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

Phone: Fax: E.mail:

+44 1865 84 3879 +44 1865 84 3987 [email protected]

General enquiries, including placing orders, should be directed to Elsevier's Regional Sales Offices - please access the Elsevier homepage for full contact details (homepage details at the top of this page).

TRANSPORT PHENOMENA IN POROUS MEDIA Volume III Edited by

D. B. Ingham & I. Pop

ELSEVIER B.V. Radarweg 29 P.O. Box 211, 1000 AE Amsterdam The Netherlands

ELSEVIER Inc. 525 B Street, Suite 1900 San Diego, CA 92101-4495 USA

ELSEVIER Ltd The Boulevard, Langford Kidlington, Oxford OX5 IGB UK

ELSEVIER Ltd 84 Theobalds Road London WCIX 8RR UK

© 2005 Elsevier Ltd. All rights reserved. This work is protected under copyright by Elsevier Ltd., and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier's Rights Department in Oxford, UK: phone (+44) 1865 843830, fax (+44) 1865 853333, e-mail: [email protected]. Requests may also be completed on-line via the Elsevier homepage (http://www.elsevier.com/locate/permissions). In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London WIP OLP, UK; phone: (+44) 20 7631 5555; fax: (+44) 20 7631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of the Publisher is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier's Rights Department, at the fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made.

First edition 2005 ISBN: 0-0804-4490-3 Printed in Great Britain.

^X^rking together to grow libraries in developing countries www.elseYier.com | www.bookaid.org | www.sabre.org

ELSEVIER

fjo^.i'!;'^

Sabre Foundation

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.

PREFACE

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.

Vlll

PREFACE

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

PREFACE

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

X

PREFACE

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.

PREFACE

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

This Page Intentionally Left Blank

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

This Page Intentionally Left Blank

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 {
= V-{Mv>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 , <^' = (¥''r+v, (1-29) or further

^' = (<^r'+v = {<^'r+v,

(1.30)

where V ' ^^^ t)e understood as either the time fluctuation of the spatial deviation or the spatial deviation of the time fluctuation. After some manipulation, we can prove that, see Pedras and de Lemos (2001a), {ipY = {TpY

or

W

(1-31)

= m \

i.e. the time- and volume-averages commute. Also, '^ = V,

{^j

(1.32)

= {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 • ({uy) + (t>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]}

{<

V -
\{uY{uY + {"u'uY

(1.53)

= V • {(/) \{uY{uY + {'u'uY + {u'Y{u'Y +{{u'Y 'u'Y + {^u'{u'YY + (V^)^] I . The fourth and fifth terms on the right-hand side contains only one space-varying quantity and will vanish under the application of volume integration. Equation (1.53) will then be reduced to

V • {(j){uuY) - V • {(/) \{uY{uY + {u'Y{u'Y + Cu'uY + ( W ) ^ ] } . Using the equivalence (1.31) and (1.32), equation (1.54) can be further rewritten as follows: V • {(l){uuY) = ^'U

[(w)^ (tip" -f {uY'{uY' + {hl^Y

+

{'U'^U'Y]

}

(1.55)

with an interpretation of the terms in equation (1.54) given later. Another route to follow to reach the same results is to start out with the application of the space decomposition in the inertia term, as usually done in classical mathematical

(1.54)

M. J. S. DE LEMOS

13

treatment of porous media flow analysis. Then we obtain

V • {(t>{uuY) - V • [(t>{{{uY + 'u) {{uY + 'u)Y] = v '[(f) {{uY{uY + {'u ^uY)],

(l.Do;

and on time-averaging the right-hand side, using equation (1.33) to express (it)* = {ny H- {u'y, this becomes:

V • [(t){{uY{uY + i'u^uY)] = V • [(l)[{{uY + {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(<^{u)0 + [ V W n ) 0 ] ^ }

(1.61)

is the macroscopic deformation tensor, (A;)* is the intrinsic average for k, and //^^ is the macroscopic turbulent viscosity assumed to be in Pedras and de Lemos (2001c) as follows:

Fluctuating velocity The starting point for an equation for theflowturbulent kinetic energy is an equation for the microscopic velocity fluctuation u'. Such a relationship can be written, after subtracting the equation for the mean velocity u from the instantaneous momentum equation, resulting in the following: p < - ^ + V • [uu' 4- u'u + u'u' - u'u']

> = -Vp' -h fiV'^u'.

(1.63)

Now, the volumetric-average of equation (1.63), using the theorem of local volumetricaverage, gives:

Pj^iHu'Y) + pv-{

(J ^^

= -V{cj>{p'Y) + i^v'{Hn'Y) + R', where

is the fluctuating part of the total drag due to the porous structure. Expanding further the divergent operators in equations (1.64) by means of the expression set (1.29), one ends up with an equation for {u'Y as follows:

P^-^iH^'Y) + pv • {0 [{uYiu'Y + (u'YiuY + {u'Y i^'Y + {'u'u'Y + Cu' 'uY + ('«' 'u'Y - (u'Yiu'Y - (VV)']} = -S7{{p'Y) + ^lV^{{u'Y) + R'. (1.66)

M.J.S.DELEMOS

It)

Another route to follow in order to obtain equation (1.66) is to start out with the macroscopic instantaneous momentum equation for an incompressible fluid given by Hsu and Cheng (1990):

p{^w«n+v.wnr(nn}

^^^^^

= - V [{(pr)] + f^V\4>{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-(<^(«rfc„) = v. dt

+ Pm- P4>^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 • (<^(%^fc)0 - pC{^IJ^) : '(Vu))^ =Gi = c , p < / . ^ ^ ^ ^ ,

(1.93)

where the constant Ck was numerically determined by fine-flow computations considering the medium to be formed by circular rods, see Pedras and de Lemos (2001c), as well as longitudinal, see Pedras and de Lemos (2001b), and transversal rods, see Pedras and de Lemos (2003). In spite of the variation in the medium morphology and the use of a wide range of porosity and Reynolds number a value of 0.28 was found to be suitable for most calculations. Comparison of macroscopic transport equations A comparison between terms in the transport equation for km and (A:)* can now be conducted. Pedras and de Lemos (2000b) have already shown the connection between

20

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

these two quantities to be: W - — y - -

^

+

^

-^m-f

.

-

(1.94)

Expanding the correlation forming the production term Pi by means of equation (1.16), a connection between the two generation rates can also be written as follows: Pi = -piu'u'Y

: \/uD

= -p {{u'Y{u'Y = Pm- pi'W'u'Y

: VUD -h {^u'^u'Y : V U D )

(1.95)

' VtXD .

We note that all the production rate of km, due to the mean flow, constitutes only part of the general production rate responsible for maintaining the overall level of {kYThe dissipation rates also carry a correspondence if we expand f\TV (Vu^y)

{eY = iy{Vu' : = v{Vu'Y

' [{"^u'Y]

+ iy{'{Vu') : ^{Vu'YY

(1.96)

- ^ V [(t>{u'Y) : [V ((/>(txOO]^ + v{^{Vu') : ^{Vu'YY 0

•

On considering the porosity to be constant, we have {tY = e^ + v{'{Vu')

: '{Vu'YY

,

(1-97)

and this indicates that an additional dissipation rate is necessary to fully account for the energy decay process inside the REV.

1.7 1.7.1

HEAT TRANSFER Governing equations

Time-average followed by volume-average In order to apply the time-average operator to equations (1.9) and (1.10), we consider the time decomposition (1.21). Thus, substituting expression (1.21) into equations (1.9) and (1.10), respectively, we obtain: {pCj,)fV'{uTf

+ uT'j+u%

+ u'rj)

-V-(A:/V(T>+T})) ,

0-V-(A:,V(T;+r;)),

(1.98) (1.99)

M.J.S.DELEMOS

21

and on time-averaging we obtain: (pCp)/V • ( i Z T V + ^ ) - V • (A:/V7>) ,

(1.100) (1.101)

0 = V'{ksVTs).

The second term on the left-hand side of equation (1.100) is known as turbulent heat flux. It requires a model for closure of the mathematical problem. Also, in order to apply the volume-average to equations (1.100) and (1.101), we first define the spatial deviations with respect to the time-averages, namely: T= {Ty + 'T,

(1.102)

u = {uy+'u.

On substituting into equations (1.98) and (1.99) and performing the volume-average operation, we obtain:

(pcp)/V • {(/) {{uYiT^y + {'u'Tfy +

i^y)} (1.103)

V • {ksV [(1 - (t>){Tsy]} - V • [ ^

I

nksTsd.s\

- ^

I

ri' ksVZdS

= 0, (1.104)

which are the macroscopic energy equations for the fluid and the porous matrix (solid) taking first the time-average followed by the volume-average operator. Volume-average followed by time-average To apply the volume-average to equations (1.9) and (1.10), we have Tz={Ty-\-'T,

(1.105)

u = {uy-^'u.

In addition, we have <^>' = ^ < ^ > ' | {uy='y{uyj

where , = | * [i-

f"*""""'

(,.106,

for the solid.

Substituting equation (1.105) into equations (1.9) and (1.10), we obtain: ipc,)fV

• {{uYiTfY

+ (uY'Tf

+ 'u{TfY + 'u%)

= V • [kfV {{TfY + %)] , (1.107)

o = v-[fc,v((r,)^ + 'r,)],

(1.108)

22

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

and taking the volume-average we obtain:

- V • [k}V {<j>{TfY)] + V • [ ^ ^ nkfT} ds] +^l^n-

kfVTf dS, (1.109)

V • {fc,V [(1 -
I

n • fc.VT, d 5 = 0.

nksTs
(1.110) The second term on the left-hand side of equation (1.109) appears in classical analysis of convection in porous media, e.g. Hsu and Cheng (1990), and is known as thermal dispersion. In order to apply the time-average to equations (1.109) and (1.110), we define the intrinsic volume-average as follows:

(1.111)

{TY = {TY + {TY , {uY = {uY + {uY ,

and substituting into equations (1.109) and (1.110), and taking the time-average, we obtain:

(1.112) V-{A:,V[(1-(/>)(I^]}-V- f ^ y "

nksTsdsl-^l

n-ksVTsdS

= 0. (1.113)

Equations (1.112) and (1.113) are the macroscopic energy equations for the fluid and the porous matrix (solid) taking first the volume-average followed by the time-average. It is interesting to observe that equations (1.103) and (1.104), obtained through the first procedure (time-volume-awcragQ) are similar to expressions (1.112) and (1.113), respectively, obtained through the second method {volume-time-avcrdLgo). 1.7.2

Turbulent thermal dispersion

Using equations (1.32)-(1.35), the third term on the left-hand side of equations (1.103) and (1.112) can be expanded as follows: {^u'TfY = {{^u-\-'u') {'Tf-\-^T')Y

= {'u'TfY

+ {'W'V^Y ,

(1-114)

23

M. J. S. DE LEMOS

and, on substituting into equation (1.103), the convection term becomes: (pcp)/V •

(Wry)

- {pc,)fV. {(/> [{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 • [){%y]}

+ V- \ ^ J

n{kfT}-ksZ)ds\.

(1.121) The model proposed in de Lemos and Rocamora, Jr (2002) for the macroscopic turbulent heat flux follows the eddy diffusivity concept embodied in equation (1.26) and becomes: -{pc,)f{^y

= Cpfl^W{T^r ,

(1.122)

where //^^ is given by equation (1.62) and ar^ is a constant. According to equation (1.122), the macroscopic turbulent heat flux is taken as the sum of the turbulent heat flux and the turbulent thermal dispersion, as proposed in Rocamora, Jr and de Lemos (2000). These two terms were related there to the components of the conductivity tensor, Kt and -'^ciisp,^ respectively, by the expression Kt + Kdisp.t = (t>Cpf^I.

(1.123)

M.J.S.DELEMOS

1.7.3

25

Local thermal equilibrium hypothesis

The local thermal equilibrium hypothesis assumes that the intrinsic averages of the timemean temperature for the fluid and solid phases are equal, i.e. (1.124)

{Tjy = {T,y ^ {TY, and substituting into equation (1.121) we obtain:

= V • {[kf(l> + ks{l - 0)] V(r)^} + V . [ ^

^

n {kfTf - ksTs) d5]

(1.125) Using the Dupuit-Forchheimerrelationship ti£) = {uY — 0(u)% we can rewrite equation (1.125) as follows: {pc,)jV-[uD{Ty) = V . {[kf + ks{l - 0)] V(r)^} + V • [ ^

^

n {kfTf - ksTs) d s ]

- (pcp)/v • [0 ((%^57)^ + (txOMT})^ + ( ^ ^ ^ r ) ] , (1.126) and the last three terms are additional unknowns coming from application of both processes of averaging, namely time- and volume-averaging. As mentioned above, they represent dispersion due to the spatial deviations, turbulent heat flux due to time fluctuations and turbulent thermal dispersion due to both time fluctuations and spatial deviations. Models for thermal dispersion and for turbulent heat flux have been applied on separate to flows through clear and porous domains, respectively. To the best of the author's knowledge, no work in the literature has proposed a general model encompassing all terms in equation (1.126). 1.7.4

Macroscopic buoyancy effects

Mean flow Focusing attention on buoyancy effects only, application of the volume-average procedure to the last term of equation (1.23) leads to:

{pgP{T - T.ef)}" = 1 ^ ^

^ ^ PgPiT - Tref) dV ,

(1.127)

26

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

and expanding the left-hand side in the light of equation (1.16), the buoyancy term becomes: {pgp{T - Tref)r = Pl34>9^ {{TY - T^ef) + P9p(t>WY , (1-128) =0

where the second term on the right-hand side is null since (*(/?)*= 0. Here, the coefficient /30 is the macroscopic thermal expansion coefficient. Assuming that gravity is constant over the REV, an expression for it based on expression (1.128) is given as follows:

Including equation (1.128) into the formulation of Pedras and de Lemos (2001a), the macroscopic time-mean Navier-Stokes (NS) equation for an incompressible fluid with constant properties is given as: pv • f ^ ^ ^ )

= - V (
+ V • {-p{ll^y)

-pl3^g(f>{{Ty-T,,t)

p(f>_

-rpUD +

CF(t>p\UD\UD

(1.130) Turbulent field As mentioned, this work extends the development in Pedras and de Lemos (2001a), in order to include the buoyancy production rate term in the turbulence model equations. For clear flows, the buoyancy contribution to the k equation is given in equation (1.28). Applying the volume-average operator to that term, we obtain: {Gkr =G',=

{-ppg-u'T'Y

^ -pplg • {u'T'^)\

(1.131)

where the coefficient ^^, for a constant value of g within the REV, is given by /J^k _ {^u' T'y /(f){pu' T'Y, which, in turn, is not necessarily equal to ^^ given by expression (1.129). However, for the sake of simplicity and in the absence of better information, we can make use of the assumption ^^ = ^^ = /3. Further, expanding the right-hand side of

M. J. S. DE LEMOS

27

expression (1.131) in light of expression (1.16) and (1.32), we have the following:

= -PPl9 • {{{u'Y{T'0 + {hlr^jY + {{u'Y'T'fY + {'w = -ppl<j>g • UuY' (TfY'M^^^Y+

t I

t

II

{T'0)

{u'Yi'T'jY + {'u'){T'jY ) ,

^^T^

^^^

(1.132) where the last two terms on the right-hand side are null since l^VjY — 0 and i^u'Y — 0. In addition, the following physical significance can be inferred to the two remaining terms on the right-hand side of equation (1.132). I Generation/destruction rate due to macroscopic time fluctuations: buoyancy generation/destructions rate of (A;)* due to time fluctuations of the macroscopic velocity and temperature. This term is also present in turbulent flow in clear (non-obstructed) domains and represents an exchange between the energy associated with the macroscopic turbulent motion and potential energy. In stable stratification, this term damps turbulence by being of negative value whereas the potential energy of the system is increased. On the other hand, in unstable stratification, it enhances (A:)* at the expense of potential energy. II Generation/destruction rate due to turbulent buoyant dispersion: buoyancy generation/destruction rate of i^kY in a porous medium due to time fluctuations and spatial deviations of both microscopic velocity and temperature. This term might be interpreted as an additional source/sink of turbulence kinetic energy due the fact that time fluctuations of local velocities and temperatures present a spatial deviation in relation to their macroscopic value. Then, additional exchange between turbulent kinetic energy and potential energy in systems may occur due to the presence of a porous matrix. A model for expression (1.132) is still necessary in order to solve an equation for (A:)*, which is a necessary information when computing //^^ using equation (1.62). As such, terms I and II above have to be modeled as a function of the average temperature, (T)\ To accomplish this, a gradient type diffusion model is used, in the form: • Buoyancy generation of (A:)* due to turbulent fluctuations: -p/3^0g • {uY\T,Y'

= pBt • V{TY ,

(1.133)

• Buoyancy generation of (fc)* due to turbulent buoyant dispersion: -/9/3>g • {iu' iVjY - pBdisp.t • V(T)^.

(1.134)

28

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

The buoyancy coefficients seen above, namely Bt and ^disp,t» are modeled here through the eddy diffusivity concept, similar to the work in Nakayama and Kuwahara (1999). It should be noticed that these terms arise only if the flow is turbulent and if buoyancy is of importance. Then using an expression similar to (1.122), the macroscopic buoyancy generation of k can be modeled as follows: Gf = -p0'^
= ^'l^P^g

• V(T)' = Beff • V ( f ) ' ,

(1.135)

where the two coefficients Bt and Bdisp.t are expressed as follows: Bt + Bdisp.t = Beflf = I3a,(j)—-g.

(1.136)

Final transport equations for (fc)' = («' • u'y/2 and (e)' = fi{Vu' : {Vu')^y/p, in their so-called high-Reynolds number form, as proposed in Pedras and de Lemos (2001a), can now include the buoyancy generation terms seen above, in the form:

pV • [uoiky) = V

M + ^ ) V im')

pW-{uD{ey)=V

^l +

P't,

+ Pi + Gi + G^i-P{ey,

(1.137)

V [m)

(1.138) + j^i [ciPi + C2Gi + CiCsGf - C2p{e)'\ , {k) where the cs are constants and G^ = Befi • V(T)* is the generation of (A:)* due to the buoyancy.

1.8 1.8.1

MASS TRANSFER Mean and turbulent fields

Mass transfer analysis using the double-decomposition concept follow similar steps applied to heat transfer as above. To apply the volume-average to equation (1.4), we have (1.139) which on substituting into equation (1.4) gives: d{{mtY

+ 'mt) dt

+ V.[((n)^ + ^n)((m,)^ + ^m,)]

- {RiY + 'Ri + D^V" {{miY + 'rnt) ,

(1.140)

M. J. S. DE LEMOS

29

where the mixture density p and the coefficient Di in equation (1.5) have been assumed to be constant. Expanding the convection term and taking the volume-average, with the help of equation (1.20), we have:

= <^{{RiY + 'Rtf

+ Bi ( V V {kmiY +

'mi))' (1.141)

or ^ ^ 1 ^ + V • [.^ [{u)\mtY

+ C«'m,)^)] =
(1.142)

where the third term on the left-hand side of equation (1.142) appears in classical analysis of mass transport in porous media, e.g. Bear and Bachmat (1967) and Bear (1972), and is known as the mass dispersion. In order to apply the time-average to equation (1.142), we define the intrinsic volumeaverage as follows: l^mtY = i^mtY + kmtY ,

W

- W

+ W

,

(1.143)

which on substituting into equation (1.142) and taking the time-average, we obtain: - ^ 5 ^ ^ + V • 0 UuY {miY + {uY' {rriiY' + {'u^rriiY) ot \ /

(1.144)

Equation (1.144) is the macroscopic mass transfer equation for the species i in the porous matrix taking first the volume-average followed by the time-average. Another route to reach an equivalent macroscopic transport equation is to invert the order of application of the same average operators. 1.8.2

I\irbulent mass dispersion

Using now equations (1.32)-(1.36), the fourth term on the left-hand side of equation (1.144) can be expanded as follows: {'u'rriiY = {{'u +'u') {mt-\-'m'^)Y

= {'u'TniY + {'u''m[Y

,

(1.145)

30

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

and, on substituting into equation (1.144), the convection term becomes: V • U{umiY)

= V • l(l)({uY{miY-}-{'u'miy

+ {uY' {ruiY' -h {'u'^m'^y)] -

t

t

t

t

I

II

III

IV

(1.146) Likewise, applying again equations (1.32)-(1.36) to the fourth term on the left-hand side of expression (1.144), we obtain: {u' m^y = {{{u'Y H- 'u') {{m[Y + ^m^))' = {u'Y{m[Y + {'W 'm[y ,

(1.147)

which, on substituting into expression (1.144), gives for the same convection term: V • UiumiY)

= V • UUuY

{miY + {'u'ruiY H- {u'Y {m'^Y + {'u'^m'^y)\

t

t

t

t

I

II

III

IV

.

(1.148) Here the order of application of both the averaging operators to the entire equation is immaterial and the proof is left to the interested reader. Further, the four terms on the right of equation (1.148) could be given the following physical significance (multiplied by p). I Convective mass flux based on macroscopic time-mean velocity and mass fraction. II Mass dispersion associated with deviations of the microscopic time-mean velocity and mass fraction. Note that this term is also present when analyzing the laminar mass transfer in porous media, but it does not exist if a volume-averaged is not performed. III T^irbulent mass flux due to the fluctuating components of both macroscopic velocity and mass fraction. This term is also present in turbulent flow in clear (non-porous) domains. It is not defined for laminar flow in porous media where time fluctuations do not exist. IV Turbulent mass dispersion in a porous medium due to both time fluctuations and spatial deviations of both microscopic velocity and mass fraction. Thus, the macroscopic mass transport equation for an incompressible flow in a rigid, homogeneous and saturated porous medium can be written as follows:

^ ^ 1 ^ 4- V • 0 [{uyimY + {'u'rfiiy -f {u'Y {m',Y + (^^7^)^) - ^{'RiY + DiV^ {^{rniy) , (1.149)

M. J. S. DE LEMOS

31

or in its equivalent form (see equation (1.32)):

d(f){mi)

(1.150)

1.9

CONCLUDING REMARKS

In this chapter we have described a new methodology for the analysis of turbulent flow in permeable media. A novel concept, called the double-decomposition idea, was detailed showing how a variable can be decomposed in both time and volume in order to simultaneously account for fluctuations (in time) and deviations (in space) around mean values. Transport equations for the mean and turbulence flow have been presented, including consideration of heat transfer with buoyancy and mass transport. The usefulness of this research might be better appreciated when studying transport over highly permeable media where turbulence flow regime occurs in the fluid phase. Analyses of important environmental and engineering flows can benefit from the derivations herein and, ultimately, it is expected that additional research on this new subject be stimulated by the work here presented.

ACKNOWLEDGEMENTS The author would like to express his thanks to CNPq, Brazil, for their financial support during the preparation of this work. Thanks are also due to the author's former and current graduate students who conducted their graduate programs under the topic here discussed.

REFERENCES Antohe, B. V. and Lage, J. L. (1997). A general two-equation macroscopic turbulence model for incompressible flow in porous media. Int. J. Heat Mass Transfer 40, 3013-24. Bear, J. (1972). Dynamics of fluids in porous media. Elsevier, New York. Bear, J. and Bachmat, Y. (1967). A generalized theory on hydrodynamic dispersion. In lASH symposium on artificial recharge and management of aquifers, Haifa, Israel, March, pp. 7-16. lASH Publication No. 72, International Association of Scientific Hydrology. de Lemos, M. J. S. (2004). Turbulent kinetic energy distribution across the interface between a porous medium and a clear region. Int. Comm. Heat Mass Transfer. In press. de Lemos, M. J. S. and Braga, E. J. (2003). Modeling of turbulent natural convection in porous media. Int. Comm. Heat Mass Transfer 30, 6\5-24.

32

DOUBLE-DECOMPOSITION FOR TURBULENT TRANSPORT

de Lemos, M. J. S. and Mesquita, M. S. (2003). Turbulent mass transport in saturated rigid porous media. Int. Comm. Heat Mass Transfer 30, 105-13. de Lemos, M. J. S. and Pedras, M. H. J. (2000). Modeling turbulence phenomena in incompressible flow through saturated porous media. In Proceedings of 34th ASME—National heat transfer conference, Pittsburgh, PA, 20-22 August. ASME-HTD-I463CD, paper NHTC2000-12120; on CD-ROM. de Lemos, M. J. S. and Pedras, M. H. J. (2001). Recent mathematical models for turbulent flow in saturated rigid porous media. ASME J. Fluids Eng. 123, 935-40. de Lemos, M. J. S. and Rocamora, Jr, F. D. (2002). Turbulent transport modeling for heated flow in rigid porous media. In Proceedings of the 12th international heat transfer conference, Kyongyu, Korea, 23-28 August, Vol. 4, pp. 791-6. de Lemos, M. J. S. and Tofaneli, L. A. (2004). Modeling of double-diffusive turbulent natural convection in porous media. Int. J. Heat Mass Transfer 47, 4221-31. Getachewa, D., Minkowycz, W. J., and Lage, J. L. (2000). A modified form of the k-e model for turbulent flow of an incompressible fluid in porous media. Int. J. Heat Mass Transfer 43, 2909-15. Gray, W. G. and Lee, P. C. Y. (1977). On the theorems for local volume averaging of multiphase system. Int. J. Multiphase Flow 3, 333^0. Hsu, C. T. and Cheng, P. (1990). Thermal dispersion in a porous medium. Int. J. Heat Mass Transfer 33, 1587-97. Ingham, D. B. and Pop, I. (eds) (2002). Transport phenomena in porous media. Vol. II. Pergamon, Oxford. Kuwahara, F. and Nakayama, A. (1998). Numerical modeling of non-Darcy convective flow in a porous medium. In Proceedings of the 11th international heat transfer conference, Kyongyu, Korea, 23-28 August, Vol. 4, pp. 411-16. Kuwahara, F., Nakayama, A., and Koyama, H. (1996). A numerical study of thermal dispersion in porous media. J. Heat Transfer 118, 756-61. Lage, J. L. (1998). The fundamental theory of flow through permeable media from Darcy to turbulence. In Transport phenomena in porous media (eds D. B. Ingham and I. Pop), pp. 1-30. Pergamon, Oxford. Lage, J. L., de Lemos, M. J. S., and Nield, D. A. (2002). Modeling turbulence in porous media. In Transport phenomena in porous media (eds D. B. Ingham and I. Pop), Vol. II, pp. 198-230. Pergamon, Oxford. Lee, K. and Howell, J. R. (1987). Forced convective and radiative transfer within a highly porous layer exposed to a turbulent external flow field. In Proceedings of the 1987 ASME/JSME thermal engineering joint conference. Vol. 2, pp. 377-86. Masuoka, T. and Takatsu, Y. (1996). Turbulence model for flow through porous media. Int. J. Heat Mass Transfer 39, 2803-9. Nakayama, A. and Kuwahara, F. (1999). A macroscopic turbulence model for flow in a porous medium. ASME J. Fluids Eng. 121, 427-33. Nield, D. A. and Bejan, A. (1999). Convection in porous media (2nd edn). Springer-Verlag, New York. Pedras, M. H. J. and de Lemos, M. J. S. (1999). On volume and time averaging of transport equations for turbulent flow in porous media. In Proceedings of the 3rd ASME/JSME joint fluids

M.J.S.DELEMOS

33

engineering conference, San Francisco, CA, 18-23 July. ASME-FED-248, paper FEDSM99-7273; on CD-ROM. Pedras, M. H. J. and de Lemos, M. J. S. (2000a). Numerical solution of turbulentflowin porous media using a spatially periodic cell and the low Reynolds k-e model. In Proceedings of CONEM2000— National mechanical engineering congress. Natal, Rio Grande do Norte, Brazil, 7-11 August. In Portuguese; on CD-ROM. Pedras, M. H. J. and de Lemos, M. J. S. (2000b). On the definition of turbulent kinetic energy for flow in porous media. Int Comm. Heat Mass Transfer 27, 211-20. Pedras, M. H. J. and de Lemos, M. J. S. (2001a). Macroscopic turbulence modeling for incompressible flow through undeformable porous media. Int. J. Heat Mass Transfer 44, 1081-93. Pedras, M. H. J. and de Lemos, M. J. S. (2001b). On mathematical description and simulation of turbulent flow in a porous medium formed by an array of elliptic rods. ASME J. Fluids Eng. 123, 941-7. Pedras, M. H. J. and de Lemos, M. J. S. (2001c). Simulation of turbulent flow in porous media using a spatially periodic array and a low Re two-equation closure. Numer. Heat Transfer, Part A 39, 35-59. Pedras, M. H. J. and de Lemos, M. J. S. (2003). Computation of turbulent flow in porous media using a low Reynolds k-e model and an infinite array of transversally-displaced elliptic rods. Numer Heat Transfer, Part A 43, 585-602. Rocamora, Jr, F. D. and de Lemos, M. J. S. (2000). Laminar recirculating flow and heat transfer in hybrid porous medium-clear fluid computational domains. In Proceedings of 34th ASME— National heat transfer conference, Pittsburgh, PA, 20-22 August. ASME-HTD-I463CD, paper NHTC2000-12317; on CD-ROM. Saito, M. and de Lemos, M. J. S. (2004). Interfacial heat transfer coefficient for non-equilibrium convective transport in porous media. Int. Comm. Heat Mass Transfer. In press. Silva, R. A. and de Lemos, M. J. S. (2003a). Numerical analysis of the stress jump interface condition for laminar flow over a porous layer. Numer Heat Transfer, Part A 43, 603-17. Silva, R. A. and de Lemos, M. J. S. (2003b). Turbulent flow in a channel occupied by a porous layer considering the stress jump at the interface. Int. J. Heat Mass Transfer 46, 5113-21. Slattery, J. C. (1967). Flow of viscoelastic fluids through porous media. AIChE J. 13, 1066-71. Takatsu, Y. and Masuoka, T. (1998). Turbulent phenomena in flow through porous media. J. Porous Media 1,243-51. Vafai, K. (ed.) (2000). Handbook of porous media. Marcel Dekker, New York. Wang, H. and Takle, E. S. (1995). Boundary-layer flow and turbulence near porous obstacles. Boundary-Layer Meteorol. 74, 73-88. Whitaker, S. (1966). Equations of motion in porous media. Chem. Eng. Sci. 21, 291-300. Whitaker, S. (1967). Diffusion and dispersion in porous media. J. Amer Inst. Chem. Eng. 13,420-31. Whitaker, S. (1969). Advances in theory of fluid motion in porous media. Ind. Eng. Chem. 61, 14-28. Whitaker, S. (1999). The method of volume averaging. Kluwer, Dordrecht.

2

HEAT TRANSFER IN BIDISPERSE POROUS MEDIA D. A. NIELD* and A. V. KUZNETSOV^ * Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland, New Zealand email: [email protected]

^Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC 27695-7910, USA email: avkuznetOeos.ncsu.edu

Abstract Recent work on heat transfer in bidisperse porous media (BDPM) is surveyed. Topics covered are the measurement of 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 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.

Keywords: bidisperse porous media, permeability, thermal conduction, dispersion, forced convection, two-velocity two-temperature model 2.1

INTRODUCTION

A standard porous medium consists of two phases, namely fluid and solid. In the case of local thermal non-equilibrium, the temperatures T/ and Tg in these two phases are not equal. A bidisperse (or bidispersed—we have opted for the shorter and more commonly used form) porous medium (BDPM), as defined by Chen et al (2000a, 2000b), is composed of clusters of large particles that are agglomerations of small particles, see Figure 2.1. Thus there are macro-pores between the clusters and micro-pores within them. Applications are found in bidisperse adsorbent or bidisperse capillary wicks in a heat pipe. 34

D. A. NIELD AND A. V. KUZNETSOV

35

Figure 2.1 Sketch of a bidisperse porous medium.

Since the bidisperse wick structure significantly increases the area available for liquid film evaporation, it has been proposed for use in the evaporator of heat pipes. A BDPM may thus be looked at as a standard porous medium in which the solid phase is replaced by another porous medium, whose temperature may be denoted by Tp if local thermal equilibrium is assumed within each cluster. We can then talk about the /-phase (the macro-pores) and the p-phase (the remainder of the structure). An alternative way of looking at the structure is to regard it as a porous medium in which fractures or tunnels have been introduced. One can then think of the /-phase as being a 'fracture phase' and thep-phase as being a 'porous phase'. Although bidisperse porous media, or something similar, have been used by chemists and chemical engineers for a number of years in processes such as chromatography, it appears that it is only very recently that their heat transport properties have been studied systematically, and then only by a few groups of research workers. Recent surveys of work on porous media by Nield and Bejan (1999), Ingham and Pop (2002), and Bejan et al. (2004) show that there has been a significant lack of progress in the area of bidisperse porous media.

2.2

DETERMINATION OF TRANSPORT PROPERTIES

Questions of interest are how one can determine the effective permeability and the effective thermal conductivity of a bidisperse porous medium. Fractal models for each of these have been formulated by Yu and Cheng (2002a, 2002b). In the first paper, the authors developed two models for the effective thermal conductivity based on the fractal theory and the electrical analogy. Theoretical predictions based on these models were compared with those from a previous lumped-parameter model and with experimental data for the stagnant thermal conductivity reported by Chen et al. (2000a, 2000b). In this paper a three-dimensional model of touching spatially periodic cubes, which are approximated by touching porous cubes, was used; Cheng and Hsu (1999) had previously used a two-

36

HEAT TRANSFER IN BIDISPERSE POROUS MEDIA

dimensional model. The effective stagnant conductivity of the micro-porous cubes is given by ^

= (1 - 7.^ - 27ci7si + 27ci7.\) + ^ I

^

^ ill ~ 7al7cl , 2(7ei7al - 7cl7al) 1 - 7al + 7alAi 1 - 7cl7al + 7cl7alAi

(2.1)

and the porosity within the cubes is given by
•

(2-2)

Here Ai = kf/kg is the fluid/solid thermal conductivity ratio, jai = cii/h and 7ci = ci /ai represent the geometric length scale ratio and contact length scale ratio, respectively, where ci, ai and /i are the contact length, the length of the sides of the cubes and the length of a unit cell at the microscale (within the clusters). Then the effective stagnant conductivity of the BDPM is given by ^

= {1-JI-

27C7S

-f 27c7.^) + ^

+ j A

kf

' " ^}^'

1 - 7a -f 7aA

+ , ^^^'^^ ~^'^'^\

(2.3)

1 - 7c7a + 7c7aAi

and the total porosity of the BDPM is given by t = l-[{l-

ZllH

+ 37e'7a] (1 - <^i) •

(2-4)

Here A = kf/kei is the fluid/solid thermal conductivity ratio, 7a = 0,2/12 and 7c = 02/02 represent the geometric length scale ratio and contact length scale ratio, respectively, where C2, 02 and I2 are the contact length, the length of the sides of the cubes and the length of a unit cell at the meso-scale (the scale of the clusters). The experiments were conducted with two samples of BDPM (with micro-pore diameter of 80 /im and macro-pore diameters of 200 fim and 400 /im) with sintered copper as the solid and three different fluids (air, FC-72 and water). For comparison, measurements were also made of the effective thermal conductivity of a monodisperse medium with a pore diameter of 800 /xm saturated with the same fluid. The measured data were found to be in good agreement with those predicted form a lumped-parameter model. The authors concluded that, when the ratio of solid/fluid thermal conductivity is greater than 100, the effective thermal conductivity of a bidisperse porous medium is smaller than that of a monodisperse porous medium saturated with the same fluid because the contact resistance at the micro-scale and the higher porosity for the bidisperse medium in comparison with the monodisperse one.

D. A. NIELD AND A. V. KUZNETSOV

2.3

37

TWO-PHASE FLOW AND BOILING HEAT TRANSFER

An experimental study of two-phase flow and boiling heat transfer in channels packed with sintered copper bidisperse porous media, with distilled water as the working fluid, was carried out by Chen et al. (2000b). Again the BDPM was formed from sintered copper and the working fluid was distilled water. The results indicated that the BDPM materials were especially effective as two-phase heat sinks since under high heat-flux conditions they have a lower flow resistance than that of monodisperse porous materials having the same pore diameter as the micro-pore diameter of the BDPM material.

2.4

DISPERSION

Dispersion in a BDPM was discussed by Moutsopolous and Koch (1999). They noted that the small grains in a BDPM have a greater influence on the permeability, while the large particles are more effective in dispersing chemical tracers. They computed the dispersion induced by a dilute array of large spheres in a Brinkman medium whose permeability is determined. They found that the effective diffusivity contains a purely hydrodynamic contribution proportional to Uaicpi and an O(C/ai0i \n{Uai/D)) contribution from the mass transfer boundary layers near the spheres. Here U denotes the mean velocity in the medium, ai and (f)i are the radii and volume fraction of the large spheres and D is the molecular diffusivity. They found that the boundary-layer dispersion is small when the Brinkman screening length K, (or the square root of the permeability) is much smaller than ai but is important for K ^ 0{ai). Moutsopolous and Koch (1999) also reported experimental results that agree with the theoretical prediction that the dispersion grows upon the inclusion of large spheres in a packed bed of small spheres. They also found that good quantitative agreement, between theoretical predictions and experimental measurements of this added dispersion, is obtained when the flow around the large spheres is modeled using Brinkman's equation with a permeability obtained from the CarmanKozeny equations.

2.5

TWO-VELOCITY MODEL

Extending the Brinkman model for a monodisperse porous medium, we propose to model the steady-state momentum transfer in a BDPM by the following pair of coupled equations foTV*f mdv*:

G = ( l ^ ) V} + C(t;} - v;) - fifV*'v} ,

(2.5)

G = ( | - ) v; + Civ; - V}) - iipW*'v;,

(2.6)

38

HEAT TRANSFER IN BIDISPERSE POROUS MEDIA

where the asterisks denote dimensional variables. Here G is the negative of the applied pressure gradient, // is the fluid viscosity, Kf and Kp are the permeabilities of the two phases, and ( is the coefficient for momentum transfer between the two phases. The quantities ftf and ftp are the respective effective viscosities. From equations (2.5) and (2.6), i;* can be eliminated to give /i//ipV*^v; -

M/(c+^)+/i,(c+|^) V*\} +

«/

(2.7)

=<-i

and Vp is given by the same equation with subscripts swapped. For flow in the x*-direction along a parallel plate channel with walls at y* = ±H the no-slip boundary conditions lead to the conditions d^t;^

(2.8)

^/-^H;i dy + ^ - ^ ^' y* = ^^We now introduce non-dimensional variables v^ KmG'

(2.9)

where Km is the mean permeability. Then we have to solve dj/4

(2.10)

B d,2+^-/=^>

where (2.11a)

HfUp.

B =

'

H^ = {flf+tlp)NH

K, 1

M V ^ / ^ ^1 H^ D = // Kp) K,

1

+: ^ + Da©

1

^ + Da^ 7 ^ + Da/Dap + ^ / ^ p j ^ = iV// UDa/ 2iVff + 1/Dap (/>Da/ + (1 - )Dap '

Mp

Da/ ' (2.11b) , (2.11c) (2.1 Id)

where in turn C^2

/sTf

Da/ = ^ ,

Dap = - ^ .

(2.12)

D. A. NIELD AND A. V. KUZNETSOV

39

For the calculation ofvp the expressions for A, B and C remain the same and the expression for D is replaced by 2iV^ + l / D a , (/)Da/ + (1 - (/))Dap * Equation (2.10) must be solved subject to the boundary conditions at y = ±l.

v = -j:^ + e = 0 dy

(2.14)

Here .= ^

.

(2.15,

The solution is readily found to be given by ^D *

^ e-XlD/C C

^2 - Af

coshXiy

^ e-XlD/C

cosh Ai

coshX2y

Af - A2

cosh A2

where

Ai = [ B + (B^ - 4AC)i/2j X2=[B-

{B^

1 /9

- AACy/^]

^

(2.17a)

.

(2.17b)

It can be easily shown that these expressions are always real. For the case of a circular tube with boundary at r* = R, and with r = r*/R, the solution is given by _D ''f-C^

e-XlD/CIo{Xir) e-XfP/C XI-XI lo(Ai) "^ XI-XI

lojX^r) Io(A2) •

^^•'''^

Here Io(a:) is the modified Bessel function of order zero. Again, Vp can be obtained by swapping subscripts / and p. (Actually only D and quantities dependent on D are changed.) For the special case of the Darcy limit one obtains „* =

(/-Z^^ + ^ Q g

(2 19)

These equations were obtained by Nield and Kuznetsov (2005b). In this case the bulk flow is thus given by G=^v\ (2.21)

40

HEAT TRANSFER IN BIDISPERSE POROUS MEDIA

where V* = 4>v} + ( 1 - )v; , i+{aiJi){Kf+K,)

(2.22) •

^^•'•'^

Thus, in this case, the effect of the coupling parameter C, is merely to modify the effective permeabilities of the two phases, via the parameter ^/jU.

2.6

TWO-TEMPERATURE MODEL

We propose to model the steady-state heat transfer by the following pair of coupled equations for Tf and Tp, on analogy with equations (2.11) and (2.12) of Nield and Bejan (1999):
(2.24)

(1 -
(2.25)

Here 0 is the volume fraction of the /-phase (if the p-phase were soUd then it would be the porosity of the medium; it is a geometrical property that can be estimated in the usual way), v'f and v* are the Darcy velocities in the two phases, kf and kp are the effective thermal conductivities of the two phases, pcp is the heat capacity per unit volume of the fluid, and hfp is the coefficient for heat transfer between the two phases (with the specific area incorporated into the coefficient). The coefficient hfp is discussed in Section 2.2.2 of Nield and Bejan (1999), where it is denoted by h. It is difficult to measure but some useful correlations are given in that reference book. We envisage that the bidisperse medium would be assembled from clusters of a normal porous medium whose permeability and effective thermal conductivity would have been measured in the normal way. It will be noted that we do not need to introduce the porosity of this medium explicitly. It should also be noted that although steady state implies local thermal equilibrium for a conduction problem, this is not so for a convection problem; see, for example, Nield (1998).

2.7

FORCED CONVECTION IN A CHANNEL BETWEEN PLANE PARALLEL WALLS

We consider fully developed steady forced convection heat transfer in a channel bounded by parallel plates at y* = ±H, there being uniform Darcy velocity in the a: *-direction, Uf in the /-phase and U* in thep-phase. (The general Darcy velocity vector v* has been specialized to the scalar Darcy velocity [/*.)

D. A. NIELD AND A. V. KUZNETSOV

41

We treat the case in which the overall Peclet number is large and so the longitudinal (axial) conduction is negligible, and we assume that kf and kp are uniform. We suppose that there is local thermal equilibrium at the boundaries, so that there Tf and T* are each equal to the wall temperature T^. The effects of tortuosity (other than those already incorporated in the transport properties of the porous medium) are assumed negligible. The effect of thermal dispersion is also neglected, for simplicity. We define e} = T}-T:,

(2.26)

e;=^T;-T:,

and, for brevity, we write

#,^

^

pep

2,1.\

O^ML^

^./^.

pcp

(2.27)

pcp

Uniform temperature boundaries: theory

When T^ is constant, equations (2.24) and (2.25) reduce to ( / 5 / | i - 7 - t / / ^ ) ^ } + 7^; = 0,

(2.28)

7 ^ / + ( ^ | ^ - 7 - C / ; ^ ) ^ ; = 0.

(2.29)

The boundary conditions are given by l9*=6>;=0

at t/* = =bF.

(2.30)

Eliminating 6^ from equations (2.28) and (2.29), we obtain -7K^P^ (2.31) 6'} = 0. Because of the assumption, already made, that the Peclet number is large, the last term on the left-hand side of equation (2.31) is negligible in comparison with the preceding one, and so the last term can be ignored. Thus the boundary conditions on 9} become ^; = | ; § = 0 at y* = ±H.

(2.32)

42

HEAT TRANSFER IN BIDISPERSE POROUS MEDIA

The method of separation of variables can then be used to obtain the solution -Ax- _ e) = AQ-^^ c o /s l ^^ l , 2H

(2.33)

e; = B e - ^ - * c o s f | | r j ,

(2.34)

where

0(1 - (f>)kfkpQ^ + hkmQ

{[4>kfU; + (1 - )KU}]Q + KU} + u;)]pcp '

(2.35)

where we have introduced the shorthand notation (2-36)

Q - ^ ' The constants A and B are related by 7

B

hfp

(2.37)

(1 - (l))kpQ + /i/p -i- AC/pVcp ' The appropriate heat transfer coefficient is given by h = - ^ , ,

{23S)

where q'' is the boundary heat flux, given by

and Tj* is the bulk mean temperature of the fluid, given in the present case by

^'=mj^l ^K

WTj + ii-m'.T;] dy*

+ -JT7 [AV} + (1 - mu;] TTU,

e-^/^* ,

(2.40)

where U*ra = Uf + (1 - )U; .

(2.41)

43

D. A. NIELD AND A. V. KUZNETSOV

The Nusselt number Nu is defined by 2Hh

Nu-

(2.42)

where the mean thermal conductivity of the bidisperse porous medium is given by km = (t>kf + (1 - (t>)kj,.

(2.43)

Equations (2.10) and (2.18)-(2.27) quickly lead to the expression Nu =

TT^ [(j>kfA^{l-)k^B]U:, 2 k,n[ct>AU} + {l-(l>)BU;] '

(2.44)

In the limiting case when the two phases are identical, the Nusselt number attains the value 7r^/2, agreeing with the well-known result for this situation; see, for example, equation (4.39) of Nield and Bejan (1999). For practical purposes, it is convenient to present results in terms of the four independent parameters 0, r/, kr and [/^, where (j) is the volume fraction already defined and (2.45)

^^ " kf U -

u; -^

(2.46)

f (2.47)

T] =

Then, from equations (2.43) and (2.41), we can write (2.48a) -1

^ ~ h

^

~

(2.48b)

rvf

(2.48c)

m

^ -1

^m

For siiortiiand, we also write

A

Ur

+ l-(t>

_ xu;{pcp)H^

(2.48d)

(2.49)

44

HEAT TRANSFER IN BIDISPERSE POROUS MEDIA

Then, in turn, we can express the Nusselt number as ' fkf + (1 - )kpiB/A) 2

4>Uf + (1 - 4>)Up{B/A)

(2.50)

where, from equation (2.37), B

A

T]

{7ry4){l-(f))kp

+ r] + A

(2.51)

and, from equation (2.35), ^^^

{7ry4)Ur[{7ry4){l-cl>)kfk,

+ rj]

(7rV4)[#/C/, + (1 - (/))fcp] + rjiUr + 1) ' and finally equations (2.48), together with equations (2.50)-(2.52), allow Nu to be expressed in terms of 0, ry, kr and Ur. If the temperature profiles are required, they can be easily computed (to within an arbitrary multiplicative constant) from equations (2.33), (2.34) and (2.37), which show that 9*j^ and 6* have the same (exponential) x*-dependence and the same (cosinusoidal) ^*-dependence. 2.7.2

Uniform flux boundaries: theory

Now q" is constant, and the wall temperature is a function of x. It is again assumed that there is local thermal equilibrium at the walls and that

with e)=9;=0

T;(X*,I/*)=T;(X*)+^}(2/*),

(2.53a)

T;{x\yn = T:{xn+0;{y),

(2.53b)

2iiy* = ±H.

When one applies the first law of thermodynamics one now obtains

ar; _ or; _ ar; _ ar^ dx*

dx*

dx*

ax*

pcpiVf -h U;)H

(2.54)

(The reader should note that for the present model the heat is convected with a velocity that is the sum of the Darcy velocities for the two phases, and not the volume average of those velocities.)

45

D. A. NIELD AND A. V. KUZNETSOV

Equations (2.28) and (2.29) are replaced by (2.55) (2.56) Equations (2.55) and (2.56) imply that

dy *4

(2.57)

dy *2

where

1/2

iWf + Pp) a— /3//?p

(2.58a)

and PfPp

(2.58b)

dx*

The boundary conditions are given by

de;

(2.59a)

^ = 0

at

, - 0 , (2.59b)

5 ^ = 0

at

,*=0, (2.60a)

61} = 0

dX y*

=H,

(2.60b)

The solution of equations (2.57), (2.59) and (2.60) is given by h (."^

ff2.,ip;dr^

b\(coshay*

\

^ ^ - 2 ^

and from equation (2.40) one then obtains

n = ^.iy''-H^)^

1

Pf\fU}dT, J J \Pf dx*

a?

cosh ay* -1 cosh aH

(2.62)

46

HEAT TRANSFER IN BIDISPERSE POROUS MEDIA

When equations (2.58a), (2.58b), (2.54) and (2.27) are used, the expression (2.60) becomes

+

(1 - <}>)kp (f>kfU; - (1 - <j>)kpU}

f/; + u;

(2.63)

cosh ay* cosh aH

The corresponding expression for 6p is given by

e: =

i^m. -ti

hr^-H^)-

IZ

^'f

cf>kfU;-{i-(t>)k,u}

U} + u;

f^m'^fp

1

01

cosh ay cosh aH

(2.64) The expressions (2.63) and (2.64) lead to equation (2.39) being satisfied identically. Equations (2.38) and (2.42) still hold, and these lead to the expression _6_ Nu

3.^(1 - 4>)ikfU; - kpU})[<j>kfU; - (1 - )kpU)] / tanh aH kmU*^{U} + u;)hfj,m V " aH (2.65)

or _6_ ^ J ^ 3.^(1 - 4>){kfUv - kUf)[kfUp - (1 - )kUf] tanh (^ _ aH Nu aH (Uf + Ur,)v V

(2.66)

where a is given by

a=

n fpf^m f rjlv'

0(1 - (t>)kfkj:

1/2

1/2

H

\ct>{l-(l>)kfkp

(2.67)

For the case of large hfp, i.e. large rj, one has local thermal equilibrium, and the Nusselt number takes the expected value Nu = 6. The Nusselt number also takes this value when kj. — Ur or when (1 - (j))kr — (l)Ur. When cj) = 0.5, Nu is less than 6 for all values of the other parameters. When 0 is not equal to 0.5, Nu can exceed 6 provided that kr/Ur and (1 — (f))kr/(l)Ur are such that one is greater than and the other is less than unity. For most practical purposes aH will be large compared with unity, so that equation (2.66) reduces to Nu = 6 1 +

30(1 - (t>){kfUp - kpUf)[(j)kfUp - (1 - 4>)kpUf]

Then equations (2.48) allow Nu to be expressed in terms of 0, rj, kr and Ur

(2.68)

47

D. A. NIELD AND A. V. KUZNETSOV

2.7.3

Uniform temperature boundaries: results

We present in Figures 2.2 to 2.4 plots of the Nusselt number Nu versus the internal heat transfer parameter rj, for various values of the thermal conductivity ratio kr. Parts (a) and (b) of each figure refer to values 0.25 and 0.5 for 0 (the volume fraction of the /-phase), respectively. Figure 2.2 is for the case where the Darcy velocities in the two phases are equal. This case is probably not typical for a practical BDPM, but it provides a baseline for later discussion because the effect of non-uniform local thermal equilibrium (LNTE) is isolated from the effect of non-uniform Darcy velocity. It is seen that the effect of the LTNE is to increase the value of Nu above the baseline value 4.93 if kp < kf (the amount of increase going through a maximum as the ratio of conductivities varies), and to decrease it below the baseline value if kp > kf. In each case the effect is larger for the smaller value of 0.

(a)

20 (> = 0.25

18 16 14 12

6 4 2 0

U,= 1

i [\^f

.K- 0.1 K = 0.01

^^r^^^^-^««

0

2

4

6

8

1

10

(b)

Figure 2.2 Plots of Nusselt number versus internal heat transfer parameter, for various conductivity ratios: uniform temperature, Ur = I, and (a) 0 = 0.25, and (b) (f) = O.b.

48

HEAT TRANSFER IN BIDISPERSE POROUS MEDIA

(a)

g

(b)

Figure 2.3 Plots of Nusselt number versus internal heat transfer parameter, for various conductivity ratios: uniform temperature, Ur =0.1, and (a) 0 = 0.25, and (b) (/> = 0.5.

Figures 2.3 and 2.4 are the companion figures for smaller values of the velocity ratio. Now the baseline value is attained at the value of kr that matches the value of Ur, and again larger values of kr lead to smaller values of Nu. 2.7.4

Uniform flux boundaries: results

Figures 2.5 to 2.7, for the case of uniform flux boundaries, match Figures 2.2 to 2.4 for the case of uniform temperature boundaries. Now the baseline value of Nu is 6, and this is attained forfc^.= 1, independent of the value of Ur. This baseline value is not exceeded for any of the parameter values employed in the figures. In other words, the variation in either direction away from A:^ =: 1 generally leads to a reduction in the value of Nu. (There are some comparatively narrow parameter ranges that provide exceptions,

D. A. NIELD AND A. V. KUZNETSOV

49

(a)

(b)

Figure 2.4 Plots of Nusselt number versus internal heat transfer parameter, for various conductivity ratios: uniform temperature, Ur = 0.01, and (a) 0 = 0.25, and (b) (/> = 0.5.

for example, when (f) — 0.25, Nu exceeds 6 when 1/3 < kr/Ur < 1.) A variation in the value of (j) leads to comparatively little change in the plots. 2.7.5

Conjugate problem

Nield and Kuznetsov (2005a) extended the above analysis to a conjugate problem, where the BDPM is bounded by slabs of solid material; see Figure 2.8 for a definition sketch, and for details of the analysis see that paper. The new feature is that the thermal boundary condition is changed from one of the first kind to one of the third kind, with a Biot number Bi appearing in that boundary condition. Some results obtained are presented in Figures 2.9 to 2.11, in each of which curves are plotted for various values of Ur, the individual curves being of Nu plotted versus r] (strictly speaking log r/) for a representative value of (/) = 0.4 (the results are not sensitive to the value of 0). The figures correspond to various

50

HEAT TRANSFER IN BIDISPERSE POROUS MEDU 6

(a)

1

•

5k 4

I

/ /

•

iC = 10

/ \ \

k^ = 100 k, = 0.1

2 1 0

•

=1 iL

3

•

1/ ^^

0

(|) = 0.25

= 0.01 2

u.= 1 4

6

8

10

(b)

Figure 2.5 Plots of Nusselt number versus internal heat transfer parameter, for various conductivity ratios: uniform heatflux,C/^ = 1, and (a) 0 = 0.25, and (b) 0 = 0.5.

values of kr (K — 0.1 in Figure 2.9, kr = 1 in Figure 2.10, and kr — 10 in Figure 2.11) and Bi (for (a) Bi == 10, (b) Bi = 1 and (c) Bi = 0.1 in each figure). The values Ur — 1, 0.1 and 0.01 are used in the presentation. The first value is probably not realistic physically, and should be regarded as a limiting case. We also point out that small values of 7] are not physically realistic. The value of Nu invariably decreases as Bi increases. Also Nu generally increases/decreases as r] decreases from large values (the case of local thermal equilibrium) to smaller values according to whether kr is less/greater than unity. The figures show an interesting variety of behavior. The value of Nu is especially sensitive to the value of r] when kr is small. In most cases Nu increases as Ur decreases, but the cases shown in Figures 2.9(b, c) are exceptions. Also in most cases there is a substantial jump as Ur changes from 1 to 0.1, but a much smaller jump as it changes from 0.1 to 0.01 (and additional calculations showed that there was then little further change as Ur decreased further). A rather dramatic change is shown in Figure 2.9(a), where the direction of variation of Nu with r] changes as Ur changes.

D. A. NIELD AND A. V. KUZNETSOV

51

(a)

(b)

:z; 3

Figure 2.6 Plots of Nusselt number versus internal heat transfer parameter, for various conductivity ratios: uniform heatflux,Ur = 0.1, and (a) 0 = 0.25, and (b) 0 = 0.5.

2.7.6

Thermal development

Kuznetsov and Nield (2005) extended the analysis from the case of fully developed convection to that in which the convection is thermally developing, see Figure 2.12). The analysis is an extension of the classical Graetz analysis, and again for details the reader is referred to the paper cited. The chief new feature is that one is now concerned with the way in which Nu varies with the dimensionless axial coordinate x = x*/Pe where Pe is the Peclet number. Some results are presented in Figures 2.13 to 2.15. Figure 2.13 was chosen to illustrate that the Nusselt number development does not depend critically on the practical values of Ur or TJ. In particular. Figure 2.13(a) shows that a variation in the value of Ur from 0.1 to 0.01 has little effect. Similarly, Figure 2.13(b) shows that the effect of local thermal non-equilibrium (as expressed by the value of rj) is relatively small in the present situation, though the developing Nusselt number does decrease as rj is reduced from infinity (the case of local thermal equilibrium).

52

HEAT TRANSFER IN BIDISPERSE POROUS MEDIA

(a)

(b)

/ N 5

k, = 0.01

^

K = 0.1^^^^-^^^

^^^^__^

4

2

3 2

/ / •/ / ^

/K^^^ * / ^

•

^

kr = 10 (|) = 0.5

k, = 100

U, = 0.01

Figure 2.7 Plots of Nusselt number versus internal heat transfer parameter, for various conductivity ratios: uniform heat flux, C/r = 0.01, and (a) 0 = 0.25, and (b) 0 = 0.5.

isothermal boundary

Figure 2.8

T = To

Definition sketch for the conjugate problem.

53

D. A. NIELD AND A. V. KUZNETSOV

(a)

logio(Tl)

(b)

1.35

1.3

u,= i

1.25 1.2 1.15 .

Ur = 0.01

(|) = 0.4

^ ^ ^ ^

Bi = l

1.1

kr = 0.1

1.05

1 0.95 0.9 y ^ 0.85

1

Ur=o.i -0'.8 -0'.6 -(J.4 -(J.2

6

0:2

0.4

0.6

0:8

1

(c)

^•^-1

-0'.8 Ah -0'.4 -(J.2—ft—0:2

0:4 0:6 0:8

logio(^)

Figure 2.9 Plots of Nusselt number versus internal heat exchange parameter for various values of the velocity ratio. Case 0 = 0.4, kr = 0.1, and (a) Bi = 10, (b) Bi = 1, and (c)Bi = 0.1.

54

HEAT TRANSFER IN BIDISPERSE POROUS MEDIA

(a)

0.8 -0.6 -ISA -0.2—0—0:2

0:4 0:6 li'X

log,o(Tl)

(b)

(|) = 0.4 Bi=l

2.6 2.4

\ ^ U r = 0.01

k,= l

2.2

I

2 1.8

\

\

^r = l

1.6 14 1

-(J.« -(J.6 -(J.4 0.2 0 0.2 loglo(Tl)

(C)

0.4

0.6

0.8

1

(t) = 0.4 Bi = 0.1 k,= l

I

-0".8 -tf.6 -0'.4 -6.1

0:2

0:4 0:6 0:8

logio(Tl)

Figure 2.10 Plots of Nusselt number versus internal heat exchange parameter for various values of the velocity ratio. Case 0 = 0.4, kr = 1, and (a) Bi = 10, (b) Bi = 1, and (c) Bi = 0.1.

55

D. A. NIELD AND A. V. KUZNETSOV

(a)

logio(Tl)

(b)

(1) = 0.4 Bi=l kr=10

4.5 4

\ \

Ur = 0.01

3 2.5

Ur = 0.1

2

^^^^^^

Ur=l

1.5 1 05

I

-C.8 -6.6 -0.4 -0.2

6

0.2

0.4

0.6

0.8

1

•ogioCn) 0.6

(c)

(|) = 0.4 Bi = 0.1 k,= 10

0.55 0.5

0.45

I

\ \

0.4

Ur = 0.01

0.35

Ur = 0.1 N N ^ ^

0.3

0.25 0.2

0.15

. „.=, ^ ^ = ^ 1

-0.8 -0.6 -0.4 -0.2

0

0.2

0.4

0.6

0.8

1

Figure 2.11 Plots of Nusselt number versus internal heat exchange parameter for various values of the velocity ratio. Case 0 = 0.4, kr = 10, and (a) Bi = 10, (b) Bi = 1, and (c) Bi = 0.1.

56

HEAT TRANSFER EST BIDISPERSE POROUS MEDIA

T*=T\ 4:

T =T

^

A ._

^ ^ bi-disperse porous medium \ "^ -> C-vilC^I' u

|-^-L;-%

0 Figure 2.12 convection.

%

•

W

H

v_

X

Definition sketch for the problem of thermally developing forced

(a)

(b)

Figure 2.13 Plots of Nusselt number versus dimensionless axial coordinate, illustrating (a) the variation with the velocity ratio, and (b) the variation with the internal heat transfer parameter.

D. A. NIELD AND A. V. KUZNETSOV

57

(a)

(b)

-3

-2.5

-2

-1.5

logio(x)

Figure 2.14 Plots of Nusselt number versus dimensionless axial coordinate, illustrating the variation with the macropore volume fraction, for the internal heat transfer parameter 7/ = 10, conductivity ratio kr = I, and velocity ratio (a) [/r =0.1, and (b) Ur = I.

The remaining results show the effect of the variation of the macroscopic volume fraction and the conductivity ratio. Figure 2.14(a) shows that variation of the macroscopic volume fraction has very little effect if Ur is small (0.1 or less) while Figure 2.14(b) shows that the effect is quite substantial (Nu increasing as (p increases) when C/^, ^ 1. The dramatic effect of the variation of the conductivity is shown in Figure 2.15. The increase of Nusselt number as kr decreases is particularly large when the effect of the local thermal nonequilibrium comes into play {r] not large).

58

HEAT TRANSFER IN BIDISPERSE POROUS MEDIA 1200

(a)

(j) = 0.4 Tl=10

1000

Ur=l 800

kr = 0.1 :!

600 400

\ 200

0

•kr

3

k,= 10

=

-2.5

-2

1.5

I

-0.5

(b)

0

0.5

1

(|) = 0.4 Ti = 100 U.= l

I

-2.5

-1.5

-1

-0.5

0.5

logio(x)

Figure 2.15 Plots of Nusselt number versus the dimensionless axial coordinate, illustrating the variation with the conductivity ratio, for the macropore volume fraction (j) = 0.4, velocity ratio Ur = 1 and internal heat transfer parameter (a) T] = 10, and (b) TJ = 100.

2.8

CONCLUSIONS

It has been noted that to date there has been little work done on heat transfer in BDPM, and efforts have been mainly confined to the measurement of permeability, thermal conductivity and dispersion. The present authors have proposed a new model involving two velocities as well as two temperatures. This model has been applied to forced convection in a channel between two parallel walls, with either uniform temperature or uniform heat flux imposed at the walls. The hydrodynamic problem has been solved for the case of the Brinkman momentum equation, but work to date on the thermal problem has been confined to the case of the Darcy equation. For the case of fully developed convection, the analysis leads to expressions for the Nusselt number as a function of properties of the BDPM, namely a conductivity ratio, a permeability ratio (which for the Darcy case is equivalent to

D. A. NIELD AND A. V. KUZNETSOV

59

a velocity ratio), a volume fraction, and an internal heat exchange parameter. For the case of uniform temperature boundaries, the analysis has been extended two directions, namely a conjugate problem and a problem involving thermally developing convection. For the conjugate problem the Nusselt number also depends on a Biot number, while for the thermally developing convection problem it also depends on a suitably scaled longitudinal coordinate. The main new feature is the way in which the Nusselt number varies with the velocity ratio. This ratio interacts in a quite complicated way with the other parameters.

ACKNOWLEDGEMENTS AVK gratefully acknowledges NSF grant # CTS-0226021 and grant # NAG3-2706 awarded to him by the NASA Office of Biological and Physical Research, Physical Sciences Division.

REFERENCES Bejan, A., Dincer, I., Lorente, S., Miguel, A. F., and Reis, A. H. (2004). Porous and complex flow structures in modern technologies. Springer-Verlag, New York. Chen, Z. Q., Cheng, P., and Hsu, C. T. (2000a). A theoretical and experimental study on stagnant thermal conductivity of bi-dispersed porous media. Int. Comm. Heat Mass Transfer 21, 601-10. Chen, Z. Q., Cheng, P., and Zhao, T. S. (2000b). An experimental study of two phase flow and boiling heat transfer in bi-disperse porous channels. Int. Comm. Heat Mass Transfer 27, 293-302. Cheng, P. and Hsu, C. T. (1999). The effect of stagnant thermal conductivity of porous media with periodic structures. J. Porous Media 2, 19-38. Ingham, D. B. and Pop, I. (eds) (2002). Transport phenomena in porous media. Vol. II. Pergamon, Oxford. Kuznetsov, A. V. and Nield, D. A. (2005). Thermally developing forced convection in a bi-disperse porous medium. ASME J. Heat Transfer. Submitted. Moutsopolous, K. N. and Koch, D. L. (1999). Hydrodynamic and boundary-layer dispersion in bidisperse porous media. J. Fluid Mech. 385, 359-79. Nield, D. A. (1998). Effects of local thermal nonequilibrium in steady convective processes in a saturated porous medium: forced convection in a channel. J. Porous Media 1, 181-7. Nield, D. A. and Bejan, A. (1999). Convection in porous media (2nd edn). Springer-Verlag, New York. Nield, D. A. and Kuznetsov, A. V. (2005a). Forced convection in a bi-disperse porous medium channel: a conjugate problem. Int. J. Heat Mass Transfer. In press. Nield, D. A. and Kuznetsov, A. V. (2005b). A two-velocity two-temperature model for a bi-dispersed porous medium: forced convection in a channel. Transport in Porous Media. In press. Yu, B. M. and Cheng, P. (2002a). Fractal models for the effective thermal conductivity of bidispersed porous media. J. Thermophys. Heat Transfer 16, 22-9. Yu, B. M. and Cheng, P. (2002b). A fractal permeability model for bi-dispersed porous media. Int. J. Heat Mass Transfer ^S, 2983-93.

FROM CONTINUUM TO POROUS-CONTINUUM: THE VISUAL RESOLUTION IMPACT ON MODELING NATURAL CONVECTION IN HETEROGENEOUS MEDIA A. A. MERRIKH* and J. L. LAGE^ * Pulmonary Research Group, Department of Internal Medicine, University of Texas, Southwestern Medical Center at Dallas, Dallas, TX 75390-9034, USA email: [email protected] ^Laboratory for Porous Materials Applications, Department of Mechanical Engineering, Southern Methodist University, Dallas, TX 75275, USA email: [email protected]

Abstract Recent studies questioning the correctness of using a low-resolution, porous-continuum model for simulating natural convection within an enclosure saturated with fluid and having discrete solid blocks uniformly distributed within it, are reviewed and expanded here. The validation of the porous-continuum results, obtained with volume-averaged equations, is established by comparison to 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), 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 modeling, 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 homogenization of the domain. General design criteria quantifying the accuracy of the porous-continuum results would require further investigation on the form-function relationship of heterogeneous structures. Keywords: natural convection, porous-continuum, heterogeneous media 60

A. A. MERRIKH AND J. L. LAGE

3.1

61

INTRODUCTION

Transport in porous media is fundamental for several industries, including oil exploration, filtering, underground water pollution control, manufacturing of medications, and electronics. The main characteristic of the processes classified as transport in porous media is the presence of more than one constituent and the complex interface between the constituents. Porous media are then said to be heterogeneous, at least in so far as interfaces between constituents are visible. Using the known balance equations from mechanics, such as the Navier-Stokes equations or the heat balance equation is the most straightforward approach to modeling transport in porous media. These equations, termed continuum equations, have evolved from the need of reducing the complexity of having to follow every molecule (or carrier) within the domain, as suggested by a molecular modeling approach. Continuum equations can be thought of as equations derived via averaging within a small elementary unit (volume) of a domain that, in fact, would represent the relationship between the carriers as well as their interaction with the domain boundaries. The amount of information necessary for studying the transport phenomena within a finite physical (or mathematical) domain can be reduced if one considers merely the net effects of all carriers within each elementary unit. These effects are defined at the continuum level by the continuum parameters such as velocity, pressure, and temperature that would represent properties of a continuum domain under the effect of all its carriers only on an average sense. The averaging process not only yields equations requiring much less information, e.g. no need to know the location of each carrier), but also providing less information as well. Observe that the information on each individual carrier is lost at the continuum level, e.g. the state of a single molecule is not known at the continuum level). Hence, if individual carrier effects are important to the transport process, then the continuum model becomes inappropriate. Assuming that the continuum approach provides the necessary information, the transport process within a porous medium can be modeled by using the continuum equations. Each constituent of the porous medium would have its unique properties and assigned a continuum equation appropriate to the transport process of that individual constituent. The interactions between different constituents take place along their interface, which would play the role of a boundary to each constituent. Compatibility conditions are then necessary, along each interface as boundaries dividing the constituents, to close the modeling. This approach, although feasible, becomes impractical once the interfaces turn out to be difficult to access. Such impracticality is not only caused by the difficulty in assessing the porous medium interior, but also by the difficulty in mapping the topologically complex interfaces common to most porous media. It is true that the interfaces of several porous media, with a small number of constituents and very regular interfaces, can be mapped without difficulty that could lead to continuum transport modeling approach. It is also true that the effect of transport processes on the transport property fields, even within somewhat complex porous media, can be investigated at the continuum level when the medium, internally, presents a periodic structure. In this case, the domain of interest becomes a single periodic cell, which normally requires

62

FROM CONTINUUM TO POROUS-CONTINUUM

much less effort to be mapped. Eidsath et al. (1983), Coulaud et al. (1988), Sahroui and Kaviany (1991), and Fowler and Bejan (1994) have followed this path. Lee and Yang (1997) modelled Darcy-Forchheimer drag for fluid flow across a bank of circular cylinders, numerically solving the momentum equations in the pore-scale making use of the Cartesian coordinates under a prescribed pressure drop. Kuwahara et al. (1996) followed the numerical approach of Arquis et al. (1991) to determine the transverse dispersion coefficient by fitting the numerical results against the similarity solution for forced convection from a line heat source derived from a porous medium macroscopic model. The numerical results were obtained for a pore-level model of a lattice of square rods. Nakayama et al. (1995) also investigated flow through a domain made of a collection of square rods. Three-dimensional analyses were conducted by Larson and Higdon (1989) for Stokesflowsthrough a lattice of spheres, and by Nakayama et al. (1995) for fully elliptic flows through a lattice of cubes. Keep in mind that the studies mentioned above involve forced convection, which facilitates the determination of appropriate boundary conditions at the boundaries of a particular periodic cell. The same cannot be said for natural convection in which, even if the medium is periodic, the boundary conditions remain unknown along the boundaries of the periodic cell. The only continuum modeling alternative in this case is to consider the entire domain, and solve the transport equations at the continuum level. Unfortunately, a large number of natural and man-made porous media are too complicated for this approach to be practical as reviewed by Cheng (1978) and Nield and Bejan (1999). There is a need then to reduce the information necessary for modeling the transport processes in structurally complex porous media. Modeling such processes at the porous-continuum level comes in response to this need. The evolution of modeling from molecular to continuum level can be considered in order to better understand the progression from continuum to porous-continuum level. Consider, for instance. Figure 3.1 where three domains representing three different resolution levels are sketched. From left to right, the domains evolve from a molecular resolution level, through which molecules can be individually seen, to a continuum resolution level and finally to a porous-continuum level in which only a homogeneous domain is visible. Below the sketches we observe the increase in length- and time-scales for observing the phenomenon taking place within each domain. The evolution from continuum to porous-continuum level involves an averaging process, similar to the one from molecular to continuum level. Formally, the applicability of a porous-continuum model becomes limited by the length- and time-scales relation shown in Figure 3.1. Unfortunately, these scales are not that precise for the porous medium model as they are based on inequalities. Hence, it becomes difficult to determine a priori the correctness of using the porouscontinuum equations to model transport processes in porous media.

63

A. A. MERRIKH AND J. L. LAGE

• • • •

• •

Continuum domain (c) {d,t)c>id,t)n

Molecular domain (m) {d, t)m

Porous-continuum domain (p) (d,0p>(^,0c

Increasing representative length d and time t Figure 3.1 Evolution from molecular to porous-continuum domain. 3.2 HORIZONTAL HEATING Figure 3.2 presents a sketch of the domain considered by Merrikh and Mohamad (2001), Merrikh et al. (2002) and Merrikh and Lage (2004), as seen at the continuum visual resolution level that consists of a fluid filled enclosure containing several conducting solid and fixed blocks disconnected and distributed uniformly within an enclosure. The enclosure is subjected to a horizontal temperature difference (applied at the walls), which in turn induces natural convection by the fluid within the domain. 3.2.1

Continuum equations

For being composed of two distinct constituents, namely the fluid and the solid, the continuum approach for the heat transfer process within the enclosure requires defining transport equations for each constituent together with appropriate compatibility conditions at the interfaces. For the fluid, the governing continuum transport equations are the continuity, momentum and energy equations, respectively: (3.1) = - Vp + tJ,V^v + pfgP(T -

Pf ipc)f(^

dT

+

vVT]=kfV'T,

Tc)j,

(3.2) (3.3)

where v is the fluid vector velocity, with components u and v, t is time, p is the pressure, T is the temperature, pf,Cf,P and kf are the fluid density, specific heat, isobaric coefficient

64

FROM CONTINUUM TO POROUS-CONTINUUM Solid ob*=<^acies

Adiabatic

Fluid

Figure 3.2

Natural convection enclosure with discrete, conducting solid blocks.

of volumetric thermal expansion and conductivity, respectively, g is the acceleration of gravity, Tc is a reference temperature, equal to the cold temperature of the enclosure right wall {Tc < Th), and j is the unit vector along the vertical ^/-direction, see Figure 3.2. Square solid obstacles, with side length d, placed inside the enclosure participate indirecdy to the transport of momentum by the fluid through their solid-fluid interfaces where a noslip condition is imposed. Their participation on the energy transport across the enclosure, however, is much more effective for being conducting solids. The energy balance equation valid within each solid block is given by

{pel

9T = ks\/^T, dt

(3.4)

where ps, Cg and ks are the solid density, specific heat and conductivity, respectively. In equations (3.1)-(3.4) the thermo-physical properties of the fluid and the solid are assumed isotropic, uniform and constant, except the fluid density variation in the buoyancy term of equation (3.2), modeled using the classical Oberbeck-Boussinesq approximation. For solving equations (3.1)-(3.4), the velocity components u and v are set to zero at all solid surfaces. The right and left walls of the enclosure are maintained at Tc and Th, respectively. The horizontal top and bottom surfaces of the enclosure are assumed to be adiabatic.

