on the diameter of the sphere. In addition, note that Eq. (3.11), Newton’s form-drag law, can also be readily expressed through Eqs. (3.13) and (3.16), with f ≡ C, the result valid in the range 102 < Re D < 105 . In other words, for the same fluid–solid pair, to increase the fluid velocity, the resistance varies from Eqs. (3.11)–(3.12). The form-drag portion of Eq. (3.12), which is linear in v, slowly changes character for increasing velocity, becoming Eq. (3.11), quadratic in v, thus gaining dominance. The viscous-drag portion of Eq. (3.12) retains its character, but loses dominance, at higher velocities.

2

Tinkering attempts like these, perhaps made Theodore von Karman (1954) quip, ‘hydraulics is the subject of variable constants’.

3.3 Physics of Flow Through Porous Media

61

Friction-factor charts ( f vs. Re D ), similar in style to the Moody diagram, have been made for flow over solid bodies as well (see for instance, Fig. 6.3-1 in page 192 of Bird et al. 1960), with the Stokes and Newton’s laws of the drag governing the two ends of the spectrum (transition from laminar to turbulent is not that apparent in the f curve on these charts). From these charts, it is evident that Eq. (3.10), in its form, is correct. Additionally, it is clear that its nature of transition from F ∼ αv to F ∼ βv 2 depends not only on the fluid velocity v, but on Re D . It is this dependency of the resistance forces on the Re D dimensionless group that Newton did not anticipate in his investigations. In other words, we can proceed (in the context of drag values) from Stokes Law, Eq. (3.12), to Newton’s law, Eq. (3.11), by simply reducing the fluid viscosity (through its temperature-dependency). Notice another subtle change in this process: The form-drag’s viscosity dependency changed character from its direct dependency in Eq. (3.12) to an indirect dependency (only through the velocity) in Eq. (3.11). That completes the discussion on the nature of the drag forces arising in porous media and their conceptual origins and clarifications. We shall proceed to discuss some of the extensions of the HDD model, to complete the momentum conservation formulation for porous medium flows.

3.4 Extensions of the HDD Model 3.4.1 Brinkman Model Following the ad hoc extension suggested in Eq. (3.9), for a homogeneous porous medium, the global Darcy equation, Eq. (3.6), can also be written in differential form as μ→ u = 0. (3.17) − ∇P − − K Similarly, the Hagen–Poiseuille differential momentum balance for fully developed laminar viscous flow is − ∇ p  + μ∇ 2 u  = 0,

(3.18)

where u  is the fluid velocity at any point and in 1D is equal to that given inside the integral of Eq. (3.1). Similarly, the pressure p  is the static pressure at each point in a fluid. By comparing the above two equations, Brinkman (1949) suggested an additional effect valid at the REV level for flow through porous media. Brinkman recognized the fluid–fluid viscous shear, ever present in a viscous flow (second term in LHS of Eq. (3.18)), as a distinct force from that of the internal fluid–solid viscous drag present in porous medium flows (second term in LHS of Eq. (3.17), identified as Darcy or viscous drag).

62

3 Fluid Flow Through a Porous Medium

Combining the above two equations by taking into account the definitions in Eqs. (3.1)–(3.3), the resulting Brinkman–Darcy equation for flow through porous media is → u − ∇ P = μe ∇ 2 −

μ− → u. K

(3.19)

Note particularly the transformation of the Laplacian velocity term of Eqs. (3.18)– (3.19). In Eq. (3.18), μ∇ 2 u is a point function defined at each point in the flow even through a pore. Employing the REV approach, it is transformed to a REV cross-sectional averaged quantity in Eq. (3.19). While doing so, the definitions in → u. Eqs. (3.1)–(3.3) are invoked to rewrite local velocity u  to the Darcy velocity − The dynamic viscosity μ is also transformed to a new quantity μe the effective viscosity of the porous medium. The pores of a porous medium are usually small (when compared to the characteristic length of the domain) and so, the fluid–fluid viscous shear is always affected by the adjacent fluid–solid viscous drag from the pore walls. Further, in a flow through interconnected pores, this viscous shear at each point is also affected by the viscous-drag influenced viscous shear of the surrounding points. This effect is identified as interference and was recognized by Brinkman. This is one reason for expecting the REV averaged Laplacian in Eq. (3.12) to be influenced by the geometry of the pores. This is enabled through the μe , believed to be a function of the geometry of the porous medium at the REV level. Another reasoning for invoking a concept of μe at the REV level of momentum conservation is as follows: It can be expected that the Brinkman effect of fluid–fluid viscous shear to manifest dominantly only when the viscous shear is comparable to the viscous drag in Eq. (3.12). This is valid only for coarser porous media with volumetric porosity φ > 0.6 as shown by Lundgren (1972) (see also Chap. 1 in Nield and Bejan 2006). One could then propose in such a media, even for small flow rates, →  μ ∇2−  e u   ≈ 1.  → (μ/K )− u

(3.20)

→ Even for pore level flows that exhibit non-slug profile, i.e. ∇ 2 − u = 0, as φ → 0, − → − → by setting μe → 0, the requirement of μe ∇ 2 u  (μ/K ) u can be met. Similarly, as → → u (μ/K )− u , leading to the reclaφ → 1, correspondingly μe → μ, while μe ∇ 2 − mation of the conventional Navier–Stokes equation that governs fluid flow without the porous medium. Based on the above arguments, μe is expected to depend on the porosity. A standard modification suggested is to bring in the volumetric porosity, μe ≈ φμ. Experimental determination of μe have been pursued in the last two decades. One of the earliest and most convincing experiments reported by Givler and Altobelli (1994) yielded inconclusive results about the value of μe . The simplest resolution of the effective viscosity is to set it as equal to the fluid dynamic viscosity, μe ≈ μ, which works in many instances because the Brinkman effect itself is negligible when com-

3.4 Extensions of the HDD Model

63

Fig. 3.4 Comparative summary of momentum conservation concepts in flow without and within porous media

pared to the viscous drag. Conclusive measurements for establishing μe as a property of a porous medium, like porosity and permeability, are yet to be carried out. Finally, taking H as a characteristic length of the domain containing the porous medium, it is useful to define a non-dimensional number by modifying Eq. (3.20) as Da =

μe K . μ H2

(3.21)

Da is the Darcy number of a porous medium and in principle provides the relative magnitude of the Darcy viscous drag term over the Brinkman viscous shear term. However, several recent practitioners of porous medium theory write Da = K /H 2 , after setting μe = μ. The value of Da, in this definition, is expected to vary between 0 and 1 for realistic porous media. Reports with Da > 1 (see for instance Lage et al. (1996)) in general consider porous media with φ > 0.6. In such porous media, the K can no longer be determined by the Darcy’s law, as the Brinkman term needs to be considered even at the REV level (see also, the discussion on p. 18, Nield and Bejan 2006).

64

3 Fluid Flow Through a Porous Medium

3.4.2 Generalized Momentum Equation A more general representation of the macroscopic momentum model, Eq. (3.4), which includes other force terms is given below.  ρf

 → 1 ∂− u 1→ 1 − μ→ → → → + − u · ∇→ u = −∇ P + μe ∇ 2 − u − ρC|− u |− u. u − − φ ∂t φ φ K

(3.22)

Equation (3.22) identified on occasion as the generalized Navier–Stokes equation is recognized as the general momentum conservation statement for porous medium flows, valid on a porous continuum. The second term on the LHS is due to the convective inertia, a result of relative motion in the fluid. The convective inertia for porous medium flows is usually negligible when compared to the other dominant drags present due to the porous solid matrix. This observation also holds for several of the biological processes that could be modelled as porous medium flows. Equation (3.22) has six physical properties of which ρ and μ can be obtained as measurements using their separate constitutive relations. Porosity φ can be determined independently, as discussed in Chap. 1. The porous medium hydraulic properties permeability K , form coefficient C and effective viscosity μe depend on the geometry of the permeable medium. Unique constitutive relations do not exist for their independent determination. Hydraulic experimental data must be matched with solutions to the above momentum equation, Eq. (3.22), for individual porous medium. The difficulty of this procedure is amplified for bio-materials, requiring internal access to organs with complex geometries modelled as porous media. A separate discussion ensues on the determination of K and C. A comparative summary of the nature of the force balances in the momentum conservation of various flows within and without porous media is given in the schematic in Fig. 3.4. Major differences between the flows within and without porous media arise in the viscous flow regimes. In the absence of the porous medium (internal walls), the origin of drag is restricted to the fluid–fluid viscous shear in the fluid continuum. In the porous continuum, the drag arises predominantly due to the solid–fluid viscous friction at each point of the porous medium, which further exhibits linear and quadratic dependence on U as speed increases. However, when the porous media coarsens (φ > 0.6) allowing larger pores, fluid–fluid viscous shear, the Brinkman effect dominates the viscous and form drags. When the fluid viscosity is zero, one reclaims the inviscid flow theory from both clear (of porous medium) fluid flow and porous media flow momentum equations. Potential flow theory should be uniformly applicable to both cases in this limit, as depicted in the LHS boxes of Fig. 3.4.

3.5 Permeability and Form Coefficient

65

3.5 Permeability and Form Coefficient The hydraulic properties characterizing (i.e. unique to) a porous medium are its permeability K , m2 and form coefficient C, m−1 . We shall discuss in this section the determination of these properties and some related issues.

3.5.1 Analytical Estimates For simple geometries, K can be obtained from analytical relationships. The Carman– Kozeny relationship is one such relation, valid for porous media comprising a bed of spherical particles or fibres. Taking Dp as the diameter of the particle or fibre and φ as the volumetric porosity of the bed, the permeability can be determined using K =

2 3 Dp∗ φ

180(1 − φ)2

,

(3.23)

where ∞ Dp∗ = 0∞ 0

Dp3 h(Dp )dDp Dp2 h(Dp )dDp

,

(3.24)

with h(Dp ) being the density function for the distribution of diameters Dp . The constant 180 is obtained from a best fit of experimental results. A similar analytical estimate is not straightforward for C, the form coefficient. Invoking the correlation proposed by Ergun (1952) for a packed column of spheres, a relationship has been proposed for C as C=

1.75(1 − φ) . Dp∗ φ 3

(3.25)

But this procedure is ad hoc as can be appreciated in light of the nature of form drag discussed in the earlier section. Readers are directed to a related discussion on p. 12, Nield and Bejan (2006) for further details. In earlier √ literature, the form coefficient is further related to the permeability as C = c F / K , where c F is a dimensionless constant that is thought of as a universal constant. This was shown not to be the case in experiments reported by Lage et al. (1997). The above analytical relations are valid only for porous media of very limited geometries (made of near spherical particles). Since the drag manifest by the solid matrix is modelled using terms that involve K and C as coefficients, for a generic porous medium, it is only sensible to determine them simultaneously in an appropriate hydraulic experiment. This is explained in a later section.

66

3 Fluid Flow Through a Porous Medium

3.5.2 Fictitious Porous Media While solving the generalized Navier–Stokes equation through numerical methods, the values of K and C must be known to provide closure. It is conventional to invoke the Carman–Kozeny and the Ergun relationships of the previous section. This comes with an important restriction. Setting μe = μ in Eq. (3.21) and substituting the Carman–Kozeny relation for permeability K result in a definition of Da as follows: Da = K /H 2 =

2 3 φ Dp∗

H 2 180(1 − φ)2

.

(3.26)

The parameter H is the characteristic length scale of the domain containing the porous medium (say, the gap between parallel plates bounding a porous medium or the side of an enclosure containing a porous medium). Numerous parametric studies in which K and φ are varied independently using numerical simulations have been carried out. Such an approach is constrained by a severe restriction even for porous media satisfying the Carman–Kozeny-type relationship. Since in Eq. (3.23) (and Eq. (3.26)), K depends on both φ and the pore scale Dp∗ , simultaneous variation in K and φ could result in Dp values that are larger than H . Table 3.1 summarizes the values of Da, φ and Dp in porous media that satisfy Eq. (3.23), the Carman–Kozeny relationship. It is apparent that the porous media formed with combinations of Da, φ and Dp values in the upper diagonal (marked ‘×’) are unrealistic (i.e. Dp ∼ H , one pore √ larger than the domain size). Only the combinations of properties denoted with ‘ ’ can form porous media that satisfy the Carman–Kozeny relationship, without resulting in pore scales Dp larger than H , the characteristic length of the global porous medium domain. The K and φ of porous media made of non-spherical solid particles are also expected to be related to the pore scale, if not by Carman–Kozeny relationship, by another empirical one. In general, it is meaningless to vary K and φ as mutually exclusive quantities in parametric studies. See a related discussion in Sect. 3.7.5.

Table 3.1 Realistic porous medium properties Da φ 0.25 0.5 0.1 0.01 0.001 0.0001

× × × √

× × √ √

0.75 × √ √ √

0.9 √ √ √ √

3.5 Permeability and Form Coefficient

67

3.5.3 Experimental Determination The model proposed by Darcy, Eq. (3.3), is a constitutive relation that defines permeability. It would seem that this equation is sufficient to determine K uniquely for a porous medium. For instance, a curve fit of the longitudinal global pressure-drop P/L, Pa/m against the average fluid speed U , m/s obtained from an isothermal, steady flow, laboratory experiment, for known fluid viscosity μ, Nsm−2 will accomplish this. To improve the accuracy of the determined K value, such experiments must be conducted at low enough fluid speeds to avoid form effects (due to shape of the solid matrix) that dominate at high speeds. To determine an acceptable fluid speed for such experiments to determine K separately, it was suggested that the porous medium pore diameter (Dp ) based Re must be maintained less than 0.1. This is discussed in Lage and Antohe (2000) and Papathanasiou et al. (2001). However, imposition of this criterion is difficult in practice, as form effects are ever present in porous medium flows. It is pertinent to note here that even Darcy’s original experimental data exhibits a quadratic relationship between pressure-drop and fluid speed as reported and analysed in Lage (1998) and Lage and Antohe (2000). The drag experienced by the fluid as it extrudes through the solid matrix of a porous medium is a single effect; it is only modelled as proportional to U and U 2 at sufficiently low and high fluid speeds. The K and C being coefficients of the corresponding drag terms (see Eq. (3.8), the global HDD model), cannot be determined separately, without causing a global momentum imbalance. This observation holds even for low permeable media. It becomes more difficult, when the Brinkman effect is comparable to the Darcy viscous drag (i.e. when φ > 0.6). In such hyper-porous media,3 none of the three K , C and μe can be determined separately. On the other hand, to simultaneously determine K and C for a porous medium, the accepted practice (see Lage et al. 2005) is to invoke Eq. (3.8), the global HDD model in conjunction with the results of a hydraulic experiment. The K and C obtained from the curve fit using Eq. (3.8), are global parameters (valid for the entire porous medium) and are, under isothermal conditions, independent of the properties and flow conditions of the fluid flowing through the porous medium. Consequently, they can be used with the HDD model, Eq. (3.8), on all subsequent occasions in the determination of the unknown global pressure-drop for a particular flow speed across a configuration using that porous medium. Such porous medium modelling have been used to understand biological processes like pathological blood flow when accumulations of fatty plaques of cholesterol and artery-clogging blood clots increase in the lumen (the cavity or channel within a tube) of the coronary artery (Khaled and Vafai 2003). We shall discuss few of these examples in Chap. 6.

3

A name suggested by Nield and Lage. See Lage (1998)

68

3 Fluid Flow Through a Porous Medium L

Porous insert whose K and C to be determined

2H P1

P2

U (m/s) Inlet

Constant Head Water Tank DC Power Supply + Digital Multimeter

Outlet

L

Test Section DPT

Vol. Flow rate Measuring Can

Fig. 3.5 Schematic of a typical hydraulic experiment to determine K and C of a porous insert

Example: Determination of K and C Figure 3.5 shows the schematic experimental set up used to conduct a typical hydraulic experiment for determining K and C of a porous medium. The experimental procedure involves inserting the porous medium in a test section of uniform cross section and known length L, placed in a closed or open hydraulic circuit. When a fluid (air or water) flows through this circuit, pressure-drop across the test section is determined using Bourdon-type static pressure gauges. The flow rate is determined by measuring the time taken to collect a known volume of the fluid in a collecting tank. From this volumetric flow rate, the average velocity U of the flow across the porous medium in the test section is determined. The experiment is repeated at different flow rates to obtain corresponding pressure-drop data, P/L, across the test section. In the given experiment, a mesh-type porous insert with porosity φ = 0.58 is used. The typical P/L versus U data across the test section carrying this insert is given in Fig. 3.6. A curve fit of this data set using Eq. (3.8), the HDD model, should yield the required K and C. The determined permeability and form coefficient were K = 4.18 × 10−8 , m2 and C = 43.59, m−1 .

3.5 Permeability and Form Coefficient

2 ΔP/L (kPa/m)

69

HDD Model Curve-fit Experimental data for 3 different porous inserts

1.5

1

0.5

0 0

1

2

U (m/s)

3

Fig. 3.6 Plot depicting typical data set from a hydraulic experiment and curve fit using the HDD model, Eq. (3.8), to determine K and C of a porous insert

Example: Estimation of Experimental Uncertainty in the Measurement of K and C Any experimental measurement involves uncertainty. In this hydraulic experiment, the uncertainty can be determined using the standard formula (from Holman 2001) wR = ±

n 

∂R 2 i=1

∂ xi

1/2 wx2i

,

(3.27)

where xi stands for the independent variable and wxi is the uncertainty associated with that variable. A sample calculation procedure for the estimation of uncertainty associated with the determination of K and C for porous media is given below. A sample data set is obtained from the plot Fig. 3.6 in the previous example as follows: Measuring can diameter d1 = 0.0625 m, height H = 0.019 m, time for collection of water in the collecting can t = 10 s, test section cross section x = 0.05 m, y = 0.03 m and length L 1 = 0.108 m across which, the typical measured pressure drop is p = 15.628 Pa. The uncertainties for the devices (in some cases, provided by the manufacturers) are Ud1 = 1.77 × 10−5 m, U y = 0.09 × 10−3 m, Uz = 0.048 × 10−3 m. For water, the thermophysical properties at the film temperature are μ = 792.4 × 10−6 Nsm−2 and ρ = 995.7 kgm−3 at 30 ◦ C. Area of measuring can, A1 = (π d1 2 )/4.

70

3 Fluid Flow Through a Porous Medium

Therefore,

∂ A1 π d1 = = 0.0981 m ∂d1 2

. Utilizing Eq. (3.27), the uncertainty in this measurement then is  2 ∂ A1 × Ud1 = ±1.736 × 10−6 m2 U A1 = ∂d1 Volume of water collected in measuring can V1 = A1 × H = 5.83 × 10−5 m3 . This implies that, ∂ V1 = H = 0.019 ∂ A1 and the uncertainty involved is  2 ∂ V1 Uv1 = × U A1 = ±3.29 × 10−8 m3 ∂ A1 Area of inlet duct A = y × z = 1.5 × 10−3 m2 . Implies, ∂A ∂A = z = 0.03, = y = 0.05 ∂y ∂z and the involved uncertainty is  2  2 ∂A ∂A UA = × Uy + × Uz = ±3.612 × 10−6 m2 ∂y ∂z For discharge, Q=

V1 = 5.814 × 10−6 m3 s−1 t

and 1 ∂ Q1 = = 0.1 ∂ V1 t and the uncertainty is  U Q1 =

∂ Q1 × UV1 ∂ V1

2

= ±3.299 × 10−9 m3 s−1

3.5 Permeability and Form Coefficient

71

Similarly, for average velocity, Q1 = 0.00387 ms−1 A

U=

∂U 1 ∂U −Q 1 = = −2.584 = 666.657, = ∂ Q1 A ∂A A2  UU =

∂U × U Q1 ∂ Q1

2

 +

∂U × UA ∂A

2

= ±9.59 × 10−6 ms−1

For pressure-drop across porous insert, p = 15.628 Pa p = Y = 144.706 Pa m−1 L1 Ur = 144.706 ×

0.25 = ±0.3617 Pa 100

Since Eq. (3.8), the HDD model, is of the form Y = γ U + βU 2 , the curve fit in Fig. 3.6 using the HDD model would result in the corresponding coefficients of U and U 2 as, γ = 4.112 × 10−6 , β = 3.566 × 10−5 . The uncertainty in the determination of Permeability, K , is thus γ =

Y − βU 2 Y = − βU U U

which, when the entire range of data set for a typical experimental run (here, one curve in Fig. 3.6 contains 26 data points) is   26  2

2 26   ∂γ ∂γ  × UU + × UY Uγ = ∂U ∂Y 1 1 Therefore, Uγ =



21.143 × 103 + 21.558 × 103 = ±206.643

U K = γγ = ±5.027 × 10−3 m2 . Similarly, the uncertainty involved in the determination of form drag coefficient is determined using: U

β=

Y γ Y − γU = 2− U2 U U

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3 Fluid Flow Through a Porous Medium

and   26  2

2 26   ∂β ∂β  × UU + × UY Uβ = ∂U ∂Y 1 1 resulting in Uβ =

 88.11 × 106 + 41.2 × 106 = ±11,362.7

and UC =

11,362.7 Uβ = = ±0.0319 m−1 β 3.65 × 105

.

3.5.4 Viscometry and Permeametry The recognition of viscosity as a fundamental property of fluids in the nineteenth century led to viscometry, the systematic measurement of viscosity for various fluids. A critical elaboration of this branch of study can be obtained from Truesdell (1974). Viscometry is the systematic measurement of the viscosity of a fluid. Permeametry is the systematic measurement of permeability, a hydraulic property of a porous medium. The importance of viscometry in engineering can be thought of as follows: If we know the density and assume it to be constant, then viscosity is the single property that is necessary to classify a fluid. This could be determined by a well-chosen experiment using one of the viscometers. Then, employing the necessary boundary and/or initial conditions, the Navier–Stokes equation can be analytically solved for any flow situation, thus completely eliminating experimental fluid mechanics. But this has not yet been possible for all flow situations. An analogous situation is the measurement of permeability of a porous medium. Similar to viscometry, in permeametry, once we perform a well-chosen isothermal, porous medium hydraulic experiment, using the HDD model, Eq. (3.8), we should, in principle, be able to permanently obtain the K and C pairs for that porous medium. Solving Eq. (3.22), the momentum conservation statement valid over a porous continuum is then essentially a mathematical or numerical venture. Experimentation in porous media would then be relegated essentially to a secondary role of comforting practitioners who doubt the results of fluid mechanics (in porous medium) theorists. This contrast highlights two things. First, the pertinence of permeametry, the experimental endeavour to accurately determine the K and C that characterize a porous medium. A detailed account of the experimental determination of permeability is provided by Bear (1988) and Dullien (1992).

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73

Secondly, the above contrast allows us to doubt the universality of the K and C pairs that we find using the porous medium hydraulic experiment. While in viscometry, myriad experiments involving different types of flow yield the same viscosity value for a particular test fluid (i.e. universality of viscosity is established), the same cannot be categorically said about the K and C pairs determined in a porous medium hydraulic experiment. For instance, even in the isothermal porous pipe flow case, for different velocity range (say, first data range is between U0 and U1 while the second is between U0 and U2 with U2 > U1 ), one can expect different K and C pairs as the outcome of the (global HDD model based) curve-fitting procedure. This aspect has been briefly discussed in Boomsma and Poulikakos (2002). Another important issue has to do with secondary effects in the determination of porous medium properties. For a chosen porous medium, one can ask whether an experiment conceived without the effects of the boundary (walls of the pipe) would yield a K and C identical to that, obtained by using the global HDD model, Eq. (3.8), to curve fit the experimental results (of P/L and U ). We shall discuss this in the ensuing section.

3.5.5 Protocol for Permeability Measurement Experiments for determining K and C are usually influenced by undesired secondary effects. This results in higher uncertainties in their determination. A protocol has been established in Lage et al. (2005) to minimize secondary effects. We shall describe here the theoretical framework to evolve the protocol. A constitutive equation defines a material property. It is necessary to reduce the number of unknowns in a balance equation. The Darcy’s law is a constitutive equation that defines permeability, K , m2 as a property of the porous medium. Rewriting it as K =

μ U, (Pv )

(3.28)

one can observe that, to determine K , measurement of K , U, (ms−1 ), usually obtained by dividing the volumetric flow rate by the cross-sectional area of the channel, and the pressure-drop Pv , Pa, is required, in the flow of a Newtonian fluid, along a length L, m of the channel occupied by a porous medium. On the other hand, the differential form of the volume-averaged momentum equation for the flow of a Newtonian fluid through a porous medium, with uniform, isotropic and constant properties is given by Eq. (3.22). → In Eq. (3.22), − u is the vectorial local fluid velocity and P, the local static pressure. The terms to the left side of the equal sign of Eq. (3.22) represent the fluid acceleration (local and convective). To the right of the equal sign, we have pressure gradient, viscous diffusion, viscous drag and form drag, respectively. The sum of the viscous and form drags constitute the total drag effect imposed on the flow by the porous medium. Their absence signals absence of a porous medium and we should reclaim

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from Eq. (3.22), the standard Navier–Stokes equation for clear (of porous medium) fluid flow. For steady, fully developed unidirectional flow, and negligible viscous diffusion and form-drag effects, the momentum balance Eq. (3.22) reduces to μ P = U, L K

(3.29)

which is almost identical to Eq. (3.28). The difference between Eq. (3.28), the constitutive equation defining K and Eq. (3.29), the momentum balance equation, is the pressure-drop. The implicit assumption when determining K from Eq. (3.28), its constitutive equation is that, once determined, the value of K would not change with changes in the porous medium length, or channel geometry, or flow rate, but only with changes in the internal structure of the porous medium. This requires that the pressure-drop Pv of Eq. (3.28) measures the pressure-drop caused by the viscous drag alone, induced by the internal structure of the solid porous matrix. The momentum balance equation version of Darcy’s law, Eq. (3.29) need not have this restriction. Obviously, the applicability of Eq. (3.28) for determining K is more restricted than that of the momentum balance equation, Eq. (3.29). To guarantee that only the viscous-drag effect is accounted for in using Eq. (3.28) to determine K (i.e. for the experimentally measured pressure-drop Pm to be identical to Pv ), the testing channel containing the porous medium must be straight and of uniform cross section. Else, Pm could be affected by acceleration caused by channel curvature and/or variations in the cross section. The effect imposed on the fluid pressure-drop by the bounding walls of the channel (boundary viscous diffusion effects) would also have to be made negligible. Obviously, this last requirement may be difficult to satisfy in practice. Two ways exist to resolve this. The first is to allow secondary effects and include it in the pressure-drop, hoping that its effect would be an insignificant raise in the total pressure-drop, while determining K . This means Eq. (3.28) can be represented as K =

μ U, (Pcv )

(3.30)

where Pcv is the channel pressure-drop that includes the pressure-drop resulting from the porous medium viscous drag (Pv ) and by the secondary wall effects (Pc ). This means that the determined K would depend on the channel wall or geometry. Further, even in the absence of a porous medium, we would measure a nonzero K from Eq. (3.30), a result of nonzero pressure-drop caused by the channel wall effects. Obviously, this method of resolving the issue leads to a maximum limit for the measured permeability—equal to that of the channel clear (of porous medium) fluid flow permeability.

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75

The other method to resolve this confusion between constitutive relation and balance equation is to subtract the secondary effects from the measured pressuredrop, while determining K . This would result in a determined K that represents only the viscous drag of the porous medium   2Po Pcv . (3.31) − K = LμU φ Dh2 Here Po = 2τw Dh /μu, is the Poiseuille number, with τw , Nm−2 the wall shear stress and Dh , m, the hydraulic diameter of the channel. The last term of Eq. (3.31) is the secondary effect by the bounding wall. This wall effect is subtracted from the measured pressure-drop, before determining K . For instance, for parallel plate channel Po = 24 and Dh = 2h and so, the last term in Eq. (3.31) would equal 12/ h 2 , a constant that must be subtracted from the first term before determining K . As defined in Eq. (3.31), K is no longer limited but can tend to infinity for a clear (of porous medium) channel because it can be shown that as the porosity φ → 1 (meaning there is no solid matrix), (Pcv /L) → (μU 2Po/Dh2 ). In an engineering situation of channel bounded porous medium flows, the significance of the above restriction is as follows. Permeability K as defined by Eq. (3.30) has a restriction for its highest value dependent on the channel cross-sectional geometry. However, when defined by Eq. (3.31), the value of K has no upper bound. Several parametric (numerical or CFD) studies involve wall-bounded channel flow through a porous medium using Da > 1/(2Po), where Da = K /Dh2 . Such parametric studies are valid only when K is defined by, and measured according to, Eq. (3.31) (and not by Eq. (3.30)). In this context, it is worth revisiting the section on fictitious porous media. To summarize, we find that two alternative constitutive equations defining K , viz. Eqs. (3.30) and (3.31) exist. They differ on the inclusion or not of the bounding channel wall viscous diffusion effect on the measured Pm . The simplicity of Eq. (3.30) makes it more appealing than Eq. (3.31) for use. But, as shown in Lage et al. (2005), when considering the simultaneous determination of K and C (as discussed in the earlier section), the use of Eq. (3.31) for determining K is preferable. A similar exercise can be done with respect to the determination of C, m−1 the form coefficient of the porous medium, which from Eq. (3.8) can be defined as C=

Pf /L , ρU 2

(3.32)

where Pf /L is the pressure-drop due to form drag alone. Observe that Eq. (3.32) can be obtained from Eq. (3.22) when acceleration, viscous-drag or boundary viscous effects are negligible in uniform flow across a porous medium of length L. In this case, the nature of the pressure-drop dependency on the fluid speed for viscous-drag and the overall viscous diffusion channel wall effect differs from the nature of the fluid speed dependency for form drag (linear vs. quadratic). The only alternative in the present case is to model the measured pressure-drop Pm as the

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sum of a component due to the form drag, Pf , and a component due to the other viscous effects. Accounting for the secondary effects, C can be determined using the relationship     Pm Pm Pcv 1 1 1 − = − μU . (3.33) C= ρU 2 L L ρU 2 L K In general, the effect of turbulence can also be accounted for in the secondary effects. However, as discussed in Lage (1998) and shown in the experiments of Wilson (2005), turbulence can become important in the porous channel flow before or after the form-drag effect becomes relevant. Turbulence effects on the channel wall are likely to be quadratic in u, the fluid speed. Hence, when turbulence is assumed to set in prior to the form-drag effects, the required pressure-drop component due to form drag Pf /L, must be written as   Pm Pc Pv Pf = − + . (3.34) L L L L Including turbulence effects, the channel wall friction effect can be written using the Darcy friction factor as 1 Pc = ρU 2 f. L 2Dh

(3.35)

The equations to determine K and C would then become  K =

1 Pm − 2 ρU f LμU 2φ Dh μ

−1

,

(3.36)

f 1 μ Pm /L . − 2 − 2 ρU 2φ Dh K ρU

(3.37)

and C=

In non-dimensional form, Eqs. (3.36) and (3.37) can be written as   f −1 Da = ζ Eu Re − Re 2 , 2φ

(3.38)

and Du = ζ Eu −

1 f , − 2φ 2 Da Re

(3.39)

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77

where Da = K /Dh2 and Du = Dh C are the Darcy and Dupuit numbers for a porous medium channel flow. The rest of the terms are aspect ratio ζ = Dh /L, Euler number Eu = Pm /ρU 2 and Reynolds number Re = ρ Dh U/μ. Equations (3.36) and (3.37), or their non-dimensional equivalent (3.38) and (3.39), are model equations for K and C involving measurable quantities. They subtract from the measured pressure-drop, the pressure-drop due to flow effects not related to the property of interest. If K and C are to be separately measured, use of (3.36) and (3.37) is recommended. When K and C are to be determined concurrently, a recommended practice, then the experiment should minimize the viscous effect of the channel wall boundary on the measured pressure-drop. This would require from Eq. (3.39), ζ >

f . 2φ 2 Eu

(3.40)

Equation (3.40) suggests a small value for f , the friction factor, which requires Re to be high but within the laminar flow regime. When K is known and C needs to be determined separately, it is optimal for the bounding channel wall and porous medium viscous-drag effects to be kept minimal. Again from Eq. (3.39), this would translate to ζ >

f 1 + . 2φ 2 Eu Eu Da Re

(3.41)

In this case, it may be necessary to increase Re even further to neutralize the effect of the last term in Eq. (3.41), resulting possibly in turbulent flow. This criterion is assisted by the fact that f is constant in a turbulent flow. In general, when K and C are to be determined concurrently, by interpolating experimental data as explained in the previous section, it should be made sure that the experimental data falling in the viscous-drag dominated regime satisfies Eq. (3.40), and the data falling in the form-drag dominated regime satisfies Eq. (3.41).

3.6 Flow Transition Knowledge of the departure from Darcy flow (linear velocity, viscous drag dominated), in terms of the global velocity U , is very important in porous medium flows. For example, this knowledge prescribes the pressure-drop from the global HDD model, Eq. (3.8), for a typical porous medium channel flow configuration with wide engineering applications.

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3.6.1 What Is Transition? For an isothermal flow through a porous medium channel of length L, let DC0 = ρC0 U 2 represent the global form drag and Dμ0 = μ0 U/K 0 , the global viscous drag. Here viscosity and density are evaluated at the ‘reference’ inlet fluid temperature, i.e. μ0 = μ(T0 = Tin ). The global HDD model, Eq. (3.8), can then be written as  μ0 P  = U + ρ0 C0 U 2 = Dμ0 + DC0 . (3.42)  L 0 K0 Here, the subscript 0 signifies that the terms are evaluated as reference values at isothermal condition, i.e. P/L|0 refers to the pressure-drop across the channel for isothermal flows and so on. It is widely understood that flow through porous media is characterized by two distinct regimes. Further elaboration on this is available in the initial chapters of Dullien (1992) and Nield and Bejan (2006). For sufficiently low speeds, as the U 2 term is much smaller than the U term, the pressure gradient on the LHS of Eq. (3.42) is likely to be balanced more by the linear velocity term, i.e. P/L|0 ∼ Dμ0 . Hence Eq. (3.42) is identical to the Darcy equation, Eq. (3.6). As the velocity increases, the balance is shifted to the quadratic velocity term, i.e. P/L|0 ∼ DC0 , hence the departure from Darcy flow. Both of these drag terms are always present, irrespective of the magnitude of the velocity. The flow situation is always best governed by Eq. (3.42). Only the strength of the individual contribution of the drag terms in the RHS changes from viscous drag to form drag, as the flow velocity increases. We will now proceed to seek parameters that help in predicting the departure (transition) from Darcy flow.

