Transport Phenomena in Porous Media

TRANSPORT PHENOMENA IN POROUS MEDIA Volume III

Elsevier Internet Homepage - http ://www.e[sevier.com Consult the Elsevier homepage for full catalogue information on all books, major reference works, journals, electronic products and services. Elsevier Titles of Related Interest Derek B. Ingham and loan Pop Transport Phenomena Porous in Porous Media 1998, 0-08-042843-6 Derek B. Ingham and loan Pop Transport Phenomena Porous in Porous Media II 2002, 0-08-043965-9 Hartnett et al Advances in Heat Transfer serial See www.elsevier.com for full details Kandlikar et al. Heat Transfer and Fluid Flow in Minichannels and Microchannels 2005, 0-08-044527-6 Related Journals: Elsevier publishes a wide-ranging portfoho of high quality research journals, including papers detailing the research and developments of heat and fluid flow in porous media. A sample journal issue is available online by visiting the Elsevier web site (details at the top of this page). Leading titles include: Applied Thermal Engineering European Journal of Mechanics B/Fluids Experimental Thermal and Fluid Science Flow Measurement and Instrumentation International Communications in Heat and Mass Transfer International Journal of Heat and Fluid Flow International Journal of Heat and Mass Transfer International Journal of Multiphase Flow International Journal of Refrigeration International Journal of Thermal Sciences All journals are available online via ScienceDirect: www.sciencedirect.com To contact the Publisher Elsevier welcomes enquiries concerning publishing proposals: books, journal special issues, conference proceedings, etc. All formats and media can be considered. Should you have a publishing proposal you wish to discuss, please contact, without obligation, the publisher responsible for Elsevier's Mechanics and Mechanical Engineering programme: Arno Schouwenburg Senior Pubhshing Editor Elsevier Ltd The Boulevard, Langford Lane Kidlington, Oxford 0X5 1GB, UK

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Preface Over recent years, fluid flow and heat transfer through porous media has seen an explosive increase in research attention and this is evident through the creation of new journals, existing journals publishing more papers, research books, edited research books, and international conferences and workshops on this topic. This rapidly increasing research activity has been mainly due to the increasing number of important applications on porous media in many modem industries, ranging from heat removal processes in engineering technology and geophysical problems to thermal insulation, chemical reactors, the underground spread of pollutant, heating of rooms, combustion, fires, and many other heat transfer processes, both natural and artificial. The physical scale of these problems range from the micro to the macro. The rapid expansion of research applications has resulted in the production of numerous new, novel and sophisticated mathematical approaches for which analytical or semi-analytical or numerical and experimental solutions have been developed. On the other hand, there are still numerous new types of applications being exploited in which new and exciting phenomena are present. Thus, it is very appropriate, interesting and timely to put some of the most recent research work in porous media in a new volume of Transport Phenomena in Porous Media, volume III. The first two volumes in this series were published in 1998 and 2002 and they were very successful and met with an excellent response by the researchers and users in the porous media community. Despite the large amount of previous research work dealing with fluid flow and heat transfer in porous media, there is still a considerable need for more comprehensive and reliable methods of accurately predicting the fluid flow and heat transfer characteristics in many problems. Thus, the present volume provides, like the previous two volumes, a thorough discussion of transport phenomena in porous media and it lays the foundation for the understanding of a wide variety of techniques used by applied mathematicians, physicists and practitioners. Each chapter begins with the theory, followed by illustrations of the way the theory can be used to obtain fairly complete solutions, and finishes with conclusions and suggestions for further research work. A broad range of technologically important applications are provided throughout all the chapters of the book. The volume contains 17 chapters and represents the collective work of 42 of the world's leading experts from 13 countries and 5 continents in the fluid flow and heat transfer in porous media. As in the previous two volumes in this book series, all the chapters of the book are very much interrelated and thus it was not easy to decide the order of the chapters. Further, some

VI

PREFACE

of the views expressed are controversial but this can only be beneficial to the development of the subject as they will no doubt provoke further new and novel research. In Chapter 1, de Lemos gives a critical review of the recently published methodologies to mathematically characterise turbulent transport in porous media. He then introduces a new concept, called double-decomposition, and the models for turbulent transport in porous media are classified in terms of the order of application of the time and volume averaging operators. Instantaneous local transport equations are reviewed for clear fluid flows before the time and volume averaging procedures are applied to them. The double-decomposition concept is presented and thoroughly discussed prior to the derivation of macroscopic governing equations. A number of natural and engineering systems can be characterised by a permeable structure through which a working fluid permeates. Turbulence models proposed for such flows depend on the order of application of the time and volume average operators. Two methodologies, following the two orders of integration, lead to different governing equations for the statistical quantities. The statistical k-e model for clear domains, used to model macroscopic turbulence effects, serves also as the basis for the heat transfer modelling. Mass transfer in porous matrices is further reviewed in the light of the double-decomposition concept. In Chapter 2, Nield and Kuznetsov review the recent work on heat transfer in bidisperse porous media (BDPM). The major topics covered are the measurement of the permeability and thermal conductivity of BDPM, dispersion in BDPM, a new two-velocity two-temperature model for BDPM, and the application of that model to forced convection in a channel between two plane parallel walls. In this application the analysis leads to expressions for the Nusselt number as a function of the properties of the BDPM, namely a conductivity ratio, a permeability ratio, a volume fraction, and an internal heat exchange parameter. For a conjugate problem, the Nusselt number also depends on a Biot number, while for thermally developing convection it also depends on a suitably scaled longitudinal coordinate. In Chapter 3, Merrikh and Lage review, expand and question some of the recent investigations on the use of a low-resolution, porous-continuum model for simulating natural convection within an enclosure saturated with a fluid and having discrete solid blocks uniformly distributed within it. The validation of the porous-continuum results, obtained with the volume-averaged equations, is established by comparison to the results obtained following a continuum model, in which balance equations are solved for each constituent together with compatibility conditions applied at their interfaces. Two configurations are considered, namely one in which the enclosure is heated horizontally by isothermal walls with the horizontal surfaces being adiabatic and the solid blocks conducting, and another in which the blocks are all at the same temperature (generating energy) that is lower than the temperature of the surfaces of the enclosure (all surfaces are at the same temperature). Although the porous-continuum model leads to a much simpler mathematical modelling, and corresponding less numerical effort, the validity of the model is restricted to cases in which the transport phenomenon at the continuum level allows the homogenisation of the domain.

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Vll

In Chapter 4, Cotta, Luz Neto, de B. Alves and Quaresma review and extend the use of hybrid numerical-analytical algorithms, based on the generalised integral transform technique. This method has been developed to handle transient two- and three-dimensional heat and fluid flow in cavities filled with a porous material. In order to illustrate the approach, specific situations of both horizontal and vertical cavities are more closely considered under the Darcy flow model. The problems are analyzed both with and without the time derivative term in the flow equations, using a vorticity-vector potential formulation, which automatically reduces to the streamfunction only formulation for twodimensional situations. Results for rectangular (2D) and parallelepiped (3D) cavities are presented to demonstrate the convergence behaviour of the proposed eigenfunction expansion solutions and comparisons with previously reported numerical solutions are critically discussed. In Chapter 5, Kim and Hyun discuss a method based on the averaging method in which the heat transfer devices are treated as a fluid-saturated porous medium. A novel method for analytically determining the unknown coefficients resulting from the averaging is presented and this represents a significant improvement over experimental and/or numerical determinations of these coefficients. The averaging method in turn yields analytical solutions for the fluid velocity and temperature distributions that are useful in the thermal analysis of heat transfer devices. The modelling technique is elucidated for thermal design and optimisation of micro-channel heat sinks and internally finned tubes. By way of these case studies, the method is shown to be a promising tool for the thermal analysis and optimisation of heat transfer devices. In Chapter 6, Rees and Pop review the local thermal non-equilibrium phenomena in porous medium convection, where the intrinsic average of the temperatures of the solid and fluid phases may be regarded as being different. There are numerous research papers that either derive or use the equations that govern the local thermal non-equilibrium in porous media and in this chapter they have compiled an exhaustive investigation of the most commonly used of these model equations. The main thrust of the chapter is then focused primarily on free and forced convection boundary layers, and on free convection within porous cavities. In Chapter 7, Nakayama and Kuwahara have reviewed and thoroughly discussed the recent investigations on three-dimensional numerical models for periodically fully-developed flow and heat transfer in anisotropic porous media. The discussion covers laminar flows around collections of spheres and cubes, laminar forced convective flows through a bank of cylinders in yaw, and turbulent flows through a bank of square cylinders in a regular arrangement. Exhaustive numerical computations have been performed to determine the macroscopic parameters, such as the permeability and the interfacial heat transfer coefficient, and the results have been compared against all the available empirical formulae. A quasi-three-dimensional calculation procedure has been proposed and this economical procedure has been used to obtain the results for three-dimensional heat and fluid flow through a bank of cylinders in yaw. Further, a large eddy simulation study for turbulence in porous media has been also performed and reported in order to elucidate the complex turbulent flow characteristics associated with porous media.