65

A. A. MERRIKH AND J. L. LAGE

Equations (3. l)-(3.4) are converted to the following non-dimensional equations: (3.5) dV

"a7

+ {V • V ) y = V P + Pr V ^ F + RaPr6li .,

dr

(3.6) (3.7)

+ v-v9 = v^e,

(3.8)

K OT

The non-dimensional representations of the boundary conditions are given by 6 = 1,

U = V = 0 at ^ = 0,

61 = 0,

U = V = 0 at ^ = A,

dO ^— = U = V = 0 at rj = 0 and rj = 1. or]

(3.9) (3.10)

At the solid-fluid interfaces, the following compatibility conditions apply:

u = v = o, e\^ = ei,dn

=

K

de_ dn

(3.11)

where n is the direction normal to each and every block boundary. The nondimensional parameters used in equations (3.5)-(3.11) are defined as: fluid velocity V = [/i -h Vj, where the velocity components are ([/, V) = {u, v)H/af, and the fluid thermal diffusivity af — kf l{pcp)f, time r = taf/H'^, pressure P = pH'^/pfa'j, Prandtl number Pr = i//a,Rayleigh number Ra = ^/?iJ3(T/,-Tc)/i/a/, temperature (9 = {T-Tc)/{Th-Tc), solid-fluid heat capacity ratio cr = (pc) 5/(pc)/, solid-fluid conductivity ration; = ks/kf, Cartesian coordinates (^, rj) — (x, y)/H, and enclosure aspect ratio A — L/H. In summary, the results of the continuum model depend on Ra, Pr, a, K, and the geometrical parameters A, D, and N, where the nondimensional block side length is D = d/H and A^ is the total number of blocks inside the enclosure. 3.2.2

Porous-continuum equations

As the resolution by which one observes Figure 3.2 is decreased, the solid blocks will eventually blur with the fluid, forming a single homogeneous porous-continuum medium. This blurring process is becomes evident, i.e. it requires a smaller decrease in resolution, when a small number of large solid bodies are placed within the domain. In this case, the transport phenomenon taking place within the enclosure can be modeled using the porous-continuum model that treats the whole structure as a porous medium. The most common porous-continuum model available for studying transport in porous media is through the volume averaging method, see Whitaker (1999). The porouscontinuum equations that govern the conservation of mass, momentum and energy, as

66

FROM CONTINUUM TO POROUS-CONTINUUM

suggested by Lage (1998) and Nield and Bejan (1999), for natural convection in an enclosure filled with a homogeneous, rigid, isotropic porous medium saturated with an incompressible fluid, and with constant thermo-physical properties except for the fluid density variation in the buoyancy term, are given by (3.12)

V'{v)=0, Pf

^

+

{{v)'V){v)

- V ( p ) + flef^V^v) + PfQp ((T) - Te) j

(3.13) ( p c ) e f f ^ + {pc)fcl>{v) . V(T) = A:effV2(T).

(3.14)

Variables within brackets represent volume-averaged parameters. The fluid velocity {v) is a pore velocity (averaged only within the total pore surface area), and the effective viscosity /Xeff is usually set as equal to the fluid viscosity /i. The last two terms of equation (3.13) represent, respectively, the linear (in fluid velocity) viscous-drag effect and the quadratic form-drag effect imposed by the solid matrix of the porous medium, see Lage (1998). The permeability K of the medium, equation (3.13), can be estimated by using the Kozeny equation, see Nield and Bejan (1999):

where d is the representative solid body length-scale, and 6 is a parameter known to depend on the morphology of the pores. The value 180 is usually set for b when the porous medium is made of particles or fibers. When the medium is made of capillaries (pores with circular section) then a better value for h seems to be 144. Finally, the form coefficient C, equation (3.13), can be estimated by using the model C — 0.55/K^/^ of Ward (1964). Although neither the Kozeny nor the Ward models are universal, they are the simplest and, therefore, the most general models available for estimating the permeability K and the form coefficient C of a porous medium. In equation (3.14), the effective volumetric heat capacity is (pc)eff — (t>{pc)f + (1 — (j))[pc)s. The effective thermal conductivity of the porous medium is estimated using the formula suggested by Nield (1991), namelyfcgff— k\~^kt, which, although not universal, seems to be the best available. Equations (3.12)-(3.14) written in dimensionless form are given by V - ( y ) = 0, ^ "^^

+ {{y) • V) {V) = - V ( P ) + P r V 2 ( y ) + P r R a ( ^ ) j p

-^J{V)-h4>''\{V)\{V),

(3.16) (3.17)

67

A. A. MERRIKH AND J. L. LAGE

a . , ^

+ 4>{V) VW=KeffV2(^).

(3.18)

The non-dimensional mathematical representations of the boundary conditions are given by {6)^1, {U) = {V) = 0 at e = 0,

m dr]

{U) = {V)=0

at r] = 0

and

T]

(3.20)

= 1.

Implicit in writing equations (3.14) and (3.18) is the assumption of thermal equilibrium between the solid and the fluid constituents of the porous medium. This assumption is more closely satisfied when the fluid and solid have the same thermal conductivities. The new parameters introduced in equations (3.16)-(3.20) are as follows: Darcy number Da = K/H'^, non-dimensional form coefficient A = CH, porosity (f), effective thermal capacity ratio (Jeff — (pc)eff/(pc)s, and effective conductivity ratio/Ceff = ke^/kf. Using equation (3.15), Da = D^(/)^/[6(l — 0)^]. Because of the model used for determining C, a unique relation exists between Da and A, namely A = 0.55 Da~^'^. Hence, the porous-continuum model results depend only on Ra, Pr, D, 0, deff, /^eff and A. 3.2.3

Heat transfer comparison parameters

A useful parameter for comparing the continuum and the porous-continuum results is, for instance, the hot-wall average heat transfer defined as /lav = Q.'lvK'^h —Tc), from which the hot-wall average heat flux can be defined as g^v — ~k{dT/dx)8,y^h' A dimensionless number, equivalent to /lav, is the hot-wall average Nusselt number, Nuav = hayH/k. Care should be taken when choosing the proper value of k to be used when determining g'av and when defining Nuav^ for the continuum model, k should be the fluid thermal conductivity /c/; however, for the porous-continuum model, k — A;efF- So, the hot-wall average Nusselt number for the continuum model is given by Nua

i^div—c-^

de\

H

^a^L=o ^ ~

H Jo

h

(3.21) dr].

The hot-wall average Nusselt number for the porous-continuum model is given by Nua ^av—pc

"•av—pc^

keS

H {T)H

- {T)c

1

f"

d{T) dx

dy x=0

r d{e) Jo d^

(3.22) dr/. 4-0

68

FROM CONTINUUM TO POROUS-CONTINUUM

Observe the final representation of Nuav-c (in terms of nondimensional temperature gradient) which seems identical to the final representation of Nuav-pc- This is deceiving because it induces a direct comparison of the Nuav value obtained from the continuum approach with the Nuav obtained from the porous-continuum approach. However, these two parameters should not be compared directly because of the different thermal conductivities used when defining Nuav and /lav for each one of the models. More clearly, if a comparison between the two models is intended, this comparison should be based on h and not on Nu. Therefore, using the original representation of Nuav, one has the following: '^av—pc '^av—c

'^^eff •'^Uav—pc ^/

-^^^av—c

•'^Uav—pc -^^^av—c

i — ^-^^^av—pc

r'X 0'2\

•'^^U.av—c

Equation (3.23) indicates that the ratio of continuum and porous-continuum heat transfer coefficients equals an equivalent Nu ratio only when /^ = 1 or 0 = 1, otherwise the Nu ratio must be corrected by the factor K,^~^. This distinction is extremely important as it may skew the conclusion if not properly accounted for. In principle, one would expect the values of the Nusselt number obtained from equation (3.21) to be identical to the values obtained from equation (3.22) corrected by the factor hi^~^, as indicated in equation (3.23). After all, both models are simulating the same phenomenon occurring within the same medium, so they should yield the same heat transfer coefficient, /lav. However, because each model considers the medium at a different resolution level, their validity becomes dependent on the sole ability of the model itself to capture the influence of the internal structure on the transport process. When the (low resolution) porous-continuum model fails to capture the details, because of its low resolution, then the model results are expected to, most likely, deviate from the results of the (high-resolution) continuum model. The determination of the cases in which this deviation occurs is done by direct comparison as detailed in the following sections. 3.2.4

Results

Steady-state results are presented for A = 1, Pr = 1, and A^ varying from 9 to 144. Furthermore, using the relationship between porosity, number and size of each block, namely D — [{1 - 0)/A/']^/^, only A^ and 0 are necessary. Also, porosity is fixed to (f) = 0.64. Finally, recall the relations Da := D'^(t)^/[b{l - 0)^], A = 0.55Da~^/^ and ACgfP l = i^^~^. Hence, the parameters governing the continuum and porous-continuum model results are limited to Ra, K, N and b. Because of the unique internal geometry of the porous enclosure under investigation, the porous-continuum model is tested for b varying from 150 and 210. The effect of fluid and solid constituent thermal conductivities on flow and heat transfer is studied by considering /^ = 0.1, 1, 10, and 100, with results presented for Ra varying from 10^ to 10^. Because of the thermal equilibrium requirement of equation (3.18), results from the porous-continuum model are considered only for /^ = 0.1, 1, and 10. A summary of the parameter values used here is presented in Table 3.1.

A. A. MERRIKH AND J. L. LAGE

69

Table 3.1 Summary of the parameter values used in the present study.

9 0.2

10^ 10^,10'^108 1 1 0.64 0.1 1 10 0.44 1 2.3 16 36 64 0.15 0.1 0.075

144 0.05

5.41 3.86 5.99 7.09

3.03 1.35 2.17 0.96 25.2 37.8 29.9 44.8

0.34 0.24 94.3 112

Ra Pr A

0 K

f^ef£

N D Da (10-4) A

b 150 210 150 210

0.76 0.54 50.5 59.8

Equations (3.5)-(3.8) of the continuum model and equations (3.16)-(3.18) of the porouscontinuum model, together with the corresponding boundary conditions, are discretized and solved using the control-volume method following the SIMPLER, see Patankar (1980), algorithm with the QUICK, see Leonard (1979), scheme. For the most stringent case, i.e. using the continuum model with the maximum number of blocks, N — 144, and the highest Rayleigh number, Ra = 10^, a uniform grid with 241 X 241 nodes yielded average Nusselt number results that were within one per cent from the results obtained when using a 301 x 301 grid. Therefore, all reported results were obtained with 241 x 241 grid nodes. Our numerical approach is validated in three ways. The first considers the enclosure clear of solid blocks (A^ = 0), for which results from the continuum model compare very well with the published results from several distinct sources, as shown in Table 3.2. The second considers the case of a single solid conducting square located at the center of the enclosure, identical to the configuration studied by House et al. (1990). Table 3.3 shows the results obtained using the continuum model again to compare well with the published results. The third validation concerns results obtained using the porous-continuum approach with A = 0, for which the code validation was reported in Merrikh and Mohamad (2000). Continuum and porous continuum results, for /^ = 1, are presented in Figure 3.3. Observe that when /c = 1, /^eff = 1 as well. In this case, the solid and the fluid have the same thermal conductivity. The first observation is that in general the porous-continuum results predict well the continuum results when Ra is small (Ra ^ 10^). Results from the porous-continuum model for Ra = 10^ under-predict the continuum results by a large margin especially when A^ is small. It is important to point out that even though the continuum and porous-continuum results for Ra = 10^ do not agree as closely as for the other Rayleigh number values, one should not disregard the porous-continuum model as a useful predicting tool for estimating natural convection phenomenon. Rather, the lack of agreement between the two models

70

FROM CONTINUUM TO POROUS-CONTINUUM

Table 3.2 Average Nusselt number comparison for a square enclosure filled only with a clear fluid (Pr = 0.71, unless otherwise noted). Ra lO'^ 10^ 10^ 10^ 10^

de Vahl Davis (1983)

Hortmann et al (1990)

House et al. (1990)

2.243 4.519 8.800

2.244 4.521 8.825

2.254 4.561 8.923

-

-

-

Lage and Bejan (1991), Pr = 1

Bejan (1995)*

Kalita et al. (2001)

-

2.418 4.715 9.194 17.927 34.954

2.245 4.522 8.829 16.52

4.9 9.2 17.9 31.8

-

Present 2.244 4.536 8.860 16.625 31.200

* Analytical estimates obtained from equation (7.100), p. 369

Table 3.3 Comparison of average Nusselt number for a square enclosure filled with fluid (Pr = 0.71) and having a conducting square solid body at the center. Ra

D

K

10^ 10^ 10^

0.5 0.5 0.9

0.2 5.0 0.2

House etal. (1990) Present 4.624 4.324 2.402

4.605 4.280 2.352

Figure 3.3 Variation of Nuav with N obtained via the continuum approach, for K = 1 and several Ra values. Also shown are results obtained via the porous-continuum approach with 6 = 150 (dotted lines) and 6 = 210 (dashed lines).

A. A. MERRIKH AND J. L. LAGE

71

indicates the inability of the porous-continuum equations to predict the natural convection phenomenon in the specific configuration being considered. That is, the inadequacy of the porous-continuum model should not be generalized but should rather be pertained to the enclosure configuration in satisfying the requirements for the successful applicability of the porous-continuum model. Another observation emerging from Figure 3.3 is the relatively small difference between the porous-continuum results when using b = 150 or b = 210, with results obtained when b = 150 being consistently higher than the results obtained when b = 210 and more in line with the continuum results. Recall, from equation (3.15), when b increases K decreases, and a decrease in the permeability hinders the natural convection flow within the enclosure, so the results seem consistent with what one would expect. The relatively small effect of b is surprising inasmuch as the effect of 6 on i^ that is linear, so one would expect a, likewise, stronger effect on Nuav, especially when the fluid speed is small (low Ra) the same as for the expected predominance of the viscous-drag effect. Finally, Figure 3.3 seems to indicate, in general, that an increase in N could lead to a decrease in Nuav; the decrease being stronger for smaller A/". This observation is consistent with the hindering effect the blocks are expected to have on the convection process inside the enclosure. Keep in mind the results presented in Figure 3.3 that were obtained for a fixed porosity. This means that as N increases the size of the blocks is reduced accordingly, so as to maintain their total volume within the enclosure the same. Hence, changes in the hindering effect are solely due to the different distributions of the amount of solid material within the enclosure. Moreover, the results demonstrate a reduced effect of the solid block distribution (i.e. size and number of blocks, maintaining the porosity constant or their total volume constant) on Nuav when Ra is high. A more careful observation of Figure 3.3 indicates a switch of the Nuav versus N curves as Ra increases. Observe how the curves for Ra = 10^ and 10^ shows a positive concavity (positive second derivative) while for Ra = 10^ this concavity switches from negative to positive when N increases beyond 50, and for Ra =10^ the concavity seems to be negative within the entire N range. Figures 3.4 and 3.5 present similar results forK = 0.1 and /€ = 10, respectively. The graphs in Figures 3.3 to 3.5 are presented using the same ranges for Nuav and N to facilitate a direct comparison of the results. The first observation is the better prediction obtained with the porous-continuum model when n equals unity. See Figure 3.4 results, for K = 0.1, and observe how the porous-continuum results are higher than the results predicted by the continuum model. The exception is for Ra = 10^, and for Ra = 10^ when N is greater than 50. On the contrary. Figure 3.5 results, for n = 10, indicate the porous-continuum results to be predominantly lower than the continuum results. The porous-continuum model seems to be more sensitive to changes in K than the continuum model. The second observation is the increase in Nuav when K increases, and the decrease of Nuav when K decreases. These effects are expected because the blocks affect the natural convection phenomenon in two distinct ways. One way is by hindering the flow due to the additional viscous resistance imposed by the blocks, an effect not influenced by K.

72

FROM CONTINUUM TO POROUS-CONTINUUM

Figure 3.4 Variation of Nuav with N obtained via the continuum approach, for K, = 0.1 and several Ra values. Also shown are results obtained via the porous-continuum approach with 6 = 150 (dotted lines) and 6 = 210 (dashed lines).

Figure 3.5 Variation of Nuav with A^ obtained via the continuum approach, for K = 10 and several Ra values. Also shown are results obtained via the porous-continuum approach with b = 150 (dotted lines) and b = 210 (dashed lines).

73

A. A. MERRIKH AND J. L. LAGE

But the other effect, on the heat that can diffuse trough the blocks within the enclosure, is certainly affected by K. Recall that, in relation to that of the fluid, the conductivity of the solid increases when K increases. This effect should enhance heat transfer across the enclosure as it would be easier for heat to diffuse through the blocks when they are more conductive. However, this is essentially a diffusion effect and so its impact should be stronger when the heat transfer process within the enclosure is mainly by diffusion. This expectation is difficult to verify by observing the graphs of Figures 3.3 to 3.5 alone. Therefore, two new graphs are prepared, one using results obtained via the porouscontinuum model, for K = 0.1, 1, 10, and for two Rayleigh numbers, namely 10^ and 10^ shown in Figure 3.6, and another using the results obtained via the continuum model, shown in Figure 3.7. Results are shown in terms of a figure-of-merit $, defined as the ratio of Nuav,K=i, or Nuav,K=io» to Nuav,K=o.i- As cxpcctcd, the strong effect of changing K for Ra = 10^ as compared to Ra = 10^ is clearly evident in Figure 3.6. An additional interesting observation follows: when Ra is small, the /^-effect is more pronounced when N increases. For Ra large, on the contrary, /c-effect is stronger when N decreases. Now, the results for the continuum model presented in Figure 3.7 are similar, except by the almost negligible effect n has on Nuav for Ra = 10^. When the natural convection is modeled at the continuum level, the heat transfer process becomes independent of K, when Ra is high. This can be attributed to the 'disconnectedness' of the solid obstacles when considering the continuum model that yields a cut-off effect by the fluid flowing between the soHd obstacles, somewhat blocking soHd-to-solid diffusion. Observe also that although present in Figure 3.7, the Ac-effect on Nuav is less also for Ra = 10^ as compared to the results from the porous-continuum model shown in Figure 3.6.

54-

$pc 3 :

--

^j^^Uav,K=lo/NUav,«=0.1

2NUav,K=l/NUav,Ac=0.1

150

100

150

N

Figure 3.6 Relative effect of changing n on Nuav, obtained via the porous-continuum approach, with b = 150, for Ra = 10^ (solid lines) and 10^ (dashed lines).

74

FROM CONTINUUM TO POROUS-CONTINUUM

54NUav,K=lo/Nu ^pc 3 -

2-

av,K=U.l

_

NUav,K=l/NUav,K=0.1 NUav,K=10/NUav,/c=0.1

1-

av,K=0.1

50

N

100

150

Figure 3.7 Relative effect of changing H, on Nuav, obtained via the continuum model, for Ra = 10^ (solid lines) and 10^ (dashed lines).

When comparing the continuum and the porous-continuum results for Ra = 1 0 ^ and Ra = 10^, the change in the Nuav versus A^ concavity (strong in the continuum case but weak in the porous-continuum case when K, is high), the stronger effect on Nuav when changing K. for low Ra, and the different sensitivity of the models to changing K, all observations made in connection with Figures 3.3 to 3.7, are the three phenomena difficult to clearly justify. However, all these observations must be related to the intrinsic difference between porouscontinuum and continuum model, which is the visual resolution. From the continuum resolution level, the internal nuances of the domain, as the fluid-solid interfaces, are precisely located. The porous-continuum model does not have this capability. Hence, the observed difference in behavior must be related to the way in which the internal structure is modeled by each method. 3.2.5

Internal structure effect

Figure 3.8 exhibits streamlines obtained with Ra = 10^, /^ = 1, and N from 9 to 64. For small number of larger blocks, i.e. N = 9, the flow is stronger along the heated and cooled walls. As the number of blocks increases, and their size is reduced (to maintain the porosity constant), the flow tends to migrate away from the walls, occupying the vertical channel between the two columns of blocks adjacent to the walls (see Figure 3.8 for AT = 36 and 64). This phenomenon seems to be a response by the system to the increased flow resistance, as the blocks get closer to the solid wall. In this case, fluid prefers the less resistive path in between the two adjacent columns of blocks. A switch in flow path, observed when A^ grows from 9 to 64, requires the buoyancy region that

75

A. A. MERRIKH AND J. L. LAGE

(a)

Figure 3.8 Streamline plots for the case Ra = 10®, K = 1, and (a) iV = 9, (b) N = 36, and (c) N = 64.

propels the fluid (region affected by the heating or cooling wall) to extend itself beyond the column of blocks adjacent to the walls. The flow path switch is revealed also by the reducedflownear the top-left and bottom-right comers of the enclosure when N increases. As a consequence, the convective heat transport is expected to get hindered as well, result of becoming restricted to an effective shorter length of the heated (or cooled) wall. Figure 3.9 presents similar results for Ra = 10^. In this case, the flow remains adjacent to the hot (and cold) walls even when iV = 36. Although this restraining flow effect caused by the proximity of the blocks to the walls is expected in this case, higher Ra yields a narrower buoyancy region (closer to the walls), hindering the flow path switch phenomenon. This certainly affects flow structure along the horizontal adiabatic surfaces of the enclosure, with a relatively stronger flow observed when Ra is high for the same number of blocks, N.

D.OQ5DCrQ:^|

Figure 3.9 Streamline plots for the case Ra and (c) N = 64.

10^ Av = 1, and (a) iV = 9, (b) AT = 36,

76

FROM CONTINUUM TO POROUS-CONTINUUM

Because the results of Figures 3.8 and 3.9 are presented for K = 1, the effect of the blocks on the temperature distribution is expected to reflect, faithfully, their influence on the flow. Figures 3.10 and 3.11 present the isothermal distribution corresponding to the streamlines for A/" =: 9 and 36 shown in Figures 3.8 and 3.9, for Ra = 10^ and Ra = 10'^, respectively. For the case Ra = 10^, see Figure 3.10, observe how the isotherms along the hot (left) wall, from bottom to top, shift further away from the wall as N increases from 9 to 64. The waviness of the isotherms observed mainly near the hot and cold walls (especially for N = 9) reflects flow channehng in between the blocks as observed in Figure 3.8. As N increases, the isotherms stratify parallel to the vertical walls indicating a reduction in the flow intensity within the thermal boundary-layer region. The isotherms of Figure 3.11, for Ra = 10^, demonstrate that the vertical temperature stratification away from the wall which represents characteristic of high Rayleigh number natural convection in an enclosure. Some waviness is still observed, but much less intense

Figure 3.10 Isotherms for the case Ra = 10*, K = 1, and (a) AT = 9, and (b) TV = 64.

(b)

UUnnnnn^

R

m

//2

-&S-&CL/

Figure 3.11 Isotherms for the case Ra = 10^, K = 1, and (a) iV = 9, and (b) N = 64.

A. A. MERRIKH AND J. L. LAGE

77

than the one observed in Figure 3.10. In this case, the flow has less channeling effect, as confirmed by the streamlines of Figure 3.8. Figures 3.8 to 3.11 confirm the hindering effect the blocks impose on the flow as A'' increases, with this effect strengthening as Ra decreases, causing the buoyancy region adjacent to the vertical walls to widen further away from the channel adjacent to the enclosure walls. The widening characteristic of the flow is exemplified by Siflow switch from a single channel adjacent to the hot or cold wall to two or more vertical channels away from the vertical walls. The flow switch phenomena already that was observed helps explain the continuum results seen in Figure 3.3 (already discussed), and the observation of Nuav variation with A^ switching as Ra increases. For instance, for Ra = 10^, Nuav decreases by only 5.2% when N increases from 9 to 16; however, when Ra = 10^ the decrease is over 31%. For Ra = 10^, the Nuav decay is appreciable only when N increases from 64 to 144. This variation is due to the narrowed buoyancy region for large Ra. An analytical prediction for the flow switch can be obtained by comparing the estimated scale of the boundary layer along a heated (or cooled) channel wall with the space between the wall and the first column of blocks in the enclosure. From Bejan (1995, p. 365), for a Pr ^ 1 fluid, the boundary layer is confined to the entire space between two isothermal wafls whenRa"^/^ = 5*, where5* is the nondimensional distance between the two walls. For a single heated wall, a system with half the buoyancy strength, Ra~^/^ = 5*/2 would be a better scale. The distance, 5, available for the fluid flowing in between a heated (or cooled) wall and the first column of blocks within the enclosure can be estimated as l-Z^iVi/2_i-(1-^)1/2 ~

2Ari/2

~

2A^i/2

•

^-"-^^^

One would expect the fluid to channel away from the wall (flow switch) only when the boundary layer (region of buoyancy effect) grows beyond the space between the wall and the first column of blocks, i.e. when 5* > S, or when ri_(i_0)i/2i2 N > ^ —^—Ra^/^ . (3.25) 16 In the present study porosity is set to 0 = 0.64, so using equation (3.25) the minimum number of blocks required for flow switch becomes Nm\n = OmRa}^'^.

(3.26)

Hence, the fluid will only channel away from the heated (or cooled) wall when TV is greater than A^min- Table 3.4 presents results for various Ra values. When N < A^min, flow will remain between the first column of blocks and the vertical walls, so the effect of increasing the number of blocks on the heat transfer process should be minor. On the other hand, when A^ > A^min flow switch takes place, and Nuav

78

FROM CONTINUUM TO POROUS-CONTINUUM

Table 3.4 Values predicted from equation (3.26) for the minimum number of blocks within the enclosure, beyond whichflowswitch takes place. Ra 10^ 10® 10^ 10^ 109

TV •

3 10 32 100 316

should then decrease with a further increase in N. Results predicted by equation (3.26) are included graphically in Figure 3.12 where results from the continuum model are replotted: for each Ra value, A^min predicted by equation (3.26) becomes the one marked by the intersection between the thick dashed line and the Ra curves. Observe that, to the right of the dashed line, A^ > A^min indicating that Nuav should decrease with an increase in N. This prediction is confirmed well by the results. The effect of the solid-fluid thermal conductivity ratio K on the flow field and heat transfer process is more difficult to predict because of two competing effects. For a fixed block distribution, when K > 1, the solid blocks near the hot or cold walls will increase the diffusion heat transfer from the wall and, by consequence, widen the effective buoyancy region, enhancing the convective process. This effect should be more prevalent when Ra is low because the original buoyancy region (thermal boundary-layer thickness) is wide

Figure 3.12 Comparison of predicted A^min (equation (3.26), dashed line) beyond which flow switch should take place causing a decrease in Nuav Results for continuum model and (j) = 0.64.

A. A. MERRIKH AND J. L. LAGE

79

in this case so the blocks adjacent to the wall have a better chance of becoming active participants on the heat transfer process. However, the blocks located away from the walls should also enhance the heat transfer rate from the top horizontal fluid stream to the bottom horizontal fluid stream, hindering the overall heat transfer process by enhancing the vertical conduction within the enclosure as discussed by Merrikh and Mohamad (2001). For K, <1 and small Ra the opposite should take place, i.e. convection along the walls should be restrained by a narrower region and the vertical heat transfer within the mid-section of the enclosure between top and bottom horizontal fluid streams (vertical conduction) should be hindered. Also, the K effect on the convection process should be minor when Ra is high (because the flow tends naturally to be restricted to the channel adjacent to the hot and cold walls). The validation of the predictions regarding K, is available in Merrikh and Lage (2004). The blockage effect on the flow channeling explains well the changes observed previously on how Nuav changes with N and AC. It also explains the behavior of the results obtained with the porous-continuum model. It is a fact that the porous continuum model lacks capability of accounting for the flow switch effect. Therefore, whenever the flow switch phenomenon takes place the results from the continuum model should differ from the results of the porous-continuum model. To reinforce this aspect. Figure 3.13 shows the isotherms for the case Ra = 10^, A^ = 64, and hi = 1 obtained from the continuum and the porous-continuum models. The minimum A^ value for this case is predicted to be 32, so the flow switch phenomenon is expected to have taken place. Indeed, it is clearly shown by the wavy isotherms from the continuum model (see Figure 3.13(a)), near the bottom-left or the top-right regions of the enclosure, that the flow is not restricted to the channel adjacent to the hot wall. On the other hand, the isotherms resulting from the porous-continuum model (see Figure 3.13(b)) are smooth indicating no flow switch.

Figure 3.13 Isotherms obtained from the (a) continuum, and (b) porous-continuum, models for the case Ra = 10^, A" = 64 and K = 1. (Observe the isotherms interval for the continuum case is half the isotherms interval used for the porous-continuum case to better show their waviness.)

80

3.3

FROM CONTINUUM TO POROUS-CONTINUUM

HEAT-GENERATING BLOCKS

Another interesting configuration, similar to the one shown in Figure 3.2, involves heatgenerating solid blocks. Studies on fluid-solid heat-generating systems are important to several practical engineering applications. In a nuclear reactor, for instance, the energy emanates from the nuclear fuel, which can be characterized as a heat-generating fluidsolid medium. Shell and tube heat exchangers can also be classified as fluid-solid heatgenerating systems. Another important application is the storage of food, e.g. grains, fruits, etc., in storage silos. In this case, the generating heat emanates from the thermal energy produced naturally by the organic reactions within the food (solid). Wager et al. (1952) performed experiments with a cylindrical storage bin filled with potatoes to obtain the maximum temperature within the container (an important parameter for determining the durability of the stored food). They found that the maximum temperature occurs at the vertical centerline of the bin. Schmidt (1955) performed a similar experimental study on wheat, for a long period of time, finding similar results. Hardee and Nilson (1977) performed a more fundamental analytical and experimental study of heat-generating porous media, considering rectangular and cylindrical containers. Their configuration had insulated bottom and sidewalls, and isothermal bottom surface. Beukema and Bruin (1983) developed an experimental model to determine the effects of heat transfer in bins filled with various simulated agricultural components. Their results, which matched well with the results of Wager et al (1952) for potatoes, revealed that the maximum temperature was located at the bottom and the center of the storage bin when conduction dominates. When convection becomes important, they found out that the maximum temperature occurred at the vertical centerline close to the top of the cylindrical enclosure. Their results indicate the limitations of conduction models for not capturing the natural convection effects. Dona and Stewart (1989) considered com silos having different cylindrical shapes. Through modeling by using porous medium equations and numerical simulations, they predicted unicellular flow within tall enclosures with small diameter, while multi-cellular flow was predominant in short and large diameter enclosures. They also found the maximum temperature within the container to decrease as the enclosure height decreases and the diameter increases. Stewart, Jr et al. (1990) performed an additional numerical study, to examine potential grain maximum and average temperatures, simulating a grain storage system as a two-dimensional rectangular enclosure. They modeled the system as a uniform heat-generating, partially insulated rectangular porous enclosure. They found that as the height-to-width ratio of the enclosure increases, the maximum temperature moves to the upper half of the enclosure along the vertical centerline. They also found the maximum temperature to be a function of the volume of the enclosure and not of the aspect ratio. Yaghoubi et al. (1991) performed a transient numerical study of a cylindrical porous medium with internal heat generation. They concluded that for short grain storage time, like 40 days, heat transfer is mainly conduction dominated, while for long periods of time (close to 90 days) temperature was found to increase and natural convection was found to be dominant.

A. A. MERRIKH AND J. L. LAGE

81

Nield (1995) performed a pioneering investigation of the onset of convection induced by nonuniform volumetric heating. He used a porous-medium model and a first-order Galerkin method for determining the critical Rayleigh number of several stability conditions. A more extensive review can be found in Nield and Bejan (1999). A configuration similar to that of Figure 3.2, where the enclosure is filled with uniformly distributed, heat-generating square solid blocks and having uniform surface temperature was studied recently by Merrikh et al. (2004) and the main findings are reviewed here. 3.3.1

Mathematical modeling

When a single-phase fluid saturates a heat-generating solid constituent, two basic configurations can be considered: (i) isothermal heat generation—when the solid constituent generates energy at a uniform temperature higher than the surrounding fluid temperature; or (ii) isoflux heat generation—when the solid constituent generates energy uniformly and independent of temperature. In terms of continuum modeling, either configuration can be handled without much problem. However, in terms of porous-continuum model, the first configuration presents a challenge: the need to know the amount of heat generated by the blocks, a necessary input to the model. Then, following the first configuration. Figure 3.14 shows the enclosure with isothermal heat-generating solid blocks. For the continuum model, the continuity, momentum and energy balance equations would be identical to equations (3.1)-(3.4), or the nondimensional equivalents (3.5)-(3.8). The dimensionless boundary conditions would also be identical, except by the fluid thermal condition at the isothermal boundaries of the enclosure, namely 0 = 0 at ^ = [0,1] and T] — [0,1], and at the fluid-solid interfaces, where the solid blocks are considered isothermal, so ^ === 1. Observe in the present case that there is no need to solve an energy equation within the blocks as they generate heat isothermally, so their temperature is known. Modeling the heat transfer process of Figure 3.14 using a porous-continuum model faces an additional challenge: when the solid constituent (the blocks) presents a known uniform temperature, the fluid and solid constituents will not be in local thermal equilibrium. So, formally a single equation for the heat transfer process should not be considered. In this case, the alternative would be a two-equation volume-averaged porous-continuum model. This alternative, however, presents another challenge: knowledge of the heat transfer coefficient at the fluid-solid interfaces needs to be known. Unfortunately, knowledge of the fluid-solid heat transfer coefficient is extremely limited and no general correlation exists. The only practical alternative is to assume the constituents of the porous medium to be in local thermal equilibrium. In this case, equations (3.12) and (3.13), and the corresponding dimensionless equations, equations (3.16) and (3.17), are still valid. The energy balance

82

FROM CONTINUUM TO POROUS-CONTINUUM Heat generating solid at T,

d

^, . , Fl^^^

Hi

:—!

/

d

y,y

i

0

/

i

1

\ >

Figure 3.14 Enclosure with 16 equally spaced, isothermal heat-generating solid square blocks, saturated with fluid.

equation, equation (3.14), is modified to account for the energy generated by the solid blocks: ( p c ) e f f ^ + {pc)fct>{v) . V(T) - k,^V^{T) + q'^', (3.27) dt where q'g is a volumetric energy generation term which is not known in the present case because the present configuration is that of an isothermal energy-generating solid constituent. However, we can obtain a suitable equivalent value for the volumetric heat generation term by using the results from the continuum model. At steady state, the total energy generated by the solid constituent must be equal to the total energy escaping from the enclosure. From the continuum model results, the total energy generated by the isothermal solid blocks can be calculated by finding the heat diffusing through the boundaries of the enclosure, namely: kfA{TH (3.28) ©TOT 5 H where A is the total boundary area of the enclosure, and the last term is defined as: % =

©TOT

=sir(-

de_ dn

dm.

(3.29)

83

A. A. MERRIKH AND J. L. LAGE

In equation (3.29), n stands for the direction normal to an enclosure's surface and m stands for the direction parallel to the same surface. The volumetric heat generation rate is obtained by dividing the energy generated by the solid, equation (3.28), by the enclosure volume (equal to H'^ in the case of a square enclosure): Qp

_ kfAjTh - Tc) = H^

(3.30)

©TOT •

In dimensionless form equation (3.27), using equation (3.30), becomes ^eff ^

+ 0(V) . V(^) - K,^VHe) + 0TOT .

(3.31)

When using equation (3.30), observe A — Hw, where w is the width of the enclosure, set as unity. Equation (3.31) is the equivalent to equation (3.18) for the heat generation configuration. The described procedure for determining the volumetric energy generation term of equation (3.27) is based on the principle that at steady state the continuum and porouscontinuum models would account for the same amount of energy generated inside the enclosure, i.e. the models are physically consistent when it comes to global energy balance. 3.3.2

Heat transfer comparison parameters

The comparison between the continuum and the porous-continuum results can be made via the Nusselt number evaluated at each surface, namely NUT, N U S , NUL, and Nui?, where the subscripts T, J5, L, and R refer to each of the four surfaces of the enclosure, namely top, bottom, left, and right surfaces, respectively. Each one can be evaluated using equation (3.21), as the defining equation for the Nusselt number for the continuum model, or equation (3.22), as the defining equation for the Nusselt number for the porouscontinuum model, adapted to each surface. An interesting aspect relates to the total Nusselt number of the enclosure, defined as (3.32)

NUTOT = NUT + NUB -f NUL -f Nui?.

When evaluated for the continuum model, hence using the Nusselt number definition equation (3.22) to evaluate the Nusselt number along each surface, equation (3.32) becomes

dd d^ C=o

NUTOT —

del

de

de

d^\ ^=1

9r] r,=0

9r]

-^ 1 I

Jo \

on

1 dm.

»,=1

(3.33)

84

FROM CONTINUUM TO POROUS-CONTINUUM

Comparing equations (3.33) and (3.29), we have N U T O T — ©TOT •

(3.34)

Hence, the total Nusselt number obtained from the continuum model equals the energy generation term to be used in the energy balance porous-continuum model equation. 3.3.3

Results

As in the study of natural convection with horizontal heating, the steady-state results from the continuum model depend on Ra, Pr, K, and the geometrical parameters A, D, and N, and the porous-continuum model results depend on Ra, Da, A, (f), /^eff» and the heat-generation term evaluated from the continuum results. From these parameters,

Figure 3.15 Streamlines, continuum model, Ra = 10^: (a) cj) = 0.96, (b) (p = 0.84, (c) 0 = 0.64, and (d) 0 = 0.36.

85

A. A. MERRIKH AND J. L. LAGE

0 = 1 - ND^, Da = D20V[6(1 - 0)2], and /^eff = /^^~^. Hence, the controlling parameters become: Ra, Pr, A, D, N, A and /Ceff. Results are presented for A = A = /^efF = 1, Pr = 7, and A^ = 16. Furthermore, the relative amount of solid constituent present in the domain is varied by altering the lengthside of the solid blocks, D = 0.05, 0.10, 0.15, and 0.20. The corresponding porosities for each of these cases are: 0 = 0.96, 0.84, 0.64, and 0.36 (for easing the comparisons, we use porosity as the characterizing parameter in the continuum model as well as in the porous-continuum model results). The objective of varying the block side length is to investigate the accuracy of the porouscontinuum model at different equivalent porosities. This is an important aspect because the transport process is more complex in the fluid constituent (because of convective effects) than in the solid constituent. Moreover, the transport complexity is hindered as

(d) ••Ni.i...Mi.i,i.i.in.p|i..u,.i...

m

1

1;;;;::::;:::; ^:::

'fm" • '"pt"— 1 i J *» 11 X I * ^

*I

I

wzzt ^:::: ! 1 1 i.

"•'••4

,

^^iiM

^%M

rf^-.*-)-

,.„*J L,... t>

:

?

1 i

1

1 I

1

11

J

,..,^1...,...Ji

^dm ^ 1 ^ ^ 1 ^ u^

Figure 3.16 Isotherms, continuum model, Ra = 10^: (a) 0 = 0.96, (b) (/> = 0.84, (c) 0 = 0.64, and (d) 0 = 0.36.

86

FROM CONTINUUM TO POROUS-CONTINUUM

the region occupied by the fluid constituent is reduced (i.e. as the porosity is reduced). Knowing how this aspect affects the porous-continuum model accuracy is important. For the continuum model simulations, Figures 3.15 and 3.16 show the steady-state streamlines and isotherms, respectively, obtained for Ra = 10^ and four different porosities. Figure 3.15 shows the flow within the enclosure to be progressively squeezed by the smaller channels as the porosity decreases. Observe also in all cases theflowpatterns consist of two counter-rotating circulations, each occupying one-half of the enclosure, a result of the heat-generating by the blocks and the symmetry of the configuration in relation to the acceleration of gravity (vertical direction). The increase in flow resistance caused by the smaller flow channels, and the subsequent reduction in flow strength, impacts the location of the maximum temperature within the enclosure as seen in Figure 3.16. As the porosity is reduced the location of maximum temperature within the enclosure moves from the top half of the enclosure towards the center.

Figure 3.17 Streamlines, continuum model, Ra = 10^: (a) 0 = 0.96, (b) (/> = 0.84, (c) 0 = 0.64, and (d) cf) = 0.36.

87

A. A. MERRIKH AND J. L. LAGE

Similar steady-state streamlines and isotherms for the case Ra = 10^ are presented in Figures 3.17 and 3.18, respectively. Comparing results of Figure 3.17 with Figure 3.15 for the same porosity, the increase in flow complexity is observed, particularly for large porosity (0 ^ 0.84). Four flow circulations are observed within the enclosure, instead of twOj when (j) increases to 0.84 and 0.96. Two small counter-rotating circulations are visible at the center top half of the enclosure. Two other circulations are presently occupying the lower portion of the enclosure, forming two ascending (in the middle) and descending (adjacent to the vertical surfaces) streams. These complicated flow patterns affect the temperature distribution within the enclosure, as seen for instance by the thermal plumes descending along the center top of the enclosure in Figure 3.18 for (/> = 0.84 and 0.96. As the Rayleigh number and the porosity increases, the heat transfer process eventually becomes time-dependent. For instance, when 0 = 0.96, a steady-state result could only be obtained if Ra is smaller than 10^. The effect of increasing the porosity is more evident in Figure 3.19, for Ra = 10^ as the porosity of the enclosure is progressively increased

(d) 1*^

'^'""""PP!^"'^'"""'""•'"•'fiw'" k;i'[

I

" ' • " ' ' "

3

— ^

m.Z

J

' ™

"^

' ™ '

*gv*

^

"

'""•""""'.iif'"""" i i «».•»?

•i..kv»

......1 L _

,...,* %..,..

»,...**'•

s _^I

r,<.>*.

::::;«

Zli

:::i

1^

^M

Figure 3.18 Isotherms, continuum model, Ra = 10^: (a) (f) = 0.96, (b) ( 0 = 0.64, and (d) (p = 0.36.

0.84,(c)

88

FROM CONTINUUM TO POROUS-CONTINUUM

0*1

O.I

0.1

Figure 3.19 Effect of porosity on the stability of total Nusselt number for Ra = 10^: (a) D = 0.09, (p = 0.87, (b) D = 0.0925, 0 = 0.86, and (c) D = 0.1, 0 = 0.84.

from 0 = 0.84 to 0.87. For 0 = 0.87 a steady-state results is no longer obtained. The determination of conditions for the stability of the natural convection process inside the enclosure is worth studying. Values for the heat generation term of the porous-continuum model are shown in Table 3.5. Keep in mind the heat-generation term is identical to the total Nusselt number for the continuum model. Observe from Table 3.5 how the shrinking of the flow channels (reduction of (/>) hinders the overall heat transfer process. For instance, when 0 == 0.84, the total Nusselt number goes from 39.52 when Ra == 10^ to 67.38 when Ra = W. On the other hand, when 0 = 0.36, NUTOT goes from 146.94 when Ra = 10^ to only 148.09 when Ra = 10'^. Table 3.6 presents a sample on the effect of varying the Kozeny parameter b, for calculating the Darcy number of the enclosure, from b — 150 to 210, and for Ra = 10^. As seen, the effect is minor, so the value b = 180 was used for the computations using the porouscontinuum model. Observe the values of NUR and NUL are identical, as expected due to the symmetry of the configuration.

A. A. MERRIKH AND J. L. LAGE

89

Table 3.5 Volumetric energy generation term, from continuum model results at steady state, for the porous-continuum model.

0

©TOT (NUTOT)

0.36 0.64 0.84 0.96 0.36 0.64 0.84 0.96 0.36 0.64 0.84 0.96

148.09 87.21 67.38

Ra

10^

10^

10^

-

146.95 65.46 48.65

-

146.94 64.10 39.52 27.62

Table 3.6 Effect of the Kozeny constant b on the surface Nusselt number of the porouscontinuum model for Ra = 10®. , ^

6=150

NUT 6=180

6 = 210

6=150

NUB 6=180

6 = 210

6=150

0.36 0.64 0.84

53.03 22.03 17.43

52.22 22.75 16.20

51.40 22.98 18.21

22.05 6.88 4.61

22.87 7.09 4.59

23.58 7.23 4.63

34.07 16.92 12.75

Nui2,NuL 6 = 1 8 0 6 = 210 35.93 17.81 13.93

34.13 16.82 12.81

Figure 3.20 shows a comparison between continuum and porous-continuum results in terms of streamlines and isotherms, for the case Ra = 10^ and 0 = 0.36. Noteworthy is the difference in the isotherm (see Figure 3.20(a,c)) distribution. In the case of the continuum model the isotherms are concentrated within the region between the surfaces of the enclosure and the blocks. On the other hand, the porous-continuum results show a more even distribution of isotherms throughout the entire enclosure. A similar effect is also observed in the case of the streamlines (see Figure 3.20(b,d)): the flow is more uniformly distributed when the porous-continuum model is utilized. These observations in regards to isotherms and streamlines reflect the smaller resolution of the porous-continuum model. That is, local effects due to structural nuances, as seen at the continuum level, are smoothed out by the porous-continuum model. Tables 3.7 and 3.8 show a comparison, in terms of % difference, between some of the continuum and the porous continuum results in respect to the Nusselt number at each individual surface for Ra = 10^ and 10^, respectively (notice Nu^ = Nu/^ due to symmetry).

90

FROM CONTINUUM TO POROUS-CONTINUUM

Figure 3.20 Comparison between the (a,b) continuum and (c,d) porous-continuum models for Ra = 10^, 0 = 0.36: (a,c) isotherms, and (b,d) streamlines.

Interestingly, the largest discrepancy between continuum and porous-continuum results is found for the top and bottom surface Nusselt numbers. This is certainly consequence of the smoothing effect the porous-continuum model has on the plumes arriving at the top surface of the enclosure (see Figure 3.20). The flow toward the top surface of the enclosure, in the case of the continuum model, is restricted by the channels formed by the solid blocks, reducing the heat transfer effect along the top surface. That is why the porous-continuum NUT values are consistently higher than those for the continuum model. The same effect leads to an inverse result when it comes to the bottom Nusselt number. In this case, the channeling formed by the blocks and the surfaces of the enclosure strengthen the downwardflow,which becomes spread out in the case of the porous-continuum model. Hence, the Nu^ values for the continuum model are higher than for the porous-continuum model.

Table 3.7 Comparison between continuum and porousxontinuum model results for Ra = lo5.

'

Continuum

NUT Porous-continuum

0.36 0.64 0.84 0.96

36.14 16.85 10.49 1.62

38.83 21.68 13.96 9.17

NUB

% Difference Continuum

5.7 28.7 33.1 20.3

36.72 15.18 9.15 5.94

Porous-continuum

32.94 10.83 5.57 3.66

% Difference

10.3 28.7 39.1 38.4

N U R ,NUL Continuum Porousxontinuum 36.74 15.32 9.66 7.03

35.76 14.99 9.50 7.04

% Difference

2.3 I .5 0.6 0.01

? ?

3m E

9 2.

U 9

r

Table 3.8 Comparison between continuum and porous-continuum model results for Ra = lo6.

' 0.36 0.64 0.84

r

%

NUT Continuum Porous-continuum 36.83 18.05 18.39

52.22 22.15 16.20

% Difference

Continuum

NUB Porous-continuum

41.8 26.0 11.9

36.64 14.33 1.36

22.87 7.09 4.59

5% Difference

37.6 50.5 37.6

N U R .N ~ L Continuum Porous-continuum

36.74 16.54 11.45

35.93 11.81 13.93

% Difference

2.2 7.6 21.9

$2

92

FROM CONTINUUM TO POROUS-CONTINUUM

Another interesting aspect is the good prediction using the porous-continuum model for the left and right surface Nusselt numbers. The homogenization brought about by the porous-continuum model does not seem to alter much the flow and heat transfer along the two vertical surfaces of the enclosure because the flow, as seen in Figures 3.15, 3.17 and 3.20 for the continuum model, is essentially parallel to the surfaces. The results of Table 3.7 show that as the porosity increases, the differences of NUT and Nu^ between the two models increase and then decrease. This is a puzzling observation because one could argue that as the porosity increases, and the medium gets further away from what one would normally consider as a porous medium, the difference between the two models should always increase. However, a high-porosity medium has less solid obstruction allowing the flow field to develop more similarly to the flow field in a homogeneous medium (as the one assumed in the porous-continuum model). Table 3.7, for the relatively weak convective effects when Ra = 10^, reveals the competition between the solid constituent homogeneity and the fluid constituent homogeneity, and how they affect the continuum model results. Table 3.8 indicates another behavior of NUT when Ra = 10^ with the difference between the two models decreasing as 0 increases. The more intense buoyancy effect (as Ra increases) enhances the heat transfer across the top surface as compared to the bottom surface. Now, the homogeneity of the fluid constituent (responsible for convection) predominates so as to have the difference between the two models decrease as the amount of fluid in the enclosure increases (increasing porosity).

3.4

CONCLUSIONS

Studies concerned with the applicability of a low resolution model based on volumeaveraging equations, called porous-continuum model, to predict natural convection within an enclosure with square discrete blocks are reviewed and expanded. Two basic configurations considered are the horizontally heated enclosure (with adiabatic top and bottom surfaces) and the isothermal heat generating blocks with isothermal enclosure surfaces. In both cases, results obtained using the porous-continuum model are compared to results obtained with the continuum model, in which balance equations are solved individually for each constituent (solid and fluid) of the heterogeneous medium. For differentially heated enclosures, the enclosure is assumed square and the amount of solid constituent within the enclosure is kept constant by fixing the porosity at 64%. The number of solid blocks varied from 9 to 144 and the fluid Prandtl number is set as unity. The study covers Rayleigh number (Ra) from 10^ to 10^ and fluid-to-solid thermal conductivity ratio, K, from 0.1 to 10. Good agreement between the continuum and porous-continuum model results, based on the hot-surface average Nusselt number, is obtained when K, — 1, or when Ra is small (less than 10^) and A^ is large (greater than 36). When comparing the continuum and the porous-continuum results, the change in the Nuav versus A^ concavity (very strong in the continuum results, but weak in the porous-continuum results, when K is high).

A. A. MERRIKH AND J. L. LAGE

93

the stronger effect on Nuav when changing K for low Ra, and the different sensitivity of the models to changing K, are observations related to the intrinsic difference between porous-continuum and continuum model, which is the visual resolution. The continuum model sees the internal nuances of the domain, as the fluid-solid interfaces are precisely located. The porous-continuum model does not have this capability. Hence, the observed difference in behavior must be related to the way in which the internal structure is modeled by each method. Results for /« = 1 indicate a switch from thefluidflowingpredominantly along the channel between the heated (or cooled) enclosure wall and the first column of solid blocks, to the fluidflowpenetrating on to the interior channels of the enclosure away from the heated (or cooled) wall as number of blocks increases. This phenomenon causes a drastic decrease in the heat transfer within the enclosure, and this decrease seems to be more pronounced (abrupt) at lower Ra. Scale analysis lead to an analytical expression for the minimum number of blocks necessary for the flow switch to take place. This analytical prediction, when compared to the numerical results, is proven to be exceptionally accurate, indicating that heat transfer across the enclosure is hindered when the number of solid blocks inside the enclosure is less than the minimum number necessary for the flow to switch, A^min, for high fluid-tosolid thermal conductivity ratio, K. For a number of solid blocks bigger than the minimum number necessary for flow switch, the heat transfer across the enclosure is enhanced as the fluid-to-solid thermal conductivity ratio K increases; the effect of increasing n becomes much more pronounced in this case. However, for A^ < A^min the degree of enhancement is inversely proportional to the Rayleigh number. Hence, the lowest K would yield higher heat transfer for N < NminIn general, the effect of increasing hi is that of enhancing the overall heat transfer process for7V>7Vn,in. For the heat-generating configuration, the number of blocks is fixed at 16 and the porosity varies from 0.36 to 0.96. Considering the strict requirements for the validity of the single-equation porous-continuum heat transfer model used in this case, the results are very satisfactory. Based on the average Nu values for each surface of the enclosure, the error between continuum and porous-continuum results varied from 0.3% ((f) = 0.96, Ra = 10^ Nu/?, NUL) to 50% (0 = 0.64, Ra = 10^ Nu^). The largest discrepancy is observed for Nu^ and NUT, a consequence of the very distinct streamline and isotherm patterns adjacent to the bottom and top surfaces of the enclosure obtained from the continuum and the porous-continuum models. Values for Nu along the sidewalls of the enclosure, obtained from each model, present excellent agreement. It is undeniable that the much smaller analytical and numerical efforts necessary to obtain results from the porous-continuum model make it a very attractive choice. A quantitative criterion for using either the continuum or the porous-continuum model is impractical as the choice must be based on each individual tolerance for error (e.g. some preliminary design in engineering might accept 50% prediction error; on the other hand, a final project might not). Generally, for Ra = 10^, the highest discrepancies are below 30% for cj) ^ 0.64 while for Ra = 10^ the highest discrepancies stay below 37.6%

94

FROM CONTINUUM TO POROUS-CONTINUUM

for (j) ^ 0.64. Although much more efficient, the porous-continuum model should be avoided when the Rayleigh number is low (Ra ^ 10^) and the porosity is high (0 ^ 0.5), and when the Rayleigh number is high (Ra > 10^) and the porosity is low (0 < 0.8). Moreover, the porous-continuum model is inappropriate when the Rayleigh number and the porosity are high (e.g. for R a = 10^ and 0 ^ 0.9, and for R a = lO*" and 0 ^ 0.8) because it fails to capture the unsteady regime observed when the continuum model is used. It must be kept in mind, however, that in dealing with the special case of an isothermal energy generating solid constituent, the determination of an equivalent energy generating parameter hinders the straightforward application of the porous-continuum model. Moreover, the absence of structural details (topology of individual constituents) in the porous-continuum model does not allow the capture of important oscillatory and transitional effects, evident in the continuum model results, limiting the application to relatively low Ra.

REFERENCES Arquis, E., Caltagirone, J. P., and le Breton, P. (1991). Determination des proprietes de dispersion d'un milieu periodique a partir de 1'analyse locale de transferts. CRAS Paris, Serie 7/313, 1087-92. Bejan, A. (1995). Convection heat transfer (2nd edn). Wiley, New York. Beukema, K. J. and Bruin, S. (1983). Three-dimensional natural convection in a confined porous medium with internal heat generation. Int. J. Heat Mass Transfer 26, 451-8. Cheng, P. (1978). Heat transfer in geothermal systems. Adv. Heat Transfer 14, 1-105. Coulaud, O., Morel, P., and Caltagirone, J. P. (1988). Numerical modelling of non-linear effects in laminar flow through a porous medium. J. Fluid Mech. 190, 393-407. de Vahl Davis, G. (1983). Natural convection of air in a square cavity: a bench mark numerical solution. Int. J. Numer. Meth. Fluids 3, 249-64. Dona, C. L. G. and Stewart, W. E. (1989). Variable porosity effects on convection in a heat generating porous medium. ASME J. Heat Transfer 111, 1100-2. Eidsath, A., Carbonell, R. G., Whitaker, S., and Herman, L. R. (1983). Dispersion in pulsed systems. III. Comparison between theory and experiment for packed beds. Chem. Eng. Sci. 38, 1803-16. Fowler, A. J. and Bejan, A. (1994). Forced convection in banks of inclined cylinders at low Reynolds numbers. Int. J. Heat Fluid Flow 15, 90-9. Hardee, H. C. and Nilson, R. H. (1977). Natural convection in porous media with heat generation. Nucl. Sci. Eng. 63,119-32. Hortmann, M., Peric, M., and Sheuerer, G. (1990). Finite volume multigrid prediction of laminar natural convection: bench-mark solutions. Int. J. Numer. Meth. Fluids 11, 189-207. House, J. M., Beckermann, C , and Smith, T. F. (1990). Effect of a centered conducting body on natural convection heat transfer in an enclosure. Numer. Heat Transfer, Part A 18, 213-25. Kalita, J. C , Dalai, D. C , and Dass, A. K. (2001). Fully compact higher-order computation of steady-state natural convection in a square cavity. Phys. Rev. E 64, 066703, 1-13.

A. A. MERRIKH AND J. L. LAGE

95

Kuwahara, E, Nakayama, A., and Koyama, H. (1996). A numerical study of thermal dispersion in porous media. ASME J. Heat Transfer 118, 756-61. Lage, J. L. (1998). The fundamental theory of flow through permeable media from Darcy to turbulence. In Transport phenomena in porous media (eds D. B. Ingham and I. Pop), pp. 1-30. Pergamon, Oxford. Lage, J. L. and Bejan, A. (1991). The Ra-Pr domain of laminar natural convection in an enclosure heated from the side. Numer. Heat Transfer, Part A 19, 21-41. Larson, R. E. and Higdon, J. J. L. (1989). A periodic grain consolidation model of porous media. Phys. Fluids A l,3S-46. Lee, S. L. and Yang, J. H. (1997). Modelling of Darcy-Forchheimer drag for fluid flow across a bank of circular cylinders. Int. J. Heat Mass Transfer 40, 3149-55. Leonard, B. P. (1979). A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comp. Meth. Appl. Mech. Eng. 19, 59-98. Merrikh, A. A. and Lage, J. L. (2004). Natural convection in an enclosure with disconnected and conducting solid blocks. Int. J. Heat Mass Transfer. In press. Merrikh, A. A. and Mohamad, A. A. (2000). Transient natural convection in differentially heated porous enclosures. J. Porous Media 3, 167-80. Merrikh, A. A. and Mohamad, A. A. (2001). Blockage effects in natural convection in differentially heated enclosures. J. Enhanced Heat Transfer 8, 55-72. Merrikh, A. A., Lage, J. L., and Mohamad, A. A. (2002). Comparison between pore-level and porous medium models for natural convection in a nonhomogeneous enclosure. AMS Contemp. Math. 295, 387-96. Merrikh, A. A., Lage, J. L., and Mohamad, A. A. (2004). Natural convection in non-homogeneous heat generating media: comparison of continuum and porous-continuum models. J. Porous Media. In press. Nakayama, A., Kuwahara, E, Kawamura, Y., and Koyama, H. (1995). Three-dimensional numerical simulation of flow through a microscopic porous structure. In Proceedings of the 4th ASME/JSME thermal engineering conference, Hawaii, Vol. 3, pp. 313-18. Nield, D. A. (1991). Estimation of the stagnant thermal conductivity of saturated porous media. Int. J. Heat Mass Transfer 34, 1575-6. Nield, D. A. (1995). Onset of convection in a porous medium with non-uniform time-dependent volumetric heating. Int. J. Heat Fluid Flow 16, 217-22. Nield, D. A. and Bejan, A. (1999). Convection in porous media (2nd edn). Springer-Verlag, New York. Patankar, S. V. (1980). Numerical heat transfer and fluid flow. Hemisphere, Washington, DC. Sahroui, M. and Kaviany, M. (1991). Slip and no-slip boundary conditions at interface of porous, plain media. ASME/JSME Therm. Eng. Proc. 4, 273-86. Schmidt, J. L. (1955). Wheat storage research at hutchinson, kansas, and Jamestown, north dakota. US Department of Agriculture Technical Bulletin 1113. Stewart, Jr, W. E., Greer, L. A., and Stickler, L. A. (1990). Heat transfer in a partially insulated rectangular enclosure of heat generating porous media. Int. Comm. Heat Mass Transfer 17, 597607.

96

FROM CONTINUUM TO POROUS-CONTINUUM

Wager, H. G., Burton, W. G., and Mann, G. (1952). The storage of ware potatoes in permanent buildings. 1. Storage in bins in an uninsulated store, 1946-7. J. Agricult. Sci. 42, 266-75. Ward, J. C. (1964). Turbulent flow in porous media. J. Hydraul Div. Amer. Soc. Civ. Eng. 90, 1-12. Whitaker, S. (1999). The method of volume averaging. Kluwer, Dordrecht. Yaghoubi, M. A., Karimi, G., and Taheri, M. (1991). Numerical modeling of thermal and hydrodynamic structure of an enclosure with porous medium and internal heat generation. ASME/JSME Therm. Eng. Proc. 4, 245-50.

4

INTEGRAL TRANSFORMS FOR NATURAL CONVECTION IN CAVITIES FILLED WITH POROUS MEDIA R. M. COTTA*, H. LUZNETQt, L. S. DEB. ALVES^ and J. N. N. QUARESMA"^ * Mechanical Engineering D e p a r t m e n t - L T T C - P O L I / C O P P E - U F R J , Universidade Federal do Rio de Janeiro, Brazil email: [email protected] ^Instituto Nacional de Tecnologia - INT, Rio de Janeiro, Brazil email: h e i t o r l u O i n t .gov.br ^Mechanical Engineering Department, University of California at Los Angeles, USA email: [email protected] "^Chemical and Food Engineering Department, Universidade Federal do Para, Belem, Brazil email: quaresma@ufpa. br

Abstract Hybrid numerical-analytical algorithms, based on the generalized integral transform technique, developed to handle transient two- and three-dimensional heat and fluid flow in cavities filled with a porous material, are reviewed. To illustrate the approach, specific situations of both horizontal and vertical cavities are more closely considered, under the Darcy flow model. The problem is analyzed 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 two-dimensional situations. Results for rectangular (2D) and parallelepiped (3D) cavities are presented to demonstrate the convergence behavior of the proposed eigenfunction expansion solutions and comparisons with previously reported numerical solutions are critically performed. Keywords: integral transforms, natural convecrion, Darcy flow, hybrid methods, cavity flow, transient flow

97

98

4.1

INTEGRAL TRANSFORMS FOR NATURAL CONVECTION IN CAVITIES

INTRODUCTION

Transport phenomena in porous media represent an important segment of the heat and mass transfer field and have a variety of applications in engineering. The effect of free convection as a result of the gravitational body force is of particular interest from both practical and theoretical points of view. Engineering applications include among others, thermal insulation, radioactive waste disposal, petroleum reservoirs recovery, solar energy collectors and geothermal energy analysis. Other applications of the porous medium heat and fluid flow modeling are discussed in Kaka9 et al. (1990) and more recently in Kaviany (1995), Nield and Bejan (1999), and Ingham and Pop (2002). The theoretical analysis of transient natural convection in porous media has appeared frequently in the literature for two-dimensional situations, while for three-dimensional problems it has been much less analyzed, especially for fully transient situations, mainly due to the sometimes prohibitive increase in computational effort associated with the most usual purely discrete solution approaches. Some relevant contributions to the present analysis, in dealing with two and three-dimensional natural convection within porous cavities, are listed below. Hoist and Aziz (1971), Caltagirone (1975), Home (1979), Schubert and Straus (1979), Straus and Schubert (1979,1981), Kimura et al. (1989), and Stamps ^r a/. (1990). In terms of the adopted modeling, a number of previous works show the importance of an adequate parametric range analysis of the different models used to represent flows in porous media. Such an analysis has been performed, for instance, by Misra and Sarkar (1995) applying the finite element method to a free convection heat transfer problem in a rectangular cavity where the Darcy, Brinkman extended, Brinkman extended with Navier-Stokes transport terms and Darcy-Brinkman-Forchheimer models were compared. In addition, although most of the available contributions in three-dimensional heat andfluidflowsimulation deal with the primitive variables formulation, the vorticity-vector potential approach has been receiving increasing attention. Aziz and Heliums (1967), in a pioneering work, have shown that the vorticity-vector potential formulation could lead to more stable and fast simulations of three-dimensional flows. This formulation was then itself originally applied to three-dimensional natural convection in porous media. Hoist and Aziz (1971). Within the last two decades, the ideas in this so-called generalized integral transform technique, see Cotta (1993), Cotta and Mikhailov (1997), and Cotta (1998), were progressively advanced towards the establishment of an alternative hybrid numerical-analytical approach, based on the formal analytical principles in the classical integral transform method, see Mikhailov and Ozi§ik (1984), for a priori non-transformable diffusion and convection-diffusion problems. The present class of problems was recently treated with this approach by Alves (2000) and Alves and Cotta (2000), where the advantages in using a software that allows for mixed symbolic and numerical computations such as the Mathematica system, see Wolfram (1999), on this analytically based approach, were demonstrated. To illustrate other contributions on this method for the specific class of problems of interest here, we can mention a number of natural convection problems in

R. M. COTTA ET AL.

99

cavities under steady and transient regimen, for both porous media, see Baohua and Cotta (1993) and Alves et ai (2001), or just fluid filled, see Leal and Cotta (1997), Leal et al. (1999, 2000), and Machado et al. (1999), two-dimensional enclosures. In all such contributions on natural convection, the streamfunction-only formulation was preferred, due to the inherent advantages in its combined use with this hybrid approach, as more closely discussed in Cotta (1993, 1998). Later on, this hybrid solution scheme was advanced to handle the three-dimensional Navier-Stokes equations based on the vector-scalar potentials formulation, see Quaresma and Cotta (1997), with similar computational advantages with respect to the two-dimensional case. This contribution then opened up the possibility for handling three-dimensional natural convection problems in porous media filled cavities, by associating the related energy equation to the vector-scalar potentials formulation, as accomplished in Luz Neto (2000) and Luz Neto et al. (2002, 2004). The present review is first concerned with the analysis of the extended Darcy flow model that includes the time derivative of the velocity in the traditional Darcy model. In general, this term is discarded by means of an order of magnitude analysis, see Whitaker (1966), Prasad and Kulacki (1984), and Rajamani et al. (1990). A comparative analysis was here performed with a selection of limiting values of the governing dimensionless parameter, see Kladias and Prasad (1989), Nield (1991), and Kaviany (1995). An integral transform solution of the transient natural convection problem was developed to obtain average Nusselt numbers for both Darcy flow and extended Darcy flow models, comparing them in the range defined by the parametric analysis, for both vertical and horizontal twodimensional rectangular cavities. The present contribution is thus also aimed at reviewing this computational tool towards the accurate solution of transient two- and three-dimensional flows in natural convection within porous media filled cavities. The Darcy flow model is adopted, together with the assumptions of constant and isotropic physical properties and linear variation with temperature of the buoyancy term, i.e. the Boussinesq approximation.

4.2

TWO-DIMENSIONAL PROBLEM

We consider two kinds of two-dimensional fluid saturated porous layers with the walls assumed to be impermeable: a vertical cavity (two isothermal vertical walls at temperatures Th and Tc, Th > Tc, and two adiabatic horizontal walls) and a horizontal cavity (two isothermal horizontal walls at temperatures Th and Tc,Th > Tc, and two adiabatic vertical walls), as presented in Figure 4.1. A positive temperature gradient in the x-direction is anticipated as a result of the imposed thermal boundary conditions. In the porous medium, Darcy's law, extended to include the time derivative term, is assumed to hold, and the fluid is assumed to be within the Boussinesq approximation limits. With these assumptions, the conservation equations for mass, momentum and energy for transient two-dimensional flow in an isotropic porous medium are recalled, where the Boussinesq approximation is introduced in the buoyancy term.

100

INTEGRAL TRANSFORMS FOR NATURAL CONVECTION IN CAVITIES

z = H

T^Tc

z=H

Porous medium x= 1

T=n

x1

Insulated x = I

C3

Figure 4.1 Geometry and coordinates system for two-dimensional problem analysis.

The streamfunction formulation is in general preferred in the application of the integral transform method, see Cotta (1993, 1998), due to the enhanced convergence behavior achieved by the eigenfunction expansions in this case. Thus, the momentum equations are first differentiated in each coordinate variable and manipulated to yield the streamfunctiononly formulation. Expressed in dimensionless form, the problem is written as follows, see Alves (2000) and Alves and Cotta (2000): Xdr

ax2

+

5^

a^ _ _

ae

az2

dQ d^ dQ dr "^ dZ dX

d^ de dX dZ

Rah

ax2 "^ dz^

30 dZ'

(4.1a) (4.1b)

where equation (4.1a) is conveniently written so as to accommodate the two cases of interest, and Ra/j = 0 for the vertical cavity and Ra„ = 0 for the horizontal cavity. The related dynamic and thermal boundary conditions are as follows: $ = 0, $ = 0,

0 = 0 on 0 = 1 on

X = 0, X = 1,

(4.1c) (4. Id)

* = 0,

^

= 0 on

Z = 0,

(4.1e)

* = 0,

^

= 0 on

Z = H.

(4. If)

The streamfunction and temperature initial conditions are given by any known spatial distributions: 0 r r 0 o , * = : * o at r = 0. (4.1g) The first step in the application of the GITT, see Cotta (1993, 1998), is thus the consideration of a filtering solution, which offers some convergence enhancement, that in the present case suffices to be applied only in the filtering of the nonhomogeneous boundary conditions. The proposed filter is essentially the steady-state heat conduction solution

R. M. COTTA ET AL.

101

given as a linear function in this case:

e (X, z, T)^efix) + e{x, Z,T) = X + e{x, z, r).

(4.2)

The resulting filtered problem with homogeneous boundary conditions is then written as follows:

xd^[dx-^ + dz^)^dx^ + dz^ = -^^ V^dx)- ^^'dz' ^'-''^ dx) dx dz ~ ax2 ^ az2'

dr '^ dz\ 'Jr = o,

^ = 0 on X = 0 and ^ = 1, dQ ^ ^ = 0 on Z = 0 and Z = i ? ,

* = 0,

* = *o

e^Q^-X,

at r = 0.

^ ' (4.3c) (4.3d) (4.3e)

The basic idea behind the generalized integral transform technique—GITT (Cotta, 1993, 1998)—in solving convection-diffusion problems is to propose eigenfunction expansions for the problem potentials in all but one independent variable, promote the integral transformation of the original partial differential equations, and finally obtain a coupled system of ordinary differential equations for the transformed potentials in that single independent variable not eliminated by the integral transformation. Thus, applying the GITT to equations (4.3), whose details are presented in Alves and Cotta (2000), one obtains the coupled ordinary differential system shown below for the transformed streamfunction and temperature potentials:

^ ^*^'^ +f,,(r) =7 f-+ A^| [u, + E f: A,,,™,J(r))

X dr' • ^ —

772=1 n=0 oo oo

^"

_j^

_

2 I \ 2 2^

00

00

2^

•^i,j,m,n"\^)

?

*? J — I5 ^5 • • • 5

(4.4a)

dr 00

i=l

CXO

j=l

00

00

0 = 1 p=0

m , n = 1,2,... , (4.4b)

102

INTEGRAL TRANSFORMS FOR NATURAL CONVECTION IN CAVITIES

where the integral transformation of the initial conditions yields ^iA^) = k3

at ^ = 0,

(4.4c)

L , „ ( r ) =/;;.,„

at r = 0.

(4.4d)

The integral coefficients that appear throughout the integral transformation process are solved for analytically, see Alves (2000), by making use of the Mathematica platform, see Wolfram (1999). Only the truncated version of such system can be actually solved. However, the direct truncation of the involved summations in this system at a specified order may not be efficient for computational purposes, since this approach can include some terms that are not important in the convergence criteria that meets the user prescribed accuracy. A way of avoiding this difficulty is to transform the double summations in single ones according to an appropriate reordering scheme, such as shown in Correa et al. (1997) and Cotta and Mikhailov (1997) for multidimensional eigenfunction expansions. Here, the criteria adopted for the ordering procedure involves the summation of the squared eigenvalues in each direction. The transformed potentials ^ij (r) and 9m,n{'^), obtained from the solution of equations (4.4), are applied to the inversion formulae written below, with the appropriate reordering scheme, see Cotta (1993, 1998) and Cotta and Mikhailov (1997), to rebuild the original potentials: oo

oo

^{X^Z^r) = X]^$,(X)$*(Z)¥,,,(r) i=i j = i N

=^

__

$,(,) (X)#;(fc) {ZWm,m

(r),

(4.5a)

k=l CO

Q{x, Z,T)

oo

_

= X + J2J2 Tm{x)r„{z)em,n{r) m = l n=0 M

_

= X + J2^^ii)iX)K(i)iZ)'Sm(i)Mi)i^).

(4.5b)

1=1

where $i(X), $*(Z), Tm{X) and r*(Z) are the eigenfunctions obtained from the eigenproblems used as the basis for the integral transformation, see Alves (2000) and Alves and Cotta (2000), and A^, M are the user selected truncation orders for the reordered eigenfunction expansions. System (4.4) is then numerically solved through specific routines for initial value problems with stiff characteristics, such as the NDSolve function in the Mathematica system, see Wolfram (1999), offering an automatic accuracy control scheme. For computational purposes, the expansions are then truncated to afinitenumber of terms, so as to reach the user requested accuracy target in the final solution. For the special case when the transient term in the Darcy model is disregarded, the transformed streamfunction potentials are

R. M. COTTA ET AL.

103

explicitly obtained as follows:

^iAr)

fu + E

= -f^ 'i "•"

J

E AiJ,mJir)

+- i ^ E

m = l 77-n m=l n=0

\

J/

''t '« ^ """

E

Bij,mJ{T).

m = 1 n.=n J3 TO=1 n=0

(4.6) Once the transformed problem is numerically solved, quantities of practical interest are readily computed, such as local and average Nusselt numbers: Nu(Z,r)=^®(^'^'^) dX

(4.7a) 1

Nuav(r) = - ^ |

/^^ 9 0 ( X , Z , r )

m{Z,r)dZ=j^l

ax

x=i

dZ.

(4.7b)

The average Nusselt number can be alternatively computed from the integral energy balance as follows: Nuav(r) M

1^

/

X

iV

M

1 / V^ ^* dgm(/),n(/)(^) "= 1 - ;^ ( Z l ^^* "^ dr dr ^ "^ XI I ] (^^,^ ~ ^h) ^i{k),j{k)Om{i)Mi) ^=1 fc=i /=i (4.7c) The details on the mixed symbolic-numerical implementation of the proposed approach are presented in Alves and Cotta (2000), including the automatic derivation of all the above cited steps in the integral transformation process and the numerical implementation of the transformed system solution using the built in routine NDSolve, see Wolfram (1999).