3.6.2 Characterization of the Transition Use of a characteristic Reynolds number as a parameter for the transition criterion indicating the departure from linear Darcy flow has been proposed in several studies. Early examples include Wyckoff et al. (1934), Ward (1964), Ahmed and Sunada (1969) and others, while more recent ones include Fand et al. (1987) and Firdaouss et al. (1997). See Lage (1998) and Lage and Antohe (2000) for an elaborate reference list. One such Reynolds number, for example, is written as Re p = U De /ν where U is the global seepage speed, De is the average particle diameter of the solid particles constituent of the porous medium. The use of particle diameter as the representative scale in the Re p is consistent with the use of packed non-consolidated spherical particles as the solid matrix of a porous medium. However, from Scheidegger (1960), we learn that the value of the

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79

Re p above which transition results ranges between 0.1 and 75, for different porous media, an uncertainty factor of 750 for a transition predicting parameter. This is partly due to the differences in the pore structure of the porous media tested. The other important reason is the characteristic length scale used by Re p which is a microscopic (fluid-continuum level) information, while the transition from Darcy equation Eq. (3.7) that this Re p attempts to predict is a phenomenon identified at the global level (i.e. above the macroscopic porous-continuum level). In this context, observe that the procedure of exchanging U with the local, pore velocity u is valid only for an isothermal flow through the porous media. Only then can we view the flow to maintain a ‘slug flow’ with u everywhere identically equal to U , the global seepage speed. This suggests the use of a global representative length scale for defining a Reynolds number. For example, using K 1/2 as the length scale, we may define a Re for the porous medium flows as Re K =

Uρ K 1/2 . μ

(3.43)

The viscosity and density in Eq. (3.43) are evaluated at a representative temperature, usually the inlet temperature of the flow configuration. Transition happens when the Re K in Eq. (3.43) is of the order of 10 or higher. This was verified experimentally, among others, by Ward (1964). Although the Re K , Eq. (3.43), is more restrictive in its numerical values, to predict transition, it is still incomplete as a parameter for establishing a transition criterion because it is devoid of the form coefficient, C, a representative parameter of the structure of any porous medium and contributes to the momentum balance (drag) in the global HDD model, Eq. (3.8) or Eq. (3.42). Clearly, any parameter that predicts the transition from viscous to form dominated flow regime should include parameters that are representative of both of the flow regimes. Based on scaling arguments using Eq. (3.42), detailed in Lage (1998), the ratio λ, of global form-drag and global viscous-drag forces along a porous channel with uniform cross section, is a much better alternative. It is given by  DC0 ρC0 K 0 form-drag U, (3.44) = = λ= viscous-drag D μ0 μ0 where K 0 and C0 are the permeability and form coefficient of the porous medium obtained from isothermal experiments and U is the cross-sectional averaged Darcy (or seepage) fluid speed. When λ > 1, the flow is said to have departed from Darcy flow, into the quadratic flow regime. Observe that the C K |0 appearing in Eq. (3.44) has the units of a length scale. Hence, λ in Eq. (3.44) can be treated as a Re number based on the C K |0 length scale. This λ (or ReC K ) is an appropriate parameter for determining flow transition, as it includes both the porous medium property that causes the drag, by considering

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the global HDD model, Eq. (3.8), in its entirety, unlike any of the transition predicting options discussed earlier. To predict the transition from the dominance of one drag to the other, it is necessary to correctly compare the relative strength of either of the drags that oppose the flow. Naturally, as noticed from Eq. (3.44), λ includes all of the parameters that should affect the transition. In general, for a given porous configuration (i.e. for a chosen set of ρ, μ0 , K 0 and C0 ), λ is a function only of U . We can expect with certainty, transition from Darcy flow to happen beyond the flow velocity where the strength of the drags is equal (λ ∼ 1). This procedure also reduces the scatter and uncertainty inherent in the use of the characteristic Reynolds numbers, discussed earlier. Ahmed and Sunada (1969) and Geertsma (1974) have previously used an alternate Reynolds number, similar to the RHS of (3.44). It was used in their studies as an effective non-dimensional parameter, along with a friction factor (recall the discussion on friction factor in an earlier section) f K , to plot pressure-drop versus flow velocity results. Citing Ahmed and Sunada (1969), later authors, like Civan and Tiab (1989), Civan and Evans (1996), have also used this Reynolds number in their work. However the interpretation of the Reynolds number as a ratio of the drags, as introduced in Eq. (3.44), is not clearly seen in any of these works. Whether it is a characteristic Reynolds number, like the one defined in Eq. (3.43) or λ, Eq. (3.44), the tacit assumption behind the use of any of these parameters for establishing the transition criterion is that the global HDD model, Eq. (3.42), is fundamentally valid for the flow configuration considered. One such situation where Eq. (3.42) is not reliable is when the porous medium channel (or region) is heated/cooled and the flowing fluid has strong temperature-dependent viscosity. This is discussed in Chap. 8, Sect. 8.2.

3.6.3 Prediction of Transition Dividing Eq. (3.8) by Eq. (3.42), we can arrive at a non-dimensional HDD model of the form

=

Dμ DC (P/L) +  = μ + C . = (P/L)|0 D μ 0 + D C0 D μ 0 + D C0

(3.45)

For the case of a fluid flowing with uniform viscosity from Eq. (3.45), P/L equals P/L|0 , thus yielding  = 1. Using Eq. (3.44), we can recast Eq. (3.45) as 0 =

λ 1 + = 1. (1 + λ) (1 + λ)

(3.46)

To understand better how these parameters predict transition, let us discuss a specific example. Using the scales in Eq. (3.46), the longitudinal pressure-drop variation

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81

Fig. 3.7 Transition from Darcy viscous drag dominated to form-drag dominated regime in porous medium flows

with average flow velocity obtained from a numerical experiment through a porous medium channel is displayed in Fig. 3.7. Following the continuous curves from left to right, the curve that starts close to value one on the y-axis and proceeds to decrease for higher velocities (λ) is the non-dimensional global viscous-drag value, μ0 , for isothermal flows. The continuous line that starts from close to zero on the y-axis and increases for higher λ values represents the corresponding global form-drag value, C0 . The pair of curves portrays the gaining dominance of the nonlinear, form-drag effect as fluid velocity U increases inside the porous medium channel (or, in general, region). The curves cross for λ = 1 (λT ), marked in Fig. 3.7 with a square, representing the equivalence in strength of the drags. Beyond this point (i.e. for all higher velocities), the global form drag dominates. For any velocity (λ), the sum of these corresponding drag values (μ0 + C0 ) will give the total non-dimensional pressure-drop experienced by the flow across the channel (0 = 1) for the no heating, constant viscosity case, represented by the horizontal continuous thick line at unity, in the y-axis. This result is a direct consequence of the scaling used in Eq. (3.46).

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3.6.4 PM Transition Flow and Turbulent Flow The global HDD model, Eq. (3.8), in its generic form would resemble P/L = αU + βU 2 , which is similar to the model that describes fully developed turbulent channel flow, without a porous medium. This superficial resemblance has erroneously led earlier researchers like Ward (1964) to believe that transition in porous medium flows from Darcy viscous drag dominated regime to the form-drag dominated regime implies transition from laminar flow to turbulent flow (marked by global, channel hydraulic diameter D based Re to be greater than 2300). The drag transition in porous medium flows is predicted by Eq. (3.42). Experimental results of Wilson et al. (2006) in rod bundle flows treated as porous media have shown using Eq. (3.42) that the flow inside porous medium channels can be form drag dominated (i.e. λT > 1) even as the global channel hydraulic diameter D based Re is less than 2300 (i.e. flow is yet to become turbulent). Similarly, fully turbulent flow based on Re D > 2300 criterion when directed through low permeable porous media can relaminarize. This has been supported by arguments made in Lage (1998) and the computational fluid flow results for compact heat exchanger treated as porous media in Wilson (2005). Establishing and sustaining fully developed turbulent flow through porous media is difficult. Turbulence in porous media is beyond the scope of the essentials discussed here. For substantial information on this topic, readers are directed to the excellent basic research articles by Antohe and Lage (1997), Kuwahara et al. (1998), Nakayama and Kuwahara (1999), reviews by Lage (1998) and the monograph de Lemos (2006).

3.7 Miscellaneous Topics 3.7.1 Hydrodynamic Boundary Layer Since a porous medium essentially comprise of interconnected pores imposing a wall shear practically in all spatial locations, the pore flow is always viscous shear flow. Hence, the pore flow inside a homogeneous porous medium does not sustain a hydrodynamic boundary layer in the traditional sense. Even when a porous medium is bounded by walls, as in a channel flow, the shear from the bounding walls does not sustain a distinct viscous sub layer near it, as the flow is always disturbed by the internal walls, i.e. pores. Therefore, in general, isothermal flow through a homogeneous porous medium is hydrodynamically fully developed plug flow, i.e. in a 2D cross section, u(y) = U . Owing to the above reason, the hydrodynamic entrance length required for any flow to become fully developed inside a porous medium channel is very small. An example is shown in Fig. 3.8. Computational simulation results of velocity vectors

3.7 Miscellaneous Topics

83

Fig. 3.8 Velocity vectors in the entrance region of a porous medium channel

for a flow at U = 0.1 m/s through a mesh-type porous medium of φ = 0.57 and K = 10−6 m2 from Wilson et al. reveal the typical entrance length not to exceed Dh , the channel hydraulic diameter.

3.7.2 Wall Channelling This effect is present in configurations with highly permeable media composed of large particles bounded by straight channel walls. Near the straight channel walls, pores are of larger volume, resulting in an increased φ(y) value that facilitates increased local flow. This phenomenon is called wall channelling. In the top row of Fig. 3.9, the porous medium made of rod bundles in (A) exhibits wall channelling. As the rod bundle-type porous medium becomes less permeable in (B), wall channelling effect also reduces. Local flow variations due to wall channelling could lead to enhanced convection heat transfer. Importantly, the concept of REV breaks down near the wall that exhibits the channelling effect. Researchers usually model wall channelling effect through exponential volumetric porosity variation near the straight bounding walls. The standard form adopted is

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Fig. 3.9 Porous media that exhibit a increased and b reduced wall channelling effect; c and d are porous media with identical porosity, φ, but different permeability, K

   y , φ = φ∞ 1 + Cexp −N dp

(3.47)

where y is the distance from the wall, dp is the particle diameter, and C and N are empirical constants. Experimental results mentioned in Nield and Bejan (2006) indicate C = 1.4 and N = 5 or 6 for a medium with φ∞ = 0.4.

3.7.3 Porosity and Permeability Consider the two porous media in the bottom row of Fig. 3.9, formed by drilling holes in solid blocks. The holes on the porous medium in (C) have diameter d = 2, mm, while those in (D) are 1 mm. But their numbers are so arranged that, both (C) and (D) have identical surface porosity φs = 0.18. Since the hole cross section does not change along the length of the insert, their volumetric porosity, φ are also identically equal to 0.18. However, when an hydraulic experiment is performed, each would exhibit different permeability, K , values as 25.6 × 10−8 , m2 and 6.5 × 10−8 , respectively. Reducing the pore size (hole diameter) while increasing the number of pores (holes), increases the solid–fluid surface area of contact, increasing the local

3.7 Miscellaneous Topics

85

viscous shear. Since K is a lumped measure of the total viscous shear (drag), it is observed to increase correspondingly. A similar case can be built for the variation in the form coefficient C even as the local shapes of the solid matrix changes in a porous medium. However, this could render the porous medium non-homogeneous, the discussion of which is beyond the scope of the current text.

3.7.4 Secondary Effect in the Determination of K and C As discussed in detail in an earlier section, permeability K and form coefficient C are the two main hydraulic properties that characterize any porous medium. Once determined, they are global parameters, i.e. valid for the entire porous medium. Under isothermal conditions, they are independent of the properties and flow conditions of the fluid flowing through the porous medium. Consequently, they can be used with the HDD model on all subsequent occasions in the determination of the unknown global pressure-drop for a particular flow speed across a configuration using that porous medium. However, this understanding has restrictions. For instance, it is valid only when the flow throughout the porous medium, across which the hydraulic experiment is conducted to determine the K and C, is of plug flow in nature. As the form coefficient C is dependent on the u(y) profile at every x along the porous medium test section, it can be different for the same porous medium, encountering non-plug flow like profiles at the inlet of the test section. Local velocity variations in porous medium flows can be initiated through variations in temperature-dependent thermophysical fluid properties under non-isothermal flow conditions. This is discussed in detail in a separate section in Chap. 8. Another situation that causes u(y) = U is wall channelling, discussed in the previous section. As a secondary effect, it can also affect K and C measurements, as reported in Wilson et al. (2006). In the top row of Fig. 3.9, the porous medium made of rod bundles exhibits wall channelling. Another way to induce local velocity variation is to impose a chosen flow profile at the inlet of a porous medium channel. Evidence for the effect of inlet velocity profiles on the values of permeability K and form coefficient C measured experimentally across a porous medium duct of square cross section were presented in Wilson et al. (2004, 2006). These results are summarized in Fig. 3.10. The numbers in the last two columns indicate the percentage variation in the measurement of K and C, when the inlet velocity profile is altered between plug flow (PF) and fully developed flow (FD). The variation in permeability K for different inlet velocity profiles is minimal, which is expected. However, the form coefficient C varies as much as 11% when the inlet velocity profile of the hydraulic experiment conducted is altered from fully developed flow to a plug flow. As discussed in the protocol for K and C determination in an earlier section, this secondary effect can be minimized by suitably choosing

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Fig. 3.10 Inlet velocity profile effect on the determination of K and C for porous media, data adapted from Wilson et al. (2006)

(increasing) the test section length for various porous media. In general, experiments seeking K and C measurements require a standardization that minimizes secondary effects.

3.7.5 Boundary and Interface Conditions The proper hydrodynamic boundary conditions required to solve the generalized porous medium momentum transport equation, Eq. (3.22), are discussed in Sect. 1.6 of Nield and Bejan (2006) in some detail. We provide here some highlights and an extended discussion on the interface conditions, including results from further developments. When the porous medium region is bounded by a solid impermeable wall, then it is assumed that the normal component of the seepage velocity v = (u, v, w) must vanish at the boundary as v = 0 at y = 0.

(3.48)

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87

Invoking the differential form of Darcy’s law, Eq. (3.7), for the flow through the porous medium region, would result in only one condition being applied at a given boundary. Therefore, components of the velocity other than the v above can have arbitrary values at y = 0; i.e. we have slip at the boundary. It is however a convention to impose the no-slip condition with u = 0 at y = 0 in computational simulations that use the volume-averaged formulation of the porous medium transport equations, as the slip magnitude is not known a priori. This usually leads to a confusion about the existence and nature of the ‘hydrodynamic boundary layer’ in the simulated porous medium flows; it is a numerical artefact. This situation can be avoided only through a ‘direct simulation’ of the flow inside the porous medium geometry, which could get tedious even for porous media of simple internal structure. When the porous medium region is unbounded, a ‘free boundary’ is usually imposed as ∂v = 0 at y = 0. ∂y

(3.49)

This boundary condition makes the inherent assumptions that the pressure along the free boundary is constant (hence its gradients along x and z are zero) and the differential form of Darcy’s law is applicable for the incompressible flow in the porous medium region. Channel flow with partially filled porous medium configuration is applicable in compact heat exchangers, electronic cooling, fuel cells, thermal insulation, oil recovery and near-compact heat exchangers. In modelling such a configuration using transport equations, the slip coefficient value at the interface between the porous medium and clear fluid flow is required for closure. A typical partially filled porous medium channel flow configuration is shown in Fig. 3.11. Considering unidirectional flow, the momentum balance statement for the porous medium flow is given by the differential form of the Darcy’s law as μ dp = − u. dx K

(3.50)

In the clear fluid flow region (without porous medium) of Fig. 3.11, the steadystate momentum balance in the x-direction is given by the Navier–Stokes equation ρ(∇ · u f )u f = −∇ p + μ∇ 2 u f .

(3.51)

The interface is subjected to normal and tangential stresses of different origin and magnitude at the top and bottom planes. Beavers and Joseph (1967) provided an interface boundary condition through the empirical relationship uf − ud ∂u |y+ = α √ , ∂y K

(3.52)

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3 Fluid Flow Through a Porous Medium

Fig. 3.11 Schematic of partially filled porous medium channel depicting the interface slip coefficient problem

where u f is the velocity in the clear fluid flow region, u D is Darcy or seepage velocity in the porous medium. The quantity α is an interface slip coefficient and Eq. (3.52), in principle, is the constitutive equation that defines this quantity. To determine α, the other quantities must be measured experimentally. It is understood from Nield and Bejan (2006) that u f and ∂u f /∂ y are evaluated at y = 0+ and u D is evaluated at some small distance from the y = 0 plane. As proposed originally by Beavers and Joseph (1967), the empirical relation for determining the interface slip coefficient, α, across a partially filled porous medium configuration inevitably requires the measurement of the Darcy velocity u D on the porous medium (PM) side, the local velocity u f and its gradient ∂u f /∂ y on the clear fluid (CF) side. The length scale across the interface over which the required measurements are to be performed for determining the slip coefficient α remains arbitrary. The arbitrariness of the locations for measuring the quantities involved in Eq. (3.52) has been a topic of research interest and remains to be resolved. From the form of Eq. (3.52), one could conjecture that the ∂u f /∂ y, u f on the clear fluid region and the u D on the porous √ medium region are to be measured at points separated by a distance not less than K , their gradient length scale. √ The simplest way to arrange these points is to bisect the K length scale by the interface. Beavers and Joseph (1967) first measured α using the above procedure for different porous media. A linear dependence of α on the porosity was observed. Experimental and analytical determination of α are reported by Taylor (1971) and Richardson (1971), where the porous medium consists of vertically grooved plates over which a moving parallel plate provides Couette flow. Ochoa-Tapia and Whitaker (1995) presented interface boundary condition results matching the Navier–Stokes equation and the Darcy equation for momentum balance. They reported a jump in the shear stress but not in the velocity, with the stress jump

3.7 Miscellaneous Topics

89

being inversely proportional to the permeability of the porous medium. These results were later used by Kuznetsov (1997, 2000) to provide analytical solutions for the flow field near the interface region, accounting for the boundary and inertial effects. The status of the theoretical interface models was reviewed in Alazmi and Vafai (2001). Several recent studies have investigated the evaluation of interface slip coefficient in partially filled porous medium configurations. Goharzadeh et al. (2005) through their experiments suggested that the transition region √ across such a interface is of the order of the grain diameter, much larger than the K length scale. Agelinchaab et al. (2006) studied using PIV experiments, the velocity measurements of flow through circular cylindrical rods installed vertically in a channel. They measured the slip velocity at the interface between the rods and the clear fluid flow, showing that the measured slip velocity depends only on the filling fraction. √ They report that the depth of the open flow into the porous medium is more than K . From the above experimental observations, it is evident that there is no clear justification for choosing √ K as the length scale across which the variables must be measured. The results of a recent study by Narasimhan et al. (2012) suggest that the REL of a porous medium is a possible length scale across which the u D , u f and ∂u f /∂ y could be measured for determining α, the slip coefficient. The fissure-type porous medium parameters are systematically varied in the porosity range 0.2 ≤ φ ≤ 0.95 √ and flow direction permeability 10−6 < K , m2 < 10−9 . When a length scale of K is used to separate the points of measurement across the interface between the PM and CF side, the determined α values show large scatter. For a REL, the α value remains invariant in the test range of porosity 0.2 ≤ φ ≤ 0.95. By using the REL of the porous medium as the length scale, the aforementioned arbitrariness appears removed. This conclusion is corroborated by both the PIV experiments and corresponding numerical √ simulations. However, The REL length scale is about one order larger than the K of the porous medium. This fundamental problem requires further experimental and theoretical investigation.

3.8 Problems (1) By proper non-dimensionalization, reduce the global porous medium Hazen– Dupuit–Darcy flow model, P/L = (μ/K )U + ρCU 2 to the form f = 1/Re + C, where f can be treated the friction factor, Re a porous medium Reynolds number and C, a constant involving porous medium properties. (2) Use the Re in the solution of the previous problem to characterize porous medium flow transition. (3) Thecross-sectional variation  of porosity in a porous medium channel is φ = φ∞ 1 + Cexp(−N y/dp ) where φ∞ is mean porosity, dp is pore diameter and C and N are appropriate constants. Taking φ∞ as fixed, plot the trend of crosssectional velocity profile u(y) for the porous medium channel subjected to

90

3 Fluid Flow Through a Porous Medium

fixed longitudinal pressure-drop for: (i) C > 0, N = 1 (ii) C < 0, N = 1 (iii) C > 0, N = −1. Which of the three situations is realistic? Why? (4) The parallel cracks formed by blocks of thickness a, separated by a distance of b and repeating endlessly in the vertical direction as shown in the figure, can be viewed as a porous medium. Determine the porosity φ of the porous medium. Assuming Hagen–Poiseuille-type ‘parallel plate channel flow’ with U m/s average velocity through each crack, derive the Darcy law for the porous medium (with u, as shown in figure, as the volume average velocity) and find an expression for the effective permeability K (m2 ). (5) Derive from first principles, a Carmen–Kozeny relationship for permeability K of a fissure-type porous medium made of uniform cross-sectional voids separated by solid blocks, similar to the figure in the previous problem.

(6) Water at room temperature is filtered through a sand bed 120 mm thick that is made of particles with mean particle diameter 0.25 mm, porosity 0.3. The depth of water above the bed is 90 mm. Estimate the steady-state flow rate of water per square metre of area at the exit of the bed. (7) A porous material with periodic geometry, is made of a REV comprising solid cubes with sides equal to b and voids made of interconnected tubes of diameter d with centrelines coinciding with all the edges of the cubes. Calculate the porosity φ and permeability K of the porous material. (8) An aquifer can be considered as a porous medium. The water pressure within such a porous medium aquifer is measured with two manometers separated by length L. Take the aquifer cross-sectional area as H 2 and the pressure gradient dP/dx = (P1 − P2 )/L. Further, the static fluid pressure across the aquifer varies with the head (height) in each manometer according to Archimedes principle; hence, the pressure difference is ρwater g(h 1 − h 2 ). The term in parentheses is the drop in head that drives the flow (say, from left to right). If a drop in head of 60 m drives a flow of 8.3−5 m3 s−1 , determine the permeability K of the aquifer, taking its cross-sectional area as 2 m2 and L as 2 km.

3.8 Problems

91

(9) Water at temperature 25 ◦ C is flowing through a tube of 30 cm length and 10 cm diameter, filled with sand grains of porosity K = 8 × 10−8 m2 /s. The tube is inclined at an angle 30◦ with the horizontal. Find the Darcy velocity and power required for the flow. (10) Calculate the flow rate of water through a filter 90 mm outside diameter and 50 mm inside diameter and 100 mm long given that the pressure on the outside is 30 kPa greater than on the inside. The mean particle diameter is 0.04 mm and the porosity is 0.4. (11) The permeability K of a porous medium comprising packed bed of spheres medium is given by the Carman–Kozeny relationship, Eq. (3.23), with a value of 180 for the ‘CK constant’ in the denominator. If pore area ‘fractal dimension’ and tortuosity ‘fractal dimension’ are given by Df and DT respectively, find an expression for the ‘CK constant’ (instead of 180) in Eq. (3.23), in terms of Df , DT and φ. Hint: Consult the reference, Yu (2001). (12) A petroleum reservoir was discovered in the Ohio river basin. Analysis by the resident petroleum engineer revealed that the reservoir was depleting because the bottom was made of a porous Stratigraphic bottom. The petroleum was seeping down to a much deeper part of the earth from where recovery would be unprofitable. Calculate the time window during which this reservoir could be used for recovering 50% of the initial oil content. Dimensions of the reservoir = 10 × 1 × 0.1 km. The depth at which the reservoir can be accessed is 3 km. The bottom stratigraphic region extends to a depth of 1 km from the bottom of the reservoir. Treservoir = 50 C ρoil = 790 kg/m3 K stratigraphic_region = 10−8 m2 μ = 0.6 × 10−2 Poise

(13) What is the pressure at which water has to be pumped into the bottom of this reservoir to force the oil through the top surface of permeability 10−9 at a mass flow rate of 100 kg/s. A schematic of the extraction process is shown in the figure. The thickness of the top surface is 50 m.

92

3 Fluid Flow Through a Porous Medium Water pumped in

Oil pipe Surface

Collection zone

2950 m

50 m

100 m

Top porous region

Oil reservoir

(14) A porous medium formed by a packed bed of spheres is introduced in a gas pre-heater of diameter 5 m. The average diameter of the packing spheres is d = 5 cm. The length of the channel is equal to 10 m. Estimate the pressuredrop across the porous bed. Compare the result with the case when the porous medium is absent. Use the Carmen–Kozeny relationship to calculate the permeability K of the porous bed. (15) The window air conditioner which is quite popular in the summer months requires regular cleaning of the air filters. This is because, the primary air filter that captures dust particles gets clogged easily which in turn loads the air conditioner’s air pump further due to higher power required for pumping. Consider the air filter as a thin one-dimensional porous medium of thickness 2 cm. The pores exist as cylindrical fissures of diameter 5 mm separated by a distance of 3 cm. The hole area reduces according to the equation Ap = Ap,0 × e−t/τ , where τ = 14 days. Estimate the increase in pumping power required 30 days after the filter is cleaned to the power required with a clean filter. (16) Consider flow through a porous medium formed by equally distributed pores, each of which has a regular rhombus cross section with side a inscribed in a square of side b. Determine the porosity φ and permeability K of the porous medium. Write a relationship for the pressure gradient for moderate Reynolds number values. Apply this relationship to find the pressure-drop for the fluid properties: ν = 0.9 × 10−6 m2 /s, Q/A = 70 mm/s, Dp = 1.5 mm, ρ = 1000 kg/m3 .

Chapter 4

Forced Convection Through Porous Medium

In this and the next chapters, we shall discuss, respectively, forced and natural convection heat transfer in a porous medium. An excellent research monograph by Nield and Bejan (2006) with extensive coverage of material segregated in relevant subtopics of convection in porous media is already available. Additionally, recent compilations by Ingham et al. (2004); Bejan et al. (2004); Vadász (2008) and the two handbooks on porous media (Vafai 2000, 2005) discuss both basic and advanced topics and ongoing research pertaining to convection in porous media. Hence, we shall restrict our discussion of convection in porous media only to the fundamentals and advanced material that are less discussed elsewhere. Readers are encouraged to consult the above texts for additional material.

4.1 Energy Equation with Flow Following the first law of thermodynamics analysis and the energy equation for heat conduction in porous medium in Sect. 2.2 of Chap. 2, one can formulate the energy equation for convection in porous medium. The additional effect is due to fluid flow that, in principle, augments the heat transfer from conduction to convection. Information about the fluid flow must be obtained by solving the momentum equation for flow through porous medium, which has been discussed in detail in Chap. 3.

4.1.1 Local Thermal Non-equilibrium (LTNE) Situation Consider the general situation of a homogeneous porous medium made of a stationary solid and mobile fluid phase as shown in Fig. 4.1. Local thermal non-equilibrium (LTNE) is assumed to prevail at the REV level, i.e. the solid and fluid phase temperatures satisfy Eq. (2.18). © The Author(s) 2023 A. Narasimhan, Essentials of Heat and Fluid Flow in Porous Media, https://doi.org/10.1007/978-3-030-99865-3_4

93

94

4 Forced Convection Through Porous Medium

Fig. 4.1 Schematic for formulating the porous medium energy equation with fluid flow

Following an analysis similar to that in Sect. 2.2, the volume averaged energy equations for the solid and fluid phase of a porous medium can be written as follows: ∂Tf → + (ρc p ) f − u .∇T f = ∂t ∇.(φk f ∇T f ) + φ Q f + μ + h s f av (Ts − T f ),

(4.1)

∂ Ts (1 − φ)(ρc)s ∂t   = ∇. (1 − φ)ks ∇Ts + (1 − φ)Q  s + h s f av (T f − Ts ),

(4.2)

φ(ρc p ) f

→ where − u is the volume averaged velocity to be evaluated from the generalized porous medium momentum equation in Eq. (3.22). The quantity h s f is a volumetric interfacial heat transfer coefficient, and av is the specific surface of the porous medium (surface per unit volume). In the two-energy model, it is crucial to determine the appropriate value of h s f . A short discussion on the modelling of h s f is already provided in Sect. 2.6.2. The rest of the terms and symbols have been explained in Sect. 2.2. As a modification to Eq. (2.45), the major additions in Eq. (4.1) here are the convection and viscous dissipation terms (second on the LHS and penultimate

4.1 Energy Equation with Flow

95

on the RHS, respectively). Viscous dissipation in porous medium is modelled as the power required for the fluid to ‘extrude’ itself over the porous solid structure. The modelling and effects of viscous dissipation will be discussed later in this chapter, in a separate section.

4.1.2 Local Thermal Equilibrium (LTE) Situation From Fig. 4.1, it is obvious that the one-dimensional porous medium convection configuration is a parallel arrangement at the REV level. For this porous medium region, as h s f → ∞, the temperature difference between the solid and fluid phases becomes negligible. Then, local thermal equilibrium satisfying Eq. (2.18) can be assumed to hold. Applying the volume averaging theory and following the procedure detailed in Sect. 2.2, Eqs. (4.1) and (4.2) can then be combined into a single equation as,   ∂T − μ +→ u · ∇T = ke ∇ 2 T + q  + . (4.3) (ρc P ) f σ ∂t K As explained in Sect. 2.2, the temperature T present in Eq. (4.3) is neither the temperature of the solid nor the fluid but is that of the porous medium. Here, σ = (ρc)s /(ρc P ) f is the ratio of the thermal capacities; effective thermal conductivity ke and the effective heat generation qe are defined, as before, by Eqs. (2.22) and (2.22). As with Eq. (2.19), the major assumptions in writing Eq. (4.3) are: (1) homogeneous porous medium, (2) LTE exists in the REV level, (3) effective properties used in the volume averaged energy equation are accurate in predicting the effects in the REV level, (4) the variables, u, P and T are volume averaged quantities defined on a porous continuum. For steady state, in the absence of heat generation and negligible viscous dissipation with constant thermal-physical properties, Eq. (4.3) reduces to → u · ∇T = ke ∇ 2 T. (ρc P ) f −

(4.4)

Observe that in Eq. (4.4), the convection term contains fluid thermophysical properties, while the diffusive term contains, ke , the porous medium effective thermal conductivity. In steady state, Eq. (4.4) is a balance between the convection of enthalpy only by the moving fluid through the pores and the heat conduction that could exist in both solid and fluid phases. Even as convection prevails through a porous medium, the heat transfer interaction between the porous medium region and its surrounding is through a solid (bounding walls), which is essentially through heat conduction. We shall now discuss solutions for the energy equation in few basic forced convection porous medium configurations.

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4 Forced Convection Through Porous Medium

4.2 Forced Convection in Porous Medium Over a Flat Plate The simplest forced convection problem is the interaction between a solid wall and the parallel flow through a porous medium bounded on one side by a solid wall. Following the analysis described in Bejan (2004), we shall briefly discuss the scale analysis solution to the problem of finding the local and average heat transfer coefficient in this situation. In 2D Cartesian coordinate system of Fig. 4.2, the steady-state governing equations are, Mass Continuity: ∂v ∂u + = 0. (4.5) ∂x ∂y Assuming Darcy’s law to be sufficient as the momentum balance statement: u=−

K ∂P , μ ∂x

(4.6)

Fig. 4.2 Schematic of forced convection flow through a porous medium bounded by a heated solid wall

4.2 Forced Convection in Porous Medium Over a Flat Plate

v=−

K ∂P . μ ∂y

97

(4.7)

The energy balance statement, assuming LTE to hold, is: u

∂T ∂T ∂2T +v = αe 2 , ∂x ∂y ∂y

(4.8)

where αe = ke /(ρc P ) f is the effective thermal diffusivity of the porous medium. It has also been assumed that ρ is constant, the boundary layer is slender, and the gravity effect is negligible. For a uniform parallel flow through the porous medium, the following boundary conditions are valid for the fluid flow u = U∞ v = 0 P(x) = −

μ U∞ x + constant. K

(4.9)

The heat transfer part of the problem is conventionally approached by considering the bounding solid wall to be at constant temperature or heat flux. Isothermal Wall: Let the temperature of the fluid-saturated porous medium be T∞ and the wall temperature downstream of some point x = 0 is T0 , as shown in Fig. 4.2. To find the heat transfer rate between the x > 0 wall and the porous medium, one should find the local heat transfer coefficient, which depends on the local wall temperature gradient. To find this, the local temperature field in the fluid-saturated porous medium must be known. Following Bejan (2004), the unknown temperature field and the heat transfer coefficient are estimated using scale analysis. It is assumed that the thermal boundary layer that affects the temperature transition from T0 to T∞ is slender, i.e. δT  x. Although the present problem is analogous to the Blasius-Pohlhausen problem of forced convection over a flat plate without the porous medium, here, the velocity boundary layer is absent (as explained in Sect. 3.7.1). Let δT be the thermal boundary layer thickness. Taking T = T0 − T∞ , the ‘scale balance’ of the energy equation (4.8) can be written as U∞ This gives,

T T ∼ αe 2 . x δT

δT ∼ Pex−1/2 , x

(4.10)

(4.11)

and hence, the local heat transfer coefficient can be found as N ux = h

x x ∼ Pex1/2 , ∼ k δT

(4.12)

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4 Forced Convection Through Porous Medium

where the Péclet number is given by Pex =

U∞ x . αe

(4.13)

From this, it can be concluded that the thermal boundary layer thickness δT increases as x 1/2 downstream from the point where wall heating begins. The local heat transfer coefficient (or the local heat flux q  ) decreases as x −1/2 . The k in the above equations usually represents k = k f because the fluid phase is the one that convects. However, since both the fluid and solid phases of the porous medium are in contact with the solid wall, for interpreting the resulting heat transfer coefficient as that of the porous medium, one can also define it to be k = ke , in which case, the N u e and the N u are related through the ratio ke /k f . Due to the assumption that δT is slender, the above N u results are valid only when > O(1), i.e. sufficiently far downstream from x = 0. Pe−1/2 x A rigorous analysis can be performed using the similarity profile method, and examples of which have been documented in the Blasius solution to fluid flow (without porous media) over a flat plate and the analysis initial regime of transient heat conduction in continuous media. The similarity variable suggested in this case by the scaling (4.11) is, y . (4.14) η = Pe1/2 x x The similarity temperature profile is T − T0 = θ (η) . T∞ − T0

(4.15)

With this notation, the energy equation and its boundary conditions become 1 θ  + ηθ  = 0, 2

(4.16)

θ (0) = 0, θ (∞) = 1.

(4.17)

and

Solving Eqs. (4.16) utilizing (4.17) (left as an exercise), it can be shown that θ = erf 

dθ dη

 η=0

The local Nusselt number is given as

η 2

,

= π −1/2 = 0.564.

(4.18)

(4.19)

4.2 Forced Convection in Porous Medium Over a Flat Plate

N ux =

99

x q  = 0.564 Pex1/2 , Tw − T∞ k

(4.20)

which agrees within a factor of order 1 with the earlier result, (4.12), obtained through scale analysis. The average heat transfer coefficient over the wall length L is Nu = h

L 1/2 = 1.128Pe L . k

(4.21)

Constant Wall Heat Flux: A similar analysis can be performed by considering the solid wall as isoflux instead of isothermal. The results for local and averaged heat transfer coefficient are N ux = Nu =

q  x = 0.886Pe1/2 x , k [T0 (x) − T∞ ] 

q  L

k T 0 − T∞

1/2

 = 1.329Pe L .

(4.22)

(4.23)

Recently, Celli et al. (2010) have analysed the effects of local thermal non-equilibrium on the boundary layer growth in steady-state forced convection inside a porous medium over an isothermal flat plate.

4.3 Forced Convection in Porous Medium Channel Next, we consider laminar forced convection flow through a porous medium channel, confined by a heat exchanging solid wall. In order to study this configuration, we first define the required basic parameters appropriate in the porous medium context (Fig. 4.3).

4.3.1 Bulk Temperature We are aware from our heat transfer texts such as McAdams (1933) and Holman (2002) that enhanced (improved) heat transfer can be achieved by reducing the thermal resistance of the transfer process. In convection heat transfer, the thermal resistance is given by 1/(h A), with h denoting the convection heat transfer coefficient. See Bergles (1988) for a historical account of this concept. The h is related to the heat power q and the driving temperature difference through q = h A (Tw − Tref ) .