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In Chapter 8, A. C. Bayta§ and A. F. Bayta§ have reviewed the effects of entropy generation with particular reference to porous cavities and channels and with various different boundary conditions and physical situations. Until recently, the research work performed on entropy generation minimisation using the second law of thermodynamics has been studied for many different applications. However, it has only been in recent years that the utilisation of the second law of thermodynamics in thermal design decision has been developed and applied to porous media. For this reason, it is very timely that this subject has received this in-depth survey so that we are better able to understand the relevant physics underlying the phenomena. In Chapter 9, Saghir, Jiang, Chacha, Yan, Khawaja and Pan introduce the phenomenon of thermal diffusion in porous media and present the theory and the numerical procedures that have been developed to simulate this process. The numerical procedure is demonstrated for both polar and hydrocarbon mixtures. Additionally, convection has a major influence on the thermal diffusion process and it is simulated and discussed for both square and rectangular porous cavities. A detailed literature review introduces a variety of techniques for the measurement of the Soret coefficient. In addition, the literature review discusses the mathematical and numerical methods for the simulation of the Soret effect in both free and porous media. This is followed by the introduction of the fundamental equations of thermal diffusion, and the equations used for the porous media and details as to how the numerical solution technique permits the solution of these equations. In Chapter 10, another aspect of double-diffusion is presented by Charrier Mojtabi, Razi, Maliwan and Mojtabi. They consider the effects of convection in porous media under the effect of mechanical vibration. The so-called time-averaged formulation has been adopted. This formulation can be effectively applied to study the vibrational induced thermo-solutal convection problem. The influence of high frequency and small amplitude vibration on the onset of thermo-solutal convection, in a confined porous cavity with various aspect ratios and saturated by a binary mixture, has been presented. Linear stability analysis of the mechanical equilibrium or quasi-equilibrium solution is also performed. A theoretical examination of the limiting case of the long-wave mode in the case of Soret driven convection under the action of vibration has been performed. The 2D numerical simulations presented allow the correlation of the results obtained from the linear stability analysis for both stationary and Hopf bifurcations. In Chapter 11, Mohamad has the main objective of reviewing the fundamentals and the applications of combustion in porous burners. Combustion in porous media is used in advanced boiler and surface burners. Also it is possible to exploit porous medium in domestic heaters, gas turbine combustion chambers, vehicle heaters, fuel cells and energy management in many industrial processes, such as furnaces and cogeneration systems. The work done on combustion in porous media is discussed in detail and the physics of combustion, the applications, the modelling of combustion, the recent developments on this topic and suggestions for further research are critically presented. In Chapter 12, Holzbecher reviews the simultaneous action of transport and biogeochemistry in porous media. Codes, which perform such computations, are implemented following different numerical and conceptual methods, of which the most important ones

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ix

are outlined in this chapter. Several examples, some hypothetical and some with practical applications, including carbonate chemistry, are presented, discussed and modelled. Simulations are performed with an in-house developed MATLAB module. It is shown that the popular operator splitting (two-step) approach has to be handled with great care. In Chapter 13, Bennacer and Lakhal present and discuss new investigations, both experimental and numerical, on the evaporation of a liquid in a confined space such as in a capillary tube. The phase change has been found to be responsible for the induced convection pattern in the liquid phase below the meniscus interface. Further, the liquid convective structure has been revealed by the use of the m-PIV technique. When extra heating is supplied to the system, the convection pattern changes and it eventually is reversed and this depends on the relative position of the heating element with respect to the liquidvapour interface. A numerical model, for the two-dimensional and axisymmetric cases, has been developed and presented in order to reproduce the experimental findings and then this has been extended to all those situations that are not experimentally accessible. The governing system of partial differential equations and boundary conditions is solved with a F^M method. An ADI approach with a block correction is used to solve the resulting system of algebraic equations and a coordinate transformation has been employed for simulating the curved meniscus interface. The numerical results are in good agreement with the experimental findings. The present study has demonstrated that the meniscus interfacial temperature profile is responsible for the thermo-capillary convection that is experimentally observed. The fundamental phenomena investigated in this chapter are strongly related to many important industrial applications, involving phase change such as heat pipes, crystal growth and glass manufacture. In Chapter 14, Peng and Wu present a series of different experimental observations and the associated theoretical investigations are conducted and/or presented in order to understand the transport phenomena at the pore scale level, including the transport phenomena both with and without phase change and chemical reaction. Special emphasis is placed on a wide range of very important practical applications. The conjugate transport phenomena with pore and matrix structures exist widely in the natural world and in a variety of practical applications. It is of critical importance to understand these phenomena that account for the dynamical processes and structure deformation taking place in the inner pores. Thus the focus of the chapter is to briefly review and discuss the transport phenomena in porous media at the pore scale level and to present some of the recently conducted and currently ongoing research on this topic that is taking place in Tsinghua University. In Chapter 15, Kimura describes a fundamental study on the ice-layer formation and the melting that occurs along a cooling surface. This surface is positioned at the top boundary of a rectangular space that is filled with water-saturated porous medium. In such conditions, the natural convection that occurs has a significant impact on the heat balance at the solid-liquid boundary that develops in the unfrozen layer. The goal of these studies is to develop a one-dimensional model that is capable of predicting the transient response of the ice-layer to a prescribed cooling temperature variation. The one-dimensional theory predicts that a higher cooling temperature frequency reduces the oscillating solid front, and that a thicker solid layer increases the phase delay of the front movement relative to the cooling temperature variation. The validity of the one-dimensional model has been

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tested against a two-dimensional numerical simulation for the dynamic response of the movement of the interface. It is shown that there is an excellent agreement between the one- and two-dimensional simulations. Further, experiments have been conducted in order to verify the numerical models. In Chapter 16, Ma, Ingham and Pourkashanian review the role of porous media in a fuel cell, which is a multi-component power generating device which relies on the chemistry rather than combustion to convert chemical energy into electricity. The key components of the fuel cell are made of porous materials through which the fuel and the oxidant are delivered to the active site of the cell where electrochemical reactions take place to generate the power, heat and water. Fuel cell technology presents a huge economical and environmental potential in the future power markets, this ranges from small portable cells to large residential power plants. However, at present, there are numerous technical barriers that prevent fuel cells from becoming commercially competitive and the fluid flow and reactant transport in the porous electrodes are major issues in the design of fuel cells. This chapter aims at providing a general introduction to the fluid flows through the porous media in fuel cells with emphasis being placed on the numerical modelling of the convective and diffusive processes of the fluid flow, species transport, heat/mass transfer and the electrical potential. The challenges and the areas that need further investigations in the modelling of fuel cells are discussed in detail. In Chapter 17, Harris, Fisher, Karimi-Fard, Vaszi and Wu describe and discuss some of the numerical techniques currently being used to model the effects of faults and fractures on fluid flow. Fault and fracture zones are often highly-complex heterogeneities that can have a significant affect on the fluid flow within petroleum reservoirs on length scales from less than one micron to more than ten kilometres. The methods used to model faults, that are partial barriers to fluid flow, and fractures, which can enhance flow, are described and discrete flow models are presented for modelling large-scale fluid flow and deriving upscaled properties. Pore structure models at the millimetre scale are created using Markov chain Monte Carlo simulations and the lattice Boltzmann method allows their permeability to be inferred. The results presented from these models can be incorporated into industry-standard production simulation models to allow better decision-making. Further, it is highlighted that production simulation models are inherently non-unique and it is sometimes difficult to elucidate whether the techniques that are being developed represent realistic improvements. During the course of the preparation of this book, we have been encouraged and supported by many researchers. First, we would like to acknowledge the contributions of Professor Adrian Bejan, Professor Peter Heggs, Dr Lionel Elliott and Dr Daniel Lesnic who have encouraged and supported us to continue this series of books. All the researchers that we contacted were very enthusiastic about the book, and also we received numerous unsolicited offers to contribute a chapter. We sincerely thank all these researchers for their enthusiasm for the book. All the authors have, at all times, been efficient in producing excellent first drafts of their chapters, and performing corrections as requested by the referees, and they always responded very enthusiastically and promptly to all our requests. We would also like to thank all the referees who produced some extremely pertinent observations on the original versions of the chapters.