4.3

THREE-DIMENSIONAL PROBLEM

Transient three-dimensional natural convection in a box-like cavity filled with a porous material saturated with a Newtonian fluid is considered. The flow is buoyancy induced by heat exchange between the fluid-porous media and the impermeable walls. The configuration considered, see Figure 4.2, is known as the horizontal cavity problem, in which boundary conditions of the first kind, To and To + AT, respectively, are imposed at the top and at the bottom walls, whilst the vertical walls are insulated. Within the validity of Darcy's model, and after invoking the Boussinesq approximation, this problem in terms of the vector potential formulation and in dimensionless form, is written as, see Luz Neto (2000) and Luz Neto et al. (2002): 86 R a — -h V^IIJX = 0 , oy -^^^Z + ^^y dx

= 0'

0 < x < M a : , 0<2/<M2/, 0 < z < l , r > 0 ,

(4.8a)

0 0 ,

(4.8b)

104

INTEGRAL TRANSFORMS FOR NATURAL CONVECTION IN CAVITIES

My

Figure 4.2 Geometry and coordinates system for natural convection in a threedimensional porous cavity.

de _de_dii;y

de_d^ de_

dr

dy dz

dx dz

dz

dx

dy

Q<x <Mx,

- ^ = ^ = 0 , ox - ^ ^'0=0, ox dy - ^

7r- = 0 on x = 0, ox T - = 0 on X = Mx, ox

=IIJ^ = 0,

i/;^='0^ = O, 6=0

^T- = ^

d^e d^e dy'^ dz'^ ' 0
r>0, (4.8c)

0
(4.8d)

0
(4.8e)

0< z <1,

(4.8f)

0
0 <x < Mx,

dy

dy il)^ =il;y -0^

d^o dx'^

o^

dy 6 = 1 on 6 = 0 on

0 < x < Mx

z = 0,

0 <x < Mx,

0
(4.8h)

z = 1,

0 <x < Mx,

0
(4.81)

at t = 0,

, 0 < z <

1,

(4.8g)

y = My,

0<x<Mx,0
(4.8j)

Once more, following the ideas in the solution methodology through the GITT approach, in order to make the boundary condition ^ = 1 on z = 0 in equation (4.8h) homogeneous, and thus enhance the computational performance of the eigenfunction expansions, a filter for the temperature field as before, which is based on the solution of the steady-state pure heat conduction problem, is proposed: 6F{Z) =

1-Z,

(4.9a)

R. M. COTTA ET AL.

(9(x, y,z,T)

= l-z-\-

105

enix, y, z, r ) .

(4.9b)

Equation (4.9b) is now employed by substituting it into equations (4.8), so that the governing equations for the filtered potential 9H are obtained, as for the two-dimensional case. The next natural step in this solution methodology consists of the choice of the appropriate eigenvalue problems, which will provide the basis for the expansions of the original potentials, i.e., the components of the vector potential and the temperature field. Following previous developments, the eigenvalue problem for the V^a;-component of the vector potential in the x-direction is chosen as follows: ^ - ^ 4 ^ -f aJAi{x) =0, '

^ ax

= 0,

^

(4.10a)

0<x<Mx, ^ ax

= 0,

(4.10b)

with the respective solution for the eigenfunctions Ai{x): - . . [1, Mx) = { , , I cos[aiX),

ai = 0, i = 0, , ai = in/Mx , z = 1,2,3,... ,

(4.10c)

the norm or normalization integral NAi is obtained from NAi= r^" A^,{x)dx=^{^''] Jo ' \MX/2,

'"^' i::. 1 , 2 , 3 , . . . ,

the normalized eigenfunction is defined as Ai{x) — Ai{x)ly/NAi symmetric kernel in the integral transform pair.

(4.10d)

which will provide a

Similar choices are made in the y- and 2;-directions, furnishing the following eigenquantities for this component of the vector potential, respectively: 5„(,) = ^ ^ , y/NBm NBm^

/

^. = ^ , My

m = l,2,3,...,

(4.11a)

B^(2/)d2/=—^,

m = l,2,3,...,

(4.11b)

and C,{z) = NCg = j

^^^^^,

j.^qTT,

C'^{z)dz=-,

9 = 1,2,3,...,

(4.12a)

q = 1,2,3,....

(4.12b)

The eigenfunctions, eigenvalues, orthogonality properties and norms associated with the V't/-component of the vector potential are given, respectively, by:

106

INTEGRAL TRANSFORMS FOR NATURAL CONVECTION IN CAVITIES

x-direction:

/•^^ ~

NDi=

Mx

Dl{x)dx

= -;^, 2

1 = 1,2,3,...,

(4.13b)

y-direction: fl/^/NE^, ^m(y) = < ^ I cos(eTO2/)/VA''£^m,

ATE™ = jT

e„=0,

m = 0,

€m=m-K/My,

m = 1,2,3,...,

Eiiy) dy = { Z' ',„

'" " : ' _

m = l,2,3,...

\My/2,

(4.14a)

(4.14b)

2:-direction: C,(z) = ^ i ^ ^ , /•I ^

7VC,-y

j,=qiT,

g = 1,2,3,...,

(4.15a)

^-1,2,3,....

(4.15b)

1

C,2(^)dz--,

For the temperature problem 6H, the eigenquantities are defined by the following equivalent vector potential auxiliary problems: x-direction—the same as for the x-direction of the -02;-component; ^/-direction—the same as for the y-direction of the V^^/-component; z-direction—the same as for the z-direction of the ipx- or ^/^j,-components. Following the formalism in the generalized integral transform technique, the triple transformations for the components of the vector potential and for the temperature field in the X-, y- and 2;-directions are obtained from the integral transform pairs below: ^a;-component: '^xima'^ Jo

/ Jo

/ Ai{x)Bm{y)Cq{z)ipx{x,y,z)dxdydz, Jo

transform, (4.16a)

CO

00

00

il)x{x,y,z) = X l X l ^M^)Bm{y)Cq{z)^^.^^,

inverse,

(4.16b)

i—O 771=1 q=l

V^2/-component: Tzr

pMx

'^yim,^ Jo

pMy

/ Jo

pi

/ Di{x)Em{y)Cq{z)il;y{x,y,z)dxdydz, Jo

transform, (4.17a)

10/

R. M. COTTA ET AL. oo

oo

oo

ipy{x,y,z) = Y,Y^

(4.17b)

inverse,

YjDii^)Em{y)C,{z)ipy.^^,

i = l 771=0

q=l

temperature field OH'rMx pMx

—

^Hirr.,(j)= I Jo C»

pMy pMy

rl pi

/ Jo

/ Jo

transform, (4.18a)

oo

CXD

^»(^) ^'"(^Z) '^«(^) ^^'-^ (^)'

OHix, y,z,T) = Y,Y.Yl i=0

Ai{x)Em{y)Cq{z)eH{x,y,z,T)Ax(\yAz,

771=0

i"^^""^^ •

(4.18b)

q=l

Applying the triple transformations (4.16a), (4.17a) and (4.18a) in the vector potential and in the temperature problems, given by equations (4.8) after filtering, results into an infinite coupled ODE system. Here also, the criteria adopted for the ordering procedure involves the summation of the squared eigenvalues in each direction as follows: a{ = ai+P'^+Yg

- ^f + C + 7 , - «i + 4 +7^

m" = n^(-A \ M x 2 + Mt/2 + q' (4.19a)

Then, the indices i, m and q are related to a single one i as follows: i = I{i),

m =

M(I)

,

(4.19b)

q = Q{i),

where I{x), M{x) and Q{x) are ordering arrays. Similarly, as the ordering criteria is the same, each set of indices j , n and r and k, p and s are, respectively, related to j and k as follows:

j = lCj).

n = M{j),

r = Q{j),

k = I{k),

p = M{k),

s = Q{k).

(4.19c) The associated triple summations are then rewritten as a single one according to equations (4.19) and the transformed ODE system is then rewritten in the more compact and computationally efficient form as follows:

dr

^ 3=1

6H. {r)+QieH, (r) = 0,

i = 1,2,3,... , (4.20a)

k=l

OHM

\^-^/2MxMy/-fg,

i ^ 0, m ^ 0, Vg, i = 0, m = 0, Vg,

(4.20b)

where (4.21a) k

108

INTEGRAL TRANSFORMS FOR NATURAL CONVECTION IN CAVITIES

af -t-e^ Q^ - df - R a - ^ - ^ y ^ ,

(4.21b)

M;.-/5.^.-^.^.'

(4-21C)

/64,,,-/6,,,-/4,,,,

(4.21d)

/3o = 0,

^0-0.

(4.21e)

System (4.20) is now in the appropriate format for numerical solution through dedicated routines for initial value problems, such as the subroutines DIVPAG or DIVPRK from IMSL, see IMSL Library (1989), which are well tested and capable of handling such situations, offering an automatic accuracy control scheme. For this computational purpose, the expansions are then truncated to NT terms, so as to reach the user requested accuracy target in the final solution. Remaining integral coefficients in equations (4.20) and (4.21) are defined in detail within Luz Neto (2000). From the solution of the system (4.20), the components of the vector potential and the temperature field, as well as other quantities of practical interest, are then readily obtained from the inversion formulae (4.16b), (4.17b) and (4.18b), in the form: Mx.y.z)

^^ e = - R a ^ -^A,(.)(x)B^(.)(2/)C^(^)(z)^i,,(r), NT

ijy{x,y,z)

—

= R a X ; ^%)(^)^„^(i)(2/)C,(i)(^)^/f, W .

(4-22b) _

ATT

0{x, y,z,T)

(4.22a)

= l - z + Yl M) (^)^m(i) (2/)C,(i) {Z)OH, ( r ) .

= l - z + eH{x,y,z,r)

(4.22c) We obtain for the overall Nusselt number: -i1

NT I .Mx

rMx PMX

pMy n

QQ

Aydx,

(4.23)

pMy

(4.24)

4.4

RESULTS AND DISCUSSION

This section illustrates some of the numerical results achieved from the computer codes constructed, both in the Mathematica system, see Wolfram (1999), for the two-dimensional

R. M. COTTA ET AL.

109

case, and in Fortran for the 3D configuration, executed under a Pentium III platform. The subroutine DIVPRK of the IMSL Library (IMSL Library, 1989) or the NDSolve function, see Wolfram (1999), were employed for the transformed system solution, always with a relative error target of 10~^. Before presenting a parametric analysis for the rectangular and parallelepiped cavity situations, aspects such as convergence analysis and covalidation are addressed. Tables 4.1(a) and (b) illustrate the convergence behavior of the overall Nusselt numbers, at steady state, for the two-dimensional situation with different values of Ra, for the vertical and horizontal cavities, respectively. Two sets of results are presented in each case, corresponding, respectively, to the direct integration of the average Nusselt number from the local expression, equation (4.20), and to the integral balance alternative, equation (4.21), when the wall temperature derivatives are obtained from integration of the energy equation. Convergence is examined for increasing truncation orders in the streamfunction {N) and temperature (M) eigenfunction expansions. It can be clearly observed that convergence to at least three significant digits, in the worst case presented, is achieved, with a noticeable advantage on convergence rates for the integral balance alternative Nuav2 • It is also evident that an increase in Ra has an effect of decreasing the convergence rates to a sensible extent, due to the increase in the convective source terms importance, but with converged results still readily obtainable within practical limits. Finally, the

Table 4.1 Convergence analysis of the average Nusselt numbers for a two-dimensional (a) vertical, and (b) horizontal, cavity (aspect ratio H = 1). (a) Ra = 50 N~M

9-12 25-30 49-56 81-90 121-132 169-182 225-240

Ra = 100

Ra = 200

NUavl

NUav2

NUavl

NUav2

NUavl

NUav2

1.9088 1.9545 1.9672 1.9744 1.9783 1.9784 1.9803

1.9287 1.9727 1.9823 1.9836 1.9839 1.9841 1.9841

2.6999 2.9011 2.9789 3.0204 3.0478 3.0511 3.0676

2.7707 2.9951 3.0878 3.1024 3.1075 3.1092 3.1104

3.5603 4.0952 4.3961 4.5296 4.6253 4.5985 4.6761

3.7134 4.3596 4.8172 4.9007 4.9374 4.9479 4.9604

(b) Ra = 100 N-M

9-12 25-30 49-56 81-90 121-132 169-182 225-240 Caltagirone (1975)

Ra = 200

Ra = 300

NUavl

NUav2

NUavl

NUav2

NUavl

NUav2

2.6127 2.6441 2.6457 2.6459 2.6459 2.6459 2.6459

2.6215 2.6461 2.6459 2.6459 2.6459 2.6459 2.6459

3.5383 3.8005 3.8016 3.8085 3.8095 3.8097 3.8098

3.5675 3.8220 3.8044 3.8093 3.8098 3.8098 3.8098

3.9440 4.4482 4.6054 4.8806 4.9480 4.9826 4.9791

3.9870 4.6651 4.7035 4.9419 4.9772 4.9848 4.9888

2.651

3.813

4.523

no

INTEGRAL TRANSFORMS FOR NATURAL CONVECTION IN CAVITIES

last row in Table 4.1(b) offers a comparison against the finite difference results reported in Caltagirone (1975). While the agreement for R a — 100 and 200 is indeed quite satisfactory, around 0.2-0.3%, the marked deviation for R a = 300 suggests that a major difference in the cellular structure within the cavity should be occurring in this case. In fact, after observing more closely the transient behavior of the horizontal cavity (in this specific case), we note that our present simulations lead to a single cell structure for the first two cases, R a = 100 and 200, but result in a three cell arrangement for R a = 300, which explains the difference from the single cell solution of Caltagirone (1975). Furthermore, our three cell solution for R a = 300 agrees very well, 0.3% relative error, with the one presented by Figueiredo and Llagostera (1999). They provided average Nusselt number results for one, three and five cell structures obtained from the steady version of Darcy law using a fully numerical finite volume formulation. For each Rayleigh number fixed in their simulations, solutions for each one of these cell structures were shown. The reason for such multiple solution behavior being that, in order to avoid convergence problems, the solution at a smaller Rayleigh number was used as initial condition for the simulation at the next higher Rayleigh number. This procedure conditioned the solution to the same cell structure for a wide range of Rayleigh numbers. The present hybrid approach is not constrained by such issues and is able to provide the stable cell structure dependence on the Rayleigh number. This change in cell structure is illustrated in Figure 4.3, where the steady-state local Nusselt number behavior for different values of the Rayleigh number is presented. One can clearly see that a transition from one to three cells in the cellular pattern takes place at 200 < R a < 300. In order to further investigate this matter, we also solved the steady Darcy law model along with the transient energy equation using the current hybrid methodology. It was found that this steady model is not able to properly predict the transient behavior of the

Ra Ra Ra Ra

= 300 = 200 = 100 = 50

\ ** ^ \ ^^^

/

7 62

-

- ^\

5

\ 1 4~

\ "

0

A

1

1

1

0.2

1

1

^ ^

1

"^"^

^ ^-^ 1

0.4

\ N

•'^.

/ 1

\

^•^.

/

\ "^

\

L. _ l

\

/

^1 [\

\

/ ""^ "/"•-

\

^3 2

\

^^ /

y-.^

'.

/

1

1

^

1

^ ^

1

^'^

«^

1

0.6

1

1

\ \

"'^1

^ N ^

1

1

\

"^

0.8

\

^

1

\

1

1

\

1

1

1

Figure 4.3 Steady-state local Nusselt number behavior for different values of Ra (2D cavity, H = 1).

Ill

R. M. COTTA ET AL.

Nusselt number. Since others have solved the same set of equations with fully numerical techniques, see Figueiredo and Llagostera (1999), we conjectured that the numerical error was the agent responsible for triggering the transition from a conductive type solution to the proper convective one, see Alves (2000). With the goal of providing further evidence to support this assumption, we introduced a small source term G in the steady Darcy law to model this numerical error and solved it using the present hybrid methodology. These transient Nusselt number results obtained for Ra = 200 and H = 1 art presented in Figure 4.4. Figure 4.4 shows that the transition to a convective solution depends strongly on the magnitude of the small parameter G. The proper transient behavior obtained from solving the fully transient equations through the present technique is shown in Figure 4.5. In this figure, the transient behavior of the average Nusselt number in the horizontal twodimensional cavity for different values of the Ray leigh number is presented, demonstrating the transition from the initial conductive regime towards the convective behavior, passing through an expected oscillatory region. The importance of the time derivative coefficient was also evaluated for the vertical cavity. Aspect ratio values equal io H = 0.5 and 2, Rayleigh numbers equal to Ra = 50, 100 and 200, and time derivative coefficient values equal to x = 10^ and lO'^ were used and results presented in Alves and Cotta (2000). A graphical adherence between the two models results is observed for all cases except for fl" = 2, Ra = 200 and x — 10^» which deviates only slightly from the other results, indicating that the parameter % can be normally discarded from the mathematical model in this case. For the three-dimensional situation, since again the average Nusselt number convergence was quite favorable, we selected to illustrate the convergence behavior through the evolution of the temperature at the cavity center, with Ra == 200. The temperature expansions were already expected to result in slower converging series in comparison with the aver-

6rF

5h

A-

r (Don

\ \

t 3H

I\

1

1

\/

'

! !

i i

1

1

!

!

{

(-^10-10 G = 10-1^

.'

1

G=0

;vj ;' 1

i

\

\/ J'

A

0

W

1 \ ''r*^^ ! \/

1

1

0.2

•\- — — •'•U 10^

A

G = io-^

i

J

1

1

1

1

1

0.4

1

1

1

1

1

0.6

1

1

1

1

1

1

1

1

1

1

0.8

Figure 4.4 Illustration of the numerical error effect on the transient average Nusselt number behavior (horizontal 2D cavity, Ra = 200 and H = 1).

112

INTEGRAL TRANSFORMS FOR NATURAL CONVECTION IN CAVITIES

Ra = 300 Ra = 200 Ra = 100 Ra^50

il.75

]l.5 _R_a_=5q_,,^^

1.25 1 10.75 '0.5

0.1

0.2

0.4 0.3 0.5 0.6 r Figure 4.5 Illustration of the average Nusselt number transient behavior for different values of Ra.

age Nusselt number expressions, which result from an analytical integration acceleration scheme. However, according to Table 4.2, the temperature convergence is also still quite reasonable, with at least three converged digits in the worst case presented, for truncation orders A^ « 300, with the underlined digits not accurate. It may also be observed that the worst convergence rates occur within the oscillatory time range for each physical position. Once the convergence behavior for the three-dimensional case has been illustrated. Table 4.3 brings an ample comparison of the steady-state average Nusselt numbers for different values of Ra in the cubic cavity. Several authors were considered in this covalidation

Table 4.2 Convergence analysis for the temperature at the cavity center (3D case, Ra = 200). T

TV = 111

N = 181

N = 251 AT = 301

0.05

0.113839 0.261620 0.247844 0.182054 0.200531 0.233117 0.249555 0.284527 0.301980 0.308412 0.310749 0.311607 0.311833 0.312108

0.113839 0.261620 0.247936 0.0822149 0.0939460 0.130204 0.154618 0.218987 0.251359 0.263501 0.267957 0.269597 0.270028 0.270554

0.113839 0.261620 0.247928 0.129088 0.145053 0.178675 0.198719 0.248412 0.274090 0.283871 0.287486 0.288820 0.289172 0.289601

0.1 0.14

0.2 0.24

0.3 0.34 0.44 0.54 0.64 0.74 0.84

0.9 2.0

0.113839 0.261620 0.247923 0.129054 0.145127 0.178260 0.198695 0.249564 0.273974 0.283349 0.286889 0.288621 0.289004 0.289403

R. M. COTTA ET AL.

113

Table 4.3 Covalidation of overall Nusselt number in cubic cavity. RI

A

B

C

D

E

F

G

50 1.3999 60 1.7201 75 2.1641

1.16 -

-

1.5^^ -

_ -

_ -

-

-

-

-

100 2.6459

2.66

-

2.6^Cf

2.6512D

_ 2.143D 2.252D 2.82D 2.93D

120 2.9491 150 3.7132 3.69 200 4.4740 4.48

-

31RG 3'7RG

45RG

250 300

5.8^«

5.0473 5.11 5.7539 5.66

5.144

3.3222D 3.3082D 44973D

4.41 4.42 5.1043D 4.95-5.0 5.30-5.43 4.5142D 5.6423D

H~ _ 1.67

2.6512

-

3.4-3.94

4972D 6.453D

5.0162D

RG-graphical reading, 2D-two-dimensional, 3D-three-dimensional A-present, B - Stamps er a/. (1990), C -Kimura e/a/. (1989), D-Strauss and Schubert (1981), E-Schubert and Strauss (1979), F-Home (1979), G-Caltagirone (1975), H-Hoist and Aziz (1971)

exercise, and a few of these results had to be obtained by direct graphical readings. Also, depending on the final flow structure either two or three-dimensional, the authors present different values for the Nusselt number. As a whole, the present computations seem to be in good agreement with the few available benchmarks. In light of the accuracy control capabilities of the present hybrid approach, it is hoped that this set of results may find usefulness in future covalidations. The influence of the aspect ratio on the heat transfer rates for the three-dimensional cavity was more closely studied by evaluating the overall Nusselt number and the temperature distributions for the specific case of Ra = 100. Different cavities were evaluated with characteristic volumes—Mx My—varying from 0.25 to 9, namely: 0.25,0.5, 0.64, 0.81, 1,2,3,4,6,8 and 9. Table 4.4 presents the numerical values of the overall Nusselt number, as obtained from equation (4.24), as well as the convective regime observed. For a better understanding of the aspect ratio effect on the convective process. Figure 4.6 shows the transient evolution of the characteristic overall Nusselt number (defined as the overall Nusselt number scaled with the volume cavity Mx My and used for visualization of the different curves) for cavities with characteristic volume greater than 1, for a unit aspect ratio in the x-direction, Mx = 1. It can be observed that cavities with high shape factor— here considered as the ratio My/Mx—present a wider transition regime, associated with the formation, movement and accommodation of convective cells. From Table 4.4, it may also be verified that cavities with the same characteristic volume, although with different regime, have presented fairly similar values for the overall Nusselt number at steady state, as for instance for the case of a characteristic volume equal to 3. It may also be noticed that the overall Nusselt numbers vary within a quite limited range

114

INTEGRAL TRANSFORMS FOR NATURAL CONVECTION IN CAVITIES

Table 4.4 Steady overall Nusselt numbers for different aspect ratios—Ra =100. Mx My

Mx

My

My/Mx

Nu

NT

Regime^

0.25 0.50 0.64 0.81

0.5 0.5 0.8 0.9

0.5 1.0 0.8 0.9

1.00 2.00 1.00 1.00

2.136 2.646 2.721 2.673

251 181 251 251

2D-1C/XZ 2D-\CrYZ 3D-1C 2D-1CA^Z

1.0

0.5 1.0

2.0 1.0

4.00 1.00

2.588 2.646

391 241

2D-3CrfZ 2D-1CA^Z

2.0

0.5 1.0

4.0 2.0

8.00 2.00

2.670 2.585

3.0

0.5 1.0 1.5

6.0 3.0 2.0

12.0 3.00 1.33

4.0

0.5 1.0 2.0

8.0 4.0 2.0

16.0 4.00 1.00

2.616

391

3D-Trans**

6.0

1.0 1.2 1.5 2.0

6.0 5.0 4.0 3.0

6.00 4.17 2.67 1.50

2.609 2.638 2.607 2.537

241 221 391 391

2D-6CA^Z 2D-6CA^Z* 2D-2C/XZ 2D-3C/XZ

8.0

1.0 1.6 2.0 2.5

8.0 5.0 4.0 3.2

8.00 3.13 2.00 1.28

2.641 2.630 2.633 2.552

271 291 211 391

2D-7CA^Z* 2D-2C/XZ 2D-3C/XZ 3D-Trans**

9.0

1.0 1.5 1.8 2.0 2.5 3.0

9.0 6.0 5.0 4.5 3.6 3.0

9.00 4.00 2.78 2.24 1.44 1.00

2.640 2.609 2.614 2.633 2.546 2.581

271 401 261 221 391 391

2D-8CA^Z* 2D-2C/XZ 3D-Trans** 2D-3C/XZ 3D-Trans** 3D-Trans**

2D-5cnrz 2D-3CA^Z 2.655 301 2D-8CA^Z 2.655 271 2D-4CA^Z 2.655 271 2D-2C/XZ 2.672 391 2D-8CA^Z* 2.627 271 2D-5CA^Z 221 391

^ Regime - two-dimensional (2D) or three-dimensional (3D), number of cells and convection plane for two-dimensional flow * -stable, still transient, ** -non-stable-fully transient

from 2.537 to 2.713, except for the cavity with characteristic volume 0.25 (Mx — 0.5 and My = 0.5) which resulted in a value of 2.136. For the sake of illustration of this aspect, Figure 4.7 summarizes the various values here obtained for this parameter, identified over isovolume lines. With respect to the evolution of the convective processes, for the cases in which the steady state has been achieved within the range of the dimensionless time considered, in only one case, namely Mx = 0.8 and My = 0.8, was a three-dimensional convective pattern observed. In all other situations the stable regimen was observed to occur under a twodimensional pattern, with flow characterized by the formation of one to eight convective cells.

R. M. COTTA ET AL.

— I — I — I — I

I

115

I

Dimensionless time Figure 4.6 Effect of the aspect ratio on the transient characteristic overall Nusselt number behavior for Mx = 1 and My = 2, 3, 4, 6, 8 and 9 when Ra = 100.

According to the methodology here employed, the spectral solutions obtained need to be truncated to a finite order, and as previously discussed, the optimum criteria for ordering the multidimensional information into a single series is somehow part of the solution process. In this sense, it is not desired to include terms that have no participation in the final numerical results, even not to overload the computational process, but on the other hand the abandoned terms have to be in fact negligible for the desired confidence on the converged results. In principle, since the actual solution is not a priori known, we have no way of guaranteeing that a simple reordering scheme based on the relative magnitudes of the squared eigenvalues in each direction, will be safe and effective for the whole domain in the time variable. This might be particularly unlikely for problems that suffer significant source term variations along the integration march, such as the present one, with the triggering of convective motions within the different configurations considered. Therefore, special care is to be exercised in the analysis of the expansions convergence, which in general requires a dynamical reordering procedure, changing the number and choice of terms along the integration process. Figure 4.8 illustrates such observation, showing the convergence behavior for the characteristic overall Nusselt number evolution in two different cavities, of characteristic volumes 8 and 9, namely Mx = 2.5, My = 3.2 and Mx = 2.5, My = 3.6, and fairly different truncation orders. A less careful implementation of the algorithm, for the present reordering scheme choice, might have concluded that the lower order results were satisfactory, essentially guided by a partial convergence pattern. However, the behavior from the actually converged situation is quite marked, with differences even in the respective steady states. Clearly, the present analysis leads to the conclusion that more robust and automatic reordering schemes are to be devised, for a reliable general purpose utilization of this approach.

116

INTEGRAL TRANSFORMS FOR NATURAL CONVECTION IN CAVITIES

Characteristic volumes • Mx My = 0.25 * MxMy = 0.5 A Mx My = 0.64 - -•• - Mx My = 1 --XMxMy = 2 - ^ - MxMy = 3 - -D - Mx My = 4 --^Mx My = 6 - -O- - MxMy = S - H- - MxMy = 9

-^2.640

C? 2.672

^2M]

<^2.6^^

^ 2.609 \

^2.609

^2.638

\ \2.630v \ "SI

H-2.614 X2.633

g.4-

>^ 2.670

H 2.627

V2.607 --K2^546

>&.^2.655

2H

it2.588

^X 2.585

• 2.646

.^•2.646 A ^2.673 2-721

• 2.136

1

Figure 4.7

^^•
^"02.655

•^O 2.5?2" - ,

^+2.581

"^2.616

Aspect ratio, Mx

Effect of the aspect ratio on the behavior of the overall Nusselt number.

25 n

Mx = 2.5, My = 3.6

B

Mx = 2.5, My = 3.2

•S 20H CO

w D

•A

A—AVA/A A A ^

§15 Tmncation orders • - NT = 391 e— NT = 161 A - ATT = 151

10 10"

10° Dimensionless time

^

Figure 4.8 Effect of the truncation order on the characteristic overall Nusselt number behavior when Ra = 100.

R. M. COTTA ET AL. 4.5

117

CONCLUSIONS

A hybrid numerical-analytical approach is proposed for the solution of transient two or three-dimensional coupled heat and fluid flow in natural convection within porous media, based on the ideas of the generalized integral transform technique and using the streamfunction-only or vorticity-vector potential formulations. The convergence behavior in predicting the transient heat transfer quantities is discussed, and validations of the proposed algorithms are undertaken. The constructed codes allowed for the qualitative analysis of the transient phenomena in terms of the Rayleigh number and aspect ratios. Another interesting feature of the present contribution is that all the required analytical steps are symbolically computed from within the Mathematica system, see Wolfram (1999), and are readily mixed with numerical computations and graphical representations.

ACKNOWLEDGEMENTS The authors would Uke to acknowledge the partial financial support provided by CNPq, FAPERJ, PRONEX and CAPES, all of them sponsoring agencies in Brazil.

REFERENCES Alves, L. S. de B. (2000). Stability analysis of transient natural convection in porous cavities: hybrid numeric-analytical solution and mixed symbolic-numerical computation. M.Sc. thesis. Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil. In Portuguese. Alves, L. S. de B. and Gotta, R. M. (2000). Transient natural convection inside porous cavities: hybrid numerical-analytical solution and mixed symbolic-numerical computation. Numer. Heat Transfer, Part A 48, 89-110. Alves, L. S. de B., Gotta, R. M., and Pontes, J. (2001). Stability analysis of natural convection in porous cavities through integral transforms. Int. J. Heat Mass Transfer ^S, 1185-95. Aziz, K. and Heliums, J. D. (1967). Numerical solution of three-dimensional equations of motion for laminar natural convection. Phys. Fluids 10, 314-24. Baohua, G. and Gotta, R. M. (1993). Integral transform analysis of natural convection in porous enclosures. Int. J. Numer Meth. Fluids 17, 787-801. Galtagirone, J. P. (1975). Thermoconvective instabilities in horizontal porous layer. J. Fluid Mech. 72, 269-87. Gorrea, E. J., Gotta, R. M., and Orlande, H. R. B. (1997). On the reduction of costs in eigenfunction expansions for multidimensional diffusion problems. Int. J. Numer. Meth. Heat Fluid Flow 7, 675-95. Gotta, R. M. (1993). Integral transforms in computational heat and fluid flow. GRG Press, Boca Raton, PL. Gotta, R. M. (ed.) (1998). The integral transform method in thermal &fluidssciences & engineering. Begell House, New York.

118

INTEGRAL TRANSFORMS FOR NATURAL CONVECTION IN CAVITIES

Gotta, R. M. and Mikhailov, M. D. (1997). Heat conduction—lumped analysis, integral transforms, symbolic computation. Wiley, England. Figueiredo, J. R. and Llagostera, J. (1999). Comparative study of the unified finite approach exponential-type scheme (unifaes) and its application to natural convection is a porous cavity. Numer Heat Transfer, Part B 35, 347-67. Hoist, P. H. and Aziz, K. (1971). Transient three-dimensional natural convection in confined porous media. Int. J. Heat Mass Transfer 15, 73-99. Home, R. N. (1979). Three-dimensional natural convection in a confined porous medium heated from below. J. Fluid Mech. 92, 751-66. IMSL Library (1989). MATH/LIB. Houston, TX. Ingham, D. B. and Pop, I. (eds) (2002). Transport phenomena in porous media. Vol. II. Pergamon, Oxford. Kakag, S., Kilkis, B., Kulacki, F. A., and Aring, F. (eds) (1990). Convective heat and mass transfer in porous media. Kluwer, Dordrecht. Kaviany, M. (1995). Principles of heat transfer in porous media (2nd edn). Springer-Verlag, New York. Kimura, S., Schubert, G., and Straus, J. M. (1989). Time-dependent convection in fluid-saturated porous cube heated from below. J. Fluid Mech. 207, 153-89. Kladias, N. and Prasad, V. (1989). Natural convection in horizontal porous layers: effects of Darcy and Prandtl numbers. J. Heat Transfer 111, 926-35. Leal, M. A. and Gotta, R. M. (1997). Steady and transient integral transform solutions of natural convection in enclosures. In Proceedings of the ICHMT international symposium on computational heat transfer, Turkey, May, pp. 418-32. Begell House, New York. Leal, M. A., Guerrero, J. S. Perez, and Gotta, R. M. (1999). Natural convection inside twodimensional cavities: the integral transform method. Comm. Numer Meth. Eng. 15, 113-25. Leal, M. A., Machado, H. A., and Gotta, R. M. (2000). Integral transform solutions of transient natural convection in enclosures with variable fluid properties. Int. J. Heat Mass Transfer A'^, 397790. Luz Neto, H. (2000). Transient three-dimensional natural convection in porous media—hybrid solutions via integral transformation. D.Sc. thesis. Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil. In Portuguese. Luz Neto, H., Quaresma, J. N. N., and Gotta, R. M. (2002). Natural convection in three-dimensional porous cavities: integral transform method. Int. J. Heat Mass Transfer AS, 3013-32. Luz Neto, H., Quaresma, J. N. N., and Gotta, R. M. (2004). Transient natural convection in threedimensional rectangular porous cavities: reference results via integral transforms. In Proceedings of the international conference on applications in porous media (IGAPM2004), GGE-UE, Evora, Portugal, 24-27 May (eds A. H. Reis and A. F. Miguel), pp. 165-72. Machado, H. A., Leal, M. A., and Gotta, R. M. (1999). A flexible algorithm for transient thermal convection problems via integral transforms. In Proceedings of the international symposium on computational heat and mass transfer. North Cyprus, Turkey, April, pp. 13-31. Mikhailov, M. D. and Ozi§ik, M. N. (1984). Unified analysis and solutions of heat and mass diffusion. Wiley, New York.

R. M. COTTA ET AL.

119

Misra, D. and Sarkar, A. (1995). A comparative study of porous media models in a differentially heated square cavity using afiniteelement method. Int. J.Numer. Meth. Heat Fluid Flow S,135-51. Nield, D. A. (1991). The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface. Int. J. Heat Fluid Flow 12, 267-72. Nield, D. A. and Bejan, A. (1999). Convection in porous media (2nd edn). Springer-Verlag, New York. Prasad, V. and Kulacki, F. A. (1984). Convective heat transfer in a rectangular porous cavity—effect of aspect ratio on flow structure and heat transfer. J. Heat Transfer 106, 158-65. Quaresma, J. N. N. and Cotta, R. M. (1997). Integral transform method for the Navier-Stokes equations in steady three-dimensional flow. In Proceedings of the 10th international symposium on transport phenomena, Kyoto, Japan, November, pp. 281-7. Rajamani, R., Srinivas, C , and Seetharamu, K. N. (1990). Finite element analysis of convection heat transfer in porous media. Int. J. Numer Meth. Fluids 11, 331-9. Schubert, G. and Straus, J. M. (1979). Three-dimensional and multicellular steady and unsteady convection in fluid-saturated porous media at high Rayleigh numbers. J. Fluid Mech. 94, 25-38. Stamps, D. W., Arpaci, V. S., and Clark, J. A. (1990). Unsteady three-dimensional natural convection in a fluid-saturated porous medium. /. Fluid Mech. 213, 377-96. Straus, J. M. and Schubert, G. (1979). Three-dimensional convection in a cubic box of fluid-saturated porous material. / Fluid Mech. 91, 155-65. Straus, J. M. and Schubert, G. (1981). Modes of finite-amplitude three-dimensional convection in rectangular boxes of fluid-saturated porous material. J. Fluid Mech. 103, 23-32. Whitaker, S. (1966). The equations of motion in porous media. Chem. Eng. Sci. 21, 291-300. Wolfram, S. (1999). The Mathematica book (4th edn). Wolfram Media, Cambridge.

5

A POROUS MEDIUM APPROACH FOR THE THERMAL ANALYSIS OF HEAT TRANSFER DEVICES S. J. KIM and J. M. HYUN Department of Mechanical Engineering, KAIST, Daejeon 305-701, South Korea email: sungjinkimQkaist.ac.kr and jmhyunSkaist . a c . k r

Abstract In this chapter, a method for modeling transport phenomena in heat transfer devices is discussed. The modeling technique is 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 or numerical determination of these coefficients. The averaging method in turn yields analytical solutions for the velocity and temperature distributions which are useful in the thermal analysis of heat transfer devices. The modeling technique is elucidated for thermal design and optimization of microchannel heat sinks and internally finned tubes. By way of these case studies, the method is shown to be a promising tool for thermal analysis and optimization of heat transfer devices. Keywords: averaging method, porous medium approach, microchannel heat sink, internally finned tube, thermal optimization

5.1

INTRODUCTION

Transport and interactions of matter in a microstructure always involve fluid flow and/or exchange of energy. Heat is transferred through irreversible transport phenomena, as can be inferred from the second law of thermodynamics, see Tien et al. (1998). There are many micro-mechanical systems for which thermal aspects are critical. These include microstructures encountered in biological reactions and processes, fuel cells, highperformance heat exchangers and cooling devices, and chemical processes, to name a few, see Ernst et al. (2002) and Chen et al. (2003). Therefore investigators have used experimental and numerical means to determine fluid flow and heat transfer characteristics of microsystems. Various numerical models have been put forth, but there exist limitations in the simulation methods. Modeling can be cumbersome due to the complex topology 120

S. J. KIM AND J. M. HYUN

121

of microstructures and complicated models for a microstructure require tedious effort and computational time. Therefore, it is highly desirable to develop a simple, but accurate, modeling method in micro-mechanical engineering. For most engineering purposes, detailed information, such as velocity and temperature distributions, is of secondary importance. Instead we are usually concerned with estimating the macroscopic aspects of a problem. Examples are the average pressure drop across a channel or average heat transfer rate from a wall. If our main interest lies in the macroscopic quantities, it is wise to average the governing equations in the direction normal to the flow. This approach can lead to considerable simplification, especially when the original problem requires a great deal of time and money for a complete solution. The simplified model enables us to identify the parameters of importance and study the effects of these parameters on the performance of heat transfer devices. For this reason, the simplified model can be effective in the thermal optimization of heat transfer devices. Many types of averages are used in engineering and these include the arithmetic mean, geometric mean, harmonic mean, logarithmic mean, and mean values of integration, etc. When a macroscopic quantity can be treated as a continuous function of the independent variable(s), the mean values can be obtained by performing integration over a certain range of the independent variable(s). The mean values of integration are classified into time averaging, line averaging, area averaging, or volume averaging, etc., depending on the independent variable(s) for which the integration is performed. Time averaging encountered in the study of turbulence and volume averaging used for flow through a porous medium are common examples of averaging methods (Ingham and Pop, 1998; Nield and Bejan, 1999). In the following sections, a modeling technique for microstructures and finned tubes is introduced. The modeling technique based on the averaging method is described, which can be used for microstructures as well as regular-sized structures. Since a heat transfer device is modeled as a fluid-saturated porous medium in this technique, it is frequently called a porous medium approach. The porous medium approach that we are going to describe has evolved out of the original work of Koh and Colony (1986). This approach is different from the other porous medium approaches in that it is based on the two-equation model for energy transport, which treats the solid phase and the fluid phase as separate entities. The two-equation model is indispensable for a thermal analysis of heat transfer devices because there exists a large difference between the thermal conductivities of the fluid and solid phases. A specific procedure for the porous medium approach with the two-equation model is elucidated by applying the procedure to a couple of heat transfer devices. A microchannel heat sink and an internally finned tube are chosen as case studies to emphasize the merits of the method in thermal design and optimization of heat transfer devices.

122 5.2

THERMAL ANALYSIS OF HEAT TRANSFER DEVICES

THERMAL ANALYSIS OF MICROCHANNEL HEAT SINKS

The trend in the electronics industry towards denser and more powerful products leads to increased power dissipation of electronic components. For reliable operation of these products, effective cooling technology is essential. A variety of cooling methods have been proposed, including the microchannel heat sink. The concept of the microchannel heat sink was introduced by Tuckerman and Pease (1981). It is based on the idea that the heat transfer coefficient is inversely proportional to the hydraulic diameter of the channel. Subsequently, a body of research has been reported for the microchannel heat sink, as summarized in the extensive reviews by Phillips (1990) and Goodling and Knight (1994). The problem under consideration concerns forced convection through a microchannel heat sink and a schematic of a microchannel heat sink is shown in Figure 5.1. The direction of the fluid flow is parallel to the x-axis. The top surface is insulated, and the bottom surface is uniformly heated. A coolant passes through the microchannels, and it takes heat away from the heat-dissipating electronic component attached below. In analyzing the problem, for simplicity, the flow is assumed to be laminar, incompressible, and thermally and hydrodynamically fully-developed, and all the thermophysical properties are assumed to be constant. In addition, the pumping power, i.e. the power required to drive the fluid through the microchannels, is assuined to be fixed. The simplest method for thermal analysis of a microchannel heat sink is to use a fin model. This is based on the assumptions of one-dimensional conduction along the fin height, a constant heat transfer coefficient, and a uniform fluid temperature. For example, Knight et al. (1991, 1992) presented a design method based on the fin model, which is widely used. Kim (2004) recently pointed out that the fin model does not predict the thermal resistance accurately when the aspect ratio of the microchannel is high. This is because the heat transfer coefficient is not constant along the periphery of the channel, and the fin efficiency based on the hyperbolic tangent profile is not valid when the heat transfer coefficient varies in the direction of the fin height. The difference between the

Figure 5.1 Schematic of a microchannel heat sink.

S. J. KIM AND J. M. HYUN

123

heat transfer coefficient along the fin height and that along the channel base increases with the increasing aspect ratio. Hence, the assumption of a uniform heat transfer coefficient, combined with incorrect values of fin efficiency, may result in errors. Then, what is the remedy? As an alternative, Koh and Colony (1986) suggested a model based on an averaging method. In this method, the microchannel heat sink is modeled as a fluid-saturated porous medium as shown in Figure 5.2, which is frequendy termed the porous medium approach. Mathematically, this is equivalent to averaging the velocity and temperature distributions in the direction perpendicular to the flow direction. This approach was applied to the microchannel heat sink by Tien and Kuo (1987) and later extended by Kim and his co-workers (Kim and Kim, 1999; Kim et al, 2000; Kim and Kim, 2004). The averaging direction is the direction of the shorter dimension, perpendicular to the flow direction. For example, the averaging can be taken in the ^/-direction for lowaspect-ratio channels and in the z-direction for high-aspect-ratio channels. The reason for this choice will be discussed later. We will first present the averaging method for high-aspect-ratio channels. 5.2.1

High-aspect-ratio microchannels

For high-aspect-ratio microchannel heat sinks, we need to define the quantities which are averaged. Since the present system has a periodic structure in the spanwise direction, the representative elementary volume (REV) for averaging can be visualized as a slender cylinder aligned parallel to the z-axis. Then, the averaged quantities over the fluid and

t t t t t t^Figure 5.2 Equivalent porous medium.

124

THERMAL ANALYSIS OF HEAT TRANSFER DEVICES

solid phases of the REV are defined, respectively, as follows:

Wc Jo

0d0,

{^r = —

(5.1)

(5.2)

(t>dz.

The governing momentum and energy equations for fully-developed flow are given by Tien and Kuo (1987):

dx

dy^ = ^

epfCf{u)f^ ^

(^- ^

)

Khigh

( ^ f c / e , h i g h ^ ) + /ii,highahigh {{TY - {T)f),

= /^'.highahigh {{Ty-

(5.4) (5.5)

(T^) ,

where 6, K, hi, a, kse, and fc/e are porosity, permeability, interstitial heat transfer coefficient, wetted area per volume, effective conductivity of the solid and effective conductivity of the fluid, respectively. The subscript 'high' is used to distinguish this case from the case of /ow-aspect-ratio microchannel heat sinks. For high-aspect-ratio microchannels, see Kim and Kim (1999), we have the following: flhigh =

2

—

,

fcse,high

= (1 - ^)ks ,

fc/e,high

= ^kf .

(5.6)

Equation (5.3) is the Brinkman-extended Darcy equation, which is the momentum equation for flow through porous media when the boundary effect cannot be neglected. Equations (5.4) and (5.5) are the energy equations based on the two-equation model, which treats the solid and the fluid as separate entities. Equation (5.4) is the energy equation for the fluid phase, and equation (5.5) that for the solid phase. The last terms in these two equations represent the thermal interaction between the two phases. The appropriate boundary conditions are as follows: (u)^-O,

{Ty = {Ty=T^

Oi^-0,

- ^ - - ^ = 0 dy dy

at

y = 0,

(5.7)

at y = H.

(5.8)

To solve the governing equations, (5.3)-(5.5), K and hi should be determined in advance. The permeability K is the measure of the flow conductance of the porous medium and it is related to the viscous shear stress caused by the fins. The interstitial heat transfer coefficient hi is the proportionality constant for the convective thermal interaction between the solid and fluid phases and related to the rate of heat transfer from the fins to the coolant. These parameters are critical parts of the porous medium approach. Successful implementation of the porous medium approach depends on the appropriate determination

125

S. J. KIM AND J. M. HYUN

of these two parameters. As pointed out by Slattery (1999), we lose some detailed information with any averaging approach—in the present case the dependence of the velocity and temperature distributions for the fluid phase in the averaging direction. In the porous medium approach, we replace the lost information with data for the permeability and the interstitial heat transfer coefficient. They are secured from either experimental investigations or numerical simulations. However, for the present configuration, these parameters can be determined analytically using an approximation devised by Kim and his co-workers (Kim and Kim, 1999, 2004). In order to explain how K and hi are determined analytically, let us begin with the classical momentum and energy equations: dp dx

(5.9)

pcu dx

(5.10)

respectively, and the boundary conditions u = V = 0, u = V = 0,

T = Tuj at y = 0,—oo
(5.11) (5.12)

Integrating equation (5.9) over the fluid region in the 2;-direction yields:

d((py

d^uY

1

dx

dy^

Wc

du /^/ ^z

f^f

du

(5.13) 2=0 >

The second term on the right-hand side of equation (5.13) represents the shear stress along the vertical solid-liquid interface. Upon comparing equation (5.3) and equation (5.13), we can express the permeability in terms of the velocity gradients at the solid walls as follows: -1 du ^high = -ewc{uy (5.14) 2=0 >

Similarly, averaging can be performed for the energy equation, which yields two equations: one for the fluid phase and the other for the solid phase. Integrating equation (5.10) over the fluid region produces:

. ^.d{Ty dx

, d''{Ty dy^

1 w.

,

dT

, dT

(5.15) z=0>

126

THERMAL ANALYSIS OF HEAT TRANSFER DEVICES

Since the convective term is zero in the solid region, the energy equation for the solid region becomes:

ay'

Wc \

dT ''' dz

oz

(5.16)

Comparing equation (5.4) and equation (5.15) leads to the following: h /,high

—

ekf t/^cahigh {Ty - {T)f

dT_

9J

(5.17) 2=0 >

Equations (5.14) and (5.17) can be used to determine K and hi analytically, provided that the velocity and temperature distributions are known. For this situation, Kim and Kim (1999, 2004) assumed that the characteristics of the pressure drop and the heat transfer from the fins are similar to those of the Poiseuille flow between two infinite parallel plates that are subject to a constant heat flux in the streamwise direction. This assumption is reasonable when the aspect ratio is of the order of one or larger because it can be easily shown by an order-of-magnitude analysis that the diffusion of momentum and energy in the fin direction is negligible compared to that in the direction normal to the fin direction. Hence, we can safely adopt this approximation in determining K and hi when the aspect ratio is of the order of one or larger. The velocity and temperature distributions for the Poiseuille flow in this configuration can be easily determined to be given by: 6^

1-

{u)f

T - (r)^ (T)/ - {TY

(5.18)

Wr

Wc

l - 2 ( -

Wc

Wc

(5.19)

Wc

Using these distributions, the permeability and interstitial heat transfer coefficient have been found to be given by: ^high -

- ^

,

NUi,oofc/

^^'^^^^ ^ ~ 2 ^ Z -

(5.20) (5.21)

where the value of Nu, oo is 10. Analytical determination of the permeability and the interstitial heat transfer coefficient just described represents a significant improvement over experimental or numerical determination of these coefficients. Previous attempts to determine these coefficients lead to either inaccurate or impractical solutions. For example, Tien and Kuo (1987) used the hydraulic diameter of the channel, instead of the channel width, in equation (5.20), for the permeability. About seven percent discrepancy of their analytical solution is attributable to the choice of the geometric length scale in the permeability. Tien and Kuo (1987)

127

S. J. KIM AND J. M. HYUN

and Kim and Kim (1999) determined the interstitial heat transfer coefficient by fitting numerical data. This method is not practical, because one needs to solve the problem numerically in order to apply the porous medium approach. The method explained here helps to overcome this difficulty, which makes the approach attractive and practically valuable. Once the permeability and the interstitial heat transfer coefficient are determined, the analytical solutions for velocity and temperature distributions can be obtained from equations (5.3)-(5.5), with the boundary conditions (5.7) and (5.8). The analytical solutions for the dimensionless velocity and temperature distributions are given by: (5.22)

Q

_

-Hiigh

Os = Ph:igh

1^\

Da cosh

T^^

Da

/

-h

(5.23)

1 - cosh(yi7Di) . ^ , /

,

sinh(yr75a)

1

smh

1

Y^ + C^Y (5.24)

where Y = es =

H

U =

{uV

^ h i g h "=

{Ty q"H/{l - e)ks

q"H/{l-e)ks'

^high ^{PV

(5.25)

diX

Dahigh =

K^i gh

eH^

(5.26)

The mathematical expression for the coefficients appearing in the above solutions can be found in Kim and Kim (1999). Figure 5.3 shows comparisons of the velocity and temperature distributions calculated by the averaging method and by numerical simulation. The numerical results are acquired by solving equations (5.9) and (5.10) by using the control-volume-based finite-difference method, and they are used as a basis for comparison. As shown in this figure, the results obtained from the averaging method are in close agreement with the numerical values.

128

THERMAL ANALYSIS OF HEAT TRANSFER DEVICES

Qis = 5

Analytical solution Shah and London (1978) —I

0.4

1

1—

0.6

(b) -5-\ Qs

=5

Numerical solution: Numerical solution: Analytical solution: Analytical solution:

-10 -15

df Os 6f ds

-20 H

S

-25 H -30

0.0

0.2

-1

0.4

'

1—

0.6

0.8

1.0

Figure 5.3 Validation of the averaging method: (a) velocity distribution, and (b) temperature distributions.

These favorable comparisons underscore the utility of the simple analytical results, v^hich are based on the porous medium approach. The variables of engineering importance are identified, making thermal optimization of microchannel heat sinks straightforward. The example of microchannel heat sink optimization is treated next. The concept of thermal resistance is useful in evaluating the thermal performance of microchannel heat sinks. The total thermal resistance is defined as the temperature difference between two points of concern per unit heat flow rate, i.e. see Knight et al. (1991),

S. J. KIM AND J. M. HYUN T-)

-^s,out

129

-^6,in

^^ r»^\

q Thus, the thermal resistance for the constant-heat-flux condition is directly proportional to the temperature difference between the heat sink base temperature at the exit and the coolant temperature at the inlet. Equation (5.27) can be decomposed into two terms: 73

/t^ tot —

-'sjOut ~" -^6,out

q

1

-^6,out ~" -^6,in

Q

•

^^ ,-,Q>.

p.zo;

Using the energy balance, we obtain ry J-s,out -n:(9,tot =

~~ J-b,out

. -t f- -:

,

^c o n \ (^-^y)

q mcf where the first term on the right-hand side is known as the convective resistance, i?^,conv» and the second term the capacitive resistance, Rg^c&p. The former is related to heat transfer from the fins to the coolant, and the latter is responsible for the temperature rise of the coolant from inlet to exit. The final goal of the thermal optimization of the microchannel heat sink is to determine the important dimensions for which the total thermal resistance is minimized when the pumping power is specified. From the definition of bulk-mean temperature, the convective thermal resistance of the microchannel heat sink can be represented by _ '''^^^^--

Of^bH (l-e)ksLW^[m^

f_l_

_ _ 1 _ \ Wcjwc + w^) Nu,,oo; kfLWH '

^^'''^^

where the bulk-mean temperature adopted in the porous medium approach is given by: ef^t,= f UOfdY. Jo

(5.31)

The second term on the right-hand side of equation (5.30) is included to account for the difference between the conventional bulk-mean temperature and the bulk-mean temperature defined in equation (5.31), as asserted by Kim et al (2002). The value for Nuoo is 8.235, which is the Nusselt number based on the bulk-mean temperature for the heat transfer between two infinite parallel plates subject to a constant heat flux. Alternatively, the convective thermal resistance can be obtained by introducing the onedimensional bulk-mean temperature of the fluid, which is defined as follows:

Jo

^dz

This one-dimensional bulk-mean temperature satisfies the following equation: '^high = -(jy^TijyJ

= 8-325^ •

(5.33)

130

THERMAL ANALYSIS OF HEAT TRANSFER DEVICES

Combining equations (5.21) and (5.33), we obtain the following:

From equation (5.34), the bulk-mean temperature of the fluid is given as follows:

r. = l / V ' d v = T. + ^ / ' . , . d y q"H

[ [i,2oi{ef-es) + es]dY T^ + (1 - e)ks Jo

(5.35)

and the convective thermal resistance of the microchannel heat sink can be represented by H ^^.conv = T. Z. w r / {l-'7)ksWL ,

[1-201(^/ - Os) + ^.] AY .

(5.36)

Figure 5.4(a) displays the optimized design variables and the corresponding thermal resistances, which are obtained by using the averaging method and by numerical simulation for various aspect ratios. The analytical results for the design variables, by using the averaging method, are in close agreement with the numerical results. The thermal resistances computed by the averaging method are consistent with the numerical results over a wide range of the aspect ratio, as illustrated in Figure 5.4(b). The averaging method described above can be extended to developing flows as well, see Kim and Kim (2004). As illustrated in this case study, the averaging method is accurate, yet considerably simpler than numerical simulations. The merits of the averaging method are apparent for the thermal design and optimization of microstructures. 5.2.2

Low-aspect-ratio microchannels

A low-aspect-ratio microchannel heat sink is sketched in Figure 5.5. As in a high-aspectratio microchannel heat sink, the direction of the fluid flow is parallel to the x-axis and all the physical conditions remain the same. The only difference in the averaging method is the direction of averaging. While the averaging is performed in the z-direction for highaspect-ratio microchannels, it is in the y-direction for low-aspect-ratio microchannels. For the present case, an averaged quantity is defined:

<«4/

H

(f>dy.

The equations for the averaged velocity and temperatures are as follows:

(5.37)

131

S. J. KIM AND J. M. HYUN

(a)

80

,A"

3 ^ 60 ~J

40

1

I

'-'''

20 H1

1 ,-'-''

g £

"T-" -

.

A'' J ,

p

200

'^ • • ,

,

400

,

•^

ly-u;, averaging approach Wc, averaging approach w^, numerical simulation Wc, numerical simulation — 1

600 H[fxm]

'

1

800

'

r-

1000

(b) 0.12

U

8 0.08

0.04

Re,tot, averaging approach Re,tot, numerical simulation 0.00

Figure 5.4

Figure 5.5

200

400

600 H\pm]

800

1000

Optimized results: (a) Ww and Wcj and (b) -R^,tot-

Schematic of a microchannel heat sink with a low aspect ratio.

132

THERMAL ANALYSIS OF HEAT TRANSFER DEVICES

for the fluid phase: d{p) _ d^u) Wdx =

Pf^fW-^

dx

^/e,low

^/ / \

(5.38) (5.39)

,low^low

dz^

and for the solid phase: (u) = 0, (T) — T

(5.40) (5.41)

conditions»are as follows:

iu)--= 0, {T) =• J- w at z = 0, {u)-- = 0,

{T) ='

-L W

at

z

-

(5.42) (5.43)

Wc

For low-aspect-ratio microchannels, see Kim and Kim (2005), we have the following: Wc Wc +

0'\ovi —

Wn,

1 H'

du dy y=H t'/,low

\y=z o

kf ,

^/e,low

-1

du 9y

)*""-

(5.44) (5.45)

y=Oj

{T))

-1

(5.46)

It is noted that the temperature gradients within the fins are neglected in equation (5.41) since the fin efficiency is high when the aspect ratio is low. To solve the governing equations, (5.38) and (5.39), the permeability Kiow and the interstitial heat transfer coefficient ^/,iow should be determined in advance. To obtain approximate values of K and hu we assume that the characteristics of the pressure drop and the heat transfer from the fins are similar to those of the Poiseuille flow between two infinite parallel plates, one of which is subject to a constant heat flux and the other to an insulated condition. The velocity and temperature distributions of the Poiseuille flow in this configuration are given by the following: (5.47)

6(»)|(1-|). T = i{T) - T„)

20y_ 7 H

20

10

7 \H)

7

\H)

+ T„

(5.48)

From equations (5.47) and (5.48), the permeability and the interstitial heat transfer coefficient are obtained as follows:

133

S. J. KIM AND J. M. HYUN K\o^ —

(5.49)

12 '

^/,low ==

(5.50)

2H'

Once the permeability and the interstitial heat transfer coefficient are determined, the analytical solutions for the velocity and temperature distributions can be secured by solving equations (5.38) and (5.39). The analytical solutions for dimensionless velocity and temperature distributions are as follows: for the fluid phase: U — -Plow ^ 1 - cosh (9 = - T T + TS; - C\ cosh

1 a^x/Daiow V2 fC'2 (\

cosh

Z e Z

cosh

c.

Dai,

1 _ Z

1

2 ~7

La5\/Daiow

(5.51)

2Q;5VDaio

Uaiow

- C2 1 cosh

1 2Qs\/Daiow, (5.52)

and for the solid phase: t/ = 0,

(5.53) (5.54)

6' = 0, where Z =

Wc + Wt,

u=

{u)

e=

(T) - T ,

_ H

Oi —

SmcQ{l/wc)J^''Udz -How —

low

efifUm

dx

_ iCiow

1- e ks —7—^low e kf

5

^ ^2

hi^iowH —

-J

(5.55)

= 1, 1 - 2a5\/Daiowtanh ( ] V2a5VDaiow/J

(5.56)

The one-dimensional bulk-mean temperature is defined as follows:

{Tr

f-H

/o "d2/

(5.57)

134

THERMAL ANALYSIS OF HEAT TRANSFER DEVICES

The interstitial heat transfer coefficient for one-dimensional bulk-mean temperature can be found from equations (5.47) and (5.48) as follows: h low

—

;

kf

dT

T^ - (T)" \ dy

= 5.385 y=Oi

kf_ 2H

(5.58)

Combining equations (5.46), (5.50) and (5.58),

e"

(T)" -T^ _ /i,,iow (T) - T, = 1.061 e*. q"H/il - e)ks /i,ow q"H/il - e)k.

(5.59)

From equations (5.52) and (5.59), the bulk-mean temperature of the fluid is given as follows:

^^^£{T)Odz_^

q"H

= T„ + 1.061 +

e(l - e)ks Jo q"H 9l

(1 - e)k Dai,

'C2

tanh ( ^ \

2a, 1 — Citanh 2a5\/Daio

Ic, (-^

Daiow 1 Daiow

- C,^ ^ 2a,

C,

1 2asx/Daio

Numerical solution Analytical solution

Figure 5.6 Dimensionless temperature distributions for QS = 0.1 (e = 0.5, kf Iks 0.00414).

(5.60)

135

S. J. KIM AND J. M. HYUN (a)

A Analytical solution for low Qg • Analytical solution for high Qg — Exact numerical solution

10^1

^. 10^1

: - * *

•'X^-

^^--r-r-^'"

"^ 10"^ i

10"^ 10"^

10-2

IQ-i

(b)

iQi

IQO

as

JQ2

JQ:

JQ2

JQ3

10^

10^

10^1

u

loN ^^*-. •

•

*

^

-

10-^1 10-3

lQ-2

lQ-1

IQO

JQl

(c)

10^ i

u

10" ^^T10-H 1010-3

10-2

IQ-

10"

W

Figure 5.7 Thermal resistances of the microchannel heat sink {Wc = Ww = ^0 //m, L = T^ = 1cm, A:^ = 148W/mK, kf = 0.613 W/mK, Ppump = 2.56 W): (a) convective, (b) capacitive, and (c) total thermal resistances.

136

THERMAL ANALYSIS OF HEAT TRANSFER DEVICES

In order to validate the porous medium approach, and the solutions based on it, the temperature distributions obtained from the analytical solutions are compared with the numerical solutions. In Figure 5.6, the analytical solutions for low-aspect-ratio heat sinks are compared with the numerical solutions and the analytical and numerical results are in satisfactory agreement. When a5 < 0.2, the analytical solutions accurately reproduce the numerical solutions. As in the case of large-aspect-ratio channels, the thermal performance of the microchannel heat sink can be evaluated by the thermal resistance. As shown in Figure 5.7, when the aspect ratio of the microchannel is smaller than about 1, the analytical solutions for lowaspect-ratio heat sinks provide accurate evaluations of the thermal resistances, while the analytical solutions for high-aspect-ratio heat sinks, calculated by Kim and Kim (1999), are less accurate in predicting thermal resistances. On the other hand, the analytical results obtained by Kim and Kim (1999) are highly consistent with the numerical results when the aspect ratio is larger than about 1. As the aspect ratio of the microchannel increases, the capacitive resistance decreases. This is due to the fact that the flow rate decreases while the cross sectional area for the fluid increases when the pumping power is fixed. When the aspect ratio is less than about 1, the convective resistance increases as the aspect ratio of the microchannel increases. This is because the heat transfer coefficient decreases while the hydraulic diameter increases. On the other hand, when the aspect ratio is greater than 1, the convective resistance has a minimum value. This is because the heat transfer area increases, but the fin efficiency decreases as the height of the microchannel heat sink becomes larger.

5.3

THERMAL ANALYSIS OF INTERNALLY FINNED TUBES

In order to enhance the rate of heat transfer, finned surfaces have been applied to cooling devices for electronic equipment and compact heat exchangers for many years. The apparent advantage offinsis that they increase the heat transfer rate by providing additional surface area. However, fins placed in a tube make the flow patterns complex and increase frictional resistance. As the number, or the height, of fins increases,flowfriction increases, and higher pumping power is required to supply the same rate of mass flow. Therefore, to design a compact heat exchanger with internally finned tubes, we should optimize the fin geometry, accounting for both fluid friction and heat transfer. Numerical and experimental studies have been made of fully developed forced convection in internally finned tubes, see Watkinson et al (1975), Masliyah and Nandakumar (1976), and Rustum and Soliman (1988b). Recently, for instance, Fabbri (1998) proposed a polynomial lateral profile for the fins. Optimization has been attempted for the geometry in the finned tube in order to enhance the heat transfer rate per unit tube length for a given weight and for a given hydraulic resistance. However, there are numerous parameters which affect the thermal performance of internally finned tubes, such as radius, length, tube material, thermal properties of the coolant, height, thickness, the number of fins, etc. As a consequence, efforts in the previous numerical and experimental studies involved

S. J. KIM AND J. M. HYUN

137

tedious numerical calculations or extensive laboratory work in order to evaluate the effects of parameters, see Patankar et al. (1979), Camavos (1980), Webb and Scott (1980), and Rustum and Soliman (1988a). On the contrary, if analytical solutions can be found, the optimization of thermal performance and the evaluation of parameters can be accomplished with ease. Only a few researchers have presented analytical solutions for finned tubes. Shah and London (1978) provided analytical solutions for both the axial velocity and temperature distributions in circular-sectored ducts for the case of an axially uniform heat input and a circumferentially uniform wall temperature at any cross section. For a constant wall heat flux at any cross section, Hu and Chang (1973) provided an analytical evaluation of the Nusselt number of internally finned tubes with zero fin thickness. Soliman and Feingold (1977) investigated theoretically the fully developed laminar flow in internally finned tubes with fin shapes approximating real fin configurations. However, these investigations did not consider conduction through the fins. They used the assumption that the temperature of the fin surface is constant. Also, no reports were given of the optimal geometry of the fins based on the flow friction and heat transfer characteristics. Due to the complex geometry of the finned tube and the conjugate heat transfer between the fluid and the fins, the conventional energy equation cannot be solved analytically. The porous medium approach has been adopted by a few researchers to model the fluid flow and heat transfer through a channel with a heat transfer augmentation device. For example, Srinivasan et al. (1994) studied fluid flow and heat transfer through spirally fluted tubes using a porous-substrate approach. The model divided the flow domain into two regions, the flute region and the core region, with the flutes being modeled as a porous substrate. Although there is a large difference between the thermal conductivities of the fluid and the internal fins, they treated the flutes and the fluid as a single entity. By contrast, a porous medium approach based on the two-equation model is expected to resolve the problems with the local thermal equilibrium assumption, which would yield more accurate results. The porous medium approach introduced for the microchannel heat sink in the previous section, which is based on the two-equation model, can also be applied to the thermal analysis of internally finned tubes. Recently, Kim et al. (2002) studied heat transfer characteristics in circular-sectored finned tubes using this approach. We are going to briefly discuss these approaches. When the circular-sectored finned tube is modeled as a porous medium, analytical solutions for both the velocity and temperature profiles can be obtained. The point of this example is to demonstrate the cost and speed advantage of the analytical solutions compared to the time-consuming numerical procedures in design applications. 5.3.1

Mathematical formulation and theoretical solutions

The problem under consideration is forced convectiveflowthrough afinnedtube, as shown in Figure 5.8(a). The direction of fluid flow is parallel to the tube axis. The tube wall is uniformly heated, and a coolant passes between the fins attached to the inside surface of the tube wall. In analyzing the problem, the flow is assumed to be steady, laminar, and

138

THERMAL ANALYSIS OF HEAT TRANSFER DEVICES

(b)

Figure 5.8 Schematic view of the configurations of interest: (a) a tube with circularsectoredfins,and (b) an equivalent porous medium.

both hydrodynamically and thermally fully developed. In addition, all the thermophysical properties are assumed to be constant. The circular-sectored finned tube is modeled as a porous medium as shown in Figure 5.8(b). The present analysis is based on using the averaging method to establish the governing equations for the velocity and temperature fields in the finned tube. For the present system, the representative elementary volume for volume-averaging can be visualized as a (dotted) ring aligned perpendicular to the flow direction as shown in Figure 5.8(a). The averaging in the present analysis is equivalent to averaging in the ^ direction because the geometry under consideration is uniform in the flow direction. To characterize the fluid flow and heat transfer, the volume-averaged momentum equation and volume-averaged energy equations for the solid and fluid phases are obtained and solved. The Brinkman-extended Darcy equation, as proposed by Vafai and Tien (1981), is used as the volume-averaged momentum equation for the present system: 1 d r dr V

d(ix)^

dr

K^

'

dx

(5.61)

where (•)'^ and (•)^ denote a volume-averaged value over the fluid region and the solid region, respectively. This modified Darcy equation is used in place of the original Darcy equation in order to account for the boundary effect. Due to the high solid-to-fluid conductivity ratio, the two-equation model is employed in the present analysis. The volume-averaged energy equations for the solid and fluid phases are expressed as follows: r dr V

tpfCf

^ dr

^h,a{{Tr-{T)^),

,„,^.,„,„_<.„,-(„.,^).

(5.62) (5.63)

S.J.KIMANDJ.M.HYUN

139

The boundary conditions to complete the problem formulation are as follows: (u)/=0,

at r = ro,

{Ty = {TY=T^

(5.64)

^

= 0, 'Jni = ^jpi = , at . = 0. (5.65) dr dr dr Now equations (5.61)-(5.63) and the boundary conditions (5.64) and (5.65) are nondimensionalized using the following dimensionless quantities: [7^M!, Um

^ ^

(r)^-r^

/;:=-,

Da=^, 12

TQ

^^^

(T).-T^

p^_i_d(p)^

(5.66) (3^7)

For the fully developed flow subject to a constant heat flux, A{Ty dx

d{Ty dx

_ dT, = constant, dx

(5.68)

and from the energy balance, ql = epfCfUmj^^

.

(5.69)

The dimensionless governing equations and boundary conditions can now be expressed as follows: 1 d / dU\ d^Os ^ IdBs ^ dr/^ T] dri

1 U h; e,-9f ^ (1 — e)ks rf-

^ + i ^ + hlOs-Of drf^ T] drj ekf rp U = 0, dU

0, = ef =0 dOf dOs

drj

dr]

^^^^^

^ 2(l-e)fc,^,^ ekf

^^^2)

at 7/ =: 1,

(5.73)

dr]

where /ij' = hiar'^ = 2rhi/{a H- p). The analytical solution to the momentum equation, equation (5.70), with the boundary conditions (5.73) and (5.74), can be found:

140

THERMAL ANALYSIS OF HEAT TRANSFER DEVICES

where the dimensionless pressure drop is determined by the relation J^ Urjdr] = 1/2. The energy equations (5.71) and (5.72), with the boundary conditions (5.73) and (5.74), can be solved to yield the dimensionless temperature profiles of the fluid and solid phases: es =

1 + 2VDa 1 + C ll-2x/D^,

7 / ^ - 7]"^

16-A

rj'

" ^ ~ 16

^\/A _

^2-hl/VB^

(2 + 1/V^f

ef =

1 + 2VDa 1 + C I 1- 2 v ^

- A

(2 +

\^4 _ ^

^2+i/VD^ _ ^

16

(2 + 1 / V D ^ ) '

1

(r]^-r]^

" C [ 16-A

(5.76)

2+1/v^a .

Vx

1/VB^)^\

_ ^2+1/VDI

'

(5.77)

" (2 + l / V ^ ) ' - A ,

A detailed derivation of these solutions can be found in Kim et al. (2002). 5.3.2

Velocity and temperature distributions

The velocity and temperature distributions given in equations (5.75)-(5.77) are incomplete since the permeability K and the local heat transfer coefficient hi are not known yet. Again, the permeability K and the local heat transfer coefficient hi are included to recover the information lost due to averaging. The lost information in the present case is the dependence of the velocity and temperature distributions in the circumferential direction. In order to determine the permeability K and the local heat transfer coefficient hi, we assume that the pressure drop and heat-transfer characteristics of the circular sector under consideration can be approximated as for the Poiseuille flow between two semi-infinite plates that meet with an angle of a and are subject to a constant heat flux, see Kim et al. (2002). For small values of a, it can be shown by an order-of-magnitude analysis that the diffusion of momentum and energy in the radial direction is negligible compared to that in the circumferential direction. Thus, the velocity and temperature distributions are obtained as follows: dp u — 2/idx kf

^

(5.78)

T

- 4 ^ + 6 ^ 4 l+T. a^

a^

(5.79)

From this velocity distribution, the relation between the pressure drop in the flow direction and the mean velocity can be calculated. After comparing this relation with the Darcy

141

S. J. KIM AND J. M. HYUN

equation, the permeability is obtained: K =

(5.80)

12

Similarly, the local heat transfer coefficient is determined: (5.81)

ra

With these approximate values of K and /i/, the analytical solutions based on the porous medium approach are compared with the corresponding velocity and temperature distributions for the conjugate heat transfer problem comprising both the solid fin and the fluid. The formulation and the numerical method for the conjugate heat transfer problem are similar to those in Patankar (1980) and therefore are not repeated here for the sake of brevity. Only the conventional energy equation is solved numerically because a closedform solution exists for the fully-developed flow in the circular sector in the following form: t/ = 2

r]^{a - t a n a ) -f (32aV7r^) En=i,3,... rf''''"^l{n^{n -h 2a/7r)(n - 2a/7r)] a - t a n a + (128aV7r^) E n = i 3

' (5.82) where it is noted that the velocity distribution is the result of the averaging in the ^ direction. Thus it can be direcdy compared with equation (5.75), which is the analytical solution based on the porous medium approach. In Figure 5.9, equation (5.75) is compared with the velocity profile of equation (5.82) for a — 45°. Similarly, in Figure 5.10, equations (5.76) and (5.77) are compared with the corresponding average temperature distributions

1.75j1 1.50-\ 1

l/[^^(^ + 2a/7r)2(n - 2a/7r)]

Porous medium approach Exact solution

1.25V

1.000.75 0.500.250.00 1 - ^ 0.0

r-^^—•

0.2

1

0.4

^

^

1

0.6

«

1

0.8

1

'

1.0

Figure 5.9 Comparison of the velocity profile obtained from the porous medium approach with the exact solution for a — 45°.