(4.24)

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4 Forced Convection Through Porous Medium

Fig. 4.3 Snapshot of local temperature profiles in a forced convection flow bounded by a heated solid wall. The analogous situation is when a porous solid matrix is inserted in the channel flow shown in the inset

For external flows (over solid walls), temperature Tref is taken as T∞ , the free stream temperature, while for internal flows (bounded by walls), Tref is taken as Tb , the bulk temperature of the confined fluid. From thermodynamics, for a static homogeneous fluid system made of a single chemical constituent, temperature can be written as T =

I , C

(4.25)

the ratio between total enthalpy, I ( = mass × specific enthalpy), and heat capacity, C ( = m × c), of the system. Observe, in doing so, enthalpy is replaced only by c P T , under the assumption of negligible local pressure changes; else, we have to include a term with pressure change and the volume expansivity of the fluid. See page 230 Bejan (1993) for additional details. Consider a forced convection channel flow with constant heat flux q˙  crossing the solid walls into the flow. Here, both the enthalpy and heat capacity become dependent on the rate of flow, but if we treat H and C as local quantities, Eq. (4.25) is still valid for every point within the channel. To retain the notion of temperature in this situation, a bulk flow model is employed to treat the channel flow as a thermodynamic analysis

4.3 Forced Convection in Porous Medium Channel

101

of an open system with the definition in Eq. (4.25) recast for a cross section of the channel, as Tb =

rate of flow of enthalpy through a cross section . rate of flow of heat capacity through a cross section

(4.26)

Although the shift to a bulk concept for temperature is apparent in Eq. (4.26), it is already implicitly present in Eq. (4.25), the way it is used in specifying a system (treated as a whole) temperature. In Eq. (4.26), we are accounting for the (cross sectional) spatial variation of temperature within a (thermodynamic) system (i.e. the channel), a better representation of the system (treated as a whole) temperature. Recognizing the mass and enthalpy flow rate across a cross-sectional element in the channel as ρu Ad A and (ρu Ad A)c P T , respectively, we can write Eq. (4.26) as,  ρuc P T d A . Tb = A A ρuc P d A

(4.27)

Notice the use of c P in Eq. (4.27) generalizes the concept to include compressible fluid flow as well. For incompressible fluids (such as water), c P can be replaced with their respective specific heat (c) (see also, page 231, Bejan 1993). With c P treated as constant and the replacement of the area integral of local velocity u by U A , the channel cross-sectional averaged velocity times the cross sectional area, Eq. (4.27) reduces to 1 uT d A, (4.28) Tb = UA A

the definition of bulk temperature, a concept frequently used in duct flow. For a 2D parallel plates channel, Eq. (4.28) takes the form 1 Tb (x) = 2HU

2H (uT )|x dy.

(4.29)

0

If we slice the channel at any cross section, at any location along its length, and allow the fluid to pour itself into an adiabatic cup, the well-mixed fluid in the cup itself can be treated as a static homogeneous fluid system. Equation (4.25) holds for this system, and the evaluated temperature of the cup is called the adiabatic mixing cup temperature. This temperature would equal the value obtained from Eq. (4.28). The concept of bulk temperature, Tb , valid for both laminar and turbulent flows, is a useful way to provide a lumped representation of enthalpy, while applying the first law of thermodynamics analysis to forced convection configurations. The definition provided in Eq. (4.29) is valid for forced convection in a porous medium channel, where, however, the concept of LTE and volume averaged properties must be invoked, resulting in a definition

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4 Forced Convection Through Porous Medium

 T b =

Ac u T d Ac



Ac u d Ac

,

(4.30)

where Ac is the flow cross-sectional area of the porous medium channel. The angle brackets are shown (only here) to indicate the volume averaged properties. Even though there are some theoretical advantages in using Tb as a reference temperature, it is important to recognize that Tb is not easy to measure accurately in a porous medium. Moreover, in most systems, Tb varies along the main flow direction.

4.3.2 Nusselt Numbers In convection heat transfer, the general relation between heat power q and the driving temperature difference is (4.31) q = h A(Tw − Tref ), where the parameters are defined in the nomenclature. In Equation (4.31), the potential driving the heat transfer is (Tw − Tref ) and the thermal resistance is 1/(h A). Equation (4.31) is the definition of the thermal resistance (1/ h A) and, specifically, the definition of h, the convection heat transfer coefficient. The q and (Tw − Tref ) are dependent on each other, which are not apparent from Eq. (4.31). At the interface between the solid wall and the convecting fluid, the relationship between q and Tw is defined by Fourier’s law ∂T , (4.32) q = −k f A ∂n where the derivative is the local fluid temperature variation along a direction perpendicular to the solid and fluid interface. Combining Eqs. (4.31) and (4.32), results in ∂T | h ∂n w . =− kf (Tw − Tref )

(4.33)

Equation (4.33) can be used as another definition of h, in this case independent of the surface area A. Multiplying both sides of Eq. (4.33) by a representative length D results in a dimensionless equation, Nu =

D ∂∂nT |w hD =− , kf (Tw − Tref )

(4.34)

defining the Nusselt number N u, the dimensionless representation of h. Now, consider the case of introducing a porous medium in the above forced convection channel configuration. The effect of increasing the surface area A of

4.3 Forced Convection in Porous Medium Channel

103

the system by inserting a porous medium must be captured in, h e , a modified local effective heat transfer coefficient of the porous medium, defined in terms of a local non-dimensional Nusselt number. This N u is defined in porous medium literature as Nu =

−D ∂∂nT |w he D . = ke (Tw − Tref )

(4.35)

Apart from the effective properties invoked, all three temperatures involved in the above equation must be understood as volume averaged quantities. For instance, although Tw is defined along the solid and porous medium interface, it is point-wise only in a porous-continuum sense. Also, Tw has two components along the porous solid interface. One is the average temperature along the region bathed by the fluid in the pores next to the interface. The other is the average temperature of the solid matrix of the porous medium in contact with the solid wall. Equation (4.35) assumes these two local temperature components to be locally the same; i.e. local thermal equilibrium (LTE) at the REV level is imposed. Another distinct definition of the Nusselt number used in porous medium literature is

 D ∂∂nT |w he D ke = − , (4.36) N um = kf kf (Tw (x) − Tb (x) which is different from Eq. (4.35) in some ways. In Eq. (4.36), fluid thermal conductivity k f replaces the ke used in Eq. (4.35). This results in the inclusion of the group ke in the right hand side of Eq. (4.36). Also, Eq. (4.36) replaces Tref with the local kf bulk temperature, Tb , of a porous medium channel flow, defined earlier in Eq. (4.30). Comparing Eqs. (4.35) and (4.36),N u e can be defined as N ue =

q  D he D ke = N um , ke (Tw (x) − Tb (x))ke kf

(4.37)

which is valid for a porous medium channel configuration.

4.3.3 Channel Filled with Porous Medium For thermally fully developed Darcy flow through a porous medium channel, the value of N u in Eq. (4.37) has been reported in literature as follows: For porous medium confined in a tube of circular cross section, from Rohsenow and Choi (1961), Nu =

D q  (x) = 5.78 (tube, Tw = constant), (Tw − Tm (x)) ke

(4.38)

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4 Forced Convection Through Porous Medium

Nu =

D q  = 8 (tube, q  = constant). (Tw (x) − Tm (x)) ke

(4.39)

For porous medium sandwiched between parallel plates with spacing D, from Hartnett and Rohsenow (1973), Nu =

D q  (x) = 4.93 (parallel plates, Tw = constant), (Tw − Tm (x)) ke

(4.40)

Nu =

D q  = 6 (parallel plates, q  = constant). (Tw (x) − Tm (x)) ke

(4.41)

4.3.4 A Measurable Nu Since Tb is difficult to measure in a local cross section inside a porous medium channel, the above local N u results are difficult to verify in an experiment. In turn, a N u definition, useful as a measure of the influence of using a porous medium in a convection situation, is required. In this context, one can define a N u averaged over the entire porous medium channel as 2H q  NuL = (4.42)

, ke Tw − Tb (in) where Tw is the channel averaged wall temperature and Tb (in) is the bulk temperature measured at the inlet of the porous medium channel. This utility of the above N u has been discussed in Lage and Narasimhan (2000), and results have been reported in Narasimhan and Lage (2001). Consider the convection through a regular confined channel of perimeter w and length L, with isoflux surfaces, and filled with an isotropic and homogeneous porous medium. Rewriting Eq. (4.36) as Tw − Tb =

Dqw , N um k f

(4.43)

where the uniform surface heat flux is qw = −ke ∂ T /∂n|w . Equation (4.43) can be integrated along the surface A of the porous medium channel as 1 A

(Tw − Tb ) dA = A

1 A



Dqw N um k f

 dA.

(4.44)

A

The integrand on the right side of Eq. (4.44) is constant when the flow is thermally fully developed because N u m is constant in this case. Furthermore, the integral of

4.3 Forced Convection in Porous Medium Channel

105

Tw along A results in the surface area averaged temperature Tw . Therefore, Tw −

1 A

Tb d A =

Dqw . N um k f

(4.45)

A

To find the surface-averaged fluid bulk temperature, the fluid longitudinal heat diffusion is neglected and the steady-state version of the first law of thermodynamics for a control volume is invoked, Tb =

(wx) qw + Tin , mc ˙ Pf

(4.46)

where w is the channel perimeter, x is the distance from the inlet to the point where Tb is measured, Tin is the fluid inlet temperature (cross-sectional area averaged temperature in the case of non-uniform Tin ), and m˙ is the fluid mass flow rate. The Tb in Eq. (4.46) is the transverse-averaged fluid bulk temperature (varying along the x-direction only). Equation (4.46) can be integrated from x = 0tox = L , multiplied by (w/A) and simplified using the relations (wdx) = d A and (wL) = A, to yield 1 wLqw Tb d A = + Tin . (4.47) A 2mc ˙ p A

Now, Eqs. (4.47) and (4.45) can be combined as Dqw ke (Tw − Tin )

= NuL = 

1 ke k f N um

+

wL ke 2D mc ˙ p

.

(4.48)

Equation (4.48) presents the relation between the Nusselt number N u m defined in Eq. (4.36) and the surface-averaged Nusselt number N u L defined in Eq. (4.42). The derivation of Eq. (4.48) assumes thermally fully developed flow and negligible longitudinal diffusion in the porous medium channel. The last term of Eq. (4.48) can be rewritten in dimensionless form using the . Reynolds number Re = m D/(μAc ) and the Prandtl number Pr = μc p /k f , as NuL = 

1 N um

  k f /ke   + 2wL Ac

1 Re Pr

.

(4.49)

The group within parentheses in the denominator of Eq. (4.49) is dependent only on the geometry of the channel. Equation (4.49) has the advantage of providing a direct verification of theoretical results predicting N u m by comparing the value of N u L from Eq. (4.49) with the experimental value. This is possible because it is easier to measure N u L than to measure N u m (which involves local values of Tb inside the channel).

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4 Forced Convection Through Porous Medium

For instance, considering the case of plug flow (Darcy flow) through a porous medium sandwiched between two parallel isoflux plates, of length L and set apart by a distance H = D, for which, using Eq. (4.41), N u m = 6ke /k f . The predicted N u L then is 1 NuL =  (4.50)  1 ,  1 L + 6 H Re Pre where an effective Prandtl number is introduced as Pre = μc P /ke . From Eq.(4.50), for high flow rate N u L → 6 as expected because Tb → Tin . From Eqs. (4.35) and (4.36), it can be verified that N u e approaches (k f /ke )N u m . If the channel surface is held at a constant and uniform temperature Twall , the local Nusselt number defined in terms of Tref is a function of x. It is derived from Eqs. (4.35) and (4.36), if the reference temperature chosen when defining N u e is Tin , as  Tw − Tb (x)  N um . (4.51) N u e (x) = k f /ke Tw − Tin For porous medium forced convection configurations where the local bulk temperature Tb is difficult to measure, an optional equation for N u L is available by invoking the first law of thermodynamics. The result, in non-dimensional form, is    Dqavg   wL 1 N um , N u L = k f /ke 1 − 2 Ac Re Pr Tw − Tin

(4.52)

 represents the surface-averaged heat flow crossing the channel surface. where qavg For a Darcy plug flow through a porous medium sandwiched between two parallel isothermal plates, of length L and separated by a distance H = D, from Eq. (4.40), one could obtain N u m = 4.93ke /k f . Therefore, N u L becomes

   Dqavg wL 1 . N u L = 4.93 1 − 2 Ac Re Pr Tw − Tin

(4.53)

4.4 Heat Transfer Enhancement Aspects Methods to modify the thermal resistance of a heat transfer apparatus now constitute an entire body of research and technology discussed under enhanced heat transfer. The book by Webb (1994) is a landmark in the area of enhanced heat transfer and provides a detailed review of enhanced heat transfer. Lage and Narasimhan (2000) provide a review of enhanced convection heat transfer using porous media. This section introduces some basic concepts of using porous media to enhance convection heat transfer.

4.4 Heat Transfer Enhancement Aspects

107

Consider the generic forced convection channel flow configuration in which the heat power q is fixed. From Eq. (4.24), it is evident that decreasing (Tw − Tref ) reduces the thermal resistance 1/(h A) by the same factor. Reducing (Tw − Tref ) could be the objective of the design engineer because a lower temperature difference results in lower interface temperature, as Tref is invariant. This in turn provides flexibility in the temperature requirement for selecting the convection fluid or the solid material for the surface. If Tw is the fixed design parameter instead, then (Tw − Tref ) becomes invariant. In this case, increasing the heat flow q would decrease the thermal resistance 1/(h A). An enhanced convection heat transfer configuration is evaluated by comparison with a reference configuration using the factor-of-merit E f , defined as Ef =

hA . (h A)ref

(4.54)

Observe in the above equation that the same Tref must be used when comparing the enhancement performance of two distinct thermal systems. The case of unimpeded fluid flow along a smooth surface is usually considered as a reference forced convection configuration. The thermal resistance (1/ h A) of a certain reference configuration can be reduced by (1) increasing the heat transfer coefficient h and (2) increasing the area A crossed by the heat flow. The convection heat transfer coefficient h is modified by changing Tw , k f or ∂∂nT |w . Changes in the heat transfer area A indirectly result in a change in h by modifying the surface temperature or the fluid temperature variation. Sometimes, the change is beneficial in increasing h; at other times, it could be detrimental. One way to increase the heat transfer area A is by including within the fluid path, a geometrically complex, permeable, solid structure, i.e. a porous medium.

4.4.1 Basic Configurations An enhanced configuration could be built by brazing a wire-mesh to the smooth surface. The thermal resistance of the surface is modified by changing the heat transfer area A because the heat can now flow into the fluid region by following the solid wire-mesh path. Moreover, the heat transfer coefficient h is also affected indirectly by modifying the flow configuration. This new flow configuration would result in a new temperature or temperature variation at the fluid–solid interface, ultimately affecting h. This effect is independent of contact between the wire-mesh and the surface. Even if the wire-mesh is not in thermal contact with the heat exchanging surface, it could affect the velocity field near the surface and the thermal resistance would be modified. However, in this case, the heat transfer area A is exactly the same as the heat transfer area of the reference configuration. It is difficult to modify A without affecting the neighbouring velocity and temperature variations. These variations affect the surface temperature and/or the temperature

108

4 Forced Convection Through Porous Medium

variation at the surface, which in turn affect h. Hence, to determine the h-variation by varying A requires, the fluid velocity and temperature fields must be known, which are obtained by solving the momentum and energy balance equations. Three basic forced convection configurations that utilize a porous medium insert for heat transfer enhancement are shown in Fig. 4.4. Tong et al. (1993) studied using numerical simulations, the enhancement in the configuration (A) of Fig. 4.4. The width s of the porous insert placed symmetrically at the centre of the channel is varied. The flow was assumed steady, laminar and hydrodynamically fully developed (unidirectional). Accordingly, they simplified the flow equation, Eq. (3.24) as the differential HDD model in 1D, similar to Eq. (3.9). The porous medium drag terms are zero only in the gap-region near the channel surfaces occupied by fluid when the porous insert does not fill completely the channel, i.e. s < H . This configuration is referred to as the partial-fill configuration, in contrast to the full-fill configuration when s = H . Taking the stress jump at the porous fluid interface as unity (see Eq. 3.52), the boundary conditions u = u and d u /dy = du/dy are imposed at the porous fluid interface (as seen earlier, angle brackets indicate volume averaged quantities).

y

(A)

isothermal or isoflux parallel plates or pipe surface

(B)

r

H

H

s

s

PM-fluid interface

x

-s

x

-H

PM-fluid interface

z x

porous insert

(C) x

W y

isoflux or isothermal solid wall porous insert

L

D

L cooling fluid flow

Fig. 4.4 Basic forced configurations that utilize porous inserts for heat transfer enhancement: a Partially filled porous medium modifying only the flow, b Partially filled porous medium in thermal contact with the heat exchanging wall, c Porous medium enhanced cold plate; side view is a channel filled with porous medium

4.4 Heat Transfer Enhancement Aspects

109

The energy equation used is   ∂ T ∂ 2 T , ρc p f u = ke ∂x ∂ y2

(4.55)

for the porous region, and similarly for the fluid region with ke replaced by k f . No distinction is made between the porous-continuum quantities of Eq. (4.55) and their fluid-continuum counterparts. The porous fluid interface boundary conditions used are: T = T and k (δT /δy) = ke (δ T /δy). The Nusselt number definition used in their work is  ⎞ ⎛  −H ∂∂Ty  he H y=H ⎟ ⎜ (4.56) =κ⎝ N um = ⎠. kf (Tw − Tb ) The parameter κ is equal to unity for the partial-fill configuration (i.e. when s < H , Fig. 4.4, (A)). Otherwise, when the porous insert fills the entire channel, i.e. s = H , then κ = ke /k f . The N u definition of Eq. (4.56) is a modified version of the one introduced in Eq. (4.36). When s is zero (no porous insert), then the surface-averaged Nusselt number becomes equal to 1.885, as well documented (see p. 298, Bejan (1993)) Tong et al. (1993) found conditions under which N u m could be maximized by filling the channel partially with the porous insert. This enhancement is the result of the extra hydrodynamic drag imposed by the porous medium, which forces the fluid to flow faster through the unimpeded channel region, i.e. the gap-region next to the isothermal channel surfaces. However, when filling the entire channel (s = H ), the porous insert is assumed to become a direct participant of the heat transfer process, with zero thermal contact resistance between the porous medium and the surfaces of the channel. Therefore, a porous insert with high effective thermal conductivity ke could yield better heat transfer coefficient when in contact with the channel surface than when partially filling the channel. When F = 0, Da = K /H 2 = 1 × 10−3 and κ < 2.5, N u m can be maximized using the partial-fill configuration (the value of s which optimizes N u m in this case varies almost linearly from 0.7 to 0.5 H as the value of decreases from 2.5 to 0.1). If κ > 2.5, the full-fill configuration yields the highest N u m . The penalty for enhancing the heat transfer is a higher pressure-drop. The case (B) of Fig. 4.4, i.e. placing a porous insert in the middle of a channel, without contacting the surfaces of the channel, would seem an inefficient heat transfer enhancing method. The increased fluid speed near the channel surface, responsible for enhancing the heat transfer in this case, can be obtained more effectively (with a smaller pressure-drop) by simply increasing the flow rate of the clear (no insert) channel. In practice, however, this might not be true because a porous insert can induce flow mixing inside the channel at a laminar flow rate, therefore, yielding a

110

4 Forced Convection Through Porous Medium

better heat transfer coefficient without the high pressure-drop penalty characteristic of an equivalent turbulent flow case in a clear channel. The case (C) of Fig. 4.4 depicts the conventional porous medium channel flow that is utilized in several forced convection configurations including the cold plates involved in the thermal management of electronics in airborne military avionics, as reported in Lage et al. (1996) and reviewed in Lage et al. (2004). The porous insert has two vital functions for this particular application: (1) to enhance the convection heat transfer mechanism and (2) to induce a uniform flow along the x-direction, leading to a uniform transverse temperature along the y-direction. The penalty for the more uniform flow is an increase of the pressure-drop from inlet to outlet of the channel, when compared against the results obtained with a plainchannel configuration. Lage et al. (1996) also found that the aluminium porous insert always enhances the cooling of the channel by reducing the maximum temperature and reducing the maximum transversal temperature difference. The identification of a suitable porous medium insert, however, is a separate exercise. For instance, based on the manufacturing method employed, same combinations of two porous material can yield different hydraulic (K and C) and thermal properties (ke ), when subjected to mechanical compression that alters their porosity and pore structure. One specific experimental study was performed by Antohe et al. (1997). Further details are available in the review by Lage et al. (2004).

4.4.2 Heat Exchangers as Porous Media It is beneficial in engineering to use porous medium analysis to understand the behaviour of any complex permeable structure. This was illustrated by Bejan and Morega (1993) by considering the cross-flow through arrays of staggered cylinders, an arrangement analysed conventionally as a porous medium, using local parameters (tube diameter, pitch, etc.). A porous medium approach often reduces the complexity by lumping local parameter variations into the porous medium hydraulic (K and C) and effective (ke ) properties. Invoking the volume averaging approach may also prove cost effective. Compact heat exchangers (CHX), with high surface-to-volume ratio α > 700 m2 / 3 m and D M ≤ 6 mm, are the best candidates for the porous medium treatment. These CHX are useful in engineering applications such as air-conditioning systems, airborne power plants and cooling applications. The hydrodynamic parameter of chief interest is the overall pressure-drop across the compact heat exchangers, as pressuredrop penalty paid for an increase in heat transfer coefficient in heat exchangers can be quite high. Next are the ‘near-compact’ heat exchangers with α ∼ 100–300 m2 /m3 and D M > 6 mm, which also find use as heat exchangers for low-grade energy, to cool nuclear reactors following a loss-of-core cooling accident, the decay of heat removal from spent fuel storage tanks and steam generators in nuclear plants. The advantage of these near-compact heat exchangers (NCHX) is the smaller overall pressure-drop.

4.4 Heat Transfer Enhancement Aspects

111

The experimental results of pressure-drop and heat transfer coefficient for several of the NCHX configurations were interpreted in Wilson et al. (2004) using the porous medium approach. The tube bank geometries were correlated to the permeability K and form coefficient C that contributed to the drag to be overcome by the incurred pressure-drop across the heat exchanger. Treating rod bundle models as a porous medium, the ‘viscous drag’ of the rod bundles was found to be a fixed value. For channel hydraulic diameter-based Re > 2300, in turbulent flow, the effects of the geometry of the rod bundles as a form resistance (through C) of the porous medium predominate the viscous resistance (caused by K ).

4.4.3 Porous Medium Inter-connectors Raju and Narasimhan (2007) investigated the heat transfer enhancement using porous medium tube-to-tube inter-connectors in cross-flow heat exchangers, which were in turn treated as porous media. Since the pressure-drop and heat transfer results of the overall system treated as a porous medium, which in turn was shown to be affected by the presence of local porous medium inter-connectors, this is an example of the bi-disperse porous medium (BDPM) concept presented in Nield and Bejan (2006). We introduced BDPM in Chap. 1 and also devote a separate section in Chap. 8 to discuss this advancement in porous medium analysis. The configuration studied by Raju and Narasimhan (2007) is shown in Fig. 4.5. Part (A) of the figure shows the typical forced convection configuration of the crossflow, NCHX with aligned tubes, while part (B) shows the tubes interconnected by partially filled porous medium. The porous medium inter-connectors are employed only in the longitudinal direction. The thickness (δ) of porous medium shown in Fig. 4.5 is equal to the diameter of the tube (D), although the reported research considers this as a parameter of study. The other geometrical configurations considered are the longitudinal pitch-to-diameter ratio (X L = SL /D), the transverse pitch-to-diameter ratio (X T = ST /D) and the number of tubes or cylinders in the direction of flow N . The fluid transport in the porous medium inter-connectors is governed by the Brinkman-Darcy flow model (similar to Eq. (3.19)), while Eq. (4.3) is used to model energy transport. The appropriate equations are given below. Continuity: ∂v ∂u + = 0. (4.57) ∂x ∂y X-Momentum Equation:     2 ∂u ∂u ∂p μ ∂ u ∂ 2u ρ u +v =− + λ1 μ + 2 + λ2 u. 2 ∂x ∂y ∂x ∂x ∂y K

(4.58)

112

4 Forced Convection Through Porous Medium

Fig. 4.5 Heat exchangers as porous media: a Cross-flow tube bank, treated as a porous medium, b Cross-flow tube bank with tube-to-tube porous medium inter-connectors. Isotherms are shown in the range 400 ◦ C (tube wall temperature) and 30 ◦ C (inlet flow temperature)

Y-Momentum Equation:    2  ∂v ∂p μ ∂ v ∂ 2v ∂v +v =− + λ1 μ + 2 + λ2 v. ρ u 2 ∂x ∂y ∂y ∂x ∂y K

(4.59)

Energy Equation:   2   ∂T ∂ T ∂T ∂2T . +v = ke ρc P u + ∂x ∂y ∂x2 ∂ y2

(4.60)

To solve the above equations, an important requirement is the set of boundary conditions to be applied on the interface between the porous medium and the clear fluid flow (as shown in Fig. 4.5). In order to accommodate a possible interface discontinuity in the diffusion flux of momentum and heat, respectively, the appropriate boundary condition at the interface between a porous medium and clear fluid flow section has been the point of interest of many researchers. In recent reviews, the merits and demerits of such interface boundary conditions and their validity are dis-

4.4 Heat Transfer Enhancement Aspects

113

cussed in detail by Alazmi and Vafai (2000) and Alazmi and Vafai (2001). Refer also to the brief discussion in Sect. 3.7.5 in Chap. 3, on the hydraulic interface conditions. The interface conditions used in this context were those proposed by Ochoa-Tapia and Whitaker (1995) for partially filled porous medium configurations, namely u f |C F = u f | P M , μeff

du f du f μf |P M − μ f |C F = β u f |interface , dy dy Da T |C F = T | P M , ke

dT dT |P M = k f |C F . dY dY

(4.61) (4.62) (4.63) (4.64)

In Eq. (4.62), β is an adjustable coefficient of the order one, which depends on the porous medium properties and, hence, has to be predetermined from individual experiments. However, experimentally determined β values for many practical porous media are not available, including for the metal foam porous media (K ∼ 10−7 ) used in the above study. For these cases, β is set to zero, for want of a better jump condition value, as proposed in Ochoa-Tapia and Whitaker (1995). As a result of β = 0, when Eq. (4.62) is used as interface boundary condition, Eq. (4.58) and Eq. (4.59) retain their Brinkman extended Darcy differential formulation of the momentum statement inside the porous layer, allowing for continuity of velocity and shear stress at the fluid-porous layer interface and a no-slip condition on the solid wall. A similar explanation is consistent with the use of Eq. (4.63) and Eq. (4.64) as interface boundary conditions for solving the volume averaged energy equation, Eq. (4.60). The above system of equations with the appropriate boundary and interface conditions was solved using numerical simulations by Raju and Narasimhan (2007). The tube walls are kept at constant temperature and exchanging heat with the cooling fluid having Pr = 0.7 under laminar flow (10 < Re < 100). The non-dimensional pressure-drop ξ is defined as ξ= and the N u as Nu f =

p/L , 2 /2D ρU∞

qD . A(Tw − Tb )k f N

(4.65)

(4.66)

To develop a correlation for the overall pressure-drop (ξ ) and N u, the NCHX is modelled as a ‘global’ porous medium with the inter-connector porous medium identified as a ‘local’ PM, whose properties (φ , K i ) influence the ‘global’ PM properties (K g ), and hence the ‘global’ heat transfer enhancement. To correlate the

114

4 Forced Convection Through Porous Medium

non-dimensional pressure-drop, a ‘global’ Darcy number Dag has been introduced and defined as K g0.5 Dag = , (4.67) D where K g , m 2 is the global permeability of the heat exchanger computed from Darcy law. Raju and Narasimhan (2007) gave a general correlation for Dag as  Dag = C1

 2δ Dai + C2 (1 − δ) . (δ + 1)

(4.68)

In the above equation, C1 and C2 are constants which subsume the effect of porosity (φ) and other geometrical parameters (D, ST , SL ) of the heat exchanger that affect the overall longitudinal pressure-drop. The correlated Dag Eq. (4.68) predicts data obtained from the numerical simulations within ±18% accuracy. The corresponding pressure-drop across the NCHX treated as porous media can be expressed in non-dimensional form as ξ = 2Re−1 Dag−2 ,

(4.69)

a form similar to that used in earlier studies such as Mohamad (2003) and Wilson et al. (2004), when the form-drag component of the porous medium model is neglected. The thermohydraulics of the global porous media (NCHX) are characterized by studying the effect of transverse thickness (δ in Fig. 4.5) and permeability (represented by Dai ) of the local metal foam type porous medium inter-connectors on the global heat transfer coefficient (N u) and non-dimensional pressure-drop (ξ ). The N u results were related to the effective properties of the heat exchanger geometry. Figure 4.6 shows the effect of the porous medium inter-connector thickness on the overall N u of the NCHX configuration. The overall NCHX N u decreases with an increase in the local interconnecting porous medium permeability (represented as Dai ), for all partially filled porous medium inter-connector (0 < α < 0.8) for all the Re tested. No significant effect of Dai was observed in the range of 10−8 < K i < 10−10 m2 , on N u, the overall heat transfer, when the porous medium inter-connector entirely fills the region around the tube bank (i.e. when α = 1). Irrespective of the value of Dai , when α = 1, the variation of N u with Re is found to be linear. These results are summarized in the correlation (4.70) N u = 0.1Re(1−0.5Dag ) + Dag . As Dag = DaC F when δ = 0 and Dag → Dai when δ = 1. In this limit, the above correlation Eq. (4.70) predicts results consistent with existing literature, as reported in Nield and Bejan (2006). The effect of spatially variable permeability of the porous medium tube-to-tube inter-connectors on the thermohydraulics of a NCHX configuration similar as above

4.4 Heat Transfer Enhancement Aspects

115

12 PM interconnector thickness

Nu

1.00

10

0.80

8 0.60

6

0.45

0.20

4

0

2

0

0

20

40

60

80

100

Re

120

Fig. 4.6 Heat exchangers as porous media: effect of porous medium inter-connector thickness on the heat transfer N u of the NCHX configuration

is reported in Raju and Narasimhan (2007). Owing to the complex local variations affecting the form or shape of the NCHX treated as the global porous medium, the analysis is presented apart from a global permeability K g , also through a global form coefficient C g . Both these parameters were influenced by the local tube-to-tube interconnector porous medium, characterized with a local permeability, K i , as already defined. The range of K i tested was between 10−10 and 10−7 m2 with a porosity variation of 0.58 < φ < 0.95. The corresponding effective thermal conductivity varied as 85.69 < ke < 10.22. In this case, the non-dimensional pressure-drop ξ is defined as before. It can be correlated to the linear and nonlinear terms as ξ=

E + F, Re

(4.71)

where E and F capture the effects of the global viscous and form drag of the global porous medium and hence include the variation in K g and C g . In other words, Eq. (4.71) can be considered as the non-dimensional equivalent of the global HDD model, Eq. (3.8), with the coefficients suitably defined as

116

4 Forced Convection Through Porous Medium

E=

2 , F = 2Du g , Dag

(4.72)

where Dag and Du g are the global non-dimensional permeability and form coefficient of the NCHX, when perceived as a global porous medium. They are defined as K g2 , Du g = C g D. Dag = (4.73) D A relationship can be sought between the global Dag and Du g [(i.e. Kg and C g through Eq. (4.73)] and the rest of the NCHX parameters such as the local K i and φi of the porous medium inter-connectors and the NCHX geometry (ST , SL , D) using which, the values for E and F in Eq. (4.72) can be evaluated. Substituting these in Eq. (4.71) would then yield a predictive correlation for the non-dimensional pressure-drop. Raju and Narasimhan (2007) report the global Darcy number Dag in Eq. (4.73) correlated to the PM inter-connector ‘local’ permeability in the following format Dag =

1 D



0.5 1  [1.07(A + B)K i ] , N

(4.74)

where A and B are given as 1 SL A= 275φ 2 D B=



 1 − (δ/D) , 1 + (δ/D)

  δ (H/D − 1)2 . D (1.73φ − 1)

(4.75)

(4.76)

The global Du g correlation reads  Du g = D

1.95φ − 0.067(δ/D) (H/D) − 0.067(δ/D)



1  −0.5 Ki , N

(4.77)

where the numerical coefficients subsume the effect of geometric parameters (ST , SL ). Combining Eqs. (4.74) and (4.77) with Eqs. (4.72) and (4.73) and substituting them in Eq. (4.71) result in the non-dimensional pressure-drop taking the form ξ 1 = + Du g , 2 0.5ReDag2 a form similar to that used in earlier studies, Wilson et al. (2006).

(4.78)

4.4 Heat Transfer Enhancement Aspects

117

For the chosen NCHX configuration with the porous medium inter-connectors made of aluminium metal foam having ks = 202 W/m.K, the overall N u in Fig. 4.6 is correlated with Re and Du g , as N u = 0.1Re0.73 + 1.5Du 0.2 g ; 20 < Re < 100.

(4.79)

The effectiveness of porous medium inter-connectors within heat exchangers was experimentally verified using air and water as coolant flows through in-line and staggered arrangement of heat exchanger tube banks in Ramana et al. (2010) and Ramana et al. (2012). The in-line configuration with the porous medium inter-connectors incurred less pressure-drop compared to the plain in-line configuration. This reduction was attributed to the absence of the pressure-drop penalty originally caused by the recirculation and vortices behind individual tubes in the in-line configuration without the porous medium. There was a maximum of 18% pressure-drop reduction observed for the range of flow considered. For staggered arrangement, this advantage does not exist, and with the porous medium inter-connector in place, the pressure-drop was systematically higher than its corresponding plain staggered configuration. The heat transfer enhancement due to the presence of the porous medium interconnectors was only marginal in the in-line configuration, while for the staggered case, a seventy per cent increase was observed.

4.5 Other Forced Convection Configurations Numerous other convection configurations in porous media have been analysed in detail in the exhaustive monograph by Nield and Bejan (2006). For instance, forced convection heat transfer results for cross-flow over a sphere and cylinder embedded in the porous matrix are given in literature as 1/2

(4.80)

1/2

(4.81)

N u D = 1.128Pe D , N u D = 1.015Pe D .

A list of such useful correlations is presented in Appendix II. Some additional forced convection heat transfer results for porous medium flows are available in the recent review compilations by Vadász (2008) and Vafai (2011).