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XI

We would also like to express our sincere thanks to Dr Julie M. Harris and Dr Simon D. Harris for the formatting of the book and the preparation of the figures. We are deeply indebted to them for all the care and attention and the patience that they have shown in both the preparation and the proof reading of the book. Finally, we gratefully appreciate the support of Pergamon Press and in particular, Amo Schouwenburg, Senior Publishing Editor, and Vicki Wetherell, Editorial Assistant. LEEDS/CLUJ MARCH,

2005

D . B . INGHAM & I. POP

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Contributors A. C. BAYTA§, The Faculty of Aeronautics and Astronautics, Istanbul Technical University, 34469-Maslak, Istanbul, Turkey A. F. B AYTA§, Institute of Energy, Istanbul Technical University, 34469-Maslak, Istanbul, Turkey R. BENNACER, L E E V A M / L E E E - I U P G C Universite Cergy-Pontoise, 5 mail Gay Lussac, Neuville sur Oise, 95031, France M. CHACHA, Department of Mechanical Engineering, UAE University, PO Box 17555, Al Ain, UAE M. C. CHARRIER MOJTABI, Laboratoire d'Energetique (LESETH), EA 810, UFR PCA, Universite Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex, France R. M. COTTA, Mechanical Engineering Department- LTTC - POLI/COPPE - UFRJ, Universidade Federal do Rio de Janeiro, Brazil L. S. DE B. ALVES, Mechanical Engineering Department, University of California at Los Angeles, USA M. J. S. DE LEMOS, Instituto Tecnologico de Aeronautica-ITA, 12228-900, Sao Jose dos Campos-SP, Brazil Q. J. FISHER, Rock Deformation Research Limited / School of Earth and Environment, University of Leeds, Leeds, LS2 9JT, UK S. D. HARRIS, Rock Deformation Research Limited, University of Leeds, Leeds, LS2 9JT, UK E. HoLZBECHER, Humboldt University Berlin, Institute of Freshwater Ecology and Inland Fisheries (IGB), Miiggelseedamm 310,12587 Berlin, Germany J. M. HYUN, Department of Mechanical Engineering, KAJST, Daejeon 305-701, South Korea D. B. INGHAM, Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, UK C. G. JIANG, Department of Mechanical Engineering, Ryerson University, Toronto, Ontario, M5B 2K3, Canada M. KARIMI-FARD, Department of Petroleum Engineering, Stanford, CA, USA M. KHAWAJA, Department of Mechanical Engineering, Ryerson University, Toronto, Ontario, M5B 2K3, Canada Xlll

xiv

CONTRIBUTORS

S. J. KIM, Department of Mechanical Engineering, KAIST, Daejeon 305-701, South Korea S. KiMURA, Institute of Nature and Environmental Technology, Kanazawa University, 2-40-20 Kodatsuno, Kanazawa, 920-8667, Japan F. KuwAHARA, Department of Mechanical Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu, 432-8561 Japan A. V. KuzNETSOV, Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC 27695-7910, USA J. L. LAGE, Laboratory for Porous Materials Applications, Department of Mechanical Engineering, Southern Methodist University, Dallas, TX 75275, USA A. LAKHAL, L E E V A M / L E E E - I U P G C Universite Cergy-Pontoise, 5 mail Gay Lussac, Neuville sur Oise, 95031, France H. Luz NETO, Instituto Nacional de Tecnologia-INT, Rio de Janeiro, Brazil L. MA, Centre for Computational Fluid Dynamics, University of Leeds, Leeds, LS2 9JT, UK K. MALIWAN, I M F T , U M R C N R S / I N P / U P S N^ 5502, UFR MIG, Universite Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex, France A. A. MERRIKH, Pulmonary Research Group, Department of Internal Medicine, University of Texas, Southwestern Medical Center at Dallas, Dallas, TX 75390-9034, USA A. A. MOHAMAD, Department of Mechanical and Manufacturing Engineering, CEERE, The University of Calgary, Calgary, Alberta, T2N 1N4, Canada A. MOJTABI, IMFT, UMR CNRS/INP/UPS N^5502, UFR MIG, Universite Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex, France A. NAKAYAMA, Department of Mechanical Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu, 432-8561 Japan D. A. NiELD, Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland, New Zealand S. PAN, Department of Mechanical Engineering, Ryerson University, Toronto, Ontario, M5B 2K3, Canada X. F. PENG, Laboratory of Phasechange and Interfacial Transport Phenomena, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China I. POP, Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania M. C. POURKASHANIAN, Energy Resources Research Institute, University of Leeds, Leeds, LS2 9JT, UK J. N. N. QuARESMA, Chemical and Food Engineering Department, Universidade Federal do Para, Belem, Brazil Y. P. RAZI, IMFT, UMR CNRS/INP/UPS N^5502, UFR MIG, Universite Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex, France D. A. S. REES, Department of Mechanical Engineering, University of Bath, Bath, BA2 7AY, UK

CONlKlBUiORS

XV

M. Z. SAGHIR, Department of Mechanical Engineering, Ryerson University, Toronto, Ontario, M5B 2K3, Canada A. Z. VASZI, Rock Deformation Research Limited, University of Leeds, Leeds, LS2 9JT, UK H. L. Wu, Laboratory of Phasechange and Interfacial Transport Phenomena, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China K. Wu, Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh, EH14 4AS, UK Y. YAN, Department of Mechanical Engineering, Ryerson University, Toronto, Ontario, M5B 2K3, Canada

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Contents 1

THE DOUBLE-DECOMPOSITION CONCEPT FOR TURBULENT TRANSPORT IN POROUS MEDIA

1

M. /. S. de Lemos

1.1 1.2 1.3 1.4 1.5 1.6 1.7

1.8

1.9

2

Introduction Instantaneous local transport equations Time- and volume-averaging procedures Time-averaged transport equations The double-decomposition concept 1.5.1 Basic relationships Turbulent transport 1.6.1 Momentum equation Heat transfer 1.7.1 Governing equations 1.7.2 Turbulent thermal dispersion 1.7.3 Local thermal equilibrium hypothesis 1.7.4 Macroscopic buoyancy effects Mass transfer 1.8.1 Mean and turbulent 1.8.2 Turbulent mass dispersion Concluding remarks References

fields

HEAT TRANSFER IN BIDISPERSE POROUS MEDIA

1 2 4 5 6 7 9 9 20 20 22 25 25 28 28 29 31 31 34

D. A. Nield and A. V. Kuznetsov

2.1 2.2 2.3 2.4

Introduction Determination of transport properties Two-phase flow and boiling heat transfer Dispersion

34 35 37 37

XVlll

CONTENTS

2.5 2.6 2.7

2.8

3

Two-velocity model Two-temperature model Forced convection in a channel between plane parallel walls 2.7.1 Uniform temperature boundaries: theory 2.7.2 Uniform flux boundaries: theory 2.7.3 Uniform temperature boundaries: results 2.7.4 Uniform flux boundaries: results 2.7.5 Conjugate problem 2.7.6 Thermal development Conclusions References

FROM CONTINUUM TO POROUS-CONTINUUM: THE VISUAL RESOLUTION IMPACT ON MODELING NATURAL CONVECTION IN HETEROGENEOUS MEDIA

37 40 40 41 44 47 48 49 51 58 59

60

A. A. Merrikh and J. L. Lage 3.1 3.2

Introduction Horizontal heating 3.2.1 Continuum equations 3.2.2 Porous-continuum equations 3.2.3 Heat transfer comparison parameters 3.2.4 Results 3.2.5 Internal structure effect 3.3 Heat-generating blocks 3.3.1 Mathematical modeling 3.3.2 Heat transfer comparison parameters 3.3.3 Results 3.4vvESPP4459_9780080445441 s Conclusion Retferences

4

61 63 63 65 67 68 74 80 81 83 84 29 94

INTEGRVAL TRANSFORMS FOR tESPP4459_9780080445441NATURA L CONVECTIO NIN CVvv AVITIE S FILLED WITH POROUS MEDIA 97 R. M. Cotta, H. Luz Neto, L S. de B. Alves and J. N. N. Quaresma 4.1 4.2 4.3 4.4 4.5

Introduction Two-dimensional problem Three-dimensional problem Results and discussion Conclusions

98 99 103 108 117

CONTENTS

XIX

References 5

117

A POROUS MEDIUM APPROACH FOR THE THERMAL ANALYSIS OF HEAT TRANSFER DEVICES

120

S. J. Kim and J. M. Hyun 5.1 5.2

Introduction Thermal analysis of microchannel heat sinks 5.2.1 High-aspect-ratio microchannels 5.2.2 Low-aspect-ratio microchannels Thermal analysis of internally finned tubes 5.3.1 Mathematical formulation and theoretical solutions 5.3.2 Velocity and temperature distributions 5.3.3 Optimization of thermal performance 5.3.4 Comments on the averaging direction Conclusions References