142

THERMAL ANALYSIS OF HEAT TRANSFER DEVICES

0 -1 -2 -3 -4 -5

0.0

0.2

0.4

^

0.6

0.8

1.0

Figure 5.10 Comparison of temperature profiles obtained from the porous medium approach with numerical solutions for a = 45° and C = 0.01.

from the numerical solutions for a — 45° and C = 0.01. As is evident in these figures, the analytical solutions based on the porous medium approach are in close agreement with the corresponding numerical results for the velocity and temperature profiles. In the range 0° -^ a ^ 45°, the error between the two sets of results is shown to increase with the angle of the circular sector a, with a maximum of 5%. This excellent agreement confirms that the permeability and the local heat transfer coefficient have been chosen appropriately to recover the information lost in the averaging process. A question may arise as to when the porous-medium model is applicable to the thermal analysis of a circular-sectored finned tube. For this purpose, the Darcy number based on equation (5.80) and the local Nusselt number based on equation (5.81) are compared with the exact solution for the Darcy number and that from the numerical solution for the Nusselt number, respectively. In Figure 5.11, the Darcy number based on equation (5.80) and the local Nusselt number based on equation (5.81) are in close agreement with the exact results. Hence the porous medium approach introduced in this chapter is valid for a < 45° with a maximum error less than 5%. These analytical solutions from the porous-medium model are helpful in identifying and studying the effects of variables of engineering importance. Therefore, the extension to more practical research, such as optimization of the finned tube, is possible without undertaking tedious numerical computations. 5.3.3

Optimization of thermal performance

As pointed out for microchannel heat sinks, optimization of the thermal performance of the internally-finned tube is identical to the minimization of the total thermal resistance. In minimizing the total thermal resistance, the physical properties of the fluid, length

143

S. J. KIM AND J. M. HYUN 0.20 Nui •

0.15

Numerical • Based on equation (5.81)

3 Nui

Da 0.10

0.05 H

Exact Based on equation (5.80)

0.00

Q [deg.] Figure 5.11 Comparison of the approximated Da and Nu/ with the exact results. and radius of the tube, and pressure drop are assumed to be given. Also, the maximum pressure difference across the tube is assigned a practical value. With these assumptions, it is shown that the total thermal resistance is a function of e and Da. Because Da — o? jVl and the fins are equally distributed, we can find the optimum values of the number of fins and 6 which minimize the total thermal resistance. The thermal resistance of the finned tube is minimized in the following example using the analytical solutions obtained from the porous-medium model. In this example, we demonstrate the cost and speed advantage of the analytical solutions in optimizing the thermal performance of a circular-sectored finned tube. The thermophysical and geometric details of the finned tube are listed as follows. The working fluid is water, the fin material is aluminum, the length and radius of the tube are 2 m and 0.1 m, respectively, and the pressure drop is 1 kPa. The thermal resistance of the finned tube is depicted in Figure 5.12 in terms of the porosity and the number of fins. As shown in Figure 5.12, the total thermal resistance has a minimum value of 0.0229 °C/W for the current example, when e is 0.9 and the number of fins is 41. 5.3.4

Comments on the averaging direction

Now, one may ask how we choose the averaging direction in the averaging approach. As demonstrated in the previous examples, the averaging direction depends on what kind of structures we are interested in. It does not matter how we choose the averaging direction as long as appropriate values for the permeability K and the interstitial heat transfer coefficient hi are used in the analysis. Once the values for K and hi are determined, we can obtain accurate velocity profiles and temperature profiles by using the averaging

144

THERMAL ANALYSIS OF HEAT TRANSFER DEVICES 1.0-

60

Number of fins Figure 5.12 Contour map of i^totai (°C/W). approach. However, in general, we suggest that the characteristic length scale in the averaging direction be smaller than that in the other direction. This is because the approximate values for K and hi can be analytically determined under this condition. For example, in the case of low-aspect-ratio microchannel heat sinks, the averaging direction is the ^/-direction because the channel height is much smaller than the channel width.

5.4

CONCLUSIONS

In the present chapter, a method of modeling transport phenomena in heat transfer devices is introduced. The modeling technique is based on the averaging method and is frequently called a porous medium approach because the heat transfer devices are modeled as a fluid-saturated porous medium. A novel method for analytically determining the unknown coefficients resulting from the averaging method is presented. This represents a significant improvement over experimental or numerical determination of these coefficients. The averaging method in turn yields analytical solutions for the velocity and temperature distributions, which are useful in the thermal analysis of heat transfer devices. This method can be used to analyze fluid flow and heat transfer in a microstructure as well as in a regular-sized structure. A microchannel heat sink and an internally finned tube are chosen as case studies to elucidate the method in the context of thermal design and optimization of heat transfer devices. The respective thermal performances of a microchannel heat sink and a tube with circular-sectored fins are optimized by modeling the heat sink and

S. J. KIM AND J. M. HYUN

145

the finned tube as fluid-saturated porous media. The porous medium approach illustrated here could be utilized in analyzing and optimizing the thermal performance of a variety of compact heat exchangers, heat sinks, and other thermal devices with complex geometry.

ACKNOWLEDGEMENTS This work was supported by KISTEP (Korea Institute of Science & Technology Evaluation and Planning) under grant number 2-578 through the National Research Lab Program.

REFERENCES Camavos, T. C. (1980). Heat transfer performance of internally finned tubes in turbulent flow. Heat Transfer Eng. A, 32-1. Chen, L., Ma, J., Tan, F., and Guan, Y. (2003). Generating high-pressure sub-microliter flow in packed microchannel by electroosmotic force: potential application in microfluidic systems. Sensors Actuators B 88, 260-5. Ernst, H., Jachimowicz, A., and Urban, G. A. (2002). High resolution flow characterization in bio-MEMS. Sensors Actuators A 100, 54-62. Fabbri, G. (1998). Heat transfer optimization in intemally finned tubes under laminar flow conditions. Int. J. Heat Mass Transfer 41, 1243-53. Goodling, J. S. and Knight, R. W. (1994). Optimal design of microchannel heat sink: a review. Optimal Design Therm. Syst. Components 279, 65-77. Hu, M. H. and Chang, Y. P. (1973). Optimization of finned tubes for heat transfer in laminar flow. ASME J. Heat Transfer 95, 332-8. Ingham, D. B. and Pop, I. (eds) (1998). Transport phenomena in porous media. Pergamon, Oxford. Kim, D. and Kim, S. J. (2004). Compact modeling of fluid flow and heat transfer in straight fin heat sinks. ASME J. Electronic Packaging 126, 247-55. Kim, D. K. and Kim, S. J. (2005). Study on the averaging approach for microchannel heat sinks encountered in electronics cooling. ASME J. Heat Transfer. Submitted. Kim, S. J. (2004). Methods for thermal optimization of microchannel heat sinks. Heat Transfer Eng. 25, 37-49. Kim, S. J. and Kim, D. (1999). Forced convection in microstructures for electronic equipment cooling. ASME J. Heat Transfer 121, 639^5. Kim, S. J., Kim, D., and Lee, D. Y. (2000). On the local thermal equilibrium in microchannel heat sinks. Int. J. Heat Mass Transfer 43, 1735-48. Kim, S. J., Yoo, J. W., and Jang, S. P. (2002). Thermal optimization of a circular-sectored finned tube using a porous medium approach. ASME J. Heat Transfer 124, 1026-33. Knight, R. W., Goodling, J. S., and Hall, D. J. (1991). Optimal thermal design of forced convection heat sinks—analytical. ASME J. Electronic Packaging 113, 313-21.

146

THERMAL ANALYSIS OF HEAT TRANSFER DEVICES

Knight, R. W., Hall, D. J., Goodling, J. S., and Jaeger, R. C. (1992). Heat sink optimization with application to microchannels. IEEE Trans. Components, Hybrids, Manufac. Tech. 15, 832-42. Koh, J. C. Y. and Colony, R. (1986). Heat transfer of microstructures for integrated circuits. Int. Comm. Heat Mass Transfer 13, 89-98. Masliyah, J. M. and Nandakumar, K. (1976). Heat transfer in internally finned tubes. ASMEJ. Heat Transfer 9H, 251-61. Nield, D. A. and Bejan, A. (1999). Convection in porous media (2nd edn). Springer-Verlag, New York. Patankar, S. V. (1980). Numerical heat transfer and fluid flow. Hemisphere, New York. Patankar, S. V., Ivanovic, M., and Sparrow, E. M. (1979). Analysis of turbulent flow and heat transfer in internally finned tubes and annuli. ASMEJ. Heat Transfer 101, 29-37. Phillips, R. J. (1990). MicroChannel heat sinks. In Advances in thermal modeling of electronic components and systems (eds A. Bar-Cohen and A. D. Kraus) (2nd edn), pp. 109-84. ASME Press, New York. Rustum, I. M. and Soliman, H. M. (1988a). Experimental investigation of laminar mixed convection in tubes with longitudinal internal fins. ASMEJ. Heat Transfer 110, 366-72. Rustum, I. M. and Soliman, H. M. (1988b). Numerical analysis of laminar forced convection in the entrance region of tubes with longitudinal internal fins. ASMEJ. Heat Transfer 100, 310-13. Shah, R. K. and London, A. L. (1978). Laminar flow forced convection in ducts. Academic Press, London. Slattery, V. C. (1999). Advanced transport phenomena. Cambridge University Press. Soliman, H. M. and Feingold, A. (1977). Analysis of fully developed laminar flow in longitudinal internally finned tubes. Chem. Eng. J. 14, 119-28. Srinivasan, V., Vafai, K., and Christensen, R. N (1994). Analysis of heat transfer and fluid flow through a spirally fluted tube using a porous substrate approach. ASMEJ. Heat Transfer 116,543-51. Tien, C. L. and Kuo, S. M. (1987). Analysis of forced convection in microstructures for electronic system cooling. In Proceedings of the international symposium on cooling technology for electronic equipment, Honolulu, HI, pp. 217-26. Tien, C. L., Majumdar, A., and Gemer, F. M. (1998). Microscale energy transport. Taylor & Francis, USA. Tuckerman, D. B. and Pease, R. F. W. (1981). High-performance heat sinking for VLSI. IEEE Electron Device Lett. 2, 126-9. Vafai, K. and Tien, C. L. (1981). Boundary and inertia effects on flow and heat transfer in porous media. Int. J Heat Mass Transfer 24, 195-203. Watkinson, A. P., Miletti, D. L., and Kubanek, G. R. (1975). Heat transfer and pressure drop in internally finned tubes in laminar oil flow. ASME Paper No. 75-HT-41. Webb, R. L. and Scott, M. J. (1980). A parametric analysis of the performance of internally finned tubes for heat exchanger application. ASME J. Heat Transfer 120, 38-43.

6

LOCAL THERMAL NON-EQUILIBRIUM IN POROUS MEDIUM CONVECTION D. A. S. REES* and I. POP^ * Department of Mechanical Engineering, University of Bath, Bath, BA2 7 AY, UK email: D.A.S.ReesQbath.ac.uk ^Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania email: [email protected]

Abstract Many papers exist which either derive or use equations which govern local thermal nonequilibrium phenomena in porous medium convection, where the intrinsic average of the temperatures of the solid and fluid phases may be regarded as being different. We compile and present the most commonly used of these model equations. Attention is then focused on describing some of the most recent research using these equations. Attention is focussed primarily on free and forced convection boundary layers, and on free convection within cavities. Keywords: local thermal non-equilibrium, convection, porous media

6.1

INTRODUCTION

In this chapter v^e consider the subject of local thermal nonequilibrium, LTNE. In most cases studied in the research literature it has been assumed that a porous matrix and the fluid which flows through it are in local thermal equilibrium, LTE. At a microscopic level, the temperature and the rate of heat flux at the interface between the solid and fluid phases must be identical, but the average temperature over a representative elementary volume, REV, does not have to yield identical temperatures for the two phases. For instance, one may consider the example of a hot highly conducting fluid flowing through an otherwise cold and poorly conducting porous matrix; in this case a certain amount of time will need to pass before the solid phase attains the same temperature as the fluid phase. We will also see that LTNE is not necessarily an unsteady phenomenon, but can also arise in steady 147

148

LOCAL THERMAL NON-EQUILIBRIUM IN POROUS MEDIUM CONVECTION

flows. An example of this would be a case where 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. One aim of this chapter is to introduce briefly those equations which have been quoted as modelling LTNE effects in porous media. We have found that it is only the simplest of these model equations which have been routinely solved for specific convective flows. The second aim is to summarise the most recent studies of applications, which are primarily in the area of free convection and external forced convection. An excellent review of all but the very latest studies of internal forced convection cases may be found in Kuznetsov (1998). We do not attempt to present the derivations of the thermal energy equations which apply when LTNE is present; instead we refer the reader to the book by Whitaker (1999), the chapters by Carbonell and Whitaker (1984), Quintard et al. (1997) and Quintard and Whitaker (2000) and the paper by Quintard (1998). An excellent review of conductive effects in a stagnant porous medium may be found in Cheng and Hsu (1998).

6.2

GOVERNING EQUATIONS

Nield and Bejan (1999) quote the following equations as the simplest way in which LTNE may be modelled: e(pc)f ^

+ {pc)fV • V7f = eV • (fcfVTf) + h{Ts - Tf), (1 - e ) ( p c ) s ^ - (1 - 6)V • (A:sVTs) + h{Tf - T^).

(6.1) (6.2)

This is sometimes called the two-temperature model. In these equations e is the porosity of the porous medium, the subscripts f and s denote fluid and solid properties, respectively, while other variables take their familiar meanings. The variable h is an inter-phase heat transfer coefficient. In equations (6.1) and (6.2) we follow the standard practice that Tf and Ts are intrinsic averages of the temperature fields, and this allows us to set Tf = T^ = To whenever the boundary of the porous medium is maintained at the temperature TQ. On the other hand, v is a superficial average; see Quintard and Whitaker (2000, p. 5). Equations similar in form to (6.1) and (6.2) but without the diffusion terms were derived by Schumann (1929) using a straightforward control volume technique. This is the first known author to consider separately the temperature fields of the solid and thefluidphases. When Ts > Tf the final term in equation (6.1) acts as a source of thermal energy into the fluid phase, while the final term in equation (6.2) is a thermal sink for the solid phase. Equation (6.1) also shows us that, when a medium is of low porosity (e <^ 1) and the solid and fluid phases exhibit the temperature difference, AT, then the rate of change of the fluid temperature is 0(/iAT/e(pc)f). As this is inversely proportional to e, there is a rapid change in the fluid temperature towards that of the solid phase to establish LTE.

149

D. A. S. REES AND I. POP

When LTE applies, then Tf = Tg everywhere, and the equations may be combined by addition to yield the one-temperature model: dT

(pc)eff ^

+ {pc)fV

. V r = V • (keffVT)

(6.3)

.

where the effective porous medium properties are given by (pc)eff = e{pc)f + (1 - e){pc)s,

(6.4)

fceff = efcf H- (1 - e)A:s •

(6.5)

Expression (6.5) is almost always quoted as being the definition of the effective conductivity of the saturated porous medium, where T may now be regarded as a superficial average. However, Nield (2002) points out that expression (6.5) corresponds to thermal resistances in series such as one would have when considering heat conduction in the direction parallel to alternating layers of fluid and solid. When conduction takes place perpendicular to such interfaces, then he suggests the following definition of /ceff: 1

e_

1-^

(6.6)

Thus, for a layered medium, such as is given by a parallel plate channel, the effective conductivities are different in the two main directions. Nield (2002) concludes that, for such a structured medium where the fluid/solid interfaces are parallel with the x-direction, the appropriate LTNE equations to be used are given by dTf e{pc)f— + {pc)fV'VTf ( i - e ) ( p 4 dt

= e dx \ ^ dx

d (^ dTi \ dy

+ dy

+ h{Ts-Tf),

^^ '^ [dx y"^ dx ) ^ dy y"'dy

(6.7)

+ HT, - Ts), (6.8)

where ACf fCg

^{ — K — eA:s -(- (1 - e)kf '

(6.9)

At the time of writing only one paper has considered Nield's model; see Kim and Kuznetsov (2003). More complex models than those given above have been proposed. Alazmi and Vafai (2000) have undertaken a comprehensive series of numerical experiments using a forced convection channel flow to determine whether the different expressions used for the various coefficients used in equations for convective flows in porous media give rise to significant differences in the sblutions of those equations. This was a general study which, in addition to LTNE effects, also considered the Forchheimer and Brinkman terms and different models for both porosity and thermal dispersion. Focussing here on the LTNE between the phases, Alazmi and Vafai (2000) use the following equations for the

150

LOCAL THERMAL NON-EQUILIBRIUM IN POROUS MEDIUM CONVECTION

Steady-State two-temperature model: {pc)fv . VTf = V • (fcf efF • VTf) -F /isfttsf(Ts - Tf),

(6.10)

0 = V • (fcseff • VJ;) - /isfasf(Ts - Tf).

(6.11)

In these equations and fcseff = (l-e)fcs

kef[ = ^kf

(6.12)

are regarded as tensor quantities, but the diffusion terms reduce to those given in equations (6.1) and (6.2) when kf andfcgare constants. The value h given in equations (6.1) and (6.2) has now been replaced by the product of /igf and Ogf» where the former is the heat-to-fluid heat transfer coefficient and the latter is the specific surface area of the solid phase. Three different combinations of expressions for these coefficients were considered by Alazmi and Vafai (2000). Case El is given by _fcf(2 + l.lPr^/^Re°-^) /isf = ^ , Up

6(1 - e) Osf = — , dp

(6.13)

where the expression for hsf is an experimental correlation obtained by Wakao et al (1979), and that for asf may be found in Dullien (1979). Case E2 is h^^ ^ 0.004 (^\

(^\

Pr^-^^Re^-^^

(Re < 75), (6.14a)

/isf = 1.064 (^\ asf =

Pr^-^^Re^-^^

(Re > 350),

20.346(1 - e)e2 ). '— , dp

(6.14b)

which may be found in Hwang et al. (1995). We note that no information is given there about the intermediate regime where 75 < Re < 350. Finally, Case E3 is , ( dpi . dp \~^ 6(1 - e) ^sf = Yrk—^TTT + TTTT ^ ^sf = — , V 0.2555 Pr^/^Re^/^A:f lO/cs/ dp

(6.15)

where the expression for /igf was derived by Dixon and Cresswell (1979). In the above expressions, dp is the particle diameter, CJNU = 4e/asf> see Alazmi and Vafai (2000), Pr — fiCf/kf is the Prandtl number, and Re — uL/u is the Reynolds number which is based on the macroscopic length scale, L. The conclusions of the comparative study by Alazmi and Vafai (2000) are very detailed, but they find that the three models give close results when the porosity is high, or when Re is large or when the particle diameters are small. The paper by Kuwahara et al. (2001) reports on a fully numerical simulation of the flow and heat transfer through a periodic array of square cylinders. The aim of the numerical study was to find a correlation for /igf as a function of the microscopic Reynolds number.

151

D. A. S. REES AND I. POP

the Prandtl number and the porosity. Using the present notation, it was found that hsf =

/ ^ 4 ( l _ - j ) \ ^ 1 ^^ _ ^)l/2j^^0.6p^l/3l

(6^6)

A slightly earlier study by Amiri and Vafai (1998) had similar aims to those of Alazmi and Vafai (2000) but concentrated on the transient development of the temperature fields. In this earlier paper the authors used the Case El expressions above, and the following expressions for the thermal conductivities in the x- and ^/-directions: ^fefFx — '^f

6 + 0.5

"'f efF 2/ "^ "'f

e + 0.1

Pr

(6.17) (6.18)

which were obtained from Wakao and Kaguei (1982). These two coefficients play the same role as do e/Cf and ekf in equation (6.7). The first terms on the right-hand sides of expressions (6.17) and (6.18) are the stagnant conductivities, while the second terms are the dispersion conductivities based on uniform flow in the x-direction. It is frequently the case that these dispersion conductivities are neglected. Other forms for the dispersion coefficients are quoted by Vafai and Amiri (1998). Recently, Fourie and Du Plessis (2003a) have derived the following version of the twotemperature model for conduction only: dTf
= V • (A:ssVrs) + V • (fcsfVTf) + Osf/isf(Tf - Ts),

(6.19) (6.20)

where we have omitted volumetric heat source terms, and where we have translated their notation from superficial to intrinsic variables. In these equations the quantities, k^ and kfs are called the fluid phase effective thermal conductivity tensor and the fluid phase coupled thermal conductivity tensor, respectively. Likewise, kss and A:sf are called the solid phase effective thermal conductivity tensor and the solid phase coupled thermal conductivity tensor, respectively. Although this model has appeared in earlier works, such as that of Quintard and Whitaker (1993), Fourie and Du Plessis (2003a) gave a formula for the conductivity tensors which may be evaluated for certain classes of porous media, as illustrated in Fourie and Du Plessis (2003b). Quintard and Whitaker (1993) also showed that A:sf = ^fs- An extension of equations (6.19) and (6.20) to convective flows has also been derived by Carbonell and Whitaker (1984) and used by Zhang and Huang (2001) for the developing thermal field in a porous channel, with additional comments by Magyari and Keller (2002) and Zhang and Huang (2002). Finally, it is necessary to consider briefly the subject of boundary conditions. For the case where a fixed temperature is imposed upon the boundary of a porous domain, then the appropriate boundary conditions which are generally used correspond to LTE locally, i.e.

152

LOCAL THERMAL NON-EQUILIBRIUM IN POROUS MEDIUM CONVECTION

one sets Tf = Tg = To on the boundary. If the bounding surface is located at x == 0, the local rate of heat transfer may then be written as (6.21) x=0

Thus a global Nusselt number may be defined in the usual way, although it is frequently the case that separate Nusselt numbers for the two phases are presented. However, when a uniform heat flux boundary condition is to be imposed, then the situation is not so straightforward. Kim and Kim (2001) considered the relative merits of two different approaches to this situation which were originally proposed by Amiri and Vafai (1994). In the first approach the imposed heat flux is split between the phases according to equation (6.21), where q!^ is now interpreted as being the imposed heat flux. In view of the fact that two second-order thermal energy equations are being solved, equation (6.21) is then supplemented by the LTE condition Tf = Tg on the surface. In the second approach it is assumed that the heat flux into the two phases are the same, i.e.

x=0

9^

(6.22)

Equation (6.22) represents two boundary conditions, and therefore the temperatures of each phase must be found as part of the final solution. Kim and Kim (2001) consider two different practical situations involving forced convective flow and conclude that the first approach is to be preferred in general. However, they make an important caveat that the thickness of the impermeable boundary is significant. When it is sufficiently thick compared with the microscopic length scale, which is a typical practical situation, then the first approach is applicable. Otherwise, when it is thin or nonexistent, the second approach must be used.

6.3

CONDITIONS FOR THE VALIDITY OF LTE

Most papers which deal with convective flows in porous media assume that LTE is valid, and it is therefore important to acquire some criteria which will indicate when this is not a good assumption. Nield (1998), using equations (6.1) and (6.2), considered steady conduction in a porous medium subject to an imposed temperature on the boundary of the domain, and where the conductivities of the phases are uniform. He showed that Tf = Tg in this case by deriving a Helmholtz equation for Tf - Tg which is subject to Tf - Tg = 0 on the boundary. Thus LTE always occurs in steady conduction problems where the boundary temperature is imposed. Then the implication is that any departure from this special state will give rise to LTNE in general, although cases where LTE pertains may always be derived as special cases.

D. A. S. REES AND I. POP

153

The present authors have, in an incomplete and as yet unpublished study, considered unsteady conduction where the plane boundary of a semi-infinite porous medium has its temperature raised suddenly. If one were to assume LTE, then the resulting temperature field may be described using the complementary error function. When LTNE is assumed, the thermal fields in the two phases at early times correspond to complementary error functions with different arguments, and therefore strong LNTE effects are observed. However, as time progresses, the thermal front slows down, as its speed is inversely proportional to ^^/^, and LTE is eventually established. In this case rapid changes in boundary conditions result in LTNE. Al-Nimr and Abu-Hijleh (2002) studied a channel flow problem using the equations of Schumann (1929), i.e. equations (6.1) and (6.2) without diffusion. At ^ = 0 the inlet temperature was raised suddenly and the authors investigated the subsequent change in the temperature field. At long times the problem becomes like that of Nield (1998) which is described at the beginning of this section, and therefore LTE is always achieved eventually. Therefore attention is focussed on the thermal relaxation time over which LTNE gives way to LTE. The final conclusions are lengthy, but it is found that the relaxation time is inversely proportional to the volumetric Biot number defined according to hL Bi = T-—ry ,

(6.23)

where U is the fluid velocity along the channel and L its length. Given the absence of diffusion, it is not surprising that the relaxation time is proportional to the imposed velocity. Similar conclusions with regard to the rapidity of changing conditions have been made by Minkowycz et al. (1999) and Al-Nimr and Kiwan (2002), although the concern of each of these authors was with the effect of oscillating boundary conditions. Minkowycz et al. (1999) considered the effect of a suddenly imposed sinusoidally varying heat flux on the lower surface of a channel. When treated as a conduction problem only, it was found that LTE is valid in general when the Sparrow number satisfies Sp > 100. In terms of the present notation, the Sparrow number is defined as Sp =

,

(6.24)

where L is the height of the channel and k is the conductivity of the porous medium. When an imposed flow field is present then LTE is valid when Sp/Pe > 100. Al-Nimr and Kiwan (2002) also used Schumann's (1929) diffusion-free model to study the effect of a sinusoidally oscillating inlet temperature to a porous channel. The criterion for the validity of assuming LTE was based on the absence of a phase lag between the solid and fluid phases and is given by h uj{pc)s{l-e)

> 20,

(6.25)

154

LOCAL THERMAL NON-EQUILIBRIUM IN POROUS MEDIUM CONVECTION

where u is the dimensional frequency of the inlet temperature. Again, we have translated the published criterion into the present notation. It is clear that the papers by Minkowycz et al. (1999) and Al-Nimr and Kiwan (2002) yield different results since only the latter uses the oscillation frequency. However, these are not necessarily contradictory for they were obtained for somewhat different configurations.

6.4 6.4.1

FREE CONVECTION BOUNDARY LAYERS General formulation

We consider the steady free convection around a two-dimensional symmetric body which is embedded in a fluid-saturated porous medium. It is assumed that the surface of the body is held at the constant temperature T^, while the ambient temperature is Too, where Tw > Too. If we assume that Darcy's law holds, that the Boussinesq approximation is valid, but that the LTE assumption does not apply, then the equations governing the two-temperature free convection flow model can be written in non-dimensional form as, see Rees and Pop (1999), du dv ^ + ^ = 0 , ox oy ^ oy

(6.26)

^ = Ra—5(x), ox oy

(6.27)

u^ + v^ = v^e + hie - 0), ox

(6.28)

oy

V20-/i7(l9-0) = 0 , where the nondimensional variables are defined as {x,y) = d{x,y),

{u,v) =

Tf = {T^u - Too)0 + Too ,

T^-

(6.29)

———{u,v), {pCp)fd

(6.30)

{Tw - TOQ)(J) -h Too •

Here x is the dimensional Cartesian coordinate which measures the distance from either the forward stagnation point along the surface of the symmetrical body or the leading edge, and y is in the normal direction. In addition the respective velocities are u and v, T is the temperature, and d is a suitably defined macroscopic length scale. The quantity S{x) depends on the shape of the heated surface with S{x) = \ corresponding to a vertical surface and S{x) — xiodi stagnation point flow such as on the underside of a hot circular cylinder. Further, h and 7 are dimensionless constants and Ra is the Darcy-Rayleigh number, which are defined as ekf '

(1 - e)A:s '

A^^f

D. A. S. REES AND I. POP

155

We introduce now a streamfunction i/i according to u = dip/dy and v = —dil^/dx. Equations (6.26)-(6.29) thus become V^t/i = Ra—S(x), oy

(6.32)

V=<,UW-.) = | ^ | ? - | i | ? . oy ox V^0 + /i7(<9-(/)) = 0 .

(6.33,

ox oy (6.34)

Let us now introduce the usual boundary-layer variables x = x,

t/ = Ra^/2j/,

^ = Ra~^/^?/)

(6.35)

= ^ ^ - ^ ^ , dy dx dx dy '

(6.37)

into equations (6.32)-(6.34) to obtain

^^HU-6) dy'^

(6.38)

^^Hj{0-(j>)=0, dy

where we have omitted terms which are asymptotically small, compared with the retained terms as Ra -^ oo. In equations (6.37) and (6.38) the dimensionless constant H is defined as h = R8iH, (6.39) where H — 0(1) a s R a —^ oo, allows the detailed study of how the boundary-layer undergoes the transition from strong thermal nonequilibrium near the body's leading edge to thermal equilibrium far from the leading edge. The boundary conditions of equations (6.36)-(6.38) are given by ^ = 0,

e = l,

(i)=l

at

2/ = 0,

x>0,

(6.40a)

-7r-->0, dy

6>->0,

(j)-^{)

as

y ^ oo,

x>0.

(6.40b)

These boundary conditions allow equation (6.36) to be integrated once to yield ^=eS{x). dy

(6.41)

The physical quantities of most interest are the fluid and solid phase Nusselt numbers, which are defined as

156

LOCAL THERMAL NON-EQUILIBRIUM IN POROUS MEDIUM CONVECTION

whereof = -kf{dTf/dy)y=o and g^s = -ks{dTs/dy)y=oand (6.35) into expression (6.42), we obtain Nuf

6.4.2

Nus

fde\

Ra^/2

Substituting variables (6.30) fd(t)\

Ra^/^

\dyJy=o'

\dy J y^o

Results for stagnation point flow

For stagnation-point flow we take S{x) = X ,

(6.44)

and write ^ - xf{y),

e = e{y),

(t> = (l>{y).

(6.45)

Substituting expressions (6.44) and (6.45) into equations (6.37), (6.38) and (6.41) leads to the following set of ordinary differential equations:

f ^e,

e" + fo' = H { e - ^ ) ,

(/>" - i f 7 ( 0 - e ) ,

(6.46)

and the boundary conditions (6.40) become /(0) = 0,

^(0) = 1,

0(0) = 1,

e(oo)-0,

0(00)-0,

(6.47)

where primes denote differentiation with respect to y. The ordinary differential equations (6.46), subject to the boundary conditions (6.47), with H and 7 as parameters were integrated numerically by Rees and Pop (1999) using the Keller-box method. The numerical results are summarized in Figure 6.1 where ^'(0) and (j)' (0) are represented for some values of H and 7. It is seen that when H is small, there is a very substantial difference between the surface rates of heat transfer of the fluid and solid phases, indicating that LTNE effects are stronger when H is small, which is not surprising since if is a measure of the ease with which heat is transferred between the phases. As H increases, the inter-phase local heat transfer becomes more effective and this means that the difference between the solid and fluid temperature fields also decreases in magnitude and LTE is recovered. Figure 6.1 also shows that the same qualitative effects are obtained as 7 decreases. Rees and Pop (1999) have also obtained asymptotic solutions of equations (6.46) for large and, respectively, small values of the parameter H. Thus, it is shown that for large values of H (^ 1), the surface rates of heat transfer for the two phases are given by

^'(0) = (1 + 1

-1/2

-1/2

n fi97=^^ 046360 1 1 0.62755- --——r^7F + --- >

"+^*

'^

J

^0.62755-riS£ + £ ^ U + ^ " (1 + 7)2

1 + 7 y if

(6.48)

157

D. A. S. REES AND I. POP

\J.\J

-0.1

iVV,..^^^^

Solid phase (/)'(0) 7 — 0.1

02

7 = 0.2 -0.3 7 = 0.5 -0.4 7= 1 -0.5

7=2 7=5

-0.6 -0.7^

'V — i n 7 lu

Fluid phase <9'(0) 2

4

^

8

6

10

Figure 6.1 The variation of 0' (0) and (f)' (0) with H for selected values of 7, as obtained by solving equations (6.46). The dotted lines correspond to the large-i7 asymptotic values given in expression (6.48).

These curves are also plotted in Figure 6.1 where excellent agreement with the full numerical solution of equations (6.46) for large values of H is observed. 6.4.3

Results for a vertical flat plate

The boundary layerflowinduced by a constant temperature heated surface has been studied by Rees and Pop (2000) and Mohamad (2001). We take S{x) = l,

(6.49)

and introduce the variables iP = X^l'^j[x,

ri),

e = e{x, 7]),

(f) = (/)(X, 7/) ,

V=

-T

/2

(6.50)

Then equations (6.37), (6.38) and (6.41) become (6.51) (6.52) (6.53)

15 8

LOCAL THERMAL NON-EQUILIBRIUM IN POROUS MEDIUM CONVECTION

subject to the boundary conditions /zzzO, 6 = 1, (j)=l ' ^ - ^ 0 , 0->O

at

77 = 0,

as

T] ^

00,

(6.54)

where primes denote partial derivatives with respect to rj. Equations (6.51)-(6.53) form a system of parabolic partial differential equations whose solution is nonsimilar due to the x-dependent forcing induced by the terms proportional to H. Such a nonsimilar set of equations is usually solved using a marching scheme. Beginning at the leading edge, where the system reduces to an ordinary differential system, the solution at each streamwise station is obtained in turn at increasing distances from the leading edge. Such solutions are typically supplemented by a series expansion for small values of x, and by an asymptotic analysis for large values of x. However, the present problem is not of this general nature since equation (6.53) cannot be solved at x = 0 while satisfying both the boundary conditions given for 0 in equation (6.54). Thus this boundary layer has the rather unusual property of having a double-layer structure near the leading edge, rather than far from it as is often the case. This complicates considerably the numerical simulation of equations (6.51)-(6.53) because it is now essential to derive the near-leading-edge solution carefully before starting the numerical work. A numerical solution which does not take this leading edge behaviour into account will exhibit numerical errors. An accurate numerical solution has been undertaken by Rees and Pop (2000) who also showed that the rates of heat transfer for the two phases are given by, as x -> 0,

drj

drj

~->/^x^/^

(6.55b)

77=0

and, as X ^ 00, de\

^^A=0

/ i\-i/2 f 1+ - J [60 + cox-i Inx -f 0(x-^)] ,

(6.56)

where ao = 1.61613, bo = -0.443748 and CQ = 0.0436897. The rate of heat transfer at the solid phase {d(j)/dr])r^=o, is identical to expression (6.56), the leading-order difference lying at 0(x~^). Having obtained the surface rates of heat transfer given by equation (6.55), it can be seen that these values rise indefinitely as x increases from zero; this would cause substantial inaccuracies in the numerical solution. Therefore, to integrate equations (6.51)-(6.53) numerically it is convenient to introduce the new variable ^ = {Hxf'

,

(6.57)

159

D. A. S. REES AND I. POP

to obtain f' = 0,

(6.58) (6.59) (6.60)

Equations (6.58)-(6.60) subject to the corresponding boundary conditions given in the paper by Rees and Pop (2000) have been integrated numerically using the Keller-box method along with the Newton-Raphson iteration scheme. The results obtained are summarized in Figures 6.2 and 6.3 for the rates of heat transfer and the isotherms for both the fluid and solid phases. It is clear, from Figure 6.2, that LTE is gradually attained as the distance from the leading edge, x is increased, and as H increases, since both are equivalent to an increase in ^. Also noticeable is the fact that the solid phase surface rate of heat transfer decreases towards zero as the leading edge is approached. This is a consequence of the fact that the thermal field in the solid phase occupies a region of finite thickness while the thermal boundary layer thickness of the fluid phase reduces to zero. This may be seen more graphically in Figure 6.3 where the isotherms for both phases are illustrated for both 7 = 1 and 7 = 10. It is also noted that the larger the value of 7, the smaller are the LTNE effects.

-0.1 | \ \ \ V ^

Solid phase (f)'{0) 7 = 0.1

|\\\

^^___^^^---'

7 = 0.2

-0.2

7 = 0.5

-0.3

" \\

A/

J^^^^^**^~~-----^^

7 = 1 7 = 2

-0 4

P/^^^

7 = 5 7 = 10

"—^Fluid phase (9'(0)

n <;

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Figure 6.2 The variation of & (0) and (\) (0) with ^ = {Hxfl'^ for selected values of 7, as obtained by solving equations (6.51)-(6.53).

160

LOCAL THERMAL NON-EQUILIBRIUM IN POROUS MEDIUM CONVECTION

(a)

Hx 1.5

Figure 6.3 Isotherms for thefluidphase (solid lines) and the solid phase (dashed lines) for (a) 7 = 1, and (b) 7 = 10. The isotherms are reconstructed from the boundary layer solutions.

One interesting aspect of Figure 6.3 is that the solid phase isotherms terminate on the horizontal axis at ^ = 0. While this might appear to be unphysical, the boundary layer is unable to give information about what happens in the close neighbourhood of the origin. Therefore it was essential to investigate that region by solving the full elliptic equations numerically; this was undertaken in Rees (2003). Figure 6.4 displays such a computation for A = 0.01 and for 7 == 0.3 and 3. For the smaller of the two values of 7 the thermal field of the solid phase extends a substantial distance below the origin, even though there is a mean upward flow due to the presence of the thermal boundary layer above. On the other hand, the thermal field of the fluid phase is confined to a: > 0. For the larger of the two values of 7, LTE is more nearly approximated since the two phases have almost identical thermal fields. 6.4.4

General comments

To our knowledge, no other papers have yet appeared which address free or mixed convective boundary layer flows when LTNE effects are strong, even though the research literature is dominated by papers for the corresponding LTE cases. The most obvious examples which remain to be considered include convection from a constant temperature (or uniform heat flux) horizontal surface, convection induced by a hot vertical cylinder.

161

D. A. S. REES AND I. POP

(a) ouur i; [ ; [

•! ' 1 — 1 — T T — ; — T T — ;

ty

;

r

r

i

500

i1 I ' 1 1

400

300

ml I i

200

1 1

i'lii '/;' /'

'/

looi iW'J: 1 \ /

^w^y\y 1 f\f\ L

0

20

40

60

80 100 120 140

y

0

20

40

60

80 100 120 140

y

Figure 6.4 Isotherms for thefluidphase (solid lines) and the solid phase (dashed lines) for (a) 7 = 0.3, and (b) 7 = 3. The value of h is 0.01. The isotherms are obtained from solutions of the full elliptic equations.

or the free convection plume. One might also extend such studies to cases where the surface temperature or heat flux is nonuniform. We think it highly likely that those LTE configurations which admit self-similarity do not retain that property when LTNE effects are strong, and that theflownear the leading edge is likely to retain the two-layer behaviour found by Rees and Pop (2000).

6.5

FORCED CONVECTION PAST A HOT CIRCULAR CYLINDER

We consider now the forced convection flow past a heated horizontal circular cylinder of radius a which is embedded in a fluid-saturated porous medium. It is assumed that the free-stream velocity is C/QO and that the temperatures of the cylinder and of the ambient fluid are T^ are T^. The basic equations for the steady forced convection flow past the cylinder under the condition of local thermal nonequilibrium between the fluid and the solid materials can be written in non-dimensional form as, see Rees et al. (2003):

ay

Idip

+

= 0,

(6.61)

162

LOCAL THERMAL NON-EQUILIBRIUM IN POROUS MEDIUM CONVECTION

Pe 1 Pe

di!^de_di^de_ + Hie - (f>)

Qj.2

J. Qj.

J.2 Q(^2

Qj.2

J. Qj.

J.2 Qop.

da dr

dr da

= H^{ct>-0),

(6.62) (6.63)

where (r, a) are the polar coordinates, and ip is the streamfunction which is defined as Id^ u = r da^

V =

"

dip dr

(6.64)

--

Further, i7 is a dimensionless constant and Pe is the Peclet number given by H =

ah

Fe =

Uoo{pCp)i'

UooCi{pCp)f

^^

ekf

(6.65)

Equations (6.61)-(6.63) are to be solved subject to the boundary conditions ^ = 0, ^ = 1, 0 = 1 ^ - > r s i n a , 0—>0, 0 - > O

on as

r = l, r—>oo.

(6.66)

The symmetry of the physical problem means that it necessary to consider only half of the physical domain, 0 ^ a ^ TT, with suitable symmetry conditions applied at a = 0, TT. For high values of the Peclet number, Pe (^ 1), the heat transfer problem reduces to a boundary-layer problem. However, the uniform flow past the cylinder is given by the solution of equation (6.61) and is, simply. 1 sm a

r

^

(6.67)

r.

for r large (r ^ 1). Substituting equation (6.67) into equations (6.62) and (6.63), we obtain the following equations for the functions 6 and (p: Pe

Qj.2

^2

J. Qj.

=

Q^2

de

1

2_ Pe

de_ - + ^ sm a + H{e-(f>), da \r r"^ J

„ , cosa-p— [•f)2

d^(j) Qj,2

1 dcj)

-f- y.-

Qj.

1 d^(j)

+ J.2

g^2

=

Hj{^-e).

(6.68)

(6.69)

Pop and Yan (1998) have shown that when the fluid and solid matrix are in local thermal equilibrium the thermal field is contained within a boundary-layer which is thin compared with the radius of the cylinder whenever Pe is large. On the other hand, solving equations (6.68) and (6.69) for the case of the two-equation model with Pe = 100, H = 0.3 and 7 = 1, we can illustrate the isotherm pattern in Figure 6.5, which is taken from the numerical study by Wong et al (2004). This figure shows that the thermal boundary-

163

D.A.S.REESANDI.POP

Figure 6.5 Computation of the temperature field for forced convection flow past a cylinder in a porous medium for Pe = 100, H = 0.3 and 7 = 1. Fluidflowsfrom left to right. The upstream (front) stagnation point corresponds to both a = TT and X = 0. The upper half shows isotherms for thefluidphase, while the lower half corresponds to the solid phase. Isotherms are drawn at intervals of 0.05.

layer thickens with distance from the upstream (front) stagnation point. Also seen is the difference in the boundary layer thicknesses between the fluid and solid phases. Having these observations in view, the following rescaled variable: (6.70) is introduced. Thus, equations (6.68) and (6.69) reduce to d^e

^

de

BO 2sma—+H(0-4>), da

= H^{4>-e),

(6.71) (6.72)

at leading order in Pe. It proves convenient to introduce the transformation

n

.sin ( I ) ,

X = cos(|),

(6.73)

where X = Q corresponds to the upstream (forward) stagnation point on the cylinder while X = \ corresponds to the downstream (rear) stagnation point where the flow detaches itself from the cylinder, see Figure 6.5. On using equation (6.73), equations (6.71) and (6.72) become

drf

H = 2X— + {0-), dr] dX 1-X2

+ 2ri-

(6.74)

164

LOCAL THERMAL NON-EQUILIBRIUM IN POROUS MEDIUM CONVECTION

97/2 ~ 1 - X2

{cf>-e).

(6.75)

and are subject to the boundary conditions 9 = 1,

= l

9-^0,

(j) ^ 0

on r? = 0,

(6.76)

as r ? ^ o o .

To this end, the local Nusselt numbers are given by Nuf =

Nus =

dr

(6.77) r-l

and, in terms of 77, these may be expressed more conveniently as It = Qs

Nuf Pel/2 sin(a/2)

89 \dri)^^^'

Pel/2 sin(a/2)

\dr))^^^'

(6.78a) (6.78b)

Also, the global rates of heat transfer are given by (6.79a) (6.79b) Equations (6.74) and (6.75) also form a parabolic partial differential system, which can be solved numerically by a standard finite-difference scheme except for the solution near X = 1, which is a singular point. At X = 1, equations (6.74) and (6.75) reduce to the following ordinary differential equations: e" + T - ^ 27/(9' = 0 ,

1+ 7

(j)" ^ 7-^27y(/>' = 0 , l-f-7

(6.80)

subject to 0(0) = 1, ^-^O,

^-^'O

(/>(0) = 1, as rj -^ oo.

(6.81)

The solution of these equations is given by 0 = (j) = erfc

7

1/2

1+ 7

and therefore LTE applies at the rear stagnation point.

rj

(6.82)

D. A. S. REES AND I. POP

165

Equations (6.74) and (6.75), subject to the boundary conditions (6.76), were integrated numerically by Rees et al. (2003) for many values of the parameters H and 7 with X in the range 0 ^ X ^ 1. Figure 6.6 shows the variation with X of the local rates of heat transfer qf and QS given by expressions (6.78) for H = 0.5, a typical case. It can be seen from Figure 6.6 that large discrepancies between the rates of heat transfer of the fluid and the solid occur especially when 7 is relatively small. This is to be expected because small values of 7 correspond directiy to cases where there is poor transfer of heat from the fluid phase to the solid phase. However, when 7 is relatively large, such as for 7 = 100, the temperature field of the solid phase is affected very strongly by the temperature of the fluid phase and the two heat transfer curves are almost coincident. Although most of the cases shown in Figure 6.6 exhibit strong LTNE effects, LTE is, as mentioned above, established at the rear stagnation point. This means that there are very rapid changes in the local values of both qf and ^s near X = 1 when 7 is small. For other values of H, the heat transfer curves are modified in the following ways. When H takes larger values the discrepancy between qf and qs reduces, while LTNE effects become even stronger when H takes smaller values. We note that Rees et al. (2003) also provide asymptotic expressions for the rates of heat transfer for both large and small values of iJ. An extension of this work to forced convection past a sphere has been undertaken by Kwan (2003).

Figure 6.6 Variation of the local rates of heat transfer around the cylinder with X for H = 0.5. The continuous curves correspond to qf and the dashed curves to qs. Curves correspond to 7 = 0.01, 0.1, 0.2, 0.5,1, 2, 5,10 and 100, and the heat transfer increases with increasing values of 7.

166

6.6

LOCAL THERMAL NON-EQUILIBRIUM IN POROUS MEDIUM CONVECTION

STABILITY OF FREE CONVECTION

The classical problem of stability is the Benard problem, which consists of a constant thickness horizontal plane layer of fluid heated uniformly from below. When the two plane bounding surfaces are filled with a porous medium we have what is called either the Darcy-Benard problem or the Horton-Rogers-Lapwood problem. A Rayleigh number is defined (see below) and it is found that the basic conduction state is stable, i.e. all possible perturbations decay, when the Rayleigh number is below a critical value. At higher values of the Rayleigh number convecting patterns form, the planform for which depend on the modelling of the porous medium, the geometry of system and the Rayleigh number, see Rees (2000, 2001). A basic linear stability analysis was undertaken by Banu and Rees (2002) who took the two-dimensional governing equations to be du dv ^ + ^ = 0 , OX oy

(6.83)

(6.84)

u = -^f., V= 6(pc)f ^

K dp ^-r -t-

II dy pfopK

iJi ox

fjL

(Tf - Too),

(6.85)

+ {pc)fu • VTf = ekfV^T, + h{Ts - Tf),

ST (1 - ^){pc)s-Q^ = (1 - e)ksV^Ts - h{Ts - Tf),

(6.86) (6.87)

where the 'hat' symbols denote dimensional quantities. Equations (6.83)-(6.87) may now be nondimensionalised using the transformations {x^y) = d{x^y),

(u,.)^^^^(u,.),

' ^ ^ ' ' ( 6 . 8 8 )

P=^^^P^

Tf = (ri-ru)e + Tu, rs = (Ti-T,)0 + ru, and by introducing the streamfunction, ijj, according iou = —^Ijy and v = ipx- Equations (6.83)-(6.87) become ^xx + '^yy = ROx , et - ^pyO, + ^l^Jy = e,x + 9yy -f H{(l) - 6), Oi(l>t = (t>xx + (l>yy + lH(e

-

(6.89a) (6.89b)

(f)) ,

(6.89c)

where 9 and 0 are the respective temperatures of the fluid and solid phases. The four governing parameters p,gm

- Tu)Kd e/i/^f

ek (1 - e)ks

„

hd' ek{

{pc)s k [pc)f ks

D. A. S. REES AND I. POP

167

are the Darcy-Rayleigh number, a porosity-modified conductivity ratio, a scaled interphase heat transfer coefficient, and a diffusivity ratio, respectively. The basic conducting state which we analyse for stability is given byxp = 0,6 = (j) = l—yWe perturb equations (6.89) about this basic solution by setting 7/; = ^ ,

e = i-y

+ Q^

0^1_2; + #

(6.91)

and linearising. Hence we obtain ^xx + ^yy = RQx , 0 , = 0^^ -f Qyy + *x + H{^ - 0 ) , a^t = ^xx + ^yy -\- lH{e

- $),

(6.92a) (6.92b) (6.92c)

which admit solutions of the form ^ = Ai sin Try cos mx ,

Q = A2 sin ny sin mx ,

^ = A3 sin ny sin mx ,

(6.93)

where m is the wavenumber and where Ai, A^ and A3 are constants which satisfy a homogeneous system of three linear equations forming an eigenvalue problem for R in terms of iJ, 7 and m. We obtain the following expression for R\ R —

7,—

TT^ -h vn? -h "yH

(6.94)

After minimising R with respect to m, we obtain the critical values of Re and rric shown in Figures 6.7 and 6.8. Figure 6.7 shows that Re rises from 47r^ when H is small to a value given by Re — 47r^(l + 7 ) / 7 when B. is large. It should be noted that R is the Darcy-Rayleigh number based on the properties of the fluid while R'yl{l -h 7) is the Darcy-Rayleigh based on the mean properties of the porous medium. For large values of if, we therefore recover the LTE result namely that i?c7/ (1+7) = 47r^ with the wavenumber mc = TT. In this case the temperature fields are almost identical. On the other hand, when H is small, the perturbation in the solid phase temperature is much smaller than that in the fluid phase, and therefore the porous medium acts in a way which is independent of the solid phase. Therefore we obtain the limiting behaviour that Re — 47r^ with rrie = n. Between these two extremes the critical Rayleigh number increases with H, while rric rises to a maximum and decays back to TT, as shown in Figure 6.8. Banu and Rees (2002) also determine detailed asymptotic representations for Re and rrie in the large-if and small-if limits, and in the large-7 and small-7 limits. The above analysis merely gives the criterion for the onset of convection. Weakly nonlinear theory then gives an indication of the planform the ensuing convection takes. The ansatz given in equation (6.93) yields a two-dimensional roll pattern, and more complicated patterns may be obtained by superimposing rolls of other orientations. For the

168

LOCAL THERMAL NON-EQUILIBRIUM IN POROUS MEDIUM CONVECTION

0.0 0.5

1.0

1.5 2.0 2.5 logio H

3.0 3.5 4.0

Figure 6.7 Variation of the critical Rayleigh number, R^ with log^o H for specific values of 7.

Figure 6.8 Variation of the critical wavenumber with logjo ^ for specific values of 7.

classical Darcy-Benard convection problem using the one-temperature model it is a fairly straightforward, if lengthy, matter to show that rolls are favoured over other patterns such as rectangles, squares or hexagons in a horizontally unbounded layer, see Rees (2001). In unpublished work the present authors have considered such a weakly nonlinear analysis for

D.A.S.REESANDI.POP

169

the above two-temperature model which concludes that rolls remain the favoured pattern for all combinations of H and 7 values. An extension of the work of Banu and Rees (2002) was undertaken by Postelnicu and Rees (2003) to allow for the additional effects of the Brinkman terms in the momentum equation. These latter authors assumed stress-free boundaries since it is then possible to solve the full equations analytically. The extra parameter is now the Darcy number, and the corresponding expression for R in terms of D, H and 7 is given by R =

TT-^ [1 +

m^

D{TT^

+ m^)]

TT^ -f m^ + H{1 -h 7) TT^ -h m^ 4- jH

(6.95)

Detailed results are, of course, different from those of Banu and Rees (2002) but share the same qualitative characteristics. We are unaware of any computations of strongly nonlinear convection using the twotemperature model except for a very early paper by Combamous and Bories (1974). In this work a finite difference solution was carried out for a unit aspect ratio cavity at a Rayleigh number of 200, which is well above 47r^. Comparisons where made between solutions corresponding to different values of the parameters which we denote by H and 7Although the following does not address the aspect of stability, there exist two papers which also consider free convection in cavities but with sidewall heating and cooling. The aim of the paper by Baytas and Pop (2002) was to describe the steady free convection in a porous two-dimensional square cavity using a the LTNE model given in equations (6.1) and (6.2). The cavity is bounded by isothermal vertical walls at different temperatures and has adiabatic horizontal walls. The governing equations in terms of the non-dimensional streamfunction and temperature fields were solved numerically using the alternating direction implicit (ADI) scheme. The reported results reveal, under others, that the values of the average Nusselt number for the solid phase are smaller than those for the fluid phase, respectively, because the solid phase temperature field undergoes less of a deformation from the uniform-gradient conduction state. However, these values tend asymptotically to the same values as both the modified conductivity ratio, 7, and the dimensionless scaled values of the volumetric heat transfer coefficient between the phases, H, increase. This is in qualitative agreement with the above-quoted results of Rees and Pop (1999, 2000). for the case of boundary layer flow over a vertical surface or near the lower stagnation point of a cylindrical surface. A similar study was carried out by Mohamad (2000) who concentrated on the validity of LTE assumption when using the Brinkman-Forchheimer extension to Darcy's law. Finally we need to mention the numerical study of Baytas (2003) who considered LTNE effects of convection in a square cavity which is induced by internal volumetric heating within the solid phase, and with the Brinkman-Forchheimer extended Darcy law for the momentum equations. Attention was focused on the steady-state cases which arises when the Rayleigh number is not too large.

170 6.7

LOCAL THERMAL NON-EQUILffiRIUM IN POROUS MEDIUM CONVECTION CONCLUSIONS

In this brief review we have concentrated on presenting the various two-equation models which have been proposed to account for LTNE in porous convection, and have described many recent studies including external boundary layers, free convection and stability. We have omitted both the derivation of these models, and their application to forced convective channel flows. Although there are an increasing number of papers available in the research, there remain many topics which are, as yet, unresearched. These include the following. (i) Free, mixed and forced convection plumes and other boundary layers, such as that induced by a heated vertical cylinder, (ii) The stability of thermal boundary layer flows, (iii) Unsteady boundary layer flows, including those set up by a sudden change in the boundary temperature or heat flux, and those induced by time-periodic forcing, (iv) The effects on leading-edge singularities. (v) Studies with the onset of unsteady convection in porous layers heated from below, (vi) Studies with convection in horizontal cylinders, (vii) Studies with application to turbulent convection. Notwithstanding the above paragraph, most of the research literature which either includes LTNE effects or examines its validity, assumes the simplest model given in equations (6.1) and (6.2). The more complicated models which exist, such as that given by equations (6.19) and (6.20), or where dispersion conductivities are included, equations (6.17) and (6.18), have not yet been applied even to classical flows such as Darcy-Benard convection and free convection boundary layer flows. However the overall aim is that an improved understanding of the fundamental convective flow processes in porous media using the LTNE model can serve to stimulate new innovations, as well as lead to improvements in the performance, reliability and costs of many existing heat transfer devices.

ACKNOWLEDGEMENTS The second author (IP) wishes to express his cordial thanks to the Royal Society forfinancialsupport. The authors wish to thank Professor D. A. Nield for his useful comments.

REFERENCES Al-Nimr, M. A. and Abu-Hijleh, B. A. (2002). Validation of thermal equilibrium assumption in transient forced convectionflowin porous channel. Transport in Porous Media 49, 127-38.

D. A. S. REES AND I. POP

171

Al-Nimr, M. A. and Kiwan, S. (2002). Examination of the thermal equilibrium assumption in periodic forced convection in a porous channel. J. Porous Media 5, 3 5 ^ 0 . Alazmi, B. and Vafai, K. (2000). Analysis of variants within the porous media transport models. Trans. ASME J. Heat Transfer 122, 303-26. Amiri, A. and Vafai, K. (1994). Analysis of dispersion effects and non-thermal equilibrium, nonDarcian, variable porosity incompressible flow through porous media. Int. J. Heat Mass Transfer 37, 939-54. Amiri, A. and Vafai, K. (1998). Transient analysis of incompressible flow through a packed bed. Int. J. Heat Mass Transferal, 4259-79. Banu, N. and Rees, D. A. S. (2002). The onset of Darcy-Benard convection using a thermal nonequilibrium model. Int. J. Heat Mass Transfer AS, 2221-8. Baytas, A. C. (2003). Thermal non-equilibrium natural convection in a square enclosure filled with a heat-generating solid phase, non-Darcy porous medium. Int. J. Energy Res. 11, 975-88. Baytas, A. C. and Pop, I. (2002). Free convection in a square porous cavity using a thermal nonequilibrium model. Int. J. Therm. Sci. 41, 861-70. Carbonell, R. G. and Whitaker, S. (1984). Heat and mass transfer in porous media. In Fundamentals of transport phenomena in porous media (eds J. Bear and M. Y. Corapcioglu). Martinus Nijhoff, Dordrecht. Cheng, P. and Hsu, C.-T. (1998). Heat conduction. In Transport phenomena in porous media (eds D. B. Ingham and I. Pop), pp. 57-76. Pergamon, Oxford. Combamous, M. and Bories, S. (1974). Modelisation de la convection naturelle au sein d'une couche poreuse horizontal I'aide d'un coefficient de transfert solide-fluide. Int. J. Heat Mass Transfer 17, 505-15. Dixon, A. G. and Cresswell, D. L. (1979). Theoretical prediction of effective heat transfer parameters in packed beds. AIChE J. 25, 663-76. DuUien, F. A. L. (1979). Porous media fluid transport and pore structure. Academic Press, New York. Fourie, J. G. and Du Plessis, J. P. (2003a). A two-equation model for heat conduction in porous media (I. Theory). Transport in Porous Media 53, 145-61. Fourie, J. G. and Du Plessis, J. P. (2003b). A two-equation model for heat conduction in porous media (II. Application). Transport in Porous Media 53, 163-74. Hwang, G. J., Wu, C. C , and Chao, C. H. (1995). Investigation of non-Darcian forced convection in an asymmetrically heated sintered porous channel. Trans. ASME J. Heat Transfer 111, 725-32. Kim, S.J. and Kim, D. (2001). Thermal interaction at the interface between a porous medium at an impermeable wall. Trans. ASME J. Heat Transfer 123, 527-33. Kim, S. Y. and Kuznetsov, A. V. (2003). Optimization of pin-fin heat sinks using anisotropic local thermal nonequilibrium porous model in a jet impinging channel. Numer. Heat Transfer, Part A 44,771-87. Kuwahara, F , Shirota, M., and Nakayama, A. (2001). A numerical study of interfacial convective heat transfer coefficient in two-energy model for convection in porous media. Int. J. Heat Mass Transfer AA, 1153-9. Kuznetsov, A. V (1998). Thermal nonequilibrium forced convection in porous media. In Transport phenomena in porous media (eds D. B. Ingham and I. Pop), pp. 103-29. Pergamon, Oxford.

172

LOCAL THERMAL NON-EQUILIBRIUM IN POROUS MEDIUM CONVECTION

Kwan, H. Y. H. (2003). The effects of local thermal nonequilibrium on forced convection past a hot sphere embedded in a porous medium. Final year undergraduate thesis, AT26/2003. Department of Mechanical Engineering, University of Bath, UK. Magyari, E. and Keller, B. (2002). Note on 'A two-equation analysis of convection heat transfer in porous media' by H. Y. Zhang and X. Y. Huang. Transport in Porous Media 46, 109-12. Minkowycz, W. J., Haji-Sheik, A., and Vafai, K. (1999). On departure from local thermal equilibrium in porous media due to rapidly changing heat source: the Sparrow number. Int. J. Heat Mass Transfer 42, 3373-85. Mohamad, A. A. (2000). Nonequilibrium natural convection in a differentially heated cavity filled with a saturated porous matrix. Trans. ASME J. Heat Transfer 111, 380-4. Mohamad, A. A. (2001). Natural convection from a vertical plate in a saturated porous medium: nonequilibrium theory. J. Porous Media 4, 181-6. Nield, D. A. (1998). Effects of local thermal nonequilibrium in steady convective processes in a saturated porous medium: forced convection in a channel. J. Porous Media 1, 181-6. Nield, D. A. (2002). A note on the modeling of local thermal non-equilibrium in a structured porous medium. Int. J. Heat Mass Transfer 45, 4367-8. Nield, D. A. and Bejan, A. (1999). Convection in porous media (2nd edn). Springer-Verlag, New York. Pop, I. and Yan, B. (1998). Forced convection flow past a cylinder and a sphere in a Darcian fluid at large Peclet numbers. Int. Comm. Heat Mass Transfer 25, 261-7. Postelnicu, A. and Rees, D. A. S. (2003). The onset of Darcy-Brinkman convection in a porous layer using a thermal nonequilibrium model. Part I. Stress free boundaries. Int. J. Energy Res. 11, 961-73. Quintard, M. (1998). Modelling local non-equilibrium heat transfer in porous media. In Proceedings of the Ilth international heat transfer conference, Kyong-ju, Korea, August, Vol. 1, pp. 279-85. Quintard, M. and Whitaker, S. (1993). One and two-equation models for transient diffusion processes in two-phase systems. Adv. Heat Transfer 23, 369-464. Quintard, M. and Whitaker, S. (2000). Theoretical analysis of transport in porous media. In Handbook of porous media (ed. K. Vafai), pp. 1-51. Marcel Dekker, New York. Quintard, M., Kaviany, M., and Whitaker, S. (1997). Two-medium treatment of heat transfer in porous media: numerical results for effective properties. Adv. Water Res. 20, 77-94. Rees, D. A. S. (2000). The stability of Darcy-Benard convection. In Handbook of porous media (ed. K. Vafai), pp. 521-58. Marcel Dekker, New York. Rees, D. A. S. (2001). Stability analysis of Darcy-Benard convection. Unpublished lecture notes from the summer school on porous medium flows, Neptun, Constanta, Romania, 25-29 June. Rees, D. A. S. (2003). Vertical free convective boundary-layer flow in a porous medium using a thermal nonequilibrium model: elliptic effects. / Appl. Math. Phys. (TAMP) 54, 437-48. Rees, D. A. S. and Pop, I. (1999). Free convective stagnation-point flow in a porous medium using a thermal nonequilibrium model. Int. Comm. Heat Mass Transfer 16, 945-54. Rees, D. A. S. and Pop, I. (2000). Vertical free convection boundary layer flow in a porous medium using a thermal nonequilibrium model. / Porous Media 3, 3 1 ^ 4 . Rees, D. A. S., Bassom, A. P., and Pop, I. (2003). Forced convection past a heated cylinder in a porous medium using a thermal nonequilibrium model: boundary layer analysis. Eur J. Mech. B-Fluids 11, 413-S6.

D. A. S. REES AND I. POP

173

Schumann, T. E. W. (1929). Heat transfer; a liquid flowing through a porous prism. J. Franklin Inst 208,405-16. Vafai, K. and Amiri, A. (1998). Non-Darcy effects in confined forced convective flows. In Transport phenomena in porous media (eds D. B. Ingham and I. Pop), pp. 313-29. Pergamon, Oxford. Wakao, N. and Kaguei, S. (1982). Heat and mass transfer in packed beds. Gordon and Breach, New York. Wakao, N., Kaguei, S., and Funazkri, T. (1979). Effect of fluid dispersion coefficients on particleto-fluid heat transfer coefficients in packed beds. Chem. Eng. Sci. 34, 325-36. Whitaker, S. (1999). The method of volume averaging. Kluwer, Dordrecht. Wong, W. S., Rees, D. A. S., and Pop, I. (2004). Forced convection past a heated cylinder in a porous medium using a thermal nonequilibrium model: finite Peclet number effects. Int. J. Therm. Sci. 43, 213-20. Zhang, H. Y. and Huang, X. Y. (2001). A two-equation analysis of convection in porous media. Transport in Porous Media 44, 305-24. Zhang, H. Y. and Huang, X. Y. (2002). Reply to the note by E. Magyari and B. Keller. Transport in Porous Media 46, 113-15.

THREE-DIMENSIONAL NUMERICAL MODELS FOR PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS WITHIN POROUS MEDIA A. NAKAYAMA and R KUWAHARA Department of Mechanical Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu, 432-8561 Japan email: [email protected] and [email protected]

Abstract Recent investigations on three-dimensional numerical models for periodically fully-developed flow and heat transfer in anisotropic porous media have been reviewed in this chapter 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 were carried out to determine macroscopic parameters, such as the permeability and the interfacial heat transfer coefficient, and the results were compared against available empirical formulas. A quasi-three-dimensional calculation procedure has been proposed and its details are presented here. This economical procedure was used to obtain the results for three-dimensional heat and fluid flow through a bank of cylinders in yaw. Furthermore, the large eddy simulation study for turbulence in porous media was carried out to elucidate complex turbulent flow characteristics associated with porous media. Keywords: heat and fluid flow, porous media, numerical model, three-dimensional computation

7.1

INTRODUCTION

Fluid particles experience complex three-dimensional motions as they pass through a microscopic porous structure. Fortunately, only macroscopic hydrodynamic and ther174

A. NAKAYAMA AND F. KUWAHARA

175

modynamic quantities, such as the pressure drop, mass flow rate and effective thermal conductivity are needed in engineering applications of porous media. Thus, in the study of porous media, a great deal of empirical effort was directed in the past towards finding a passage from complex three-dimensional phenomena in a pore scale to macroscopic transport phenomena in an engineering scale, e.g. Darcy (1856) and Ergun (1952). Comprehensive review articles may be found elsewhere, e.g. Ingham and Pop (1998, 2002) and Nield and Bejan (1999). Despite its three-dimensional geometrical complexities, the macroscopic hydrodynamic and thermodynamic behavior in porous media can be obtained from the direct application of first principles to viscous flow and heat transfer at a pore scale. However, in reality, it is hardly feasible to resolve such three-dimensional details of the flow and heat transfer fields within a real porous medium, even with the most powerful super-computer available today. Instead of dealing with real porous media, we may model a porous medium in terms of obstacles arranged in a regular pattern, and solve the set of the microscopic governing equations, upon exploiting periodic boundary conditions. The microscopic numerical results obtained at such a pore scale can be processed to extract the macroscopic hydrodynamic and thermal characteristics in terms of the volume-averaged quantities. We may carry out such direct numerical computations for various macroscopic flow angles and take ensemble averages of the numerical results. Such averaging over the macroscopic angles should lead to the macroscopic values close to those in real porous media of randomly oriented structure. In this way, we can determine the empirical constants, such as the permeability and dispersion coefficients, without any empiricism. It is quite likely that more than one unit may be involved in recurring patterns. Exploitation of periodic boundary conditions excludes such recurring patterns. However, the effect of such flow motions extending over more than one unit on the macroscopic quantities is believed to be only secondary, and may well be neglected. A number of numerical and analytical models, based on periodic microstructures, were proposed to formulate explicit expressions for the macroscopic pressure gradient, thermal dispersion and interfacial heat transfer coefficient. Eidsath et al. (1983), Coulaud et al. (1988), Sahraui and Kaviany (1991), and Fowler and Bejan (1994) have carried out twodimensional numerical simulations for laminar flows across banks of circular cylinders, whereas Kuwahara et al. (1994) investigated a collection of square rods to cover a wide range of porosity, virtually from zero to unity. Only a limited number of three-dimensional numerical computational models are available in the literature. Larson and Higdon (1989) conducted three-dimensional flow analyses assuming Stokes flows through a lattice of spheres, whereas Kuwahara et al (1994) and Nakayama et al (1995) numerically treated fully elliptic laminar flows through a lattice of spheres and cubes to study not only the Darcy contribution but also the porous inertial contribution to the macroscopic pressure drop. Recently, a great deal of effort has been directed towards investigations of three-dimensional characteristics of flow and heat transfer in anisotropic porous media, with a view to applying the volume-averaging theory developed in porous media to manmade structures

176

PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS

such as in complex heat exchangers, see Nakayama et al. (2002). Turbulent flows in porous media are also of great interest since such flows encountered in engineering applications are most likely to be turbulent, even in the presence of manmade porous structures. Nakayama and Kuwahara (1999) proposed macroscopic turbulence transport equations based on the conventional two-equation turbulence model. However, in order to understand fully the three-dimensional characteristics of laminar and turbulent flows in porous media, either direct numerical simulations or large eddy simulations of flow in porous media are definitely required, instead of appealing to the Navier-Stokes equations or their Reynolds-averaged version. In this chapter, we shall review some of our recent investigations associated with threedimensional numerical models for periodically fully-developed flow and heat transfer in anisotropic porous media, including large eddy simulations of turbulent flow in porous media. Firstly, three-dimensional laminar flows around collections of spheres and cubes are discussed, and compared against available empirical formulas. Secondly, a quasithree-dimensional calculation procedure is presented, and this economical procedure is used to obtain the results for three-dimensional heat and fluid flow through a bank of cylinders in yaw. Finally, the large eddy simulation study for turbulence in porous media is carried out to elucidate complex turbulent flow characteristics associated with porous media.

7.2

THREE-DIMENSIONAL NUMERICAL MODEL FOR ISOTROPIC POROUS MEDIA

In this first section, we shall consider one of the most fundamental geometrical configurations for representing isotropic porous media, namely, a collection of cubes in a simple cubic arrangement. A three-dimensional numerical procedure proposed by Nakayama et al (1995) is described below, to determine the permeability of the isotropic porous medium, from a series of numerical experiments. 7.2.1

Numerical model

We consider a macroscopically uniform flow, meandering through a three-dimensional array of obstacles, namely, an infinite number of cubes placed in a regular fashion in an infinite space, as shown in Figure 7.1. Only one structural unit of size H x H x H, as shown in Figure 7.2 may be taken as a calculation domain in the consideration of the geometric periodicity. Here, for simplicity, we shall focus on the case of cubes. Essentially the same procedure can be taken to treat the case of spheres. The direction of the microscopically uniform flow is expressed in terms of the orthogonal unit vectors (/,7n,n), as illustrated in Figures 7.2 and 7.3, such that the macroscopic velocity field follows: (u) — \{u)\ (cosaZ -l-cos^m -l-cos7n),

(7.1)

A. NAKAYAMA AND F. KUWAHARA

177

^Zm^ Figure 7.1 Arrangement of cubes representing a porous medium.

Figure 7.2 Structural unit for the computational domain. where the directional cosines are used and {u)=

Jv

u dV

(7.2)

is the volume-averaged velocity vector averaged over a structural volume element V, i.e. the Darcian velocity vector or apparent velocity vector. The directional cosines of the volume-averaged velocity satisfy the obvious relationship, namely, cos^ a -h cos^ /3 + cos^ 7 = 1.

(7.3)

17 8

PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS (0°,0°)..

^ - - ' ^ ^ ^ ^

(45°, 0°)

1-^-

(45°, 35.26°

^

^ ( 4 5 ° , 35.26°)

H|H

^ __^^ ^ ---

•(45°,0°)

(0°,0°)

Figure 7.3 Typical macroscopicflowdirections for computations. This relation may be rewritten equivalently using the cross flow angle a' projected onto the x-y plane, see Figure 7.2, as follows: cos a = sin 7 cos a',

cos ^ = sin 7 sin a'

(7.4)

such that {u) = \{u)\ (cos a' sin7i + sin a' sin7 m + c o s 7 n ) . 7.2.2

(7.5)

Governing equations and periodic boundary conditions

In the numerical computations, periodic boundary conditions are often used to obtain the velocity and pressure fields within manmade periodic structures, such as banks of tubes, arrays of fins, and conduits with periodically-shaped wall surfaces. Patankar et al. (1977) prescribed the pressure drop over one structural unit to attack the problem of fully-developed flow and heat transfer in ducts having streamwise-periodic variations of cross-sectional area, while Nakayama et al. (1995) and Kuwahara et al (2001) chose to prescribe the mass flow rate (rather than the pressure drop) to obtain the fully-developed velocity and pressure fields within three-dimensional periodic arrays. We shall follow the latter procedure. Usually, the local control volume V for the volume averaging is much smaller than a macroscopic characteristic length and can be taken as H^ for this periodic structure. Due to the periodicity of the model, only one structural unit, as indicated in Figure 7.2, may be taken as a calculation domain. The governing equations for the detailed flow field, namely, the equations of continuity and momentum are given as follows: V " a = 0, V 'uu = — V p + z/V^u . P The boundary and compatibility conditions are given as follows:

(7.6) (7.7)

A. NAKAYAMA AND F. KUWAHARA

179

u = 0,

(7.8)

on the solid walls: on the periodic boundaries: '^\x=-H/2

-

'^\x=H/2

(7.9a)

U y=-H/2

-

'^\y=H/2

(7.9b) (7.9c)

'^\z=H/2

'^\z=-H/2

where the origin of the Cartesian coordinates (x, y, z) is set in the center of the structural unit ( - i f / 2 ^ X ^ H/2, -H/2 ^y ^ H/2, -H/2 ^ z ^ H/2). Furthermore, the mass flow rate constraints are given by rH/2

I

J-H/2

H/2

/

H/2

nH/2

/

J-H/2

udydz x=-H/2

/

nH/2

-H/2

J-H/2

H/2

nH/2

-H/2 -H/2

fH/2 n/z

dxdz y=-H/2

/

/

nH/2

udydz

J-H/2 J-H/2

nH/2 nn/z

H/2 -H/2

J-H/2 J-H/2

H/2 n/z

nnH/2

= H"^ COS x=H/2

dxdz

a{\u\),

(7.10a) ^H'^ COS I3{\u\),

y=H/2

(7.10b) /

-H/2

/

J-H/2

wdxdy Z=:-H/2

/

H/2 J-J-H/2 -H/2

=

wdxdy

H^cosj{\u\).

z=H/2

(7.10c) We shall define the Reynolds number, based on the Darcian velocity \{u)\ and length of structural unit H, as \{u)\H RCH = (7.11) The porosity is varied changing the ratio of the size of cube D to the unit cell size H as 3

€=

7.2.3

1

(7.12)

Method of computation

The governing equations (7.6) and (7.7), subject to the foregoing boundary and compatibility conditions (7.8)-(7.10), were numerically solved using the SIMPLE algorithm proposed by Patankar and Spalding (1972). The pressure-velocity coupling based on the SIMPLE algorithm was adopted to correct both the pressure and velocity fields simultaneously. The calculation starts by solving the three momentum equations and subsequently the estimated velocity field is corrected by solving the pressure correction equation reformulated from the discretized continuity and momentum equations such that the velocity

180

PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS

field fulfils the continuity principle. As the w, f and w velocity fields are established, the remaining scalar transport equations, if any, are solved. Convergence was measured in terms of the maximum change in each variable during an iteration. The maximum change allowed for the convergence check was set to 10~^, as the variables were normalized by appropriate references. The hybrid scheme was adopted for the advection terms. Further details on this numerical procedure can be found in Patankar (1980) and Nakayama (1995). For the three-dimensional laminar flow cases, all computations have been carried out for a one structural unit H x H x H using non-uniform grid arrangements with 45 x 45 x 45, after comparing the results against those obtained with 6 1 x 6 1 x 6 1 f o r some selected cases, and confirming that the results are independent of the grid system. All computations were performed using the computer system at Shizuoka University Computer Center. 7.2.4

Macroscopic pressure gradient and permeability

Typical velocity vector plots obtained on the z = 0 plane for the cases of 7 = 7r/2 and Re/f = 10 are presented in Figure 7.4 for two different macroscopic flow angles, namely, a = 0 (horizontal flow) and a — n/i (diagonal flow). Both the velocity vector plots exhibit parabolic profiles, as in a fully-developed channel flow. Hence, it is expected that the viscous force is more significant than the inertial contribution. Thus, the pressure drop is mainly due to the viscous force in this range of comparatively small Reynolds number. The macroscopic pressure gradient, measured along the macroscopic flow direction, may be evaluated by substituting the microscopic results into the following expression:

Figure 7.4 Velocity vector plots for a unit structure when Ren = 10.