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4 Forced Convection Through Porous Medium

4.6 Viscous Dissipation Effects The viscous dissipation effect is an irreversibility that must be accounted for in the first law of thermodynamics formulation of the energy balance statement for any thermodynamic system that involves flow. It is understood as a local production of thermal energy through the mechanism of viscous stresses. Naturally, it is also present in convection flows through porous media. In the absence of flow, viscous stresses are absent; hence, there is no dissipation of energy by this mechanism. Hence, it neither appears in the energy equation Eq. (2.19) nor appears in the irreversibilities accounted in the S˙  term in Eq. (2.26) for heat conduction through stagnant porous medium in Chap. 2. Viscous dissipation effects in convection have been discussed in detail in the excellent review by Magyari et al. (2005). In this section, we shall introduce the basic equations involved. In a convection situation, viscous dissipation is ever-present and is represented by the symbol , as can be seen in Eqs. (4.1) and (4.3), the respective energy equations valid for the fluid flow inside a porous medium, when LTNE and LTE conditions are assumed. For a plane boundary layer flow or a unidirectional flow,  is written for clear fluids and for Darcy flow through a porous medium, as 

clear

∂u ≡μ ∂y

2 and Darcy ≡

μ− → u 2. K

(4.82)

In Eq. (4.82), μ is the dynamic viscosity and K is the permeability. Importantly, → the − u signifies that the velocity is a volume averaged quantity defined on a porous continuum. Within a porous medium, the fluid is ‘extruded’ through the pores of the solid matrix. This renders the local flows as 3D even as the overall macroscopic flow is uniform and unidirectional (in a volume averaged sense, as explained in Chap. 3). Therefore, due to the omniscient presence of the viscous stresses at the pore level, the rate of heat generation due to viscous dissipation in porous medium flows is important. By contrast, for uniform forced convection flows in clear fluids (u=const. ≡ u ∞ ), no heat is generated by viscous dissipation, due to internal frictional forces. In porous media, the heat generation rate increases quadratically with u. The nature of the terms involved in the model for  depends on the momentum equation chosen to model the flow in a porous medium convection configuration. When the momentum balance in a porous medium flow is represented by a differential form of Darcy’s law, Eq. (3.17), the viscous dissipation can be written as =

 μ 2 u + v 2 + w2 . K

(4.83)

This form should be used for isotropic media and is independent of the orientation of the coordinate axes, as pointed out in Nield (2000a). When the momentum balance in a porous medium flow is represented by the HDD model, Eq. (3.9), viscous dissipation can be written as

4.4 Heat Transfer Enhancement Aspects

=

119

cfρ μ− → → u ·− u + 1/2 K K

→ − − → u ·→ u. u −

(4.84)

The paradox that a term that is independent of the viscosity may contribute to the viscous dissipation was resolved in Nield (2000). Under such conditions, the advective inertia terms in the Navier–Stokes equations are not negligible. Wake formation and boundary layer separation occur at pore-particle length scales in a porous medium. Hence, pore-level (u  ) velocities are altered, generating heat by viscous dissipation. In this context, the discussion of the nature of form drag and the influence of viscosity on it, provided in Sect. 3.2, may be revisited. When the momentum balance in a porous medium flow is represented by the Brinkman model, Eq. (3.19), the viscous dissipation can be written as =

μ− → → → → u ·− u − μ− u · ∇2− u, K

(4.85)

where  μ is an effective viscosity. Al-Hadhrami et al. (2003) using the work done by frictional forces presented an alternate model, as  μ 2 u + v 2 + w2 K   2  2    ∂u ∂v ∂w 2 +2 μ + + ∂x ∂y ∂z  2      ∂u ∂u ∂v ∂v ∂w 2 ∂w 2 + + + μ . + + + ∂y ∂x ∂z ∂x ∂z ∂y

 =

(4.86)

When Brinkman model, Eq. (3.19), governs the flow, both these formulae yield the correct form for  in the limit of small permeability. When φ → 1, only Eq. (4.86) matches the  for a clear fluid, Eq. (4.82). Nield (2000b) presented an analysis to understand when viscous dissipation would be significant in porous medium flows, by comparing the orders of magnitude of the dissipation terms with the thermal diffusion terms in the energy equation. Using U ,L and T to denote representative values of velocity, length and temperature drop, the orders of magnitude of the thermal diffusion and viscous dissipation terms in the energy equation, Eq. (4.3) are, respectively, k T L2

and

μU 2 . K

(4.87)

Mixed and forced convection flows present a velocity scale. Therefore, the viscous dissipation effects are negligible when 

μU 2 k T



L2 Br =  1. K Da

(4.88)

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4 Forced Convection Through Porous Medium

Here, Br and Da are the Brinkman and Darcy numbers where the Brinkman number is the term in brackets in Eq. (4.88). Further discussions on viscous dissipation models are available in Al-Hadhrami et al. (2003), Nield (2000a, b) and in the review by Magyari et al. (2005).

4.7 Problems (1) Consider a fluid-saturated porous medium with u = C x n near a solid impermeable wall of temperature T0 = T∞ + Ax b . The temperature T∞ is the porous medium temperature sufficiently far away from the wall outside the thermal boundary layer. (i) Starting from suitable governing equations, prove that a similarity temperature profile exists if n = b. (ii) Derive the similarity form of the boundary layer energy equation when n = b = 0 and find its solution. (Hint: use the error function notation). (2) From second law of thermodynamics, one can write the expression for the entropy generation rate in a convection configuration as Sgen =

 qi   ∂ SC V − − (ms) ˙ + (ms) ˙ ≥0 ∂t Ti inlet ports outlet ports

Derive the local entropy generation rate formula for a homogeneous porous medium by applying the above equation to the one-dimensional convection model used to derive the volume averaged energy equation. (Hint: integrate Sgen expression for solid and fluid over respective volumes and then average over total volume; use when necessary, the mass and energy conservation equations, relation between entropy, enthalpy and internal energy). (3) Air at a pressure of 6 kN/m2 and a temperature of 300 ◦ C flows at a velocity of 10 m/s over a flat plate at 27 ◦ C. What is the local and plate length average heat transfer coefficient and heat removal rate for this configuration, when the region over the plate is (a) clear (of any porous medium) and (b) with a porous medium (brazed to the plate) made of aluminium mesh with porosity φ = 0.3. (Hint: identify and use suitable N u correlations). (4) Consider a temperature-dependent viscosity flow through a parallel plate channel (of length L, plates separated by gap 2H ), embedded with a homogeneous low-permeability porous medium. Uniform heat flux q  (W/m2 ) crosses into the channel from the top and bottom walls. (a) The local and channel length averaged Nusselt numbers for the above configuration are given as Nu = and

2H q  (1) ke [Tw (x) − Tb (x)]

4.6 Viscous Dissipation Effects

121

NuL =

2H q  (2) ke [T¯w − Tb (in)]

where T¯w is the average channel wall temperature (averaged over L); Tb (x) is the bulk temperature. Assuming fully developed flow, find a general expression relating N u L and N u by applying the first law of thermodynamics for the convection configuration, neglecting longitudinal diffusion. Take λ = DC /Dμ , the ratio of the form and viscous drags and Pr = ν/αe , the Prandtl number. State clearly, all other assumption made. (Hint: start with the N u equation above and apply first law to find an expression for T¯w .) (b) Draw the schematic longitudinal pressure-drop (y) versus average velocity (x) graph for this configuration, when the heat flux supplied at the top and bottom walls progressively increase from 0 to 100 MW/m2 in steps of 50 MW/m2 assuming strong temperature dependency of viscosity. Draw also the corresponding local velocity profile u(y) schematic at inlet (x = 0), midsection (x = L/2) and exit (x = L) of the channel. (5) A porous medium formed by a packed bed of spheres is introduced in a gas preheater of diameter 5 m. The average diameter of the packing spheres is d = 5 cm. The length of the channel is equal to 10 m. Calculate the heat transfer augmentation achieved. Hint: use φ j¯H = 2.06Re−0.575 D St Pr 2/3 = j H (6) On a cold day, a person wears a woollen shirt of thickness 4 mm. Assuming the skin temperature is to be maintained at 30 ◦ C and the heat rejection from the body to be 37.4 W/m2 , find the minimum ambient temperature upto which the person can venture out safely on calm day. Make suitable assumptions. (7) Consider a transdermal drug delivery system as shown in the figure. Drug is being injected with an average velocity U from the patch through a microneedle array. Treat the tissue + blood as a homogeneous porous medium. From first principles, determine the position of moving interface of drug in tissue as a function of time. Neglect the effect of mass diffusion from the moving interface.

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4 Forced Convection Through Porous Medium

Hint: Consult the reference, Lv, Y., Liu, J., Gao,Y. and Xu, B., 2006, Modelling of transdermal drug delivery with a micro-needle array. Journal of Micromechanics and Microengineering 16, 2492. (8) In the previous question, consider a ‘local concentration non-equilibrium’ between solid and liquid (blood) parts of the tissue. Develop governing equations for mass transport in valid in the tissue porous medium and discuss the pertinence of each term. (9) Consider a flow through a channel filled with a homogeneous porous medium, which is governed by the Brinkman-Darcy model. Constant heat flux q  is provided from the top and bottom bounding walls. Let the solution for the momentum equation be of the form U α(cosh α − cosh αη) = Uα α cosh α − sinh α h where η = hy and α = √(k/ε) . Determine the solution for the velocity and temperature profiles. Hint: consult the reference, 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’. International journal of heat and mass transfer 23, 291–295.) (10) Consider a flow, governed by the HDD model, through a porous medium bounded by an isothermal, hot, flat plate. Let the temperature variation of viscosity for the fluid be given by μ1 = μ1∞ [1 + γ (T − T∞ )]. Derive the solution for the velocity and temperature profiles. Hint: consult the reference, Seddeek, M. (2005), ‘Effects of non-Darcian on forced convection heat transfer over a flat plate in a porous medium temperature dependent viscosity’, International communications in heat and mass transfer, 32, 258–265.

Chapter 5

Natural Convection Through Porous Medium

The monograph on convection in porous media by Nield and Bejan (2006) provides elaborate cataloguing and exposition of research in natural convection configurations for several decades. Repeating the analysis is redundant, and hence, only the adopted basic analysis is provided in this chapter. Appendix II provides a list of heat transfer solutions and correlations, with relevant references. For advance subject material, Nield and Bejan (2006) may be consulted.

5.1 Natural Convection Boundary Layers 5.1.1 Vertical Wall The heat transfer between a vertical heated surface and a fluid-saturated semi-infinite porous reservoir is one of the basic problems studied in detail in porous medium literature. We present the simplest boundary layer model for natural convection in porous media, first published by Cheng and Minkowycz (1977). An account of this analysis is provided in the exposition by Bejan (2004), which we adopt here. The gravitational acceleration g is oriented in the negative y direction. Assuming Darcy’s law to govern the momentum in the vertical natural convection flow in Fig. 5.1, the governing equations for the system are: Momentum:   K ∂P K ∂P ,v=− + ρg , (5.1) u=− μ ∂x μ ∂y where, the Oberbeck–Boussinesq approximation ρ  ρ0 [1 − β(T − T0 )],

© The Author(s) 2023 A. Narasimhan, Essentials of Heat and Fluid Flow in Porous Media, https://doi.org/10.1007/978-3-030-99865-3_5

(5.2)

123

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5 Natural Convection Through Porous Medium

g

y

vertical velocity profile

impermeable wall temperature profile

T

porous medium

x

unheated length

Fig. 5.1 Natural convection along a heated vertical wall in a porous medium (adapted from Bejan 2004)

with β = (1/ρ∞ )(∂ρ/∂ T ) P , is invoked to write the ρ in terms of the local T . Also, the gravity vector is pointing in the negative y direction. Energy balance, assuming LTE to hold: u

 2  ∂ T ∂T ∂2T ∂T , +v = αe + ∂x ∂y ∂x2 ∂ y2

(5.3)

where αe = ke /(ρc P ) f is the effective thermal diffusivity of the porous medium. Following standard boundary layer analysis, P can be eliminated by crossmultiplying to yield a single momentum equation, ∂v K g ∂ρ ∂u − = . ∂y ∂x μ ∂x

(5.4)

Satisfying the mass continuity equation, the stream function ψ is defined as u=

∂ψ ∂ψ ,v = − . ∂y ∂x

(5.5)

Using ψ, the momentum equation Eq. (5.4) becomes ∂ 2ψ ∂ 2ψ K g ∂ρ . + = 2 2 ∂x ∂y μ ∂x

(5.6)

5.1 Natural Convection Boundary Layers

125

Taking the boundary layer of Fig. 5.1 as slender, and combining the momentum and energy equations by substituting for T in the expression for ρ, the system of equations reduces to K gβ ∂ T ∂ 2ψ =− , (5.7) 2 ∂x ν ∂x ∂ψ ∂ T ∂2T ∂ψ ∂ T − =α 2. ∂y ∂x ∂x ∂y ∂x

(5.8)

Case 1: Uniform Wall Temperature Let the temperature of the vertical impermeable wall be uniform, T0 , to yield the boundary conditions: T = T0 , ψ = 0 at x = 0 T → T∞ , ∂ψ → 0 as x → ∞. ∂x

(5.9)

Using scale analysis, from Eqs. (5.7) and (5.8), one could write the scales ψ K gβ T ∼ , 2 ν δT δT

(5.10)

ψ T T ∼α 2 , yδT δT

(5.11)

where δT is the boundary layer thickness (the x scale) and T = T0 − T∞ . Combining expressions (5.10) and (5.11) yields δT ∼ Ra −1/2 , ψ → α Ra 1/2 y y , y Nuy = h

y y ∼ ∼ Ra 1/2 y , k δ

(5.12) (5.13)

where the heat transfer coefficient scales as h ∼ q  / T ∼ k/δT , and Ra y is the Darcy-modified Rayleigh number Ra y =

K gβy T . αe υ

(5.14)

The similarity formulation of the isothermal vertical wall problem yields a rigorous analytical solution. As before (see for another instance in Sect. 4.2), the similarity variable is first defined, as, x η = Ra 1/2 (5.15) y . y

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5 Natural Convection Through Porous Medium

Using the similarity profiles T − T∞ ψ = f (η) = θ (η) , T αe Ra 1/2 0 − T∞ y

(5.16)

the problem statement is recast as f  = −θ  ,

(5.17)

f θ  = 2θ  ,

(5.18)

θ (0) = 1, f (θ ) = 0, θ (∞) → 0, f  (∞) → 0.

(5.19)

Cheng and Minkowycz (1977) performed numerical integration that resulted in Nuy =

y q  = 0.444Ra 1/2 y . T0 − T∞ ke

(5.20)

The average value (over a wall of height H ) is Nu = h

H 1/2 = 0.888Ra H . ke

(5.21)

The scale analysis results compare well with the above within an order. Case 2: Uniform Wall Heat Flux When the vertical wall is maintained as a constant isoflux surface, following an analysis similar to above, the heat transfer results are obtained as follows: Defining the Rayleigh number as K gβy 2 q  , αe νke

(5.22)

y q  = 0.772Ra 1/3 ∗y , T0 (y) − T∞ ke

(5.23)

Ra ∗y = the local wall heat transfer is given by Nuy =

and the average heat transfer is Nu =

q  T 0 − T∞

H 1/3 = 1.03Ra∗H . ke

(5.24)

5.1 Natural Convection Boundary Layers

127

5.1.2 Horizontal Wall Another basic natural convection configuration is the one that happens adjacent to a horizontal wall bounding a saturated porous medium. A similar procedure as above, with minor variations, can be applied to analyse this situation, which is explained in detail in Nield and Bejan (2006). The interesting result is that the corresponding N u varies as Ra 1/2 , unlike the above vertical wall case, where N u is seen to vary as Ra 1/3 . A collection of heat transfer solutions and correlations is provided in the Appendix II.

5.2 Natural Convection with Vertical Thermal Gradient 5.2.1 Onset of Natural Convection Another well-investigated configuration is the natural convection heat transfer in a saturated homogeneous porous layer heated from below. The general version of the problem would be the instability of a semi-infinite saturated porous layer subjected to a vertical differential heating, called the Horton–Rogers–Lapwood convection, the equivalent of the classical Rayleigh–Benard convection in fluid flow without porous medium. The complete derivation and discussion of the linear stability analysis is available in Sect. 6.1 of Nield and Bejan (2006). A basic understanding of the analysis and the results are provided here. Consider a saturated porous layer placed in a Cartesian reference between z = 0 and z = H along the vertical direction and extending from x = y = 0 along the other two directions. The governing equations of the situation are: → ∇ ·− u = 0, μ− → → u + ρf− g, K

(5.26)

∂T − +→ u · ∇T = αe ∇ 2 T. ∂t

(5.27)

0 = −∇ P − σ

(5.25)

Observe that the momentum equation is treated steady as the LHS is set to zero. The → transient momentum term cs ρ0 ∂ − u /∂t −→ 0 as K /H 2 → 0, which is valid for most practical porous media with low permeability values. See Sect. 6.1 of Nield and Bejan → (2006), for a discussion on the transient term of the present problem. The − u , P, T are volume averaged quantities, with σ = (ρc P )e /(ρc P ) f and αe = ke /(ρc P ) f . Finally, the Boussinesq approximation, Eq. (5.2), is invoked to rewrite ρ f in terms of the local temperature variation.

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5 Natural Convection Through Porous Medium

Let this porous layer be heated from below with the boundary conditions T = T0 + T at z = 0; T = T0 at z = H.

(5.28)

Seeking to know of the stability of the porous layer and its heat transfer, subject to the above boundary conditions, it can be readily seen that there is a steady-state solution available for Eqs. (5.25) to (5.27) that satisfies the boundary conditions Eq. (5.28) as follows: − → (5.29) u b = 0,  z Tb = T0 + T 1 − , H  2   z 1 Pb = P0 − ρ0 g z + β T − 2z , 2 H

(5.30)

(5.31)

where the subscript b signifies the baseline case. The above solution signifies the heat conduction regime of the considered porous layer, heated from below. The question that could be asked now is, at what higher temperature, or T value, is the above conduction regime forever ‘disturbed’ to a convection regime. This is answered in linear stability theory through a perturbation analysis. The original governing equations of the system, Eqs. (5.25) to (5.27), are subjected to a small ‘disturbance’ or perturbation and the disturbed mode is checked for a permanent instability that grows with the perturbation parameter. The variables are written as follows: → − → → u  , P = Pb + P  , T = Tb + T  . u =− u b+−

(5.32)

Substituting these into Eqs. (5.25) to (5.27) and neglecting the second order variations, the governing equations can be recast in terms of the perturbation variables as follows: → (5.33) ∇ ·− u  = 0, 0 = −∇ P  − σ

μ− → → u  − βρ0 T  − g, K

∂T  + T H w  = αe ∇ 2 T  , ∂t

(5.34)

(5.35)

→ where, − u  = (u  , v  , w  ). By defining the non-dimensional variables X=

− → αe t x u H ∗ ; τ= ; u = , H σ H2 αe T K P T∗ = , ; P∗ = T μαe

(5.36)

5.2 Natural Convection with Vertical Thermal Gradient

129

the perturbation equations, Eqs. (5.33)–(5.35), can be recast in the non-dimensional form as (5.37) ∇ · u ∗ = 0, ˆ 0 = −∇ P ∗ − u ∗ + RaT ∗ k,

(5.38)

∂T ∗ − w∗ = ∇ 2 T ∗ , ∂τ

(5.39)

where kˆ is a unit vector along the z direction. The other non-dimensional variables appearing in the above equations are: Ra =

ρ0 gβ K H T , αe μ

Pre =

μ , ρ0 αe

(5.40)

where Ra is the Darcy number times the regular fluid Rayleigh number and Pre is the overall effective Prandtl number for the porous medium. The system of equations Eqs. (5.37)–(5.39) is solved to seek a solution in terms of the critical value of T , i.e. the critical value of Ra in Eq. (5.40) beyond which, instability persists. The solution is given as Ra =

( j 2 π 2 + α 2 )2 , α2

(5.41)

where α is the wave number. When j = 1 and α = π , Ra is a minimum, i.e. the critical Rayleigh number associated with the critical wave number αc = π is Rac = 4π 2 = 39.48.

(5.42)

For higher-order modes, when j = 2, 3, ..., and αcj = jπ , Ra j = 4π 2 j 2 . When Ra < 4π 2 , the porous layer heated from below is subjected to only heat conduction. When Ra  40, the instability of natural convection prevails as a cellular roll. Further details on this onset problem and other variations of the convection in porous layers subjected to vertical thermal gradient are discussed in Chapter 6 of Nield and Bejan (2006). Banu and Rees (2002) studied using a two-energy model, the influence of local thermal non-equilibrium (LTNE) on onset of Darcy–Bènard convection in a horizontal porous medium cavity heated from the bottom. The critical Rayleigh number and wave number were different from the corresponding LTE values due to LTNE effects.

130

5 Natural Convection Through Porous Medium

5.2.2 Experimental Results A wealth of experimental results is also available for this configuration of a porous layer heated from below. Cheng (1978) compiled as many as nine sets of experimental and numerical simulation data set of N u variation with Ra within a range 10 < Ra < 4000. Wang and Bejan (1987) introduced a non-dimensional group Pr p = Pre /Da based on a scale analysis of the non-Darcy effects and used it to suggest a correlation for the experimental data set as follows  Nu =

Ra 40

1/n

n + [c(Ra Pr p )1/2 ]n

,

(5.43)

where the empirical constants n = −1.65 and c = 1896.4. Further details are available in Nield and Bejan (2006).

5.3 Natural Convection with Horizontal Thermal Gradient 5.3.1 Isothermal Side-Wall Heated Configuration A more restricted version of the heated porous layer problem would be to consider the saturated porous medium bounded inside an enclosure of arbitrary cross section, with two basic situations of heating from the bottom wall and heating from the side wall. Of these, heating from the side wall necessitates natural convection more immediately than the heating from the bottom wall. The schematic for the natural convection inside the side-wall heated porous medium enclosure is given in Fig. 5.2. The governing equations are as follows ∂v ∂u + = 0, ∂x ∂y u=− σ

K ∂P , μ ∂x

v=−

K μ

∂T ∂T ∂T +u +v = αe ∂t ∂x ∂y





(5.44)  ∂P + ρg , ∂y

∂2T ∂2T + ∂x2 ∂ y2

(5.45)

 ,

(5.46)

where, as before, u, v, P, T are volume averaged quantities, with σ = (ρc P )e /(ρc P ) f and αe = ke /(ρc P ) f . The momentum equations can be combined as before, by cross multiplying to eliminate P and by invoking the Boussinesq approximation for local ρ variation to be recast as T variation. The resulting single equation would be

5.3 Natural Convection with Horizontal Thermal Gradient

131

Fig. 5.2 Heat transfer inside a porous medium enclosure heated from the side, with the plot on the RHS (adapted from Bejan 1984) showing the four heat transfer regimes

∂v K gβ ∂ T ∂u − =− . ∂y ∂x ν ∂x

(5.47)

The above energy and momentum equations are to be solved subject to the following boundary conditions u = 0, T = T /2 at x = 0 u = 0, T = − T /2 at x = L (5.48) v = 0, ∂∂Ty = 0 at y = 0, H. The scale analysis of this situation, discussed in Bejan (2004), is provided now. From Fig. 5.2, the growth of the thermal boundary layer provides a scale for the local wall heat transfer as ∂T T ∼ . (5.49) ∂x δ From the energy equation, the initial regime of heat transfer near the hot wall can be deduced to be governed by a balance between the thermal inertia and the local diffusion. The scale balance would be σ

∂T ∂2T ∼ αe 2 , ∂t ∂x

(5.50)

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5 Natural Convection Through Porous Medium

resulting in a scale for the thermal boundary layer as  δ∼

αe t σ

1/2 ,

(5.51)

as expected. When sufficient time has passed, one can expect this diffusion dominant heat transfer to change into a buoyancy driven convection dominant heat transfer, at which times, the energy equation is dominated by a balance between thermal inertia and convection terms. For t 0, the energy equation scales as σ

T T ∼v . tf H

(5.52)

From the above, the final time scale, t f , beyond which the convection dominance persists can be found as σH , (5.53) tf ∼ v where v is the unknown velocity scale for the buoyancy driven thermal boundary layer flow. When there is a buoyancy-driven convection flow inside the boundary layer δ with a vertical component of velocity v, it can be expected to have a flow rate vδ(t) scaled as (5.54) vδ(t) ∼ t 1/2 . To find the unknown value of v, both mass and momentum conservation statements have to be utilized. The scale from the steady-state mass continuity would be u/H ∼ v/δ



δ H

2 .

(5.55)

From this, it can be concluded that since the boundary layer is slender, i.e. δ H , u/H v/δ. Invoking this and using the temperature gradient scale obtained earlier in the momentum equation, the unknown velocity scale for v can be obtained as v∼

K gβ T. ν

(5.56)

Substituting this value, the final time scale t f can be estimated. Using the t f scale, since δ ∼ (αe t/σ )(1/2) , the thickness of the boundary layer at this final time, δ f , can be found as −1/2 (5.57) δ ∼ H Ra H ,

5.3 Natural Convection with Horizontal Thermal Gradient

133

where Ra H is defined as K gβ H T , αe ν

Ra H =

(5.58)

the enclosure heated wall-height-based porous medium Rayleigh number (observe the use of K in the definition). Using the scale for δ the scale for ∂ T /∂ x can be estimated. Substituting this in the definition for heat transfer coefficient, its non-dimensional version, N u, for enclosures with H == L can be obtained as Nu ∼

L Ra 1/2 . H

(5.59)

Weber (1975) presented a rigorous solution for high Ra porous medium enclosure convection as L (5.60) N u = 0.577 Ra 1/2 . H Bejan and Tien (1978) updated this and proposed two correlations Nu ∼ 1 +

1 120

N u ∼ 0.508

 Ra H

H L

2 as

H → 0, L

L 1/2 Ra H as Ra H → ∞. H

(5.61)

(5.62)

Both these expressions agree in an order of magnitude sense with the scale analysis result. Based on the size of the enclosure, i.e H × L, Bejan (1984) has enunciated four steady-state heat transfer regimes for understanding natural convection. The pictorial representation of these regimes for porous medium convection is shown in Fig. 5.2. The results of heat transfer in these fours regimes can be obtained as the following scales (I) Pure conduction q  ∼ (ke H T ) /L , ˜ (ke H T ) /L , (II) Tall layers q  > (5.63) (III) High RaH convectionq  ∼ (ke H T ) /δ f ,  ˜ (ke H T ) /δ f , (IV) Shallow layers q < where the scale of δ f is already derived. The detailed scale analysis to obtain the above results is discussed in Sect. 7.1.1 of Nield and Bejan (2006) and is suggested here as an end of chapter exercise. As noted in that reference, a more detailed classification of natural convection regimes in porous layers heated from the side is provided by Blythe et al. (1983).

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5 Natural Convection Through Porous Medium

5.3.2 Isoflux Side-Wall Heated Configuration The heat transfer results for a porous medium enclosure subjected to a constant heat flux are given by   1 L 4/5 2/5 Ra∗H , (5.64) Nu = 2 H where Ra∗H is the Rayleigh number based on heat flux Ra∗H =

gβ K H 2 q  , αe νke

(5.65)

5.3.3 High Ra Results Ramanathan et al. (2011) employed the lattice Boltzmann method (see Sect. 8.5) for a discussion on this method applied to porous medium flows) to simulate steady natural convection at high fluid Rayleigh numbers (106 ≤ Ra ≤ 1010 ) in a square enclosure saturated with a homogeneous porous medium. New heat transfer results (N u values) at high porous medium modified Rayleigh numbers (Ram ≥ 104 ) were presented. Figure 5.3 shows the variation of N u with Ram for Da = 10−6 , 10−4 and 10−2 and a comparison of the predictions with Weber’s correlation N u = 0.577Ram0.5 . As expected, at high Ra values (Ram ≥ 104 ), the variation between log N u and log Ram is found to be nonlinear. Since this is the effect of the form drag on the flow field due to the porous medium, the fluid Rayleigh number Ra was modified based on the form-drag coefficient C as RaC . The conventional porous medium Ram and the form-drag extended RaC are defined as follows Ram =

gβ (T − Tm ) KL , χm ν

RaC =

gβ (T − Tm ) KCL2 . χm ν

(5.66)

In order to capture the effect of flow nonlinearity on the heat transfer at high Ram , the correlation for N u is expressed using the form-drag modified Rayleigh number RaC as (5.67) N u = 0.85Da 0.33 RaCn , 10−6 ≤ Da ≤ 10−2 , where n = 0.4 − 0.05 log Da − 0.02 log RaC . The coefficient of determination (R 2 ) and root mean square error of the fit are 0.9716 and 0.09399, respectively. This correlation is valid in the range 10−6 ≤ Da ≤ 10−2 and 10 ≤ Ram ≤ 108 . It correctly predicts an exponent close to 0.5 for Ram = 108 at low permeability (Da = 10−6 ). This corresponds to the asymptotic solution as Ram → ∞ given by Bejan (1979), N u = 0.212Ram0.5 in the Darcy regime. It also correctly predicts an

5.3 Natural Convection with Horizontal Thermal Gradient

135

Fig. 5.3 N u variation with Ram and Da (adapted from Fig. 10 in Ramanathan et al. 2011)

exponent close to 0.25 for Ram = 108 at high permeability (Da = 10−2 ) which corresponds to the clear fluid limit. Equation (5.67) can be seen as an extension of Eqs. (5.60) and (5.62), as it covers N u prediction for a wider range 10 < Ram < 108 .

5.4 Heatline Visualization The concept of heatline for visualizing convection was introduced in Kimura and Bejan (1983). A brief introduction to this concept in porous medium context is provided here. Streamlines visualize two dimensional steady fluid flow. They are the pictorial representation of the stream function defined using the equations u=

∂ψ ∂ψ and v = − , ∂y ∂x

(5.68)

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5 Natural Convection Through Porous Medium

such that the mass conservation is satisfied as follows ∂v ∂u + = 0. ∂x ∂y

(5.69)

Here u and v are the fluid velocity components in the Cartesian x and y direction, and ψ is the stream function. From the stream function definition it is evident that no flow happens orthogonal to the streamline (ψ = constant lines). A plot of the stream function ψ from its minimum to maximum value visualizes the pattern of steady fluid flow. A heatline is similar to a streamline, but visualizes net energy flow in a convection or conduction heat transfer situation. The energy conservation equation for an incompressible steady flow with zero heat generation and negligible viscous dissipation can be written in two dimension in Cartesian form as   2 ∂2T ∂T ∂T ∂ T . (5.70) + u +v =α ∂x ∂y ∂x2 ∂ y2 Here α is the thermal diffusivity of the fluid in m2 /s and T is the temperature. Rearranging Eq. (5.70) as ∂ ∂x



∂T ρc P uT − k ∂x



∂ + ∂y

  ∂T ρc P vT − k = 0, ∂y

(5.71)

one can define a function H from Eq. (5.71) such that

and

  ∂H ∂T = ρc P u(T − Tref ) − k , ∂y ∂x

(5.72)

  ∂H ∂T = ρc P v(T − Tref ) − k . ∂x ∂y

(5.73)

Equation (5.72) represents the net energy flow by convection and conduction in x direction, while Eq. (5.73) represents the same for y direction. Similar to the stream function formulation, across each H = constant line, the net flow of energy by conduction and convection is zero. The difference between two successive heatline values (say, H1 and H2 ) should equal the total energy flow rate per depth of separation of the heatlines. The Tref in Eqs. (5.72) and (5.73) is an arbitrary constant. By convention (and by necessity when heat generation is involved), it is taken as the lowest temperature in the heat transfer configuration. Equations (5.72) and (5.73) can be carried over smoothly to a porous medium context, having the properties u, v, T volume averaged and replacing k with ke , theeffective thermal conductivity. The resulting equations that define H , to depict

5.4 Heatline Visualization

137

energy flow through a porous medium subject to local thermal equilibrium (LTE) in 2D are   ∂T ∂H , (5.74) = ρc P u(T − Tref ) − ke ∂y ∂x   ∂H ∂T = ρc P v(T − Tref ) − ke . ∂x ∂y

and

(5.75)

Figure 5.4 are heatlines of natural convection flow of water inside a square enclosure stuffed with a porous medium. The solid matrix of the porous medium can be thought of as uncompressed wire crimps of permeability K = 0.01 m2 , contained in a square box of length 10 cm. Observe that the heatlines are not perpendicular to the isotherms indicating the presence of convection (instead of pure conduction). Costa (2003) has presented a unification method of streamlines, heatlines and masslines.

Adiabatic wall 3.50 2.30

TH

-1.60

-0.28

2.90 1.60

-0.35

0.99

TC -0.92

Adiabatic wall Fig. 5.4 Heatlines in steady-state natural convection inside a porous medium enclosure for Gr = 105 , Da = 0.1

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5 Natural Convection Through Porous Medium

5.5 Non-Darcy, LTNE and Heat Generation Effects In important technological applications like nuclear reactor cooling, the reactor bed can be modelled as a cylindrical cross-sectioned heat generating porous medium quenched by a convection flow, as shown in Suresh et al. (2005) and Reddy (2010). Such modelling is useful in assessing the coolability of heat generating porous debris bed in post severe accident scenario and predicting the solid-phase temperature, an important safety parameter. A two-energy model accords a more realistic prediction in such cases. Studies of local thermal non-equilibrium (LTNE) effects have been presented in the reviews, Kuznetsov (1998) and Rees and Pop (2005). The problem of natural convection inside a heat-generating porous medium annulus subjected to differential heating using two-energy model and generalized modified Navier–Stokes equation is reported in Narasimhan and Reddy (2010). An analysis of this is presented in this section to discuss in succession, the effects of departure from Darcy flow, LTNE condition and heat generation effects.

5.5.1 Governing Equations Figure 5.5 shows the schematic of a fluid-saturated vertical porous annulus with a heat generating porous matrix of equal length and height. The vertical walls are subjected to differential heating, i.e. hot left wall (θh ) and cold right wall (θc ) while the horizontal walls are kept adiabatic. The present analysis is carried out for a steady, laminar, incompressible natural convection fluid flow saturating an annulus filled with a heat-generating solid matrix of the porous medium. Axisymmetric annular flow domain is considered. The porous medium is treated as homogeneous and isotropic with uniform constant porosity, while the effects of viscous dissipation, thermal dispersion and thermophysical property variation are considered negligible. However, the convecting fluid and the heat-generating solid matrix are considered not in local thermal equilibrium (LTE). The Boussinesq approximation, Eq. (5.2), is assumed to be the cause of the buoyancy force in the convecting fluid. The dimensional conservation of mass, momentum balance and energy equations that govern the natural convection flow are non-dimensionalized by using the following terms r∗ =

r ∗ z vr L ∗ vz L ∗ ( p − pref )L 2 ν , z = , vr∗ = , vz = ,p = , Pr = , L L αf αf αf ρα 2f

γ =

K T − (Th + Tc )/2 gβ(Th − Tref )L 3 ks , Da = 2 , θ = , Ra E = , kf L Th − Tc να f

Ra I =

5 gβ Q  s L , α f νk f

(5.76)

5.5 Non-Darcy, LTNE and Heat Generation Effects

139

Fig. 5.5 Schematic of the porous annulus geometry

where the reference values for length (L), velocity, pressure ( pref ) and temperature (Tref ) have been taken as the width of the annulus (ro − ri ), (α f /L), atmosphere pressure and temperature, respectively. Observe that Ra E is the external Rayleigh number that defines the natural convection through the temperature gradient across the walls Th − Tref , while Ra I is the internal Rayleigh number that does it through the temperature gradients that arise due to internal heat generation Q  s . The non-dimensional governing equations are written as: Continuity equation: ∂ 1 ∂ ∗ ∗ (r vr ) + ∗ (vz∗ ) = 0. ∗ ∗ r ∂r ∂z

(5.77)

Momentum equations: In r ∗ -direction ∂v ∗ vr∗ r∗ ∂r

∂v ∗ +vz∗ r∗ ∂z

=

∂ p∗ −φ 2 ∗ ∂r

 2 φ Pr cF − + φ2 √ Da Da

   1 ∂ ∗ ∗ ∂ ∂ 2 vr∗ + φ Pr (r vr ) + ∗2 ∂r ∗ r ∗ ∂r ∗ ∂z

vr∗2 + vz∗2 vr∗ (5.78) 

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5 Natural Convection Through Porous Medium

In z ∗ -direction vr∗

∗ ∂vz∗ ∂ p∗ ∗ ∂vz +v = −φ 2 ∗ + φ 2 Pr Ra E θ f z ∗ ∗ ∂r ∂z ∂z     ∂ 2 v∗ ∂v ∗ 1 ∂ +φ Pr ∗ ∗ r ∗ ∗z + ∗2z r ∂r ∂r ∂z  2

φ Pr cF + φ2 √ − vr∗2 + vz∗2 vz∗ . Da Da

(5.79)

The energy conservation equation considered when the porous medium configuration is modelled with LTE existing between the fluid and the solid is      1 ∂ ∂ 2θ Ra I ∂θ ∗ ∂θ ∗ ∂θ r + + (1 − φ) + v = γ (1 − φ) + φ . z ∂r ∗ ∂z ∗ r ∗ ∂r ∗ ∂r ∗ ∂z ∗2 Ra E (5.80) When modelling LTNE effects, i.e. when (θs = θ f ), energy equations for the solid and the fluid are individually solved with an additional term for local heat transfer closure. These equations for the solid and the fluid phases respectively are Solid phase: vr∗

 0 = γ (1 − φ)

   ∂ 2 θs Ra I 1 ∂ ∗ ∂θs r + + (1 − φ) + H(θ f − θs ). (5.81) r ∗ ∂r ∗ ∂r ∗ ∂z ∗2 Ra E

Fluid phase: vr∗

    ∂ 2θ f ∂θ f 1 ∂ ∗ ∂θ f ∗ ∂θ f r + + H(θs − θ f ). + v = φ z ∂r ∗ ∂z ∗ r ∗ ∂r ∗ ∂r ∗ ∂z ∗2

(5.82)

The porous matrix is modelled as spherical balls of uniform diameter, d p , for which, the Darcy number value is calculated as Da =

φ 3 d 2p 150(1 − φ)2 L 2

.