120 122 123 130 136 137 140 142 143 144 145

LOCAL THERMAL NON-EQUILIBRIUM IN POROUS MEDIUM CONVECTION

147

5.3

5.4

6

D. A. S. Rees and I. Pop 6.1 6.2 6.3 6.4

6.5 6.6 6.7

7

Introduction Governing equations Conditions for the validity of LTE Free convection boundary layers 6.4.1 General formulation 6.4.2 Results for stagnation point 6.4.3 Results for a vertical flat plate 6.4.4 General comments Forced convection past a hot circular cylinder Stability of free convection Conclusions References

flow

THREE-DIMENSIONAL NUMERICAL MODELS FOR PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS WITHIN POROUS MEDIA

147 148 152 154 154 156 157 160 161 166 170 170

174

A. Nakayama and F. Kuwahara 7.1

Introduction

174

XX

CONTENTS

7.2

7.3

7.4

7.5

8

Three-dimensional numerical model for isotropic porous media 7.2.1 Numerical model 7.2.2 Governing equations and periodic boundary conditions 7.2.3 Method of computation 7.2.4 Macroscopic pressure gradient and permeability Quasi-three-dimensional numerical model for anisotropic porous media 7.3.1 Periodic thermal boundary conditions 7.3.2 Quasi-three-dimensional solution procedure for anisotropic arrays of infinitely long cylinders 7.3.3 Effect of cross flow angle on the Euler and Nusselt numbers 7.3.4 Effect of yaw angle on the Euler and Nusselt numbers Large eddy simulation of turbulent flow in porous media 7.4.1 Large eddy simulation and numerical model 7.4.2 Velocity fluctuations and turbulent kinetic energy 7.4.3 Macroscopic pressure gradient in turbulent flow Conclusions References

ENTROPY GENERATION IN POROUS MEDIA

176 176 178 179 180 182 182 184 187 188 190 190 192 196 198 199 201

A. C. Bayta§ and A. F. Bayta§ 8.1 8.2 8.3

8.4

8.5

9

Introduction A short history of the second law of thermodynamics Governing equations 8.3.1 Continuity equation 8.3.2 Momentum balance equation 8.3.3 Energy equation 8.3.4 Entropy generation Entropy generation in a porous cavity and channel 8.4.1 Entropy generation in a porous cavity 8.4.2 Entropy generation in a porous channel Conclusions References

THERMODIFFUSION IN POROUS MEDIA

201 202 204 204 205 206 206 207 207 218 223 224 227

M. Z. Saghir, C. G. Jiang, M. Chacha, Y. Yan, M. Khawaja and S. Pan 9.1 9.2

Introduction Literature review 9.2.1 Measurement techniques of the Soret coefficient

227 228 228

9.3

9.4 9.5 9.6 9.7

9.8

CONTENTS

XXI

9.2.2 Mathematical and numerical techniques Fundamental equations of thermodiffusion 9.3.1 Haase model 9.3.2 Kempers model 9.3.3 Firoozabadi model Fundamental equations in porous media Numerical solution technique Mesh sensitivity analysis Results and discussion 9.7.1 Comparison of molecular and thermodiffusion coefficients for water alcohol mixtures 9.7.2 Calculation of molecular and thermodiffusion coefficients for hydrocarbon mixtures 9.7.3 Convection in a square cavity 9.7.4 Convection in a rectangular cavity Conclusions References

230 233 234 234 235 236 237 239 241

10 EFFECT OF VIBRATION ON THE ONSET OF DOUBLE-DIFFUSIVE CONVECTION IN POROUS MEDIA

241 242 243 249 257 258 261

M. C. Charrier Mojtabi, Y. P. Razi, K. Maliwan and A. Mojtabi

10.1 Introduction 10.2 Mathematical formulation 10.2.1 Direct formulation 10.2.2 Time-averaged formulation 10.2.3 Scale analysis method 10.2.4 Time-averaged system of equations 10.3 Linear stability analysis 10.3.1 Infinite horizontal porous layer 10.3.2 Limiting case of the long-wave mode 10.3.3 Convective instability under static gravity (no vibration) 10.4 Comparison of the results with fluid media 10.5 Numerical method 10.5.1 Vertical vibration 10.5.2 Horizontal vibration 10.6 The onset of thermo-solutal convection under the influence of vibration without Soret effect 10.6.1 Linear stability analysis 10.7 Conclusions

262 263 264 265 266 267 268 269 274 274 275 276 277 277 280 280 283

XXll

CONTENTS

References 11 COMBUSTION IN POROUS MEDIA: FUNDAMENTALS AND APPLICATIONS

284

287

A. A. Mohamad

11.1 Introduction 11.2 Previous works 11.3 Characteristics of combustion in porous media 11.4 Applications 11.5 Porous burners 11.6 Mathematical modeling 11.7 Results and discussion 11.8 Radial burner 11.9 Conclusions 11.10 Possible future work References 12 REACTIVE TRANSPORT IN POROUS MEDIA—CONCEPTS AND NUMERICAL APPROACHES E.

12.1 12.2 12.3 12.4

12.5

12.6 12.7 12.8

287 289 290 291 293 294 297 298 301 301 302

305

Holzbecher

Introduction Quantitative geochemistry Analytical description of reactive transport Examples 12.4.1 Equilibrium example 1 12.4.2 Equilibrium example 2 12.4.3 Equilibrium and kinetics example 1 12.4.4 Equilibrium and kinetics example 2 Numerical approaches 12.5.1 Speciation calculations 12.5.2 Transport modelling 12.5.3 Transport and reaction coupling Numerical errors Implementation in MATLAB Example models 12.8.1 Three-species model 12.8.2 Calcite dissolution test case (ID) 12.8.3 Two-dimensional modelling

306 307 310 313 313 314 316 316 318 318 319 320 322 324 325 325 329 333

CONTENTS

12.9 Conclusions References 13 NUMERICAL AND ANALYTICAL ANALYSIS OF THE THERMOSOLUTAL CONVECTION IN AN ANNULAR FIELD: EFFECT OF THERMODIFFUSION

xxiii

336 337

341

R. Bennacer and A. Lakhal

13.1 Introduction 13.2 Mathematical model 13.2.1 Numerical solution 13.3 Analytical solution 13.4 Results and discussion 13.5 Conclusions References 14 PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA

342 342 345 346 350 362 363 366

X. F.Peng and H.LWu

14.1 Introduction 14.2 Conjugated transport phenomena with pore structure 14.2.1 Conjugated phenomena in sludge drying 14.2.2 Effect of inner evaporation on the pore structure 14.3 Transport-reaction phenomena 14.3.1 Reaction in a porous solid 14.3.2 Experimental investigation 14.4 Boiling and interfacial transport 14.4.1 Experimental observations 14.4.2 Static description of primary bubble interface 14.4.3 Replenishnient and dynamic behavior of the interface 14.4.4 Interfacial heat and mass transfer at pore level 14.5 Freezing and thawing 14.5.1 Experimental facility 14.5.2 Sludge agglomerates during freezing 14.5.3 Botanical tissues during freezing 14.6 Two-phase flow behavior 14.6.1 Experimental observation 14.6.2 Critical diameter 14.6.3 Transport of small bubbles 14.6.4 Transport of big bubbles

366 367 368 371 374 374 376 379 379 381 382 382 385 385 385 387 390 390 392 393 395

XXIV

CONTENTS

14.7 Conclusion References 15 DYNAMIC SOLIDIFICATION IN A WATER-SATURATED POROUS MEDIUM COOLED FROM ABOVE

396 396

399

S. Kimura 15.1 Introduction 15.2 Mathematical formulation 15.2.1 Two-dimensional model 15.2.2 A reduced one-dimensional model 15.3 Numerical results 15.3.1 Development of a solid layer and convecting flow 15.3.2 Amplitude and phase lag of the oscillating solid-liquid interface 15.4 Experimental results 15.4.1 Experimental apparatus and procedure 15.4.2 Ice-layer thickness at steady state 15.4.3 Average Nusselt number and vertical temperature variation at steady state 15.4.4 Oscillating cooUng temperature and the response of ice-layer 15.4.5 Amplitude and phase lag against oscillating cooling temperature 15.5 Conclusion References 16 APPLICATION OF FLUID FLOWS THROUGH POROUS MEDIA IN FUEL CELLS