181

A. NAKAYAMA AND F. KUWAHARA

cos a

ds

(H-D)/2

J-(H D)/2

+

cos/? COS7

r^"-'D)/2

i-{H-D)/2

L

-{H-D)/2 Ati-i r(H-D)/2

D)/2 J-{H- D)I2 D)/2 MH-iD)/2 n"-'

+ H{m-D^)J_^HJ-(H-

D)I2 J-(HJ-{H-D)/2

{p\x=-H/2-P\x=H/2)dydz {P\y=-1H/2-P\y=H/2)dzdx H/2-

P\z=H/2)^^^y^

(7.13)

where

{pY

4/v/ dV

(7.14)

is the intrinsic average pressure of the fluid. The macroscopic pressure gradients, obtained for various flow angles and Reynolds numbers, show that the gradients, when divided by the product // |(tx)|, remains constant for all data within the range of Ren < 10, where the viscous force is the major cause for the pressure drop, namely, d{p)f 1 ^ 1_ (7.15) ds fi\{u)\ K' This corresponds to the inverse of the directional permeability K, and clearly suggests that Darcy's law holds for this range of small Reynolds number. The directional permeability determined for various macroscopic flow angles a is presented in Figure 7.5 for the case of e = 0.5 and 7 = 7r/2. It is rather surprising to note that the directional permeability stays constant, irrespective of the flow angle a. This, together with other sets of data, reveals that the directional permeability does not depend on either a or 7. Hence, the

Figure 7.5

Directional permeability for a collection of cubes when e = 0.5 and 7 = 90°.

182

PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS

three-dimensional model of a simple cubic arrangement may be regarded as one of the numerical models faithfully representing an isotropic porous medium. Exhaustive three-dimensional numerical computations were conducted for low Reynolds numberflowsover a wide range of the porosity e, using collections of cubes, see Nakayama et al. (1995), and spheres, see Kuwahara et al. (1994), in a simple cubic arrangement. The resulting expressions for the permeability run as follows: ^3

^ = TTTTT 2 ^^ 152 ( T - 6 ) '

for ^^t)es,

^3

K =

^TD^ for spheres. 147 ( 1 - e ) '

(7.16a) (7.16b)

For the case of spheres and cubes, the symbol D stands for the diameter of the sphere and for the side length of the cube, respectively. Both expressions (7.16a) and (7.16b) closely follow the empirical expression obtained by Ergun, namely, ^3

K=

7.3

^i?2 150(1-6)^

(7.17)

QUASI-THREE-DIMENSIONAL NUMERICAL MODEL FOR ANISOTROPIC POROUS MEDIA

The volume-averaging theory, e.g. Cheng (1978), Vafai and Tien (1981), Quintard and Whitaker (1993), and Nakayama (1995), developed in the field of porous media may be exploited to attack a number of complex heat andfluidflowswithin heat transfer equipment in engineering applications. The heat transfer equipment may consist of banks of tubes, arrays of fins and conduits with periodically-shaped wall surfaces, which may well be modeled as anisotropic porous media. Naturally, numerical models for such anisotropic porous media are of great interest in engineering applications. In this section, we study three-dimensional heat and fluid flow through anisotropic porous media, so as to develop a versatile numerical procedure to deal with complex heat and fluid flow in heat transfer equipment. 7.3.1

Periodic thermal boundary conditions

The prescription of the periodic boundary conditions for the velocity field, or the pressure field, is rather straightforward, as we have already studied in the preceding section. The velocity profiles at both the upstream and downstream boundaries must be identical. However, that of the temperature field requires some consideration when the surface wall temperature is kept constant. The energy conservation equation is given by pCp^V ' {uT) = kfV^T,

(7.18)

183

A. NAKAYAMA AND F. KUWAHARA

and the solid wall boundary condition is at the temperature (7.19)

T = T^

Naturally, the temperature difference between thefluidand solid wall becomes vanishingly small at the fully-developed stage, as in the case of thermally fully-developed tube flow with a uniform surface temperature. Our literature survey has revealed that no explicit thermal boundary conditions for the periodic boundaries are available for analyzing threedimensional flow and heat transfer within a three-dimensional periodic structure. In this section, we shall obtain explicit thermal boundary conditions for the case of fully threedimensional flow and heat transfer within a three-dimensional periodic structure with uniform surface temperature, appealing to the volume-averaging theory, extensively used in the field of porous media. The volume-averaged version of the energy equation (7.18) under a macroscopically steady and uniform velocity field with negligible macroscopic longitudinal conduction reduces to PCpf

\{u)\

= -hfaf {{ry - (Ty)

ds where s is the coordinate measured along the macroscopic flow direction, and (T) /,« _

(7.20)

(7.21)

TdV

Vf,s Jv,.,

such that (T)^ and (T)* (= T^) denote the intrinsic averaged temperatures of the fluid and that of the structure, respectively. Note that Vf and Vg are the volumes of fluid and solid, respectively, within the structural elementary volume V (— Vf + Vj). Moreover, the interfacial heat transfer coefficient on the right-hand side of equation (7.20) is defined by hf

(7.22)

{TY - {T)f

where rij is the unit vector normal to the interface pointing from the fluid side to the solid side. The net heat transfer between the fluid and solid is given by hfaf {{T)^ — {TY), where a/ is the specific interfacial area, i.e. the interfacial area per unit volume. Since the surface temperature of the structure {TY is constant, equation (7.20) naturally yields the macroscopic temperature field as follows: (T)^-(T)^ = ((r)^-(r)^)

.exp

afhf [ PCpf 1(^)1

{s - Sref)

(7.23)

Naturally, the correct set of the periodic thermal boundary conditions must lead to the microscopic temperature field which complies with the macroscopic temperature field as given by the expression. In other words, the resulting microscopic temperature field, when averaged spatially, must yield the macroscopic temperature field as given by equation (7.23). The detail of this argument may be found in Nakayama et al. (2004), which finally

184

PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS

reveals the following thermal boundary conditions for the three-dimensional periodic boundaries: = r^^°^-/(^^<'»"+^<=°^'5+^=°^^) (T - T„)|^^_^/2 >

(7-24a)

{T - T^)\y=H/2 = r^^°'^/(^^<'»"+^^°»'3+Mcos7) (T - T^)|^=_^/2 ,

(7.24b)

(T - T „ ) | , ^ ^ / 2 = r^^°^^/(^-^"+^<=°=^+^^°^T) (T - T „ ) U _ ^ / 2 ,

(7.24c)

(T -

T;.)L=L/2

where _

( ^ ~ ^'t^)lx=:L/2,y=i//2,z=M/2

2_.

and the origin of the Cartesian coordinates {x,y,z) is set in the center of the structural unit whose volume is L x H x M ( - L / 2 ^ x ^ L/2, -H/2 ^y ^ H/2, -M/2 ^ z ^ M/2). 7.3.2

Quasi-three-dimensional solution procedure for anisotropic arrays of infinitely long cylinders

Having established appropriate periodic boundary conditions, the continuity, NavierStokes and energy equations can be solved using a standard numerical algorithm such as SIMPLE. Even dealing with a single unit, exploiting the periodic boundary conditions, such three-dimensional computations require substantial memory space and CPU time. However, the governing equations and corresponding boundary conditions may greatly be simplified for the case of the three-dimensional heat and fluid flow through a twodimensional periodic structure, such as a bank of cylinders in yaw, as illustrated in Figure 7.6, and more specifically in Figure 7.7 to show the cross-sectional plane of the square cylinder bank. All square cylinders in the figure, which may be regarded as heat sinks (or sources), are maintained at a constant temperature T^ {= {T)^), which is lower (or higher) than the temperature of the flowing fluid. Since the cylinders are infinitely long, the set of the governing equations reduces to a quasi-three-dimensional form, in consideration of the limiting case, namely, the case of vanishing spanwise length M —>^ 0, see Nakayama et al. (2004): du dv ^ + ^ - 0 , (7.26) ox ay

dx \

dx J

dy \

dy J

pdx ^

d (

9^\ , 9 f 2

^^A

^ ^P

dx \

dxj

dy J

pdy'

dy \

d f dw\ d ( dw\ — Uzi;-i/— + — U ^ - z / — ax V dx J dy \ dy J

/ dw ^^ f ^ d P , Afluid JP,^, dn V

.^ ^^^ (7.29)

A. NAKAYAMA AND R KUWAHARA

185

Figure 7.6 Bank of circular cylinders representing an anisotropic porous medium.

n • DD n

D n n n nD Figure 7.7 Bank of square cylinders (cross-sectional view).

where P is the coordinate along the wetted periphery whereas n is the coordinate normal to P pointing inward from the peripheral wall to the fluid side, i4fluid is the passage area

186

PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS

of the fluid, and ^ f

Q

T

"

^^

COS 7 In To u cos 7 In To {T-T^)l=,, Pr/ L cos a + if cos P J L cos a-\- H cos p

(7.31)

since M cosy/{L COS a-^HCOSP-\-M COS j) _ i

^rp

^

= (T - T.)|,=„ lim

M

{T{x,y,0)-T^)cosj Lcosa + H cos fi

(7.32)

In To,

where ^0 = rl ^0 =

{T'Tw

)

[T

\x=L/2,y=H/2,z=^0

(7.33)

Tw)\x=-L/2,y=-H/2,z=0

The boundary and compatibility conditions for the periodic planes are given by {ui + i'j)|^=_H/2 = (ui + vj)\^^H/2

(7.34a)

{ui + «^i)ly=_i//2 = (ui + •"3)\y=H/2

(7.34b)

\x=~H/2 = rH/2

/ udy J-H/2

^\y=-H/2

W |x=H/2 x=

= x=-L/2

rH/2

udy

-^-^/^

= ^\y=H/2

— ii'cosa(|n|) ,

(7.35a)

— LcosP {\u\) ,

(7.35b)

x=L/2

rL/2

rL/2

/ vdx J~L/2

— I rH/2

y=H/2

rL/2 J-L/2

\L - J-w)\x=L/2

L cos a/^^i--COS a/{L cos a-TJTi oc-\-Hc ucos 1^ COS s p (5) ; /rp rp \\ — ^0 V^ ~ -'•w)\x=-L/2

(rp

_

7^ rp

'\ I \\

\J- - lw)\y=H/2—^Q

(7.35c)

wdxdy — Lif cos7(|t/|) ,

\

J-H/2

vdx

^-^/2

V^-H/2

\

/rp

(7.34c)

'

H COS 13/{L COS a-\-H

COS (3) (rp_rp

V^

\\

•^w)\y^-H/2

^

(7.36a) (7.36b)

In this way, all derivatives associated with z can be eliminated. Thus, only two-dimensional storages are required to solve equations (7.26)-(7.30). (Note that both equations (7.29) and (7.30) may be treated as two-dimensional scalar transport equations.)

187

A. NAKAYAMA AND F. KUWAHARA

7.3.3

Effect of cross flow angle on the Euler and Nusselt numbers

Some researchers, including Grimison (1936, 1937), Omohundro et al (1949), Bergelin et al. (1950, 1952) and Zukauskas (1987), have carried out extensive experimental investigations for the heat transfer from a bundle of tubes in crossflow,and provided useful experimental data and correlations for designing cross flow heat exchangers. However, these experimental data are limited to a certain class of geometrical configurations, such as tube banks for aligned and staggered arrangements, subject to a limited number of sets of transverse and longitudinal pitches. As we consider the effect of the cross flow angle a', we fix the value of the yaw angle to 7 = 7r/2, such that a' = a. Zukauskas (1982) assembled the experimental data for the fully-developed pressure drop across the tube and presented a chart for the Euler number, which is defined by Eu = 2Ap/pfu'^^^, where Ap is the pressure drop per tube row and Umax = 1(^)1 H/{H — D) is the maximum velocity around the structure. His inlinesquare arrangement (despite the difference in the cross-sectional shape) corresponds to the present arrangement with a — Q,^ — 7r/2 and H/D — 2. It is also noted that, in reality, the macroscopic flow direction rarely coincides with the principal axes, since even small disturbances at a sufficiently high Reynolds number make the flow deviate from the axis. Thus it is understood that the chart provided by Zukauskas gives only the average levelofthepressuredropwithinarangeofsmall valuesof a, say 0° < a < 5°. The Euler number obtained by Zukauskas (1982) from his pressure drop measurement for the case of 7 = 7r/2 and EjD = 2 is plotted against the Reynolds number Re/) max = u^axDI^ in Figure 7.8, where the numerical results from our numerical experiment based on the quasi-three-dimensional calculation procedure are drawn together for the three cross flow angles, namely, a := 0°, 5° and 45°, so as to show the effect of the cross flow angle a on the Euler number. We observe that the predicted curve obtained at a = 5° follows closely along Zukauskas' experimental curve.

a = 0° - - - a = 5° . . . . a = 45°

10'

— .... Zukauskas

Eu

*^^V

10"

1 '*'*^iL^^

•*

0-' : 1 1 m i

10'

10^ ^^D

Figure 7.8

10^

10"

max

Effect of Reynolds number on the Euler number.

188

PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS

The microscopic temperature results have been processed using equation (7.22) to obtain the interfacial heat transfer coefficient hf. In Figure 7.9, the heat transfer results obtained at a = 0°, 5° and 45° for the cross flows, i.e. 7 = 7r/2, are presented in terms of the interfacial Nusselt number Nu^ = hfD/kf as a function of the Reynolds number Rep max- The figure suggests that the lower and higher Reynolds number data follow two distinct limiting lines for the cases of non-zero a. Unlike the Euler number, the interfacial Nusselt number is fairly insensitive to the (non-zero) cross flow angle a. The lower Reynolds number data stays constant for the given array and flow angle, whereas ,0.5~0.6 the high Reynolds number data vary in proportion to Re£^ As already pointed out in connection with the pressure drop, it is quite unlikely to have the macroscopic flow aligned perfectly with the principal axes. Therefore, the curve predicted for a = 0 should rarely be realized. The experimental correlation proposed by Zukauskas (1987), for the heat transfer from the circular tubes in staggered banks, are compared with the present results obtained for the case of a = 7r/4, 7 = 7r/2 and H/L = 1. (Note Nu/ = Nup and Re/ = Re^max in equation (39) of Zukauskas (1987) since H/D = 2.) The present results follow closely along the experimental correlation of Zukauskas as the Reynolds number increases. 7.3.4

Effect of yaw angle on the Euler and Nusselt numbers

The pressure drop naturally decreases with decreasing the yaw angle 7 from 7r/2. The correction factor, namely, the ratio of the Euler number at a certain yaw angle 7 to that obtained at 7 = 7r/2, i.e. Eu(7)/Eu(7r/2), is often introduced for engineering use with the cross flow angle a' being fixed. It should be noted that the ratio becomes insensitive to increasing Re^ max- The curve of the ratio of the Euler number generated by changing the yaw angle 7 for the case of a' = 5° is compared against the data obtained from the

1

1

r • • nil

-•-

10^

"•

«

1

1 < lll|

1

a = 45° Zukauskas

^ . . .

10°

I 1 1 IM|

1

1

1 1 1 Mil

a = 0°

t

10'

1

10'

10^

.

lO^

^

A 10"

max

Figure 7.9

Effect of Reynolds number on the interfacial Nusselt number (Pr = 1).

189

A. NAKAYAMA AND F. KUWAHARA

experiment by Zukauskas (1982) in Figure 7.10, in which a good agreement can be seen. (Note that Zukauskas' in-line arrangement may correspond to the case of a' = 5°, as we suggested from Figure 7.8.)

1.0

•

\

0.8

^

Present prediction

1l \ ^ \

^

•

•\

Zukauskas

0.4

. 90

80

Figure 7.10

70

60 50 7 [deg.]

40

^^\ 30

Effect of the yaw angle on the Euler number.

I.OA

Present prediction 0.9 CM

§ 0.8

I

0.7

Zukauskas #

Staggered

Jk. Inline 0.6 90

Figure 7.11

80

70

60 7 [deg.]

50

40

30

Effect of the yaw angle on the interfacial Nusselt number.

190

PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS

Zukauskas (1982) investigated the effect of the yaw angle on the interfacial heat transfer rate. He varied the yaw angle 7 for both staggered and aligned arrangements, and compared the corresponding heat transfer rates for the same Reynolds number. He pointed out that the data, when normalized by the value obtained at 7 = 7r/2 for all staggered and inline arrangements, namely, Nu£)(7)/Nui)(7r/2), can be approximated by a single curve, irrespective of the Reynolds number. His data for both staggered and inline arrangements are plotted together with our prediction in Figure 7.11. As in the case of the Euler number ratio, the Nusselt number ratio becomes insensitive to increasing Re£)niax» and may be expressed as a function of 7 alone. The agreement between the experimental data and the prediction is fairly good, which again indicates the validity of our quasi-three-dimensional calculation procedure.

7.4

LARGE EDDY SIMULATION OF TURBULENT FLOW IN POROUS MEDIA

In this last section of this chapter, we shall present an LES (large eddy simulation) study conducted for one of the standard numerical models for a porous medium, namely, a flow through a periodic array of square cylinders. The LES results are processed to extract the macroscopic results, such as the macroscopic turbulent kinetic energy and the macroscopic pressure gradient. These macroscopic results are compared against those obtained using conventional models of turbulent kinetic energy and its dissipation rate, so as to examine the validity of extending the conventional two equation models of turbulence to theflowin porous media. The spectrum of turbulence will also be examined to appreciate the onset of turbulence. 7.4.1

Large eddy simulation and numerical model

Kuwahara et al. (1998), Nakayama and Kuwahara (1999) and de Lemos and Pedras (2001) have conducted numerical experiments for turbulent flows through a periodic array of cylinders using the conventional two-equation turbulence model, see Launder and Spalding (1974), based on the Reynolds-averaged Navier-Stokes (RANS) equations. The pressure results from their numerical experiment showed a good agreement with the Forchheimerextended Darcy law. This prompted Nakayama and Kuwahara (1999) to conclude that the Forchheimer extended-Darcy law holds even in the turbulent flow regime in porous media. However, the application of conventional two-equation turbulence models to such complex unsteady turbulent flows, as in porous media, must be justified for a further discussion related to turbulence in porous media. Thus the validity of the models based on RANS needs to be confirmed for flows through porous media by comparing the results obtained against more reliable results. In contrast to statistical turbulence models, direct numerical simulations (DNS) require no assumptions, but will not be available for engineering applications for some time because of the large memory and CPU time requirements. Large eddy simulation (LES) methods

A. NAKAYAMA AND F. KUWAHARA

191

can be described as a compromise between the DNS and the simulation based on the RANS equations. For the LES, the three-dimensional unsteady Navier-Stokes equations are filtered in order to separate the large scale eddies from the small scale ones. Thus the main flow structure is resolved directly, while only the small scale eddies are modelled by a subgrid scale model. The physical model for a porous medium is illustrated in Figure 7.12, where a periodic array of square cylinders is shown. In this LES study, only one structural unit is chosen for a calculation domain, and along its boundaries, the periodicity of the flow is assumed. In this way, we intentionally neglect the eddies larger than the scale of the porous structure, since such large eddies cannot survive long enough to be detected in the porous medium. Furthermore, the renormalization group (RNG) subgrid scale model is used for modeling non-resolvable subgrid scale eddies. As for the advection terms, the QUICK scheme is introduced. The code used to solve the filtered equations is based on a three-dimensional finite volume method. The Reynolds number Ren — UDH/V, based on the center-to-center distance iJ, was varied from 100 to 5 x 10^, while the porosity 0 = 1 - (D/H)^ was changed from 0.3 to 0.9. (Note that, in this section, the symbol e is reserved for the dissipation rate of the turbulent kinetic energy fc, whereas 0 is assigned for the porosity.) Computations were carried out using grid systems which guarantee that the grid spacing close to the wall is small enough to satisfy the condition Ay < (z/^/e)^/^. A typical grid system for computation consists

Flow direction ^^' ^-^^f^^'^-i

Spanwise direction

Domain of integration Figure 7.12

Numerical model for the LES.

192

PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS

of 161 X 91 X 35 = 512 785 nodes with highly non-uniform grid spacing to cover the domain of integration 2H x H x 0.5H. Preliminary calculations were made to compare the results against those obtained using 243 x 137 x 35 == 1165 185 nodes for some selected cases. In this way, the originally used grid resolution is found to be sufficient. Moreover, in order to investigate the effect of its spanwise size on the LES results, the domain of integration was doubled in the spanwise direction by doubling the number of grid points, while keeping the spanwise cell size constant. This modification yielded no significant changes in the LES results, proving that the originally used domain is fully sufficient. The time step was set small enough to satisfy the Courant condition, namely. At < min (Ax, Ay, Az) / {u)^ (where {u)^ is the time-averaged intrinsic velocity), after confirming that any further decrease in the time step does not alter the results significantly. A typical time period advanced in the LES was 20H/ (u)^, which was found to be long enough for the flow to develop periodically. 7.4.2

Velocity fluctuations and turbulent kinetic energy

Figure 7.13 shows a series of the spanwise velocity oscillations at a selected point behind the cylinder for the case of 0 = 0.64. The spanwise oscillations are indicative of three-dimensional velocity fluctuations, i.e. turbulent motions, are already appreciable at Reo = UDD/U = 80. The turbulent kinetic energy for Ren = 400 is illustrated in the spectrum in Figure 7.14, where the - 5 / 3 law for the inertial subrange may be confirmed. In Figure 7.15, the time-averaged velocity, pressure and turbulent kinetic energy fields for Re£) = lO'^, based on the LES (presented on the left of Figure 7.15), are compared against those based on the RANS with a low Reynolds number version of the k-e model proposed by Abe et al. (1992) (presented on the right of Figure 7.15). The figure shows that the mean velocity and pressure fields from both the LES and RANS are in reasonable agreement, while the turbulent kinetic energy field of the LES is quite different from that of RANS with the k-e model. It is well known that conventional k-e models, coupled with the effective viscosity formulation, tend to overestimate the level of turbulent kinetic energy around the stagnation point. The reason for this may best be illustrated by writing the production rate of k for the k-e model as follows:

P^u,^(?^ + ^], ' dxj \ dxj

dxi

(131)

where ut and Ui are the eddy diffusivity and time-averaged velocity vectors, respectively. The foregoing k production term contains such a predominant term as {dujdxY that attains a quite large value for a decelerating flow around the front stagnation point, yielding an extremely high production rate of k in this vicinity. This shortcoming, inherent to the conventional k-e models, has been remedied by the LES, which provides a reasonable turbulent kinetic energy distribution. It can clearly be seen from thefigurethat the turbulent kinetic energy is produced exclusively within the shear layers above the lateral surfaces of the cylinder, where the mean strain rate is quite high, due to flow separation.

193

A. NAKAYAMA AND F. KUWAHARA (a) 10

0

(b)lOr

100 200 300 400 500 600 700 800

0

100 200 300 400 500 600

{x,y,z) = (50,20,0)

500

600 700 800 900 1000

t[s]

50

100

150 200 250 300

Figure 7.13 Spanwise velocity oscillations for 0 = 0.64 when (a) Ren = 40, (b) ReD = 80, (c) RBD = 160, and (d) Re^ = 400.

A series of computations carried out for various sets of porosity and Reynolds number are presented in Figure 7.16 in terms of the macroscopic turbulent kinetic energy (k)^, i.e. the volume-averaged turbulent kinetic energy, normalized by the square of the Darcian velocity uj^ (= \ {u) \ ). As found in the previous study, Nakayama and Kuwahara (1999), the normalized (A;)*' increases with increasing Rej^, and stays constant for large Re/p, say, ReD > 3000. The figure again suggests that the turbulence may appear at comparatively low Reynolds number especially in the case of small porosity. Since uj^ = {(f) (u)-^)^ can be quite small, the normalized {k)^ exceeds unity for comparatively small 0. (Note that (k) /({uy)^ never exceeds 0.5.) The constant values of {k)^ for large Reo are plotted in Figure 7.17, together with those obtained using RANS with the low Reynolds number version of the k-e model. The

194

PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS

10" 10-

101010r l 7 o

OH

101010-

0.01

Figure 7.14

0.1

Frequency [Hz]

1

10

Spectrum of the turbulent kinetic energy at Rep = 400.

Figure 7.15 Macroscopic (a) velocity, (b) pressure (isobars), and (c) turbulent kinetic energy fields at Reo = 10'* (LES on the left and RANS with k-e model on the right).

195

A. NAKAYAMA AND F. KUWAHARA

•-

.^*

/'."'

10"i

f

A f

/•

• --= 0.3 • 0 == 0.5 • -- = 0.64 T 4>-- = 0.84

A' /

10"

10' Rec

10^

10'

10^

10=

Figure 7.16 Effect of the Reynolds number on the macroscopic turbulent kinetic energy.

lOh

— , — —,—,—p. LES - Low Re k-e model

1

1 •

1 r—r-T

1

/ «

n

;

o ( •

10": 10"

o m

• ^'^'='''if-o 1

1

1

1 1 r—r-T

1

1

10"

(1 - 4>)/V4'

Figure 7.17 Effect of the porosity on macroscopic turbulent kinetic energy. present results from the LES follow the correlation proposed by Nakayama and Kuwahara (1999), namely, {k)f^3.7^ul.

(7.38)

Despite the shortcoming associated with the production term, the low Reynolds number version of the k-e model gives a reasonable level of macroscopic turbulent kinetic energy. Thus, the conventional two-equation turbulence model, based on Reynolds-averaged Navier-Stokes (RANS) equations may well be used to estimate the macroscopicfieldof the turbulent flow in porous media.

196 7.4.3

PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS Macroscopic pressure gradient in turbulent flow

The results of the macroscopic pressure gradient calculated using the LES results are presented in a dimensionless form as a function of Ren for the case of 0 = 0.84 in Figure 7.18, where the previous results based on the low Reynolds number version of the k-e model are also plotted for comparison. Agreement between the results based on the LES and those based on the k-e model appears to be excellent, which substantiates the validity of the simulation using the RANS with a two-equation turbulence model. The functional relationship of the Forchheimer drag is investigated by plotting the dimensionless pressure gradient (—d(p)-^/dx) [D/pu^), using only the high Reynolds number results obtained at Re^ > 3000. Ergun's empirical equation, accounting for the Forchheimer drag in packed beds of particle diameter dp, is given by cl(p)^ dx

150(1-0)2 / V

0

P^D

+ 1.75-

uodf^

6 1-0 ^ ^ 1.7503 ^ 03 (Red^ > 3000). (7.39)

This prompts us to correlate the foregoing dimensionless pressure gradient with (1 - 0) /0^, anticipating a linear relationship with (1 — 0)/0^. Thus the pressure gradient results are re-processed and plotted in Figure 7.19, which, in fact, substantiates a linear relationship as follows:

AW [ D \^^l

(7.40)

(Rep > 3000).

03

dx

This value of 2 for a periodic array of square cylinders of size D is slightly higher than 1.75 in Ergun's equation (7.39) for a packed bed with a particle diameter dp = D. The foregoing high Reynolds number results may be coupled with the low Reynolds number results previously established by Kuwahara et al. (1994) and Nakayama et al. (1995)

10^

n O •

Laminar Low Re k-e model LES

A

y^ \

o^

^

'10^ lOjoO

10'

10^

Rec

10^

10"

10^

Figure 7.18 Effect of the Reynolds number on the macroscopic pressure gradient.

197

A. NAKAYAMA AND F. KUWAHARA 10' j

J

dx

\X d

^ 10^ H

3 •^

1

y^

• O

LES Low Re k-e model

j

1^ 10"

' ' ' 1

'

10^

10"

10'

Figure 7.19 Effect of the porosity on the Forchheimer drag. for laminar Darcy flow, conducting the numerical experiment for the same geometrical array, namely, d(p)^ _ 120(1 - 0)2 -\XUD (Darcy flow), (7.41) dx 032)2 so that the corresponding law for the relationship between the friction coefficient and Reynolds number is given by d(p)/ / p ( ( « ) / ) ' _ 51.2 -f 1.85, dx / 2c?e,

(7.42)

where l^^rfea =

=

—

"32" (j) D 150 1 - 0

(7.43)

is the equivalent Reynolds number which transforms the Ergun equation (7.39), for dp D, into Poiseuille's expression:

_ -^eq

d{p)f

/p{wy

dx

2d,eq

64 -hl.62. Red,,

(7.44)

Finally, equation (7.42) obtained for the array of square cylinders is compared with Ergun's equation (7.44) in Figure 7.20 where Aeq is plotted against Re^^^. We observe that both results are in quite close agreement. This indicates that Ergun's equation for the relationship between the pressure and Reynolds number, namely, equation (7.44) with (7.43), is such a universal equation that it can be used for most two- and three-dimensional periodic structures over a wide range of porosity.

198

PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS

10^

Red,,

10^

Figure 7.20 Universal laws for the macroscopic pressure gradient.

The present LES study revealed that the Ergun equation (7.44) may well describe the drag relationship for the turbulentflowin porous media. Furthermore, it suggests that turbulence may appear in porous media at comparatively low Reynolds number. However, it must be pointed out that the characteristics of turbulence in porous media may differ significantly from those in a clear fluid flow, because of the comparatively high turbulence intensity usually encountered in such porous media. Further numerical experiments (preferably DNS) are definitely required to explore how unsteady laminar flows develop into turbulent flows and how the onset of turbulence influences the drag characteristics.

7.5

CONCLUSIONS

Three-dimensional numerical models for both isotropic and anisotropic porous media have been reviewed in order to estimate the macroscopic hydrodynamic and heat transfer characteristics, assuming periodically fully-developed flow and heat transfer in a single structural unit. As for isotropic porous media, laminar flows around collections of spheres and cubes in a regular arrangement were considered, whereas, for anisotropic porous media, laminar flows through a bank of square cylinders were investigated changing the yaw angle, porosity and Reynolds number. An economical quasi-three-dimensional calculation procedure was proposed and exploited to obtain the results for three-dimensional heat and fluid flow through a bank of cylinders in yaw. Exhaustive numerical computations were conducted so as to determine macroscopic parameters, such as the permeability and the interfacial heat transfer coefficient. The results based on these numerical models are found to agree very well with the available empirical formulas. Furthermore, LES solutions were obtained for unsteady flows through periodic arrays of square cylinders with various porosities. Both the velocity oscillation and its kinetic energy in the spectrum

A. NAKAYAMA AND F. KUWAHARA

199

predicted by the LES indicate that the onset of turbulence takes place at comparatively low Reynolds number. The macroscopic results based on the LES solutions are in good accord with the previous results based on RANS with the two-equation turbulence model. This substantiates the validity of the simulation procedure using RANS with a two-equation turbulence model, for obtaining the macroscopic quantities associate with turbulent flow in porous media.

REFERENCES Abe, K., Nagano, Y., and Kondoh, T. (1992). An improved k-e model for prediction of turbulent flows with separation and reattachment. Trans. JSME, Ser. B 58, 3003-10. Bergelin, O. P., Brown, G. A., and Doberstein, S. C. (1952). Heat transfer and fluid friction during viscous flow across banks of tubes. IV. A study of the transition zone between viscous and turbulent flow. Trans. ASME 74, 953-60. Bergelin, O. P., Brown, G. A., Hull, H. L., and Sullivan, F. W. (1950). Heat transfer and fluid friction during viscous flow across banks of tubes. III. A study of tube spacing and tube size. Trans. ASME II, 881-8. Cheng, P. (1978). Heat transfer in geothermal systems. Adv. Heat Transfer 14, 1-105. Coulaud, O., Morel, P., and Caltagirone, J. P. (1988). Numerical modeling of nonlinear effects in laminar flow through a porous medium. J. Fluid Mech. 190, 393-407. Darcy, H. (1856). Les fontaines publiques de la ville de Dijon. Victor Dalmont, Paris. de Lemos, M. J. S. and Pedras, M. H. J. (2001). Recent mathematical models for turbulent flow in saturated rigid porous media. J. Fluids Eng. 123, 935^0. Eidsath, A., Carbonell, R. G., Whitaker, S., and Herman, L. R. (1983). Dispersion in pulsed systems. III. Comparison between theory and experiment for packed beds. Chem. Eng. Sci. 38, 1803-16. Ergun, S. (1952). Fluid flow through packed columns. Chem. Eng. Progress 48, 89-94. Fowler, A. J. and Bejan, A. (1994). Forced convection in banks of inclined cylinders at low Reynolds numbers. Int. J. Heat Fluid Flow 15, 90-9. Grimison, E. D. (1936). Heat transfer and flow resistance of gases over tube banks. Trans. ASME 58, 381-92. Grimison, E. D. (1937). Correlation and utilization of new data on flow resistance and heat transfer for cross flow of gases over tube banks. Trans. ASME 59, 583-94. Ingham, D. B. and Pop, I. (eds) (1998). Transport phenomena in porous media. Pergamon, Oxford. Ingham, D. B. and Pop, I. (eds) (2002). Transport phenomena in porous media. Vol. II. Pergamon, Oxford. Kuwahara, F., Kameyama, Y., Yamashita, S., and Nakayama, A. (1998). Numerical modeling of turbulent flow in porous media using a spatially periodic array. J. Porous Media 1, 47-55. Kuwahara, F , Nakayama, A., and Koyama, H. (1994). Numerical modelling of heat and fluid flow in a porous medium. In Proceedings of the 10th international heat transfer conference, Brighton, UK, Vol. 5, pp. 309-14.

200

PERIODICALLY FULLY-DEVELOPED HEAT AND FLUID FLOWS

Kuwahara, R, Shirota, M., and Nakayama, A. (2001). A numerical study of interfacial convective heat transfer coefficient in two-energy equation model for convection in porous media. Int. J. Heat Mass Transfer 44, 1153-9. Larson, R. E. and Higdon, J. J. L. (1989). A periodic grain consolidation model of porous media. Phys. Fluids A h3S-46. Launder, B. E. and Spalding, D. B. (1974). The numerical computation of turbulent flow. Comp. Meth. Appl Mech. Eng. 3, 269-89. Nakayama, A. (1995). PC-aided numerical heat transfer and convective flow. CRC Press, Boca Raton, PL. Nakayama, A. and Kuwahara, F. (1999). A macroscopic turbulence model for flow in a porous medium. Trans. ASME J. Fluids Eng. 121,427-33. Nakayama, A., Kuwahara, E, and Hayashi, T. (2004). Numerical modeling for three-dimensional heat and fluid flow through a bank of cylinders in yaw. J. Fluid Mech. 498, 139-59. Nakayama, A., Kuwahara, E, Kawamura, Y., and Koyama, H. (1995). Three-dimensional numerical simulation of flow through a microscopic porous structure. In Proceedings of the 4th ASME/JSME thermal engineering conference, Hawaii, Vol. 3, pp. 313-18. Nakayama, A., Kuwahara, E, Naoki, A., and Xu, G. (2002). A volume averaging theory and its sub-control-volume model for analyzing heat andfluidflowwithin complex heat transfer equipment. In Proceedings of the 12th international heat transfer conference, Grenoble, Vol. 1, pp. 851-6. Nield, D. A. and Bejan, A. (1999). Convection in porous media (2nd edn). Springer-Verlag, New York. Omohundro, G. A., Bergelin, O. P., and Colbum, A. P. (1949). Heat transfer and fluid friction during viscous flow across banks of tubes. Trans. ASME 71, 583-94. Patankar, S. V. (1980). Numerical heat transfer and fluid flow. Hemisphere, Washington, DC. Patankar, S. V. and Spalding, D. B. (1972). A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transfer 15, 1787-806. Patankar, S. V., Liu, C. H., and Sparrow, E. M. (1977). Eully-developed flow and heat transfer in ducts having stream wise-periodic variations of cross-sectional area. J. Heat Transfer 99, 180-6. Quintard, M. and Whitaker, S. (1993). One and two equation models for transient diffusion in two-phase systems. Adv. Heat Transfer 23,169-464. Sahraui, M. and Kaviany, M. (1991). Slip and no-slip boundary condition at interface of porous, plain media. ASME/JSME Therm. Eng. Proc. 4, 273-86. Vafai, K. and Tien, C. L. (1981). Boundary and inertia effects on flow and heat transfer in porous media. Int. J. Heat Mass Transfer 24, 195-203. Zukauskas, A. (1982). Konvektivnyi perenos v teploobmennikakh (Convective transfer in heat exchangers). Nauka, Moscow. Zukauskas, A. (1987). Heat transfer from tubes in crossflow. Adv. Heat Transfer 18, 87-159.

8

ENTROPY GENERATION IN POROUS MEDIA A. C. BAYTA§* and A. R BAYTA§t *The Faculty of Aeronautics and Astronautics, Istanbul Technical University, 34469-Maslak, Istanbul, Turkey email: baytasQitu.edu.tr ^Institute of Energy, Istanbul Technical University, 34469-Maslak, Istanbul, Turkey email: [email protected]

Abstract Recently, research on entropy generation minimization using the second law of thermodynamics has been studied for many different situations. Further, the utilization of the second law of thermodynamics in thermal design decision has been developed and applied for porous media. For this reason, these lines of inquiry have been largely motivated by a desire to obtain an in-depth understanding of the relevant physics underlying this phenomena. Therefore, in this chapter, entropy generation has been reviewed for porous media and the recent studies related to entropy generation have been investigated for porous cavities and channels for different boundary conditions and physical situations. Keywords: entropy generation, porous media, free convection, forced convection, hydromagnetic effect, Darcy flow

8.1

INTRODUCTION

Convective heat and mass transfer in porous media has been studied extensively over the past two decades. Using a variety of numerical, experimental, and analytical methods, results have been published for many engineering applications regarding forced and free convection in porous channels and cavities which are in geometrically simple enclosures, e.g. rectangular cavities, cylindrical containers and also in complex geometries. This significant interest in entropy generation has been motivated by its importance in many natural and industrial applications. Prominent among these applications are heat exchangers, migration of moisture through air contained in fibrous insulation, energy efficient drying processes, underground spread of pollutants, packed-bed nuclear reactors, cooling 201

202

ENTROPY GENERATION IN POROUS MEDIA

of radioactive waste containers, microelectronic devices during their operation, the effects of electrically conducting fluids such as liquid metals, the presence of a magnetic field on the flow and heat transfer, high volumetric heat generation effects in electronic equipment, etc. In the aforementioned investigations in porous media, the quantities to be calculated are usually the temperature and velocity fields, pressure, etc., but these investigations rarely contain entropy properties using the second law of thermodynamics. Principally, only heat transfer was reported in these free and forced convection studies. The necessity for the utilization of the second law of thermodynamics in thermal design decisions for porous media has been clearly investigated over the last four years and the concept of irreversibility is based on the second law of thermodynamics. It has been shown in theory and practice that the second law of thermodynamics can serve to optimise the design of thermal, fluid and energy systems. The theories of reversibility and irreversibility are important in thermodynamics and crucial to the exergy method. Understanding the nature of irreversibility and how to minimize them in practice is essential for an engineering thermodynamist. Contemporary engineering thermodynamics uses the rate of entropy generation as a parameter to quantify the significance of these irreversibilities. The local values of entropy generation due to viscous and thermal effects can be mapped to detect, by inspection, the key areas that require a design modification. This emerging technology is viewed to have a meaningful potential for improving thermal system designs. The studies on entropy generation minimization using the second law of thermodynamics have been intensively researched and it is a well-established subject. In thermal design decisions, the utilization of the second law of thermodynamics is presented in Bejan (1979, 1994,1996) and Poulikakos and Bejan (1982). Drost and White (1991) have numerically studied the local entropy generation map in an impinging jet. San et al (1987) have studied entropy generation in convective heat and mass transfer within a smooth channel under some specific thermal boundary conditions. Cheng et al. (1994) have presented a numerical study of entropy generation for mixed convection in a vertical channel with a transverse fin array. For the first time, Baytas (1997) has studied the optimization in an inclined enclosure filled with a clean fluid for minimum entropy generation in natural convection in an externally heated vertical and inclined square box containing an internal energy source.

8.2

A SHORT HISTORY OF THE SECOND LAW OF THERMODYNAMICS

Sadi Camot was the son of a minister of war under Napoleon. He formed the opinion that one cause of France's defeat had been her industrial inferiority and the contrast between England's and France's use of steam epitomized the difference. He saw that taking away England's steam engine would remove the heart of her military power. Camot perceived that whoever possessed efficient steam power would not only be the industrial and military master of the world but also the leader of a social revolution far more universal than the one France had so recently undergone. Camot saw steam power as a universal motor. So, the first theory on the conservation of heat into mechanical work was due to Sadi Camot in 1824. Camot based his analysis on the assumption that the quantity of heat

A. C. BAYTA§ AND A. F. BAYTA§

203

was conserved, and that work was generated by the engine because the fluid flowed from a hot thermally *high' source to a cold, thermally 'low' sink. He subscribed to the then conventional theory that heat was some kind of massless fluid or a caloric fluid. Joule's experiments, performed in the 1840s, confirmed that heat was not conserved. Joule showed that work could be converted quantitatively into heat. This was the birth of the concept of the mechanical equivalence of heat that work and heat are mutually interconvertible, and that heat is not a substance like water, see Atkins (1994). His work was on the conservation of energy. Lord Kelvin (William Thomson) (1824-1907) and Joule (1818-1889) met at the Oxford meeting of the British Association for the Advancement of Science in 1847. From that meeting Kelvin returned with an unsettled mind. He was reported as being astounded by Joule's refutation of the conservation of heat. He believed that Camot's work would be overthrown if heat were not conserved, and if there were no such thing as a caloric fluid. Kelvin went on to develop the view that in some sense the work of Camot could survive without contradicting the work of Joule. So, 'thermodynamics', the theory of the mechanical action of heat, emerged. Rudolf Clausius (1822-1888) has shown that the case of Camot vs Joule could be, to some extent, resolved if there were two underlying principles of Nature. He refined Camot's principle, and rid the world of caloric, but he did go on to speculate on how heat could be explained in terms of the behaviour of the particles of which matter is composed. Clausius also realized in 1865 that he had discovered a new thermodynamic property, and he chose to name this property entropy. Camot was bom in 1796 and died of cholera in 1832. Camot, Joule, Kelvin and Clausius were bom in the period 1818-1824 and their generation thrust thermodynamics onto the intellectual stage. The name thermodynamics is a term denoting the study of heat, but now it has been extended to include the study of the transformation of energy in all its forms, see Atkins (1994). There are four laws of thermodynamics. The first law of thermodynamics states that one form of energy, e.g. kinetic, potential, electrical energy thermal, etc. can be converted into another without any loss of energy. The second law states that thermal energy, or heat, is special among the types of energies: all the forms of energy can be converted into heat, but in a way that is not reversible; it is not possible to convert the heat back fully into its original form. The second law of thermodynamics asserts that energy has quality as well as quantity. The first law is concemed with the quantity of energy and the transformation of energy from one form to another with no regard to its quality. Preserving the quality of energy is a major concem to engineers, and the second law provides the necessary means to determine the quality, as well as the degree of degradation of energy during a physical process. The second law leads to the definition of a new property called entropy. Entropy is a somewhat abstract property, and it is difficult to give a physical description of it without considering the microscopic state of the system. The second law of thermodynamics is important to engineers because it provides a way to determine the quality, as well as the amount of degradation of energy during a process. It is also used to determine the theoretical upper limits for the performance of many

204

ENTROPY GENERATION IN POROUS MEDIA

engineering systems, such as refrigerators, internal combustion engines and chemical reactors. Nowadays, the utility of the second law of thermodynamics is not only in heavy industry and machines but also in improving the performance of micro electronic devices. We scrutinize the performance of engineering devices and systems in the light of the second law of thermodynamics. Entropy generation reflect very interesting the physical phenomena and this manifested in the second law of thermodynamics.

8.3

GOVERNING EQUATIONS

In principle, the equations in thermal science describe the various transport phenomena and may be written at the microscopic level. The description and solution of a transport problem at the microscopic level is not practical and sometimes also impossible. A different level of description is needed, namely the macroscopic level, at which measurable, continuous and differentiable quantities may be determined and boundary value problems can be stated and solved. The representative elementary volume, or REV, is a conceptual space unit. When the measuring volume is at least of REV, measurable characteristics of the porous medium become continuum quantities. Volume averaging is a method that makes the measurable quantities continuum properties based on the REV concept. The continuum or macroscopic, governing equations are derived based on the microscopic governing equations. The porosity is the most important property of a porous medium and it affects most of the physical properties of the medium. Measurement of porosity is made by using several techniques, such as imbibition, mercury injection and gas injection methods give an effective porosity value. The true porosity can be measured by using direct, optical and gamma-ray attenuation methods, see Kaviany (1995). The local porosity can be measured directly along a porous column in the gamma-ray attenuation method, see Baytas and Akbal (2002) and Ishakoglu and Baytas (2002). Using the definitions of the porosity and the volume average of a quantity and the local (microscopic) quantity of fluid (see Liu and Masliyah, 1999), the volume-averaged velocity vector, v, and the intrinsic phase-averaged velocity vector, Ve, are related by the expression v = eVe which is also known as the Dupuit-Forchheimer relationship, see Ingham (2004). Volume averaging of the governing equations can be performed by averaging over the REV, term by term. 8.3.1

Continuity equation

The continuity of mass equation can be written as follows, see Bejan (1995): ,|^ + V . ( H = 0 ,

(8.1)

where p is the density of the fluid and e is the porosity. One can observe that the volume averaged continuity equation is very similar to the continuity equation for a clear fluid.

A. C. BAYTA§ AND A. F. BAYTA§

8.3.2

205

Momentum balance equation

The volume averaged Navier-Stokes equations can be written as follows: ^dv edt

1, €2

^,

= - V P + /ieV^v --^v-cp K

\v\ v-hpg,

(8.2)

where P is pressure, K is permeability of porous medium, c = CQ/VK and Cg is the formdrag constant, g is gravity, t is time, /i is the viscosity of fluid, and /Xg is the effective fluid viscosity and it depends on the geometry of the permeable medium. The permeability, K, is the measure of the flow conductance of the porous medium and it is defined by the Darcy law and can be written by using porosity as follows, see Ergun (1952):

The third and fourth terms on the right-hand side of equation (8.2) are the viscous drag and form drag, respectively. Equation (8.2) is the volume averaged Navier-Stokes equations for homogenous fluid flow in an isotropic porous media and it is known as the HazenDupuit-Darcy model, see Lage (1998). For steady Newtonian fluid flow in a porous medium of constant porosity, equation (8.2) can be reduced to the famous Darcy equation by neglecting the inertial and viscous forces and the form drag terms, as follows: v = --{VP

+ pg),

(8.4)

where the terms v and V P are the Darcy velocity and the pressure gradient vectors, respectively. For anisotropic porous medium, the permeability, K, is a general secondorder tensor. Henry Darcy (1856) introduced a one-dimensional empirical model, a modem refinement (popularized by Muskat (1937)) of which is equation (8.4), for single phase Newtonian fluid flow in porous media based on the unidirectional water permeation in a fountain. When the flow is weak, or at low discharge fluid rates, the pressure drop is linearly related to the flow discharge rate. The Darcy model ignores the boundary effects on the flow and this assumption is not valid when the boundaries of the porous medium are taken into account. To overcome this problem, then the Brinkman model is usually employed, see Ingham and Pop (1998, 2002) and Nield and Bejan (1999). In the presence of a magnetic field, a body force term has to be added to equation (8.2) as follows:

(8.5)

206

ENTROPY GENERATION IN POROUS MEDIA

where 7 is the electrical conductivity of the fluid and BQ is the constant externally imposed magnetic field. For more details, see, for example, Nield (1999), Tasnim et al. (2002) and Ingham (2004). 8.3.3

Energy equation

The energy equation for an homogeneous porous medium can be derived by using the first law of thermodynamics and for local thermal equilibrium and isotropic porous media it is given as follows: ipcp)f [a^

+ t; • V T ) = fcV^T + q'" + ^v'

,

(8.6)

where q'" is the volumetric heat source strength and the last term on the right-hand side of equation (8.6) is the viscous dissipation effect appropriate for the Darcy model. For more details, see Bejan (1995), Kaviany (1995), Hossain and Pop (1997), Nield (2000, 2002) and Ingham (2004), and for the thermal non-equilibrium model Baytas (2003). Also, in equation (8.6), k is the thermal conductivity of the porous medium and it is a combination of the conductivities of the two constituents, namely k = ekf + {l-

(8.7)

e)ks,

where the subscripts / and s represent the fluid and solid phases, respectively, a is the capacity ratio of two constituents and it is given by the following: a =

—— {pCp)f

.

(8.8)

If the porous media is subjected to a hydromagnetic effect, then a new energy production term due to the magnetic effect has to be added to the equation (8.6), namely we obtain the following: ipcp)f [a^+v-WT\=

8.3.4

kV'T + q'" + ^v'

+ jB^^v •

(8-9)

Entropy generation

Non-equilibrium conditions, due to the exchange of energy and momentum within the fluid and at the solid boundaries, cause a continuous entropy production in the flow field. This entropy generation is due to the irreversible nature of heat transfer and viscosity effects within the fluid and at the solid boundaries. The property entropy is a measure of the molecular disorder or randomness of a system, and the second law of thermodynamics states that entropy can be created but it can never be destroyed. Based on the increase of entropy principle for any system, the local entropy generation rate per unit volume can be estimated writing the second law of thermodynamics in a differential form as an open

A. C. BAYTA§ AND A. F. BAYTA§

207

system as follows, see Arpaci and Larsen (1984): (8.10)

SZn = 7i;2i'^T)' + ^^+'i-,

where To is the absolute boundary temperature and $ is the viscous dissipation. The first term on the right-hand side of equation (8.10) represents the entropy generation due to the heat transfer irreversibility, while the second term is the entropy generation associated with fluid friction, and the last term is the entropy generation due to volumetric heat generation. The second law of thermodynamics can be applied to the homogeneous porous medium to yield the volumetric entropy generation rate, S'^^^^, as follows:

5;:. = i(vr)' + ^ . H | * + l f . ,

(8.n,

where the second term on the right-hand side of equation (8.11) represents the viscous dissipation (irreversibility) term for porous media and it is important for the Darcy flow model, and the third term on the right side of equation (8.11) is the extra viscous dissipation term for the non-Darcy flow model and the last term on the right-hand side of the equation represents the entropy generation due to hydromagnetic effects. On the Brinkman model, the extra viscous dissipation as used by some authors, $, can be written in a two-dimensional Cartesian coordinate system given as follows: du\

f dv^

^ = 2 1 ( ^ 1 + ( ^ 1 l + ( | + |^)

.

(8.12)

where u and v are the fluid velocity components in x- and y-directions, respectively.

8.4 8.4.1

ENTROPY GENERATION IN A POROUS CAVITY AND CHANNEL Entropy generation in a porous cavity

Natural convection heat transfer and fluid flow in cavities has received considerable attention in the past few decades. The interest in such problems comes form their importance in numerous technical and engineering applications. The issue of entropy generation in a tilted saturated porous cavity for laminar natural convection heat transfer has been analysed by solving numerically the mass, momentum and energy balance equations, using Darcy's law and the Boussinesq-incompressible approximation, see Baytas (2000). He considered an inclined and externally (differentially) heated square porous cavity, see Figure 8.1, where ^p is the inclination angle of the cavity. The horizontal walls are considered to be adiabatic and the vertical walls are maintained at constant temperature TH and Tc (TH > Tc), respectively. The effect of the inclination angle on the flow and heat transfer

208

ENTROPY GENERATION IN POROUS MEDIA

Figure 8.1 Physical model of the two-dimensional inclined porous cavity studied by Baytas (2000).

characteristics and entropy generation was studied by varying the inclination angle from 0° to 360° and the dimensionless Rayleigh number from 10^ to 10^. The isotherms, the streamline patterns and their corresponding entropy generation maps, the variation of entropy generation due to heat transfer and fluid friction irreversibility versus inclination angle, (f, for different Rayleigh numbers were presented. Darcy flow model (8.4) can be rearranged for an inclined cavity as follows:

u^~\^^+P9[l-p{T-Tc)\sm^Y (8.13)

and the pressure gradient term in these equations (8.13) can be eliminated by crossdifferentiation and subtraction. The set of non-dimensional governing equations, in terms of the stream function, ip, and the dimensionless temperature, 6, were obtained from equations (8.13) and (8.6) for an inclined porous cavity as follows, see Baytas (2000, 2004): V V' = Ra I —- cos if — ae -^r- sin ifi BY dY

de

dr

+ vW9 = V^9.

(8.14) (8.15)

A. C. BAYTA§ AND A. F. BAYTA§

209

These equations have to be solved subject to the following initial and boundary conditions:

^:=0, i/; = 0, ^ = 0,

0 = 0.5, 9 = -0.5, 89 gy=0.

O^y^l, O^y^l,

r>0, r>0,

O^X^l,

r>0,

(8-16)

and the non-dimensional variables are defined by

XY

= —^ L

"^ = a

U V =-^— a/L

where r, 0, [/, F and jp are the dimensionless time, temperature, x and y velocity components and stream function, respectively. The dimensionless governing equations (8.14) and (8.15) were discretized by the two-dimensional finite-difference control volume method of Patankar (1980), along with boundary conditions given in equation (8.16). The power-law scheme was adopted for the discretization of the convection-diffusion term. The alternating direction implicit (ADI) method was chosen for the unsteady energy equation and a tridiagonal matrix algorithm was used for solving the discretized equations. The stream function equation (8.14) was solved by the successive over relaxation (SOR) procedure. The non-uniform finite-difference mesh systems were utilized and grid refinement near the boundary was used to reduce the computing time. The chosen grid is the similar grid used by Baytas (1996). From the known temperature and velocity fields, volumetric entropy generation were computed using equation (8.11) which can be rearranged by utilising the dimensionless variables and then the defining the local entropy generation number, N, for a two-dimensional enclosure to be given by iv = ( v ^ ) 2 - f ( / ) ( v ^ ) ^ HTI

where 0 = {ijiTo/k){a^/[K{AT)'^])

(8.18)

FFI

is the irreversibility distribution ratio.

In this study, the dimensionless Bejan number, Be, was used as the alternative irreversibility distribution number for an inclined porous cavity from equation (8.18) as follows:

The Be number is clearly a measure of the relative magnitude of the heat transfer and the fluid friction irreversibilities. For this reason, the Be number is useful in the understanding of the thermodynamic optimization of thermal systems. Here Be = 1 is the limit at which all the irreversibility is due to heat transfer, Be == 0 is due to the fluid friction, and Be — 1/2 is then the heat transfer and fluid friction entropy generation rates are equal.

210

ENTROPY GENERATION IN POROUS MEDIA

(a)

(b)

(c)

(d)

^ = 0°

ip = 40'

ip = 90°

ip = 130'

if = 330°

Figure 8.2 (a) Isotherms, (b) streamlines, (c) entropy generation due to heat transfer, and (d) the local entropy generation (N), at different inclined angles for Ra = 10^. Baytas (2000). Reprinted with permission from Elsevier.

211

A. C. BAYTA§ AND A. F. BAYTA§

for more details, see Baytas (2000). To obtain the entropy generation number in the entire enclosure volume, the local entropy generation number, see equation (8.18), has to be integrated over the entire domain as follows: N.,

-ff

N

dxdy.

(8.20)

Jo Jo

In Baytas (2000, 2004), the entropy generation distribution within an inclined porous square cavity is mainly considered by using the second law of thermodynamics for inclination angles from 0° to 360° and Ra varying from 10^ to 10"*. The results presented here are based on the steady-state results obtained from the transient solution of equations (8.14) and (8.15). In Figure 8.2, isotherms, streamlines, entropy generation due the heat transfer and local entropy generation number, equation (8.18), are shown graphically for Ra = 10^. Results for other Rayleigh numbers and all inclination angles can be found in Baytas (2000). In Figure 8.2(c) for (p = 0°, it is clear that the entropy generation is higher at high temperature gradients. This is due to the heat transfer irreversibility because large heat transfer is confined to these locations. As it is clear from Figure 8.2(c) for (/? = 0°, entropy generation is mainly confined to the lower and upper comers for the left and right walls, respectively. This entropy generation length along the wall increases as
0

n ^ \ r~n "—i ^n • i • i • i—^-n—' i • i 30 60 90 120 150 180 210 240 270 300 330 360

Figure 8.3 Variation of entropy generation due to heat transfer (HTI) and average Nusselt number versus inclined angle. From Baytas (2000). Reprinted with permission from Elsevier.

212

ENTROPY GENERATION IN POROUS MEDIA

entropy generation covers almost the whole domain for Ra == 10^, while it covers only part of the domain and it reduces as the Rayleigh number increases, see Baytas (2000). The variation of the heat transfer irreversibiUty (HTI) and the average Nusselt number, Nua, for different angular positions are shown in Figure 8.3. The conduction regime is dominant for all Rayleigh numbers between the inclined angles 240° and 300°. This is due to the fact that buoyancy is no longer available between these angles. The HTI and average Nusselt number, Nua, have a minimum at about 90° for Ra = 10^ and Ra = 10^, whereas they were a minimum at about 80° for Ra = 10^. In Figure 8.4, the variation of Be versus the inclination angle is shown as an alternative irreversibility distribution parameter as described in equation (8.19). As seen in Figure 8.4, Be = 1.0 at about (p = 270° is the limit at which the heat transfer irreversibility dominates. As the Rayleigh number decreases, the heat transfer irreversibility is dominated around (p = 270°. For high Rayleigh number, fluid friction irreversibility dominates for a porous cavity, except around ip = 270°, see Figure 8.4. As shown in Figure 8.4, the Bejan number changes more rapidly when Ra and z increase after (p = 180°. The Bejan number takes a small value for inclination angles between about 30° to 60° and 120° to 170°, see in Figure 8.4, and it is seen that the heat transfer and fluid friction contribution to the irreversible losses were not comparable in these flow cases. The main findings of Baytas (2000) can be outlined as follows: the distribution of entropy generation in 2D laminar natural convective flows for a saturated tilted porous cavity

\3

0.1

0.01

Be 0.001

0.0001 i Ra = 10 0.00001

I—I

0

30

I

I

60

I—I

90

•

I

<

I

<—I

"

I

'

I

'—r-"—1—^^—r-"—I

120 150 180 210 240 270 300 330 360

Figure 8.4 Variation of Bejan number, Be, with inclined angle for Ra = 10 , 10 and 10^. From Baytas (2004). Reprinted with permission of Kluwer.

213

A. C. BAYTA§ AND A. F. BAYTA§

had been studied numerically using the finite-difference method and the second law of thermodynamics. The influence of the physical parameters, Ra, Be and ^p have been evaluated. Results showed that as Ra decreases, the heat transfer irreversibility begins to dominate the fluid friction irreversibility and Be changes rapidly between if — 150° and 270^ The entropy generation contains two physical levels: at a local level (from equation (8.18)) in Figure 8.2, it shows that not only where irreversibilities are present, but also to what extent they were sensitive to the design changes according to different inclination angles; at an integral level (from equation (8.20)) in Figures 8.3 and 8.4, the entropy generation gives a measure of the 'degree of irreversibility' of the convective flow in the enclosure. The problem of entropy generation in a fluid saturated porous cavity for laminar magnetohydrodynamic natural convection heat transfer was analyzed by Mahmud and Fraser (2004). They assumed that the Darcy law held, the fluid was a normal Boussinesqincompressible fluid and inertial effects were neglected. The boundary conditions on the surface of the cavity, which is illustrated in Figure 8.5, are similar to the previous study as shown in Figure 8.1. The magnetic field was assumed to act along the direction of the gravity, see Figure 8.5. The governing equations can be obtained by the modification of equation (8.4) for the porous cavity with the magnetic field in the x- and ^/-directions as follows: u = V —

-I

dx + dP_ dy

+

(8.21)

pg[l-P{T-Tc)]y

The set of non-dimensional governing equations in terms of the stream function, * , and the dimensionless temperature, 6, were obtained from equation (8.21) and equation (8.6)

dT/dy = 0 Bo

'' ' ' '

TH\

9

1

^ •

Tc

dT/dy = 0 Figure 8.5 Physical model of the 2D square porous cavity studied by Mahmud and Fraser (2004).

214

ENTROPY GENERATION IN POROUS MEDIA

by neglecting the viscous dissipation term as follows:

dr ^ dY dx

dx dY~

dx^'^d^'

^^ ^

where the dimensionless variables are defined in equation (8.17) and Ha is the dimensionless Hartmann number that is defined by (8.24) Equations (8.22) and (8.23) have to be solved subject to the initial and boundary conditions:

* = 0, * = 0, * = 0,

e = l, 9 = 0, 89 ^ - 0 ,

O^Y^l, O^y^l,

r>0, r>0,

O^X^l,

r>0.

(8-25)

They also expressed the entropy generation equation, in which the hydromagnetic effect was included. Their equation is similar to equation (8.11) and can be defined in dimensional form for a porous enclosure as follows: ^g^n = ^

[(V.T)^ + ( V , r ) ^ ] + j ^ y + v') + ^ u ' ,

(8.26)

where the last term represents the entropy generation due to hydromagnetic effects. They investigated the effects of magnetohydrodynamic free convection and entropy generation for a square porous cavity on the flow and temperature fields in terms of the Rayleigh, Hartmann, Prandtl and Bejan numbers. In the absence of any magnetic force, i.e. Ha = 0, the usual convective distortion of the isothermal lines occurred with two thermal spots: one at the bottom of the hot wall and another at the top of the cold wall. They found that on the introduction of a magnetic field acting along the direction of gravity tended to retard the fluid motion inside the cavity. The strength of the circulation inside the cavity reduces as the Hartmann number increases for a constant value of Ra. As the Hartmann number increases, a large portion of the fluid in the central portion of the cavity becomes almost motionless. Distortion of the isothermal lines, as well as the thermal spots also starting to disappear as the value of Ha increases. As Ha increases, the isothermal lines inside the cavity approach more and more towards the conduction-like distribution. For large values of Ha, say 10, the convection was almost suppressed, and the isotherms were almost parallel to the vertical wall, indicating that a quasi conduction regime has been reached. At Ha = 0, the entropy generation spreads all over the cavity, see Baytas (2000).

A. C. BAYTA§ AND A. F. BAYTA§

215

It is evident that the entropy generates at a higher magnitude near the cavity walls and this is because of the strong concentrators of irreversibility due to higher values of the near wall velocity components and temperature gradient. In the absence of the magnetic field, a region along the diagonal that connects the top comer of the cold wall and the bottom comer of the hot wall showed high heat transfer irreversibility. In this situation, a large portion of the fluid in this region was either stagnant or slower in moving leaving a negligible contribution offluidfriction irreversibility on the overall entropy generation rate. However, heat transfer irreversibility spreads all over the domain after the introduction of the magnetic field. The main findings of Mahmud and Fraser (2004) may be summarised as follows: for Ha = 0 and 1, the average entropy generation rate increases as the Rayleigh number increases. At low and moderate Rayleigh numbers and with a magnetic field, conduction dominates. Most of the contribution to the overall entropy generation comes from the heat transfer irreversibility. The increase in the value of Ha, i.e. the magnetic field, has a tendency to retard the fluid motion inside the cavity. In the absenqe of a magnetic field, the entropy generation rate is high near the two vertical walls. Further, the entropy generation rate decreases in magnitude as the magnetic field increases. Baytas and Pop (2000) have studied the steady entropy generation due to laminar free convection from a sector of an annulus cavity filled with a porous medium. They used an enclosure that is shown in Figure 8.6, and assumed that the flow is given by the Darcy model and that the fluid is a normal Boussinesq fluid. Under these assumptions, the Darcy

Figure 8.6 Schematic diagram of the porous cylindrical annular sector studied by Baytas and Pop (2000).

216

ENTROPY GENERATION IN POROUS MEDIA

and energy equations were given in their study as follows: &^

19;$^

1 ^ _ R

( ' A^

cos(t)de\

Qj,2

J, Qj,

j,2 Q^2

y

r

Qj.

d(j)) ^

de_ ifd^de__m_de\_^

ide_ ^ 9 ^

dr

r dr

r \d(f) dr

dr dcj))

dr"^

r^ 90^ '

where r, (j) are the non-dimensional polar coordinates, r is the non-dimensional time, 6 is the non-dimensional temperature and Ra = gK/3{Ti — To)ri/iya is the Rayleigh number for the porous medium. The non-dimensional variables are defined by

.= '

* = S. « = I ^ ,

. = ifL, J., = ^ ,

(8.29,

where Ti and To are the inner and outer constant boundary temperature, respectively, and Ti > To, while the radial side walls are considered to be adiabatic. In this study, the governing equations were solved numerically for a porous cylindrical annular sector with if (the angle between the centreline of the annular sector and the vertical) varying from 15° to 180° and for a range of values of Ra up to 900 using the finite-difference method as mentioned before, for more details, see Baytas (1996, 2000) and Baytas and Pop (2001). The resulting algebraic equations were solved by the alternating direction implicit (ADI) method and the results were given in terms of the isotherms and streamlines patterns. In Figure 8.7(a,b), isotherms and streamlines are given for Ra = 500 and (/? = 15°, 45° and 75°, and Figure 8.7(c,d) shows the entropy generation due to the heat transfer and the local entropy generation number, N, (equation (8.18)) for the irreversibility distribution ratio parameter of 10~^, where Ra = 500. It is evident from Figure 8.7(c) that the entropy generation was higher at high temperature gradients; see Figure 8.7(a). This is due to the heat transfer irreversibility, because the large heat transfer is confined to these locations. The distribution of the local entropy generation due to the heat transfer and fluid flow is shown in Figure 8.7(d) and it is seen that the local entropy generation due to heat transfer and fluid friction irreversibility spreads over the whole of the cavity walls because of the heat transfer and fluid friction irreversibility. The variation of the entropy generation number, TVg, as a function of (p is shown in Figure 8.8 for Ra = 300, 500 and 900 and it can be seen that the flow is turned into a conduction regime from (p — 120° to 165° for all the values of the Rayleigh number considered. This is due to the fact that the buoyancy is no longer available at these angles. It is observed that the entropy generation number has a maximum at approximately 45° and from Figure 8.8 we observe that Ra increases as Ng increases. Baytas and Pop (2000) used the Bejan number, see equation (8.19), and the local entropy generation number for cylindrical coordinates to examine the entropy generation due to convection heat transfer as follows: Ns=

\ J\

i 'iVrdrdc/), J ifi

(8.30)

A. C. BAYTA§ AND A. F. BAYTA§

217

(c)

Figure 8.7 (a) Isotherms, (b) streamlines, (c) heat transfer irreversibility, and (d) the map of the entropy generation number, N, for Ra = 500 and for different values of (p from top to bottom: 15°, 45° and 75°.

where A = Vo/vi. Another finding is that as (p increases Be approaches 1; Be = 1 is the limit at which all the irreversibility is due to heat transfer. The value of Be = 0 is the limit to which all the irreversibility is due to fluid friction for (p about 0° and 15°. The value of Be = 1/2 is the case in which the heat transfer and the fluid friction entropy generation rates are equal for ^ about 165° and Ra == 300.

218

ENTROPY GENERATION IN POROUS MEDIA lOU-

• A •

Ra = 900| Ra = 500 Ra = 30o|

120 <

N, 80-

40-

0-— 1 — i — 1 — 1 — 1 — 1 — I — I — 1 — 1 — 1 — 1 — 1 — 1 — 1 15

Figure 8.8 and 900.

8.4.2

30

45

60

75

90

105

120

|T^ 1 1 W

135

150 165

Variation of the entropy generation number, Ns, with (p for Ra = 300, 500

Entropy generation in a porous channel

Tasnim et al. (2002) have performed an analytical study, using the first and second laws of thermodynamics, on the characteristics of flow and heat transfer inside a vertical channel comprising of two parallel plates embedded in a porous medium under the action of a transverse magnetic field, see Figure 8.9. In their study, combined free and forced convection inside the channel is considered and the flow is assumed to be steady, laminar, fully developed, electrically conducting and with a heat-generating/absorbing fluid. Also, the fluid is assumed to be incompressible with constant properties, except the density in the buoyancy term in the momentum equation. Both vertical walls are kept at isothermal temperature which may be the same or different. Since the flow is fully developed, the fluid velocity components no longer depend on the axial distance x. This yields that the velocity component in the ^-direction is zero everywhere in the channel and mass continuity gives du/dx = 0. The magnetic Reynolds number was assumed to be small, so that the induced magnetic field can be neglected and the Hall effect in the magnetohydrodynamics was assumed to be negligible. In this investigation, the governing equations were based on the usual conservation laws of mass, linear momentum and energy modified to take into account the combined boundary and inertia effects of the porous media, buoyancy effects and hydromagnetic and heat generation/absorption effects. The momentum equation was a reduced form of the Brinkman-Forchheimer equation with the additional body forces (buoyancy and magnetic).

219

A. C. BAYTA§ AND A. F. BAYTA§ W

•]

9

Bi

Bo

^

T3

O

TR

TL

I I I Flow Figure 8.9 Schematic diagram of the problem studied by Tasnim et al (2002).

For steady Newtonianfluidflowin a porous channel of constant porosity, equation (8.5) can be rearranged for fully-developed flow to the modified Brinkman equation by neglecting the inertial forces and the form drag terms, as follows:

''•0-(f + ^«»)" = l^-'"'«^-^")'

(8.31)

where Too is the ambient temperature and equation (8.9) can be modified for the fullydeveloped and steady-state flow in a porous channel with hydromagnetic effect as follows: k ^

+ Qo{T - Too) + jB^u^ = ^P^P^''^

'

(8.32)

where Qo (W m~^ K~^) is the heat generation or absorption term. Equation (8.32) was simplified to a linear form in order to obtain an analytical solution by Tasnim et al. (2002). They neglected the temperature gradient in the x-direction and magnetic dissipation effect in equation (8.32) in order to find the analytical solution of the governing equation. Equations (8.31) and (8.32) are formed dimensionless using suitable scaling parameters. The axial distance x is scaled with M^Gr, where Gr is the Grashof number (= glS/STW^v"'^) and the transverse distance y is scaled with the width, W. The velocity u is scaled with uGi/W and the dimensionless temperature 9 is defined as (T — TOO)/(TR — Too). The momentum and energy equations are written in dimensionless form and the boundary conditions in their studies are as follows:

'JE-iA^M')u = §-e. ± ViHd - 0,

(8.33) (8.34)

220

ENTROPY GENERATION IN POROUS MEDIA

f/(0) = 0,

d{Q) = 9L,

t/(i) = o, e{i) = eR,

^"- "^

where 9 is the dimensionless fluid temperature, A is Da~^, where Da denotes the Darcy number, M is the Hartman number, Y is the dimensionless transverse distance, Pr is the Prandtl number and H is the heat generation/absorption coefficient. The form of the analytical solution for 6 is different for a heat-generating fluid (positive sign in equation (8.34)), than for a heat-absorbing fluid (negative sign in equation (8.34)). The second law of thermodynamics analysis is achieved by using equation (8.11) in this investigation. Also, the entropy generation number, A^^, that is the dimensionless form of the entropy generation rate is used as follows: ^s={^\

+ZpU\

(8.36)

where Zp = ^Too{vGvY jkK/ST'^ is the irreversibility distribution ratio (group parameter). The entropy generation number was also used for the two different forms of the heat-generating and heat-absorbing fluids. Tasnim et al (2002) presented their results in the form of the dimensionless velocity and temperature profiles at different values of the Hartmann number in the range 0 to 20. Also, the entropy generation number was plotted for different group parameters and the dimensionless modified velocity, the dimensionless temperature and the entropy generation number were plotted as a function of the dimensionless transverse distance. The results showed that an increase in the Hartmann numbers tends to slow down the movement of the fluid in the porous channel. This is because the application of the magnetic field creates a resistive force similar to the drag force that acts in the opposite direction to the fluid motion, thus causing the velocity of the fluid to decrease. In addition, for a positive value of the heat generation/absorption parameter, the entropy generation rate is higher than for the negative value of the same magnitude. The irreversibility distribution ratio, Zp, has a significant effect on the entropy generation. As the group parameter increases, the entropy generation increases, see Figure 8.10, where r is the porous medium inertia coefficient {— c^jeVFGrK""^/^) and 6L and OR are the left and right wall temperatures, respectively. The influence of the inverse of the Darcy number {A) on the entropy generation profile is shown in Figure 8.10(a) and it seen that it has an insignificant effect on the entropy generation. A large variation of A causes a small variation in the rate of entropy generation. In Figure 8.10(b), the velocity is zero at the walls because of the no-slip boundary conditions and therefore, at the walls the entropy generation is entirely dominated by the heat transfer and the entropy generation rate is the same for all values of Zp. For large values of Zp, say greater than about 30, the value of the entropy generation passes through a maximum at F = 0.5. Mahmud and Fraser (2003) have recently examined analytically the effects of radiation heat transfer on magnetohydrodynamic mixed convection through a vertical channel packed with fluid saturated porous materials and for this aim both the first and second laws of thermodynamics were applied to analyze the problem. A steady laminar flow of an

221

A. C. BAYTA§ AND A. F. BAYTA§

(b) 0.175 r 0.15 r 0.125

.^

bv^- 1>^

0.1 t- M L

0.075 r"

•

0.05 F.

"

-^^ ^

^

"

"

"

^

^

^

-^^""^yy^ '^^'^yv/ ^8--Vj/

b 0.025 F_

0,EuL 1

''^^—---—^ -IS

'

1 1 1 »

0.25

v6

0.5 Y

0.75

Figure 8.10 Entropy generation number Ns as a function Y at (a) different values of A for an asymmetric temperature at the boundary when Zp = 10, and (b) different values of Zp for a symmetric temperature at the boundary when A = 10. Here dP/dX = 0.1, M = 0, i7 = 1, Pr = 0.7, r = 0, ^L = 1 and OR = 1. From Tasnim et aL (2002). Reprinted with permission from Elsevier.

incompressible viscous fluid was assumed to flow through the channel with a negligible inertia effect and the fluid was considered to be an optically thin gas and the effects of electrical conduction was investigated. The schematic diagram of this problem is similar to that shown in Figure 8.10. Both walls are isothermal and kept at the same constant temperature (TR = TL) or different temperatures {TR > TL). The right and left walls contain the radiative heat flux. Radiation heat transfer is considered and an optically thin gas was assumed. Mahmud and Eraser (2003) considered that the flow was hydrodynamically and thermally fully developed. The governing equations for fully-developed flow, in Cartesian coordinates, are written for the steady-state condition and solved analytically. The radiative heat flux is presented in the energy equation as follows: k

dy

(8.37)

where q^ is the radiative heat flux. A simplified approximation for calculating the radiative heat transfer term dq^/dy was used in order to analytically solve the problem, see Arpaci et al. (2000). In this investigation, the influence of different dimensionless parameters, such as the radiation parameter, porous-magnetic parameter, (Gr Ri)°-^ (Grashof number x Richardson number)^'^, and the Brinkman number was investigated on the calculated velocity, temperature and local and average entropy generation number.