(5.83)

In Eqs. (5.81) and (5.82), H is the local volumetric solid–fluid Nusselt number and is defined as H=

h s f av L 2 av L 2 hL 2 = = N us f . kf kf dp

(5.84)

The specific surface of the porous medium av evaluated from geometric considerations of spherical balls is given by Bird et al. (2002) as av = 6(1 − φ)/d p . The local solid–fluid Nusselt number N u s f evaluated by taking the sum of the two resistances namely, internal and external to particles, is given by Kuznetsov (1998) as

5.5 Non-Darcy, LTNE and Heat Generation Effects

141

(lc /d p ) 1 1 = + , N us f N ud γ

(5.85)

where lc , the characteristic length (d p /10 for spheres, from Dixon and Cresswell (1979)), and particle diameter-based Nusselt number N u d can be found using the existing correlation Wakao et al. (1979) as 

 N u d = 2 + 1.1Pr

1/3

Re0.6 p

.

(5.86)

In Eq. (5.86), Re p is the particle length-scale-based Reynolds number.

5.5.2 Boundary Conditions The boundary conditions for Eqs. (5.77) to (5.82) with respect to the geometry shown in Fig. 5.5 are At the hot wall r ∗ = ri /L :

vr∗ = vz∗ = 0,

θs = θ f = 0.5.

(5.87)

vr∗ = vz∗ = 0,

θs = θ f = −0.5.

(5.88)

vr∗ = vz∗ = 0,

∂θ f ∂θs = ∗ = 0. ∂z ∗ ∂z

(5.89)

At the cold wall r ∗ = ro /L : And at the upper and lower walls z ∗ = 0, H/L :

The heat transfer rate is predicted in terms of Nusselt number and is evaluated for different conditions as (a) LTE: Once the flow and temperature distribution are obtained in the flow domain, the average Nusselt numbers are calculated as

Nu = −

∂θ ∗ dz . ∂r ∗

(5.90)

N u calculated using Eq. (5.90) at the hot and cold wall must satisfy the relation in the annulus i.e. N u h = κ N u c . (b) LTNE: The average solid–fluid volumetric heat transfer in the porous medium, quantified using H and the average temperature difference between the phases θ s f , are given by

1 1 H=

∗

Hdz dr 0

0

∗

1 1 and

θsf = 0

0

θs f dz ∗ dr ∗ .

(5.91)

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5 Natural Convection Through Porous Medium

By considering the total energy balance at the side walls, q  = (1 − φ)qs + φq f , the average Nusselt numbers at the hot and cold walls are written as  

1

1 ∂θ f ∗ ∂θs ∗ −1 Nu = dz + φ dz . (1 − φ)γ (1 − φ)γ + φ ∂r ∗ ∂r ∗ 0

(5.92)

0

The fluid-based average Nusselt number at hot wall N u h, f can be related with N u h as  N u h, f = N u h (1 − φ)γ + φ . (5.93) Similarly, we can define the fluid-based average Nusselt number at cold wall N u c, f as  (5.94) N u c, f = N u c (1 − φ)γ + φ .

5.5.3 Departure from Darcy Flow Effects To bring out the importance of an appropriate momentum formulation, annulus natural convection using the generalized momentum equations, Eqs. (5.78) and (5.79), and all of its simplified versions (reported in Chap. 3) that are valid for porous medium flows have been simulated in Narasimhan and Reddy (2010). Both LTE (single-energy model) and LTNE (two-energy model) formulations are employed while solving the energy equations, in tandem with different momentum equation formulations for predicting the flow. The results are plotted in Fig. 5.6 in the form of variation of N u c, f , Eq. (5.94), for increasing modified porous Rayleigh number (Ra ∗E = Ra E Da) through Da. For both LTE and LTNE formulations, the simplified Darcy formulation is seen to overestimate the heat transfer for higher Ra ∗E . Beyond about Ra ∗E = 102 , using the generalized momentum equation yields N u c, f values that reflect the appropriate expenditure of buoyancy forces (in Eq. (5.79)) required to overcome all the drags present including the form effects. The difference of N u values between Darcy and non-Darcy models is more significant with LTE than LTNE condition. The reason is the infinite interfacial heat transfer between the solid and fluid phase in LTE model, which in turn, increases the local fluid velocity.

5.5.4 Heat Generation Effect Figure 5.7 shows streamlines and isotherms for natural convection inside the porous annulus, with and without heat generation effects, defined through the internal

5.5 Non-Darcy, LTNE and Heat Generation Effects

143

600 ____

Nu c,f

450

300

101

10 2

Ra *E

10 4

Fig. 5.6 Effect of Non-darcy terms on fluid Nusselt number (N u c, f ) for various Darcy numbers using LTE and LTNE models at Ra E = 108 , Ra I = 1010 , Pr = 5, γ = 15, φ = 0.4 and κ = 11

Rayleigh number, Ra ∗I = Ra I Da. The dashed lines indicate the results predicted by LTE model and continuous lines are LTNE model results. In Fig. 5.7a, heat generation is absent (Ra ∗I = 0). Nevertheless, natural convection is present in the annulus due to the temperature difference maintained across the annulus through the Ra E , as already mentioned. Stronger convection due to the hotter fluid is observed as Ra ∗I is increased in Fig. 5.7b. Interestingly, in (b), Ra ∗I = 106 , the unicellular convection splits into a bicellular roll. This is because of the increased heat generation (Ra ∗I ∼ 105 ) that shifts the θmax of the configuration from the left wall (θh = 0.5, for Ra ∗E = 103 ) into the annulus. As a consequence, the heat flux vector changes direction at the hot (left) wall of the annulus. Near the hot wall, the temperature gradients change direction for higher heat generation (Ra ∗I ), due to the formation of secondary convection roll. For quantifying the local wall heat transfer, a fluid-based N u c, f for the cold wall  of the enclosure can be defined as N u c, f = N u c (1 − φ)γ + φ . Here, for brevity, the subscripts LTE and LTNE are removed in Nusselt number.

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5 Natural Convection Through Porous Medium

Fig. 5.7 Streamlines and isotherms at Ra E = 108 , Da = 10−5 , Pr = 5, γ = 1, φ = 0.4, κ = 11 for internal heat generation magnitude, a Ra ∗I = 0 and b Ra ∗I = 106

5.5.5 Prediction of Critical Internal Heat Generation From the isotherms, it is evident that when heat generation is high enough, even the heat transfer at the hot wall could reverse direction, i.e. the N u, could change sign at the hot wall. This necessitates a total N u to be defined, to account for the overall heat transfer coming out from annulus, as N u t = N u c, f + N u h, f .

(5.95)

When Ra ∗I = 0, the heat transfer out of the hot wall (to the surrounding) is zero, i.e. N u h, f = 0. For Ra ∗I > 0, the N u c, f accounts for the steady state heat transfer (coming out of the wall) from the enclosure cold wall and the N u h, f continues to be zero. However, when Ra ∗I > Ra ∗I,cr (∼ 8 × 104 ), N u h, f > 0, the porous medium is hotter than the hot wall, resulting in heat release from the enclosure to the surrounding, through the hot and cold walls. Obviously, the critical value of Ra I that sets in a bicellular roll thereby reversing the heat transfer at the hot wall depends strongly on the prevailing external natural

5.5 Non-Darcy, LTNE and Heat Generation Effects

145

Fig. 5.8 Variation of Ra ∗I,cr with external porous Rayleigh number (Ra ∗E ) for different thermal conductivity ratio (γ ) values (adapted from Fig. 3 in Narasimhan and Reddy (2010))

convection due to Ra E and also on the ke of the porous medium. The variation of Ra ∗I,cr with Ra ∗E for different thermal conductivity ratio (γ = ks /k f ) values is presented in Fig. 5.8. For a fixed Ra ∗E , Ra ∗I,cr value increases with γ values. When γ < 1, the increment in Ra ∗I,cr is smaller than the increment as γ > 1. This is due to higher thermal conductivity of solid material (in this case, aluminium) than that of the fluid (water), which causes more heat conduction to the walls, which requires a higher Ra ∗I,cr to transform the single cell to bicellular mode.

5.5.6 Correlation for Nu in Porous Annulus A correlation has been proposed in Narasimhan and Reddy (2010) to predict the cold end, fluid-based, Nusselt number N u c, f , Eq. (5.94), for the natural convection in the porous annulus with internal heat generation, as follows N u c, f =

a Ra bE

    Ra I m 1+ A Da c (c1 + c2 γ )d κ e . Ra E

(5.96)

In Eq. (5.96), the coefficients are a = 0.05, b = 0.32, c = 0.07, d = 0.77, e = 0.4, c1 = 0.7 and c2 = 0.3 when heat generation is absent, i.e. Ra I = 0. When the porous annulus is heat generating, i.e. 0 < Ra I ≤ 5 × 1010 , the coefficients take the values

146

5 Natural Convection Through Porous Medium

a = 0.09, A = 0.0132, m = 0.94, d = 0.25, e = 0.84 with b, c, c1 and c2 remaining as before. The above correlation is valid for the parameter range: 500 ≤ Ra ∗E ≤ 104 , 5 × 108 ≤ Ra I ≤ 5 × 1010 , 10−6 ≤ Da ≤ 10−4 , 0.01 ≤ γ ≤ 100 and 5 ≤ κ ≤ 17. It predicts the corresponding N u c f data set generated by numerical simulations within ±20% with a correlation coefficient of 0.987. Beyond the critical heat generation value discussed earlier, the heat transferred from the annulus to the surrounding is no longer restricted to the cold wall. Hence, the above correlation is not valid for such high heat generation values, greater than 5 × 1010 . Prasad and Kulacki (1984) proposed a similar correlation when heat generation is absent, the results of which are predicted within ±15% by Eq. (5.96). Furthermore, the predictions of the proposed correlations, Eq. (5.96), agree within ±5% with that proposed in Bejan (1987). While the correlations by Prasad and Kulacki (1984) and Bejan (1987) are valid only for Darcy flow with LTE assumed in a vertical annulus, in the absence of heat generation, Eq. (5.96) is more general and includes heat generation and LTNE effects for natural convection in a porous annulus.

5.6 Viscous Dissipation The models presented in Sect. 4.6 for viscous dissipation, , i.e. Eqs. (4.83)–(4.86), are also valid in the context of natural convection. To estimate when viscous dissipation effects would be significant in natural convection flows in porous media, one can repeat the order of magnitude analysis presented in Sect. 4.6, with minor alterations. In forced convection, there is a natural velocity scale present in the form of the free stream velocity or the channel cross-sectional averaged velocity. Such a natural length scale is not available in natural convection. One choice is to use the vertical velocity scale of Eq. (5.56). Obviously. this velocity scale can be expressed as U∝

α  e

L

Ra 1/2 ,

(5.97)

where Ra is given by Eq. (5.58) with H replaced by L here, and αe is the effective thermal diffusivity. The above expression for U is substituted, as before (see Sect. 4.6), into 

μU 2 k T



L2 Br =

1, K Da

(5.98)

to yield Ge, the Gebhart number, Ge =

gβ L

1, c p, f

(5.99)

5.7 Problems

147

as a condition when viscous dissipation is negligible in porous medium natural convection. From Eqs. (5.58) and (5.98), it is clear that viscous dissipation is likely to be significant when velocities are high and length scales are large. Nield (2000b) cites flows within geologically sized regions and particle bed nuclear reactors as examples where viscous dissipation should not be neglected. Further discussion is available in Magyari et al. (2005).

5.7 Problems (1) Consider unsteady natural convection inside a two-dimensional porous medium enclosure, heated differentially from isothermal side walls while the top and bottom walls are adiabatic. Using order of magnitude analysis of the conservation equations, derive a criterion for distinct thermal boundary layers (ceases to grow). Find the scale for the thickness of this boundary layer. (2) Consider an enclosure of arbitrary length and height (H = L) saturated by a porous medium. Estimate using scale analysis, the wall heat transfer for all the four regimes discussed in the Sect. 5.3 and shown in Fig. 5.2. By comparing the scales for the convection heat transfer between the vertical walls and the crossconduction between the horizontal walls, derive the criteria for distinct vertical and horizontal thermal boundary layers respectively for shallow (H L) and tall (H L) enclosures.

Chapter 6

Porous Medium Aspects of Biological Systems

This chapter introduces porous medium concepts used in modelling fluid flow and heat and mass transfer in biological systems. With the advancements in biology, health and medicine becoming interdisciplinary, physical and mathematical concepts are being increasingly invoked to model biological processes. Porous medium modelling of bio-fluid and heat flows is an interesting and useful approach to simplify and understand biological phenomena.

6.1 Introduction to Bio-thermofluids Biothermology or heat and mass transfer in biological systems is a subdivision of bio-engineering that applies the threefold approaches of heat transfer, viz. insulation, enhancement and control of temperature, to gain insights into natural processes. Bioheat transfer phenomenon is ubiquitous in nature and is discussed in detail in the review by Charny (1992). Representative examples in the human body are: (1) Thermoregulation, the process of heat management in the human body through evaporation, convection and radiation to maintain body temperature within normal range (2) Thermogeneration, the process of metabolic heat generation arising from exercise, food and other involuntary actions like shivering (3) Heat transfer in muscles and tissues that accompany blood flow (perfusion) (4) Burning of skin through accidents (5) Fever and hypothermia (6) Thermal comfort resulting from convective and conductive heat transfer through clothing and other artificial fittings like contact lens. Similarly, mass transfer studies, within the purview of biology, help in understanding several bio-mass transfer processes that can be identified in nature. For the human system, mass transfer is found in: © The Author(s) 2023 A. Narasimhan, Essentials of Heat and Fluid Flow in Porous Media, https://doi.org/10.1007/978-3-030-99865-3_6

149

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6 Porous Medium Aspects of Biological Systems

(1) The flow of blood and the dynamic equilibrium between blood and inhaled oxygen (2) Metabolism, the diffusive oxygen transfer in tissues (3) Membrane barriers to bulk flow, diffusive and ionic flows through membrane channels and porous medium models of capillaries and tissues (4) Liquid diffusion in tissues, drug delivery to local regions inside body and diffusion of gastric juice in the stomach. The complexity of experimental studies in bio-engineering makes computer modelling and simulation a useful tool in understanding natural phenomena. Advances in simulation and modelling aids, hardware and software improvements and interdisciplinary knowledge have enabled application of computational methods to biological systems. Bio-engineering computational models are typically either bionic (designing bio-engineering systems) or mechano-biological (understanding basic processes). In computational bio-thermal studies, modelling of bio-fluid dynamics is an inherent part of the subsequent bio-heat transfer, when convection or conjugate heat transfer is investigated. The porous medium modelling approach, hitherto well-established in various engineering applications as can be observed from the discussion in earlier chapters, has made forays into bio-mechanical engineering in the past two decades due to possibility of viewing many biological systems as porous media. An obvious example of the applicability of porous medium approach to bio-mass transfer is the diffusion of nutrients and other macro-molecules (drugs, lipids, etc.) across and within biological tissues, which are classic three-dimensional porous media composed of dispersed cells and fibres separated by connective voids, through which the interstitial fluids circulate. The porous medium theory is applicable at various scales in biological systems—organ, tissue, macro-, micro-, nano- and cells can all be represented as porous media. The review by Khaled and Vafai (2003) describes research carried out before 2002 on porous medium models used in specific biological and biomedical applications such as tissue generation in scaffolds, transport in brain tissues, MRI applications, liquid chromatography, transport of macro-molecules in aortic media, blood flow through muscles and interstitial fluid flow in axisymmetric soft connective tissue. The review by Narasimhan (2011) updates the purview of research literature pertaining to porous medium modelling in bio-thermofluids. Other recent reviews on this growing topic are Nicholson (2001), Khanafer and Vafai (2006) and the collection of review chapters in Vafai (2011), the handbook edited by Prof. Vafai. Basic porous medium models and their bio-applications mentioned in these reviews are introduced in the ensuing sections.

6.2 Porous Medium Modelling in Bio-heat Transport

151

6.2 Porous Medium Modelling in Bio-heat Transport Living tissues are complex structures, and the heat transfer in them primarily involves heat conduction in tissue, convection heat transfer between blood and vessel and blood perfusion, which cannot be understood using simplistic models. The living tissue ensemble can, however, be conceived as a fluid-saturated porous medium including the effects of blood perfusion to help understand their heat transfer. A generic region of biological tissue irrigated by blood flow can readily be perceived to fit our definition of a porous medium comprising a stationary solid (tissue) matrix saturated by fluid (blood) flow, with identifiable interfaces at a resolution level. The heat transport in such biological tissue region can be modelled as convection in porous media with internal heat generation (Zhang 2009). From this perspective, it is apparent that the investigation of heat transfer processes in such a tissue blood region would require an energy conservation statement similar to the porous medium energy conservation equation, discussed earlier.

6.2.1 Chen and Holmes Bio-heat Model A heat transfer equation similar in purport was proposed by Chen and Holmes in 1980, although, without the claim of being a porous medium model. Chen and Holmes (1980) divided the control volume occupied by the tissue and blood vessels into two separate volumes: one consisting of solid tissue only and the other of blood in the vascular space within the blood vessels. While the term ‘porous medium’ was not explicitly used in this work, such a division of the system into the stationary tissue phase and the fluid blood phase fits our definition of a classic porous medium with identifiable interfaces. The proposed ‘continuum model’ assumes that heat transfer between blood and surrounding tissue occurs along the circulatory network after the blood flows through the terminal arteries and before the arterioles and that there is no significant heat transfer between tissue and blood within the capillary region until the blood reaches the terminal veins. The Chen and Holmes model can be written as

(ρc)e

∂ Te = ke ∇ 2 Te + kp ∇ 2 Te + V jm (ρcp )b (Ta − Te ) ∂t  → v · ∇Te + Q m . − (ρcp )b −

(6.1)

Here, kp is the perfusion conductivity. The first two terms on the RHS of the equation are the effective heat conduction and apparent conductive heat transfer enhancement due to blood flow, respectively. The V jm (ρcp )b (Ta − Te ) is the local → v · ∇Te is blood perfusion term at the jth branching level of blood vessel. (ρcp )b −

152

6 Porous Medium Aspects of Biological Systems

the convection transport term to account for the effect of blood flow direction within  tissue, and Q m is the volumetric metabolic heat generation. The averaging of the temperature (Te = f (Tt , Tb ) where Tt and Tb are tissue and blood temperatures) and related thermophysical properties of tissue and blood (k and c) carried out when writing Eq. (6.1) is similar to the volume averaging procedure discussed in Sect. 2. However, the models used for ke and ce are ad hoc and not based on any assumed local porous structure for the biological tissue region. Hence, Eq. (6.1) is defined not on a REV-based porous continuum that porous medium energy balance, even when local thermal equilibrium exists between tissue and blood. Neglecting the local (point-wise) effects of blood flow (dispersion or perfusion conductivity effect), the thermophysical properties take values for the tissue and setting Te = Tt , and the Chen and Holmes (1980) model, Eq. (6.1), can be reduced to the Pennes (1948) bio-heat transfer equation, the earliest and rudimentary bio-heat model that first accounted for the blood perfusion effect, as (ρc)t

∂ Tt  = −kt ∇ 2 Tt + Q m + (ωρcP )b (Ta − Tt ), ∂t

(6.2)

where ω is the Pennes blood perfusion rate, a global quantity used ad hoc inside the differential bio-heat equation. The Ta is the arterial temperature dependent on human anatomy (Pennes used it for human forearm, for which he had experimental data). The Pennes bio-heat transfer equation is commonly used to model energy transport in biological systems. Although the Pennes bio-heat equation recognizes the different temperatures between the tissues and blood, the blood temperature is assumed to be a constant throughout the heat transfer domain. It is also not based on the porous medium approach.

6.2.2 Porous Medium Bio-heat Model with LTNE A logical extension of the Chen and Holmes model is a two-energy heat transfer model, suitable when local non-thermal equilibrium (LTNE) prevails between the solid tissue and the fluid blood phase of a bio-material. A suitable connecting interphase heat transfer relationship for h fs , similar to those discussed in Chaps. 2 and 4 must be developed for providing closure in this situation. A LTNE-based porous medium bio-heat transfer model has been derived from first principles by Nakayama and Kuwahara (2008). The individual macroscopic energy equations were written for the blood and tissue phases as follows

6.2 Porous Medium Modelling in Bio-heat Transport

153

For blood phase, ∂T  ∂ ∂ + ρf cp f u j T  f = ερf cpf ∂t ∂x j ∂x j

  ∂T  f ∂T  f εkf + εkdis jk ∂x j ∂ xk

− af h f (T  f − T s ) − ρf cpf ω(T  f − T s ). (6.3) In the above equation, the LHS is the macroscopic convection term, while the four terms on the RHS correspond to the macroscopic conduction, thermal dispersion, interfacial convective heat transfer and blood perfusion, respectively. For the tissue phase,

(1 − ε) ρs cs

∂ ∂T s = ∂t ∂x j

  ∂T s (1 − ε)ks + af h f (T  f − T s ) ∂x j

+ ρf cpf ω(T  f − T s ) + (1 − ε)Sm .

(6.4)

In Eqs. (6.3) and (6.4), the angle brackets signify average quantities, and the average is taken over either solid (tissue) or fluid (blood) volume at the REV level. In the tissue, the LHS represents the thermal inertia term, and the terms on the RHS correspond to the macroscopic conduction, interfacial convective heat transfer, blood perfusion heat source and metabolic heat source, respectively. Combining the energy equation for the tissue phase with the blood and assuming local thermal equilibrium to hold between tissue and blood at the REV level, one could arrive at the porous medium form of the bio-heat equation, which would resemble the Chen and Holmes model without the global blood perfusion term and the perfusion or dispersion conductivity term.

6.2.3 Other Recent Models Nield and Kuznetsov (2008) proposed an illustrative model for bio-heat transfer and provided an analytical solution for forced convection in a parallel plate channel occupied by a layered saturated porous medium with counterflow, the dominant feature that distinguishes bio-heat transfer from other forms of heat transfer. The case of asymmetrical constant heat flux boundary conditions is considered, and the Brinkman model was employed for the porous medium. Belmiloudi (2010) studied the effects of blood perfusion rate and the porosity parameters on the transient temperature of biological tissues for use in thermal diagnostics applications such as laser surgery and thermotherapy often used in the treatment of cancer. A generalized transient bio-heat transfer-type model has been introduced, and the existence, uniqueness and regularity of the solution of the state equation are established.

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Fourier’s heat conduction model has been found to be sufficient for most engineering applications. However, it is insufficient for accurately predicting temperature distribution in non-homogeneous materials like meat as shown in Mitra et al. (1995). Since the human biological tissue has a structure that is similar to porcine tissue, the study of non-Fourier energy transport in human tissues is important. An example of such an application would be laser irradiation of the retina where a high heat flux is typically applied for a short period of time. The constitutive relation between heat flux and temperature gradient, to account for a finite heat propagation velocity, was proposed by Cattaneo (1958). It states that there is a time lag τq between the heat flux and temperature gradient, i.e. a temperature gradient ∇T at time t causes a heat flux −k∇T to flow at time t + τq . It was later extended to include a time lag τT for the temperature gradient as well. Such a model is known as the dual phase lag constitutive relation between heat flux and temperature gradient, written as, q + τq

∂q ∂(∇T ) = −k∇T − kτT . ∂t ∂t

(6.5)

Retinal laser irradiation has been analysed by Narasimhan and Sadasivam (2012) using the bio-heat dual-phase lag model for heat conduction. Both one- and threedimensional models of the human eye were used to simulate retinal eye surgery. The laser heating was modelled as a volumetric heat source and the respective magnitudes calculated based on the absorptivities of the various layers. The retinal pigmented epithelium (RPE) is a highly pigmented layer of about 10 µm thickness where bulk of the heat absorption takes place. The sclera, choroid and the RPE were modelled as a porous medium, with the choroidal blood flow modelled as Darcy flow. Numerical simulations were performed to compare temperature distributions obtained from the dual-phase lag model with corresponding results from an earlier Fourier model in Narasimhan et al. (2010). Based on a non-equilibrium heat transfer model in the living tissue obtained by performing volume average to the local instantaneous energy equations for blood and tissues, Zhang (2009) obtained the dual-phase lag bio-heat equations with blood or tissue temperature as sole unknown temperature by eliminating the tissue or blood temperature from the non-equilibrium model. In this model, the phase lag times were expressed in terms of the properties of blood and tissue and the interphase convective heat transfer coefficient and blood perfusion rate. It was found that the phase lag times for heat flux and temperature gradient for the living tissue are very close to each other. Further, recent developments on the applications of porous medium modelling in bio-thermofluids are available in the two recent reviews Vafai (2011) and Narasimhan (2011).

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6.3 Modelling Drug Delivery Drug administration is a critical part of modern medicine and hinges on the effective release of active pharmaceutical ingredients from delivery vehicles (pills, injections, transdermal patches, etc.) to the target tissues. Mathematical modelling can help understand drug release and can enable accurate control of dosage and administration. The delivery of a drug from its carrier is mediated by rate-controlling release mechanisms such as diffusion, erosion/chemical reactions, swelling and osmosis. The drug is often distributed in a carrier that can either be non-porous/homogeneous or be porous/granular. In the case of a porous carrier, diffusion of the drug is restricted to pores in an otherwise impermeable material. Diffusion is usually the rate determining step in such systems, but drug release can also occur through processes such as matrix swelling and erosion. In transdermal patches, drug transport is predominantly through convection. An interesting area of drug delivery is the transscleral drug delivery in the human eye, where the medication is introduced at the posterior section of the eye to treat maladies such as age related macular degeneration. Accurate delivery of the drug in the eye is critical given the sensitivity of the sense organ and its vulnerability to accidental damage, and computational studies help in predicting optimal delivery patterns. Pharmacokinetic models can be used to explain the transfer rates of drug through the posterior eye tissues. Ranta and Urtti (2006) reported a pharmacokinetic simulation model based on the scleral permeability coefficient Ps , which accounts for circulation loss and predicts the overall permeation flux through the sclera. Balachandran and Barocas (2008) reported a 3D porous medium approach using finite element method for studying transscleral drug delivery, which accounted for the diffusion and convection losses, assuming linear effect of choroidal blood flow on the drug delivery.

6.3.1 Transscleral Porous Medium Model A porous medium model of sclera and choroid has been recently developed in Narasimhan and Ramanathan (2012) to study the effect of choroidal blood flow on transscleral delivery of the drug anecortave desacetate, to the retina. The permeation of the drug through the direct penetration pathway has been modelled as a diffusion process and studied using Fick’s second law of diffusion in conjunction with an effective diffusivity for the porous media. Using the porous medium model, the transient mean plasma concentration C of the drug anecortave desacetate in the choroid has been predicted. The effect of choroidal blood flow on the transient peak mean plasma concentration C max has been studied and compared with available experiments by Dahlin and Rahimy (2007).

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6.3.2 Governing Equations The schematic of the cross section of human eye is provided in Fig. 6.1. The sclera and the choroid are treated as homogeneous isotropic porous media. The conservation equations for porous media are solved to obtain the concentration distribution within the domain. The volume averaged mass conservation and momentum and species transport equations for porous media in the sclera and the choroid are written as, ∇ · vb = 0,  ρb

(6.6)

 ∂vb φc μb + (vb · ∇) vb = −∇ P + μb ∇ 2 vb − vb , ∂t Kc ∂C = Ds ∇ 2 C, ∂t

(6.8)

∂C + (vb · ∇) C = Dc ∇ 2 C, ∂t

(6.9)

φs φc

(6.7)

Vitreous humor Retina Sclera Ciliary muscle

Choroid Blood Vessels

Cornea

Fovea Optic disc

Iris Aqueous humor Pupil

Optic nerve

Hyaloid canal Fig. 6.1 Schematic of the human eye cross section. The drug is to be delivered across the sclera and choroid into the retina

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Fig. 6.2 Sequence of images obtained on processing a TEM image of sclera for porosity determination. a Is the TEM image, b cleaned image and c close-up of the section of sclera used for determining the φs

where C is the drug concentration in kgm−3 , D is the effective diffusion coefficient of the drug in m2 /s, K and φ are the permeability and surface porosity of the regions shown partly in the TEM images of Fig. 6.2. The subscripts s and c stand for sclera and choroid.

6.3.3 Determination of Porosity and Permeability of the Sclera Since direct measurement of the porous medium properties like porosity and permeability is not possible for the sclera and choroid of the human eye, indirect means must be adopted to determine them. The procedure explained here can be generalized for most bio-materials with micro-size internal structure that is difficult to access for direct measurement. The surface porosity of the sclera φs is determined from its transmission electron microscopy (TEM) images. Digitized TEM images of the sclera are obtained from various sources, viz. Kanai and Kaufman (1972), Komai and Ushiki (1991), Marshall (1995), Watson and Young (2004). The images are in the form of two-dimensional

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arrays of pixels with grey level 0 ≤ g ≤ 255. A sample TEM image is shown in Fig. 6.2, on the LHS. Light regions of the image correspond to pores, and dark regions correspond to the solid matrix. A threshold grey level of 128 is used to distinguish pores from the solid. Annotations are removed from the image prior to the analysis. The visual resolution level of all TEM images was roughly equal and of µm order. The surface porosity φs from each image is computed as the ratio of the area of light pixels (g > 128) to the area of dark pixels (g ≤ 128). Using this method, the mean surface porosity of the sclera was obtained as φs = 0.39 ± 0.09 (n = 9). The porosity of the choroid φc is assumed to be equal to the porosity of the sclera φs . The permeability of the choroid K c is also assumed to be equal to the hydrodynamic permeability of the sclera. The scleral permeability may be determined empirically by assuming sclera to consist of randomly packed monodisperse fibres. Using the minimum diameter of such a fibre, dmin = 25 nm (Komai and Ushiki 1991), a lower limit on the permeability is given as (Rahli et al. 1997) K =

φ3d 2 , 4kk (1 − φ)2 (2 + 1/r )2

(6.10)

where r the fibre aspect ratio is assumed to be extremely large and kk for perpendicular flow is given by (see Chap. 3) −1  1 1 − (1 − φ)2 2φ 3 ln − . kk = (1 − φ) 1−φ 1 + (1 − φ)2

(6.11)

Using φ = 0.39, as determined from the earlier TEM images, and d = 25 nm, we obtain K = 1.2 nm2 . The scleral permeability K s may also be obtained from experiments, by measuring the hydraulic conductivity of the sclera and curve fitting the data with the Darcy’s law, as performed by Fatt and Hedbys (1970). From this procedure, at normal hydration of the sclera, it is found that K s = 1.3 nm2 at 37 ◦ C. This result is comparable to the value obtained by the ‘pixel-porosity’ method explained earlier.

6.3.4 Boundary Conditions At the interface between the sclera and the choroid, a no-slip boundary condition is used for the blood velocity. The concentration and its gradient normal to the surface are assumed to be continuous. An impermeable boundary condition (∇C) · n = 0 is used at the interfaces between the sclera and the cornea and the sclera and the optic nerve, respectively. Blood flow is simplified in the presented model and is assumed to enter the choroid through the interface between the choroid and the optic nerve with a physical or fluid inlet velocity of Ub and exit through the interface between the choroid and the cornea.

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At the interface between the choroid and the optic nerve, the concentration of the drug is assumed to be zero. A transient concentration profile is assumed to exist on the outer surface of the sclera, written as   (x − xD )2 + (y − yD )2 , (6.12) C(x, y, t) = C0 e−λt exp − σ2 where C0 is the initial depot concentration of the drug and (x D , y D ) are the Cartesian coordinates of the point of injection. This location is seen in Fig. 6.3 on the top surface of sclera at the tail of the arrow mark. Due to loss of drug from periocular spaces, C0 < C0 , the dose concentration. The initial depot concentration C0 is obtained from the relation, C0 = β10 C0 ,

(6.13)

where β10 is the dimensionless periocular loss coefficient. β10 is related to the bioavailability of the drug in the vitreous BAv . For a given β10 , the bio-availability BAv of the drug is the ratio of the average concentration gradients normal to the surface of the sclera obtained when the simulation is performed with depot concentrations C0 and C0 , respectively. BAv is usually very low (∼0.2%) for subconjuctival injection of prednisolone in rabbits (Tsuji et al. 1988). The boundary condition on the retinal side of the choroid surface is written as,  λ C(x, y, t), (6.14) (∇C) · n = β20 Dc where β20 is the dimensionless vitreous coefficient.

6.3.5 Anecortave Acetate Case Study The simulations presented in Narasimhan and Ramanathan (2012) have been performed for a posterior juxtascleral depot (PJD) administration of anecortave acetate (MWt. = 386.48 g/mol) using a 19-gauge needle (d = 0.686 mm). Assuming that an observable concentration corresponds to 1% of the initial dose, it is estimated that t1/2 for the 3 mg dose is about 2.1 days. On the other hand, fitting a linear model for the half-life versus dose concentration gives t1/2 = 2.3 days. The mean of the two estimates, t1/2 = 2.2 days is used. The mean plasma concentration of the drug C is reported as the average concentration over the choroid. Immediately, following PJD administration C increases from 0 to a maximum value C max . The peak mean plasma concentration C max indicates the therapeutic effectiveness of a drug. For a given dose concentration, it indicates whether the drug is bio-available at the retina. Due to the loss of drug through peri-

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Sclera

Drug depot

Retina Choroidal Blood flow Ub = 0.01 cm/s

Ub = 10 cm/s

Fig. 6.3 Drug concentration profiles in the sclera and choroid at t = tmax for a choroidal blood flow of Ub = 0.01 cm/s (left side) and Ub = 10 cm/s (right side)

ocular, vitreous and circulation routes, the mean plasma concentration C eventually decreases and ultimately becomes 0 again. The duration of time for which a traceable concentration of anecortave desacetate exists in the choroid varies from 2 to 6 weeks.

6.3.6 Effect of Choroidal Blood Flow on Drug Diffusion The conservation equations can be numerically solved employing the finite volume method (FVM), ans the details of which are provided in Narasimhan and Ramanathan (2012). The position and time-dependent concentrations of the drug in the sclera and the choroid can thus be predicted, and the relative magnitudes of the periocular, vitreous and circulation losses can been compared for various blood flow velocities. At t = 0, a steep concentration gradient exists just near the depot which causes the drug to diffuse passively through the sclera. As the drug reaches the choroid, the transport is through both convection and diffusion. The mass diffusion Péclet number Pe given by, Pe =

φ s U b ts , Ds

(6.15)

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indicates the relative magnitude of convection over diffusion. Assuming that the blood flow velocity Ub is uniform and equal to the average of the PSV and the EDV, the Péclet number is determined as Pe = 1.48 × 105 , which indicates that convection is the dominant mode of transport. Figure 6.3 shows the concentration profiles of the drug at t = tmax , in the sclera and choroid, before it is deposited onto the retinal surface. The effect of choroidal blood flow is evident by comparing the left and right side profiles. As the choroidal blood flow is increased from Ub = 0.01 to 10 cm/s, the drug is ‘washed’ or convected strongly in the choroid before it reaches the retinal surface. As the choroidal blood flow varies for different humans both by person and in time, an estimate of the expected deposition quantity of the drug on the retina, as revealed through these modelling, is essential. The simulations can also predict the transient mean plasma concentration of a drug in the choroid and the effect of choroidal blood flow on the peak mean plasma concentration. The peak mean plasma concentration C max is found to decrease by about 70% as Ub increases from EDV to PSV. Decreasing Ub would lead to higher concentration levels of anecortave acetate in the choroid and may achieve bio-available delivery of the drug at the retina. A fundamental assumption has been made on the flow configuration of blood in the choroid. The anastomosis of choroidal vessels has been ignored, and the complex flow pattern through the vessels and capillaries has been replaced by a Darcy flow through a porous continuum. Moreover, the blood flow within the choroid has been assumed to be uniform. In reality, blood flow pulsates between the PSV and the EDV at a frequency related to the heart rate.