400 401 401 405 407 407 408 409 409 410 411 413 413 415 416 418

L. Ma, D. B. Ingham and M. C. Pourkashanian

16.1 Introduction 16.2 Operation principles of fuel cells 16.3 Governing equations for the fluid flows in porous electrodes 16.3.1 Equations for the fluid flow and mass transfer in fuel cells 16.3.2 Heat generation and transfer in fuel cells 16.3.3 The electric field in fuel cells 16.4 Multicomponent gas transport in porous electrodes 16.4.1 Convective transport 16.4.2 Diffusive transport 16.5 CFD model predictions of fuel cells 16.6 Concluding remarks References

419 419 423 423 425 427 427 428 429 431 438 439

CONTENTS

17 MODELLING THE EFFECTS OF FAULTS AND FRACTURES ON FLUID FLOW IN PETROLEUM RESERVOIRS

XXV

441

S. D. Harris, Q. J. Fisher, M. Karimi-Fard, A. Z Vaszi and K. Wu

17.1 Introduction 17.2 Single and multiphase flow 17.3 Modelling flow in petroleum reservoirs where faults act as barriers 17.3.1 Numerical modelling of the permeabihty of fault rocks 17.3.2 Modelling flow in complex damage zones 17.3.3 Incorporation of fault properties into production simulation models 17.3.4 Knowledge gaps and future directions 17.4 Modelling flow in reservoirs where faults and fractures act as conduits 17.4.1 Overview of existing discrete fracture models 17.4.2 Technical description of the methodology 17.4.3 An example of flow simulation in a fractured reservoir 17.5 Discussion and conclusions References

442 443 446 447 453 460 461 463 464 466 469 471 472

This Page Intentionally Left Blank

1

THE DOUBLE-DECOMPOSITION CONCEPT FOR TURBULENT TRANSPORT IN POROUS MEDIA M.J. S.DELEMOS Institute Tecnologico de Aeronautica-ITA, 12228-900, Sao Jose dos Campos-SP, Brazil email: [email protected]

Abstract Environmental impact analyses as well as engineering equipment design can both benefit from reliable modeling of turbulent flow in porous media. A number of natural and engineering systems can be characterized by a permeable structure through which a working fluid permeates. Turbulence models proposed for such flows depend on the order of application of time- and volume-average operators. Two methodologies, following the two orders of integration, lead to different governing equations for the statistical quantities. This chapter reviews recently published methodologies to mathematically characterize turbulent transport in porous media. A new concept, called double-decomposition, is here discussed and models for turbulent transport in porous media are classified in terms of the order of application of the time- and volumeaveraging operators, among other peculiarities. Within this chapter instantaneous local transport equations are reviewed for clear flow before time- and volume-averaging procedures are applied to them. The double-decomposition concept is presented and thoroughly discussed prior to the derivation of macroscopic governing equations. Equations for turbulent transport follow, showing detailed derivation for the mean and turbulent field quantities. The statistical k-e model for clear domains, used to model macroscopic turbulence effects, serves also as the basis for heat transfer modeling. Mass transfer in porous matrices is further reviewed in the light of the double-decomposition concept.

Keywords: turbulence, porous media, modeling, nonlinear effects, doubledecomposirion 1.1

INTRODUCTION

It is well established in the literature that modeling of macroscopic transport for incompressible flows in porous media can be based on the volume-average methodology, see Whitaker (1999), for either heat, see Hsu and Cheng (1990), or mass transfer, see Whitaker 1

2

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

(1966, 1967), Bear and Bachmat (1967), and Bear (1972). If the fluid phase properties fluctuate with time, in addition to presenting spatial deviations, there are two possible methodologies to follow in order to obtain macroscopic equations: (i) application of time-average operator followed by volume-averaging, see Masuoka and Takatsu (1996), Kuwahara et al (1996), Takatsu and Masuoka (1998), Kuwahara and Nakayama (1998), and Nakayama and Kuwahara (1999), or (ii) use of volume-averaging before time-averaging is applied, see Lee and Howell (1987), Wang and Takle (1995), Antohe and Lage (1997), Getachewa et al (2000). In fact, these two sets of macroscopic transport equations are equivalent when examined under the recently established double-decomposition concept, Pedras and de Lemos (1999, 2000a, 2000b, 2001a, 2001b, 2001c, 2003). Recent reviews on the topic of turbulence in permeable media can be found in Lage (1998) and Lage et al. (2002). Advances in the general area of porous media are found in edited books devoted to the subject such as Nield and Bejan (1999), Vafai (2000) and Ingham and Pop (2002). The double-decomposition idea was initially developed for the flow variables in porous media and has been extended to non-buoyant heat transfer, see de Lemos and Rocamora, Jr (2002), buoyant flows, see de Lemos and Braga (2003), mass transfer, see de Lemos and Mesquita (2003), non-equilibrium heat transfer, see Saito and de Lemos (2004), and double-diffusive transport, see de Lemos and Tofaneli (2004). The problem of treating macroscopic interfaces bounding finite porous media, considering a diffusionjump condition for the mean, see Silva and de Lemos (2003a, 2003b), and turbulence fields, see de Lemos (2004), have also been investigated under the concept first proposed by Pedras and de Lemos (1999, 2000a, 2000b, 2001a, 2001b, 2001c, 2003). A general classification of all proposed models for turbulent flow and heat transfer in porous media has been recently published, see de Lemos and Pedras (2001). Here, a systematic review of this new concept is presented.

1.2

INSTANTANEOUS LOCAL TRANSPORT EQUATIONS

The steady-state local or microscopic instantaneous transport equations for an incompressible fluid with constant properties are given by: V'U = 0, pV •{uu) = -Vp-\-pV^u-^pg, (pCp)V • (nT) - V . (AVT),

(1.1) (1.2) (1.3)

where u is the velocity vector, p is the density, p is the pressure, p is the fluid viscosity, g is the gravity acceleration vector, Cp is the specific heat, T is the temperature, and A is the fluid thermal conductivity.

M. J. S. DE LEMOS

3

In addition, the mass fraction distribution for the chemical species t is governed by the following transport equation: V -{pumt^Jt)

(1.4)

= pRi,

where m^ is the mass fraction of component £, u is the mass-averaged velocity of the mixture, u — Yl,i '^i'^i^ and ui is the velocity of species t. Further, the mass diffusion flux Ji is due to the velocity slip of species i and is given by Ji = pi{ui -u)

(1.5)

= -pDiVmi,

where Di is the diffusion coefficient of species £ into the mixture. The second equality in equation (1.5) is known as Pick's Law. The generation rate of species t per unit of mixture mass is given in equation (1.4) by Ri. If one considers that the density in the last term of equation (1.2) varies with temperature, for natural convection flow, the Boussinesq hypothesis reads, after renaming this density PT'-

PT = P [ 1 - ^ ( T - T r e f ) ] ,

(1.6)

where the subscript 'ref' indicates a reference value and ^ is the thermal expansion coefficient defined by 1 dp\ (1.7) ^ pdT Further, substituting equation (1.6) into equation (1.2), we obtain: pV • {uu) = -{VpY

+ liV'u - pgPiT - Tref),

(1.8)

where (Vp)* = Vp — pg is a. modified pressure gradient. When equation (1.3) is written for the fluid and solid phases with heat sources it becomes: fluid: solid (porous matrix):

{pCp)fV . (uTf) = V • (XfVTf)

-f Sf ,

(1.9)

0 = V • (A^ VT^) + 5^,

(1.10)

where the subscripts / and s refer to each phase, respectively. If there is no heat generation either in the solid or in the fluid, we obtain: Sf = Ss = 0.

(1.11)

As mentioned, there are, in principle, two ways that one can follow in order to treat turbulent flow in porous media. The first method applies a time-average operator to the governing equations (1.1)-(1.4) before the volume-average procedure is applied. In the second approach, the order of application of the two average operators is reversed. Both techniques aim at derivation of suitable macroscopic transport equations. Volume-averaging in a porous medium, described in detail in Slattery (1967), Whitaker (1969, 1999), and Gray and Lee (1977), makes use of the concept of a representative

4

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

elementary volume (REV) over which local equations are integrated. In a similar fashion, statistical analysis of turbulent flow leads to time-mean properties. Transport equations for statistical values are considered in lieu of instantaneous information on the flow.

1.3

TIME- AND VOLUMEAVERAGING PROCEDURES

Traditional analyses of turbulence are based on statistical quantities, which are obtained by applying time-averaging to the flow governing equations. As such, the time-average of a general quantity ^ is defined as follows: ^ = ^ /

V'dt,

(1.12)

where the time interval A^ is small compared to the fluctuations of the average value, Tp, but large enough to capture turbulent fluctuations of ip. Time decomposition can then be written as follows: ^^Tp + if', (1.13) with ^' = 0, where ip' is the time fluctuation of (p around its average value Jp. The volume-average of a general property ^p taken over an REV in a porous medium can be written, see Slattery (1967), as follows:

The value {(py is defined for any point x surrounded by an REV of size AV. This average is related to the intrinsic average for the fluid phase as follows:

{vsY^HfjY.