222

ENTROPY GENERATION IN POROUS MEDIA

The volumetric rate of entropy generation for this problem is as follows:

- € ( | ) -"-»).r„ /^

S'" =

2 U .

(8.38)

For a non-Darcy model of a porous media, the first two terms in equation (8.38) are used to calculate the local entropy generation rate, i.e. n — \. When the Darcy law holds in the porous channel then the entropy generation due to the fluid friction can be evaluated using the third term, i.e. n = 0, on the right-hand side of equation (8.38). The second term is also more important for high porous medium. The dimensionless form of equation (8.38) is given by ^, (dey^ Bv fduy^ ,, ,:Br* (8.39) u'

n

where Br and 11 are the Brinkman number and the dimensionless temperature difference (AT/To), respectively. The ratio Br/II is termed the group parameter and the modified Brinkman number is given by Br* = BVW/VK. The results showed that the radiation parameter introduces a non-linearity into the temperature profiles, and the radiation and mixed convection parameters have a more dominating influence on the entropy generation rate than the porous-magnetic and group parameters. Figure 8.11(a) shows the variation of A^s,av with the radiation parameter, jR^, for different mixed convection parameters, (Gr Ri)^^ and n = 1. The reason behind choosing the parameters (Gr Ri)^*^ and Rd was due to their practical importance and strong contribution to the entropy generation. For a particular value of (GrRi)^^, Ns^av decreases as Rd increases, and A^s,av shows its minimum entropy generation number at a particular value of Rd and then it increased.

(Gr Rir=

- 100 - 50 -20 10

1010"

I I ml

10^

I

R.

1 MINI

10^

Figure 8.11 Variation of the average entropy generation with the mixed convection parameter at (a) n = 1 (G = 0.1, ^ = 0.5 and Br/n = 1), and (b) n = 0 (G = 0.1, ^ = 0.5 and BrVn = 1).

A. C. BAYTA§ AND A. F. BAYTA§

223

For the selected range of values of (Gr Ri)°-^, the A^s,av profiles merge with each other after Rd « 15. For n = 0, Figure 8.11(b) shows the distribution of A^s,av with Rd while keeping all the other parameters at their same values as those used in Figure 8.11(a). The behaviour of the average entropy generation A^s,av with Rd, see Figures 8.11 (a,b), is similar but the variations of the magnitude of A^s,av are different for non-Darcy flow model (n = 1) and Darcy flow model (n = 0). The corresponding magnitude of the radiation parameter at which the average entropy generation becomes a minimum, see Figure 8.11, is termed the optimum radiation parameter, Rd,opt- The optimum radiation parameters for different values of (Gr Ri)°-^ is plotted in Figure 8.11(a) forn = 1 and Figure 8.11(b) for n = 0. The behaviour of the system's irreversibility in terms of entropy generation minimization (EGM) and the variation of the irreversibilities with the channel parameters ((GrRi)°•^Br/^,i^rf) may be obtained using Figure 8.11.

8.5

CONCLUSIONS

In this chapter, a brief overview has been given so as to evaluate the current state of knowledge on entropy generation in a porous cavity and a porous channel. In this review, the first study was related to the inclined and externally heated square porous cavity. The effect of the inclination angle on the flow and heat transfer characteristics and entropy generation was studied by varying the inclination angle from 0° to 360° and for different Rayleigh numbers. The influence of the Rayleigh number, Ra, Bejan number, Be (alternative irreversibility distribution number), and the inclination angle, (p, were evaluated. Results show that as Ra decreases, heat transfer irreversibility begins to dominate the fluid friction irreversibility. The entropy generation contained two physical levels: at a local level, it showed not only where the irreversibilities were present but also to what extent they were sensitive to the design changes at different inclination angles; the total entropy generation gave a measure of the * degree of irreversibility' of the convective flow in the enclosure. The problem of entropy generation in a fluid saturated porous cavity was described for laminar magnetohydrodynamic free-convection heat transfer. The effect of the Rayleigh and Hartmann number was tested on the average Nusselt, entropy generation and Bejan numbers. It has been shown that increase the value of Ha, or the magnetic field, had the tendency to retard the fluid motion inside the cavity. In the absence of a magnetic field, the entropy generation rate was relatively higher in magnitude near the two vertical walls. The entropy generation decreased in magnitude as the magnetic force was introduced and strengthened. The local Bejan number distribution showed a diagonal dominance and the average Bejan number distribution with Ra or Ha showed the opposite behaviour when compared with the average entropy generation number. The studies on entropy generation for a complex geometry are also important because there are many applications in nature and industry of complex geometries. Therefore the next described topic was about entropy generation for a porous cylindrical annular sector filled with a porous medium. It was inclined with an arbitrary angle from the vertical and

224

ENTROPY GENERATION IN POROUS MEDIA

for a range of values of the Rayleigh numbers up to 900, and the enclosure tilt angles were 0° to 180°. The variation of the entropy generation number depended on the tilt angle and showed that the flow was turned into the conduction regime from about 120° to 165° for all values of the Rayleigh number investigated. The entropy generation number had a maximum at approximately 45°. The other study presented in this chapter was related to the entropy generation in a porous channel including the hydromagnetic effect. The flow was assumed to be steady, laminar, fully developed of electrically conducting and a heat generating/absorbing fluid. The group parameter, Z^, has a significant effect on entropy generation rate. As the group parameter increases the entropy generation increases, and the inverse of the Darcy number has an insignificant effect on the entropy generation. The last study presented in this chapter was for the entropy generation of the mixed convection-radiation interaction in a vertical porous channel. The influence of different dimensionless parameters, such as the radiation parameter, porous-magnetic parameter, Grashof number, Richardson number, and Brinkman number have been investigated on the computed velocity, temperature and local and average entropy generation number. The entropy generation number was characterized by a concave shaped profile and it was symmetrical about the channel centreline for a symmetrical temperature boundary condition. The radiation and mixed convection parameters had a more dominate influence on the entropy generation rate than that of the porous-magnetic and group parameters. For the entropy generation minimization, the optimum radiation parameters were determined and they increase with increasing values of (Gr Ri)°^ and Br/II (or Br*/n). It should be mentioned that further work is required to be performed on the entropy generation in porous media. The utilization of the second law of thermodynamics is very important in thermal design decisions and for improving thermal system designs. The principal aim of computing the entropy generation should find a thermodynamic optimization criterion using the idea of entropy generation minimization. The studies relate to the entropy generation in porous media have been investigated for constant physical properties and an isotropic porous media with smooth geometries. Further research on the entropy generation should be performed for complex geometrical configurations, anisotropic porous media, variable thermal physical properties of porous media and variable porosity, permeability and tortuosity.

REFERENCES Arpaci, V. S. and Larsen, P. S. (1984). Convection heat transfer. Prentice Hall, Englewood Cliffs, NJ. Arpaci, V. S., Selamet, A., and Kao, S. H. (2000). Introduction to heat transfer. Prentice Hall, New York. Atkins, P. W. (1994). The second law. Scientific American Books, New York. Baytas, A. C. (1996). Buoyancy-driven flow in an enclosure containing time periodic internal sources. Heat Mass Transfer 31, 113-19.

A. C. BAYTA§ AND A. F. BAYTA§

225

Baytas, A. C. (1997). Optimisation in an inclined enclosure for minimum entropy generation in natural convection. J. Non-Equilibrium Thermodyn. 22, 145-55. Baytas, A. C. (2000). Entropy generation for natural convection in an inclined porous cavity. Int. y. Heat Mass Transfer 43, 2089-99. Baytas, A. C. (2003). Thermal non-equilibrium natural convection in a square enclosure filled with a heat-generating solid phase, non-Darcy porous medium. Int. J. Energy Res. 27, 975-88. Baytas, A. C. (2004). Entropy generation for free and forced convection in a porous cavity and a porous channel. In Emerging technologies and techniques in porous media (eds D. B. Ingham, A. Bejan, E. Mamut, and I. Pop), pp. 259-70. Kluwer, Dordrecht. Baytas, A. C. and Pop, I. (2000). Entropy generation due to free convection from a trapezoidal cavity filled with a porous medium. In Recent advances in transport phenomena. Proceedings of the 12th international symposium on transport phenomena, Istanbul, Turkey, pp. 237-40. Baytas, A. C. and Pop, I. (2001). Natural convection in a trapezoidal enclosure filled with a porous medium. Int. J. Eng. Sci. 39, 125-34. Baytas, A. F. and Akbal, S. (2002). Determination of soil parameters by gamma-ray transmission. Radiation Measurements 35, 17-21. Bejan, A. (1979). A study of entropy generation in fundamental convective heat transfer. J. Heat rran5/er 101, 718-25. Bejan, A. (1994). Entropy generation through heat and fluid flow. Wiley, New York. Bejan, A. (1995). Convection heat transfer (2nd edn). Wiley, New York. Bejan, A. (1996). Entropy generation minimization. CRC Press, Boca Raton, PL. Cheng, C. H., Ma, W. P., and Huang, W. H. (1994). Numerical predictions of entropy generation for mixed convective flows in a vertical channel with transverse fin arrays. Int. J. Heat Mass Transfer 21,519-30. Darcy, H. P. G. (1856). Les fontaines publiques de la ville de Dijon. Victor Dalmont, Paris. Drost, M. K. and White, M. D. (1991). Numerical predictions of local entropy generation in an impinging jet. J. Heat Transfer 113, 823-9. Ergun, S. (1952). Fluid flow through packed columns. Chem. Eng. Progress 8, 89-94. Hossain, M. and Pop, I. (1997). Radiation effect on Darcy free convection flow along an inclined surface placed in a porous media. Heat Mass Transfer 30, 149-53. Ingham, D. B. (2004). Governing equations for laminar flows through porous media. In Emerging technologies and techniques in porous media (eds D. B. Ingham, A. Bejan, E. Mamut, and I. Pop), pp. 1-11. Kluwer, Dordrecht. Ingham, D. B. and Pop, I. (eds) (1998). Transport phenomena in porous media. Pergamon, Oxford. Ingham, D. B. and Pop, I. (eds) (2002). Transport phenomena in porous media. Vol. II. Pergamon, Oxford. Ishakoglu, A. and Baytas, A. F. (2002). Measurement and evaluation of saturations for water, ethanol and a light non-aqueous phase liquid in a porous medium by gamma attenuation. Appl. Radiation Isotopes 56, 601-6. Kaviany, M. (1995). Principles of heat transfer in porous media. Springer-Verlag, New York. Lage, J. L. (1998). The fundamental theory of flow through permeable media from Darcy to turbulence. In Transport phenomena in porous media (eds D. B. Ingham and I. Pop), pp. 1-30. Pergamon, Oxford.

226

ENTROPY GENERATION IN POROUS MEDIA

Liu, S. and Masliyah, J. H. (1999). Non-linear flows in porous media. J. Non-Newtonian Fluid Mech. 86, 229-52. Mahmud, S. and Fraser, R. A. (2003). Mixed convection-radiation interaction in a vertical porous channel: entropy generation. Energy 28, 1557-77. Mahmud, S. and Fraser, R. A. (2004). Magnetohydrodynamic free convection and entropy generation in a square porous cavity. Int J. Heat Mass Transfer 47, 3245-56. Muskat, M. (1937). Flow of homogenousfluidthrough porous media. McGraw-Hill, New York. Nield, D. A. (1999). Modelling the effects of a magnetic field or rotation on flow in a porous medium: momentum equation and anisotropic permeability analogy. Int. J. Heat Mass Transfer 42, 3715-18. Nield, D. A. (2000). Resolution of a paradox involving viscous dissipation and nonlinear drag in porous medium. Transport in Porous Media 41, 349-57. Nield, D. A. (2002). Modelling fluid flow in saturated porous media and at interfaces. In Transport phenomena in porous media (eds D. B. Ingham and I. Pop), Vol. II, pp. 1-19. Pergamon, Oxford. Nield, D. A. and Bejan, A. (1999). Convection in porous media (2nd edn). Springer-Verlag, New York. Patankar, S. V. (1980). Numerical heat transfer and fluid flow. Hemisphere, Washington, DC. Poulikakos, D. and Bejan, A. (1982). Fin geometry for minimum entropy generation in forced convection. J. Heat Transfer 104, 616-23. San, J. Y., Worek, W. M., and Lavan, Z. (1987). Entropy generation in convective heat transfer and isothermal convective mass transfer. J. Heat Transfer 109, 647-52. Tasnim, S. H., Shohel, M., and Mamun, M. A. H. (2002). Entropy generation in a porous channel with hydromagnetic effect. Exergy, Int. J. 2, 300-8.

9

THERMODIFFUSION IN POROUS MEDIA M. Z. SAGHIR*, C. G. H A N G * , M. C H A C H A t , Y. Y A N * , M. KHAWAJA* and S.PAN* * Department of Mechanical Engineering, Ryerson University, Toronto, Ontario, MSB 2K3, Canada email: [email protected], [email protected], [email protected], [email protected] and [email protected] ^Department of Mechanical Engineering, UAE University, PO Box 17555, Al Ain, UAE email: m. chachaQuaeu. a c . ae

Abstract The chapter introduces the phenomenon of thermal diffusion in porous media and presents the theory and the numerical procedure which have been developed to simulate this process. The numerical procedure is demonstrated for both polar and hydrocarbon mixtures. Additionally, convection as a major influence on the thermal diffusion process is further simulated and discussed for both square and rectangular porous cavities. Keywords: porous media, thermal diffusion, molecular diffusion, control volume, separation ratio, Soret coefficient

9.1

INTRODUCTION

The thermal diffusion process, also known as the Soret effect, is the tendency of a convection-free mixture to separate under a temperature gradient. For binary mixtures, the Soret effect is measured by the Soret coefficient, ST, that is, the ratio of the thermal diffusion coefficient, DT, to the molecular diffusion coefficient D. However, for multicomponent mixtures, the thermal diffusion coefficient is more commonly used as a measure of the Soret effect. The Soret effect is an important phenomenon for the study of the compositional variation in hydrocarbon reservoirs. It also plays a crucial role in the hydrodynamic instability of mixtures, mineral migrations and mass transport in living matters. In low-pressure gaseous mixtures and ideal liquid mixtures, the magnitude of 227

228

THERMODIFFUSION IN POROUS MEDIA

the thermal diffusion coefficient may be small. In contrast, in non-ideal liquid mixtures, particularly close to the critical points, the thermal diffusion coefficient becomes large and strongly depends on the energetic interactions, the size and shape of the molecules and thermodynamic conditions. For years, thermal diffusion has triggered active research in both theoretical and experimental studies. A large part of such research activities is on thermal diffusion in porous medium, see Ingham and Pop (1998,2002) and Nield and Bejan (1999). A porous medium is usually defined as material that consists of a solid matrix with an interconnected void. This void, also known as pores, allow for the flow of one or more fluids through the material. The distribution of the pores in a natural porous medium with respect to the size and shape is irregular. This chapter reveals the complication of the thermal diffusion process in porous media. A detailed literature review introduces a variety of techniques for measurement of the Soret coefficient. As well, the literature review discusses the mathematical and numerical methods for the simulation of the Soret effect in both free and porous media. Followed by the introduction of the fundamental equations of thermal diffusion and equations used for the porous media, this chapter explains the numerical solution technique that allows these equations to be solved. With the aforementioned techniques, the thermal diffusion process for various cases, namely square and rectangular porous cavities, are then simulated and the effect of the convection on thermal diffusion is further discussed.

9.2 9.2.1

LITERATURE REVIEW Measurement techniques of the Soret coefficient

In general, the Soret coefficient is evaluated based on the differentiation in the cell. Especially when the Soret coefficient is negative, such an evaluation is an extremely delicate process. Several ground based techniques have been developed to measure the Soret coefficient. Two techniques are commonly used for binary mixtures with highmelting points. The first is the long capillary technique, where the samples are processed in capillary tubes placed in a gradient furnace. The liquid vein is then quenched once a steady-state separation is attained. The second is the shear cell technique, where there is a superposition of discs with veins running through them. The temperature gradient is created between the bottom disc and the very top disc in the shear cell. Once the process is completed, discs are rotated around the central axis in order to divide the liquid veins, hence, isolating the sections for further analysis. The benefit of this technique is that it does not require the quenching process, which can cause volume contractions, convection, and phase segregation and hence influence the values obtained. To measure the Soret coefficients of binary systems with low melting points, numerous methods have been developed. Flow cell, for example, is one technique in which the separation process occurs in a very thin flowing layer in Poiseuille regime. A sharp edge at the outlet separates the flow in upper and lower halves. The difference of concentration

M. Z. SAGHIR ET AL.

229

between the two samples is analyzed by using refractometry or densitometry. Another commonly used technique is the Benard cell configuration. Here, the flow is heated from below and the onset of convection is determined by the Schmidt-Milverton plot technique, Schmidt and Milverton (1935). Additionally, diaphragm cell is a technique in which a porous medium is used to separate the flow into two chambers at different temperatures. Finally, the thermogravitation column is a common technique, where a temperature gradient is imposed horizontally to a vertical channel. This creates a differential accumulation of different species at the top and bottom. The Soret coefficient is then evaluated from the obtained separation. Optical techniques were explored in the Soret coefficient measurement, i.e. thermal diffusion forced Rayleigh scattering (TDFRS) technique, Kohler and Muller (1995), and laser-Doppler velocimetry (LDV) technique, Kohler and Wiegand (2002). According to the principle of TDFRS, a grating created by the interference of two laser beams is written in a sample. A small amount of dye present in the sample, converts the intensity grating into a temperature gradient. This in turn causes a concentration gradient by the effect of thermal diffusion. Both gratings contribute to a combined refractive index grating, which is read out by diffraction of a third laser beam. Upon analysis of the time dependent diffraction efficiency, the transport coefficients are then obtained. In the LDV technique, the experimental cell consists of two horizontal copper plates that are maintained at different yet constant temperatures. The imposed temperature difference is then gradually increased in small steps. Enough time is allowed between two successive temperature increments to allow steady state to establish. The time dependent velocity amplitude is recorded, and on the onset of convection, the critical temperature difference and the dimensional frequency corresponding to the oscillation of the velocity amplitude is used to calculate the Soret coefficient. As Kohler and Wiegand (2002) found, the LDV technique is more accurate than the Schmidt-Milverton plot technique in measuring the Soret coefficient. Convection has a major influence on the accuracy of the Soret measurements. Utilization of porous media might help in reducing the distortion caused by the convection because the fluid moves slowly in porous media. Costeseque et al. (2004) conducted experiments in both a free fluid and a porous medium. They utilized a cell with two horizontal steel plates maintained at different temperatures in order to create a vertical temperature gradient. Six couples of small holes along the horizontal sides were used to retrieve small liquid samples. These couples were then analyzed using a high resolution refractometer to determine the concentration difference. Two experiments were conducted in the same cell with the same thermal gradient; the first experiment representing the free fluid case and the second representing the porous medium case. It was found that, when the thermal conductivities of the fluid and solid matrix are of the same magnitude, the Soret coefficients do not differ significantly from the case of the porous medium to that of the free fluid. These findings indicate that the contribution of the solid matrix in a porous medium on the two mass fluxes—namely the isothermal mass flux caused by the gradient of chemical potential, and the thermodiffusive mass flux caused by the temperature gradient—is the same as those in free fluid. Therefore, it can be concluded that the Soret coefficient is the same in a free fluid as in a porous medium.

230

THERMODIFFUSION IN POROUS MEDIA

Several researchers have published values of the Soret coefficient for organic molecules, polymers, and even electrolytes solutions. However, few data is available from other techniques on the same systems. In summer 1999, research groups from various universities started a ground-based measurement campaign. The goal was to establish a reliable database of Soret and isothermal mass diffusion coefficient for three binary mixtures composed of dodecane, isobutylbenzene, and 1,2,3,4-tetrahydronaphtalene. As well, there was a clear desire to establish benchmark values for the Soret, thermal diffusion, and diffusion coefficients so that they could be used to compare Earth results with results obtained through other means such as experiments in microgravity or numerical simulation, Flatten et al (2003). The values obtained from researchers were in strong agreement with each other; their differences did not exceed more than 7%. The Soret coefficient for three binary mixtures of (i) 1,2,3,4-tetrahydronaphtalene-dodecane with mass fraction of 50% for each component, (ii) 1,2,3,4-tetrahydronaphtalene-isobutylbenzene with mass fraction of 50% for each component, and (iii) isobutylbenzene-dodecane with mass fraction of 50% for each component were found to be (9.5 ± 0.5) x 10"^ 1/K, (3.3 ± 0.3) x 10"^ 1/K and (3.9 ± 0.1) X 10~^ 1/K, respectively at the mean temperature of 25 °C. 9.2.2

Mathematical and numerical techniques

Modeling of the Soret coefficient in free columns The transport coefficient data for binary mixtures is sparsely available. Moreover, for nonideal multi-component mixtures, neither diffusion coefficients nor a theoretical framework exist for the estimation of thermal coefficients. This may significantly differ in binary mixtures. It is already known that the variation of hydrocarbon reservoirs, a non-ideal multi-component mixture system, cannot be determined using results obtained from binary mixtures. Furthermore, the sign of thermal diffusion factors is immaterial in defining the direction of the component segregation in the hot or cold regions. Shukla and Firoozabadi (1998) developed a model for thermal diffusion factors in multicomponent non-ideal mixtures. This model was based on the thermodynamics of irreversible processes where the effects of both equilibrium and non-equilibrium properties are incorporated. The equilibrium properties, such as partial internal energies and fugacities, were estimated using the volume-translated Peng-Robinson equation of state. On the other hand, the non-equilibrium properties, such as viscosity, were accounted for by incorporating the energy of viscous flow. The model was validated with the measured compositional data for a ternary mixture nC2i/nCi6/nCi2. They further predicted the thermal diffusion behavior for a six-component mixture of Ci /C3/nC5/nCio/nCi6/C2 at various temperatures and pressures and found that when the system was far from the critical point, the thermal diffusion factor was not sensitive to temperature or pressure. Firoozabadi et al (2000) further examined the performance of this model for three different binary systems; the hydrocarbon system, i.e. C1/C3, C1/C4, C7/C12 and C7/C16, the non-hydrocarbon system, i.e. Ar/C02, N2/CO2, H2/N2 and H2/CO2, and the hydrocarbon-nonhydrocarbon system, i.e. C1/N2 and C1/CO2. These three systems were used to model the binary mixtures of reservoir fluids. The model was compared with

M. Z. SAGHIR ET AL.

231

experimental data and two other models: the Rutherford and Kempers models. When detailed comparison is made between the theoretical results with experimental data, the Firoozabadi model is superior in describing experimental results for the thermal diffusion coefficients in binary mixtures. In particular, the sign of thermal diffusion coefficient is consistent with the experimental data. Modeling of the Soret coefficient in packed columns The thermal diffusion in packed columns (TPC) refers to an instance when a porous medium is present in the column between two thermal sources. The advantage of studying the porous configuration is that it allows the investigation of thermogravitation effect in natural systems which tend to exist in porous medium such as oil reservoirs. Additionally, the porous packing governs the coupling conditions between the vertical and horizontal transport. Since the mass separation in a thermal diffusion column depends on the balance between two rates—the rate of mass flux (due to the basic thermal diffusion effect in the horizontal direction) and the convection velocity in the vertical direction—anything that decreases the convection velocity, such as packing, increases the relative importance of the horizontal flux, and thus increases the steady-state mass separation in the batch column. A study by Lorenz and Emery, Jr (1959) shows that the maximum magnitude of separation or the optimum coupling between buoyancy and thermal diffusion is obtained at a particular value of the permeability. They also found that the packed column offered the advantage of using columns constructed with fewer intricacies. Jamet et al. (1996) presented a computer model of thermal diffusion in packed columns. This was used to determine the governing transport coefficients, especially the thermal diffusion coefficient. Based on a scientific principle, due to the intricacy of the coupled processes occurring in the TPC column, the diffusion coefficient could not be ascribed as a priori value. Rather, according to this principle, the diffusion coefficient needed to be considered as an implicit parameter. Based on this methodology, a model which uses a value of diffusion coefficient approximated from the hydrodynamic dispersion theory was created. This model provided values that were close to the experimental values. Therefore, Jamet et al. (1996) concluded that an a priori approach of the diffusion coefficient can in fact reduce the deviation between the experimental and numerical methods. Modeling of the thermal diffusion process in porous medium The thermogravitational diffusion effect has been used in thermal diffusion cells containing a porous medium to control convection. In these devices, mixtures are submitted to a horizontal thermal gradient and the Soret coefficient has been obtained for different species in binary mixtures. Benano-Melly et al. (2001) investigated numerically the phenomenon of thermogravitation in binary fluid mixtures. In their study, a two-dimensional cell with two vertical walls set at constant yet different temperatures was used. The binary mixture filling the cell porous medium was initially homogenous. Additionally, the twodimensional cell's boundary condition was noflowand no mass flux through the cell walls, and no heat flux through the horizontal walls. The flow analysis showed that the mixture undergoes convection due to the presence of thermal gradient with higher convection

232

THERMODIFFUSION IN POROUS MEDIA

velocity near the cell vertical sides. When the solutal Rayleigh number is high enough, and the Soret number is positive, the solutal and thermal buoyancy force combine their actions to enhance convection. On the other hand, when the Rayleigh number is high and the Soret number is negative, the counteracting solutal and thermal buoyancy force weaken the convection. It was also observed that the solutal buoyancy force had negligible, or no effect, on the convection for solutal Rayleigh numbers lower than 0.1. Benano-Melly et al. (2001) further investigated the solute migration and distribution behavior and found that the amplitude of separation mainly depends on the thermal Rayleigh number. Also, Benano-Melly et al. (2001) found that decreasing Lewis number induces a decreasing separation ratio. The thermal diffusion process can occur in both liquid and gaseous mixtures. Studies based on the thermodynamics of irreversible processes have shown that thermal diffusion, along with natural convection, can in fact have a significant effect on the compositional variation in the horizontal and vertical directions in hydrocarbon reservoirs. According to such studies, along with natural convection, thermal diffusion can either enhance or weaken the separation in mixtures. Riley and Firoozabadi (1998) presented a model to investigate the effects of natural convection and diffusion (thermal, pressure andfickian)on a single-phase binary mixture methane-n-butane in a horizontal cross-sectional reservoir in the presence of a prescribed linear temperature field. The compositional distribution in the reservoir under both horizontal and lateral heating conditions was carefully examined. It was found that increasing the permeability increases the horizontal compositional variation. As well, it was found that there exists a local maximum and/or minimum in the compositional gradient as a function of the permeability. As the permeability approaches infinity, the compositional gradient lessens monotonically to zero. It was also noticed that the boundary layers developed at the right and left boundaries of the reservoir, decrease in width as the permeability increases. Although there is a rapid variation in the compositional gradients within these boundaries, the effect of the boundary in the larger remaining region in between is insignificant. Modeling of double diffusion in free columns undermicrogravity conditions The evaluation of transport coefficients by ground-based techniques is difficult. In particular, in multi-component systems, such an evaluation is complex since the number of couplings between the thermal and composition transport—and consequently the number of transport coefficients to be known—increases rapidly. When convection is coupled to thermal and solute transport, the phenomenon is known as double diffusion. One possible remedy is to minimize the buoyancy by performing experiments in microgravity on free flying platforms, such as the International Space Station. In microgravity conditions, the convection induced by the gravity is eliminated. Chacha et al (2003) simulated the double diffusion process under several thermal boundary conditions in order to better understand the influence of these variations during a microgravity experiment. These included radiating horizontal walls, smooth change in the lateral thermal boundary condition, and, even more drastic, the abrupt change in lateral thermal boundary condition. They observed that when there is a strong convection, secondary recirculation eddies form along the comers

M. Z. SAGHIR ET AL.

233

of the cavity and a clockwise rotating cell is present in the center of the cavity. This formation indicates the strength of convection inside the cavity. The temperature profile near the hot and cold walls was distorted due to the presence of convection. As well, the heat transfer due to convection is stronger and faster than the heat transfer due to conduction. They therefore concluded that the transport of heat in the boundary layers as well as in normal gravity is driven by the convective flow. It was observed that in the case of radiating horizontal walls, a stronger radiative heat loss, or, a high emissive or a very weak temperature difference between the end walls, could possibly lead to the break up of the entire cavity into different thermodiffusion sub-cells. This would hence result in greater uncertainties in the Soret coefficient measurements. Further, it was observed in the event of power outage during an experiment in space that the system responds with some delay to the modification of the boundary conditions. Due to the absence of buoyancy-induced convection, the experiments on the microgravity environment aboard platforms, such as the International Space Station (ISS), may lead to accurate measurement of the Soret coefficient for multi-component mixtures. However, like other space laboratories, the ISS experiences a steady, and/or residual time dependent acceleration called gravity jitter or g-jitter. Such acceleration may be caused by the movement of the crew, thruster firing for altitude adjustment, or, operation of equipment. Therefore, it is important to consider the effect of g-jitter in the diffusion-dominated fluid science experiments on the ISS. Theoretical work in this area has been reported by several authors such as Gershuni and Zhukhovitsky (1981), Savino et al. (1998), and so on. Recently, Chacha et al. (2002) investigated the role of thermal diffusion phenomena on compositional variation in a binary mixture of methane and n-butane in the presence of g-jitter. They found that the presence of g-jitter causes mixing and overcomes the Soret effect in a cavity, therefore making it difficult to measure the Soret coefficient accurately in space.

9.3

FUNDAMENTAL EQUATIONS OF THERMODIFFUSION

Although the thermal diffusion coefficient can be measured through different techniques, the measurement condition has to be controlled strictly to avoid the possible occurring effect of thermal convection, which is very difficult in ground conditions. For years research has been endeavored to generate reliable thermal diffusion coefficient models. Several theoretical approaches have been proposed for binary mixtures with various degrees of success. These include the phenomenological theory of irreversible thermodynamics by Haase (1969), the kinetic theory of irreversible thermodynamics by Rutherford and Drickamer (1954), Dougherty and Drickamer (1955) and Shukla and Firoozabadi (1998), the elementary transition state theory by Mortimer and Eyring (1980), the kinetic theory of dense hard spheres and its revisions by De Haro et al (1983) and Kincaid et al. (1983), the partial excess enthalpy and activity coefficient by Guy (1986), the Brownian motion and heat of transport by Bearman et al. (1958), and the maximization of the partition function of two idealized bulbs by Kempers (1989). It should be kept in mind that in all of the abovementioned models, partial molar properties—derived from the equation of state

234

THERMODIFFUSION IN POROUS MEDIA

(EOS)—are required. Therefore, for any particular mixture, the accuracy of the model not only relies on the model itself, but also on both the EOS of choice and, the numerical method utilized in the calculation. Below three models are listed and described. For further reading on other models, refer to the original papers. 9.3.1

Haase model

The Haase (1969) model is based on the phenomenological approach. Such an approach uses the phenomenological equations of irreversible thermodynamics to derive the following equation for binary systems in terms of the net heat of transport and chemical potential: xi[dixildxi) where ar is the thermal diffusion factor in a binary mixture, xi is the mole fraction of component 1, /ii is the chemical potential of component 1, and Q\ is the net heat of transport of component i (z == 1,2). The net heat of transport is then interpolated with the molar enthalpy in a mass conserved system. With this assumption, the net heat of transport is expressed in the following format with the thermostatic values of enthalpy: ^^2

^ 1 - Mixi + M2X2 \M2

^^-^^

MJ'

where Hi is the partial molar enthalpy of component i (i = 1,2) and Mi is the molecule weight of component i (i = 1,2). The expression of the thermal diffusion factor in the Haase model then becomes H^ MiiJ2 - M2H1 ""^ (Mixi + M2X2) xi (dfii/dxi)

'

^ '^

For ideal gas/liquid at a reference temperature, equation (9.3) gives ^^

{Mixi + M2X2) RT '

^^'^^

where R is the gas constant. Then the Haase model may be rewritten, with regard to a reference state, as follows: ,, ^ a^i?T(MiXi + M2X2) + Mi(ff2 - H^2) " ^ 2 ( ^ 1 - H^) ""^ (Mixi + M2X2) xi {dfii/dxi) ' 9.3.2

^ '^

Kempers model

The Kempers (1989) model is based on a statistical description of a non-equilibrium twobulb system, which is similar to a system with afixedvolume at a uniform pressure without

M. Z. SAGHIR ET AL.

235

external forces. This assumption leads to the expression of the net heat of transport as follows:

where Vi is the partial molar volume of component i (z = 1,2) and Hi is the partial molar enthalpy of component i (i = 1,2). Combining equations (9.1) and (9.6), the thermal diffusion factor in the Kempers model is then given by: (Fixi 4-1/22:2) xi

(dfii/dxi)

9.3.3 Firoozabadi model Based on the thermodynamics of irreversible processes, Shukla and Firoozabadi (1998) presented a model for binary mixtures of reservoir fluids. In this model, the net heat of transport is related to the energy of detaching a molecular from its neighbors in the region of the mixture, and the energy given up in that region when one molecule fills a hole. Expressed in terms of the partial molar internal energy and the ratio of the energy of vaporization to the energy of viscous flow of the component, the net heat of transport is given by Ql = Wm - M^iWm + X2WH2), Q2 =

WH2

-

MXIWHI

+ X2WH2),

with WHI^-—, Ti

WH2 = - — , r2

(9.9)

where Ui is the partial molar internal energy of component i (2 = 1,2), and r^ is the ratio of the energy of vaporization, AC/^^P, and the energy of viscousflow,AC/^^^^: At/7^p In equation (9.8), tpi represents the volume fraction of molecules moving into a hole left by a molecule of type i in the mixture, which can be expressed as a function of molar fractions Xi(i = 1,2) and partial molar volumes Vi (i = 1,2) as follows: Vi V2 ^1 = —17— r r , ^2 = —77— 77-, a:iVi+a:2V2 X1F1+X2F2 with the constraint of the Gibbs-Duhem relation XiV^l -hX2^2 = 1.

(9.11)

(9.12)

236

THERMODIFFUSION IN POROUS MEDIA

Substituting equations (9.8)-(9.12) into equation (9.1), the thermal diffusion factor is thus given by: ^F ^ Ui/ri-U2/r2

^ {V2-Vi){xiUi/n+X2U2/r2)

^^ ^^^

The thermal diffusion factor in the Firoozabadi model is an explicit function of partial molar internal energies, volumes, the chemical potential and the ratio of the energies of vaporization and viscous flow.

9.4

FUNDAMENTAL EQUATIONS IN POROUS MEDIA

In natural porous media such as beach sand, rye bread and wood, the distribution of pores is irregular with respect to the shape and size of the media. However, in experimental and theoretical studies, the quantities to describe the flow are usually treated in a regular manner with respect to the space and time. The standard procedure of deriving the laws governing the macroscopic variables is to start with the equations that the fluid obeys, namely, the mass conservation, momentum and energy equations, and then obtain the macroscopic equations by taking average over volumes or areas containing many pores. Many books, such as Nield and Bejan (1999), have discussed these equations in details. The diffusive mass flux in the porous medium, J, as an essential part to form the mass conservation equation, can be generally expressed as follows: j = -p (DVC + DTVT + DpVp), (9.14) where D, DT, Dp are the mass diffusion, thermal diffusion and pressure diffusion coefficients of the porous medium, respectively, p is the density of the fluid mixture, C is the concentration (mass fraction) of a mixture component, T is the temperature, and p is the pressure. The Soret coefficient, ST, can be calculated as follows: DT

(9.15)

ST = ^ -

In the equations of momentum, mass and energy, the density of the fluid mixture is a function of the temperature, pressure and concentration. Therefore an equation of state (EOS) is required to complement the calculation. The following expression may be used when the mixture density depends linearly on the temperature, concentration and pressure: P^PO[1-I3T{T-

TO) - PC {C - Co) -Pp{p-

Po)],

(9.16)

where po is the density of the mixture at a reference state with the temperature To, concentration Co and pressure po- Here /3T, PC and /?p are the thermal expansion, solutal

M. Z. SAGHIR ET AL.

237

expansion and compressibility coefficients, respectively, defined as follows:

P

\9PJT,C

For more complicated cases, cubic EOS may be used to calculate thermodynamic properties including the density. Cubic EOS, such as the Ping-Robinson (PR) EOS and the cubic-plus-association (CPA) EOS are widely used in simulations of oil reservoirs. It has been demonstrated that PR-EOS can provide great accuracy for light and heavy hydrocarbons. As well, the CPA-EOS may be used to determine the density and other equilibrium properties for polar mixtures such as water-methanol and water-ethanol.

9.5

NUMERICAL SOLUTION TECHNIQUE

Computational fluid dynamics, CFD, is a common analytical tool to tackle various fluid flow problems including the thermodiffusion phenomenon. Before solving a particular flow problem, the geometry of the region of interest must be clearly defined. Additionally, since the accuracy of a CFD solution is sensitive to the mesh size, the mesh must be generated properly and tested. Generally, the finer the mesh is, the more accurate the solution. However, a fine mesh will incur a high cost in terms of the computer hardware and calculation time. Therefore, in practice, there is a need to do mesh sensitivity analysis for an optimal mesh size. Using an example for methane-n-butane subject to the lateral heating condition, the next section will discuss, in detail, the importance of finding an optimal mesh size. A series of conservation equations (mass, momentum, energy) and equation of state, together with the appropriate boundary and initial conditions, must be solved numerically. There are several numerical solution techniques, such as finite element and finitedifference. Among these, the finite volume method, or, control volume method, is one of the well-established and thoroughly-validated general purpose CFD technique. Its numerical algorithm consists of three components. The first is integrating the governing equations of fluid flow over all the control volumes of the solution domain. The second component involves converting the governing equations into discrete equations at its nodal points by applying finite-difference-type approximations for the terms. This represents flow processes such as convection, diffusion and sources in the integrated equations. Lastly, the third component involves solving the discrete equations using appropriate iterative methods. The control volume is constrained by the law of conservation, i.e. for a general flow variable ^ such as a velocity component or enthalpy the rate of change of ^ within the finite control volume with respect to time equals to the sum of the net flux of ^

238

THERMODIFFUSION IN POROUS MEDIA

due to both convection and diffusion into the control volume and the net rate of creation of ^ inside the control volume. The problem of thermodiffusion is associated with nonlinearities in the equation set and is strongly coupled since the density is related to the pressure, temperature and composition. Such a problem can be resolved by adopting an iterative solution strategy such as the SIMPLE algorithm of Patankar and Spalding (1972). In this algorithm, the iteration process is started by using guessed initial pressure and velocity fields. Then, by solving the continuity equation, a pressure correction field is obtained, which, in turn, is used to update the velocity and pressure field. This process is iterated until the velocity and pressure fields are converged. For two-dimensional cases, a pressure correlation equation may be expressed as follows: p-p*-fp', (9.20) where p* is a guessed (or 'incorrect') pressure field andp' is the pressure correction. The same method can be applied to the velocity components: u = u*-\-u',

v = v*+v',

(9.21)

where u* and t^* are the solutions of the momentum equations corresponding to the guessed pressure field p* and u' and v' are the responses to the pressure corrections to produce the exact velocity field represented by u and v. Assuming u' = -K^-^,

v' = -Ky-i-

(9.22)

and substituting the resulting expressions for u and v into the continuity equation yields the equation for the pressure-correction p', namely d_

dx

pKa

m

d_ dy

I+£(''"*)+1(^^*)' ''•'''

where Kx and Ky are simply coefficients that result from the discretization scheme. Further details in the derivation of the pressure and velocity corrections approach may be found in Patankar (1980), Peyret and Taylor (1983), and Hirsch (1990). Many researchers, such as Faruque et al. (2004), Chacha and Saghir (2003), and Jiang et al (2004), have used SIMPLE in their studies. For example, Chacha and Saghir (2003) applied this algorithm in simulating double-diffusion process for a binary mixture methane-n-butane. They also summarized the main steps of the numerical procedure derived from the SIMPLE algorithm: Step 0 Set problem configuration, boundary and working/initial conditions. Step 1 Determine the average density of the mixture p by solving the EOS of choice for the working condition defined by the set {pr^, T^, (7o). The obtained average value is kept unchanged during the course of the calculation. Step 2 Increment the time: t — {k -\-\) y. A^. Step 3 Guess the pressure field p* = pr^.

M. Z. SAGHIR ET AL.

239

Step 4 Determine the mass diffusion coefficient D, the thermal diffusion coefficient DT and the viscosity /i. Formulation for these coefficients may be referred to Chacha et al. (2002) and the viscosity to Faruque et al. (2004). The values of the transport coefficients at the interface between two control volumes are harmonic means of the values calculated at the adjacent (main) grid points. Step 5 Solve the momentum equations to obtain the velocity components u and v by applying the boundary conditions; these are the starting values for u* and v*, respectively. Relaxation factor may be introduced in the algebraic equations before solving. Compute 0^ and 0^;. Step 6 Solve the pressure-correction equation (9.23) for p'. Step 7 Calculate p by adding p' (part or whole) to p*, i.e. p = p* -f app'. Here ap is the pressure relaxation factor. Step 8 Correct u and v from their starting values using the velocity-correction formulas (9.21). Step 9 Solve the energy equation to update the temperature profile T by applying appropriate boundary conditions. Compute (t)TStep 10 Solve the species mass conservation equation to update the mass fraction C subject to boundary conditions. Compute ^ c Step 11 Use p as the new guessed pressure p*, return to step 3 and repeat the whole procedure until a converged solution is obtained. The solution convergence is achieved at each time step once the maximum of the average relative errors in n, v, c and T throughout the mesh is less than a given value, i.e. max(0^i, (t>v, (t>T, (t>c) < 10~^, say, for two successive iterations. The average relative error is defined by ij i-\

j=l

rpk,s-\-l

ij

(9.24)

where F stands for the unknown u, v, T or C, k denotes the time step, and s is the iteration number. The grid point is specified by its mesh coordinates (i, j ) . Step 12 Return to step 2 to calculate for the next time step or end the program.

9.6

MESH SENSITIVITY ANALYSIS

To simulate a physical flow problem, it is crucial to create a model of the flow domain. The accuracy of the solution for the flow problem is dependent on the finite element mesh. Fundamentally, the finite element mesh is an idealization of the model geometry. It involves defining a discrete point called the nodes, and collections of points called the elements, which the numerical procedure then uses to solve the appropriateflowequations. The accuracy of the results is often dependent on the quality of the finite element mesh. While having a coarse mesh may lead to erroneous results, having a very fine mesh can be very memory intensive. Thus, it is imperative to choose a suitable mesh size. There are several ways of determining such suitability. These include comparing the steady-state

240

THERMODIFFUSION IN POROUS MEDIA

average Nusselt number, for example, or, any other required output, such as the ratio of final concentration to the initial concentration at different mesh sizes. Once the error between the two output values is less than an acceptable value, say, 1% or the value asymptotes to a particular value, the optimal mesh size is then selected. In the case of a porous medium, both the flow and the natural convection are usually very weak. As a result, the Nusselt number does not change very much, thus making it very difficult to determine the optimal mesh size based on the Nusselt number. Alternatively, the ratio of final concentration, C, to the initial concentration, Co, of one of the components may be used to accomplish the mesh sensitivity. Take methane-n-butane as an example. The mixture is subject to a lateral heating condition. The cold wall is at the temperature of 334 K and the hot wall at 344 K. The cavity is 5 m long by 5 m wide with porosity 0.5, permeability 10 mD and the initial concentration of methane is 0.2 (molar fraction). The final concentration of methane refers to its concentration at the center line of the cavity along the width of the cavity when the process has reached steady state. Figure 9.1 shows the concentration ratios of methane at different mesh sizes. It is apparent that the mesh 40 x 40, 50 x 50 and 80 x 80 give nearly identical curves compared to coarser ones of 5 x 5 , 1 0 x 1 0 and 20 x 20. With both accuracy and cost-effectiveness in mind, it was decided that the mesh 40 x 40 was in fact the optimal mesh size. The same procedure has been applied to the case studies presented in the Section 9.7.

1.005

-^ — -^ -^ 1.003 J -^ -^

1.004-1

1.002H

5x5 10x10 20x20 40x40 50x50 80x80

2

Width [ml

3

Figure 9.1 Mesh sensitivity analysis.

M. Z. SAGHIR ET AL.

9.7

241

RESULTS AND DISCUSSION

9.7.1

Comparison of molecular and thermodiffusion coefficients for water alcohol mixtures

In Saghir et al. (2004), Firoozabadi's thermal diffusion model combined with CPAEOS was applied to calculate the thermal diffusion coefficient and molecular diffusion coefficient for methanol-water and ethanol-water mixtures at temperature 310.65 K and pressure 1 bar with 10% mass fraction of water in each mixture, respectively. The results were compared with experimental data obtained by Flatten (2002) as well as theoretical predictions with other models. Firoozabadi model achieved a better agreement to the experimental data compared to Haase and Kempers models. Tables 9.1 and 9.2 show the comparison between the theoretical and experimental values. The temperature of the experiment by Flatten (2002) is 37.5 °C or 310.65 K at normal atmosphere pressure. As it can be seen, all of the three mentioned models predict the Soret coefficient at a much higher value than the experimental one with the Firoozabadi model giving the closest match. The reason for this is due to the very special thermodynamic properties of water and the big difference between water and hydrocarbons in their chemical structures. An accurate EOS for water related mixtures is required to generate satisfactory results in thermal diffusion research. However, this is a big challenge and the obstacles are yet to have been overcome.

Table 9.1 Comparison of theoretical and experimental results for water-methanol mixtures. Water-methanol Experiment by Flatten Firoozabadi model Haase model Kempers model

Density (kg/m3) 804.2 804.5

-

D DT ST (10 -^^m^ 7s) (10-- i 2 K - i m Vs) (10-3 K - i ) 39.1 32.39

4.87 4.237

-

-

1.24 1.31 35.09 45.71

Table 9.2 Comparison of theoretical and experimental results for water-ethanol mixtures. Water-ethanol Experiment by Flatten Firoozabadi model Haase model Kempers model

Density (kg/m3)

D (10--'^m^

7s)

DT

ST

1.85 1.75 52.86 61.59

(10-- i 2 K - i m Vs) (10-3 K - i )

802.8 792.65

15.3 16.68

2.48 2.924

-

-

-

242

THERMODIFFUSION IN POROUS MEDIA

9.7.2

Calculation of molecular and thermodiffusion coefficients for hydrocarbon mixtures

Rutherford and Roof (1959) have conducted an experiment to measure the thermal diffusion of the methane-n-butane system in the critical region. They utilized the steady-state thermodynamics to interpret the thermal diffusion behaviour of methane-n-butane. The assumption is made that the fluxes of heat and matter are linearly related to the thermodynamic forces. Rutherford and Roof (1959) utilized a two-chambered cell that consisted of a well-mixed reservoir maintained at different temperatures and connected by a porous diaphragm or by a capillary tube to make the measurements. A coolant was circulated through a coil of hypodermic tubing placed inside the cold chamber to keep the temperatures at the two chambers different yet constant. The cell was mounted horizontally in a constant-temperature air-bath. When the cell was filled with the fluid mixture of methane-n-butane, the coolant and air-bath were set at proper temperatures, hence, creating a temperature gradient. Once the pressure was set to a desired value, the experiment started. After a sufficient amount of time, the steady state was reached. The sample was then taken and flashed through sampling valves into the evacuated chambers of the interferometer, where the difference in refractive index, and hence, the difference in composition was recorded. The steady-state separation was used to calculate the thermal diffusion factor, Q:T- The measurements of the thermal diffusion factor were made on the methane-n-butane system at 0.4 molar fraction of methane in the composition in the pressure range of 90 to 210 bar and at temperatures of 320 K, 344 K, 377 K and 394 K.

0.04

Theoretical at 320 K Experimental at 320 K Theoretical at 344 K Experimental at 344 K Theoretical at 377 K Experimental at 377 K

0.03

S 0.02 H

o

GO

O.OH

0.00

90

13o"

Pressure [bar]

170

210

Figure 9.2 Variation of Soret coefficient with pressure and temperature (relative error bar 15%).

M. Z. SAGHIR ET AL.

243

Shukla and Firoozabadi (1998) predicted the thermal diffusion factors of a methane-nbutane system with an analytical model. Details of the model were presented earlier in the Section on Firoozabadi model. Figure 9.2 shows the comparison between the experimental and theoretical data. It was found that the model performed a good prediction at low temperatures, such as 320 K and 344 K. When the temperature increases to 377 K and 394 K, the modeled values were found to deviate considerably from the experimental values, especially for pressures below 170 bars, which is due to the methane-n-butane system approaching to the critical point. Nevertheless, the average deviation does not exceed 15% for the range of pressures considered. It is therefore concluded that this model can produce good performance in various non-ideal conditions away from the critical region. 9.7.3

Convection in a square cavity

Faruque et al. (2004) studied the effect of thermal diffusion, pressure diffusion and buoyancy force on both the flow and the concentration in a square porous cavity with a dimension of iJ = L = 1.5 m at different heating conditions. Table 9.3 is the thermophysical properties of methane-n-butane used in the calculation. Figure 9.3 shows the physical model and boundary conditions. For the combined heating case. Figure 9.3(c), the boundary conditions are expressed as functions of x and y: h {y) = To + Tyy , fsiy) = To + Tyy + T^L,

/2(x) = To + T,x, U{x) = To + TyH + T^x,

Table 9.3 Thermophysical properties of the mixture modelled. Molar fraction of fluid mixture Average density: po {po, To, Co) Thermal expansion coefficient: pr Solutal expansion coefficient: /3c Compressibility: /3p Reference pressure: po Reference temperature: To Porosity: (j) Specific heat of fluid: Cpf Conductivity of fluid: kf Density of porous medium: pp Specific heat of porous medium: Cpp Thermal conductivity of porous medium: kp Mass diffusion coefficient: D{po,To,Co) Thermal diffusion coefficient: DT (po, To, Co) Pressure diffusion coefficient: Dp{po, To, Co) Dynamic viscosity: /x(po 5 To, Co)

CH4(20%) + nC4Hio(80%) 515.8 kg/m^ 3.25 X 10-3 K - i 0.56 - 8 . 5 5 X 10-9 P a - i ll.lSMPa 339 K 0.2 2.75kJ/kgK 0.095 W / m K 2050 kg/m^ 1.84kJ/kgK 1.5W/mK 3.305 X 10-10 m V s -2.996 X 10-13 m V s K 1.518 X 1 0 - i ^ m V s P a 2 X lO-^kg/ms

244

THERMODIFFUSION IN POROUS MEDIA {^)

U = V = jy = 0

(b) u = v = jy =0,Tc = 334K

,

I

w CO CO

II 8.^ o'

II II II 3

a 1-J

II H)

L= 1.5m

''

••jy

(c)

=

0

u = v = jy=0,Th=Z4AK

u = v = jy=0,T = f4{x)

Y II

9

II

X

II 5

u = v = jy =0,T =^ f2{x) Figure 9.3 Physical model and boundary conditions (square cavity) for (a) lateral heating, (b) bottom heating, and (c) combined heating.

where Tx and Ty are the temperature gradients in the X and Y directions, respectively, and in this case they have values Tx = 6.6667 K/m and Ty = —6.6667 K/m. Lateral heating The physical parameters, such as the Soret coefficient, viscosity and density, vary as functions of temperature, pressure and concentration. Since the cavity height is small, i.e. i? = 1.5 m, the pressure is approximately constant in the cavity. Figure 9.4 presents the concentration, streamlines, density and temperature variation in the presence of the Soret effect. Figure 9.4(a) shows the temperature variation in the cavity, and in this case the average thermal Rayleigh number Rax is 50, the Darcy number is 1.5 x 10~^^ and the aspect ratio (AR) is 1. The calculated thermal characteristic time TT is 9.1 x 10^ s and the flow characteristic time r/ is 9.7 x 10^ s. Since the flow characteristic time is 10 times shorter than the thermal characteristic time, the isothermal lines are distorted due to the influence of convection. Figure 9.4(b) shows the streamline contours in the cavity. As expected, a singular convective cell is rotating in a counterclockwise direction. The

245

M.Z.SAGHIRETAL.

-0.00508 0.0

0.3

0.6 0.9 1.2 X direction fml

1.5

0.0

0.3

0.6 0.9 1.2 7 direction [m]

Figure 9.4 Lateral heating in a square cavity with R a r = 50, R a c = 11.2, Da = 1.5 X 10~^^, AR = 1 : (a) temperature contours, (b) streamline contours, (c) density contours, (d) methane concentration contours, (e) Soret variation in the X direction, and (f) Soret variation in the Y direction.

1.5

246

THERMODIFFUSION IN POROUS MEDIA

density variation of the fluid mixture is illustrated in Figure 9.4(c), where high density is accumulated near the cold wall and low density near the hot wall. Following the convention from thermodynamics of irreversible processes, Firoozabadi model, the Soret number has a negative sign when a light component moves to the hot wall. Therefore, one may expect that, due to the Soret effect, the methane—being the light component—will migrate to the hot end of the cavity. Figure 9.4(d) displays the light component profile of the mixture. Indeed, it highlights that the methane, in fact, migrates towards the cold end of the cavity. Moreover, see Figure 9.4(e), the Soret coefficient is negative and decreases along the length of the cavity. This reverse phenomenon may be explained as a result of the long diffusion characteristic time TM = 2.2 x 10^ s in this case. Thus, the flow reverses the concentration variation. The deformed S shape of the Soret coefficient as depicted in Figure 9.4(f) in the Y direction is mainly due to the convection pattern, which rotates counterclockwise. Bottom heating Figure 9.3(b) shows the boundary conditions of the bottom heating. This case is similar to the previous one with the average thermal Raleigh number Ra^ set to 50 and the Darcy number Da to 1.5 x 10~^^. Figure 9.5(b) shows two convective cells rotating in opposite directions. It is also observed the deformation of the temperature lines with a minimum at the centre of the cavity. Figure 9.5(a). These isothermal lines are deformed due to the much longer thermal characteristic time TT (9.1 x 10^ s) than the flow characteristic time TJ (9.7 x 10^ s). Figure 9.5(d) shows the concentration profile of the lighter component methane, which accumulates towards the hot side of the cavity. A very strong convection influence is observed on the concentration profile due to the fact that the diffusion characteristic time TM = 2.2 x 10^ s is about 1000 times longer than the flow characteristic time. The variation of the Soret coefficient along the middle lines of the cavity in both directions is depicted in Figures 9.5(e,f). The strong influence of buoyancy convection is clearly evident upon examination of Figure 9.5(e). In the X direction, the Soret coefficient shows a positive slope when the cell rotates clockwise. In contrast, the Soret coefficient changes to a negative slope when the cell is rotating counterclockwise. Upon examination of the Soret coefficient variation in the Y direction, a continuous increase of the coefficient is observed. This indeed indicates a high concentration of methane in the hot wall of the cavity when the Soret coefficient is the lowest. Thus, the dominance of the Soret effect is highlighted in this particular case. Such an occurrence stands in stark contrast to the lateral heating condition where the buoyancy effect is the dominant force in the cavity. Combined heating In this case, the average Rayleigh number Rar is 100 and the Darcy number Da is 1.5 X 10~^^. A single convective cell is observed rotating counterclockwise. Figure 9.6(b). This flow pattern implies that the lateral heating dominates the flow in this case. Since the flow characteristic time is much lower than the thermal characteristic time, a deformed temperature profile is observed in Figure 9.6(a). The deformation of the

247

M.Z.SAGHIRETAL.

(e) -0.004980

- -0.004984-1

^ -0.004988

-0.004992 0.0

0.3

0.6 0.9 1.2 X direction [m]

1.5

0.0

0.3

0.6 0.9 1.2 Y direction [m]

Figure 9.5 Bottom heating in a square cavity with R a r = 50, R a c = 16.3, Da = 1.5 X 10~^^, AR = 1 : (a) tenaperature contours, (b) streamline contours, (c) density contours, (d) methane concentration contours, (e) Soret variation in the X direction, and (f) Soret variation in the Y direction.

1.5

248

THERMODIFFUSION IN POROUS MEDIA

(a)

f

r^A^

^ ^ ^ ^ ^

519.809

K

(e)

vC?{

\ W 515.375 ^

" ^

\ \ \ 510.941

-0.00490

^ -0.00495 -O.OOSOOi

-0.00505 i -0.00510

0.3

0.6 0.9 1.2 X direction [m]

0.3

0.6 0.9 1.2 7 direction [m]

Figure 9.6 Combined heating in a square cavity with R a r = 100, Rac = 10.9, Da = 1.5xl0~^^,Ai?= 1: (a) temperature contours, (b) streamline contours, (c) density contours, (d) methane concentration contours, (e) Soret variation in the X direction, and (f) Soret variation in the Y direction.

M. Z. SAGHIR ET AL.

249

methane contour lines is also evident with high methane concentration in the cold region of the cavity (upper left comer). This indicates a suppressed Soret effect. In Figure 9.6(e), the Soret coefficient decreases linearly along the horizontal direction of the cavity, yet, deforms in the vertical direction. Figure 9.6(f). The dominance of the flow due to the lateral heating can also be understood by comparing the average solutal Rayleigh number Rac at different heating conditions. In bottom heating, when the Soret effect is effective, the average solutal Rayleigh number is 16. In both lateral and combined heating cases, however, the average solutal Rayleigh number was found to be 11. This indicates that these two cases are indeed physically similar with each other. 9.7.4

Convection in a rectangular cavity

As a comparison, the thermal diffusion phenomena in a rectangular porous cavity (0.5 m long by 5 m high) have been studied. The cavity is filled with the binary mixture methanen-butane and subject to the lateral, bottom and combined heating, respectively. The boundary conditions for the lateral and bottom heating cases are exactly the same as those in the square cavity studied earlier. Figures 9.3(a) and 9.3(b), respectively. For the combined heating case, the boundary condition is represented by equation (9.25) with the temperature gradient in the X direction, Tx — 20 K/m, and the temperature gradient in the Y direction, Ty — —2 K/m. All the walls are assumed to be solid, and therefore zero velocities are maintained. It is also assumed that there is no heat generation, no chemical reaction, and no interactive superficial forces acting between the porous medium particles and the liquid mixture. The thermophysical properties are listed in Table 9.3. Lateral heating The thermal diffusion process, as well as the concentration distribution and the separation ratio were investigated for various permeabilities by Jiang et al. (2004). Figure 9.7(b) shows the vertical distribution of the Soret coefficient. When the permeability is varied between 0.001 mD and 0.1 mD, the Soret coefficient along the vertical direction is constant. Methane being the lighter component does not displace in the vertical direction. When the permeability varies between 0.1 mD and 100 mD, a substantial variation of the vertical Soret coefficient is shown, in particular at a permeability of 10 mD. This suggests that beyond about 10 mD, the convection plays a more important role and transports methane to the top horizontal wall near the hot wall. As the permeability increases further, we observe a disturbance near the horizontal wall, and this is due to a large temperature gradient at the horizontal walls. Figure 9.7(a) shows the variation of the Soret coefficient along the horizontal direction. It is evident that, as the permeability increases, the convection becomes effective in reducing the horizontal variation of the thermal diffusion in the cavity. In terms of the fluid composition variation, it is found that when the permeability varies between 0.001 mD and 0.1 mD, the convection flow is weak and the Soret effect is the dominant force. Therefore, the horizontal concentration distribution is linear with a gradient proportional to the gradient of the horizontal temperature distribution. Accordingly, the vertical concentration distribution is constant, see Figure 9.8. In this range of the

250

THERMODIFFUSION IN POROUS MEDIA

-

-0.00471

0.001 mD 0.1 mD lOmD lOOmD lOOOOmD

0.001 mD 0.1 mD lOmD lOOmD lOOOOmD

- - -0.0049

^^^^^^i^^i^^i!^^:^

Z -0.005 H o

-0.0053

0

2 3 F direction [m]

Figure 9.7 Variation of the Soret coefficient with permeability along the centre line of the cavity (lateral heating): (a) horizontal distribution, and (b) vertical distribution.

permeability, the transport of the methane is shown to be effective in the horizontal direction and weak in the vertical direction. As the value of the permeability increases from 0.1 mD to 100 mD, the concentration of methane at the right top of the cavity increases first then decreases. The largest variation of the concentration is found at lOmD and the separation of the methane is at its maximum in the horizontal and vertical direction. As well, the maximum Soret effect is observed with a large amount of methane migrating to the hot wall, see Figure 9.8(b). This phenomenon is also evident in the horizontal direction, see Figure 9.8(a). Beyond the point of lOmD permeability, buoyancy convection becomes dominant. When the value of permeability increases from 100 mD to 10 000 mD, the Soret effect disappears gradually. At the permeability of lOOOOmD, a flat methane concentration distribution in both the vertical and horizontal directions is observed.

251

M. Z. SAGHIR ET AL.

(a)

0.205

0.203

B o o 0.200

-

0.001 mD 0.1 mD lOmD lOOmD lOOOOmD

I

-3 0.198

0.195

0.0

(b)

0.22n

0.18

0.1

0.2 0.3 X direction [m]

0.4

0.5

3 y direction [m]

4

5

0.001 mD 0.1 mD lOmD lOOmD lOOOOmD

1

2

Figure 9.8 Methane distribution with permeability along the centre line of the cavity (lateral heating): (a) horizontal distribution, and (b) vertical distribution.

To better understand the phenomena, the separation ratio and its relationship to the permeability is discussed here. The separation ratio, q, is defined as Q =

Ch/{l-Ch) Cc/{l-Cc)

(9.26)

where Ch and Cc are the solute concentrations at the cavity hot and cold walls, respectively. Figure 9.9 shows the variation of the separation ratio as a function of the permeability. When the permeability is in the range of 0.001 mD to 0.1 mD, the separation ratio remains constant, which is about 1.05. In this range, the convection effect is negligible and the contribution of the thermal and molecular diffusion expressed in terms of the Soret effect dominates. For a range of permeability between 1 mD and lOOmD, a peak value of the separation ratio, q^nax = 1-29, is observed at about K = 10 m D , i.e. Darcy number

2b2

THERMODIFFUSION IN POROUS MEDIA

Permeability [mD] Figure 9.9 Separation ratio as a function of the permeability.

3.91 X 10~^^. As the permeability further increases, the separation ratio decreases rapidly. When the permeability is 10"^ mD, the separation ratio is close to 1. The fluid thus mixes and separation does not take place. This is because the convection becomes dominant and the Soret effect is suppressed. The separation ratio is a useful parameter in evaluating the effect of the convection flow on thermal diffusion. It can be derived either from numerical calculation of the concentration distribution, equation (9.26), or from an analytical method as shown in equation (9.27), Lorenz and Emery, Jr (1959): ALK,

where A, B and C are constants, and K is the permeability. In equation (9.26), the solutal buoyancy force is neglected and constant molecular and thermal diffusion coefficients are assumed. Benano-Melly et ai (2001) showed that a maximum value for the separation ratio exists at the permeability /^^, given by f^m —

gpT^TLp

(9.28)

For this lateral heating case, equation (9.28) predicts that the maximum separation occurs at tirn = 11.9 mD, whereas Figure 9.9 shows that the separation occurs at Kr„ = 10 mD. The discrepancy between the analytical method, equation (9.28) and the numerical modeling. Figure 9.9, is justified by the fact that in the numerical model the solutal buoyancy is included and both the thermal and molecular diffusion coefficients are functions of the temperature, fluid mixture and pressure. For the same reason, the maximum separation ratio from the analytical expression equation (9.27), which is 1.2, differs slightly from the numerical result, which is 1.29.

253

M.Z.SAGHIRETAL. Bottom heating

The boundary conditions are shown in Figure 9.3(b). This case is very similar to the lateral heating case except for the fact that the heat is applied at the bottom of the cavity, hence, creating a vertical temperature gradient of - 2 K / m . A wide range of permeabilities have been studied to reveal its effect on the flow pattern inside the cavity. It was observed that at a very low permeability, such as in the range of 0.1 mD up to the value of lOmD, the influence of buoyancy convection is extremely weak. However there is a complete stratification of flow and heat transfer does take place strictly because of conduction. Yet, as the permeability further increases, a convective cell rotating in the clockwise direction develops in the vertical cell and intensifies with the increased permeability. This convective cell is created due to the strong buoyancy convection caused by the vertical

(a)

-0.004983

--

^ -0.004985

0.01 mD lOmD lOOmD 500mD lOOOmD 5000 mD lOOOOmD

^ -0.004987 o o -0.004989

-0.004991

0.0

(b)

-0.0048-1

-0.0049

0.2 0.3 X direction [m]

0.1

--

0.01 mD lOmD lOOmD 500 mD lOOOmD 5000 mD I lOOOOmD

-0.0050

o

00

-0.0051

-0.0052

0

1

2

3 f direction [m]

Figure 9.10 Variation of the Soret coefficient with permeability along the centre line of the cavity (bottom heating): (a) horizontal distribution, and (b) vertical distribution.

254

THERMODIFFUSION IN POROUS MEDIA

temperature gradient and this is in contrast to the two convective cells that appear in the square cavity. The influence of the convection is further examined with the Soret coefficient variation along the median of the cavity in both directions, Figure 9.10. For the permeabilities investigated here the Soret coefficient along the X direction decreases, which is similar to the lateral heating case. On the contrary, the Soret coefficient increases almost linearly along the Y direction, which is opposite to the findings for the lateral heating case. This indicates a stronger dominance of Soret effect along the Y direction compared to the lateral heating case. The influence of the Soret effect is further evident in the concentration variation. Figure 9.11. When the permeability is small, methane accumulates towards the hot side of the cavity in the Y direction. Figure 9.11(b). With permeability increased to 5000 mD, methane concentration does not change very much along the Y direction. By comparing the findings at different permeabilities, it can be

(a)

0.2002

0.200 H

0.01 mD lOmD lOOmD 500 mD lOOOmD 5000 mD lOOOOmD

&

0.1998

0.0

(b)

0.1

0.2 0.3 X direction [m]

0.205

0.4

0.5

0.01 mD lOmD lOOmD 500 mD lOOOmD 5000 mD lOOOOmD

0.203-^ 0.20 H

0.199

s

0.197^ 0.195

2 3 y direction [m]

Figure 9.11 Methane distribution with permeability along the centre line of the cavity (bottom heating): (a) horizontal distribution, and (b) vertical distribution.

255

M.Z.SAGHIRETAL.

concluded that at a very low permeability, the influence of buoyancy is negligible and the Soret effect is in primary dominance, thus causing the stratification of the flow. However, as the permeability increases, the influence of the buoyancy increases and creates a convective cell in the center of the vertical cell. Nevertheless, the Soret effect remains more influential compared to the buoyancy. Combined

heating

The boundary conditions are shown in Figure 9.3(c). In this case, both lateral and bottom heating are applied, hence creating a horizontal temperature gradient of 20 K / m and a vertical temperature gradient of - 2 K / m . Various permeabilities were utilized to investigate its effect on the rectangular cavity.

(a)

-0.0048

- 0.001 mD * lOmD - lOOOOmD

^ -0.0049

g -0.0050

-0.0052 0.0 (b)

0.2 0.3 X direction [m]

-0.0048

-0.0052

-

0

1

0.001 mD lOmD lOOOOmD

2 3 Y direction [ml

Figure 9.12 Variation of the Soret coefficient with permeability along the centre line of the cavity (combined heating): (a) horizontal distribution, and (b) vertical distribution.

256

THERMODIFFUSION IN POROUS MEDIA

When the permeability has a value of 0.001 mD, it was observed that a single convective cell rotating in the clockwise direction appears. The flow pattern is indicative of the fact that bottom heating is dominant. This fact is even more evident by examining the Soret coefficient variation along the median of the cavity in both directions, Figure 9.12. Akin to the bottom heating case, the Soret coefficient has a negative slope and continuously decreases along the X direction, see Figure 9.12(a). In contrast, the Soret coefficient increases continuously along the Y direction indicating the Soret effect in the cavity as evident, Figure 9.12(b). The dominance of Soret effect is further evident by examining the concentration variation along the median of the cavity. Figure 9.13. The lighter component—methane—has a higher concentration along the hotter end of the cavity.

(a)

0.206 0.204

0.194

0.0

(b)

0.221

0.18

-

0.001 mD

•*• l O m D

-

lOOOOmD

0.1

0.2 0.3 X direction [m]

0.4

0.5

- 0.00 ImD - lOmD ^ lOOOOmD

2 3 Y direction [m]

Figure 9.13 Methane distribution with permeability along the centre line of the cavity (combined heating): (a) horizontal distribution, and (b) vertical distribution.

M. Z. SAGHIR ET AL.

257

When the permeability is increased to 10 mD, the pattern changes and indicates that the lateral heating case is more dominant. A single cell rotating in the counterclockwise direction appears. When the Soret coefficient along the median is examined, it shows a negative slope in both directions, Figure 9.12, indicating that the buoyancy convection becomes more dominant and the Soret effect is suppressed. The concentration variation shown in Figure 9.13 also shows that the methane accumulates at the cooler side of the cavity. This further illustrates that buoyancy convection indeed has more influence. When the permeability increases to lOOOOmD, a single convective cell rotating in a counterclockwise direction appears. This flow pattern indicates that the lateral heating is more influential than the bottom heating in this case. The dominance of buoyancy convection again can be evaluated from the variation of Soret coefficients along the median of the cavity. Figure 9.12(a) shows that the Soret coefficient decreases along the horizontal direction. However, as Figure 9.12(b) shows, the vertical direction of the Soret coefficient is deformed as a result of the strong buoyancy convection. Thus, methane is distributed uniformly in the cavity since the convection causes mixing of the flow. Close to the vertical walls there exists a very small (concentration) gradient. Figure 9.13. From the above analysis, it can be concluded that the Soret effect is more influential when the permeability is small. When it increases to a certain value, i.e. lOmD in this case, the buoyancy convection becomes more dominant. Further increase in the permeability will cause the Soret effect to be partly or completely suppressed, thus making the separation of mixtures difficult.

9.8

CONCLUSIONS

Thermal diffusion, or the Soret effect, in porous medium is a very active research topic due to its wide application in practice. In this chapter, we have briefly reviewed the experimental and numerical work done in this area, and discussed the theory and numerical methodology which is commonly used in the thermodiffusion studies. Due to the difficulty in measuring thermal diffusion and mass diffusion coefficients accurately, it is hoped that numerical models could provide reliable results and thus reduce the burden of costly experiments. Although many models have been presented in the past, one may be superior to others for certain cases. When and how to apply these models depends on the specific fluid mixture of interest and whether an appropriate EOS is available since most of the available models require partial molar properties calculated from the equations of state. For hydrocarbon mixtures, modeling with Firoozabadi model with Peng-Robinson EOS may be able to produce satisfactory results. However, for polar mixtures, such as mixtures comprised of water, a big challenge is still ahead. In reality, thermal diffusion rarely occurs alone. The buoyancy convection has a major influence on this process, thus distorting the mixture concentration distribution that would result without the convection. Due to a relative small velocity, convection in porous medium is generally not as strong as that in free columns. However, in some cases, it is still very pronounced depending on different heating conditions and permeability of

258

THERMODIFFUSION IN POROUS MEDIA

the porous cavity. Therefore, in this chapter, we have further discussed several scenarios in Section 9.7 with an attempt to understand the complexity beneath. Whether from the numerical simulation or analytical study point of view, the parameters, i.e. separation ratio and characteristic times, are valuable in analyzing the effect of the convection flow on the thermal diffusion process. In general, when the flow characteristic time, r/, is larger than the thermal characteristic time, rth, then the Soret effect is the dominant force for the mixture species separation in the cavity. Maximum separation is thus reached when Tf = Tth- When Tf is much less than rth, the buoyancy convection becomes dominant and the separation of the mixture is difficult to achieve. In porous medium, the maximum separation ratio is a function of the permeability of the cavity.

ACKNOWLEDGEMENTS The authors acknowledge thefinancialsupport of the Canadian Space Agency, CRESTech and the Natural Science and Engineering Research Council.