6.3.7 Transdermal Drug Delivery Transdermal drug delivery, where the drug is delivered in small doses through skin using patches, is another area for porous medium modelling. Second-generation transdermal drug delivery systems use physical and chemical enhancers to improve drug permeation through the skin. Low-frequency ultrasound has been used as transdermal drug delivery enhancer. Tezel et al. (2003) used a modified porous pathway model to achieve a detailed understanding of the pathways responsible for hydrophilic permeant delivery under ultrasound. The results of this study showed that low-frequency sonophoresis creates pathways for permeant delivery with a wide range of pore sizes.

6.4 LDL Transport Across Arterial Tissues The transport of low-density lipoprotein (LDL) among other macro-molecules across the arterial wall causes plaque, the build up of fatty substances, cholesterol, cellu-

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lar waste products, calcium and other materials on arterial walls. This results in atherosclerosis and other serious cardiovascular problems. The transport of macromolecules across arterial walls is a classic case of transport in porous medium since the wall is a porous material, comprising muscle cells, extracellular matrix, proteins, elastin and collagen, in a lamellar architecture. Modelling of the arterial wall as porous medium during computational studies of transport in arterial walls is important to understand the pattern of accumulation of species in the wall itself, as seen in plaque build-up in arterial walls. Studies on macromolecular transport across arterial walls fall into three categories: wall-free models, fluid wall (or homogeneous wall) models and multilayer models. A homogeneous wall model by Sun et al. (2006) used the Navier–Stokes and advection–diffusion equations to model flow and species transport in the lumen and Darcy’s law to solve the transmural flow in the arterial wall. The mass transport within the artery was coupled to transmural flow to obtain the species distribution across the arterial wall. Olgac et al. (2009) also used a homogenized model in a patient-specific threedimensional simulation of LDL accumulation in a human left coronary artery in its healthy and atherosclerotic states. They showed that in the diseased state, the site with high-LDL concentration shifted distal to the plaque, which is in agreement with the clinical observation that plaques generally grow in the downstream direction. The early wall-free and homogeneous porous medium models are simplistic but are not a realistic situation as the wall is made of three (four, on counting the endothelium) layers that differ in their structure and properties, as seen in Fig. 6.4. The internal elastic lumina (IEL) alone is a complex layer comprising elastic fibres that allow

artery lumen

elastica interna elastica externa

Fig. 6.4 LDL transport across multilayered arterial walls

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transport of macro-molecules into the wall. Thus, the flow through the arterial wall involves the flow through lumen, the lumen–endothelium interface, endothelium, endothelium–intima interface, intima, intima–internal elastic lumina (IEL) interface, IEL, IEL–media interface and media (which is the thickest part of the arterial wall). Prosi et al. (2005) used a multilayer model composed of endothelium, intima, IEL and media to solve the Navier–Stokes equation and the advection–diffusion equation to obtain the flow and concentration fields within the lumen. Darcy’s law was coupled with a species equation for the transport process. The transport processes in the lumen, intima and media are coupled by the flux across the endothelium and IEL, which is traditionally modelled using the Kedem–Katchalsky equations. Ai and Vafai (2006) adapted an advection–diffusion porous media formulation to specify the properties of each of the four layer and studied the effect of hypertension on LDL transport across the arterial wall. A further complication in modelling macro-molecule infiltration through arterial walls is that the boundary conditions considerably influence filtration velocity and concentration profiles. Simplistic models use smooth and straight walls with axisymmetric geometries. This is not the case in reality since many diseased regions of the artery show changes in geometry that affect wall shear stresses. Poorva et al. (2012) and Gundiah et al. (2011) have described a porous medium model for arterial LDL transport, improved from earlier versions by Ai and Vafai (2006). The tri-layered arterial wall is a porous medium with interconnected voids that allow the transport of macro-molecules through it. A representative volume consisting of non-uniform voids can be considered in modelling LDL flow through arterial walls and all parameters averaged over this volume element. The fraction of void space to total volume—the porosity and the path taken by the macro-molecule through the arterial wall—its tortuosity can be obtained. The porosity of biological tissues is usually around 0.0005–0.15 and is significantly lower than inorganic or engineered material; sand, in comparison, has a porosity of 0.37–0.5. Hydraulic and osmotic pressures drive the macro-molecular flux through the arterial wall. The porosity of the tissue is determined separately. The steady-state version of the generalized porous medium momentum equation, Eq. (3.22), without the convection terms on the LHS, is used as the governing equation to model fluid transport into arterial walls. The Brinkman viscous shear stress term of Eq. (3.22) is significant in this context only in thin layers that lie within K from the solid arterial wall where K is the Darcy permeability that depends on the pore size, the porosity and on the detailed geometry. Mass transport equations in arterial lumen may be written as ∂c + v∇c − D∇ 2 c = 0, ∂t

(6.16)

where c is the average species concentration in the fluid, D is the diffusion coefficient and all parameters are local volume averaged quantities. Since the arterial wall is a porous medium through which there is diffusion of mass, the equation must be modified by introducing a Steverman filtration coefficient (γ )

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to account for selective permeability of specific macro-molecules through biological tissues. Thus, the above equation becomes

∂c + ∇ · (γ vc − τ D∇c) = 0, ∂t

(6.17)

where τ is the tortuosity given as L e /L (L e the length of path connecting the pores that the particle traverses and L the linear distance between the start and end point of the particle). The diffusion of multiple species causes incomplete mixing in the absence of external forces. This diffusion occurs in a confined space and is impeded by tortuosity and variation in porosity. The dimensionless Péclet number, Pe, defined similar to Eq. (6.15), that relates the rate of advection of flow to its rate of diffusion thus becomes an important parameter in these models. When Pe > 1, convection mass transport dominates.

6.5 Porous Medium Modelling in Bio-mass Transport 6.5.1 Mass Transport in Tissue Regeneration Three-dimensional porous scaffolds are being increasingly used as the substrate for cell culture and attachment and tissue organization in bone regeneration—an in vivo ‘bio-reactor’. The properties of tissue-engineered scaffold matrices are critical for safety and long-term clinical success of the scaffold. An ideal scaffold should provide a suitable environment for nutrient delivery, waste removal and mechanical stimulation, thus necessitating high porosity and permeability. The scaffolds may be considered porous media though which there is a flow (‘perfusion’) of homogeneous fluid and may be related to flows with simple geometries, such as flow over a porous layer or flow through a porous plug. Thus, the porous medium approach can be used to understand nutrient flow and resulting cell growth or the in vivo degeneration characteristics of the scaffold itself. Fluid perfusion through scaffolds can be modelled considering cell-scaffold constructs as porous media and using a volume average method, to elucidate the distribution of nutrient (e.g. glucose, oxygen) concentration and its corresponding effect on cell growth. For this, the system can be assumed to consist of two phases: the void phase, which contains the nutrient fluid and some polymer matrix, and the cell phase, which includes the cells, nutrient fluid, extracellular matrix and some polymer matrix. For tissue generation, the cells are seeded into the bio-degradable polymer scaffold, and the entire set-up is immersed in nutrient medium. The porosity of the polymer matrix allows the nutrients to easily reach the cells and also provides space for the cells to grow and multiply.

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To develop a mathematical model for the nutrient profile for the system, the porosity of the scaffold is assumed to be large enough to not inhibit nutrient transport. Another assumption is that the polymer degrades much slower than the cell growth; thus, there is no deformation of the scaffold during cell growth. The model consists of two differential equations, one for the diffusion/consumption of the nutrient whose concentration is given by c and the other for cell proliferation, given by the cell number density n. The nutrient is assumed to have a constant diffusion coefficient D and is consumed by the cells at a rate Q. In the simplistic situation, the cells are assumed to be immobile in the scaffold but grow in number at a rate G. The rate of cell death is also assumed to be negligible. The scaffold is assumed to be initially seeded with a uniform density of cells, n 0 . The nutrient concentration is taken to be continuous across the boundary of the scaffold and equal to that in the surrounding medium, c0 . In addition, there is no flux of oxygen at the centre of the scaffold to account for scaffold symmetry. The model is given as ∂c ∂ 2c = D 2 − Q, ∂t ∂t

(6.18)

∂n = G, ∂t

(6.19)

with c = c0 at x = 0, ∂c = 0 at x = L , n = n 0 at t = 0. ∂t Given that there is no cell death, G is equal to βcn where β is a positive constant. Furthermore, assuming that all the nutrients are consumed for cell growth, Q = αβcn where α is also a positive constant. Such models help design effective experiments by estimating the range of flow rates required to achieve a desired shear stress distribution and ensure sufficient nutrient and waste transport. These models can also predict how often the culture medium needs to be replenished during the culturing. Whittaker et al. (2009) developed a simple mathematical model based on Darcy’s law, Eq. (3.17), for forced flow of fluid (culture medium) through a porous scaffold. In this model, porous-walled hollow fibres penetrate the scaffold and act as additional sources of culture medium. Similarly, Yu et al. (2009) applied numerical simulations to predict the fluid dynamics and oxygen transport inside a scaffold structure, and cells attached on the scaffold were considered the porous medium. The velocity around the top surface of the scaffold was higher at a higher Reynolds number and the flow approached the scaffold more perpendicularly. High porous flow was also observed within the scaffold with an increase in Reynolds number. When a scaffold is implanted inside the body, it gradually starts to degrade, along with cell growth so that finally, when the tissue is fully regenerated, the scaffold has completely degraded. The degradation of the porous scaffold with time can also be modelled as a porous medium problem. The degradation of the scaffold can be assumed to occur through hydrolysis. The mechanical properties of the scaffold material, such as stiffness and strength, are generally affected by the molecular

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weight W . As a result of the decrease in molecular weight caused by hydrolysis, the mechanical integrity of the scaffold decreases. For a simple model, Young’s modulus of the porous material of the scaffold E s is assumed to be linearly related to the molecular weight: E s (W (T )) = E s0

Wt . W0

(6.20)

Here, E s is a constant and W0 is the initial molecular weight. Poisson’s ratio VS is assumed to remain constant during the regeneration process. The rate of decrease in molecular weight due to hydrolysis can be assumed to depend on the local water content c (0 ≤ c ≤ 1), W (c) ˙ = −βc,

(6.21)

where the coefficient β(> 0) is a material constant. The morphology and surface area of the scaffold affect the rate of degradation; large surface areas accelerate the diffusion of water molecules into the bulk of the polymers. Here, the rate of water contents is assumed to be governed by the diffusion equation c˙ = α∇ 2 c,

(6.22)

where α > 0 is the diffusion coefficient in the bulk of the polymers and the water contents on the scaffold surface are maintained constant as c = 1. Solving the above three equations provides a measure of the change in local Young’s modulus of the porous scaffold caused by the process of degradation, which becomes a monotonous decreasing function in time. Masood et al. (2005) developed a fused deposition modelling (FDM) rapid prototyping technology to design and manufacture scaffold structures. The theoretical expression for the porosity of a horizontal layer of a cylindrical model was derived as follows: Vpore layer = 1 − (1 − initial ) Vpart layer  n=N

 r D −1 × RW Wn Nr + (Nr − 1)RG × π t , Nr n=0

theoretical =

initial = 1 −

Wmodel ρmaterial × NL × t × RW

n=Nr n=0

 , Wn N r

(6.23)

(6.24)

where D is the diameter, H , the height of the cylindrical model, Wmodel is the average weight of the model when an RG setting is taken as zero, RG is the raster gap, RW

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is road width, t is the layer thickness, ρ material is the density of the material, Wn is the width of nth road from the centre of the cylindrical layer, Nr and NL are the number of raster lines and the number of layers. Nr and NL can be defined as follows D + RG RW + RG H NL = . t

Nr =

(6.25)

A comprehensive mathematical model of convection and diffusion in a perfusion bio-reactor, combined with cell growth kinetics, was developed by Coletti et al. (2006). The model included time-dependent porosity and permeability changes due to the cell density. Navier–Stokes equations for incompressible fluids were used to describe the fluid dynamics of the medium flow inside the bio-reactor, and Brinkman’s extension of Darcy’s law for porous media was used to model convection through the scaffold. The scaffold porosity (xi , t), which decreases from its initial value 0 (xi , 0) as the cell density increases, was given as follows: (xi , t) = (xi , 0) − Vcell ρcell (xi , t),

(6.26)

where Vcell is the single cell volume. As recommended by Perry and Green (1997), tortuosity was modelled as a function of porosity as  τ=

2−

2 .

(6.27)

The functional form was used for permeability K, as mentioned in Koponen et al. (1996) K =

3 , qτ 2 s 2

(6.28)

where, s is the pore surface per unit volume of porous material and q is a structural parameter of the scaffold.

6.5.2 Porous Medium Model for Lung Diffusion Given the porous structure of various organs, the porous medium approach has been successfully used to model organs. The obvious example is the lung that looks like a sponge under microscopic examination and possesses anatomic complexities such as multiple bifurcations and microscopic cavities that make it a classic porous medium. The airflow and gas exchanges within a lung may be considered examples of fluid

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flow through porous medium. The fact that the lung has a finite boundary condition dictated by the visceral pleura makes it an ideal case for numerical simulations using porous medium models. Koulich et al. (1999) developed a mathematical model based on the volume averaging technique to simulate the diffusion process within the alveolar region of the lung. The steady-state solution of the macroscopic model was used to obtain the lung effective diffusivity, with known lung diffusing capacity. They later developed a macroscopic gas transport model to simulate the three-dimensional, unsteady respiration process within the alveolar region of the lungs. The simulations mimicked the single-breath technique for measuring the lung diffusing capacity for carbon monoxide and predicted effect of red blood cell (RBC) distribution on the lung diffusing capacity, as shown in Kulish et al. (2002). The macroscopic diffusion equation is based on the averaging of the diffusion equation within a REV that represents the constituents of the alveolar region that includes the membrane, interstitial fluid, capillary membrane, plasma, RBCs, etc. The diffusion equation resulting from the averaging process within an REV is ∂P = De ∇ 2 P, ∂t

(6.29)

where t is the time, P is the REV averaged partial pressure of the gas being considered during the diffusion process. P is the average of the diffusing gas partial pressure P within an REV of volume V and is given by  1 P = PdV. (6.30) V V

De is the effective diffusivity of the REV and is defined as ⎡ ⎤    c c−1 Vi ⎢ (D − Di+1 ) − −→β dA⎥ , Di + De = n i,i+1 ⎣ ⎦ i V V i=1 i=1

(6.31)

Ai,i+1

where c is the total number of constituents present within the REV, Vi and Di are the −→ is the unit vector normal volume and diffusivity of each constituent within V , − n i,i+1 to the interface with surface area Ai,i+1 between constituents i and i + 1, and βi is the pseudo-diffusivity tensor of constituent i. Once Deff is known, suitable boundary conditions can be applied to solve the equation without the need to describe the internal structure of the alveolar region which is understood from the area integral term. However, precise mapping of the boundaries of each constituent in the alveolar region is required for mathematical determination of Deff . The effective diffusivity of the medium is a function of the structure and properties of each constituent, and this complicates mathematical determination of Deff . The lung diffusing capacity can be used to determine Deff without mapping the

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169

topography of the alveolar regions of the lungs. The lung diffusion capacity can be derived from the Krogh equation: Pv (t) = Pv0 e−(DL Pref /VA t) ,

(6.32)

where Pv is the result of volume averaging P over the entire REV domain and Pv0 is the value of Pv at t = 0. Pref is the total pressure of dry gases, and VA is the alveolar lung volume, inclusive of the residual lung volume and inspired volume. The above three-dimensional, unsteady diffusion model is solved using numerical simulations, and the results are presented in Kulish et al. (2002). The alveolar region of the lungs is modelled as a porous medium, and the gas exchange process inside it is successfully simulated. Imposing the single-breath experimental technique as a transient condition, the results of the numerical simulation lead to an effective diffusivity value equivalent to the measured lung diffusing capacity. With this effective diffusivity, a quantity independent of the positioning of red blood cells (RBC) within the lung domain, several RBC distributions were investigated. The lung diffusing capacity is clinically measurable. However, it is affected by the RBC distribution within the alveolar region. Therefore, it must also depend on the domain used for the theoretical analysis. The three-dimensional distribution of the red cells, characterized by the distribution radius, has a fundamental impact on the lung diffusing capacity. An interesting understanding that can be obtained through modelling the lung as porous medium is the justification of the level of tracheal bifurcation required for effective transfer of oxygen and carbon dioxide between the red blood cells and the external air. The human lung has 23 levels of bifurcation—is there any sanctity to this number? A two-medium treatment for the air convection and the diffusion in its surrounding wall tissue and the oxygen mass transfer between the inhaling air and the tissue along with the effects of the blood perfusion on the mass transfer within the tissue can be analysed using the porous medium model. Kuwahara et al. (2009) analysed such a model and justified the need for 23 bifurcation levels to achieve minimum overall mass transfer resistance for the mass transport from the external air to the red blood cells—an observation consistent with Bejan’s constructal law (‘for a flow system to persist in time, it must evolve in such a way that it provides easier access to its currents’, Bejan 1997). The heart muscle, with its intracoronary vasculature and the kidney, whose porosity is an essential feature for the function of blood filtration, are other organs that can be modelled as porous media. Cimrman and Rohan (2007) presented a numerical model to describe diffusion-deformation processes in the heart muscle and the kidney. The model consisted of the equilibrium equation and a number of mass conservation equations, incorporating the Darcy law of fluid diffusion. In the heart case, the model helped understand the distribution of the perfusing blood during the cardiac cycle. In the case of kidneys, the model aimed at simulating extreme dynamical loads on the kidney during accidents.

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6 Porous Medium Aspects of Biological Systems

6.5.3 Porous Medium Models of Microbial Transport Another area of practical importance that porous medium modelling can help is in understanding microbial transport in the environment. Microbes are constantly being released into the subsurface of the earth as a result of human activities or natural causes. The subsurface of the earth is a classic porous medium, and understanding the fate of the microbe in this porous medium is useful in understanding the pathways of contamination of land and water. Sim and Chrysikopoulos (2000) developed a numerical model for one-dimensional virus transport in homogeneous, unsaturated porous media. Consider a microbe (virus) transported vertically in a one-dimensional porous medium. The governing equation for the microbial transport can be given as    ∂  ∂C ∂ ∂C ∗ ∂ ♦ + θm C = Dz θm [θm C] + ρ ∂t ∂t ∂t ∂z ∂z ∂ − [qC] − λθm C − λ∗ ρC ∗ − λ∗ ρC ∗ − λ♦ θm C ♦ , ∂z

(6.33)

where C(t, z) is the microbial concentration in the liquid phase, C ∗ (t, z) is the adsorbed virus concentration at the liquid–solid interface, C ♦ (t, z) is the adsorbed virus concentration at the air–liquid interface, q is the specific discharge (Darcian fluid flux, as governed by an equation similar to Eq. (3.17)); θm is the moisture content (moisture volume divided by the total volume of the porous medium); λ, λ∗ and λ♦ are the inactivation rate coefficients of the microbe in liquid phase, those at the air–liquid interface and those sorbed at the air–liquid interface, respectively. These three coefficients are time and temperature dependent. ρ is the bulk density of the solid matrix, t the time and z the spacial coordinate in vertical direction. The vertical hydrodynamic dispersion coefficient Dz is related to the effective molecular diffusion coefficient and vertical dispersivity of the unsaturated porous medium. Observe that the above model accounts for virus sorption onto liquid–solid and air–liquid interfaces as well as inactivation of viruses suspended in the liquid phase and viruses attached at both interfaces. The effects of the moisture content variation on virus transport in unsaturated porous media were investigated using this model. For this, the model is solved using numerical simulations, subjecting to the initial and boundary conditions

6.5 Porous Medium Modelling in Bio-mass Transport

C(0, z) = C ∗ (0, z) = C ♦ (0, z) = 0, ∂C ∗ (t, 0) − Dz θm + q(t, 0)C(t, 0) = q(t, 0)C0 0 ≤ t ≤ tp ∂z =0 tp < t ∂C ∗ (t, ∞) = 0, ∂z

171

(6.34)

where C0 is the pulse-type source concentration and tp is the duration of the pulse. The first condition establishes that there is no initial liquid phase and adsorbed virus concentrations within the porous medium. The next is a flux-type boundary condition for pulse injection, which implies concentration discontinuity at the ground surface (inlet) and leads to material balance conservation. The last one is that the downstream boundary condition ensures concentration continuity for a vertical, semiinfinite porous medium. The model simulation results from Sim and Chrysikopoulos (2000) indicate that virus transport in unsaturated porous media is highly sensitive to interstitial moisture variability. At low moisture content levels, the transport of viruses is significantly affected by the irreversible sorption of viruses onto air–liquid interfaces. Results also indicated that virus sorption is greater at air–liquid than liquid–solid interfaces. Bhattacharjee et al. (2002) developed a two-dimensional model for virus transport in physically and geochemically heterogeneous subsurface porous media. Microbial transport in porous media can be described by the advection–dispersion equation with terms that represent virus attachment, release and inactivation ∂n = ∇ · (D) · ∇n − (V n) − Ra − ki n, ∂t

(6.35)

where n is the microbe number concentration in the solution phase, t is the time, D is the dispersion tensor, V is the interstitial fluid velocity, and ki is the inactivation rate constant of viruses in the dispersed (solution) phase. The source/sink term Ra comprises rate expressions governing virus attachment, release and surface inactivation. The study reports two surface inactivation models for the fate of attached inactive viruses and their subsequent role on virus attachment and release. Geochemical heterogeneity was modelled as patches of positively charged metal oxyhydroxide coatings on collector grain surfaces. Physical heterogeneity was represented as spatial variability of hydraulic conductivity. The upstream weighted multiple cell balance method was employed to numerically solve the governing equations of groundwater flow and virus transport. Subsurface layered geochemical and physical heterogeneity was found to significantly affect virus mobility. Random distributions of physical and geochemical heterogeneity were also found to influence virus transport behaviour. Large virus release rates resulted in extended periods of virus breakthrough over significant distances downstream from the injection sites.

Chapter 7

Radiation Heat Transfer in Porous Medium

In this short chapter, the radiative transport equation (RTE) valid for a porous medium is first presented. The volume-averaged energy equations when LTE and LTNE conditions prevail, introduced in Chaps. 2 and 4 are reintroduced with the radiative flux term present. The coupling between the RTE and the energy equation is discussed briefly. Various effective properties, relevant to radiative energy transport in porous media, that must be determined to solve the RTE are also mentioned. The content is largely based on the review chapter by Prof. Howell, in the Handbook of Porous Media, 1st Edition, ed. Vafai (2000).

7.1 The Radiative Transfer Equation (RTE) As we have seen until now, for modelling purposes, the porous medium can be treated as a porous continuum, over which the conservation equations involving volume-averaged quantities can be operated to predict flow and heat transfer results. Under radiative mode of heat transfer inside a porous medium, the standard radiative transfer equation (RTE) describes the propagation of radiative intensity. The RTE can be written as ∂i λ () = Iλ () − i λ (). ∂τλ

(7.1)

The above equation indicates that the attenuation of radiation intensity along a given direction depends on the difference between the source function Iλ and the local intensity i λ . Each of these quantities depends on the wavelength. The variables are defined as

© The Author(s) 2023 A. Narasimhan, Essentials of Heat and Fluid Flow in Porous Media, https://doi.org/10.1007/978-3-030-99865-3_7

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7 Radiation Heat Transfer in Porous Medium

dτλ = (aλ + σλ )ds = κλ ds;



Iλ () = (1 − βλ )i λb + βλ

βλ =

σλ κλ

i λ (i ) (i , )di ,

(7.2)

i =4π

where dτλ is the differential optical thickness, which indicates the ability of the medium to attenuate the intensity across a differential length ds as caused by the medium attenuation coefficient κλ . Also, κλ = aλ + σRλ , where aλ is the absorption coefficient and σRλ , the scattering coefficient. The quantity βλ = σRλ /(aλ + σRλ ) is the single scattering albedo of the medium. It measures the fraction of attenuation that is due to scattering. Of the two terms that the source function Iλ is made of, the first is the contribution to intensity by radiation emission from the medium and the second is the contribution from intensity travelling in other directions and then scattered into directions within the solid angle . The RTE can be integrated resulting in τλ i λ (τλ , ) = i λ (0, ) exp (−τλ ) +

     Iλ τλ∗ ,  exp − τλ − τλ∗ dτλ∗ . (7.3)

0

Solving either Eq. (7.1) or Eq. (7.3) would require the value of local temperature of the porous medium to be known, so that the local black body intensity i λb can be specified in the RTE, allowing solution for the local intensity i λ . The temperature distribution of the porous medium is known only by solving the appropriate energy equation, along with the RTE.

7.2 The Energy Equation with Radiation The simplest way to account for radiation heat transport inside a porous medium is to add a radiative flux term to the steady-state version of the energy conservation equation, Eq. (2.19), without the source or sink terms. Doing so, results in ∇ · qR + ∇ · qC = 0.

(7.4)

Obviously, the above equation holds only when the porous medium fluid is stagnant, with only heat conduction present in addition to radiation (no flow = convection is absent). Also, the LTE assumption, Eq. (2.18), is implicit. The next level in complexity for the energy equation, is to include flow inside the porous medium. The simpler case in this is to have the flow rate of the fluid sufficiently large, such that, the volumetric heat transfer coefficient is large, allowing one to invoke the LTE assumption, Eq. (2.18), even in this situation. This results in an energy equation of the form

7.2 The Energy Equation with Radiation

ρu

1 d dT − dx cp dx

175

  1 d dT 1 ke + (qR ) + S = 0. dx cP dx cp

(7.5)

Here, S is the volumetric source strength that results from combustion or other internal energy source. As seen earlier, here ke is the effective thermal conductivity of the fluid-saturated porous medium (see Chap. 2 for various models) and the rest of the properties are defined in a volume-averaged sense, as before. As can be seen from the first and third terms of the above equation, the temperature is now affected both by flow (u) and radiation (qR ). The next general energy equation can be written only by considering local thermal non-equilibrium (LTNE) to prevail between the solid and fluid phase at the REV level, thus writing separate equations for solid and fluid phase of the porous medium. These would be: Fluid phase energy equation ρu

1 d dTf − dx cp dx

  1 dTf 1 kf + S + h v (Tf − Ts ) = 0. dx cp cp

(7.6)

Solid phase energy equation d dx

  d dTs ks − (qR ) + h v (Tf − Ts ) = 0. dx dx

(7.7)

Notice here that the fluid is considered transparent to radiation, which then allows the radiative flux divergence to appear only in the equation for the solid phase. However, the rest of the thermophysical properties are ‘effective’, i.e. they depend upon the structure of the medium for the solid and the flow configuration for the fluid. Importantly, due to the local coupling between Eqs. (7.6) and (7.7) at the REV level, through h, radiation will affect both the solid and fluid temperature distributions, even though the radiation term appears only in the energy equation for the solid. h v is the volumetric heat transfer coefficient and S is the volumetric source strength that results from combustion or other internal energy source. If multiple species are present, as in a combustion problem, additional terms for species diffusion must be included in the fluid phase equation. The radiative flux divergence appearing in the energy equations, Eqs. (7.5)–(7.7), ∇ · qR , can be determined from the local radiative intensity by the relation ∞ ∇ · qR = λ=0

⎡ ⎣



⎤ i λ ()μd⎦dλ.

(7.8)

=4π

Keep in mind that the RTE and the porous medium energy equation (any one of the above three) are fully coupled. They must be solved simultaneously to find the porous medium temperature distribution when LTE holds, or the solid temperature

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7 Radiation Heat Transfer in Porous Medium

distribution when LTNE prevails, which in turn is used in the fluid energy equation along with the local h to yield the fluid temperature distribution. Further details of the methods of solution of the RTE for this class of problems are available in the standard radiation texts such as Siegel and Howell (1992) and Modest (1993).

7.3 Radiative Property Measurement One of the major difficulty in the application of RTE solution methods to porous media is in determining the appropriate property values to use in the solutions. The availability and accuracy of appropriate properties often determine the method of solution of the RTE. For instance, the radiative properties under the continuum assumption are the spectral absorption and scattering coefficients of the medium (aλ and σR,λ ) and the single-scattering phase function (i , ). However, data on the spectral behaviour of (i , ) is usually unavailable. The simple models invoked, assume it to be independent of wavelength. Almost all radiative transfer applications in porous media require the radiative properties at wavelengths in the infrared region of the spectrum. For instance, in applications like pebble bed heaters, bed temperatures are of the order of 1500 K or below, and the important radiative transfers are at wavelengths of 2 µm and greater. One exception is the porous media used as absorbers in solar collectors, where a very-high-temperature Sun is the source, wherein values of properties at visible or near-IR wavelengths is required. Here, the spectral dependence of the radiative properties must be considered and averaged properties using Kirchhoff’s law are inappropriate. In a typical porous medium, when the solid matrix particles are large in number, they crowd near each other (low volumetric porosity). The radiation path from one particle is hindered and scattered by the adjacent particles, identified as the nearfield effect. In addition, the radiation from one particle can engage in destructive interference with the radiation from another particle, however far it is, identified as the far-field effect. For a porous medium composed of spherical particles, Kaviany and Singh (1993) have made recommendations on the effect of dependent scattering. According to them, independent scattering can be assumed when the criterion C + 0.1dp >

λ , 2

(7.9)

is met, where C is the inter-particle clearance distance. This criterion can also be written as ξC + 0.1ξ >

π , 2

(7.10)

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177

where the clearance parameter ξC = πC/λ. If this criterion is not met, the more detailed dependent scattering approach should be used. This result is derived for porosities typical of rhombohedral packing, ≈ 0.26; however, they observe that dependent scattering effect in packed beds dominate even for bed porosities as high as 0.935 (most pronounced for opaque particles).

7.4 Solving the RTE The most straightforward simplification of the RTE is achieved when the porous medium is rendered ‘optically thick’, whereby the radiation travels across a short mean free path, before being interacted by other regions of the solid matrix. In this case, the radiation mode of heat transfer can be ‘condensed’ or lumped as an equivalent diffusion process. The local radiative energy flux then is expressed as qR (S) = −

∂T 16n 2 σ T 3 ∂ T = −kR , 3κ ∂S ∂S

(7.11)

where κ is the extinction coefficient that depends on the material of the solid matrix, kR is the ‘radiative conductivity’ of the porous medium and n is the refractive index. The condition that bed optical thickness τ = κ L >> 1, where L is the smallest dimension of the bed, should be met for using the form of Eq. (7.11) for modelling qR . Obviously, the most difficult case is when the radiative mean free path is in the order of the overall bed dimensions. In this case, complete solution of the RTE may be required. The standard methods as briefed in Howell (2000), include the P − n method, in which the intensity in the differential form of the RTE is expanded in a series of spherical harmonics; the discrete ordinates method, in which the sphere of solid angles about an element is divided into discrete solid angular increments, and the integrated RTE is solved along each ordinate direction; Monte Carlo technique, which simulates the behaviour of the radiation by sampling a large set of radiative energy, packets throughout their lifetime. These methods are discussed in detail in Siegel and Howell (1992) and Modest (1993). In this context, it is worth mentioning that if the structure of the porous medium is known (packed bed of spheres etc.), then it is possible to use conventional surface–surface interchange analysis or employ bruteforce numerical simulation in a computational geometry, to determine the radiative transfer.

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7 Radiation Heat Transfer in Porous Medium

7.5 Coupling of RTE with Other Heat Transfer Modes When convection mode of heat transfer is absent, i.e. fluid saturating the porous medium is stagnant, and when the porosity is very low, the radiative flux can be expressed by Eq. (7.11). In such a porous medium, Eq. (7.4) is valid as the total energy equation, even when heat conduction exists. When LTE prevails, taking into account Eq. (7.11) for radiation, Eq. (7.4) can be expressed as ∇ · qR + ∇ · qC = ∇ · (qR + qC ) = −∇ · (ktot ∇T ) = 0,

(7.12)

which can be solved with conventional numerical tools for handling conduction problems with temperature-dependent thermal conductivity. In Eq. (7.12), the total thermal conductivity ktot = (kR + ke ) accounts for both radiation and conduction heat transfer. It is highly nonlinear because of its dependence on T 3 , as gathered from Eq. (7.11). The value of ke has to be determined from one of the models discussed extensively in Chap. 2. This approach simplifies the energy equation terms involving conduction and radiation when compared to solving the complete RTE for determining the radiative flux divergence. This approach is used in majority of multimode heat transfer studies in porous materials. However, the difficulty with the use of a kR is in its accurate prediction for porous materials of varying internal complexity, even when porosity is low. kR would depend on particle size and spacing, on the surface emissivity of the particles in the bed, and probably on the solid thermal conductivity ks , which affects the temperature distribution within the solid matrix. For optically thick porous media with 0.4 < φ < 0.5, composed of opaque solid spherical particles with surface emissivity R and thermal conductivity ks , Singh and Kaviany (1994) provide an estimate of the radiative conductivity kR as

kR = 4dσ T

3

 0.5756 R tan

−1



ks 1.5353 4dσ T 3

0.8011 

 + 0.1834 . (7.13)

This formulation seems accurate for both diffuse and specular particles and is weakly dependent on bed porosity.

Chapter 8

Advanced Topics

This chapter introduces a few current research topics that involve porous medium modelling discussed in the preceding chapters. Each section is treated as a separate case study, providing details on the fundamental equations and modelling involved. Most of the topics chosen are based on the author’s involvement in the associated research and should by no means be considered either exhaustive or superior to other ongoing research. The selection should enthuse those readers who wish to pursue research in transport in porous media.

8.1 Phase Change in Porous Media In this section, we shall consider a case study that involves melting and solidification of one constituent of a two-constituent porous medium.

8.1.1 Porous Medium LHTES Device Latent Heat Thermal Energy Storage (LHTES) devices find a wide variety of applications including storage of solar energy, passive energy storage in architecture, heating of sanitary hot water, medical applications such as transporting blood, etc., as detailed in the reviews Zalba et al. (2003) and Farid et al. (2004). A useful application, for instance, is to charge (heat) a LHTES device during off-peak hours of electricity usage (hence lower cost) and use the device in heating applications during peak hours of electricity usage (higher cost). The use of a phase change material (PCM) such as paraffin, eicosane, to store energy as latent heat in the LHTES device has several restrictions including very low thermal conductivity of the PCM causing large temperature irreversibilities during the charging and discharging of the device, and structural stability issues. © The Author(s) 2023 A. Narasimhan, Essentials of Heat and Fluid Flow in Porous Media, https://doi.org/10.1007/978-3-030-99865-3_8

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8 Advanced Topics

To offset these restrictions, a base material (BM) of very high thermal conductivity (metal) is usually used along with the PCM inside the TES device. Since TES devices store energy as enthalpy, they are susceptible to many irreversibilities associated with the large temperature gradients generated while receiving energy as heat (charging) and releasing the heat (discharging) back. The low thermal conductivity of PCM results in temperature gradient related irreversibilities while charging and discharging. Bejan (1978) proposed the analysis of TES devices using availability (exergy) as an index of performance. But his analysis is restricted only to the sensible heat storage. As two materials (PCM and BM) of widely varying thermophysical properties are used with possibly different geometric structure of the BM and PCM combination, the LHTES can be analysed using a volume averaged porous medium model. Since a porous medium provides large interface area between the BM and PCM, it is reasonable to expect that the PM modelling of the LHTES would predict a ceiling performance, which is asymptotically approached by other LHTES designs—such as the finned LHTES. Furthermore, an optimum porosity (quantity of PCM to be used), that does not impeded heat transfer while storing maximum enthalpy, can be estimated from the analysis.