(1-15)

where 0 = A V / / A y is the local medium porosity and AV) is the volume occupied by the fluid in an REV. Furthermore, we can write (^ = ((/p)^ + V ,

(1-16)

with (V)* — 0- In equation (1.16), V is the spatial deviation of (p with respect to the intrinsic average {ipY. For deriving theflowgoverning equations, it is necessary to know the relationship between the volumetric-average of derivatives and the derivatives of the volumetric-average. These relationships are presented in a number of works, e.g. Whitaker (1969,1999), being known

M. J. S. DE LEMOS

as the theorem of local volumetric-average. They are written as follows: {Vipy = V {y) + ^ j

(1.17) n-vdS,

(1.18)

where Ai, ui and n are the interfacial area, the velocity of phase / and the unit vector normal to Ai, respectively. Also, the local volume-average theorem can be expressed as, see Gray and Lee (1977):

(1.20)

The area Ai should not be confused with the surface area surrounding volume AV. To the interested reader, mathematical details and proof of the theorem of local volumetricaverage can be found in Slattery (1967), Whitaker (1969,1999), and Gray and Lee (1977). For single-phase flow, phase / is the fluid itself and Ui = 0 if the porous substrate is assumed to be fixed. In developing equations (L17)-(1.19), the only restriction applied is the independence of A F in relation to time and space. If the medium is further assumed to be rigid, then AV/ is dependent only on space and not time-dependent, see Gray and Lee (1977).

1.4

TIME-AVERAGED TRANSPORT EQUATIONS

In order to apply the time-average operator to equations (1.1), (L2) and (1.8), we consider: u = u + u',

T = T + T',

p = p + p'.

(1.21)

Substituting expression (1.21) into equations (1.1), (1.2) and (1.8) we obtain, after considering constant flow properties: \/'U

(1.22)

= 0,

pV • {uu) = -{VpY

+ fiV^u + W • i-pu'u')

{pCp)V • {uT) = V • (fcVT) -h V • {-pCpvJr)

.

- pg/3 (T - T^ef),

(1.23) (1.24)

6

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

For a clear fluid, the use of the eddy diffusivity concept for expressing the stress-rate of strain relationship for the Reynolds stress appearing in equation (1.23) gives: -pu'u'

.^

2

= ^it2D - -pkl, o

(1.25)

where D = [Vu + (Vt/)^]/2 is the mean deformation tensor, k — u' - u' 12 is the turbulent kinetic energy per unit mass, /i^ is the turbulent viscosity and / is the unity tensor. Similarly, for the turbulent heat flux on the right-hand side of equation (1.24) the eddy diffusivity concept reads: Mt -pCpU'V = Cp-^VT,

(1.26)

where ar is the turbulent Prandtl number. The transport equation for the turbulent kinetic energy is obtained by multiplyingfirstthe difference between the instantaneous and the time-averaged momentum equations by u'. Thus, applying further the time-average operator to the resulting product, we obtain: pV • (uk) = - p V • u'

[--{-q

+ fiV^k-\-Pk+Gk-pe,

(1.27)

where Pk = —pu'u' : Vu is the generation rate of A; due to gradients of the mean velocity and Gk^-ppg-^^^ (1.28) is the buoyancy generation rate of k. Also, q = u' - u'/2.

1.5

THE DOUBLE-DECOMPOSITION CONCEPT

The double-decomposition idea, herein used for obtaining macroscopic equations, has been detailed in Pedras and de Lemos (1999, 2000a, 2000b, 2001a, 2001b, 2001c, 2003). Here, a general overview is presented. Further, the resulting equations using this concept for the flow, see Pedras and de Lemos (2001a), and non-buoyant thermal fields, see de Lemos and Rocamora, Jr (2002), are already available in the literature and because of this they are not reviewed here in great detail. As mentioned, extensions of the doubledecomposition methodology to buoyant flows, see de Lemos and Braga (2003), to mass transport, see de Lemos and Mesquita (2003), and to double-diffusive convection, see de Lemos and Tofaneli (2004), have also been presented in the open literature. Basically, for porous media analysis, a macroscopic form of the governing equations is obtained by taking the volumetric-average of the entire equation set. In that development, the porous medium is considered to be rigid and saturated by an incompressible fluid.

M.J.S.DELEMOS

1.5.1

7

Basic relationships

From the work in Pedras and de Lemos (2000a) and de Lemos and Rocamora, Jr (2002), one can write for any flow property ip combining decompositions (1.16) and (1.13):

{^Y = {TpY + {ip')\ ^ = ( ^ r + v , v = v + v , Y',

or '= A ^ /

^ ^ ^ = Air /

{^ + ^')^V

= {TpY^y)\

V = v + V' = V + V ,

(1.33) (1-34)

so that (^'= ((^')^ + V ' ,

V = V + V',

where

> ' = ^ ' - (^')'= V " V-

(1-35)

Finally, we can have a full variable decomposition as follows: ^ = iW + {^y + v + V ' = W + M ' ' + V + V ' ,

(1-36)

(p^(^)^ + ((p')^ + V + V = M^ + V+(^)' + V •

(1-37)

or, further.

'

^

'

'

^

'

Equation (1.36) comprises the double-decomposition concept. The significance of the four terms in expression (1.37) can be reviewed as follows. (i) ((^)\ is the intrinsic average of the time-mean value of (/?, i.e. we computefirstthe timeaveraged values of all points composing the REV, and then we find their volumetric mean to obtain ( ^ ) \ Instead, we could also consider a certain point x surrounded by the REV, according to equations (1.14) and (1.15), and take the volumetric-average at

8

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

different time steps. Thus, we calculate the average over such different values of {(fY in time. We get then ((/?)* and, according to expression (1.31), (^)* = ((/?)*, i.e. the volumetric- and time-average commute, (ii) If we now take the volume-average of all fluctuating components of (/?, which compose the REV, we end up with ((/^')^ Instead, with the volumetric-average around point x taken at different time steps we can determine the difference between the instantaneous and a time-averaged value. This will be {(pY that, according to expression (1.32), equals {i^'YFurther, on performing first a time-averaging operation over all points that contribute with their local values to the REV, we get a distribution of Tp within this volume. If now we calculate the intrinsic average of this distribution of ^, we get ( ^ ) \ The difference or deviation between these two value is ^Tp. Now, using the same space decomposition approach, we can find for any instant of time t the deviation V- This value also fluctuates with time, and as such a time mean can be calculated as V- Again, the use of expression (1.32) gives '^Tp — ^ip. Finally, it is interesting to note the meaning of the last term on each side of equation (1.37). The first term, *((/?'), is the time fluctuation of the spatial component whereas (V)' means the spatial component of the time-varying term. If, however, one makes use of relationships (1.31) and (1.32) to simplify expression (1.37), we finally conclude that V - V

(1.38)

and, for simplicity of notation, we can write both superscripts at the same level in the format V'- Also, ( V ' ) ' = V = 0.

Figure 1.1 General three-dimensional vector diagram for a quantity (p.

M.J.S.DELEMOS

9

With the help of Figure 1.1, the concept of double-decomposition can be better understood. The figure shows a three-dimensional diagram for a general vector variable (p. For a scalar, all the quantities shown would be drawn on a single line. The basic advantage of the double-decomposition concept is to serve as a mathematical framework for analysis of flows where within the fluid phase there is enough room for turbulence to be established. As such, the double-decomposition methodology would be useful in situations where a solid phase is present in the domain under analysis so that a macroscopic view is appropriate. At the same time, properties in the fluid phase are subjected to the turbulent regime, and a statistical approach is appropriate. Examples of possible apphcations of such methodology can be found in engineering systems such as heat exchangers, porous combustors, nuclear reactor cores, etc. Natural systems include atmospheric boundary layer over forests and crops.

1.6 1.6.1

TURBULENT TRANSPORT Momentum equation

Mean flow The development to follow assumes single-phase flow in a saturated, rigid porous medium (AVf independent of time) for which, in accordance with expression (1.31), time-average operation on the variable ip commutes with the space-average. Application of the doubledecomposition idea in equation (1.37) to the inertia term in the momentum equation leads to four different terms. Not all of these terms are considered in the same analysis in the literature. Continuity The microscopic continuity equation for an incompressible fluid flowing in a clean (nonporous) domain was given by equation (1.1) and using the double-decomposition idea of expression (1.37) gives: Vu

= \/' {{uY + {u'Y -h % -f V ) = 0.