REFERENCES Bearman, R. J., Kirkwood, J. G., and Fixman, M. (1958). Statistical-mechanical theory of transport processes: the heat of transport in binary liquid solutions. Adv. Chem. Phys. 1, 1-13. Benano-Melly, L. B., Caltagirone, J. R, Faissat, B., Montel, F., and Costeseque, P. (2001). Modeling Soret coefficient measurement experiments in porous media considering thermal and solutal convection. Int. J. Heat Mass Transfer A^, 1285-97. Chacha, M. and Saghir, M. Z. (2003). Influence of thermal boundary conditions on the doublediffusive process in a binary mixture. Phil. Mag. 83, 2109-29. Chacha, M., Faruque, D., Saghir, M. Z., and Legros, J. C. (2002). Solutal thermodiffusion in binary mixture in the presence of g-jitter. Int. J. Therm. Sci. 41, 899-911. Chacha, M., Saghir, M. Z., Van Vaerenbergh, S., and Legros, J. C. (2003). Influence of thermal boundary conditions on the double-diffusive process in binary mixture. Phil. Mag. 83, 2109-29. Costeseque, R, PoUak, T, Flatten, J., and Marcoux, M. (2004). Simultaneous evaluation of Soret and Fick coefficients in a free and a packed vertical gradient Soret cell. In 6th international meeting on thermodiffusion, Varenna, Italy. De Haro, M. L., Cohen, E. G. D., and Kincaid, J. M. (1983). The Enskog theory for multicomponent mixtures. 1. Linear transport—theory. J. Chem. Phys. 78, 2746-59. Dougherty, E. L. and Drickamer, H. G. (1955). Thermal diffusion and molecular motion in Uquids. J. Phys. Chem. 59, 443-9. Faruque, D., Chacha, M., Saghir, M. Z., and Ghorayeb, K. (2004). Compositional variation considering diffusion and convection for binary mixture in porous media. J. Porous Media 7, 1-19. Firoozabadi, A., Ghorayeb, K., and Shukla, K. (2000). Theoretical model of thermal diffusion factors in multicomponent mixtures. AIChE J. 46, 892-900.

M. Z. SAGHIR ET AL.

2b9

Gershuni, G. Z. and Zhukhovitsky, E. M. (1981). Convective instability of a fluid in a vibration field under conditions of weightless. Fluid Dyn. 16, 498-504. Guy, A. G. (1986). Prediction of thermal diffusion in binary mixtures of nonelectrolyte liquids by the use of nonequilibrium thermodynamics. Int. J. Thermodyn. 7, 563-72. Haase, R. (1969). Thermodynamics of irreversible processes. Addison-Wesley, London. Hirsch, C. (1990). Numerical computation in internal and external flows. Wesley, New York. Ingham, D. B. and Pop, I. (eds) (1998). Transport phenomena in porous media. Pergamon, Oxford. Ingham, D. B. and Pop, I. (eds) (2002). Transport phenomena in porous media. Vol. II. Pergamon, Oxford. Jamet, Ph., Costeseque, P., and Fargue, D. (1996). Determination of the effective transport coefficients for the separation of binary mixtures of organic compounds into packed thermal diffusion columns. Chem. Eng. Sci. 51, 4463-75. Jiang, C. G., Saghir, M. Z., Kawaji, M., and Ghorayeb, K. (2004). Two-dimensional numerical simulation of thermo-gravitational convection in a vertical porous column filled with a binary fluid mixture. Int. J. Therm. Sci. 43, 1057-65. Kempers, L. J. T. M. (1989). A thermodynamic theory of the Soret effect in a multicomponent liquid. J. Chem. Phys. 90, 6541-8. Kincaid, J. M., De Haro, M. L., and Cohen, E. G. D. (1983). The Enskog theory for multicomponent mixtures. 2. Mutual diffusion. / Chem. Phys. 79,4509-21. Kohler, W. and MuUer, B. (1995). Soret and mass diffusion coefficients of toluene/n-hexane mixtures. J. Chem. Phys. 103, 4367-70. Kohler, W. and Wiegand, S. (2002). Thermal nonequilibrium phenomena influidmixtures. Springer, Berlin. Lorenz, M. and Emery, Jr, A. H. (1959). The packed thermal diffusion column. Chem. Eng. Sci. 11, 16-23. Mortimer, R. G. and Eyring, H. (1980). Elementary transition-state theory of the Soret and Dufour effects. Proc. Nat. Acad. Sci. USA-Phys. Sci. 77, 1728-31. Nield, D. A. and Bejan, A. (1999). Convection in porous media (2nd edn). Springer-Verlag, New York. Patankar, S. V. (1980). Numerical heat transfer and fluid flow. Hemisphere, New York. Patankar, S. V. and Spalding, D. B. (1972). A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transfer 15, 1787-806. Peyret, R. and Taylor, T. D. (1983). Computational methods in fluid flows. Springer, New York. Platten, J. (2002). The published microgravity results on Soret coefficients of some water-alcohol systems are false: new and correct results are presented. In 5th international meeting on thermodiffusion, Lyngby. Platten, J. K., Bou-Ali, M. M., Costeseque, P., Dutrieux, J. P., Kohler, W., Leppla, C , Wiegand, S., and Wittko, G. (2003). Benchmark values for Soret, thermal diffusion and diffusion coefficients of three binary organic liquid mixtures. Phil. Mag. 83, 1965-71. Riley, M. and Firoozabadi, A. (1998). Compositional variation in hydrocarbon reservoirs with natural convection and diffusion. AIChE J. 44,452-64.

260

THERMODIFFUSION IN POROUS MEDIA

Rutherford, W. M. and Drickamer, H. G. (1954). Theory of thermal diffusion in liquids and the use of pressure to investigate the theory. J. Chem. Phys. 22, 1157-65. Rutherford, W. M. and Roof, J. G. (1959). Thermal diffusion in methane n-butane mixture in the critical region. J. Phys. Chem. 63, 1506-11. Saghir, M. Z., Jiang, C. G., Derawi, S. O., and Kawaji, M. (2004). Comparison between theoretical and experimental data for the Soret coefficient for water-alcohol mixtures. In 6th international meeting on thermodiffusion, Varenna, Italy. Savino, R., Monti, R., and Piccirillo, M. (1998). Thermovibrational convection in a fluid cell. Comp. Fluids 27, 923-39. Schmidt, R. J. and Milverton, S. W. (1935). On the instability of the fluid when heated from below. Proc. Roy. Soc. Lond., Series A 152, 586-94. Shukla, K. and Firoozabadi, A. (1998). A new model of thermal diffusion coefficients in binary hydrocarbon mixtures. Ind. Eng. Chem. Res. 37,3331-42.

10

EFFECT OF VIBRATION ON THE ONSET OF DOUBLE-DIFFUSIVE CONVECTION IN POROUS MEDIA M. C. CHARRIER MOITABI*, Y. R RAZI^ K. MALIWAN+ and A. MOJTABI+ *Laboratoire d'Energetique (LESETH), EA 810, UFR PCA, Universite Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex, France email: [email protected] +IMFT, UMR CNRS/INP/UPS N°5502, UFR MIG, Universite Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex, France email: [email protected], mkittinaOme.psu.ac.th and [email protected]

Abstract In this chapter we consider the instability of double-diffusive 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 thermosolutal 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 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 carried out. The 2D numerical simulations of the problem are presented which allow us to corroborate the results obtained from the linear stability analysis for both stationary and Hopf bifurcations. Keywords: porous media, vibration, double-diffusive convection, Soret effect, linear stability, long-wave mode, Hopf bifurcation

261

262

10.1

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

INTRODUCTION

In recent years, the effect of mechanical vibration on the stability threshold of thermal systems has been the subject of numerous studies. In thermo-vibrational convection, the energy of mechanical vibration in the presence of a temperature or a concentration gradient can be used to control the onset of convective motion. This type of convective motion, in which the buoyancy force may be thought of as time dependent, has attracted the attention of many researchers. Theoretical studies concerning linear and weakly nonlinear stability analysis of the Rayleigh-Benard convection subjected to a sinusoidal acceleration modulation have been conducted by several researchers, e.g. Gershuni et al. (1970), Gresho and Sani (1970), Biringen and Peltier (1990), and Clever et al (1993). Relative to the classical problem of the Horton-Rogers-Lapwood which has been documented in many books, see, for example, Ingham and Pop (1998, 2002) and Nield and Bejan (1999), only a few works have been devoted to the onset of convection under the action of harmonic vibration. Among existing thermo-vibrational studies in porous media saturated by a pure fluid we mention the works of Zen'kovskaya (1992) and Zen'kovskaya and Rogovenko (1999) in an infinite layer heated from below or above, Khallouf et al (1996) and Sovran et al (2000) in a rectangular cavity heated differentially, Bardan and Mojtabi (2000) in a rectangular cavity heated from below. Also, Jounet and Bardan (2001) consider the thermohaline problem in a rectangular cavity. Finally Sovran et al (2002) consider the effect of vibration on the onset of Soret driven convection in a rectangular cavity. In addition, Rees and Pop (2000,2001, 2003) have recently reported the effect of g-jitter on some boundary-layer problems. It should be emphasized that vibration-induced natural convection may exist even under weightlessness. This phenomenon is in contradiction with the common belief that natural convection cannot exist in space. Further research showed that a spacecraft in orbit is subject to many disturbing influences, see Nelson (1994). These influences result in the production of residual accelerations, which are commonly referred to as 'g-jitter'. It is important to note that these accelerations occurring on microgravity platforms may induce disturbances in space experiments that deal with liquids in the presence of density gradients. The construction of the international space station has increased interest in the influence of g-jitter on convective phenomena. The term g-jitter is used to describe a residual acceleration of 10~^ g fluctuating with a frequency that varies from 10~^ to hundreds of hertz. This acceleration is the result of crew activity as well as machinery on board the space station. One of the aims of the space station is to perform experiments under zero gravity conditions, i.e. without natural convection. It is well known that the g-jitter can produce drastic disturbances in space experiments as, for instance, in solidification processes during which mushy zones modelled as porous media may appear. With reduced gravity, other forces, which are normally masked on earth, may play a dominant role in buoyancy-driven convection. For a better understanding of the g-jitter effects, it was suggested that harmonic oscillations might be used to model this phenomenon, see Alexander (1994).

M. C. CHARRIER MOJTABIET AL.

263

The objective of this chapter is to study the effect of this vibrational mechanism on doublediffusive convection in a porous medium with or without Soret effect. We first present the so-called time averaged formulation applied to the double-convective oscillation in a porous medium in the framework of a Darcy-Boussinesq approximation. This formulation, restricted to the limiting case of high-frequency and small-amplitude vibration, can be effectively applied to investigate thermo-soluto-vibrational convection. We will later show, by using the scale analysis method, what is meant by high frequency and small amplitude. Under the Soret effect, the temperature gradient can produce mass flux in a multicomponent system, see for example De Groot and Mazur (1984). The influence of vibration on the thermo-solutal convective motion with Soret effect has been studied in an infinite horizontal fluid layer, see Gershuni et al. (1997,1999). The governing equations were described by a time-averaged formulation, which can be adopted under the condition of high-frequency and small-amplitude vibration. Their results showed that vibrations could drastically change the stable zones in the stability diagram. Generally, vertical vibrations (parallel to the temperature gradient) increase the stability threshold of the conductive mode. Smorodin et al. (2002) studied the same problem under low frequency vibration. They concluded that the synchronous mode has a stabilizing effect on the onset of convection. In this chapter, we describe the problem of vibrational double-diffusive convection and write down their basic system of equations in the framework of the standard DarcyBoussinesq approximation. The system for the mean field is obtained by applying the averaging technique. In the first part of the chapter, the thermo-convective motion in an infinite horizontal layer and confined cavity saturated by a binary mixture is studied. A linear stability analysis is carried out. The influence of the direction of vibration on the stability threshold is investigated. Stationary and Hopf bifurcations are investigated and the corresponding convective structures under the combined effects of vibration and gravitational accelerations are examined. In the second part, a numerical simulation has been carried out which allow us to corroborate the results obtained from the linear stability analysis.

10.2 MATHEMATICAL FORMULATION The geometry of the problem is a rectangular cavity filled with a porous medium saturated by a binary mixture, as shown in Figure 10.1. The aspect ratio is defined as A = L/H, where H is the height and L is the length of the cavity. The boundaries of the cavity are rigid and impermeable, the horizontal ones can be heated from below or above while the lateral ones are thermally insulated and impermeable. The governing equations are written in a reference frame linked to the cavity. We suppose that the porous medium is homogeneous and isotropic and that Darcy's law is valid. The binary fluid is assumed to be Newtonian and to satisfy the Oberbeck-Boussinesq approximation. The vibration frequency is large and the amplitude of movement is small enough for the averaging method

264

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

H

T = T2, Jm-n = 0,

Vn

=0

6T _ ac dx ~ dx

dx

T = Ti,

J ^ - n = 0,

dx

y.n =0

Figure 10.1 Geometry and coordinate system.

to hold. Other standard assumptions, local thermal equilibrium, negligible heating from viscous dissipation, etc., are made. The density p of the binary fluid depends linearly on the local temperature and local concentration of the denser component: p = pref [1 - I3T{T

- Tref) - Pc{C

- Cref)] ,

where the subscript *ref' is taken as the reference state, and the coefficient of volumetric expansion with temperature is represented as PT and for concentration /3c which are assumed constant. It is noted that the expansion coefficient pr is usually positive while 13c can be positive or negative. The entire system, including the cavity and the porous medium oscillates along the axis of vibration e = cos a i -h sin a j , where a = (i, e) is the angle of vibration, following the displacement law b sin cut e, where h is the displacement amplitude and cj is the angular frequency. The gravitational field is then replaced by the sum of the gravitational and the vibrational accelerations in the Darcy equation: g —^ g — huP" smut e. 10.2.1 Direct formulation Using the filtration velocity V, the pressure P , the temperature T and the concentration C as independent variables, the governing equations with Darcy-Boussinesq model can be written as:

v - y = o, PodV W 6*" V " - ' ^ ^ - / ^ o [ / 3 T ( T - T r e f ) - ^ c ( C - C , e f ) ] {g - boj'shiut e) ^V, 8T {pc).-^ + {pc)fV-VT

=

X.^^T,

(10.1)

M. C. CHARRIER MOJTABIET AL.

265

In equation (10.1), (pc)* is the effective heat capacity per unit volume of the medium, {pc)f is the heat capacity of fluid per unit volume, A* is the effective thermal conductivity of porous medium, D^ is the effective mass diffusivity coefficient, a' thermo-diffusion ratio, po density of fluid, and the porosity and permeability of the porous system filling the cavity are e* and X, respectively. The velocity is defined by V = U{t,x,y)i-\-V{t,x,y)j. Then the boundary conditions are given by: V{t,x,y

= 0) = 0,

V{t,x,y = H) = 0,

r(i,x,y = 0 ) = T i ,

T{t,x,y = H)=T2,

dT — {t,x = 0,y) = Uit,x = 0,y) = 0 , dT — {t,x = L,y) = U{t,x = L,y) = 0,

dC dT — + a ' — = 0 ,

^ + "'^=0. dC -^it,x dC —{t,x

(10.2)

= 0,y) = 0, = L,y) = 0.

10.2.2 Time-averaged formulation In order to study the averaged behaviour of the system (10.1), we use the time-averaged method. This method is used under the condition of high-frequency and small-amplitude vibration. The application of the averaging method only allows the mean velocity, concentration and temperature to be solved, see Zen'kovskaya and Simonenko (1966). The basic idea consists of treating the non-stationary flow with an approach similar to that used in the study of turbulent flows. This technique is widely used in different areas of physics and mechanics. According to this method, each field (velocity and temperature) is divided into two parts: the first part varies slowly with time (i.e. the characteristic time is large with respect to the vibration period) and the second part is r periodic and varies rapidly with time, i.e. the characteristic time is of the order magnitude of the vibration period. Under these conditions, it is shown mathematically that two different time scales exist, which make it possible to subdivide the fields into two parts; fast ( V , T', C", P') and slow ( V , r , C , F ) :

y(M, t) = y(M, t) + v'{M, ujt),

r(M, t) = T[M, t) -h T\M,

P{M,t) = P{M,t) -\-P'{M,ujt),

C{M,t) = C{M,t)^

ut),

C(M,oot).

For a given function / ( M , t), the average value is defined by

f{M,t):=-

1 /'*+^/2 ^ Jt-r/2

f{M,s)ds.

On substituting expressions (10.3) into equations (10.1) and boundary conditions (10.2), and averaging the resulting system over the vibrational period, a system for averaged field is obtained. This system is dependent on the average of the product of the oscillating

266

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

parts:

- ^

e dt

= - V P + po [/3r(T - T2) - ^c(C - Cef)] pfe

!ff

{pc).-^

_

_

+ {pc)fV • VT + {pc)fV' • VT' = A.V^T,

f^'n

_

_

e * — + y . VC + ( p c ) / V . VC - iP^CV^T 4- a ' V ^ C ) . On subtracting equation (10.4) from equation (10.1) and using expressions (10.3), we find the oscillating system, which depends on the averaged parts:

v-y' = o, PodV = -WP' + po [0T{T - T2) - /3c{C - Cref)] bu^ sinute e dt + Po WTT' - PcC] [gj + bu^sinute) - pobuj\^TT-l3cC')smujte

dV — (pc)*— + {pc)^V'. VT + {pc)fV'. VT' + {pc)fV • VT' - {pc)fV'' v r =

- ^V

,

Kv^r,

e * ^ 4 - V ' - V C + (pc)/V'-VC' + {pc)fV ' VC" - {pc)fV' 'VC - D,{V'^C' + a'V^T'). (10.5) It is evident that equations (10.4) and (10.5) are coupled, and the coupling terms are related to po^^^ (i^rT' - ^cC") sin a;te, {pc)fV' • VT' and {pc)fV' • V C which appear following the averaging procedure. The key step in resolving the closure lies in expressing the oscillating (fast) fields ( V , T', C") in terms of the scalar slow fields (T, C). For this reason, an order of magnitude analysis can be carried out on the system of equations having a rapid time evolution. 10.2.3 Scale analysis method In order to obtain the order magnitude of the relevant terms in equation (10.5), we perform a scale analysis method. This method of pedagogical importance has been successfully used in predicting the boundary-layer approximations, optimal geometries and critical parameters, see for example Bejan (1995, 2000, 2003) and Bejan and Nelson (1998). We

M.C.CHARRIERMOJTABIETAL.

267

use the following scales in the oscillatory system of equations: 0(T - Tret) « Ti - T2 = A T ,

0{C - Cref) ^ Ci - C2 = AC ,

o(ffi)..(.,, . dz n HBy introducing these scales into the momentum equation governing the rapid evolution (10.5), and assuming further for the oscillating temperature and concentration fields that T' < AT and C < AC, we obtain: v' ^ e*6a;(^TAr - ^ c A C ) .

(10.6)

Equation (10.6) has been obtained from the consideration that under a high frequency, the buoyancy terms, containing A T and A C are balanced by the inertia term. The domain of validity of this assumption can be obtained from the following inequality (inertia > friction): ^ « l .

(10.7)

By adopting the same procedure in the oscillatory energy and concentration equation, the scales of the oscillating temperature and concentration fields are found: T'«^^^(/3TAT-/3CAC),

(10.8)

C ' « ^ ( ^ T A T - ^ C A C ) .

Equation (10.8) gives the criteria for defining the small-amplitude vibration. The above scales are valid under the following conditions (transient term ^ diffusive terms):

As at the start of our analysis, we assumed that the buoyancy terms containing AT and AC are the dominant convective generating mechanisms, the final step is to validate this assumption. Utilizing expressions (10.8) and with some easy manipulations, we obtain: a;2 > A(e/3^AT - /3cAC). Under condition (10.10), the term po [/3T{T -T2) dominant buoyancy force in the momentum equation.

PC{C

(10.10) - Cref)] buj'^ sinute

is the

10.2.4 Time-averaged system of equations By applying the assumptions (10.7), (10.9) and (10.10) to the oscillatory system of equations, we may simplify the system of equations (10.5). Then by using Helmholtz's decomposition, we eliminate the oscillatory pressure. This allows us to obtain the oscillatory velocity, temperature and concentration, which upon substitution into equation (10.4),

268

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

provides us with the time-averaged system of equations. The dimensionless governing equations for the mean flow, averaged over the vibration period, can be written as follows:

B ^

-\-V = -VP + Ra(T + V^C)j + R v ( W T - h ^ W c ) • V (T-\--CJ

sinaj),

'

dt fir 1 e ^ + V -VC = :^iV^C ot Le V - W T - 0 ,

(cosai +

-V^T),

V - W C = 0,

where WT and W c are the solenoidal vectors corresponding to the temperature and concentration, respectively. The corresponding boundary conditions are given by: VTT

• n == W c • n = 0,

J ^ - n = 0, J^-n-0,

r-Ti T-r2

dT dC ^ — = ^- = 0

at t / - 0 , at 2/ = l , at

(10.12)

x-0,A.

System (10.12) depends on eight parameters: the thermal Rayleigh number Ra = Kg^/STH/va^, the vibrational Rayleigh number Rv = i?^Ra^, the separation factor ^ — —Ci{l — Ci){Pc/PT)DT/D*, the normalized porosity e (e = e*/a) where a = {pc)^/{pc)f, the Lewis number Le (Le = a* /D* in which a* is the effective thermal diffusivity and D* is the effective mass diffusivity), the coefficient of the unsteady Darcy term in the momentum equation B = Da/aePr* (in porous media B « 10~^ and Da represents the Darcy number Da = K/H^), and finally a the direction of vibration with respect to the heated boundary.

10.3 LINEAR STABILITY ANALYSIS When the direction of vibration is parallel to the temperature gradient, i.e. a = 7r/2, there exists a mechanical equilibrium, for both an infinite horizontal layer and a confined cavity, which is characterized by: Vo^O,

To = 1 - 2 / ,

Co = constant - y ,

WTO = 0,

WCO = 0.

(10.13)

269

M. C. CHARRIER MOJTABIET AL.

However, for other directions of vibration, we may obtain a quasi-equilibrium solution only for the infinite horizontal layer. This is characterized by: Vo = 0,

To = l - y ,

Wroy = 0,

VKTO,

Co = ci-y,

=C2-ycosa,

Wcoy = 0 .

Wco. =C3-ycosa,

(10.14)

It should be noted that, for a confined cavity, upon considering the boundaries conditions we conclude that the solenoidal fields are not equal to zero for a ^ 7r/2. 10.3.1

Infinite horizontal porous layer

The fields are perturbed around the (quasi-) equilibrium state in order to investigate the stability of the conductive solution. Then, after linearization, the disturbances are developed in the form of normal modes. It is assumed that the perturbation quantities are sufficiently small and the second-order terms are neglected (rj = c — 6):

= R a — [(1 + iP)e + VT?] + Rv(W^To. + -\-ipWcoJ

-{WTO.

dxdy [\

^WcoJg-^

eJ

e

H)

+ I1+ - ^ ^

sma

cos a

1+

-lyiWro^+Wcojl

. '0

cos a

[(1 + i^)FT + i^Fr,] sina

H)

d^ [{1 + ip) FT+ il^Frj] cos a. dxdy

(10.15)

In equation (10.15), the stream functions are defined as follows: W^Tx

=

dFr dy dF„

^-nx — Q

WTy

,

'W'ny —

dFr dx OF^ dx

WCx

dFc dy

'^Cy

=

d_Fh dx

- •

_ ^ dx

For the energy equation, we have: dl dt

^ _ y2^ dx

(10.16)

270

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

For the concentration equation, we obtain: dt

(10.17)

Le

dx

For Helmholtz decomposition, we find: V^FT

dy cos adr] V'^Fn = -^cosa dy

de . 7— dx sin a , df]

dx

(10.18)

sin a .

The corresponding boundary conditions are given by: dx dgx,!) dx

' ^ '

dy dx ^ dr]{x,l) ^ dFTJx,!) dy

dx ^ 9F,(x,l)

dx

(10.19) = 0.

dx

The disturbances of the normal mode are introduced as follows: (^(x, y, t),e{x, y, t),r]{x, y, t),FT{x, y, t),Frj{x, y, t)) = (ay), 0{y)My). FT{y),Fr,{y)) exp{Xt + Ikx), (10.20) where P = —1. We may obtain the following amplitude equations. For the momentum equation (D = d/dy): - {XB + 1) {D^ - e) I = IA:Ra [(1 + ^ ) ^ + il^i)] sin a

l + e^V+^r)' / e cos a

-lkD{WTo.^Wco.)

1 +el^V+^^' cos a e

\k{WTo.+WcoJD

;fcMi + ^ lk{l

(1 + V)^T + i^Fr, Sin a +

i, -]D

(1 + IP)FT + rpFr,

)sa > . (10.21)

For the energy equation: A^ + ifc| = (D^ - k'^)e.

(10.22)

M. C. CHARRIER MOJTABIET AL.

271

For the concentration equation: Le For the Helmholtz decomposition: {D'^ -k'^)FT

= D§ cos a - IkO sin a ,

{D'^ ~ k'^)Fr^ = Dfj cos a - Ikfj sin a .

(10.24)

In the above equations, WTO^ and Wco^ are defined as follows:

The corresponding boundary conditions for equations (10.21)-( 10.24) are given by: |(x,0) = e{x,{)) - Dfi{x,Q) = FT{X,0)

= F^(x,0) = 0,

| ( x , 1) = e^(x, 1) = Dfi{x, 1) = FT{X, 1) = F^(a;, 1) = 0. The systems (10.21)-( 10.24), along with the boundary conditions (10.25) correspond to the spectral amplitude problem with the decay rate, A, as an eigenvalue and with the amplitude as the eigenvector components. The characteristic value of the decay rate depends on all the parameters of the problem namely A = A(Ra, Rv, ^ , a, e, A;, Le) and, generally, the decay rate A is complex, i.e. A = Ar + lA^, because the spectral amplitude problem is not self adjoint. If A^ = 0 then the stability boundary is determined by the condition A = 0, i.e. the stationary bifurcation. If Af ^ 0, then the stability boundary is determined by the condition A^ = 0. In this case \i = Vth which is the frequency of the neutral oscillation. It should be emphasized that in accordance with the timeaveraged formulation this frequency {Vth) is smaller than the frequency of vibration. The stream function perturbation is introduced as 0, the temperature perturbation as 6 and the mass fraction perturbation as c. The stream function perturbations are designated as (j)e and 0c for corresponding solenoidal fields WT and Wc. To facilitate our study, the transformations 7] = c — 0 and ^p^^ = ip^ — ipo are used. For an infinite horizontal porous layer, we introduce disturbances of the normal mode in the following form: N

(f = 22 ^^ sin(i7ry) exp{at -h Ikx), i=l N-l

T] = 2_] ^i cos{i7ry) exp{at 4- Ikx), i=:0 N

(frj = / J 9i sin{i7ry) exp(crt + Ikx), 2=1

N

6=^

bi sm{i7ry) exp{at -\- Ikx),

i=l N

(pe = y j di sm{i7ry) exp{at + Ikx), i=l

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

272

where k is the wave number, a is the decay rate and (a^, bi,Ci,di,gi) are the amplitudes. The corresponding linear stability problem is solved using the Galerkin method. Horizontal vibration Figure 10.2 shows the stability domain for different vibrational parameters in the (Ra, ^p) stability diagram. This diagram is characterized by stationary and oscillatory bifurcations. For ^ > 0 the bifurcation is always of the stationary type, while for ^ < 0, we may obtain oscillatory (Hopf) or stationary bifurcations. The computations are performed for e* = 0.3, B = 10~^ (usual values used in porous media) and Le = 10. From the results we conclude that horizontal vibration has a destabilizing effect on both stationary and Hopf bifurcations. One of the interesting features of Figure 10.2 is that we may obtain long-wave mode instability in the regions which, under static gravity conditions, are infinitelyAinearly stable. The existence of these regions is due to the vibrational mechanism. Vertical vibration The aim of this section is to provide a qualitative picture of the flow and temperature fields to complete the results of our stability analysis. In order to study the effect of vibration on the convective pattern, we set Le — 2, -0 = 0.4, A = 1 and Ra = 30 and changed the R = 0.1^

Rac "^.^

- -A-o---A- -0-

45-

R = 0.5k 35\

Rdics {R = 0.1) Raco {R = 0.1) Racs (i? = 0.1, Arc = 0) Raics{R = 0.5) Raco (i? = 0.5) Racs {R = 0.5, kc = 0)

25-

'A. ••i..

^ 'J

\

15

•©•-9. -0.1

-0.08

-0.06

-0.04

0.02

-0.02

...®

004 ,-•0-

006 .«--e-

-0--9 O08 Ol' .e--9--9

.0*

-15 Figure 10.2 Stability diagram for stationary and oscillatory convection for Le = 10, e = 05 and ^ = 10~^

M. C. CHARRIER MOJTABIET AL.

273

value of the vibrational Rayleigh number Rv. The calculations are performed for e = 0.5 and 0.7 and results are presented in Table 10.1. These values are chosen according to the results of the stability analysis. We conclude from Table 10.1 that, for the selected values of Le, e, i/;, A and Ra, vibration reduces the Nusselt number and we may obtain a conductive solution. In addition we find that for the combination of Rv, V^ and e we have the interesting relation Rvc(l -h i/^/e) = constant. For the case under investigation this constant is 31.5. The linear stability analysis is carried out for different sets of parameters 0 < Rv < 100, 2 < Le < 100, ^ = - 0 . 2 and e = 0.5, 0.7. The results for Le = 2 and e = 0.5 are presented in Table 10.2. As can be observed from this table, two types of bifurcations; namely stationary and Hopf bifurcations, may be distinguished. For the stationary bifurcation, the principle of the exchange of stability is valid, i.e. a G M, and the marginal state is determined (cr = 0). For the Hopf bifurcation (a = ar -\- lujo)'-, the marginal state corresponds to cr^ = 0. In the case of the layer heated from below the Hopf bifurcation is present only for negative separation factors. In this case the Hopf bifurcation is formed before the stationary bifurcation, i.e. Raco < Racs. It can be concluded from Table 10.2 that vertical vibration has a stabilizing effect; it increases the critical value of thermal Rayleigh number for the onset of convection. This conclusion is true for both the stationary and the Hopf bifurcations. In addition, vertical vibration reduces the critical wave number (fcc, kco) and the Hopf frequency (ojo)-

Table 10.1 Effect of mechanical vibration on the Nu number for A = 1, a = 7r/2, Le = 2, V' = 0.4, Ra = 30, and (a) e = 0.5, and (b) e = 0.7. (a) (b) Rv

Nu

Rv

Nu

5 10 15 16 17 17.5 17.53

1.204 1.1483 1.0789 1.0604 1.0359 1.0124 1.0084

5 10 15 16 19 19.7 20.02

1.2057 1.1528 1.0942 1.0604 1.0329 1.0156 1.0034

Table 10.2 Effect of vibrations on the stationary and Hopf bifurcation (Le = 2, ^ -0.2 and e = 0.5). Rv

rC3/cs

0 10 50 100

153.19 157.53 173.63 193.60

r^CS

4.75 4.73 4.65 4.54

iXiOiQQ

95.43 91.1% 107.1 117.8

i^CO

2.59 2.56 2.41 2.26

CJo

10.78 10.75 10.50 10.26

274

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

10.3.2 Limiting case of the long-wave mode The results of the previous section indicate that the long wave mode (k = 0) is the dominant mode of Soret-driven convection under the effect of mechanical vibration in binary liquids. For this reason, we study the special case of long wave mode theoretically. In some related studies, see Gershuni et ai (1997, 1999), this analysis results in a closed form relation for the stability threshold. To obtain such a relation, a regular perturbation method with the wave number as the small parameter is employed: OO

OO

CO

„=0

„=0

„=0

OO

OO

OO

n=0

n=0

n=0

Vxvy.

v//

By substituting expressions (10.26) in the amplitude equations resulting from linear stability analysis, we find for zeroth-order: ^0=0,

^0 = 0,

r]o = constant,

(pQ^ = 0,

(p^^ = 0,

CTQ = 0.

For the first-order: ^1 = ^1=0,

I [Ra -

RYD{WTO.

7/1 == constant,

+ ^pWco^) cos a] V^/yo ,, , 2 ^^ ~ ^^' ipe^ = 0,

iprj^ = 0,

ai = 0.

After invoking the solvability condition, for the second-order we find: (T2

1 eLe

eRa-^ -I- Rvi/;(1 -h ip) cos^ a 12e

It is clear that (72 G R, which means that instability is of a stationary type; consequently for marginal stability (cr2 = 0) the following relation is obtained: ^^^(l + ^)Rvcos^a^^

(10.27)

We may distinguish different physical situations. 10.3.3

Convective instability under static gravity (no vibration)

In order to check the validity of relation (10.27), the classical case of convective instabifity due to static gravity is studied first. For this situation, we set Rv — 0, which results in:

Ra = 4 r - ,

(10.28)

M. C. CHARRIER MOJTABIET AL.

275

which is identical to the published results in the literature; see for example Schopf (1992) and Sovran ^r ^/. (2001). Convective instability under microgravity conditions In this case the instability excitation is due to vibrational mechanism. In order to determine the stable domain under microgravity conditions or weightlessness (g = 0), the following relation may be used: '0(1 -hV^jLecos^a From equation (10.8), we conclude that instability exists in V^ > 0, corresponding to positive Soret effect. By increasing the direction of vibration a from zero to 7r/2, the stable domain increases. In addition in the interval of -0 G [—1,0], the mono-cellular convection is not possible. Also, it can be noted that increasing the Le number decreases the stable domain. In the case of a = 7r/2, the long-wave mode (mono-cellular convection) does not exists under high frequency and small amplitude. Convective instability under the simultaneous action of vibration and gravitation In this case both the instability mechanisms, vibrational and gravitational, are in action. The stability boundary can be determined from the following relationship:

~

2^(1 + 0)Le cos2 a R?

'

^ ^ ^

When the direction of vibration is not parallel to the temperature gradient i.e. a 7^ 7r/2, there is a possibility of mono-cellular convection in all regions of the stability diagram (Ra, 0) except in the region 0; G [—1,0]. However, for the case of vertical vibration in which the direction of vibration is parallel to the temperature gradient, for stability domain we have: Ra== - ^ . (10.31) xpLe

Relation (10.31) demonstrates that under vertical vibration, the critical value of the thermal Rayleigh number for the onset of mono-cellular convection does not depend on the vibrational parameter. But it should be noted that the mono-cellular convection appears from smaller values of separation factors.

10.4

COMPARISON OF THE RESULTS WITH FLUID MEDIA

The objective of the present section is to compare the results of long wave mode in the case of fluid layer with those of porous layer. The results of long wave mode for the onset of mono-cellular thermo-solutal convection in an infinite horizontal layer under variable

276

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

Table 10.3 Comparison of the results of long-wave mode for the onset of convection in the presence or in the absence of vibration. . ^' .

No vibration

Microgravity conditions

Fluid layer

rv-3, — -77— V'Le

-K/V —

^^^^"^

Ra = - i ^

Rv =

layer

V'Le

, ,. ,—TT-; ^— V(14-'/')Lecos^ a

i2e

Simultaneous action of vibration and gravitation rv,a —

-V>Le±v/(V>Le)2-)-2880i/>(l+V>)Lecos2Qfl2 2V;(H-V')Lecos2aH2

p

-eV>Le±i/(eV;Le)2-f48e-0(l-f-t/')Lecos2afl2

_

V(l+V')Lecos2 a

orientation of vibration is reported by Razi ^r al. (2002, 2004). The results are presented in Table 10.3, the comparison of the results reveals that the Darcy model can provide us with a very good approximation for the physical behavior of the fluid system.

10.5 NUMERICAL METHOD The numerical simulations for a confined cavity are performed for vertical and horizontal vibration. The calculations are made for different aspect ratios A — \ and A — 10. The 27 X 27 collocation points are used for A — \, while 63 x 27 collocation points are used for A — 10. In order to solve the system (10.11) with the corresponding boundary conditions (10.12), the projection diffusion algorithm is used, see Azaiez ^r a/. (1994). The linear (viscous) terms are treated implicitly using a second-order Euler backward scheme, while a second-order semi-explicit Adams-Bashforth scheme is employed to estimate the nonlinear (advective) terms. We apply this method to an advection-diffusion equation such as (F is a general coefficient):

g + (n.v/) = rvV,

(10.32)

which can be discretized as follows: (3/2)/"+^ - 2 / " + (1/2)/ n - l

At

py2^„+i _ [2(„ . v / ) " - (u • V / ) " - i ] .

(10.33) Equation (10.33) may be written in the form of the following Helmholtz equation:

(v^-ft)/ n+1

_

(10.34)

A high-accuracy spectral method, namely, the Chebyshev collocation method with the Gauss-Lobatto zeros as collocation points, is used in the spatial discretization of the operators. The successive diagonalization method is applied to the inverse of these operators, see Azaiez et al. (1994).

M. C. CHARRIER MOJTABIET AL.

277

10.5.1 Vertical vibration In order to demonstrate the effect of vibration on the convective structure under different aspect ratios, we begin with the study of convection under gravitational acceleration: A = 10, Le = 2, e = 0.5, ip = 0.4, Racs = 13 and Rv = 0. The streamlines corresponding to this case, which are characterized by six convective rolls, are shown in Figure 10.3. The numerical result for the onset of convection is in good agreement with the result obtained from the linear stability analysis of an infinite horizontal layer heated from below, Racs = 12.95 andfcc= 1.94. It should be added that, based on our numerical simulations, we conclude that .4 = 10 provides a good approximation of an infinite layer. Figure 10.4 represents the effect of vertical vibration. It is clear that vibration changes the convective structure dramatically. The numerical results are in good agreement with the linear stability analysis, Racs = 15.04 andfees= 0.01. The result of the Hopf bifurcation for the temporal evolution of velocity for Le = 2, e = 0.5, Rv = 100 and ip = -0.2 is presented in Figure 10.5. As mentioned earlier, the Hopf bifurcation appears for negative separation factors. The numerical result shows that the critical values corresponding to the Hopf bifurcation are: Raco = 118.5, kco = 2.2 and ujo = 10.08. These are in good agreement with stability analysis results, Raco = 117.83, kco = 2.26 and OUQ — 10.25. 10.5.2 Horizontal vibration In this case, we set A = 1, Ra = 6, Le = 2, e = 0.5, if) = 0.2 and R = 0.3. The value of Ra is set to such a value so that only the vibrational mechanism is in action. Figure 10.6 shows the corresponding fluid flow structure and temperature distribution; the stream functions are characterized by symmetrical four-vortex rolls. This structure is a typical example of an imperfect bifurcation as was observed earlier in convection under microgravity conditions, see Gershuni et al (1982). The sum of stream functions is zero in this case. If we further increase the thermal Rayleigh number to Ra = 13.15, gravitational

Figure 10.3 Onset of stationary convection for A Rv = 0 and Racs = 13.

10, Le = 2, e = 0.5, t/^ = 0.4,

Figure 10.4 Effect of vertical vibration on the onset of convection for A = 10, Le = 2, e = 0.5, ip = 0.4, Rv = 20 and Rac^ = 15.7.

278

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

0.0002

O

3 7 5 0.0001 [ 3 1 -1

S

0

-0.0001

'

0

I

2

1

1

4

6

,|i|||||lli|||||||||||||

[

-0.0002 170

190

Time

210

230

Figure 10.5 Onset of oscillatory convection for A = 10,^ = —0.2,e = 0.5, Rv = 100, Raco = 118.5 andcjo = 10.08.

(b)

Figure 10.6 (a) Stream functions, and (b) isotherms, for A = 1, Le ip = 0.2, Ra = 6 and jR = 0.3.

2, e = 0.5,

acceleration will also be in action, this value is chosen according to linear stability analysis results: for Le = 2, e = 0.5, '0 = 0.5, R — 0.3, the critical Rayleigh number Racs « 14. The intensity of the convective motion will be accordingly increased and the sum of stream functions at all points in the domain is a good criterion for representing the intensity of convective motion. This case is shown in Figure 10.7. As can be seen from the figures, a symmetry breaking structure is obtained. This is explained by the coalescence of the two rolls with the same sign in the diagonal direction and the existence of two separate offdiagonal rolls with weaker intensity. If the thermal Rayleigh number is further increased

M. C. CHARRBER MOJTABIET AL.

279

(b)

Figure 10.7 (a) Stream functions, and (b) isotherms, for A = 1, Le = 2, e = 0.5, il) = 0.2, Ra = 13.15 and R = 0.3.

Figure 10.8 (a) Stream functions, and (b) isotherms, for A = 1, Le = 2, e = 0.5, ip = 0.2, Ra = 15 and i^ = 0.3.

to Ra = 15, a single convective roll appears, which means that the gravitational effect is more important than the vibrational effect, see Figure 10.8. For A = 10, the case corresponding to Le = 2, e = 0.5, ip = 0.2 and i? = 0.1 and Ra = 15 is considered. The typical four-vortex structure is presented in Figure 10.9. Further increase in Ra results in the appearance of a multi-cellular convective regime, see Figure 10.10.

280

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION •*'<"

I

0^

7

10

Figure 10.9 Stream functions for A = 10, Le = 2, e = 0.5, tp = 0.2, Ra = 15 and R = 0.1.

Figure 10.10 Stream functions for A = 10, Le = 2, e = 0.5, xp = 0.2, Ra = 17 and R = 0.1.

10.6

THE ONSET OF THERMO-SOLUTAL CONVECTION UNDER THE EVFLUENCE OF VIBRATION WITHOUT SORET EFFECT

In this section, the onset of double-diffusion convection, thermohaline convection, is analyzed. Although the realization of this problem experimentally faces us with practical problems, it has the advantages that we may obtain closed form analytical relations for the onset of convection. Thus, in the present section, in order to gain better understanding of this complex problem, the effect of high-frequency and small-amplitude vibration on the onset of convection in an infinite horizontal porous layer is studied. The direction of vibration is parallel to imposed temperature and concentration gradients. 10.6.1 Linear stability analysis The linear stability analysis procedure is the same as in Section 10.3. For the vertical vibration, the mechanical equilibrium is possible. In order to perform linear stability analysis, the temperature, concentration, velocity and solenoidal field are perturbed around the equilibrium state. Using the linearization procedure, we find the amplitude equation, which admits exact solutions of sinusoidal form. Stationary bifurcation For the onset of stationary double-diffusion convection, the stability domain is obtained from: 1 (P+7r2)2 /^ N\^ e Ra.ct = (10.35) A;2 •^•^•^7j^^fc^ + .^ l-t-ATLe

(Rv = R^Ral).

281

M. C. CHARRIER MOJTABIET AL.

If we set dRast/dfc^ =: 0, the critical Rayleigh and wave numbers can be found from the following simultaneous system of equations: R^stc =

7r2(H-ArLe)fc2

'

(10.36)

7r^(7r^-fcg)(l + iVLe) R' = (fc2+7r2)(27r2-A:2)2(H-iV/e)

Figure 10.11 illustrates the effect of vibration on the stability threshold for Le = 4 and e = 0.5. As can be seen from this figure, vibrations increase the stability domain in the (Rac, N) plane. From a convective pattern formation point of view, vibrations reduce the critical wave number as can be observed in Figure 10.12. Hopf bifurcation For the onset of Hopf bifurcation we obtain the following relations: 1 + eLe (fc2+7r2)2 (e + N)Le k^

-1 =

/ \

iV\ eJ

P k^ -\-7r^

(10.37)

(A:2+7r^)^(l + 6iVLe^)

150

100

Rac

50h

R = 0.12 R = 0.1 R = 0.08 R=0 -0.2

-0.1

0.1

N

0.2

0.3

0,4

0.5

Figure 10.11 The effect of the vibration parameter on the onset of convection for Le = 4 and e = 0.5.

282

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

R=0 R = 0.08 i? = 0.12

..^MR

= 0.16

Figure 10.12 The effect of vibration on the critical wave number for Le = 4 and e = 0.5.

Following the same procedure as for stationary bifurcations, we may obtain the following simultaneous system of equation for obtaining the critical parameters (Raco, kco)' '^^co

—

R' =

7r^Le{e + N) en'Le{7r'-kl)

k^

(10.38)

It should be added that the Hopf bifurcation appears for negative values of N and the location of co-dimensional point is given by: A T ^ - -

1 eLe2

•

(10.39)

It is obvious from relation (10.39) that vibrations do not change the location of the codimensional point. Table 10.4 shows the influence of the vibration on the critical values of the Rayleigh number, the wave number and the Hopf frequency. In this study Le = 4, e = 0.5 and A^ = - 0 . 3 . We deduce from Table 10.4 that vibration increases the stability domain of the Hopf bifurcation and it reduces the critical wave number and the Hopf frequency.

M. C. CHARRIER MOJTABIET AL.

283

Table 10.4 The influence of vibration on the onset of oscillatory convection. R

LV^CO

rZco

0 0.01 0.08 0.1

148 148.4 191.2 249.6

3.140 3.136 2.698 2.220

CJo

18.46 18.43 16.04 13.85

Double-diffusion convection under microgravity conditions In this situation, in the interval of N e ]—e, —1/Le[ the onset of stationary convection is possible. The critical wave number and critical vibrational Rayleigh number are found as follows: kc = v27r, 1 277r2 ^ - ^ - - - ( l + iVLe)(l^iVA)-l--

(10.40) iV6]-e,-l/Le[.

Under microgravity conditions and high-frequency and small-amplitude vibrations, there is no possibility of the Hopf bifurcation.

10.7

CONCLUSIONS

In this chapter two-dimensional thermo-solutal convection both with and without Soret effect, under the influence of a mechanical vibration has been studied analytically and numerically. The time-averaged formulation is used. The influence of the direction of the vibration for different cavity aspect ratios has been studied for the case of Soretdriven convection and the corresponding fluid flow structures explained. For an infinite horizontal layer, linear stability analyses of equilibrium and quasi-equilibrium states have been performed. Our results show that, depending on the orientation of vibration, different effects on the onset of convection may be expected. It is found that, when the direction of vibration is considered parallel to the temperature gradient, vibration has a stabilizing effect on both the stationary and the Hopf bifurcation. The action of vibration reduces the number of convective rolls and the Hopf frequency. However, when the direction of vibration is perpendicular to the temperature gradient, vibration has a destabilizing effect. New instability regions appear in the bifurcation diagram (Ra, V^) in which the preferred pattern is the mono-cellular convective roll. In the limit of long wave mode, the regular perturbation method is used to determine the stability boundary through an analytical relation. A comparison with the long wave mode in a horizontal fluid layer filled with a binary mixture under the action of vibration is given. For a confined rectangular cavity with different aspect ratios, numerical simulations using a spectral method are performed which corroborate the results of the stability analysis. It is

284

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

shown that the vertical vibration can reduce the number of convective rolls. For horizontal vibrations, thefluidflowstructures are sought. For afixedvalue of the vibrational Rayleigh number, we increase the thermal Rayleigh number from a value much less than the critical value corresponding to the onset of convection in an infinite layer. A symmetrical fourvortex structure is observed first, then a diagonal dominant symmetry breaking structure and finally a mono-cellular structure. These results are similar to the results obtained in a cavity filled with a pure fluid under the action of vibration in weightlessness. The interesting result of this study is that, by appropriate use of the direction of residual acceleration in the microgravity environment, significant enhancement in heat and mass transfer rates may be obtained. As the problem depends on several parameters, obtaining closed form relations are of highest importance. The linear stability analysis of an infinite horizontal porous layer under a vertical vibration with imposed temperature and concentration gradients is performed, the Soret effect is neglected. This analysis provides us with closed form relations for the stationary and Hopf bifurcation. It is shown that under microgravity conditions, the onset of the Hopf bifurcation is not possible. This study contributes to the ongoing research in order to define the appropriate microgravity environment.

REFERENCES Alexander, J. I. D. (1994). Residual gravity jitter onfluidprocesses. Micro. ScL Tech. 7, 131^. Azaiez, M., Bemardi, C, and Grundmann, M. (1994). Spectral method applied to porous media. East-West J. Numer. Math. 2, 91-105. Bardan, G. and Mojtabi, A. (2000). On the Horton-Rogers-Lapwood convective instability with vertical vibration. Phys. Fluids 12, 1-9. Bejan, A. (1995). Convection heat transfer (2nd edn). Wiley, New York. Bejan, A. (2000). Shape and structure, from engineering to nature. Cambridge University Press. Bejan, A. (2003). Simple methods in convection in porous media: scale analysis and the intersection of asymptotes. Int. J. Energy Res. 11, 859-74. Bejan, A. and Nelson, R. A. (1998). Constructal optimization of internalflowgeometry in convection. ASMEJ. Heat Transfer 120, 357-64. Biringen, S. and Peltier, L. J. (1990). Numerical simulation of 3D-Benard convection with gravity modulation. Phys. Fluids A 2, 754-64. Clever, R., Schubert, G., and Busse, F. H. (1993). Two-dimensional oscillatory convection in a gravitationally modulatedfluidlayer. J. Fluid Mech. 253, 663-80. De Groot, S. R. and Mazur, P. (1984). Non equilibrium thermodynamics. Dover, New York. Gershuni, G. Z., Kolesnikov, A. K., Legros, J. C, and Myznikova, B. I. (1997). On the vibrational convective instability of a horizontal binary mixture layer with Soret effect. J. Fluid Mech. 330,25169.

M. C. CHARRIER MOJTABIET AL.

285

Gershuni, G. Z., Kolesnikov, A. K., Legros, J. C , and Myznikova, B. I. (1999). On the convective instability of a horizontal binary mixture layer with Soret effect under transversal high frequency vibration. Int. J. Heat Mass Transfer 42, 547-53. Gershuni, G. Z., Zhukhovitskii, E. M., and Yurkov, Yu. S. (1982). Vibrational thermal convection in a rectangular cavity. Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza. 4, 94-9. Gershuni, G. Z., Zhukovitskii, E. M., and lurkov, S. (1970). On convective stability in the presence of periodically varying parameter. J. Appl. Math. Mech. 34,470-80. Gresho, P. M. and Sani, R. L. (1970). The effects of gravity modulation on the stability of a heated fluid layer. /. Fluid Mech. 40, 783-806. Ingham, D. B. and Pop, I. (eds) (1998). Transport phenomena in porous media. Pergamon, Oxford. Ingham, D. B. and Pop, I. (eds) (2002). Transport phenomena in porous media. Vol. II. Pergamon, Oxford. Jounet, A. and Bardan, G. (2001). Onset of thermohaline convection in a rectangular porous cavity in the presence of vertical vibration. Phys. Fluids 13, 1-13. Khallouf, H., Gershuni, G. Z., and Mojtabi, A. (1996). Some properties of convective oscillations in porous medium. Numer. Heat Transfer, Part A 30, 605-18. Nelson, E. (1994). An examination of anticipated g-jitter on Space Station and its effects on material processes. NASATM 103775. Nield, D. A. and Bejan, A. (1999). Convection in porous media (2nd edn). Springer-Verlag, New York. Razi, Y. P., Maliwan, K., Charrier Mojtabi, M. C., and Mojtabi, A. (2004). Importance of direction of vibration on the onset of Soret-driven convection under gravity or weightlessness. Eur Phys. J. £15,335-41. Razi, Y. P., Maliwan, K., and Mojtabi, A. (2002). Influence de la direction des vibrations sur la naissance de la convection thermosolutale en presence d'effet Soret. Societe Frangaise Therm. 10, 285-90. Rees, D. A. S. and Pop, I. (2000). The effect of g-jitter on vertical free convection boundary-layer flow in porous media. Int. Comm. Heat Mass Transfer 21, 415-24. Rees, D. A. S. and Pop, I. (2001). The effect of g-jitter on free convection near a stagnation point in a porous medium. Int. J. Heat Mass Transfer^, 877-83. Rees, D. A. S. and Pop, I. (2003). The effect of large amplitude g-jitter vertical free convection boundary layer-flow in porous media. Int. J. Heat Mass Transfer 46, 1097-102. Schopf, W. (1992). Convection onset for a binary mixture in a porous medium and in a narrow cell: a comparison. J. Fluid Mech. 25, 263-78. Smorodin, B. L., Myznikova, B. I., and Keller, I. O. (2002). On the Soret-driven thermosolutal convection in a vibrational field of arbitrary frequency. In Thermal non-equilibrium phenomena in fluid mixtures, Lecture notes in physics, vol. 584 (eds W. Kohler and S. W. Wiegand), pp. 372-88. Springer, Berlin. Sovran, O., Bardan, G., Mojtabi, A., and Charrier Mojtabi, M. C. (2000). Finite frequency external modulation in doubly diffusive convection. Numer Heat Transfer, Part A 37, 877-96. Sovran, O., Charrier Mojtabi, M. C , Azaiez, M., and Mojtabi, A. (2002). Onset of Soret driven convection in porous medium under vertical vibration. In Proceedings of the I2th international heat transfer conference, Grenoble.

286

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

Sovran, O., Charrier Mojtabi, M. C , and Mojtabi, A. (2001). Naissance de la convection thermosolutale en couche poreuse infinie avec effet Soret. CRAS Paris, Serie lib 4, 287-93. Zen'kovskaya, S. M. (1992). Action of high-frequency vibration on filtration convection. Prikl Mekh. Tekh. Fiz. 32, 83-8. Zen'kovskaya, S. M. and Rogovenko, T. N. (1999). Filtration convection in a high-frequency vibration field. 7. Appl MecK Tech. Phys. 40, 379-85. Zen'kovskaya, S. M. and Simonenko, I. B. (1966). Effect of high frequency vibration on convection initiation. Izv. Akad. NaukSSSR, Mekh. Zhidk. Gaza. 1, 51-5.

11

COMBUSTION IN POROUS MEDIA: FUNDAMENTALS AND APPLICATIONS A. A. M O H A M A D Department of Mechanical and Manufacturing Engineering, CEERE, The University of Calgary, Calgary, Alberta, T2N 1N4, Canada email: amohamadQenme. u c a l g a r y . ca

Abstract Porous medium is a good candidate for enhancing the combustion efficiency and reducing pollution formation, such as soot, CO and NOx- It is possible to obtain a super adiabatic temperature in the porous medium due to the passive preheating of the combustible gases. Moreover, uniform radiative heat flux can be obtained by utilizing porous media in combustion systems, which is essential for many industrial applications, such as paint and paper drying and glass tampering processes. The power rate can be varied without compromising the efficiency of the combustion. Combustion in porous media is utilized in advanced boiler and surface burners. Further, it is possible to exploit porous medium in domestic heaters, gas turbine combustion chambers, vehicle heaters, fuel cells and energy managements in many industrial processes, such as furnaces and cogeneration systems. The main objectives of the chapter are to discuss fundamentals and applications of combustion in porous burners. Keywords: porous media, combustion, low NOx burner

11.1

INTRODUCTION

There is no doubt that increasing the efficiency and reducing pollution of any system are the ultimate goals of thermal scientists and environmentalists. The results have a great impact on the economics and life styles of the present and future generations. Most power production and energy systems rely, in one way or another, on the combustion of fossil fuels, which will stay with us for years to come. Combustion of premixed and nonpremixed flame has been extensively studied, yet the problem of pollution and efficiency needs more effort. The combustion of fossil fuels produce CO2 and there is no feasible way to eliminate CO2 formation as far as the combustion of hydrocarbons is involved. Therefore there is no need to discuss this issue. The major concern is the formation of other 287

288

COMBUSTION IN POROUS MEDIA: FUNDAMENTALS AND APPLICATIONS

pollutants, such as NOx, SOx, CO and UHC (unbumed hydrocarbons). The formation of these products mainly depends on the type of fuel and physical and chemical mechanisms of the combustion. Different strategies were developed in controlling the pollution, which can be classified into three categories, namely, pre-, in- and post-combustion processes. Fuel treatment before combustion, such as sulfur scrubbing, is called preprocessing. Removing pollutants from flue gases, such as using a catalytic converter or scrubber, is called post-processing. Controlling the mechanism of combustion is the in-process pollution control, and it is the subject of this chapter. Hence, understanding the mechanisms of combustion may lead to the control and reduction of the formation of those pollutants mainly, NOx, CO and UHC. Combustion in porous media is a means to alter the physical and chemical mechanisms of the combustion. One of the essential mechanics of NOx formation is thermal, and this mainly depends on the flame temperature and the oxygen availability. Therefore, controlling the physical process of combustion, by avoiding high temperatures is essential to reduce the NOx formation. Also, controlling the chemical process of the combustion by the redistribution of the oxygen supply may help in reducing the NOx formation. The essential mechanism of the formation of UHC and CO depends on the temperature and oxygen availability. High temperatures that are required to crack down the UHC and convert CO into CO2. Also, the formation of UHC and CO is related to the efficiency of the combustion process. Hence, there is a trade off between NOx formation, on the one hand, and the formation of CO and UHC, on the other hand. The ultimate choice is to minimize the NOx formation without sacrificing the efficiency of the combustion. This may be achieved by engineering the thermo-physical properties of the combustion medium. Free flame combustion takes place in gases and, in general, the thermal conductivity of gases is very low, of the order of 0.03 W / m K . Furthermore, most gases have poor radiative properties (low radiative emission, absorption and scattering). Also, the heat transfer from the flame and combustion products to the load is mainly by convection mechanisms, which have diverse effects on the uniformity of the temperature. In drying processes, glass tempering and painting, uniformity of the source temperature is essential on the quality of the products. Combustion in porous media offer many advantages over free flame combustion and overcome most of the mentioned shortcomings of the free flame combustion. Porous material defined in this chapter as a material with connected voids, where flow can easily penetrate through the medium. For combustion applications, the porous material must withstand high temperatures, and therefore alumina, ceramic, cordierite, silicon carbide and zirconia are used for combustion applications. The following paragraphs review work done on this topic, discuss the physics of combustion in porous medium, applications, modeling of combustion, recent developments in combustion in porous medium and suggestions for further research works.

A. A. MOHAMAD

289

11.2 PREVIOUS WORKS This section is not intended to review all the research work published on combustion in porous media. However, only work that may help us to build a background on the topic will be mentioned. Certainly there are many excellent published works left without citation due to page limitation, but without any intention. Flame propagation through porous media has been investigated by several authors, e.g. Korzhavin et al. (1982) and Babkin et al. (1991). Different flame propagation regimes were classified, i.e. low velocity, high velocity, rapid combustion, sound velocities, low velocity detonation and normal detonation. In most applications mentioned previously, the flame speed is in the range of low and high velocity regimes, which range from 0 to lOm/s. Historically, combustion on stationary combustion in inert porous media has been reported by Bone (1912), Lucke (1913) and Hays (1933). Bone designed the first boilers and muffle heaters while Lucke constructed radiant room heaters, crucible furnaces and cooking stoves. Experiments performed by Jensen et al (1989) and Andersen et al. (1990) confirmed that the porous radiant burners have many advantages over conventional free flame burners. The higher efficiency due to preheating of the upstream air-gas mixture, lower emission of NOx due to lower flame temperature, elimination of hot spots in the heat transfer devices as a result of heat conduction and radiation, uniform heating, compact design, wide power modulation rate and low noise are the main advantages of the porous burners. Combustion in a porous medium has been modeled by Lawson and Norbury (1985), where the radiative heat transfer is modeled as a diffusive process, i.e. the Rosseland approximation. Detailed studies of thermal radiation have been reported by Tong and Sathe (1988) and Andersen (1989), where the combustion is modeled as a heat source. However the model predictions were not comparable with the experimental data. Further, results indicated that the correct radiation mode of heat transfer is an important factor. Experimental results of Golombok and Shirvil (1990) suggested that the effective emittance of a porous burner is a concept that should be used with caution because the spectral emittance is a function of the temperature gradient in the fibrous layer near the burner surface. A more realistic model is reported by Chen et al. (1987), Yoshizawa et al. (1988), and Sathe et al. (1989, 1990). Mohamad and Viskanta (1989) modeled radiation and convection heat transfer in a single module of a surface combustor heater. Two energy equations were used for the gas and solid phases. The analysis revealed that there are a large number of parameters that control the thermal performance of the porous heater. Detailed radiative heat transfer in porous media were modeled by considering emission, absorption and scattering, see Tien and Drolen (1987), Singh and Kaviany (1993), and Mohamad et al. (1994). Williams et al. (1992) performed experiments on the combustion of premixed methane/air mixtures in a porous matrix burner. They investigated pollutant productions, such as NOx, CO and CO2 formations and in addition to the radiative performance of the burner. The maximum level of NOx reported was about 22ppm. Numerical analysis of bi-layered porous burners was considered by Kulkami and Peck (1996) and Younis et al. (2001). The effect of thermophysical and radiative properties of this type of burner is considered for different thicknesses of the porous layers. Also, researches on using

290

COMBUSTION IN POROUS MEDIA: FUNDAMENTALS AND APPLICATIONS

porous material for liquid fuel combustion have been reported by Haack (1993), Kaplan (1994), and Itaya et al (1995). Flame stability in a porous burner has been extensively studied experimentally and numerically at the University of Erlangen, see Trimis and Durst (1996), Durst and Trimis (1997), and Trimis (2000). Furthermore, research work on catalytic burners has been reported by Rumminger et al. (1999).

11.3 CHARACTERISTICS OF COMBUSTION IN POROUS MEDIA One may ask, what makes porous material special in combustion. To answer such a question there is a need to list unique features of combustion in porous media which can be summarized as follows. • The flow is turbulent due to small vortices generated by a solid porous matrix, see Figure 11.1, which enhances momentum heat and mass transfer and stabilizes the flame. Furthermore, the surface area per unit volume of porous matrix is very large, of the order of 10"* m~^ and higher. This results in an extensive inter-phase energy and momentum transfer between the fluid and solid phases. • The thermophy sical properties of the porous medium, such as the thermal conductivity and thermal capacity, can be engineered according to the application. For instance, increasing the thermal conductivity of the porous medium enhances the heat conduction from the flame zone to pre- and post-flame zones. Accordingly, the NOx decreases due to the decrease in the flame temperature. Also, directional thermophysical properties, such as thermal conductivity, can be engineered using anisotropic materials, such as fibers. • Radiative heat transfer inside the porous medium is much higher than that of the gases. Hence heat transfer increases within a porous medium and reduces the flame temperature, i.e. low NOx formation. In other words, the heat transfer can be managed for the efficient operation of the combustor with less pollution. Also radiative properties of the material, such as the scattering albedo and optical thickness can be managed for the best performance of the burner. • Pressure drop, through a porous medium, can be controlled by changing the permeability of the porous material. In most applications, high permeable porous media can be used, where the pressure drop becomes insignificant.

Figure 11.1 Schematic of theflowthrough porous medium.

A. A. MOHAMAD

291

Figure 11.2 Mechanisms of heat transfer in a porous burner. • Adsorption/desorption process may take place on the solid surfaces as well as catalytic effects, which depends on the kind of the solid material. Figure 11.2 illustrates the heat transfer mechanisms in porous medium. These mechanisms can be controlled by using different porous material, or by changing the topology of the material.

11.4

APPLICATIONS

Porous burners can be used for industrial and domestic applications. For instance, combustion in a porous media can be used in cogeneration systems and gas turbines. In manufacturing processes, such as glass tempering and for steel, aluminum and ingot heating, the heat losses to the ambient can be reduced by 20-30% by redirecting the heat to the load, see Figure 11.3. Porous burners are used in paper and paint drying, where a uniform radiative heat flux can be obtained by using inert porous materials. Domestic boilers can be improved and reduced in size by utilizing the concept of combustion in porous media, see Figure 11.4. Such a boiler has the ability of a wide power modulation,

Figure 11.3 Energy management in furnaces.

292

COMBUSTION IN POROUS MEDIA: FUNDAMENTALS AND APPLICATIONS Flue gases

Cooling tubes (heat exchanger) Flame

Fuel-air mixture Figure 11.4 A low NOx combustor.

when the demand to the hot water changes on a daily and seasonal basis, the boiler can be modulated without scarifying efficiency and yet with less pollution. Flame stability is a detrimental factor in some applications, such as in gas turbines and porous burners are ideal for such applications. Also, incinerations of low heating value fuels, such as biomass and waste materials, can be performed efficiently with porous burners. It is possible to obtain super-adiabatic combustion in porous medium. In general, the super-adiabatic flame temperatures occur whenever heat is recirculated from the hot products to the unbumed reactants, see Weinberg (1971, 1986). The first observation of super-adiabatic temperatures was reported by Egerton et al. (1963), when they examined the mechanism of smoldering in cigarettes. Also, super-adiabatic temperatures occur in metallurgical applications, including the self-propagating high temperature synthesis process, see Merzhanov and Khaikin (1988). Kennedy et al (1995) reviewed the work done on super-adiabatic combustors and discussed new types of reactors. Super-adiabatic burners can be utilized in plasma application and direct thermal-electric conversion, ceramic spay technology, etc. As far as the author is aware, the first work which used liquid fuel combustion with porous medium is the analysis and experimental work of Tseng and Howell (1994, 1995) who studied super adiabatic flame prediction for an equivalence ratio less than 0.7. Also, the flame stability limits for liquid fuel combustion in porous medium is wider than conventional combustion. The measured CO and NOx emissions were about 10 and 20ppm, respectively. Kaplan and Hall (1995) measured CO and NOx on a porous burner using heptane as a fuel and obtain results that were consistent with those obtained by Tseng and Howell (1994, 1995). Also, Itaya et al. (1995) combusted kerosene in porous medium as an application of the operation of a Stirling engine, and gas turbines using liquid fuels. Drops of kerosene are combusted on the surface of a porous medium. The combustion was complete with

A. A. MOHAMAD

293

Flue gases

Infra-red cells

Burner ^

Fuel-air Figure 11.5 Suggested cogeneration system. input loads of 674-3879 kW/m^. The turn-down ratio was over 5.8 and the flammability limit appears until 0.1 in lean equivalence ratio. The NOx was about 60 ppm for most of the range of operating conditions. Further, Trimis et al. (2001) have developed an oil burner with high power modulation ranges of 1:10 and reaching a low power of 2.5 kW. Hence, efficient liquid burners can be developed for many domestic and industrial applications. Experimental works on combustion in a porous layer, with swirl effects, are proceeding in our laboratory (Kamal, 2004). The results indicate that it is possible to reach ultralow NOx (of the order of one digit) emission. Furthermore, experiments are proceeding on non-premixed combustion in porous medium, where radiation spectral analysis and emission concentrations are documented. It is possible to develop a cogeneration system by incorporating infra-red cells with radiant porous burners. Infra-red photo cells convert radiant energy directly to electric current and hot flue gases can be used for heating processes, see Figure 11.5.

11.5 POROUS BURNERS Most work done has been performed on axial flow (flat) burners, where the fuel/air mixture ignited in a porous layer and flame usually stabilized at the outlet. Therefore, the radiation

294

COMBUSTION IN POROUS MEDIA: FUNDAMENTALS AND APPLICATIONS

flux at the outlet of the burner is maximized. These types of burners are extensively studied for different operating conditions both experimentally and numerically. Ceramic or metallic porous material is usually used, where the combustion takes places in pores between the solid matrix. The surface temperature is about 1100 °C and no flame is visible, indicating that combustion takes place just below the burner surface, see Williams et al. (1992). Also, a double layer porous matrix is used to ensure a wide range of flame stability; accordingly two layers of different porosity are used. In general, low porous material is used to filter the fuel/air mixture and combustion takes place in a high porous layer. Therefore, the flash back possibilities are reduced. Another type of porous burner has been studied by Mohamad (2002) and it is a radial flow burner. This type of burner offers wide flame stability limits and high surface area per unit volume.

11.6 MATHEMATICAL MODELING The schematic diagrams of three porous burner configurations are shown in Figure 11.6(ac). In the flat plate burner, air flows axially through a constant area duct filled with a porous layer of thickness L. In the cylindrical and spherical burners, the air flows radially through an annular porous matrix. The thermophysical properties of the air (density, thermal conductivity and specific heat) are assumed to be functions of the temperature and species concentration. Usually, the pressure drop through the porous burner is not that high and its effect on the thermophysical properties can be neglected. In general, the properties of the solid phase may be assumed to be constant and it is assumed that there is thermal non-equilibrium between the gas and the solid phase. Consequently,

(b)

(a)

Flue gases

Premixed gas-air

Premixed gas-air Figure 11.6 Schematic diagrams of the burners: (a) axialflow,(b) radialflow,and (c) spherical flow.

A. A. MOHAMAD

295

there are two energy equations to model the energy transport in the system. The porous solid is assumed to be gray and to emit, absorb and scatter radiant energy. The radiant heat transport is approximated by the Rosseland diffusion equation, see Chen et al (1987) and Siegel and Howell (1992). Gaseous radiation is assumed to be negligible compared to the solid radiation. The energy equations for the gas and solid phases are, respectively, given as follows: g-^{(pPgCpgTg) + —-g^{(t>CpgPgr''vTg) (11.1)

= ^Q-r (^^^'^ V j " (^ ~ ^^'^"^^^ " ^^) "^ '^^^'^'^ ' 16CTT3

3/3

dr

-K{Ts-Tg),

(11.2)

where (j>, p, Cp, T, v, k, hy, AHc, S/g, a and 0 are the porosity, density, specific heat, temperature, velocity, thermal conductivity, volumetric heat transfer coefficient, enthalpy of combustion, rate of fuel consumption per unit volume, Stefan Boltzmann constant and extinction coefficient, respectively. Subscripts g and s refer to gaseous and solid phases, respectively. The conservation equation for the mass fraction of the fuel is given as follows: JtiP9mf) + ^§:iPor-vmf) = ^ l

(r^D^BP,^)

- 5/, ,

(11.3)

where m/ and DAB are the fuel mass fraction and diffusion coefficient, respectively. The n value is set to 2, 1 and 0 for spherical, radial and axial flow burners, respectively. A single-step Arrhenius type chemical kinetic equation is adopted in modeling the combustion, i.e. the following equation is used: Sfg = fplrrifmo^ exp \-^]

.

(11-4)

where / , moa, E and R refer to pre-exponential factor, oxygen mass fraction, activation energy and gas constant, respectively. Zhou and Pereira (1998) have compared four combustion models for the simulation of premixed combustion in an inert porous media, namely, the full mechanism (49 species), a skeletal mechanism (26 species), a four-step reduced mechanism (9 species) and a singlestep global mechanism. Their results illustrated that a single-step model prediction of the gas and solid temperature is comparable with the detailed kinetics. The single-step model slightly over predicted the temperature profile, but the results are within the experimental error. Further, the single-step model prediction of the flame speed is consistent with the experimental data for the excess air ratio in the range of 1 to 1.4. The performance of a single-step model is much better than the skeletal model in predicting the mass fraction of O2, H2O and CH4 along the burner. Hence, for the purpose of the comparison between

296

COMBUSTION IN POROUS MEDIA: FUNDAMENTALS AND APPLICATIONS

axial and radial burners and to get a global understanding of the radial burner purposes, the single-step model is assumed to be sufficiently accurate. It should be noted that the radiation transport in porous medium is modeled as a diffusive term and the validity of this assumption is questionable. The porous material can be assumed as a scattering, emitting and absorbing medium. Therefore, radiative transport equation can be written as follows, see Mohamad and Viskanta (1989): (11.5a)

V-F = -P{l-uj){G-AEt), where u is the single scattering albedo, and the irradiance G is governed by

(11.5b)

V^G = 7]^{G-4Et).

The parameter rj'^ = 3/3^(1 - UJ){1 — goj), where E^ is the Planck black body emitted flux, aT^, F is the radiative flux, F = Fxi-\- fyj, P is the extinction coefficient, and g is an asymmetry factor. An alternative approach is to use the discrete ordinate method for solving the radiative transport equations, see Mohamad (1996). Since radiative fluxes are available from the solution of the radiative transport equation, therefore the energy equation (11.2) for the solid phase should be modified as follows: d_{psCsTs) = dt

^l[r-ks

K{Ts-Tg)-V'F.

dr

(11.6)

The following boundary conditions are adopted for the gas, solid and species: gas:

T,U,^^=Tin dr

= 0

at r = ru (11.7)

at r = Tout,

solid: 4 / l i n ( T , , i n - T , U , ^ J + ( 7 6 i n ( T ; 'in,amb

m dr

rp^ I

s r=rinj

at

r=rin r = Tin ,

(11.8a) /iout (^out,amb " '^s\r=rouJ

~^ ^^out (^-^Dut,amb

^s \r=zraut) ~

^

on dr

at

species:

rrif = rrif^in at r = n^ , dnif = 0 at r = To dr

r=rout

r = rout, (11.8b)

(11.9)

297

A. A. MOHAMAD

The control volume approach, or the finite-difference method, can be used to solve the governing equations. The solution is advanced in time by using a fully implicit technique and this was necessary due to the stiffness of the governing matrix of the problem. Also, it is necessary to use an adaptive grid, or a very fine grid, to insure the accuracy of the solution.

11.7

RESULTS AND DISCUSSION

Typical results for the axial and cylindrical burners are discussed in the following paragraphs. The performance of the axial (flat) porous burner has been extensively studied, as mentioned in the introduction section, but results for the axial burner are presented in the following paragraph for the purpose of comparison with the radial burner. Figure 11.7 shows the gas and solid phase temperature along the axial burner for the stoichiometric air-fuel ratio, i.e. without excess air for different inlet velocities. As the inlet velocity decreases, the combustion temperature decreases, in other words, as the input power decreases the combustion temperature decreases. The flame stabilizes near the inlet of the burner. It was difficult to stabilize the flame for an inlet velocity of 0.4 m/s, where the flame moved towards the exit of the burner as the time proceeded until blow out occurred. Also, the flame temperature increases as the inlet flow rate increases (more input power). In other words, the flux increases as the input power increases because the axial burner has a constant cross section, and this is not the situation for a cylindrical burner as we will discuss later. As expected, the solid phase temperature is higher than the gas temperature at the upstream of the flame due to heat transfer by conduction and radiation from the flame zone to the solid phase.

25001

Fin = 0.3 m / s l)>c—

If/'

Vi„=0.2m/s

2000 T

yin=0.1m/s

If/ ^

1500

^ 1000

500

0.025

0.05 X [m]

0.075

0.1

Figure 11.7 Gas (solid lines) and solid (dashed lines) temperature variations along the axial burner for different inlet velocities (Vin) when the excess air ratio is Ex = 0.

298

COMBUSTION IN POROUS MEDIA: FUNDAMENTALS AND APPLICATIONS 2000

Ex:^0

1500

Ex = 0.2,0.4,0.6,0.8

1000

0.01

0.02 0.03 X [m]

_x_ 0.04

0.05

Figure 11.8 Gas (solid lines) and solid (dashed lines) temperature variations along the axial burner for different excess air ratios (Ex) when the inlet velocity is Vin = 0.1 m/s.