8.1.2 The Enthalpy Model and Governing Equations The primary objective in melting and solidification problems is to determine the phase change or melt front velocity. This holds even when the melt front has to traverse through a solid matrix in a porous medium, as in the PM-LHTES device being discussed. An enthalpy formulation is usually adopted to write the governing equations of the problem, which is essentially the diffusion energy equation extended to include enthalpy terms to account for latent heat of phase change. The method involves assuming that the melting occurs across a narrow range of temperatures (358–360 K in this case) instead of at a discrete temperature, say TM . The material properties of the PCM are suitably altered in this range (the specific heat in this case) to account for the latent heat of fusion. This method allows the usage of a fixed grid in computational heat transfer and is easy to implement. Since the specific heat capacity is modified in the melting range of temperatures, the problem reduces to the solution of the diffusion equation with variable material properties. This means that the moving front need not be monitored. Figure 8.1 shows the axisymmetric schematic of the PM-LHTES device. The geometry is cylindrical and is assumed to be surrounded by an ambient at a uniform temperature T∞ with a uniform heat transfer coefficient h ∞ . The design considered here is a pipe covered with a coaxial porous medium made of PCM (n-eicosane) and BM (aluminium metal foam). The PCM is considered to be dispersed in a consolidated BM matrix, forming a saturated, isotropic, homogeneous porous medium. When the LHTES device discharges, the heat exchanging fluid (liquid water in this

8.1 Phase Change in Porous Media

181

Fig. 8.1 Schematic of the PM-LHTES device with the sample solidification front

study) enters the pipe at T∞ and exits at T∞ + To . Therefore, it is expected to be symmetric across all values of θ , the angle of the radius. The 2D (in r and z) version of the unsteady energy equation used to model the PM-LHTES domain is, ke

∂2 T ∂2 T ∂T ke ∂ T + k . + = ρe ce e 2 2 ∂t r ∂r ∂z ∂t

(8.1)

Local thermal equilibrium (LTE) is assumed between the PCM and the BM, resulting in the single energy equation above, involving effective properties. Suitable models are required to evaluate these effective properties, as a combination of those of the base material and PCM.

8.1.3 Effective Properties and Boundary Conditions The porosity of the LHTES device, φ is defined as the ratio of the volume of PCM to the total volume. The thermal conductivity of PCM, kPCM is usually three orders of magnitude less than the conductivity of the BM, kBM , which is usually a metal. Mass and energy conservation equations can be employed to obtain ρe , L e and ce . However,

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8 Advanced Topics

estimating ke requires an effective conductivity model. As the porosity increases, the composite PCM-LHTES ke would decrease, creating more irreversibility at high values of φ. The effective energy storage density, however, would increase at high values of φ. A satisfactory model for ke is proposed in Bauer (1993) suitable for a porous material with the pores (comprised of PCM) having very low thermal conductivity, i.e. kPCM Tliq then ce is determined by the equation above. However, when Tsol < Ti j < Tliq , then: Le ρPCM cPCM + ρBM cBM (1 − φ) + . (8.6) ce = Tliq − Tsol ρe The solution to the energy equation using the above properties is achieved using numerical simulations and is reported in Akhilesh (2005). Adiabatic conditions are imposed on three sides of the TES device in Fig. 8.1. The h value on the interior surface that exchanges heat with the cooling fluid is determined using the Dittus– Boelter equation for turbulent flow through a duct, after Incropera et al. (2007) as 0.4 N u D = 0.023Re0.5 D Pr ,

(8.7)

where N u D is the Nusselt number defined with the diameter of the pipe as the length scale, and Re D is the Reynolds number defined with the same length scale. Temperatures in the flow are assumed to be constant with respect to r and their variation along z is calculated using h. The To is the temperature of the heat exchanging fluid at z = D. At the inlet, z = 0, T = Tr , the room temperature. A sample evolution of the melt/solidification front profiles is shown in Fig. 8.2. As φ = 0.9 → 1, the melt front evolves parallel to the flow (along the z direction). This behaviour provides the basis for the scale analysis discussed in the ensuing section.

8.1 Phase Change in Porous Media

183

Fig. 8.2 Results of evolution of the solidification front of the PCM, when the PM-LHTES discharges enthalpy to the cooling fluid flow. The simulation results are for the PCM porosity φ = 0.9. Adapted from Akhilesh (2005)

8.1.4 Exergy Criterion and Optimum φ The transient performance of such a PM-LHTES system is analysed here during the discharging cycle. Since the irreversibilities of the system are to be minimized, availability defined as (8.8) A = (h − h ∞ ) − T∞ (s − s∞ ), can be used as a norm in this analysis. When the PM-LHTES device discharges, the heat exchanging fluid (water) enters the pipe at Ti = T∞ and exits at T∞ + To . In this case, the availability is equal to multiplying the enthalpy of flow, m˙ f cf To with the Carnot efficiency of the energy transfer process between the temperatures To and T∞ . This results in AP =

m˙ f cf To2 . T∞ + To

(8.9)

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8 Advanced Topics

Therefore, for fixed values of parameters such as the geometry of the PM-LHTES device, mass flow rate of the heat exchanging fluid BM/PCM combination at a given initial uniform discharging temperature, the cumulative available energy at the end of a transient discharging process can be maximized as ∞ AE = 0

m˙ f cf To2 dt. T∞ + To

(8.10)

Let the energy storage density, E D , be defined as the energy that can be stored in LHTES per unit volume. Observe from the expressions for effective properties that increase in E D is always linear with an increase in φ, while the decrease in ke is not linear with φ. The decrease in the value of ke with φ depends on the nature of the distribution of the PCM inside the BM. As φ increases, the increase in E D tends to increase AE ; on the other hand, the decrease in ke decreases AE due to the formation of large temperature gradients. Owing to these conflicting effects, an optimum porosity φ for the PCM to be used in the TES can be expected.

8.1.5 Scale Analysis The estimation of the optimum configuration of the PM-LHTES is complicated due to the non-linear effects of irreversibility. We shall present an order-of-magnitude analysis here and compare its predictions with the simulation results that yielded the solidification fronts earlier. The optimum φ configuration is to be seen as a compromise between larger latent heat storage and the corresponding irreversibility encountered while releasing energy to the heat exchanging fluid owing to the lower effective conductivity at high values of φ. When the porosity is high, the solidification front almost becomes parallel—as can be seen when φ = 0.9. This fact is used to arrive at a scale analysis which is valid at high values of φ. At the solid–liquid interface,  ∂ T  drb . = ρe L e (2πrb D) − ke (2πrb D)  ∂r interior dt

(8.11)

If rb is the mean solidification radius, and Ta is the average temperature of the heat exchanging fluid (measured over T∞ ) for solidification profiles parallel to the z axis (as can be seen at φ= 0.9), the equation can be scaled to ke

Tm − Ta d(rb − ri ) . ∼ ρe L e rb − ri dt

(8.12)

8.1 Phase Change in Porous Media

185

Or, on applying boundary conditions (at t = 0, rb = ri )  rb ∼ ri +

2ke (TM − Ta )t . ρe L e

(8.13)

The boundary condition imposed by the flow is    ∂ T  T0 − ke (2πrb D) = m˙ f cf T0 , = h(2πri D) Ta − ∂r interior 2

(8.14)

T0 is measured from T∞ , and the above equation is valid for all time instances during the solidification cycle. This equation, when estimated by scales, becomes   Tm − Ta T0 ∼ m˙ f cf T0 , ∼ h(2πri D) Ta − ke (2πrb D) rb − ri 2

(8.15)

which when rearranged and substituted for rb eliminates Ta and results in a scale estimate for the exit temperature of the heat exchanging fluid, T0 (measured from T∞ ) TM T0 ∼ . (8.16)  2TM t ke L e ρe m˙ f cf m˙ f cf  +1 2πri Dh + 2π D 2ke TM t ri + L e ρe Note that the above expression for T0 has the following features: • The maximum temperature at the outlet T0 is always less than TM (since the melting point temperature TM = initial temperature). This is true because all the terms in the denominator are positive. • The role of ρ, not apparent, is subsumed in the effective properties of the PCMLHTES, viz. L e , ρe and ke . • The temperature falls rapidly to a certain asymptotic value and does not fall further. In reality, this would not happen, since solidification would conclude at some finite time. To capture that effect, the expression for T0 must be complemented with a time scale. However, the asymptotic value of T0 , can be estimated by making t → ∞, as TM   . (8.17) T0 ∼ 1 m˙ f cf 1 +1 + 2π D ri h ke Since the objective is to carry all the latent heat of the PM-LHTES in the enthalpy of the heat exchanging fluid at a given flow rate, a time scale can be arrived at by

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dividing the latent heat energy stored in the heat sink by the enthalpy (W ) of the flow π(ro2 − ri2 )DL e ρe . (8.18) τ∼ m˙ f cf T0 • Since T0 is defined to be the difference between the temperature at the outlet and T∞ , the available energy is: ∞ AE = 0

m˙ f cf T02 dt. T0 + T∞

(8.19)

By using the asymptotic scale for T0 and the scale for AE , an expression for available energy independent of time is obtained for the complete solidification of the TES device as   T0 2 . (8.20) AE ∼ π(ro − ri ) L e ρe D T0 + T∞ This is equal to the enthalpy stored as latent heat in the LHTES multiplied by a Carnot efficiency based on T0 , which is a temperature intermediate between T∞ and T∞ + TM as determined earlier. The plot in Fig. 8.3 between AE (the integrated version of AP ) and φ compares the results obtained from the above scale analysis and numerical simulations. It shows that an optimum is indeed present when φ ∼ 0.7 and that the scale estimate locates the optimum φ with reasonable accuracy. However, since the scale analysis that predicted the optimum does not consider sensible heat at all, the optimum is expected to be under-predicted. The scale analysis also uses a lower asymptotic expression for the temperature of the fluid, which may also under-predict the optimum.

8.2 Variable Viscosity Porous Medium Flows 8.2.1 Temperature-Dependent Viscosity Effects What does variable viscosity or more precisely, temperature-dependent viscosity do in convection? For simplicity, we shall ask this question in the context of a liquid flow through a parallel-plate channel in Fig. 8.4. The pressure dependency of a liquid’s viscosity, found to be negligible, is not considered in our discussion. Also, for liquids, μ decreases with increase in temperature, while μ increases for gases with an increase in temperature. We consider liquids in this discussion, but the purport of the theory discussed also holds for gases.

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Fig. 8.3 Variation of cumulative available energy AE with quantity of PCM φ used in the PMLHTES: comparison between scale analysis prediction and numerical simulation results. Adapted from Akhilesh (2005)

Heating (or cooling) the liquid flow inside the channel makes the local temperature vary everywhere inside the channel. The temperature distribution of this process is obtained, in principle, by solving the general energy transport equation. To solve the energy equation, a priori knowledge about the local velocity distribution is essential, which is obtained usually from the general momentum transport equation. While solving the momentum equation (for the convection process), the local viscosity variation is taken into account, as it is a function of the local temperature. In essence, we are to solve a system of coupled partial differential equations. The word coupled represents a more intricate coupling here than that already existing, due to the appearance of velocity in the energy equation. Local viscosity variation creates a two-way coupling so that neither of the conservation statements can be solved separately. Assuming μ a constant (in the former case) allows us to solve the momentum equation separately. Figure 8.4 explains the situation with a collage in the bottom. The solution to the momentum equation is often known, for instance, it can be Poiseuille laminar flow in the above example of parallel-plate channel. For nonisothermal flows (channel is heated or cooled), since the local viscosity is spatially varying, the pressure-drop is found by evaluating the viscosity at suitably averaged channel representative temperatures, to capture the μ(T ) effect.

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Fig. 8.4 Schematic of a heated, parallel-plate, porous medium channel (top); ‘Drawing Hands,’ 1948, by Maurice Escher (Courtesy: Cordon Art B. V.-Baarn-the Netherlands) and coupling of energy and momentum equations due to temperature dependency of viscosity (bottom)

8.2.2 Variable Viscosity and HDD Model Following the line of thinking for clear fluid flow, we may ask what happens to viscosity variation if the channel in Fig. 8.4 is filled with a porous medium that is saturated by the fluid flowing through it. Narasimhan and Lage (2001) have shown that the Darcy’s law. Equation (3.19), or its extension, the global HDD model, Eq. (3.22), repeated here for clarity, μ (T ) P = U + C0 ρU 2 , L K0

(8.21)

does not predict correct pressure-drop values for non-isothermal liquid flow through the porous medium channel. The subscript ‘0’ in K and C in the above equation signify that these properties of the porous medium are obtained from the results of isothermal experiments. Even a correction for the viscosity in the viscous-drag term, (μ/K )U , through its evaluation

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at one of the representative average temperature values (like film temperature), does not yield correct predictions. To understand better, why the HDD model, Eq. (8.21), fails for temperaturedependent viscosity, let us retrace its evolution from the differential counterpart and in the process, suggest a modification of Eq. (8.21) that incorporates the temperaturedependent viscosity effects into it. The general macroscopic differential momentum conservation statement for the porous channel flow of Fig. 8.4 is,  μ (T ) u − ρC0 |u| u. 0 = −∇ p − K0 

(8.22)

As before, the subscript ‘0’ on the porous medium properties K and C denote that these quantities are obtained experimentally under isothermal condition, where the fluid viscosity is uniform throughout the channel, μ(T ) = μ(Tin ) = μin . The absence of the convection inertia and Brinkman terms in the momentum equation, Eq. (8.22) is in accordance with the low permeability (K 0 ) and high form-coefficient (C0 ) porous medium assumption made earlier. For isothermal porous medium channel flows, the velocity profile (local velocity variation along y direction, u(y), in Fig. 8.4) would resemble a slug or uniform flow profile. Hence, barring the ‘entrance effects’ that normally subside within a short channel length, the local velocity u(y) everywhere inside the channel is identical to U, the channel cross-sectional averaged longitudinal velocity. This fact allows us to integrate the differential HDD model, Eq. (8.22), easily for the entire channel, resulting in the global HDD model, Eq. (8.21). However, while doing so, the local viscosity is assumed to be uniform everywhere inside the channel. In other words, the viscosity in Eqs. (8.21) and (8.22) are the same, evaluated at a suitable reference temperature, usually the inlet temperature. However, when the channel is heated/cooled, the spatial variation of fluid viscosity distribution distorts the velocity profile in the x-direction as the fluid flows along the channel, thus affecting the energy transport equation. The resulting altered temperature profile from the energy equation n turn affects the local fluid viscosity, owing to the coupling between energy and momentum transport equations, as explained earlier in Fig. 8.4.

8.2.3 Limiting Case of the HDD Model Observe that in Eq. (8.21), μ is present only in the linear term. During the forced convection experiments, for higher heating, one can expect the μ(T ) variation to affect only this linear viscous-drag term. Therefore, for liquids (like water, PAO),  one may infer that when q → ∞, viscosity μ → 0 and hence the viscous-drag term in the HDD model, Eq. (8.21), may be neglected to read as follows,

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P = ρC0 U 2 . L

(8.23)

Equation (8.23) is for a limiting case. The physical situation akin to this model is a flow with small, nonzero positive local viscosity that renders, in Eq. (8.21), the global viscous-drag term negligible in comparison with the global form drag. It is important to remember that when viscosity actually is zero (μ = 0), there is no drag on the flow because of the porous medium. The fluid is ideal, and hence the flow is inviscid. Recall, inviscid flow requires (μ∇ 2 u = 0); this can happen even for μ = 0, with ∇ 2 u = 0 and should not experience any drag, form or otherwise.

8.2.4 Modified HDD Model Obviously, the local viscous-drag term, second term in the RHS of Eq. (8.22) is affected when the fluid is heated and the velocity profile is no longer flat. Moreover, the local velocity influenced by the local viscosity also alters the local form drag as it depends on the local velocity. Particularly, when u is only a function of y, the unidirectional differential Eq. (8.22) written with uniform viscous and form coefficients α0 = μ0 /K 0 and β0 = ρC0 , becomes ∂ p(y) = − ∂x



 μ(T ) α0 u(y) + β0 u(y)2 , μ0

(8.24)

with μ0 being the fluid dynamic viscosity evaluated at the inlet temperature T0 . An algebraic representation is obtained by the cross-sectional averaging of Eq. (8.24), the details of which are available in Narasimhan and Lage (2001). The final result, the Modified-HDD model reads,   μ0 P = ζμ U + ζC (ρC0 ) U 2 , (8.25) L K0 where the coefficients ζμ and ζC are lumped parameters, representing ζμ = ζC =

1 L

1 L

L 1 H 0

H

0

μ(T ) u(y)dy μ0

U L 1 H μ(T ) 0

H

0

μ0

U

 dx 

,

u(y)dy dx

.

(8.26)

The new algebraic model presented in Eq. (8.25) retains the same form of the original HDD model, Eq. (8.21) describing the transport of fluids with temperaturedependent viscosity through porous media. The coefficients ζu and ζC represent the lumped local effect of temperature dependent viscosity and the effect of viscosity on

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the fluid velocity profile, respectively. Obviously, for uniform viscosity (no heating), ζu = ζC = 1 and Eq. (8.21) is recovered. Comparing Eq. (8.25) with Eq. (8.21), it is apparent that the inappropriateness of the global HDD model, Eq. (8.21) is because it is unable to capture the indirect viscosity effect on the global form-drag term, a term originally believed to be viscosity-independent.

8.2.5 M-HDD Model Coefficients Figure 8.5 presents ζu and ζC for several heat flux values. Observe by following the  circles that for increasing heat flux, ζu reaches zero asymptotically, beyond q ∼ 2 0.5 MW/m (white circles). The region beyond these heat flux values, where ζμ ∼ 0, is referred as the ‘null global viscous-drag regime’, as shown in Fig. 8.5. As it is difficult to precisely identify the switch by ζμ , from nonzero (positive) to zero value, it is presented as a transition region. Based on these results, predictive empirical relationships for correcting the viscous- and form-drag terms, complementing the algebraic global (M-HDD) model,

Fig. 8.5 Coefficient ζμ and ζC of the M-HDD model, for several heat fluxes

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were obtained by Narasimhan and Lage (2001, 2004), as functions of the surface heat flux and inlet temperature, ⎡ ζμ = ⎣1 −



μ(Tin ) μ(Tr ) 

ζC = 2 +

0.15



Q 1 + Q 

μ(Tin ) μ(Tr )

0.28 ⎤  ⎦

1 1 + Q 

45 ,

(8.27)

0.2

5 Q



− ζμ−0.01 ,

(8.28)

with Tr , a suitable reference temperature based on the particular fluid, and the non dimensional Q given by    dμ    .  Q =

 dT  ke T0 μ 0 K 0 C0 

q˙



(8.29)

In Eq. (8.29), the viscosity, μ0 , and its derivative are evaluated at the inlet temperature T0 . In other words, the parameters necessary to estimate the dimensionless group in Eq. (8.27), using Eq. (8.29), are already known, once the isothermal pressure drop experiments to determine K 0 and C0 are performed.  Therefore, for a heat flux input q , by using Eqs. (8.27) and (8.29), we can estimate the viscosity variation effects from the M-HDD model, Eq. (8.25), on the total pressure-drop along the channel.

8.2.6 Hydrodynamics of μ(T ) Flows A summary of the fundamental implications of temperature-dependent viscosity effects on the global porous media flow models is presented in Fig. 8.6. The uppermost curve is for a non-heating (uniform viscosity) configuration, where the HDD model, Eq. (8.21), is fully valid. When the heat flux is progressively increased (following the block arrow) the ‘viscous-drag and form-drag regime’ is reached first. Due to the non-uniformity of the velocity profile (a result of spatially varying local viscosity), both the global viscous- and form-drag terms are affected. That is, the coefficients ζμ and ζC of the M-HDD model, Eq. (8.25), take nonzero positive values (ζμ < 1 and ζC > 1). This regime is followed by a transition beyond which a form-drag regime is achieved where only the form-drag term is affected by the viscosity variation. The interesting aspect of this transition is that it is similar but not identical to the transition from Darcy flow achieved even in unheated flows (constant viscosity) by merely increasing the flow speed, as discussed in Sect. 3.6. Both of these transitions make the flow form-drag dominant, i.e. the pressure-drop is balanced (more) by global form-drag.

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Fig. 8.6 Summary of the hydrodynamics of temperature-dependent viscosity flows through heated porous medium channels

However, the transition in the μ(T ) liquid flow achieved by heating makes it even more so, because it almost entirely obliterates the viscous-drag term. In the constant viscosity case, the viscous-drag term is only neglected (but is always present) in comparison with the form-drag term that gains magnitude for higher velocities. Moreover, transition by heating can happen for a particular velocity (observe in Fig. 8.6, the curve drops vertically) even well within the Darcy flow limit of the constant viscosity flow. This suggests that a Darcy flow can be made to become formdrag dominant at practically any speed, by merely heating the fluid sufficiently. This provides an insight into the departure from Darcy flow in the case of μ(T ) liquids: the departure to non-Darcy flow happens at λ < 1, i.e. an earlier value compared to λ ∼ 1, the transition criterion introduced in Sect. 3.6. Narasimhan and Lage (2002) have shown that by performing a heuristic scale comparison of the drag terms of the M-HDD model for the case of μ(T ) flows, the corresponding transition criterion for departure from Darcy’s flow can be predicted. Doing such a comparison (8.30) ζμ Dμ0 ∼ ζC DC0 would result in λT |μ(T ) =

ζμ . ζC

(8.31)

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Equation (8.31) gives the λT |μ(T ) , beyond which the flow becomes form-drag dominant for flows with temperature-dependent viscosity effects. Since ζμ < 1 and ζC > 1 always, the transition point for temperature-dependent viscosity flows is always less than that for the constant viscosity case (i.e. λT = 1).  In addition, for uniform viscosity, i.e. for isothermal flows (q = 0), ζμ and ζC are identically equal to unity, as seen earlier. This makes the prediction of Eq. (5.36) consistent with the previous result, i.e. λT |μ(T ) = λT = 1. Proceeding with Fig. 8.6, finally, when the heat flux is large enough, the viscosity effect on the form-drag becomes negligible and the fluid velocity profile becomes uniform again. At this limit, the flow is essentially independent of viscosity effects and the plot of global pressure-drop versus average fluid-speed reaches a minimum. Further heating will have no hydrodynamic effect through viscosity. This limit (at which ζC = 1 and ζμ = 0) is termed ‘heating form-drag limit’, as the global formdrag becomes independent of the viscosity effect. For this limiting case, the longitudinal pressure-drop will be equal to the form drag when the channel is not heated. In other words, this limit can be predicted by the simple equation, Eq. (8.23). Moreover, this result is fundamental in nature—true for all fluids with viscosity inversely dependent on temperature—and of great practical importance, as it sets an upper bound for the magnitude of the reduction in the global pressure-drop achievable by heating a fluid with temperature-dependent viscosity. The last assertion, Eq. (8.23), puts a limitation on the analogy between Hagen– Poiseuille flow through capillary beds and flow through porous media, for flows with temperature-dependent viscosity. In Hagen–Poiseuille flow, the pressure-drop decreases without limit, with an increase in the heat flux. In porous medium flow, the decrease in pressure-drop by increasing the heat flux is limited by the ever-existing pressure-drop caused by the form drag.

8.3 Flow and Convection in Bi-disperse Porous Media As introduced in Chap. 1, a porous medium whose solid matrix itself is another porous medium can be considered as a bi-disperse or bi-dispersed porous medium (BDPM). Chen et al. (2000) defined such a BDPM to be composed of clusters of large particles that are agglomerates of small particles. So the macro-porous medium is made of a solid matrix region which in turn is made of a micro-porous medium. In between such micro-porous clusters, there are definite macro-pores. Likewise, the micro-porous medium could contain micro- or even smaller-sized pores. Usually, these micro- and macro-pores are connected and the same fluid pervades through all of them. Nield and Bejan (2006) consider a BDPM as a porous medium in which fractures or tunnels have been introduced. The ‘porous medium’ here is the microporous medium, and the fractures or tunnels are the macro-pores that runs through them, resulting in a BDPM. Studies on mass diffusion in such BDPM have been reported in 1980s in chemical engineering literature, where chemical reactors can be modelled as BDPM. Nield

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195

and Kuznetsov have worked extensively on modelling convection in such BDPM, which are discussed in their review, Nield and Kuznetsov (2005) and also in Nield and Bejan (2006). Some applications of such BDPM modelling are in PEM fuel cells, porous wicks in heat pipes, modelling granular mixtures and chemical reactors. In this context, the heat exchangers with interconnected porous medium, discussed in Chap. 4, can be viewed as a BDPM. The cross-flow heat exchanger tube banks is a macro-porous medium, and the interconnecting porous medium is the micro-porous medium. In this section, we shall discuss modelling convection in such BDPM and its usefulness in thermal management design of electronics and data centres.

8.3.1 Laminar Flow Through BDPM Channel Narasimhan and Reddy (2011) have analysed forced convection cooling of distributed heat generating electronics by modelling the configuration as a BDPM channel. The BDPM channel considered is a two-dimensional porous block array, as shown in Fig. 8.7. Each porous block of size D ∗ × D ∗ can be viewed as an array of n × n (in figure, n = 5) regularly arranged square electronic components, each electronic component having dimensions of d ∗ × d ∗ . The heat generating porous blocks are distributed uniformly in the channel. The generated heat is removed by the fluid flowing through micro- and macro-pores of the BDPM channel. The bi-dispersion effects, such as the permeabilities of the micro- and macroporous medium (Da I and Da E respectively) and porosities (again, φ I and φ E respectively), can be expected to influence the forced convection thermohydraulic performance of the BDPM channel flow. The fluid flow and heat transfer behaviour are studied using the 2D versions of the mass, momentum and energy conservation equations.

8.3.2 Governing Equations and Boundary Conditions The system of governing partial differential equations is represented in nondimensional form in Narasimhan and Reddy (2011) using the following terms x y u v T − Tin , y∗ = , u∗ = , v∗ = , θ =  2 , H H u in u in Q s H /k f ( p − pref ) ν KI p∗ = , Pr = , Da I = 2 , 2 α H ρu in keff ρu in H Re H = , γ = , μ kf

x∗ =

(8.32)

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Fig. 8.7 Bi-disperse porous medium (BDPM), its equivalent ‘blocks’ type macro-porous medium model with each block being a micro-porous medium

where pref and Tin are the reference pressure and temperature and are taken as the atmosphere pressure and inlet temperature, respectively. Using the above terms, the non-dimensional governing equations for mass, momentum in the x ∗ and y ∗ directions and heat transport in porous media are written as ∂v ∗ ∂u ∗ + =0 (8.33) ∂x∗ ∂ y∗

u∗

 2 ∗  ∗ ∗ ∂ u ∂u ∗ φI ∂ 2u∗ ∗ ∂u 2 ∂p + v = −φ + + I ∂x∗ ∂ y∗ ∂x∗ Re H ∂ x ∗2 ∂ y ∗2   1/2 φ 2I c F φ I  ∗2 ∗2 − +√ v + u u∗, Re H Da I Da I

 ∂ 2 v∗ ∂ 2 v∗ u +v = + ∗2 ∂x∗ ∂ y∗ ∂ x ∗2 ∂y   1/2  2 φI cF φ − +√ I v ∗2 + u ∗2 v ∗ , Re H Da I Da I ∗ ∂v

∗

∗ ∂v

∗

∂ p∗ −φ 2I ∗

φI + ∂y Re H

(8.34)



(8.35)

8.3 Flow and Convection in Bi-disperse Porous Media

∂θ ∂θ γ u + v∗ ∗ = ∂x∗ ∂y Re H Pr ∗



197

∂ 2θ ∂ 2θ + ∂ x ∗2 ∂ y ∗2

 +

(1 − φ I ) , Re H Pr

(8.36)

where γ (keff /k f ) is the ratio of effective thermal conductivity of the porous medium to the thermal conductivity of a flowing fluid. One major assumption made by Narasimhan and Reddy (2011) while analysing this problem is that the convecting fluid and the porous matrix are in local thermal equilibrium. Inside the macro-pore region, Eqs. (8.34) and (8.35) are solved by setting φ I = 1 and Da I = infinity. Equation (8.36) is solved by setting ke f f = k f . The associated boundary conditions for Eqs. (8.33)–(8.36) with respect to the geometry shown in Fig. 8.7 are At the inlet u ∗ = 1, v ∗ = 0 and θ = 0. (8.37) x∗ = 0 : At the exit x ∗ = (L ∗in + L ∗ + L ∗out ) :

∂ 2u∗ ∂ 2θ ∗ = v = = 0, ∂ x ∗2 ∂ x ∗2

(8.38)

where L ∗in is the upstream length and L ∗out is the downstream length of the BDPM channel (Fig. 8.7). At the upper and lower walls y ∗ = 0, W ∗ :

u ∗ = v∗ = 0

and

∂θ = 0. ∂ y∗

(8.39)

At the interface between the macro-pores and the porous blocks, in line with the discussion on interface conditions in Chaps. 3 and 4, the following conditions are imposed   ∂u ∗  μeff ∂u ∗  ∗ ∗ u ∗cf = u ∗pm , pcf = ppm , = , (8.40) ∂n  μ ∂n  cf

cf

pm

and θcf = θpm ,

  ∂θ  ∂θ  −  = −γ  , ∂n cf ∂n pm

(8.41)

where n denotes the direction normal to the corresponding wall of each porous block. The results from the solution of the above conservation equations along with the boundary conditions using numerical methods are analysed using the following nondimensional variables. The non-dimensional pressure-drop p ∗ and Nusselt number N u are defined as

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1 p = ∗ W ∗

 W ∗

∗

W ∗

∗

p dy |x ∗ =L ∗ − 0

and

 p ∗ dy ∗ |x ∗ =2L ∗ ,

(8.42)

0

 N 2 D∗ D∗ Nu =

i=1 0

where N u x ∗ y∗ =

N u x ∗ y ∗ dx ∗ dy ∗ , (N 2 D ∗2 ) 0

2 Q  1 s H = . (Tx,y − Tin )k f θx ∗ y ∗

(8.43)

(8.44)

The performance parameter of a uniformly heat generating BDPM channel, χ , is defined as the maximum ratio of the dissipated heat to pumping power and is written as Nu , (8.45) χ= p ∗ Re3 where Re is the Reynolds number based on the hydraulic diameter of the channel (Re = Re H × (Dh /H )). For a given fluid (Pr = 0.7), BDPM channel volume (W ∗ × L ∗ × 1) and temperature difference, the dissipated heat is proportional to Nu which is dependent on the area (W ∗ × L ∗ ) of the BDPM channel. The pumping power is proportional to the p ∗ Re3 . Observe that for the BDPM geometry (with width and length W ∗ = L ∗ ) in Fig. 8.7, once the macro-pore volume fraction φ E and the number of porous blocks N 2 are known, individual block size D ∗ and macro-pore width δ can be determined uniquely by using the relations, (8.46) φ E = 1 − N 2 D ∗2 , and δ=

1 − D∗ N , N

(8.47)

where Eq. (8.46) is obtained by observing that the channel length or width has the same length as N macro-pore widths and N porous blocks.

8.3.3 Effects of Internal Permeability ( Da I ) For Re = 1000, N 2 = 25, φ I = 0.75 and φ E = 0.64, the effects of internal permeability Da I of the porous blocks on the streamlines and isotherms are shown in Fig. 8.8. The heat generating micro-porous blocks are shown as thin continuous wall squares. Streamlines and isotherms in Fig. 8.8 for Da I = 10−7 can be considered as the Mono-Disperse Porous Medium (MDPM) limit, where the blocks are solid

8.3 Flow and Convection in Bi-disperse Porous Media

199

Fig. 8.8 Streamlines and isotherms for 5 × 5 porous blocks at internal permeability Da I values (top) 10−7 and (bottom) 10−2 [Re = 1000, φ E = 0.64 and φ I = 0.75]. Adapted from Narasimhan and Reddy (2011)

(Da I → 0). Convection is restricted mostly to the macro-pores, around the microporous blocks. From this MDPM limit, as Da I increases, bi-dispersion effects appear. Fluid permeates the micro-pores as well, resulting in more uniform distribution of streamlines as seen in Fig. 8.8 for Da I = 10−2 . Convection increases significantly, resulting in a corresponding local maximum temperature decrease (Fig. 8.9). The effects of Da I on the Nusselt number N u (Eq. 8.43) and the non-dimensional pressure-drop p ∗ (Eq. 8.42) in the BDPM domain are shown in Fig. 8.10. Increasing Da I causes a reduction in BDPM channel pressure-drop and increase in forced convection. However, the variation in N u and p ∗ with Da I is not linear. Beyond Da I > 10−5 , the enhancement in N u and the corresponding reduction in p ∗ is phenomenal. This can be attributed to the increased permeation of flow inside the micro-porous blocks due to the reduced local viscous drag. The BDPM channel accommodates more uniform mass flow in a cross section, reducing p ∗ and the block temperature θ (thus increasing N u). It can be surmised that, when compared to a MDPM channel, the effect of bidispersion by increasing the Da I is to provide large heat transfer enhancement (∼ 4 times) and corresponding large reduction (∼ 8 times) in pressure-drop. Distributing heat generating electronics that comprise the micro-porous blocks by suitably

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Fig. 8.9 Streamlines and isotherms for 5 × 5 porous blocks at macro-pore volume fraction φ E values (top) 0.2 and (bottom) 10.86 [Re = 1000, φ E = 0.64 and φ I = 0.75]. Adapted from Narasimhan et al. (2011)

choosing Da I provides thermohydraulic performance enhancement using BDPM approach.

8.3.4 Effects of Macro-pore Volume Fraction (φ E ) Alternate ways to induce bi-dispersion effect is by varying macro-pore volume fraction φ E and number of blocks N 2 in the BDPM channel. This has been studied by Narasimhan et al. (2011). Here we shall explain the effect of φ E on the BDPM channel performance. Streamlines and isotherms in Fig. 8.9 reveal the effect of increasing φ E from 0.2 to 0.86, without altering the number of blocks. Observe that for fixed N 2 , φ E can be changed by varying the size of micro-porous blocks D ∗ and macro-pore gaps (Eq. 8.46). For fixed Da I = 10−3 and N 2 = 25, the effects of φ E on N u and p ∗ are shown in Fig. 8.11. An increase in φ E (or reduction in porous block size D ∗ ) causes an increase in the effective permeability of the BDPM channel. This reduces the macropore related viscous drag and hence, for fixed Re = 1000, a monotonous reduction in p ∗ is observed in Fig. 8.11 as φ E increases.

8.3 Flow and Convection in Bi-disperse Porous Media

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Fig. 8.10 Effect of internal permeability Da I on Nusselt number (N u) and pressure-drop ( p ∗ ) for 5 × 5 porous blocks [Re = 1000, φ E = 0.64 and φ I = 0.75]. Adapted from Narasimhan and Reddy (2011)

Unlike the earlier situation of varying Da I , when φ E is increased, the microporous block size reduces when N 2 are fixed. This reduction in volume, for constant  volumetric heat generation Q s , correspondingly reduces the magnitude of available energy in the BDPM domain. Hence, the enhancement in the heat transfer N u in Fig. 8.11 as φ E increases, is sustained across a smaller temperature difference, which is a beneficial design for cooling electronics. For φ E < 0.2 a limit is reached where the macro-pore channel size δ approaches or is equal to the micro-pore size (that determines Da I and φ I ) of the porous blocks. At this limit where macro-pore and micro-pore are of the same length scale, the BDMP domain becomes MDPM. Using the BDPM approach, modelling heat generating electronics as microporous blocks separated by macro-gaps is advantageous. The geometry and distribution of such a BDPM channel can be suitably tuned by changing the bi-dispersion parameters. This is done by changing the micro-porous medium φ I and Da I or the macro-porous medium Da E by varying φ E and N 2 . This provides flexibility in the thermal design to augment heat transfer and reduce pressure-drop.