(1.39)

On applying both a volume- and time-average gives: V- {(f){uy) =0. For the continuity equation, the averaging order is immaterial.

(1.40)

10

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

Momentum—one average operator The transient form of the microscopic momentum equation (1.2) for a fluid with constant properties is given by the Navier-Stokes equation as follows: = - Vp + /iV^tx + pg.

(1.41)

Its time-average, using u = u-\-u', gives du ^ , ' ^ + V . ( n « ) - -Vp + juV^u + V • {-pu'u')

+ pg,

(1.42)

where the stresses, -pu'u', are the well-known Reynolds stresses. On the other hand, the volumetric-average of equation (1.41) using the theorem of local volumetric-average, equations (1.17)-(1.19), results in the following: dt

{4>{uy) + V • (pg + R,

(1.43)

where R represents the total drag force per unit volume due to the presence of the porous matrix, being composed by both viscous drag and form (pressure) drags. Further, using spatial decomposition to write u = {uY + ^u in the inertia term we obtain the following:

|(0(u)O+V-(0(.x)^(u)O

(1.45)

= - V {(l>{py) + pV^ [HuY) - V • {(t>{'u'uY) +cPpg + R. Hsu and Cheng (1990) pointed out that the third term on the right-hand side represents the hydrodynamic dispersion due to spatial deviations. Note that equation (1.45) models typical porous media flow for Re^ < 150-200. When extending the analysis to turbulent flow, time-varying quantities have to be considered. Momentum equation—two average operators The set of equations (1.42) and (1.45) are used when treating turbulent flow in clear fluid or low-Rep porous media flow, respectively. In each one of those equations only one averaging operator was applied, either time or volume, respectively. In this work, an investigation on the use of both operators in now conducted with the objective of modeling turbulent flow in porous media.

11

M.J.S.DELEMOS

The volume-average of equation (1.42) gives for the time-mean flow in a porous medium: dt

l(b(uV) + V • {(biuu)-] I

(1.46)

- V {0(p)*) + M V {(f>{uY) + V • {-pcPiu'u'Y)

+,j)pg + R,

where

" f n- (V«) ^S-^

~

f npdS

(1.47)

is the time-averaged total drag force per unit volume ('body force'), due to solid particles, composed by both viscous and form (pressure) drags. Likewise, applying now the time-average operation to equation (1.43), we obtain: — {(piu + u'Y) + V • {(j){{u + u'){u + u')y)

(1.48)

= -V{<j){p + p'y) + MV^ ({ur) + V . {4>{uuy) dt

(1.49)

where (1.50) Ay

/ JAi

n • (Vu) d 5 - - \ - / ^ ^

np(\S.

JAi

Comparing equations (1.46) and (1.49), we can see that also for the momentum equation the order of the application of both averaging operators is immaterial. It is interesting to emphasize that both views in the literature use the same final form for the momentum equation. The term jR is modeled by the Darcy-Forchheimer (Dupuit) expression after either order of application of the average operators. Since both orders of integration lead to the same equation, namely expression (1.47) or (1.50), there would be no reason for modeling them in a different form. Had the outcome of both integration processes been distinct, the use of a different model for each case would have been consistent. In fact, it has been pointed out by Pedras and de Lemos (2000b) that the major difference between those two paths lies in the definition of a suitable turbulent kinetic energy for the flow. Accordingly, the source of controversies comes from the inertia term, as seen below.

12

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

Inertia term—space and time (double) decomposition Applying the double-decomposition idea seen before for velocity (equation (1.37)) to the inertia term of equation (1.41) will lead to different sets of terms. In the literature, not all of them are used in the same analysis. Starting with time decomposition and applying both average operators, see equation (1.46), gives:

V•{(j){uuY) ^v'{(j){{u + u'){u-^u')Y) = v• [(i){{uuy + {u'u'y)]. (1.51) Using spatial decomposition to write u = {uY + ^u we obtain:

V • [(j> {{uuY + (ii^Y)] = V • {0 [{{{uY + %) {{uY + %))' + i^i^Y]} - V • {0 [{uYiuY + {'u'uY +

(i^Y]}.

(1.52) Now, applying equation (1.30) to write u' = {«')' + ' u ' , and substituting into expression (1.52) gives:

V • {4> [{uYiuY + i'u'uY + {u'u'Y]}

{
' + (^0') + {'u'uY]] = V • {0 [{uYiuY -h {u'Yiu'Y + {'U'UY] }. (1.57) With the help of equation (1.34) one can write *u = *tx + *u' which, inserted into expression (1.57), gives:

V • U UuYiuY + {u^Y{y''Y + WW]} = V'U \{uY{uY + {u'Yiu'Y + W^T^WT^Y]

I

(1.58)

= V- l(j)\{uY{uY -f {u'Yiu'Y + i'um + ^^u^u' + 'u' ^u + 'u' 'u'y] | . Application of the time-average operator to the fourth andfifthterms on the right-hand side of equation (1.58), containing only one fluctuating component, vanishes it. In addition, remembering that with expression (1.32) the equivalences ^u — ^u and {u'Y = {uY are valid, and that with expression (1.31) we can write {uY — (u)% we obtain the following alternative form for equation (1.58):

V- [(t)[{uY{uY + {^u^uY)] = V • U({uY{uY + {u'Y{u'Y + {'u'uY + {'u'^u'Y^}, t t t t I

II

III

IV

(1.59) which is the same result as expression (1.54). The physical significance of all four terms on the right-hand side of (1.59) can be discussed as follows. I Convective term of macroscopic mean velocity. II l\irbulent (Reynolds) stresses divided by the density p due to the fluctuating component of the macroscopic velocity. III Dispersion associated with spatial fluctuations of microscopic time-mean velocity. Note that this term is also present in the laminar flow, or say, when Re^ < 150. IV l\irbulent dispersion in a porous medium due to both time and spatial fluctuations of the microscopic velocity.

14

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

Further, the macroscopic Reynolds stress tensor (MRST) is given in Pedras and de Lemos (2001a) based on equation (1.25) as follows: -p<j>{u'u'Y = fit,2{Dr

- -Mkyi,

(1.60)

where (Dy = \ {V({nr) + P9-^- Wn'uY)

+ R,

where R was given earlier by expression (1.44) and the term {^u^uY is known as dispersion. The mathematical meaning of dispersion can be seen as a correlation between spatial deviations of velocity components. Making use of the double-decomposition concept given by equations (1.36), expression (1.67) can be expanded as follows:

= - V [')] + ct>P9 + R, (1.68) which results, after some manipulation, in the following:

+ V • [{W + (p'Y)] + MV^ [{{nY + {u'Y)] + {uY) + V • {(/. [{uY{uY + {u'Yiu'Y + (%%)' + ( V ^ ) ' ] } I = - V {m')+P''^\{uY) where P- I n-{Vu)AS--—

«=Ar

JAi

I npdS ^V

+ 4>pg + R, (1.70) (1.71)

JA.

represents the time-averaged value of the instantaneous total drag given by equation (1.44).

16

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

An expression for the fluctuating macroscopic velocity is then obtained by subtracting equation (1.70) from (1.69) and this results in the following:

P^liH^'Y) + pv. {cf>[{uy{ur + {uriuY + {uriu'y + (% V)^ + Cu' 'uY + Cu' 'u'Y - {u'Y{u'Y -

I^^^H^Y] }

= -yWY)+^^'^\{u'Y) + B!. (1.72) where R is also given by expression (1.65) such that equation (1.72) is the same as equation (1.66). Turbulent kinetic energy As mentioned, the determination of the flow macroscopic turbulent kinetic energy follows two different paths in the literature. In the models of Lee and Howell (1987), Wang and Takle (1995), Antohe and Lage (1997), and Getachewa et al. (2000), their turbulence kinetic energy was based on km = {u'Y ' {u'Yf^- They started with a simplified form of equation (1.66) neglecting the 5th, 6th, 7th and 9th terms (dispersion). Then they took the scalar product of it with {u'Y and applied the time-average operator. On the other hand, if one starts with equation (1.63) and proceeds with time-averaging first, one ends up, after volume-averaging, with {kY — {u' • u'Yl'^- This was the path followed by Masuoka and Takatsu (1996), Takatsu and Masuoka (1998), and Nakayama and Kuwahara (1999). The objective of this section is to derive both transport equations for km and {kY in order to compare similar terms. Equation for km = {u'Y ' {u'Y 1'^ From the instantaneous microscopic continuity equation for a constant property fluid one obtains:

V • {(j>{uY) = 0

^

V • [0 [{uY + {u'Y)] = 0,

(1.73)

and with time-average: V-((/)(it)^)=0.

(1.74)

From equations (1.73) and (1.74) we obtain: V-(0(txO')=O.

(1.75)

Taking the scalar product of equation (1.64) with {u'Y, making use of equations (1.73)(1.75) and time-averaging it, an equation for km will have for each of its terms (note that (j) is here considered as independent of time):

p ( « ' ) - I W«'>^) = P ^ ^ ,

(1-76)

M.J.S.DELEMOS

17

p{u'y • {V • {(i){uu'Y)} = p{u'Y • {V • {(t){uY{u'Y + (i){'u^u'Y)] = ^V . {(l>{uYkm) + p{u'Y ' {V ' {(l>{'u'u'Y)] ,

p{u'Y • {V • {(i){u'uY)] = p{u'Y' {V • {(j){u'Y{uY + (/>(^tx'^ti)*)} = p(j){u'Y{u'Y : V(tx)' H- p{u'Y • {V • (0(^16' ^uY)} , (1.78) / o X ) ' • {V • {(j){u'u'Y)} = p(ii')' • {V • {^{u'Y{u'Y + 0 ( ^ t t ' V ) 0 } / (u'V • iu'V\ = pV • f (j){u'Y^ 2 ) "^ '^^^'^'' ^ ^ ' ^'^^''^'''"'^'''^ ' (1.79) p{u'Y • {V • [-(t){u'u'Y)] = 0, (1.80)

-{u'Y' V {(t>{p'Y) = - V • {(t>{u'Y{p'Y). /i(tx')^ • V2 {cj>{u'Y) = fiV\(l>km) - P(t>em , {u'Y-R' = 0,

(1-81) (1-82) (1.83)

where em = iyV{u'Y - (V(u')*)^. In handling equation (1.81), the porosity 0 was assumed to be constant only for simplifying the manipulation to be shown next. However, this procedure does not represent a limitation in deriving a general transport equation for Another important point is the treatment given to the scalar product shown in equation (1.83). Here, a different view from the work in Lee and Howell (1987), Wang and Takle (1995), Antohe and Lage (1997), and Getachewa et al, (2000) is considered. The fluctuating drag form R' acts through the solid-fluid interfacial area and, as such, on fluid particles at rest. The fluctuating mechanical energy represented by the operation in equation (1.83) is not associated with any fluid particle movement and, as a result, is here considered to be of null value. This point is further discussed later in this chapter. The final equation for km gives: P^^-^pV^iHuYkm)

+ fiV\(t>km) - pcf>{u'Y{u'Y : V(tl)^ - p(t)em - Dm , where Dm = piu'Y • {V • [0{{^u'u'Y + {'W'uY + {'u''^''Y)]}

(1-85)

represents the dispersion of km given by the last terms on the right-hand sides of equations (1.77), (1.78) and (1.79), respectively. It is interesting to note that this term can be both negative and positive.

18

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

The first term on the right-hand side of equation (1.84) represents the turbulent diffusion of km and is normally modeled via a diffusion-like expression resulting for the transport equation for km, see Antohe and Lage (1997) and Getachewa et al. (2000):

p^^+pv-(^m - Dm , (1-86)

where (1.87)

P„ = -p<j>{u'Y{u'Y : V{uY

is the production rate of km due to the gradient of the macroscopic time-mean velocity

{u)\ Lee and Howell (1987), Wang and Takle (1995), Antohe and Lage (1997), and Getachewa et al. (2000) made use of the above equation for km considering for R! the DarcyForchheimer extended model with macroscopic time-fluctuation velocities {u'Y. They have also neglected all dispersion terms that were grouped into Dm, see equation (1.85). Note also that the order of application of both volume- and time-average operators in this case cannot be changed. The quantity km is defined by applying first the volume operator to the fluctuating velocity field. Equation for {ky = {u' • u'Y/2 The other procedure for composing the flow turbulent kinetic energy is to take the scalar product of equation (1.63) by the microscopic fluctuating velocity u'. Then apply both time and volume-operators for obtaining an equation for {ky = {u' • u'y/2. It is worth noting that in this case the order of application of both operations is immaterial since no additional mathematical operation (the scalar product) is conducted between the averaging processes. Therefore, this is the same as applying the volume operator to an equation for the microscopic k. The volumetric-average of a transport equation for k has been carried out in detail by de Lemos and Pedras (2000) and Pedras and de Lemos (2001a), and only the final resulting equation is presented, namely:

{(i>{ky) + v-{uD{ky) V-

/^+^)v(0(fcr)

where Pi = -p{u'u'y

: VUD ,

(1.88) -{-Pi + Gi- p(t>{ey,

{ky\uD\

Gi - Ckpcp

K

(1.89)

are the production rate of {ky due to mean gradients of the seepage velocity UD and the generation rate of intrinsic k due the presence of the porous matrix, respectively. Also, in equation (1.89) K is the medium permeability and Ck is a constant. As mentioned, equation (1.88) has been proposed by Pedras and de Lemos (2001a). Nevertheless, for the

M.J.S.DELEMOS

19

sake of completeness, a few steps of such derivation are here reproduced. Application of the volume-average theorem to the transport equation for the turbulence kinetic energy k gives:

^-pV-{(t>{u'[^

+ k))

}-\-piV^ {(t>{ky) - p(j){u'u' : WuY - p(t>{eY , (1.90)

where the divergence of the right-hand side can be expanded as follows:

V . [ct>{uky) = V • [0 [{uy{kY + {'u'kY)],

(1.91)

where the first term is the convection of {ky due to the macroscopic velocity whereas the second is the convective transport due to spatial deviations of both k and u. Likewise, the production term on the right-hand side of equation (1.90) can be expanded as follows: -p(j)(^

: Vuy = -pcj) [i^y

: {Vuy -f- {'(vJvJ) : \Vu)y] .

(1.92)

Similarly, the first term on the right-hand side of equation (1.92) is the production of (A:)* due to the mean macroscopic flow and the second is the {ky production associated with spatial deviations of flow quantities k and u. The extra terms appearing in equations (1.91) and (1.92), respectively, represent extra transport/production of {ky due to the presence of solid material inside the integration volume. They should be null for the limiting case of clear fluid flow, or say, when (j) -^ 1 ^ K -^ oo. Also, they should be proportional to the macroscopic velocity and to

{ky. In Pedras and de Lemos (2001a), a proposal for those two extra transport/production rates of {ky was made as follows:

V • ( [{uYiT'fy + Cu%y ^ {u')Hr^Y-^

{h^y)}.

(1.115) Substituting expression (1.114) into equation (1.112) gives for the same convection term:

{pc,)fV • [Wry) t

t

t

t

I

II

III

IV

(1.116) Comparing equations (1.115) and (1.116), in the light of equations (1.31) and (1.32), we conclude that equation (1.103) is, in fact, the same as equation (1.112). This demonstrates that the final expanded form of the macroscopic energy equation for a rigid, homogeneous porous medium saturated with an incompressible fluid does not depend on the averaging order, i.e. both procedures lead to the same results. Further, the four terms on the right-hand side of equation (1.116) could be given the following physical significance. I Convective heat flux based on the macroscopic time-mean velocity and temperature. II T\irbulent heat flux due to the fluctuating components of the macroscopic velocity and temperature. III Thermal dispersion associated with deviations of the microscopic time-mean velocity and temperature. Note that this term is also present when analyzing laminar convective heat transfer in porous media. IV T\irbulent thermal dispersion in a porous medium due to both time fluctuations and spatial deviations of both the microscopic velocity and temperature. Thus, the macroscopic energy equations for an incompressible flow in a rigid, homogeneous and saturated porous medium can be written as follows: fluid:

{pcp)fV • [(/> [{uy{T^y + {'u%y^{u'y{rfy

+ {Hl^y)] = (1.117)

solid:

V • {ksV [(1 - 0)(T,)']} - V • [ ^ I nhTs dS (1.118) fc.VT. d 5 = 0.

24

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

Further, adding equations (1.117) and (1.118), a global macroscopic energy equation can be obtained as follows:

(pcp)/v • [4> [{uYiT^y + {'u'T}y + {u'y{T}y + {hI^y)] = V • {fc/V imy) + k,w [(1 - mTsy]} + -^1

n-(fc/VT7-fc,VT7)d5, (1.119)

where the applicable boundary conditions on the surface Ai are given by: Tf=Ts 1 \ , A n-{kfVTf)=n-{k,\/Ts)\

'\r\Ai.

(1.120)

In view of these boundary conditions we verify that the last term on the right-hand side of equation (1.119) vanishes (due to the heat flux continuity at the fluid-solid interface). Thus, we can write:

(pcp)/v • [Da/ + (1 -