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Fig. 8.11 Effect of macro-pore volume fraction φ E on Nusselt number (N u) and pressure-drop ( p ∗ ) for 5 × 5 porous blocks [Re = 1000, φ E = 0.64 and φ I = 0.75]. Adapted from Narasimhan et al. (2011)

8.3.5 Natural Convection in BDPM Enclosure In a separate study, Narasimhan and Reddy (2010) have investigated the bi-dispersion effects on the steady-state natural convection inside a BDPM enclosure. The enclosure configuration is similar in construct to the BDPM made of blocks, shown in Fig. 8.7, with all sides bounded by adiabatic walls. Natural convection is ensured with a temperature gradient across the side walls. Similar to the earlier discussion for BDPM forced convection, bi-dispersion effect in natural convection can also be studied by varying the internal Darcy number Da I of the micro-porous blocks and the external Darcy number of the macro-pores Da E . Streamlines and isotherms inside the BDPM enclosure for Ra = 107 are shown in Fig. 8.12. The micro-phase porous blocks are shown as thin continuous-walled squares in these figures. At the mono-disperse limit when the control volume forming the micro-phase porous blocks are solid (Da I → 0), the convection flow is restricted to the macro-pores. Since for the BDPM, γ = 1, the wall-to-wall heat transfer is dominated by convection flow. However, this highly restricted flow at Da I → 0 results in very little convection heat transfer as shown by the isotherms. As Da I increases, the resulting bi-dispersivity allows flow through the micro-porous blocks.

8.3 Flow and Convection in Bi-disperse Porous Media

203

Fig. 8.12 Streamlines and isotherms for BDPM enclosure with a 4 × 4 porous block array at Ra = 107 , Da E = 2.53 × 10−4 , Pr = 1, γ = 1, φ E = 0.64 and φ I = 0.5, for Da I → 0 solid blocks (top) and Da I = 10−3 (bottom)

Bi-dispersion effect operates between the two asymptotes of Da I → 0, where the enclosure with the solid blocks reduces to a mono-disperse porous medium, and Da I → ∞, where the enclosure is without the porous medium. As Da I increases, more flow ensues through the micro-pores. Correspondingly, for all Ra, the Nu increases between Da I → 0 and Da I → ∞. Bi-dispersivity effect on Nu progressively reduces as Ra increases. Narasimhan and Reddy (2010) have summarized their results in the form of a correlation to predict the average Nusselt number, N u h , for the BDPM enclosure as a function of Rayleigh number, Ra φ , and external Darcy number Da E , both defined by Ra φ = 0.144Ra 1.208 where Ra = and Da eq is calculated from

and

Da E = 1.005Da 1.208 eq ,

gβ(Th − Tref )L 3 , να

(8.48)

(8.49)

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8 Advanced Topics

Da E =

KE 1 φ E3 D ∗2 = , L2 180 (1 − φ E )2

(8.50)

where Da E is the non-dimensional representation of the permeability of the solid block-filled enclosure when treated as a porous medium. The correlation is written as N u = 0.577(Ra φ Da E )0.5 Nu =

0.07 0.37Ra 0.25 φ Da E

10 ≤ Ra φ Da E ≤ 200, 200 < Ra φ Da E ≤ 105 ,

(8.51)

where the Da E is separately modelled as a function of number of blocks, N , and the internal Darcy number Da I and is written as Da E =

4.1 × 10−3 + 0.007Da 0.32 I . N2

(8.52)

The above correlation is valid for fixed macro-pore volume fraction (φ E = 0.64) and micro-pore volume fraction (φ I = 0.5), within the parameter range: 3 ≤ N ≤ 8 and 0 ≤ Da I ≤ 10−7 . It predicts with a correlation coefficient of 0.98, the N u h data set generated by numerical simulations within ±15% and ±9% for 10 ≤ Ra φ Da E ≤ 200 and 200 < Ra φ Da E ≤ 105 respectively.

8.4 Two-Phase Flow Through Porous Media Modelling two-phase convection in porous media using two-phase mixture model is a subject of recent interest. Wang and Beckermann (1993) introduced a two-phase mixture model applying Darcy’s law and a one-energy model for porous media based on Local Thermal Equilibrium (LTE) of the solid, vapour, and liquid phases. The LTE assumption deviates in the following cases: (a) during step change, (b) at high heat flux and high flow velocity, (c) when the particles or pores size are not small enough, (d) when the thermal properties of the solid and liquid phases are vastly different from each other, (e) when there is significant heat generation in any of the phases and (f) in rapid phase change situations such as rapid evaporation and condensation. Yuki et al. (2008) studied numerically, the thermo-fluid flow characteristics in porous medium using a ‘modified two-phase mixture model and a two-energy model’. An LTNE formulation of the mixture model is required for analysing many applications. For instance, such a formulation is required to understand two-phase convection in light water reactors (LWRs), in the post severe accident scenario, wherein heat generating particulate debris beds are formed when corium melt comes in contact with water. In these situations, nuclear decay heat is removed by boiling of the water and convection.

8.4 Two-Phase Flow Through Porous Media

205

A two-phase mixture model, applying Darcy’s law and the two-energy model valid when LTNE prevails, is presented in this section. The typical configuration studied in nuclear reactor applications, where the present problem is relevant, is a fluid-saturated and heat generating vertical cylindrical porous matrix. Most of the boundary conditions are identical to the one discussed earlier in Sect. 5.5. The outer surface of the annulus is impermeable and bottom and top ones, permeable.

8.4.1 Mixture Model Governing Equations The flow is governed by the two-dimensional version of the conservation equations: mass, momentum (Darcy’s law) and energy transport subject to Local Thermal Non-Equilibrium (LTNE) between solid and fluid (mixture) phases. The Oberbeck– Boussinesq approximation, where the local density varies only with temperature and only within the buoyancy term, is invoked in subcooled liquid and superheated vapour regions. The convection flow is incompressible and laminar, and the Newtonian fluid is assumed to have constant physical properties in each phase, but the properties of the two-phase mixture vary with the relative proportion of each phase. The porous medium is of constant porosity, homogeneous and isotropic with constant thermophysical properties. The thermal dispersion effect in fluid is neglected. With the above assumptions, the governing equations for fluid mixture and solid are given below and a complete derivation of these equations and corresponding supplementary relations can be obtained from Wang and Beckermann (1993), Wang (1997), Yuki et al. (2008). Equation of continuity: ∂ρ + ∇ · (ρu) = 0. (8.53) φ ∂t Momentum equation: ρu = −

K (∇ p − ρk g). ν

(8.54)

Energy equations for fluid mixture and solid phases under LTNE (i.e. T f = Ts ) between the phases, are written separately, as follows φ

  K ρh fg ∂H + ∇ · (γh uH ) = ∇ · (h ∇ H ) + ∇ · f (s) g + Q sf , ∂t νv (1 − φ)(ρc)s

  ∂ Ts  = ∇ · ks,e ∇Ts − Q sf + (1 − φ)Q . ∂t

(8.55)

(8.56)

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8 Advanced Topics

8.4.2 Mixture Variables and Coefficients In Eqs. (8.53)–(8.56), the mixture variables and properties are defined as Density: ρ = ρl s + ρv (1 − s).

(8.57)

Velocity: ρu = ρl u l + ρv u v .

(8.58)

Enthalpy h and volumetric enthalpy H are given as ρh = ρl sh l + ρv (1 − s)h v

and

Pressure: pl + pv 1 p= + 2 2

H = ρ(h − 2h vsat ).

pc [λv (ξ ) − λl (ξ )]dξ.

(8.59)

(8.60)

0

Kinematic viscosity: ν=

1 . krl krv + vl vv

(8.61)

Relative mobilities for liquid and vapour phases are: λl =

ν K rl νl

and

λv =

ν K rv . νv

(8.62)

Kinematic density: ρk = ρl [1 − βl (T − Tsat )]λl + ρv [1 − βv (T − Tsat )]λv .

(8.63)

Hindrance function: f (s) = K rv λl .

(8.64)

Effective diffusion coefficient by capillary force can be written as D(s) =

K λl λv [− pc ]. ν

(8.65)

Advection correction coefficient is given by    (ρv /ρl )(1 − s) + s (λl + 1)h vsat − λl h lsat   γh = . (2h vsat − h lsat )s + (ρv /ρl )h vsat (1 − s)

(8.66)

8.4 Two-Phase Flow Through Porous Media

207

The effective diffusion coefficient may be written as h =

h fg dT D(s) + k f,e . ρl h fg + (ρl − ρv )h vsat dH

(8.67)

8.4.3 Constitutive Relations A list of constitutive relations is required to supplement the above mixture model. These relations are collected from literature and presented here, to complete the formulation. Permeability: φ 3 d 2p K = . (8.68) 150(1 − φ)2 Relative permeabilities of liquid and vapour phases, K rl = s 3

and

K rv = (1 − s)3 .

(8.69)

Capillary pressure function, J (s) = 1.417(1 − s) − 2.120(1 − s)2 + 1.263(1 − s)3 . Capillary pressure: pc (s) = √

σ J (s). K /φ

(8.70)

(8.71)

Liquid saturation: ⎧ H ≤ −ρl (2h vsat − h lsat ) ⎪ ⎨1 +ρv h vsat −ρ s = − ρl h fgH+(ρ l (2h vsat − h lsat ) < H ≤ −ρv h vsat l −ρv )h vsat ⎪ ⎩ 0 H > −ρv h vsat .

(8.72)

Temperature: ⎧ H + 2ρl h vsat ⎪ ⎪ H ≤ −ρl (2h vsat − h lsat ) ⎪ ⎪ ρl cl ⎨ T = Tsat −ρl (2h vsat − h lsat ) < H ≤ −ρv h vsat ⎪ ⎪ h H + 2ρ ⎪ v vsat ⎪ H > −ρv h vsat . ⎩Tsat + ρv cv

(8.73)

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8 Advanced Topics

k f,e , the effective thermal conductivity of fluid in Eq. (8.66), and ks,e , the effective thermal conductivity of solid in Eq. (8.56) are modelled with φ(kl s + kv (1 − s)) and (1 − φ)ks + kr , respectively. Radiation is expected to play an important role in reality because the temperature inside the dry zone will be very high and the bed is porous. Modified thermal conductivity based on the Rosseland approximation (see Sect. 7.5) is used to account for the contribution by radiation. The local heat transfer between the solid and fluid phases is an important parameter that affects the growth of the two-phase region. The corresponding relationships to find the local heat transfer in single- and two-phase regions have been presented in Wakao et al. (1979) and, Rohsenow and Choi (1961), respectively and are written as follows: When Ts < Tsat : (8.74) Q sf = h sf av (Ts − T f ), 1

where, local heat transfer coefficient, h sf = (k f /d p )(2 + 1.1Pr 3f Re0.6 p ) and specific surface of the porous medium, av = 6(1 − φ)/d p , respectively. When Ts ≥ Tsat : Q sf is found according to the degree of liquid saturation using the equation below Q sf = av sqall + av (1 − s)h sf (Ts − T f ).

(8.75)

The first term on the right-hand side includes influences of pool boiling and forced convection heat transfer of a fluid with degree of subcooling, qall = qboil + qconv . qboil and qconv are calculated as follows:  qboil = μh fg and

g(ρl − ρv ) σ

1/2 

cpf (Ts − T f ) Csf h fg

3 

1 Pr f

5.1

    ρ|u|d p (Pr f )−2/3 (Ts − T f ), qconv = 0.019 cpf ρ|u| μf

,

(8.76)

(8.77)

where Csf is a constant decided by the combination of fluid and porous matrix properties. For the water and stainless steel combination Csf = 0.008. In addition to the above conservation equations and constitutive relations, the following equations are used to calculate the individual phase velocities from a fluid mixture: ρl ul = jl + λl ρu

and

ρv uv = −jl + λv ρu,

(8.78)

where jl is diffusive flux of liquid and is written as   λl λv K ∇ pc + (ρl − ρv )g . jl = ν

(8.79)

8.4 Two-Phase Flow Through Porous Media

209

Fig. 8.13 Comparison of volumetric vapour fraction predicted by present numerical results with experimental results for porous enclosure height 0.1048

8.4.4 Sample Result The finite volume formulation of Eqs. (8.53)–(8.56) can be solved using numerical simulations for suitable boundary conditions—similar to the one discussed in Sect. 5.5. Sample results from such a simulation are compared with published experimental data by Sundergeld and Turcotte (1977) in Fig. 8.13. Further results using the mixture model for two-phase flow in the above heat generating porous annulus example, like the LTNE influence on the temperatures and the prediction of dry-out heat flux for several inlet mass flow rates, are available in Reddy (2010).

8.5 LBM Formulation for Porous Medium Flows 8.5.1 Introduction In recent years, the lattice Boltzmann method (LBM) has emerged as a powerful tool in computational fluid dynamics (Chen and Doolen 1998). Since the advent of

210

8 Advanced Topics

discrete lattice kinetic theory and in particular lattice Boltzmann method (LBM) for solving the Navier–Stokes equations, several isothermal flow simulations have been performed using LBM. They have been discussed in the excellent monograph by Succi (2001). Compared to traditional methods, LBM has the advantages of being intrinsically parallel and easier to implement. LBM solves the discrete Boltzmann equation for the particle density distribution function f , as propounded by He et al. (1996). The zeroth and first moments of f recover the macroscopic density and velocity, respectively. Dixit and Babu (2006) have studied natural convection in a square enclosure at high fluid Rayleigh numbers using the thermal formulation of the LBM. The universal structure of the turbulent boundary layer was accurately predicted by their LB calculations. Guo and Zhao (2002) proposed an LB model to simulate convection in porous media. Seta et al. (2006) simulated natural convection in a porous square cavity for Ra m ≤ 104 . Pan et al. (2006) have used the multiple relaxation time model to study the dependence of permeability on fluid viscosity for flow through porous media. Babu and Narasimhan (2010) simulated incompressible, 2D flow around a porous square cylinder placed in an infinite stream using the d2q9i model of the LBM. Ramanathan et al. (2011) have investigated using LBM, natural convection inside square porous medium enclosures at high porous medium modified Rayleigh numbers (Ra m ≥ 105 ). The correlations from this study have been discussed earlier in Sect. 5.4. In this section, we present briefly the LBM formulation valid for porous medium flows. Since a porous medium in its basic form comprises at least two constituents, it offers two approaches of simulation, which is valid also for LBM. The first one is to treat the solid matrix as a complex ‘internal wall’ and simulate only the flow by considering the LBM formulation of the Navier–Stokes equations. Obviously, except for simple, periodic, porous medium geometries, this ‘direct simulation’ approach is computationally intensive and tedious. The second method is to invoke the volume averaging procedure, discussed throughout this textbook, to homogenize the porous medium and employ the LBM formulation on the generalized momentum equation valid on the porous continuum. This is the approach favoured by most researchers.

8.5.2 d2q9 Model In this d2q9 model, each node of the lattice is populated by three kinds of particles—a rest particle that resides in the node, particles that move in the coordinate directions and particles that move in the diagonal directions. The total number of particles in each node in this model is 9. The speed of the particles is such that they move from one node to another during each time step. These speeds can be written as

8.5 LBM Formulation for Porous Medium Flows

211

⎧ 0, i =0 ⎪ ⎪ ⎨ c(cos((i − 1)π ), sin((i − 1)π )), i = 1, 2, 3, 4 ei = √ 2c(cos((i − 5)π/2 + π/4), ⎪ ⎪ ⎩ sin((i − 5)π/2 + π/4)), i = 5, 6, 7, 8

(8.80)

Here c = δx /δt , where δx and δt are the lattice spacing and the time step, respectively. In the traditional LB method, the particles at each node undergo collision followed by advection. In terms of distribution functions, this can be written as f i (x + ei δt , t + δt ) − f i (x, t) =  1 (eq) f i (x, t) − f i (x, t) + δt Fi , − τ

0 ≤ i ≤ 8,

(8.81)

where τ is the dimensionless collisional relaxation time and is related to the kinematic viscosity ν of the lattice fluid as ν=

(2τ − 1) δx2 , 6 δt

(8.82)

and Fi is the force term that accounts for drag effects due to the presence of the porous medium. This can be written as    ei · F 1 9 (ei · u)(ei · F) 3 u·F , 3 2 + − Fi = wi ρ 1 − 2τ c φ c4 φ c2

(8.83)

where φ is the porosity. Here, F is the body force due to the presence of the porous medium and is given as F=−

φc F φν u − √ |u|u + φG, K K

(8.84)

where K is the permeability and G is the body force due to gravity. The formulation above is equivalent to the differential form of the HDD model, Eq. (3.9), with C = cK −1/2 . Obviously, the first term in the above equation is the viscous-drag term and the second term, which is quadratic in velocity, is the formdrag term. When the velocity is low, i.e. for Darcy flow, the second and the third terms in the above equation can be neglected. Thus,   φν 1 Fi = − wi ρ 1 − K 2τ   ei · u 9 (ei · u)2 3 u·u , 3 2 + − c φ c4 φ c2

0 ≤ i ≤ 8,

(8.85)

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8 Advanced Topics

Equation (8.81) above describes the evolution of the LB automaton. The equilibrium functions that appear in the right-hand side of this equation can be evaluated as follows:   ei · u 9 (ei · u)2 3 u·u (eq) , 0 ≤ i ≤ 8, (8.86) − f i = wi ρ 1 + 3 2 + c 2φ c4 2φ c2 where w0 = 4/9, wi = 1/9 for i = 1, 2, 3, 4 and wi = 1/36 for i = 5, 6, 7, 8. Also, ρ is the density and u is the velocity of the fluid. These are given as ρ=

8 

fi ,

(8.87)

i=0

ρu =

8 

ei f i +

i=1

δt ρF. 2

(8.88)

If we substitute for F from Eq. (8.84), we get ρu =

8  i=1

ei f i −

δt φν ρu. 2 K

(8.89)

8.5.3 d2q9i Model In the lattice Boltzmann method, the pressure is calculated using an equation of state. Pressure and density are related through the speed of sound as p = cs2 ρ,

(8.90)

√ where cs = c/ 3 is the speed of sound. The entire governing equations can be written in terms of pressure rather than density using the above equation so that the pressure field can be obtained directly. This is the d2q9i model. Further details of this model are available in He et al. (1996) and Baskar and Babu (2004). Another advantage of this approach is that the incompressible Navier–Stokes equations can be recovered from the LB equations to within O(Ma 2 ) where Ma is the Mach number, in contrast to the d2q9 model which recovers the incompressible N–S equations only to within O(Ma). The derivation of the working equations for the d2q9i model follows.

8.5 LBM Formulation for Porous Medium Flows

213

Upon multiplying Eq. (8.81) by cs2 and defining pi = cs2 f i , we get pi (x + ei δt , t + δt ) − pi (x, t)  1 (eq) pi (x, t) − pi (x, t) + δt cs2 Fi , =− τ

0 ≤ i ≤ 8.

(8.91)

Since the flow is incompressible, the density is essentially constant but with small fluctuations, viz., ρ = ρ0 + δρ. The quantity ρ0 can be thought of as a reference density and it has been conveniently taken to be equal to 1 in the present work. The exact value used for ρ0 is immaterial as the solution obtained is independent of this value. It should be noted that δρ is O(Ma 2 ) in the incompressible limit. The equilibrium functions that appear in the right hand side of Eq. (8.91) can be evaluated by first multiplying both sides of Eq. (8.86) by cs2 and then substituting for ρ. After neglecting product terms like δρ(u/c) and δρ(u/c)2 , which are actually O(Ma 3 ) and O(Ma 4 ), we get (eq)

pi

  3ρ0 ρ0 (ei · u)2 − u·u , = wi p + ρ0 (ei · u) + 2φ 2φ

0 ≤ i ≤ 8,

(8.92)

where, p is the pressure as defined above. In a similar manner, it is easy to show from Eqs. (8.85), (8.87) and (8.89) that   1 φν Fi = − wi ρ0 1 − K 2τ   ei · u 9 (ei · u)2 3 u·u 3 2 + , − c φ c4 φ c2 and p=

8 

pi ,

0 ≤ i ≤ 8,

(8.93)

(8.94)

i=0

and ρ0 u =

8 1  δt φν ρ0 u. eii pi − cs2 i=1 2 K

(8.95)

To solve the above lattice models using computational methods, particularly in the ISLB method, the lattice Boltzmann automaton is assumed to reside on a uniform lattice with spacing equal to δx . During each time step, the particles on this lattice undergo collision followed by advection as dictated by the lattice-BGK model. However, the particle distribution functions f i (and hence the density ρ and velocity u) are evaluated on the nodes of a computational grid overlaid on the lattice. The nodes of the computational grid need not coincide with the nodes of the lattice and after advection on the lattice, the particles can be off-node. The particle distribution

214

8 Advanced Topics

function at each node of the computational grid can then be determined by using second-order accurate Lagrangian interpolation, as shown in He et al. (1996). When the configuration involves mass inlet/outlet (for instance, porous medium channel flow), at the inlet, the particle distribution functions are set equal to the corresponding equilibrium distributions. At the exit, the particles are simply allowed to leave the domain. On the top and bottom boundaries (channel walls), slip boundary condition is imposed using the particle reflection rule. Further details of the formulation and computational solutions are available in Babu and Narasimhan (2010) and Ramanathan et al. (2011).

8.6 Combustion in Inert Porous Media Combustion in inert porous media finds use in many engineering applications, especially with growing demand for high power density, high-power dynamic range, less-polluting power production devices. Combustion within a porous medium differs from open combustion in that, the porous medium induces different heat transfer mechanisms, since, for the same volume, the emissive and absorptive power of thermal radiation of the porous media are much higher than those of gas. The concept of introducing a porous medium into burners may have originated following a suggestion by Weinberg in 1971 that pollutant emission reduction can be achieved by heat recirculation without dilution in a burner. Porous medium combustion is an internally self-organized process because the well-developed inner surface of the solid component of the porous medium causes efficient heat transfer between the combustible medium and the inert solid. The interconnected pores of the medium enable easier diffusion of species and heat transfer between phases, thus causing a regenerative internal heat feedback mechanism that results in free-burning flame, higher burning velocities, low emission of pollutants and the ability to burn fuels with a low energy content (Fig. 8.14). Porous medium combustion is analysed by the stationary or transient system approaches. The former is typically used to study radiant burners and surface combustion heaters where the solid component of the porous medium exhibits high radiative emissivity. The transient approach is used where an unsteady reaction zone freely propagates as a combustion wave in the downstream or upstream direction due to positive and negative energy fluxes. The resultant ‘excess enthalpy combustion’ theory is the basis for high-efficiency, low-polluting reactors that use high reactant velocity. In porous medium, the line between the two types of combustion—deflagration and detonation—is blurry. The stabilization and propagation of flame in a porous medium is given by the modified Péclet number. Separate energy equations for the solid and fluid (gas) components are used in modelling studies. Conduction and radiation equations, described in earlier chapters, are incorporated in the solid phase equation, as is the convection between solid and fluid. The radiant heat transport can be approximated using the Rosseland diffusion equation. The basic RTE and energy

8.6 Combustion in Inert Porous Media

215

Combustion products Heat transfer Reaction zone: PM with large pore size

Heat loss (conduction and radiation from solid, convection and dispersion in fluid)

Internal heat feedback

Preheating zone: PM with large pore size

Flame stabilization

Reactants: Fuel + air Fig. 8.14 Schematic of two-layer, premixed, porous medium burner with the heat transfer interactions

equations discussed in Chap. 7 are adequate in modelling radiation heat transport during combustion. More complex radiation models that are being developed seek to consider emission, absorption and scattering. When there is flow, the fluid phase energy equation includes terms for conduction and chemical energy release. Applying single-step reaction kinetics is a simplistic route to PMC modelling, but multiple reaction kinetics is incorporated for more realistic modelling. Terms for the heat flux due to chemical potential gradient (Dufour effect) and mass diffusion due to temperature gradient (Soret effect) are also incorporated. A detailed account of the mathematical formulation for PMC are given in Chap. 11 of ‘Transport Phenomena in Porous Medium’. A good modelling of porous medium combustion is seen in Chen et al. (1988), where the energy and species equations are used to model porous medium burners. This model used a multi-step mechanism for methane that included 17 species and 55 reactions. The effect of flame speed on temperature profiles was obtained from parametric variations of the thermal conductivity of the solid, volumetric heat transfer coefficient and radiative properties. A one-dimensional medium was modelled with no scattering effects, and a one-energy equation was solved assuming a very high local heat transfer coefficient and temperature invariance between solid and fluid. Similarly, Zhou and Pereira (1997) modelled one-dimensional combustion and heat transfer of methane/air fuel in a two-region burner with a small-pore size upstream considering a more detailed reaction mechanism describing formation and destruction of nitrogen oxides, which includes 27 species and 73 reactions. The effect of

216

8 Advanced Topics

excess air ratio, thermal power, solid conductivity and radiative heat transfer to the temperature profiles and the emission of CO and NO were analysed, and it was found that NO and CO emission depend mainly on the excess air ratio and thermal power. The peak flame temperature was reduced with the reduction of the solid conductivity resulting in a decrease of NO emission—thus, NO emission can be decreased through the use of porous medium with small solid conductivity. A detailed account of modelling studies pertaining to PMC is found in reviews by Mujeebu et al. (2010).

Suggested Reading

Dynamics of Fluids in Porous Media by J. Bear, Dover, New York, 1998. Theory of Porous Media by R. de Boer, Springer-Verlag, Berlin, 2000. Turbulence in Porous Media: Modeling and Applications by M. de Lemos, Elsevier, San Diego, 2006. Porous Media: Fluid Transport and Pore Structure by F. A. L. Dullien, Academic Press, San Diego, 1992. Porous Media Theory, Experiments and Numerical Applications, Eds. W. Ehlers and J. Bluhm, Springer-Verlag, Berlin, 2002. Transport Phenomena in Porous Media, Eds. D. B. Ingham and I. Pop, Elsevier, Oxford, 1998. Emerging Technologies and Techniques in Porous Media, Eds. Ingham, D. B., and Bejan, A, and Eden Mamut and Ioan Pop, Kluwer Academic, Netherlands, 2004. Heat Transfer in Porous Media by M. Kaviany, 3rd Edition, Wiley, 1995. Convection in Porous Media by D. A. Nield and A. Bejan, 3rd Edition, Springer, 2006. Emerging Topics in Heat and Mass Transfer in Porous Media: from Bioengineering and Microelectronics to Nanotechnology, Ed. P. Vadàsz, Springer-Verlag, Berlin, 2008. Handbook of Porous Media, First Edition, Ed. K. Vafai, Marcel Dekker, New York, 2000. Handbook of Porous Media, Second Edition, Ed. K. Vafai, Taylor & Francis, Florida, 2005. Porous Media: Applications in Biological Systems and Biotechnology, Ed. K. Vafai, CRC Press, Florida, 2011. The Method of Volume Averaging by S. Whitaker, Kluwer Academic, Netherlands, 1999.

© The Author(s) 2023 A. Narasimhan, Essentials of Heat and Fluid Flow in Porous Media, https://doi.org/10.1007/978-3-030-99865-3

217

Appendix A

Porosity and Permeability Values

(Major source: Nield and Bejan (2006); range of values pertaining to applications are collected from research literature) See Tables A.1 and A.2.

Table A.1 Properties of common porous materials Material φ Agar-agar Black slate powder Brick Cigarette Cigarette filters Coal Concrete (bituminous) Copper powder (hot-compacted) Cork board Granular crushed rock Hair (on mammals) Leather Limestone (dolomite) Sand Sandstone (oil sand) Silica grains Silica powder Soil Spherical packings (well shaken) Wire crimps

0.57–0.66 0.12–0.34

K , (cm 2 ) 2 × 10−10 −4.4 × 10−9 4.9 × 10−10 −1.24 × 10−9 4.8 × 10−11 −4.4 × 10−9 1.1 × 10−5

0.17–0.49 0.02–0.12 0.09-0.34

1 × 10−9 −2.3 × 10−7 3.3 × 10−6 −1.5 × 10−5 2.4 × 10−7 −5.1 × 10−7

0.45 0.95–0.99 0.56–0.59 0.04-0.10 0.37–0.50 0.08–0.38 0.65 0.37–0.49 0.43–0.54 0.36–0.43 0.68–0.76

9.5 × 10−10 −1.2 × 10−9 2 × 10−11 −4.5 × 10−10 2 × 10−7 −1.8 × 10−6 5 × 10−12 −3 × 10−8 1.3 × 10−10 −5.1 × 10−10 2.9 × 10−9 −1.4 × 10−7

3.8 × 10−5 −1 × 10−4

© The Author(s) 2023 A. Narasimhan, Essentials of Heat and Fluid Flow in Porous Media, https://doi.org/10.1007/978-3-030-99865-3

219

220

Appendix A: Porosity and Permeability Values

Table A.2 Properties of common porous materials Material φ Application

K , (cm2 )

Geothermal energy exploitation Ground water aquifer

0.4 – 0.6

10−16 – 10−3

0.03 – 0.21

Porous wick in heat pipe Drying process-food Catalysts and adsorbents Nuclear safety analysis PEM Fuel cells Copper wire mesh

0.4 – 0.6 0.1 – 0.6 0.65 – 0.75 0.3 – 0.4 0.3 – 0.7 0.58

4.9346 × 10−8 – 4.9346 × 10−6 10−4 – 10−3 1.974 × 10−10 – 2.27 × 10−7 2.18 × 10−10 – 7.78 × 10−9 10−7 – 10−5 10−8 – 10−7 4.183 × 10−7

Appendix B

Convection Results and Correlations

Results and recent correlations pertaining to forced and natural convection inside porous media (Chaps. 4 and 5) are collected here. A major source of reference is Nield and Bejan (2006). Forced Convection Plane wall with Prescribed temperature—Bejan (1984) x q  = 0.564 Pex1/2 Tw − T∞ km

N ux =

where N u x is local Nusselt number, Tw is temperature at surface, T∞ is the far-field temperature of saturated porous medium, q  is the local heat flux, km refers to thermal conductivity of the porous medium, Pex is the Peclet number based on U∞ and x, given by U∞ x/αm . The overall Nusselt number based on the heat flux q¯  averaged from x = 0 to a given plate length x = L is NuL =

L q  1/2 = 1.128 Pe L Tw − T km

Plane wall with constant Heat flux—Bejan (1984) N ux =

x q  = 0.886 Pex1/2 Tw (x) − T km

The overall Nusselt number that is based on the average wall temperature T w (specifically, the temperature averaged from x = 0 to x = L) is

© The Author(s) 2023 A. Narasimhan, Essentials of Heat and Fluid Flow in Porous Media, https://doi.org/10.1007/978-3-030-99865-3

221

222

Appendix B: Convection Results and Correlations

NuL =

L q  1/2 = 1.329 Pe L Tw − T km

Sphere and Cylinder: Boundary Layers—Cheng (1982) The distributions of heat flux around the sphere and cylinder were determined by Cheng. Local peripheral Nusselt number N u θ is given by: Sphere: 

ur0 θ N u θ = 0.564 αm Cylinder:

1/2 



3θ 2

ur0 θ N u θ = 0.564 αm

1/2

 sin θ 2

1 2 cos3 θ − cos θ + 3 3

1/2

1/2 (2θ )1/2 sin θ (1 − cos θ )1/2

Pèclet number based on the swept arc r0 θ , is given by Peθ = ur0 θ/αm . N uθ =

r0 θ q  Tw − T∞ km

Equations above are valid when the boundary layers are distinct (thin), i.e. when the boundary layer thickness r0 Pe1/2 is smaller than the radius r0 . This requirement can also be written as Pe1/2  1, or N u θ  1. Sphere: 1/2 N u D = 1.128 Pe D Cylinder:

1/2

N u D = 1.015 Pe D

In these expressions, the Nusselt and Peclet numbers are based on the diameter D = 2r0 , D q  uD NuD = , Pe D = Tw − T km αm Confined Flows In cases where the confining wall is a tube with the internal diameter D, the relation for fully developed heat transfer can be expressed as a constant Nusselt number (Rohsenow and Choi 1961): NuD =

D q  (x) = 5.78 (tube, Tw = constant) Tw − Tb (x) km

Appendix B: Convection Results and Correlations

NuD =

223

D q  = 8 (tube, q  = constant) Tw (x) − Tb (x) km

When the porous matrix is sandwiched between two parallel plates with the spacing D, the corresponding Nusselt numbers are (Hartnett and Rohsenow 1973) NuD =

D q  (x) = 4.93 (parallel pates, Tw = constant) Tw − Tb (x) km

NuD =

D q  = 6 (parallel plates, q  = constant) Tw (x) − Tb (x) km

The forced convection results are valid when the temperature profile across the channel is fully developed, i.e. sufficiently far from the entrance x = 0. Effects of Thermal Dispersion:External Flow—Lai and Kulacki (1989) N u x = 0.886(1 + C Ped )Pex1/2 where Ped = U∞ d p /αm in which d p is the mean particle or pore diameter and C is a numerical constant. External Natural Convection Vertical plate- Cheng and Minkowycz (1977) N ux = −θ  (0) Ra 1/2 x where the local Nusselt number is defined by N u x = hx/k and where h is the local heat transfer coefficient q  /(Tw − T∞ ), θ (η) = (T − T∞ )/(Tw − T∞ ), and η is given 1/2 by (y Rax )/x, Rax refers to Rayleigh number given by gβ K (Tw − T∞ )x/ναm . Boundary fiction effects—Kim and Vafai (1989) Kim and Vafai found that the local Nusselt number N u x is given by 1/4 N u x = 0.5027Dax−1/4 Ra 1/4 x = 0.5027(Ra f ϕ)

where Ra f is the standard Rayleigh number for a viscous fluid (independent of permeability), as expected for a very sparse medium. Horizontal plate—Cheng and Chang (1976) ηT δ = x Ra 1/3 x where ηT is the value of η at the edge of the boundary layer.

224

Appendix B: Convection Results and Correlations

N ux = −θ  (0) Ra 1/3 x Nu Ra

1/3

3(1 + λ)4/3 [−θ  (0)] (1 + 4λ)

=

for the thermal boundary layer thickness δ, the local Nusselt number N u x and the overall Nusselt number N u. Scaling analysis shows, for a horizontal plate, N u ∼ Ra 1/3 Horizontal cylinder—Merkin (1978) N uϕ 1/2 Ra D

= 0.628

sin ϕ (1 − cos ϕ)1/2

where N uϕ = and Ra D =

qw D km (Tw − T∞ )

gβ K (Tw − T∞ )D ναm

with D denoting the diameter of the cylinder, ϕ the angle that the y-axis makes with the downward vertical. The average surface heat flux in dimensionless form is given by Nu 1/2

Ra D

= 0.565

Sphere-Merkin (1979) Nu 1/2

Ra D

= 0.724

Internal Natural Convection Heating from below—Wang and Bejan (1987)  Nu =

Ra 40

1/n

n + [c(Ra Pr p )1/2 ]n

Appendix B: Convection Results and Correlations

225

where n and c are two empirical constants, n = −1.65 and c = 1896.4 where, 2 Pr p = Pre HK Pre is effective Prandtl number given by Pre =

Pr f k f C F km



K H2

1/2

where Pr f refers to fluid Prandtl number, C F is Forchheimer coefficient, K is permeability and H is vertical dimension. Vertical cylinder or annulus—Wooding (1957) Rac = λ21

H2 H2 = 3.390 r02 r02

where λ = Ra 1/2 . ˜ c = 3.390, where This can be written as Ra 2 2 ˜ = Ra r0 = gβ K r0 T Ra H2 H ναm

Horizontal Cylinder or Annulus—Storesletten and Tveitereid (1987) Rac was found to be 46.265 Ra =

gβT K r0 ναm

where r0 is the radius of the cylinder and T is the temperature difference across the vertical diameter. Internal Natural Convection: Heating from the Side Darcy Flow between Isothermal Side Walls—Bejan (1985) Ra =

Nu =

gβ K H T ναm

L q km T Ra 1/2 ∼ Ra 1/2 ∼ qc km H T /L H

in which qc = km H T /L is the true heat transfer rate in the pure-conduction limit (i.e., in the absence of convection).

226

Appendix B: Convection Results and Correlations

The four regimes that characterize the heat transfer through a porous layer heated from the side. The following heat transfer scales may be adopted in the different regimes (Daniels et al. 1982). I. Pure conduction (no distinct boundary layers): T Nu ∼ = km H = 1, q  ∼ L II. Tall layers (distinct horizontal boundary layers only): N u >1, q  > km H

T L

III. High-Ra convection (distinct vertical and horizontal boundary layers): T L Ra 1/2 , q  ∼ km H H H

Nu ∼

IV. Shallow layers (distinct vertical boundary layers only): N u