The effect of the excess air on the flame location, the gas and solid phase temperatures is illustrated in Figure 11.8. The combustion temperature decreases as the amount of excess air increases, and this is due to the energy needed to heat the excess air. Also, it is evident that the flame location moves downstream as the excess air increases. For instance, the flame stabilizes at about x = 0.4 cm for Ex = 0 and at about x = 1.4 cm for Ex = 0.8. It was difficult to stabilize the flame inside the burner for Ex = 1, where the flame blows out. It should be mentioned that the results are consistent with the published results of others, such as Zhou and Pereira (1998).

11.8 RADIAL BURNER The main geometrical feature of the cylinder is that the surface area of the cylinder increases as its radius increases. Such a geometrical property gives the cylindrical burner an advantage of a wide flame stability range compared with the axial burner. Since the flame stabilizes in a region where the flame speed is equal to the flow speed, therefore the velocity distribution through the porous layer is related to the flame stability and to the rate of power modulation. The velocity profile in the porous layer can be easily calculated from the continuity equation. For constant density fluid flow, the velocity distribution through the cylindrical porous layer is inversely proportional to r. While for the axial flow burner (flat burner) the flow velocity is constant along the porous layer. Accordingly the cylindrical burner offers a wide range of flame stability and rate of power modulation than does the axial flow burner. It is known that the range of flame stability in the axial flow porous burner is much wider than that of a free flame. Hence, it is expected that the cylindrical burner will offer a much wider range of flame stability limit and power modulation than do the free flame and axial porous burners.

A. A. MOHAMAD

299

It is possible to stabilize the flame inside the cylindrical porous burner for velocities up to about 8 m/s for a stoichiometric fuel-air mixture (Ex = 0), as the results revealed in Figure 11.9, i.e. more that 25 times greater than that of the axial burner. Of course, the range of the flame stability depends on the outer and inner radius of the burner. As it is expected that the solid phase temperature is higher than that of the gas phase temperature upstream of the flame due to the conduction and radiation heat transfer as mentioned before. The gas temperature became higher than the solid phase temperature at the location where the temperature reaches about 1800K, regardless of the magnitude of the inlet velocity. Figure 11.10 shows the gas and solid temperatures for different flow rates where the excess air is set to be 1, i.e. 100% excess air. For such a high excess air, it was not possible to hold the flame inside the porous layer for an inlet velocity of 1.8 m/s or higher. The maximum flame temperature is not a function of the inlet velocity, as it is for the axial flow, and the reason is that increasing the inlet velocity moves the flame to a position of larger surface area, r increases and therefore the flux is almost constant. This is an advantage compared with an axial burner, where increasing the inlet velocity increases the flame temperature. Since the NOx formation is directly related to the flame temperature, therefore increasing the power input (inlet velocity) does not have an effect on the flame temperature and NOx thermal formation, which is not the case for an axial burner. Hence, V/r is almost constant for a radial burner and calculations showed that for Ex = 0, the ratio of V/r is about 60 s~^ and for Ex = 1, V/r is about 18 s~^, especially for y > 1 m/s. Figure 11.11 shows the results for a porous burner of inner radius of 1cm, and for different amounts of excess air, where the inlet velocity is fixed at 1 m/s. It is found that Vinrin/r is equal to a constant for a given excess air ratio. Therefore, it is possible to predict the flame location for burners of different inner diameters. For example, for Yin = 1,1.4,1-8,2 m/s

0.025

0.05 0.075 r - Tin [m]

Figure 11.9 Gas (solid lines) and solid (dashed lines) temperature variations along the radial direction for different inlet velocities and for a stoichiometric fuel-air ratio (Ex = 0).

300

COMBUSTION IN POROUS MEDIA: FUNDAMENTALS AND APPLICATIONS yi„=0.2 0.4 0.6 1 1.2 1.4 1.6m/s

0.025

0.05

r - Tin fml

0.075

0.1

Figure 11.10 Gas (solid lines) and solid (dashed lines) temperature variations along the radial direction for different inlet velocities and for an excess air ratio of 100% (Ex =1). Ex = 0

2500 h

Ex = 0.2 Ex = 0.4 Ex = 0.6 Ex = 0.8 EX=:1 Ex =1.2

2000

1500 lOOOh

500 0.025

0.05

r - Tin fml

0.075

0.1

Figure 11.11 Gas (solid lines) and solid (dashed lines) temperature variations along the radial direction for a range of excess air ratios when the inlet velocity is Wn = 1 m/s.

Ex = 0.2 and for a burner of inner radius of 1 cm, the flame stabilizes at locations about r = 0.03 + 0.01 = 0.04 m, hence MnHn/r = 1 x 0.01/0.04 = 0.25 m/s. Therefore, for a burner of inner radius of 2 cm and for Ex = 0.2, the flame stabilize at 0.02/0.25 = 0.08 m. The gas or solid temperature is not a function of the inner radius for the same excess air. The effect of the inlet velocity is studied for an extra lean mixture, typically Ex = 1.4. It is found that the flame can be stabilized inside of the burner for inlet velocities up to about 0.8 m/s and the flame temperature is less than 1400 K, see Figure 11.12. It should be noted that in most burners for a lean mixture then it is difficult to stabilize. Also, flame temperatures less than 1400K imply that the NOx formation is very low and may be of the order of lOppm.

301

A. A. MOHAMAD 1400 E-

//

\

Mn = 0.5m/s/

f/

1/

/

1200

/

11

/ Vin = 0.8m/s

ll

1

1000

K

r

/

1

1

E-

il 7

600

//V y y

400

0.025

'f ^Mn =0.6m/s

11

ij

^ 800

^^Ni^

1

1 1

1

1

1

if

11

//

11 7 A/

J

1

L__J

0.05 r - r-in [ml

l_

1

1

i__j

0.075

1

1 1

0.1

Figure 11.12 Gas (solid lines) and solid (dashed lines) temperature variations along the radial direction for an excess air ratio of Ex = 1.4.

11.9

CONCLUSIONS

Finally, it is concluded that porous medium has the potential application for an efficient and less pollutant burner. Also, the combustion in the porous medium widens the flame stability limits compared with convectional burners. The results of the mathematical models presented in this chapter for radialflowburners are discussed. Further, temperature and velocity distributions through the porous layer are discussed and compared for both radial and axial flow burners. For the sameflowrate and heat release, the peak temperature for the axial flow burner is higher than that of radial flow burner. This implies that the NOx formation for cylindrical burner is lower than that of a flat burner. Also, the velocity distribution in the cylindrical burner decreases as the radius increases, which ensures a wide range of flame stability and power modulation. The preliminary results presented in this chapter indicate that the cylindrical burner is superior to the flat burner as far as the thermal performance and the NOx formation are concerned.

11.10 POSSIBLE FUTURE WORK Combustion of liquid fuels and non-premixed in a porous media requires more attention and fundamental understanding. In conventional combustion, a spray of liquid fuel droplets introduced into the stream of air where the combustion takes place, first by vaporization of these droplets and the reaction of the gaseous (vapor) of the fuel with the oxidant. The evaporation process takes place by convective heat transfer mechanisms, which is in the range of 100 to 500 W/m^ K, depending on the velocity of the flow. In general, the combustion of the liquid fuel produces a high rate of pollutants, which mainly depends on the evaporation rate and residence time of the drops. Usually, very fine drops

302

COMBUSTION IN POROUS MEDIA: FUNDAMENTALS AND APPLICATIONS

are utilize to enhance the combustion process. It is anticipated that the combustion of a hquid fuel in the presence of a porous material enhances the evaporation of the fuel by two mechanisms, namely, the droplet impinging on a porous surface breaks droplets into several drops, which depends on the ratio of the drop size to the porous pore size, and the rate of heat transfer by direct contact between the drops and the hot porous media is several orders of magnitude higher than that of the convection mechanism. Furthermore, liquid droplets absorb infrared radiation emitting from the porous surfaces, which increases the temperature of the liquid drops, and accordingly increases the rate of evaporation. The subject is not fundamentally understood and hence in the first stage of these investigations, the mechanism of non-reactive drops impinging on a hot porous surface have to be studied. This step would help us to understand the physical process of drop evaporation and the effect of drop size to the porous material pore size for a range of hot surface temperatures. In the second stage, a reactive drop should be used and the mechanism of drop evaporation, combustion and pollution production studied. Furthermore, pollution control from flare would have to be explored by utilizing a porous medium. As far as the author is aware, this subject has not been explored in the past. Since, our understanding is that the interaction of a cold air stream with flare is the major cause of particulate pollution, then utilizing porous medium may reduce the effect of cold air mixing with combustible gases. A premixed flame is less efficient than that of premixed flame. The air entrainment into the gas is mainly by a diffusion mechanism, which is a very poor mixing process. The mechanism of combustion in porous medium is different from that of a free flame (without a porous media) due to the change in the thermal properties of the medium, which enhances the rate of heat transfer from the flame zone. Accordingly, the entrained air is preheated before the diffusion process, which may enhance the combustion efficiency. Also, the combustion in porous media takes in volumes between pores rather than through a thin layer of gas, as is the case in free flame combustion. Also, a better understanding of pore level (macro-analysis) combustion in a porous medium is required.

REFERENCES Andersen, F. (1989). A low NOx, low temperature oxygen-fuel burner. In Industrial combustion technologies (ed. M. A. Lukasiewicz), pp. 329-34. American Society for Metals. Andersen, F, Andersen, M., Jensen, J., and Hadvig, S. (1990). Ceramicfiberplate asflameholder. In Conference on ceramics in energy applications, Sheffield, England, April. Babkin, V. S., Korzhavin, A. A., and Bunev, V. A. (1991). Propagation of premixed gaseous explosionflamesin porous media. Combustion and Flame 87, 182-90. Bone, W. A. (1912). Surface combustion. J. Franklin Inst. 173, 101-31. Chen, Y. K., Matthews, R. D., and Howell, J. R. (1987). The effect of radiation on the stmcture of premixedflamewithin a highly porous inert medium. ASME HTD 81, 35-42.

A. A. MOHAMAD

303

Durst, F. and Trimis, D. (1997). Combustion by free flames versus combustion reactors. In Proceedings of the 4th international conference on technologies and combustion for a clean environment, Lisbon. Egerton, A., Gugan, K., and Weinberg, F. J. (1963). The mechanism of smouldering in cigarettes. Combustion and Flame 7, 63-78. Golombok, M. and Shirvil, L. C. (1990). Radiation characteristics of surface combustion burners. In Proceedings ofEurotherm seminar no. 7 on heat transfer in radiating and combustion systems, Cascais, Portugal, 8-10 October, pp. 7-13. Haack, D. P. (1993). Mathematical analysis of radiatively enhanced liquid droplet vaporization and liquid fuel combustion within a porous inert medium. M.Sc. thesis. University of Texas, Austin. Hays, J. W. (1933). Surface combustion process. US Patent, No. 2095065. Itaya, Y., Suzuki, T, Hasatani, M., and Saotome, M. (1995). Combustion characteristics of a liquid fuel in a porous burner. In Proceedings of the ASME/JSME thermal engineering conference. Vol. 3, Maui, Hawaii, 19-24 March, pp. 99-104. Jensen, J., Krighaar, M., Andersen, F., and Hadvig, S. (1989). Analysis of natural gas fired ceramic fiber-burner. Part I. Experimental study and practical applications. In Proceedings of the Xlth international symposium on combustion processes, Miedzyzdroje, Poland, 27 September. Kamal, A. M. (2004). Personal communication. University of Calgary. Kaplan, M. (1994). The combustion of liquid fuels within a porous media radiant burner. M.Sc. thesis. University of Texas, Austin. Kaplan, M. and Hall, M. J. (1995). The combustion of liquid fuels within a porous media radiant burner. Experim. Therm. Fluid Sci. 11, 13-20. Kennedy, L. A., Fridman, A. A., and Saveliev, A. V. (1995). Superadiabatic combustion in porous media: wave propagation, instabilities, new type of chemical reactor. Int. J. Fluid Mech. Res. 11, 1-26. Korzhavin, A. A., Bunev, V. A., Abdullin, R. Kh., and Babkin, V. S. (1982). Flame zone in gas combustion in an inert porous medium. Combustion Expl. Shock Waves 18, 628-32. Kulkami, M. R. and Peck, R. E. (1996). Analysis of a bilayered porous radiant burner. Numer Heat Transfer, Part A 30, 219-32. Lawson, D. A. and Norbury, J. (1985). Porous medium combustion. In Numerical methods in heat transfer (ed. R. W. Lewis), Vol. Ill, pp. 173-93. Wiley, New York. Lucke, C. E. (1913). Design of surface combustion appliances. J. Ind. Eng. Chem. 5, 801-24. Merzhanov, A. G. and Khaikin, B. I. (1988). Theory of combustion waves in homogeneous media. Progress Energy Combustion Sci. 14, 1-98. Mohamad, A. A. (1996). Local analytical discrete ordinate method for the solution of the radiative transfer equation. Int. J. Heat Mass Transfer 39, 1859-64. Mohamad, A. A. (2002). Numerical simulations of a cylindrical porous medium. In Proceedings of ASME^Energy resources technology conference, Houston, TX, February. ETC2002/CAE-29017. Mohamad, A. A. and Viskanta, R. (1989). Combined convection-radiation heat transfer in a surface combustor-process heater. In Simulation of thermal energy systems (eds R. F. Boehm and Y. M. El-Sayed), pp. 1-8. ASME, New York. ASME-HTD-124. Mohamad, A. A., Ramadhyani, S., and Viskanta, R. (1994). Modelling of combustion and heat transfer in a packed bed with embedded coolant tubes. Int. J. Heat Mass Transfer 37, 1181-97.

304

COMBUSTION IN POROUS MEDIA: FUNDAMENTALS AND APPLICATIONS

Rumminger, M. D., Hamlin, R. D., and Dibble, R. W. (1999). Numerical analysis of a catalytic radiant burner: effect of catalyst on radiant efficiency and operability. Catalysis Today 47, 253-62. Sathe, S. B., Peck, R. E., and Tong, T. W. (1989). A numerical analysis of combustion and heat transfer in porous radiant burner. In Proceedings of ASME—National heat transfer conference, pp. 93-109. ASME-HTD-106. Sathe, S. B., Peck, R. E., and Tong, T. W. (1990). A numerical analysis of heat transfer and combustion in porous beds. Int. J. Heat Mass Transfer y^, 1331-8. Siegel, R. and Howell, J. R. (1992). Thermal radiation heat transfer. Hemisphere, Washington, DC. Singh, B. P. and Kaviany, M. (1993). Radiation in porous media. In Advances in heat transfer (eds J. P. Hartnett and T. F. Irvine), Vol. 23, pp. 369-464. Academic Press, New York. Tien, C. L. and Drolen, B. L. (1987). Thermal radiation in particulate media with dependent and independent scattering. In Annual review of numerical fluid mechanics and heat transfer (ed. T. C. Chawla), pp. 1-32. Hemisphere, Washington, DC. Tong, T. W. and Sathe, S. B. (1988). Heat transfer characteristics of porous radiant burners. ASME HTD 104, 147-55. Trimis, D. (2000). Stabilized combustion in porous media—applications of the porous burner technology in energy and heat engineering. In Proceedings ofAIAA—Fluid 2000 conference and exhibition, Denver, CO, 19-22 June. Trimis, D. and Durst, F. (1996). Combustion in a porous medium—advances and applications. Combustion Sci. Tech. 121, 153-68. Trimis, D., Wawrzinek, K., Hatzfeld, O., Lucka, K., Rutsche, A., Haase, F., Kriiger, K., and Kuchen, C. (2001). High modulation burner for liquid fuels based on porous media combustion and cool flame vaporization. In Proceedings of the 6th international conference on technologies and combustion for clean environment. Vol. 2, Porto, Portugal, 9-12 July, pp. 1-8. Tseng, C.-J. and Howell, J. R. (1994). Liquid fuel combustion within porous inert media. In Heat transfer with combined modes, pp. 63-9. ASME, New York. ASME-HTD-299. Tseng, C.-J. and Howell, J. R. (1995). Experimental stability limits and CO/NOx emissions of heptane combustion within porous ceramic burners. In Proceedings of ASME—National heat transfer conference, Vol. 2, pp. 11-17. ASME-HTD-304. Weinberg, F. J. (1971). Combustion temperatures: the future. Nature 233, 239-41. Weinberg, F. J. (ed.) (1986). Advanced combustion methods. Academic Press, London. Williams, A., Woolley, R., and Lawes, M. (1992). The formation of NOx in surface burners. Combustion and Flame 89, 157-66. Yoshizawa, Y, Sasaki, K., and Echigo, R. (1988). Analytical study of the structure of radiation controlled flame. Int. J. Heat Mass Transfer 31, 311-19. Younis, L. B., Mohamad, A. A., and Wierzba, I. (2001). Numerical analysis of premixed combustion in a double-layered radiant porous burner. In Advances in computational heat and mass transfer, ICHMT symposium. Palm Cove, Australia, 2-25 May (eds G. de Vahl Davis and E. Leonardi), pp. 397^04. Zhou, X. Y and Pereira, J. C. F. (1998). Comparison of four combustion models for simulating the premixed combustion in inert porous media. Fire Mater 22, 187-97.

1 2 REACTIVE TRANSPORT IN POROUS MEDIA—CONCEPTS AND NUMERICAL APPROACHES E. HOLZBECHER Humboldt University Berlin, Institute of Freshwater Ecology and Inland Fisheries (IBM), Muggelseedamm 310, 12587 Beriin, Germany email: [email protected]

Abstract The distribution of most chemical species in a porous environment is generally determined by both transport and biogeochemical processes. When a system in equilibrium is disturbed by the addition of species, the system reacts in a complex way to approach a new equilibrium. The time scale of that adjustment depends on the kinetics of the slow reactions. Such a representation is given in many experiments or technical devices. In the natural environment surface water penetrates into the adjacent sediments and may reach pumping wells for drinking water after a subsurface passage in a porous medium. Downstream from landfills, contaminated sites and mine tailings the situation is particularly crucial, as toxic or otherwise harmful substances contribute to the pool of chemical species. In order to obtain an understanding of the complex interaction of the different processes involved, computer simulations are a tool without alternative. In modelling studies the simultaneous action of transport and biogeochemistry has to be taken into account. Codes, which perform such computations, are implemented following different numerical and conceptual methods, of which most important ones are outlined in this chapter Several example cases, some hypothetical, some with practical relation including carbonate chemistry are presented, discussed and modelled. Simulations are performed with a self-developed MATLAB module. It is shown that the popular operator splitting (two-step) approach is to be handled with care. It may produce severe errors for increasing kinetic rate constants, for which the direct approach produces reliable results. Keywords: multi-species, reacrive transport, biogeochemistry, speciarion, chemical equilibrium, kinetics, speciation calculation, operator splitung

305

306

12.1

REACTIVE TRANSPORT IN POROUS MEDIA

INTRODUCTION

The chemical characteristic of a fluid in a porous medium is determined by various processes of physical, chemical, biochemical and geochemical type. Fundamental physical processes are advection, the movement in aflowfield,and dispersion, which is generalized diffusion valid in flow fields. Biogeochemical processes can basically be divided into slow and fast reactions; the first characterised by their kinetics, the second by their equilibrium. It can also be distinguished between heterogeneous and homogeneous reactions. Although this small list is in no way complete in gathering processes, which are relevant for the distribution of chemical species in porous media, it contains the most fundamental processes, which are in the focus of this chapter. In many applications some kinetic and some equilibrium reactions both contribute to the biogeochemical characteristic of the pore fluid and the porous medium. The simultaneous interaction of kinetic and equilibrium processes not only complicates the understanding of a real situation—in a laboratory experiment, a technical facility or in the field. The mathematical tools, which enable abetter understanding of the system, also face difficulties to deliver reliable results. Already the mathematical analytical description of both types of reactions is fundamentally different. It is clear that the numerical treatment is different also. The aim of this chapter to introduce the basic mathematical concepts—analytical and numerical—on which all approaches to gain an improved understanding of the complex interaction of processes in porous medium are based. In groundwater studies there are two important problem fields, in which the mentioned process constellation is particularly important, illustrated schematically in Figure 12.1. One problem field concerns landfills, contaminated sites and mine tailings, where components are introduced into a natural system. Before due to typical long timescales the ambient groundwater usually had the ability to reach a quasi steady state equilibrium chemistry, but is now confronted with a pool of new substances and common substances in very different concentrations. In the first phase, in the vicinity of the source of polluted water, kinetic processes play an important role. Some recent studies, in which mathematical models are applied for an understanding of such systems, should be mentioned. Barry et al. (2002) examine the fate of oxidisable organic contaminants in groundwater, which could be subject to some remediation or attenuation concept. Wang et al. (2003) more specifically are concerned with bioremediation of uranium-contaminated aquifers, Eckert and Appelo (2002) with benzene, toluene, ethylbenzene and xylene. Bacon and McGrail (2003) deal with a shallow subsurface disposal system for low-level glass waste. Kallvenius and Ekberg (2003) are concerned with the safety of a nuclear waste disposal site in fractured rock. Similarly Browning et al. (2003) set up a reactive transport model for the unsaturated zone. The other field of application concerns problems related to the interaction between groundwater and surface water. Where fluid discharges from rivers and lakes into the aquifers, a similar situation is given. The presence of the porous medium itself changes the chemistry of the water. Organic matter is the feeding substance for several biochemical processes, in which bacteria are directly or indirectly involved. Most important are redox processes.

E. HOLZBECHER

307

Figure 12.1 Scheme indicating regions of groundwaterflow,and the relative importance of kinetic and equilibrium reactions ('-f' stands for relatively high importance, *—' for low importance).

which are usually kinetic, see Holzbecher et al. (2002) and Kim and Coracpcioglu (2002). Moreover the different chemistry of surface water and of ambient groundwater initiates chemical reactions, see Doussan et al. (1997) and Stuyfzand (1998). The technology of riverbank filtration, pumping high quality water in the vicinity of surface water bodies, relies on the these natural attenuation processes, see Hiscock and Grischek (2002) and Melin (2003). In von Gunten et al. (1991) seasonal biogeochemical cycles in riverbome groundwater are identified. Keating and Bahr (1998) examine redox conditions induced by river water infiltrating the connected aquifer. The field of applications is so vast that not all aspects of reactive transport can be outlined in this chapter. Not considered are any interactions between flow on one side and transport and biogeochemistry on the other side, i.e. for the considered systems there is no relevant change of density, viscosity, permeability or porosity. Clogging or any other process that changes the pore space is not considered. Sorption is a process, for which several different simulation approaches have been proposed, which could not be included in this study. When transport is nonlinear, additional difficulties may arise, which are also excluded.

12.2 QUANTITATIVE GEOCHEMISTRY Here a system of Ns chemical species is considered, which interact with each other by Nr independent reactions. The most compact formulation for such a system is given by the

308

REACTIVE TRANSPORT IN POROUS MEDIA

reaction matrix S: (5ii

S=

S2I

5i2

•• •

SIN,

S22

'• •

S2N,

\

(12.1)

Each row of S represents a reaction, each column a specie. The number of rows of S is equal to the number of reactions A^^., and the number of columns is equal to the number of species Ns. In each row the involved species are represented by nonzero entries according to the stoichiometry of the reaction. Reactants have negative entries and reaction products have positive entries. Those species, which do not participate in a reaction have a zero entry in the corresponding row. Examples of reaction matrices are given in the next section. In many applications both fast and slow reactions are to be considered. Rubin (1983) makes the more precise distinction into 'sufficiently fast' and reversible reactions on one side and 'insufficiently fast' and/or reversible reactions. The attributes have to be understood in relation to the velocity of the transport processes, which can be very different from one field of application to the other. Clearly the time scale for combustion processes is much smaller than for geological problems. When a reaction is fast compared to the other processes in the system, it can be assumed that an equilibrium concerning that reaction is reached at all locations and at all instances. Such a reaction is thus referred to as an equilibrium reaction. Otherwise the temporal development of the reaction, its kinetic, has to be considered explicitly and it is called a kinetic reaction. The distinction between equilibrium and kinetic reactions is substantial, as the mathematical analysis is quite different and different numerical algorithms are needed for finding solutions on a computer. For the following it is assumed that all equilibrium reactions are gathered in the reaction matrix Seq and the kinetic reactions are gathered in the reaction matrix 5kin- Formally both could be connected to yield the total reaction matrix:

The mathematical treatment of chemical systems in equilibrium is based on thermodynamics, see Pitzer (1995) and Rau and Rau (1995). For each reaction the equilibrium state is characterized by its Gibbs energy change AG^, in which the chemical potential of all involved species is added, weighted by the stoichiometric numbers, see Pitzer (1995), i.e.: AG' = -RTlog

'"'^^^''^f'" , ^rl^r2

(12.3)

'' '

where the product in the nominator extends over all reaction products and the product in the denominator extends over all reactants, 5i,52,... as well as ^1,^2,... denote stoichiometric numbers, R is the universal gas constant, T the absolute temperature, and as denote the species' activities. In some ion-exchange applications a chemical potential can be used alternatively, see Lorente et al. (2004). In the equilibrium situation, the

E. HOLZBECHER

309

product on the right-hand side is a characteristic of the chemical reaction; it is called the equilibrium constant K for that reaction: ^sl ^s2 . . .

K = Ti If

,

(12.4)

where K depends on temperature and pressure. The connection between the equilibrium constant and Gibbs energy change is thus given by AG^ = -RT\ogK,

(12.5)

For all reactions of a multi-species system, formulae (12.3) can be rewritten in a more compact form in terms of the reaction matrix S^q'. AG° = -RT ' 5eq • loga ,

(12.6)

where AG^ denotes the vector of standard Gibbs energy change for the considered system and a the vector of activities. Here and in the entire paper vectors and matrices are denoted in bold, scalars in italic letters, and log a denotes the vector of logarithms of elements of a to the basis of 10. One can also use the vector of equilibrium constants K and write instead of (12.6): logii: = 5 e q - l o g a . (12.7) There are several data bases in which equilibrium constants for reactions are collected, for example from the Lawrence Livermore National Laboratory, see Delany and Lundeen (1990). Various codes use such databases for speciation computations. PHREEQC, see Parkhurst (1995), and Geochemists Workbench, see Bethke (1996), and CHESS, see van der Lee (1998) for example, calculate the species concentrations and activities based on data from chemical analyses. Some details concerning the numerics of speciation calculations are given below. Activity a and concentration c of a specie are connected by the formula a = jc, (12.8) with the activity coefficient 7 for a specie. Most speciation codes allow the user to choose between different approaches concerning the activity coefficients. Activity coefficients depend on ionic strength /i of the solution, which is defined as follows:

f^= i;Yl^i^h

(12.9)

where the sum is to be extended over all charged components, and Zi denotes the charge of specie i. The simplest formulation for the logarithm of the activity coefficient is an explicit formula: log7--A-z2^. (12.10) For very low values of /i (< 10~^-^; Sigg and Stumm, 1989), it is valid in water at 25 °C with a value of A = 0.51, see Krauskopf and Bird (1995). A more extended formula is

310

REACTIVE TRANSPORT IN POROUS MEDIA

proposed by Davies: 2 I VE_ where the coefficient A depends on temperature only. Thus 7 depends on temperature, ionic strength, and on the charge of the specie. Another formulation for the relation between activity coefficients and ionic strength is given by the Debye-Hiickel expression, see Bethke (1996) or Parkhurst (1995):

logTi = -T—^4^

+ M,

(12.12)

with coefficients B, a^ and hi. The coefficient B depends on temperature only. The ionsize parameter a^, as well as 6^, are species dependent, see Parkhurst (1995). Formulation (12.12) is the extended version as used in the speciation code PHREEQC. The Davies formula and the classical Debye-Hiickel formula, i.e. equation (12.12) without the last term, deliver very similar values up to /x « 0.1 (molal). For higher values of ionic strength, both deviate significantly: values from Debye-Hiickel decrease further, while values from the Davies formula start to increase again, see Bethke (1996). For solutions with such high ion content different formulae have to be used, as delivered by the B-dot model, see Bethke (1996), or by virial methods, see Bethke (1996). In application cases with low concentration of ions, activity coefficients are neglected, i.e. concentrations and activities are assumed to be identical. In the examples of this chapter this assumption is also made.

12.3 ANALYTICAL DESCRIPTION OF REACTIVE TRANSPORT Transport is a general term for mainly physical processes. Advection is the migration with the fluid flow. Diffusion and dispersion are processes that describe the tendency to equalization of concentration gradients. While molecular diffusion as a process is present in all fluid systems, dispersion takes additional effects into account, which appear due to the structure of inhomogeneities in fluid flow through a porous medium formation. For a single component transport in 1, 2, or 3 dimensions is described by a partial differential equation for the concentration c (see for example Nield and Bejan, 1999): dc (^—= V - D V c - t ; - V c ,

(12.13)

where (p denotes porosity, D the dispersion tensor and v the Darcy velocity vector. As a generalization of (12.13) a multi-species system in the aqueous phase can be noted as follows: tp—c^iS/-DV-v-V)c + S'^r. (12.14)

E.HOLZBECHER

311

The column vector c gathers the concentrations of all unknown species in the fluid phase. The matrix product of S^ (the transpose of the reaction matrix) with the rate vector r represents all chemical reactions which are to be considered. The reaction term couples the partial differential equations, given in each element of equation (12.14). In the formulation (12.14) kinetic and equilibrium type of reactions can be taken into account. One may split the reaction term into two parts, characterised by the reaction type + 5 e V e q + Sjin^kin •

^ -C={V-DV-VV)C

(12.15)

There is a crucial difference in the equilibrium reactions and the kinetic reactions. If a reaction is kinetic, the rate is known in principle; it usually is a function of some of the components. A rate law that fits to the equilibrium equation (12.4) is given by: ricin - r {KlXU

' " - Kal\all

• • •) .

(12.16)

In the case of equilibrium, the net exchange rkin for that reaction becomes zero, while in the case of non-equilibrium the deviation from the equilibrium determines the size of the exchange, and r denotes the rate coefficient for that reaction. For biogeochemical processes, the rates included in rkin, are typically given by a product of the form: rkm,i = rmax,2 ^

]_

] ] [ T<^.. 1

'

(12.17)

where rmax denote the maximum rates, and Ki and Kij are reaction-specific parameters. The first ratio in expression (12.17) is the so-called Monod term, which describes the growth (increase of concentration) up a certain limit; the second term is a product of inhibition terms, in which all those species are present, which are inhibitors to the concerned process. An example concerning the behaviour of oxygen and organic matter is presented by Kaluarachchi and Morshed (1995b). Such an approach is utilized by van Cappellen and Wang (1996) for redox kinetics in aquatic sediments, and by Holzbecher and Homer (2002) for redox chemistry in groundwater. A similar notation can also be used when bacteria, instead of chemical species, oppose the process. The terms Ci then represent bacteria populations. There are various other mathematical formulations of the kinetics, which do not fit into the form given in expression (12.17). The particular form of the kinetic term is not relevant for the procedure described in the following, which is valid in all cases, in which equilibrium and kinetic processes are present. For equilibrium reactions, the rates r^q are not known. Instead equation (12.15) is complemented by the equilibrium laws (12.7), which are algebraic equations. In order to obtain a set of differential equations, which can be handled by a numerical solution method, a transformation of the component space is needed. Mathematically such a transformation is described by a full-rank matrix C7eq with the property t/gq • Sj^ = 0, see Saaltink et al. (1998). Note that such a matrix can always been found, and that t/gq is not unique. Following Saaltink et al. (1998), the matrix J7eq

312

REACTIVE TRANSPORT IN POROUS MEDIA

can be obtained from 5eq and for that purpose Seq is split into two parts: 5eq = {Si S2) ,

(12.18)

where Si is a submatrix with Ng ~ Nr columns and 52 is a square submatrix with Nr rows and columns. The species, corresponding with the first Ng — Nr columns of S, i.e. with 5 i , are the so-called pnmary species, while those corresponding with the last A^^, columns, i.e. with 52, are the so-called secondary species. Parkhurst (1995) prefers the term master species. The matrix C7eq is then given by: Ueq={lN.-N. i-S^'Si)'^),

(12.19)

where IN denotes the unit matrix with N rows and columns. Reading the last columns of Ueq in columns yields the secondary species in terms of the primary species, which Lichtner (1996) calls the canonical form. The given procedure can only be applied, when the matrix 52 is invertible. Moreover, Saaltink et al. (1998) present their derivation for the situation, where all relevant species are involved in equilibrium reactions. The procedure has to be extended, if there are species participating in kinetic reactions only. In that case there are columns in the matrix 5eq with only zero entries. In such a situation the corresponding components have to be omitted from 5eq before the calculation of a matrix Ueq. Otherwise the matrix Ueq cannot be calculated according to formula (12.19). If an only-kinetic specie is primary, it is not possible to transform the corresponding 0 column into a column of the unity matrix by elementary linear operations on the rows. If the only-kinetics specie is secondary, the matrices 52 and Sj are not invertible. In a second step, the matrix Ueq has to extended by sets of columns and rows from the unity matrix. When the only-kinetic species are placed at the end of the specie vector, the extended matrix U is given by:

with Jex representing a unit matrix, and the Os representing zero matrices of appropriate size. The subscript 'ex' indicates the number of species, which participate exclusively in the kinetic reactions. Multiplication of equation (12.15) with U from the left leads to a new set of differential equations: if-u

= {V'DV~V'V)u

+ t75j;,rkin

(12.21)

and the unknown is the vector of so-called total concentrations u = U'C.

(12.22)

Examples for that procedure are given below in the equilibrium and kinetics examples. The set of partial differential equations (12.21), together with the equilibrium equations

E.HOLZBECHER

313

(12.7) provides a complete description of a system which can account for transport, equilibrium and kinetic reactions. As there are two types of equations—partial differential equations and algebraic equations, for the complete description the term differential algebraic equations (DAE) is used in the literature quite often. After the following examples, which illustrate the above derivations, the numerical solution of DAE systems will be described. Altogether in the system (12.21) the number of equations is reduced in comparison to the original system (12.14). The size of the vectors is reduced by the number of equilibrium reactions; there remain Ns — Nr equations for example, if none of the reactions is slow. The same number of algebraic equations, which have to be considered additionally, complete the system.

12.4

EXAMPLES

12.4.1 Equilibrium example 1 Three species A, B and C are connected by two simple reactions: (12.23)

B ^ A . C ^ B . The reaction matrix for the two equilibrium reactions is given by: 5 = Se<,= ( j

"/

_°j).

(12.24)

The matrices Si and ^2 are given by:

The inverse of 52 is given by:

^2"'=(_J \ )

(12.26)

and the matrix U = t/eq, given by equation (12.19), reads: C/eq=(l

1

1).

(12.27)

The total concentration is thus given by: u = A-hB-\-C.

(12.28)

According to the described transformation the solution for the entire system can be found by solving a single transport equation without any source or sink terms for the total concentration u. At each location, where species concentrations are to be known, these

314

REACTIVE TRANSPORT IN POROUS MEDIA

can be computed by a speciation calculation in dependence of the value for u at that position, which is delivered by the transport solver. Details concerning the speciation calculation are given below in Section 12.5. 12.4.2 Equilibrium example 2 As an example, a chemical system of species and reactions is considered which may be used to describe calcite dissolution and/or precipitation. The reactions and equilibrium constants are given in Table 12.1. A similar system, without the specie CaHCOs, was described in detail by Steefel and MacQuarrie (1996). There are 9 species involved in the reactions. These are ordered in the species vector as follows: (H+,HCO^,Ca2+,CaC03,H20,OH-,H2C03,CO^",CaHCO+).

(12.29)

The first four species are convenient primary species for the representation of the total concentrations for the elements involved in the system: H, C, Ca and 0 . The following reaction matrix corresponds with the reaction system:

^ — ^eq —

(1 1 0 2

0 -1 1 0

\0

0

0 0 0 0 1 0 0 0 1 -1

-1 0 0 0 0

o\

1 0 0 0 0

0 0 0 -1 0

0 1 0 0 -1 1 0 1 0/

1 0 0 0 0

0 0 0 -1 0

0 1 0 0 -1 1 0 1 0/

(12.30)

1 and S2 are given by:

/I 0 0 0 \ 1 - 1 0 0 1 1 0 Si = 0 2 0 0 0 \p 0 1 -V

5

/-I 0 0 S2 = 0

\o

o\

Table 12.1 Reactions of the carbonate system including calcite. equilibrium constants (log) for T = 25 °C. H+ + OHH+ + CO^Ca2+ + HCO^ 2H+ + CO^" Ca2+ -f CO^-

-14 f^HaO -10.329 ^HC03 0 CaHCO+ -1.106 -16.7 0 H2CO3 -8.48 ^ CaCOa

(12.31)

Reactions and

315

E. HOLZBECHER

With species order (12.29), the matrix S2 becomes singular. However, for the derivation of U, as noted in equation (12.19), it is necessary to have a regular S2, which can be inverted. A reordering of species usually suffices to reach that goal. Here the fourth and fifth species, CaCOs and H2O, can be exchanged. Then it follows:

/o0

/I 0 0 -1\ 1 - 1 0 0 Si = O l i o 2 0 0 0

vo

0

0 0

1 0y

V-i

1 0 0 0 0

0 0 0 -1 0

0 1 0 1 1

o\ 0 -1 0

(12.32)

oy

The inverse of S2 is given by:

c-i

"^2

/O 1 0 0 -1\ 1 0 0 0 0 0 1 0 - 1 0 0 1 0 0 0 yo 0 - 1 0 0 /

(12.33)

and the matrix U — Ueq, given by equation (12.19), becomes: /I 0 0 0 0 1 0 0 Ue^ = 0 0 1 0 \0 0 0 1

-1 1 -1 0 1 1 0 0 0 1 0 0

0\ 1 1 0/

(12.34)

Thus the total concentrations are given by: /TotH\ Tote TotCa V Tot 0 /

cor

/

H+ - CaCOs - O R - + H2CO3 HCO^ + CaCOs + H2CO3 + c o r + CaHCO^ Ca^+ + CaCOs + CaHCO^ H2O + OH" \ /

(12.35)

In most cases, some of the species activities can be assumed to be not changed during an application. The concentration of the fluid, mostly H2O, does not change and its activity in the equilibrium formula (12.4) is taken as 1. Also the activity of pure phases have an activity equal to unity, by definition, see Rau and Rau (1995) and Saaltink et al. (1998). These species can be simply omitted in the equilibrium equations. The constant activity species neither have to be considered in the mass balance equations and this simplifies the mathematical analytical description. Saaltink et al. (1998) show that there are different ways to deal with constant activity species. When both H2O and CaCOs are omitted as constant activity species, the following set of total concentrations is obtained: TotH TotC

cor

H+ + Ca2+ - OH" + H2CO3 - V.W3 + CaHCO^ HCO^ - Ca2+ + H2C0s + C0^

(12.36)

316

REACTIVE TRANSPORT IN POROUS MEDIA

12.4.3 Equilibrium and kinetics example 1 Here we return to the three-species system introduced above as the equilibrium example 1. It is assumed that the first reaction is fast and can be modelled as equilibrium type, while the second is slow and has to be modelled as a kinetic The matrix Seq, which contains the equilibrium reactions only, is given by: Seq=(l

(12.37)

-1),

in which only the species A and B are considered, as C is involved in the kinetic reaction exclusively. For the matrices Ueq and U results: (12.38a)

f ^ e q = ( l 1), ^1 1 0 u = ,0 0 1

(12.38b)

The total concentration u is thus given by the sum of A and B. In contrast to the equilibrium situation, described above, two differential equations remain, one foru = A + B and one for C. The entire system is thus described by: d_ 'Pdt

\c

= (V • P V - w • V)

+ rkh

1

(12.39a) (12.39b)

System (12.39) consists of two partial differential equations for u and for C, and one algebraic equation K = B/A. In addition, an explicit formulation of the kinetic exchange term r^in needs to be given, also, in order to be able to solve the system. 12.4.4

Equilibrium and kinetics example 2

It is assumed that in the carbonate system, the first four reactions can be treated as fast reactions, while the last, for calcite dissolution and precipitation, is slow. The first four reactions will thus be treated as equilibrium type, while the last is of kinetic type. Calcite has been put at the very end, as it participates in the kinetic reaction only. The reaction matrix is the same as given in equation (12.30) above. The splitting of the reaction matrix into one part representing the equilibrium reactions and one representing the kinetic reactions is given by:

'-'eq —

/I 0 0 0 - 1 1 1 - 1 0 0 0 0 0 1 1 0 0 0 \2 0 0 0 0 0 -

Skin = ( 0 0 1

-10

0 0

0 0 0 1 0 0 1 1 1 0)

0\ 0 - 1 0/

(12.40a) (12.40b)

317

E. HOLZBECHER

When the fourth column, related to the kinetic-only specie calcite, is omitted from 5eq and the matrices Si and 52 are given by: 0 0 -1 \ (I 1 - 1 0 0 Si = 0 1 1 0 ! \2 0 0 0 y

S2 =

0 0 0 -1

(I 0 0

lo

0 0\ 1 0 0 -1 1 0/

(12.41)

The inverse of 5^2 is given by:

s- =

/I 0 0 0 1 0 0 1 0 \0 0 - 1

o\

-1 0

(12.42)

)

oy

and thus the matrix C/eq is given by: /I 0 0 0 0 1 0 0 f/eq = 0 0 1 0 \0 0 0 1 For U follows:

/I 0 0 0 1 0 U = 0 0 1 0 0 0 0 0

Vo

-1 0 0 1

0 0 -1 0 0 0 0 0 0 1 0 0 0 1 1

:I 1 0 0

--1 1 0 0

1 -1 1 1 0 0 0 0 0 0

0\ 1 1

(12.43)

0/ 0>V 1 1 0

(12.44)

o>/

The first three species are convenient primary species for the representation of the total concentrations for the elements involved in the system: H, C and Ca. The vector of total concentrations is obtained easily by reading the first three rows of U and gathering the species related to the columns: TotH TotC ,TotCa;

H+-CO^--f-H2C03-OHHCO^ + H2CO3 -f CO^- + CaHCO+ Ca2+ + CaHCOj

(12.45)

The last two rows are neglected as they correspond to the constant activity species CaCOs and H2O. Note that a reordering of the secondary species does not change the combination to total concentrations. In the given case, the same constellation of total concentration is obtained, when the two constant activity species CaCOa and H2O are omitted from the start. Then Si differs

318

REACTIVE TRANSPORT IN POROUS MEDIA

from the form given in equation (12.41), i.e. 1 0 1 -1 0 1 ^2 0

0' 0 1 0

(12.46)

but 52 and its inverse, given by equation (12.42), are identical. The resulting matrices [/eq and U can be obtained from equation (12.43) by deleting the fourth column (which is related to the constant activity specie H2O) and the last row: l/eq = t / =

rVo I

0 0 -1 1 0 0 0 1 0

1 -1 1 1 0 0

0 1 1

(12.47)

Clearly the matrix U in expression (12.47) yields the total concentrations, given in expression (12.45). The entire system now consists of three differential equations for the total concentrations Tot H, Tot C and Tot Ca, which are coupled by a kinetic term, and of four equilibrium conditions for the four homogeneous reactions in the fluid phase.

12.5

NUMERICAL APPROACHES

There are various different ways in which the system of differential equations (12.21) can be solved numerically, see Saaltink et al. (1998). One major difference concerns the variables taken as unknowns in the equations. The original species c can be used, which has the disadvantage that the differential equations are coupled in each term by the matrix U. It is clearly more convenient to solve the system for the total concentrations u. Then the coupling is restricted to the last term. Moreover the number of total concentrations is lower than the number of species, and it is of advantage, when there is a lower number of variables. In summary, if the modeller uses the total concentrations as new variables, there are less unknowns and the nonlinear coupling is less severe. For that reason, such an approach will be implemented here. 12.5.1

Speciation calculations

Most speciation calculations are based on equations (12.7) for the equilibrium constants and (12.22) for total concentrations, which altogether forms a system of nonlinear equations for the activities a or the concentration c. These are the so-called stoichiometric algorithms, in contrast to the non-stoichiometric algorithms, which work directly with Gibbs' free energy, as given by equation (12.6). The Newton-Raphson technique for finding zeros of a multi-values function F is the standard method for the solution of such systems. A modified form of the Newton-Raphson iteration is implemented in Geo-

E.HOLZBECHER

319

chemists Workbench, see Bethke (1996), and in PHREEQC, see Parkhurst (1995), for example. Here the basic technique is outlined only for the case of small concentrations, where activity coefficients can be assumed to be near 1. Then activities a are equal to the concentrations c and the nonlinear system is solved for the components of the vector c. The species concentrations have to be calculated from the total concentrations. The relation between both is given by equation (12.22), which for an equilibrium system can be restated as follows: l/eqC-U = 0. (12.48) These are Ng — Nr equations for Ng unknown concentrations. The resulting Nj. equations stem from the equilibrium constants (12.7), which can be restated as: 5 e q - l o g a - l o g i i : = 0.

(12.49)

Both equations (12.48) and (12.49) can be gathered in a functional F{c) which operates on the space of species and for which the following is required:

The standard numerical algorithm for the calculation of the zeros for such a nonlinear functional is the Newton-Raphson method. This is an iterative method, in which a current approximation Coid is adjusted in each iteration step by computing the following expression: c = Cold - DF{c)-^F{c), (12.51) where DF denotes the Jacobi matrix of F. For practical use in speciation calculations, the standard method given by equation (12.51) is usually modified. Often solutions are determined first without taking activities into account, i.e. by setting a = c in equation (12.50), and are then corrected in a second step considering ionic strength /x, according to equation (12.9) and activity coefficients. Moreover it has to be taken into account that for concentrations only non-negative values are reasonable. This can be achieved by a simple modification, which transforms the vector c only within the space of positive values, see Bethke (1996), or by using the logarithms as unknowns. A discussion of these modification is given by Brassard and Bodurtha (2000). The latter finally favour an algorithm based on the Levenberg-Marquard scheme, in which the algorithm switches from a more robust, slow convergence method to the fast converging Newton method during solution finding. Such a strategy is also recommended by Tripathi (1986). 12.5.2

Transport modelling

For transport simulations, a variety of methods and tools are available nowadays, and an overview is given by Holzbecher and Sorek (2005). Well-established methods like finite differences, finite elements or finite volumes all have been applied successfully for

320

REACTIVE TRANSPORT IN POROUS MEDIA

transport modelling in 1, 2 and 3 space dimensions. Several powerful codes are available for modelling transport processes in porous media; perhaps the most used is MT3D, see Zheng (1990). When one of the mentioned numerical methods is applied for the transport equation (12.13), numerical errors may disturb the solution calculation significantly. Unrealistic oscillations or numerical dispersion can be observed, if the temporal and spatial discretisations are not small enough. Courant, Peclet and Neumann criteria have to be fulfilled in order to reach accurate solutions, see Holzbecher and Sorek (2005). In fact the error related to diffusion/dispersion decreases with decreasing time steps, while the error related to advection may increase instead. In that respect the operator splitting (OS) method improves the results significantly without the need to select very small time steps. The numerical treatment of diffusion and advection are split. Appelo and Postma (1993) provide a very good description of the simplest implementation, i.e. for the one-dimensional situation in a constant flow field. In such a situation an analytical solution for advection is coupled with a standard finite-difference or finite-element technique for diffusion. Then the module for advection does not contribute additionally to the total error of the numerical procedure. For higher-dimensional cases the generalization of the simple idea of OS is given by the method of characteristics (MOC). In most common groundwater codes, MOC, see Konikow and Bredehoeft (1978), and MT3D, see Zheng (1990), this technique is implemented and applied in numerous models. In all cases the diffusion/dispersion part is solved by one of the usual methods, while advection is treated in a Lagrangian frame of reference, i.e. in a moving flow field, which explains the naming as Lagrangian methods. It turns out that for the common problems of standard transport simulations, numerical dispersion or oscillations, can be overcome by the operator splitting procedure, see Holzbecher and Sorek (2005). In mathematical software tools solvers for partial differential equations are included, which also deliver transport solutions with high accuracy. These follow a different solution strategy, as the time step is reduced automatically, when it becomes necessary. In MATLAB (2003) for example, internally a solver for ordinary differential equations is called, which is even capable of solving stiff problems, see Ashino et al. (2000). This feature is exploited in the reactive transport module used for this paper, which is described below. 12.5.3 Transport and reaction coupling Models for reactive transport can follow different strategies. A basic critical evaluation of different strategies was given by Yeh and Tripathi (1989), in which the distinction between a direct approach and a sequential approach is made. While in the direct approaches (in the following referred to as DSA: direct substitution approach) all processes are treated together, in the sequential approaches processes are modelled separately in the simulation procedure. For the sequential design two different cases have to be distinguished: the

321

E. HOLZBECHER

non-iterative approach (SNIA) and the iterative approach (SIA). The scheme in Figure 12.2 illustrates the differences in a data flow chart. The sequential approach is closely connected with the concept of operator splitting (OS, see above). The idea of separating the numerical treatment of the different processes involved in reactive transport is nearby. The reaction simulation is separated from the simulation of the transport processes, for which a splitting technique as noted above could be applied as well. Models of reactive transport with the PHREEQC code, see Parkhurst (1995), are examples in which all different processes are treated separately. Khan and Liu (1995) also treat advection, dispersion and kinetic reactions differently. With reference to a transport step and a reaction step the term two-step approach can also be found, see Herzer and Kinzelbach (1989). For equilibrium processes, the application of the OS approach is even more justified. In case of no kinetic reactions, the last term in equation (12.21) vanishes and a set of linear uncoupled equations for the unknown total concentrations u remains. When u is determined by a usual transport solver, a speciation calculation (see above) can be started for each node in a separate step. In fact for such a situation the splitting technique does not produce any additional errors, even when applied as SNIA, without iterations. Clearly the operator splitting is a nearby procedure, evolving from splitting the problem into a set of uncoupled transport equations and into nonlinear speciation calculations. Given the noted circumstances, the so-called two-step approach can be followed without any errors. The original system of transport and equilibrium geochemistry, consisting of a set of coupled nonlinear partial differential equations is decoupled into two tasks, which can be performed sequentially. The first task consists of the solution of standard transport

DSA:

SNIA: 1

T" 1 rsnSj^ui V

'

Reaction"

|

SIA: Transport i

Reactloti

i

Figure 12.2 Schematic view of data-flow in reactive transport models, illustrating the three different solution strategies direct substitution approach (DSA), sequential noniterative approach (SNIA) and sequential iterative approach (SIA).

322

REACTIVE TRANSPORT IN POROUS MEDIA

equations for a smaller system of unknowns—a system, which is uncoupled and linear. The second task is solving the non-linear speciation problem at each node. If there are kinetic reactions, as they are usually introduced by biogeochemical processes, at least some equations of the system (12.21) are still coupled in the last term. Thus some of the advantages of the transformation, noted above, are lost, when there are kinetic processes additionally. The entire system is not de-coupled, it still stays coupled by the kinetic reactions. In order to improve the simple two-step method, an iteration can be introduced within each time step simulation. In the scientific literature the procedure is known under the term SIA (sequential iterative approach), while the simple two-step, described before, is called SNIA (sequential non-iterative approach), see Yeh and Tripathi (1989) and Regnier et al (2002). Other improvements are described by Tebes-Stevens and Valocchi (2000). An alternative to the SIA or SNIA methods, which both can be regarded as operator splitting methods, see Tebes-Stevens and Valocchi (2000), are the DS A (direct substitution approach) methods, see Yeh and Tripathi (1989). In the DSA the partial differential equations are solved as they appear, by direct substitution of the reaction terms. The DSA can be expected to reach a higher accuracy than any OS method, as the interaction of the processes at the time step scale is still considered. Moreover, in the implementation, which is described below, the DSA allows an automatic reduction of the time step, when it is necessary. Nevertheless computational costs are usually much higher for the DSA. The numerical methods differ concerning the system of variables. It is possible to apply the DSA and the OS methods for the original species c, or for the total concentrations u, see Saaltink et al. (1998). Performing the iterations for the total concentrations has the advantage that a coupling between the differential equations is given only in the last term of equation (12.21). Moreover, the number of partial differential equations to be solved is smaller than in the formulation with species concentrations. Such a system can be expected to be solved more easily. OS in the form of the SNIA is the most used technique for reactive transport hitherto. The PHT3D code for 3D reactive transport modelling is an SNIA coupling of the popular PHREEQC, see Parkhurst (1995), speciation code and the MT3D, see Zheng (1990), transport code, see Prommer et al. (1999) and Prommer (2002). Its popularity has two reasons. First available codes for transport or for speciation often can be linked with each other. It is even an advantage when only one of the two tasks can be performed by an existing code, as only one other module has to be taken or developed to be connected. The second reason is that for 2D or even for 3D application cases most implementations of the DSA require extremely high computer resources.

12.6 NUMERICAL ERRORS The mentioned two-step approach is often used as a numerical method in the case with kinetics. A solution is obtained by taking one transport step first and one reaction step

E. HOLZBECHER

323

second during one time step simulation. It is clear that such a procedure leads to errors, when a kinetic process is fast, but not fast enough to be treated as an equilibrium process. When the time step is relatively long compared to the characteristic time of a kinetic process, a false simulation of the behaviour can be expected because the interaction between transport and reactions during the time step is not taken into account. The steps are not coupled in the sense that they influence each other within the time step. The numerical error for the operator splitting method applied to reactive transport was first examined by Valocchi and Malmstead (1992). The reaction in that study reduces to a simple first-order decay term for a single specie. However, using an analytical approach, the study showed that the OS method introduces a mass balance error, even when there are no numerical errors in the transport and in the decay step. The error thus stems from the incomplete coupling of the different processes, which act simultaneously in the system. Moreover the study shows that the error depends on the predefined time step A^ and that the OS solution 'lags behind' the exact solution. Later Barry et al. (1996) extended the examination for more general boundary conditions and other reaction terms: zero-order production, radioactive decay, interphase mass transfer and linear retardation. They found temporal errors of first- or second-order in At in all cases, and suggested modellers to be careful with the OS approach. Barry et al. (1997) extend the former work, showing the relevance of temporal errors also for a two-species system, where two cations are coupled by a fast exchange reaction. Also the publication of Kaluarachchi and Morshed (1995a) is a generalization of the results of Valocchi and Malmstead (1992) for different initial and boundary conditions, characterizing the timelag error as an inherent error of the splitting technique independent of the discretization. In a second publication Kaluarachchi and Morshed (1995b) extend their analysis to Monod kinetics for a single specie and a two-specie setup. Saaltink et al (2001) compare the SI A and the DSA methods for several test cases, one of which is the calcite dissolution model introduced above. They show that for more complex models, an increased number of time steps is required by the SIA, accompanied by an increased requirement of CPU time. Without a detailed examination of errors, they state that 'the SIA particularly gives problems for at least two types of cases: cases with high kinetic rates and cases with high numbers of flushed pore volumes'. In two very recent publications, Kanney et al (2003) deal with the time-lag error for nonlinear reactive terms and Carrayrou et al (2004) with mass balance errors of different operator splitting schemes. Aside from mathematical analysis of the numerics, it can expected that methods which work with fixed time steps have difficulties modelling kinetics with a certain reaction scale. When a kinetic reaction has a temporal dynamic, which lets significant changes appear with the scale of a time step, this can hardly be simulated using a fixed rate within a time step. For slower kinetics the method may work well, when the method is able to capture the change of the reaction term. For faster kinetics the modeller can solve the problem by changing the reaction type from kinetic to equilibrium. An example for such behaviour is given in the example models below.

324 12.7

REACTIVE TRANSPORT IN POROUS MEDIA

IMPLEMENTATION IN MATLAB

The SNIA and DS A procedures are implemented using MATLAB (2003), as application of the DSA method for the variable u. Speciation calculations are performed using a modified Newton-Raphson method, implemented also within MATLAB. For the solution of the partial differential equations the MATLAB solver pdepe is used. Only one-dimensional transport can be treated, yet, as MATLAB offers one-dimensional partial differential equation solvers only in the basic version of the software. However, the procedure is similarly applicable for higher dimensional problems, to be solved by the appropriate toolbox or by FEMLAB (2003). More details of the implementation of both methods are shown in the process chart in Figure 12.3. For DSA the speciation calculation and computation of the kinetic term is performed within ihQ pdepe solver, i.e. every time the coefficients of the partial differential equation are updated. In contrast, for the SNIA the speciation calculation and kinetics computation are performed outside of the pdepe solver. An array of source terms is available after this, which is used for the solution of the partial differential equation. It is important to note that for the SNIA, the specification of a time step is necessary, while it is obsolete for the DSA. In fact the call of iht pdepe solver requires a time vector, but this is only for model output and does not specify the internal time step used for the solution of the partial differential equation. The transport solution within the SNIA can thus be expected to be of high accuracy in the MATLAB model, even when the Peclet, Courant, or Neumann criteria are not fulfilled. The errors for the SNIA, presented below, thus stem DSA:

ocessing I I Pre-processing

f liiitialo0ii(itious J Speciation calculation |

I Post-processing I SNIA:

L

Kinetic terms

Pre-processing [Speciation calculation

t = t + At

Initial oonditions

Kinetic terms pdepe Post-processing

^

:

pde coeMeients

Figure 12.3 Software process chart for the DSA and SNIA modules implemented using MATLAB.

E. HOLZBECHER

325

from the incomplete coupling due to OS only. Note also that in most applications the SNIA is implemented for fixed time steps, although it could in an appropriate software environment be performed with time steps adjusted to the need of the kinetic. In MATLAB that seems to be difficult, as the internal time step is not known by the modeller.

12.8 EXAMPLE MODELS 12.8.1 Three-species model A simple test model is examined, which is based on the three-species example A, B and C, introduced above. Equilibria for the reactions, specified by the two rows of S are given by 10~^ and 10^. The first equation is assumed to be in equilibrium, the second kinetic. The numerical approaches SNIA and DSA are compared for this hypothetical test case. Both are implemented in MATLAB. For the solution of the transport equation in both cases the pdepe module of MATLAB for the solution of partial differential equations is applied. For the DSA the speciation calculation is performed, whenever the solver needs an update of the coefficients of the transport equation. For the SNIA the speciation is performed alternating outside the solver for the partial differential equation: after each call of pdepe the Newton-Raphson procedure is performed for the speciation calculation, htioxQ pdepe is called again for the next time step. The transport parameters used for the three-species example are shown in Table 12.2. The number of blocks is 50, each with 1 m length; the number of time steps is 20 or 40, which is relevant for SNIA only. The criteria concerning the three dimensionless numbers, which are relevant for the numerical performance of transport codes, the Courant number Cou, the grid-Peclet number Pe and the Neumann number Neu are fulfilled. Figures 12.4 to 12.6 depict comparisons between the SNIA and the DSA results, for different values of the kinetic transfer coefficient r.

Table 12.2 Transport parameters for the three-species example model. Length Maximum simulated time Porosity Darcy velocity Diffusivity Initial concentjation u Initial concentration C Inflow concentration u Inflow concentration C Kinetic transfer coefficients

Value

Unit

50 20 0.2 0.2 1 -2 -2 0 0 0.001-0.02

m a

-

m/yr m^/yr log mol/1 log mol/1 log mol/1 log mol/1 mol/(lyr)

326

REACTIVE TRANSPORT IN POROUS MEDIA

10^

A, SNIA B, SNIA C, SNIAl ^,DSA 5,DSA C,DSA

\ ' ^ l T l i M I MMI Ml i Ml i i Mi I M i II i l l M M M I

lo- 0

5

10

15

20 25 30 Distance [m]

35

40

45

50

Figure 12.4 Comparison of the SNIA and DSA methods for the three-species system at t = 20 yr; for the kinetic transfer coefficient r = 10"^ mol/(lyr) and time step A* = 1 yr in case for the SNIA; results for the SNIA after thefirstmetres deviate from the more accurate results, obtained with DSA. The rate parameter r was changed between r = 10"^ and 0.2mol/(lyr). For r < 0.09mol/(lyr) the deviation between the SNIA and the DSA method is marginal. But SNIA starts to have problems, when r = 10"^ mol/(lyr) is approached, see Figure 12.4. Two types of errors are apparent: there is an 'undershooting' for the lower concentration species, when the steady level is approached. Moreover the level at the end of the travel path is not predicted correctly: it is overestimated for the lower concentration species, and underestimated for the higher concentration specie. The performance of the SNIA method becomes worse, when the rate parameter is increased further. Figure 12.5 depicts the results for both the methods for r = 0.02mol/(lyr). Clearly the SNIA method breaks down completely after the first metres along the flow path. The concentrations of species A and B fall by several orders of magnitude, recover sporadically and fall again. The high concentration specie is not affected first, due to the fact that concentrations of both others species are several orders of magnitude lower. But at end of the travel path the concentration of this specie becomes wrong also. The final remark for the three-species system concerns the time step. The DSA method does not depend on the time step, as the pdepe solver reduces the time steps, when necessary, i.e. when the dynamics of the kinetics is high within a time step. The SNIA

E. HOLZBECHER

327

20 25 30 Distance [m] Figure 12.5 Comparison of the SNIA and DSA methods for the three-species test case; for the kinetic transfer coefficient r = 0.02 mol/(l yr) and time step At = 1 yr in case of the SNIA; the SNIA clearly breaks down completely. does not have such an internal control, as a time step size has to be selected by the modeller with the other input values. Nevertheless, in case of problems the SNIA user can choose a reduced time step, in order to improve the simulation results. This has been done for the input parameters, used to produce the results of Figure 12.6. When the time step is reduced to At = 0.5 yr, the results for r = 0.01 mol/(l yr) of the SNIA and the DSA coincide. For r = 0.02mol/(lyr) the results improved, see Figure 12.6, but still deviate significantly from the accurate DSA output. Figures 12.7 and 12.8 show the development of errors in time for the three reaction rates r = 0.0025, 0.005 and 0.01 mol/(lyr). The absolute error is highest for specie C with highest concentration, and appears, when the front has just passed the inlet, after the first few time steps. The error of C increases with r. The error of the less abundant species A and B shows less expected behaviour. The errors are highest for the fastest kinetic (r = 0.01), but second highest for the slowest kinetic (r = 0.0025), while for the intermediate kinetic the SNIA performs best. In order to explain such unexpected behaviour, one has to take into account that the concentrations of A and B are several orders of magnitude smaller than the concentrations for C. A false performance of an algorithm concerning C may be hardly be recognizable, but induce high errors for the species with smaller concentrations.

328

REACTIVE TRANSPORT IN POROUS MEDIA 10^

A, SNIA B, SNIA C, SNIA .4,DSA 5,DSA C,DSA

©ee^O000009<

10'l^#

X X XX XXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXX

o

u 10"

10^ {•••ij j..|.^.|„|--|.4-4^.4,4-.H-f-4-fJ {• { M f ) . { I } f j f .{ { f - f - H H - f - H H - f

10-

10

15

20 25 30 Distance [m]

35

40

45

50

Figure 12.6 Comparison of the SNIA and DSA methods for the three-species test case; for the kinetic transfer coefficient r = 0.02 mol/(l yr) and time step At = 0.5 yr in case of the SNIA; the SNIA values improved significandy due to the reduced time step, but are still far from being satisfactory. ^ ^ ^ , r = 0.0025 ->^ B,r = 0.0025 -^ C,r = 0.0025 A, r = 0.005 B,r = 0.005 C,r = 0.005 ^ , r = 0.01 -O- B,r = 0.0\ - e - C r = 0.01

Time step Figure 12.7 SNIA errors (linear scale) in their development in time for the three-species system; curves for species A, B and C, and for three different reaction rates r.

329

E. HOLZBECHER

10 12 Time step Figure 12.8 SNIA errors (logarithmic scale) in their development in time for the threespecies system; curves for species A, B and C, and for three different reaction rates

12.8.2

Calcite dissolution test case (ID)

The test case for the model was taken from Saaltink et al. (2001), denoted as CAL, describing dissolution of calcite by sub-saturated infiltrating water. Initially the water is saturated with calcite. Saaltink et al. (2001) distinguish four different scenarios, which differ by the kinetic transfer coefficient rkin • ^ki]

=

a{l-a),

(12.52)

where a denotes the transfer coefficient and a the saturation index for calcite. Transformed into the units, used here, the values are approximately 10"^, 10"^, 10~^ and 10~^ mol/(l yr), representing cases CAL-1,..., CAL-4. CAL-1 has the lowest, CAL-4 the highest dissolution rate, and is thus nearest to the equilibrium case. Transport parameters for the test case are listed in Table 12.3, reactions are given above in the kinetics and equilibrium example 2 and their equilibrium constants are given in Table 12.1. For the calculations the system with seven non-constant activity species is chosen (see above). All results, presented in the following, are for a maximum simulated time f = 5 yr. A comparison between the original results by Saaltink et al. (2001) and a MATLAB model (using DSA) were already published by Holzbecher (2004), showing differences caused by the inflow boundary condition, while along the flow path the differences vanished. Here it is the aim to compare results of SNIA and DSA, both implemented in MATLAB, in order to avoid differences from the spatial discretization method.

330

REACTIVE TRANSPORT IN POROUS MEDIA

Table 12.3 Parameter for calcite dissolution test case CAL, see Saaltink et al. (2001). Length Maximum simulated time Porosity Darcy velocity Diffusivity Initial concentration Tot H Initial concentration Tot C Initial concentration Tot Ca Boundary concentration Tot H Boundary concentration Tot C Boundary concentration Tot Ca Kinetic transfer coefficients

Value

Unit

100 5 0.1 2 200 -7.978 -3.018 -3.019 -5.496 -5.421 -4.398 9.939 X 10-4(1)7

m a

-

m/yr m^/yr log mol/1 log mol/1 log mol/1 log mol/1 log mol/1 log mol/1 mol/(lyr)

All CAL cases of Saaltink et al (2001) were simulated using both DSA and SNIA approaches, as described above, and two additional cases. The time step for the SNIA was chosen as 2 months, i.e. At = 0.167 yr. Figure 12.9 depicts the results for pH from five selected runs for the final time, showing the behaviour of the numerical schemes. The inlet is located at position 0 on the left-hand side of the figure. H+ ions are consumed after calcite dissolution by COg" ions to forming HCO^. The gradient from the left to

9.5

pH 8.5

7.5 0

SNIA, r = 3x10-" DSA, r= 3x10^ SNIA & DSA, r = 10^ (CAL-3) DSA,r=10"^(CAL-4) 10

20

30

40

50 60 Distance [m]

70

80

90

100

Figure 12.9 Results for the CAL test case, calculated with the MATLAB model.

331

E. HOLZBECHER

the right appears due to the fact that the inflowing water has a different characteristic than the water initially present. For low and medium kinetic rates, cases CAL-1, CAL-2 and CAL-3, the results from both numerical approaches are identical. Results for CAL-3 are shown in Figure 12.9. For the fastest kinetic, case CAL-4, the DSA method delivered good results, which are also depicted, while the SNIA started, but broke down due to an internal error. In order to examine the behaviour further, an intermediate rate value of r = 3 x lO""* mol/(lyr), was chosen, for which results are given in Figure 12.9 also. While the results for both approaches differ only slightly near the inlet, the SNIA shows an unrealistic behaviour after a distance of 60 m. The temporal development of the errors is examined for the reaction rate r = S x 10~^mol/(lyr), see Figure 12.10. Clearly the position at which the SNIA becomes unreliable, moves along the flow path during the simulation, in fact it moves with the concentration front. Between the inlet and the position of the front, the calculated concentrations become excellent during the course of the simulation. The reason for that positive behaviour is that the front has passed through and the concentrations are almost adjusted to the chemistry of the inflowing water. At the front of the inflowing water the changes

- e - SNIA,^=1 - e - DSA,r=l - ^ SNIA, ^ = 2 -*- DSA,r = 2 f SNIA,/ = 3 -~4™ DSA,r = 3 -^- SNIA,/ = 4 - ^ DSA,/ = 4

40

50 60 Distance [m]

70

80

90

100

Figure 12.10 Results for the CAL test case with r = 3 x 10~^ mol/(l yr) for * = 1, 2, 3, 4 yr, and At = 2 months.

332

REACTIVE TRANSPORT IN POROUS MEDIA

during each time step are much higher than behind the inlet. Therefore the errors can be expected to be higher at the front. A second argument explains the behaviour of the errors, illustrated in Figure 12.10. The highest error can be expected to occur in the very first time period straight after the inflowing fluid started to penetrate the system. It can be expected that the numerical error is also high. Due to the flow within the system, the error-laden values are transported through the system with the flow velocity. The errors at intermediate times are clearly higher, when the curves in Figure 12.10 are compared with the corresponding curves at the final time t = 5 yr in Figure 12.9. Results for a reduced time step have a much better performance, see Figure 12.11. In Figure 12.12 the development of the errors in both SNIA calculations are depicted. After each time step the maximum difference between the SNIA and the DSA results, taken for all blocks along the flow path, is plotted. The two SNIA runs with a coarse time step of 2 months and with a refined time step of 1 month, for which results are already depicted in Figures 12.10 and 12.11; for the refined At, two steps are made before the comparison, making times on the x-axis coincide. Figure 12.12 demonstrates how the error develops with time. Clearly, for the coarse time step simulation the high error follows a rather chaotic regime, while for the refined time step model the error decreases. -e-e-^ -^ -t-H™ -^ -^

9.5 F

9h

SNIA,^=1 DSA, t = 1 SNIA, ^ = 2 DSA, ^ = 2 SNIA, ^ = 3 DSA,r = 3 SNIA, r = 4 DSA,r = 4

pH 8.5

7.5

0

10

20

30

40

50 60 Distance [m]

70

80

90

100

Figure 12.11 Results for the CAL test case with r = 3 x 10 '^ mol/(l yr) for t = 1, 2, 3, 4 yr, and At = 1 month.

333

E. HOLZBECHER

15 Time step Figure 12.12 Errors in pH simulation using SNIA for the CAL test case with r = 3 x 10-^mol/(lyr).

Figure 12.13 shows the temporal development of the errors for different rate coefficients r. Clearly the SNIA codes have the biggest problems directly after the start of the simulation, when the front is near the inlet. In all simulation runs results become more accurate after that initial phase. However the intercomparison of the results from runs with varying kinetics does not follow a clear structure. Over the entire time period, none of the runs shows a monotonic increase or decrease in the errors. The simulation for the lowest rate value is most accurate for most of the simulated time, but near the end of the time period it becomes the worst. All other runs differ mostly in the initial stages, where some show a dramatic decrease in the error, but only for it to increase afterwards. In an intermediate region, the errors for four of the r values are almost identical. 12.8.3 Two-dimensional modelling A 2D transport model is presented in order to demonstrate the described approach for a two-dimensional flow field. Here the flow through a porous fracture of length L and height H is considered. It is assumed that the permeability of the porous fracture follows a quadratic function with a maximum at the fracture centreline and it is zero on the impermeable walls. A velocity profile with a quadratic shape is given by: V{y) = ^max i'^-

4y^ Jp

(12.53)

334

REACTIVE TRANSPORT IN POROUS MEDIA

10^

r=10r = 3 10-^ r = 5 10-^ r = l 10"^ r = 5 10-^ 10"

X 10r 2

10"

10

15

Time step

20

25

30

Figure 12.13 Temporal development of errors in pH simulation using SNIA for the CAL test case for different values of the calcite dissolution rate r.

with a maximum Darcy velocity i;max and where the vertical coordinate extends from y = 0 at the centreline to y = ±H/2 ai the upper and lower boundary. The dispersion tensor D for a horizontal flow field is given by Bear (1972): D =

' aLV 0

0 arVj

(12.54)

where it is assumed that the diffusivity can be neglected with respect to dispersivity, and aL and ar denote longitudinal and transversal dispersion lengths. The differential equation (12.13) becomes:

dc

d

('-^')|^-^|"^^-(^-^')

dc dy

-Vn

(i-v=)l^ dx '

(12.55) Using the dimensionless parameter combinations K — aTJOLL and Pe = Hjlaj,, the differential equation (12.55) can be written as follows: dc

,

2\ 9^^

9 .

2\ 9^

^ /. 9\ dc Pe(l-.^)^,

(12.56)

where x and t now denote dimensionless variables, which are obtained from the original space and time variables by the transformation. Initial and boundary conditions are

335

E. HOLZBECHER

Table 12.4

Additional parameter for 2D calcite dissolution test case.

Length L Width H Maximum simulated time Maximum velocity Longitudinal dispersion length Transversal dispersion length

Value

Unit

10 2 5 0.1 1 0.01

m m a m/yr m m

adopted from the ID calcite dissolution test case treated above, see Table 12.3, and additional parameters for the 2D example are given in Table 12.4. Within the modelled fracture, an equilibrium with calcite is assumed. The system of chemical species is thus transformed as described in the equilibrium example 2. Total concentrations are given by equation (12.36). Two transport equations have to solved. As these are independent, it suffices alternatively to solve a single partial differential equation for a normalized total concentration u, when the results are transformed to the real scale as post-processing task using the values for initial and boundary conditions. The differential equation is solved using the FEMLAB (2003) software package, designed for finite element simulations of multiphysics applications. In FEMLAB it is possible to enter the differential equation explicitly or to use pre-defined equations. The pre-defined 'convection-diffusion' type is appropriate for transport simulations. The finite element grid consists of 3817 nodes and 7168 elements. Results for pH are depicted in Figure 12.14, where the flow is from left to right. High velocities are on the bottom, no flow at the top. The plot clearly visualizes the influence

9.8

wi 19.6 19.5 19.4 I9.3 I9.2 I9.I Figure 12.14

pH distribution for the 2D test case.

336

REACTIVE TRANSPORT IN POROUS MEDIA

from the high speed velocity component at the bottom, where the front of the inflowing fluid penetrates faster than at the no-flow boundary at the top.

12.9

CONCLUSIONS

A mathematical analysis is presented, which is appropriate to describe problems in porous media, which include transport, equilibrium geochemistry and kinetic reactions, see Ingham and Pop (1998, 2002). The proposed procedure has the advantage that it leads to a smaller set of unknown variables in the solution of the differential equations, if there is at least one equilibrium process. In the case of no kinetic processes, the method reduces to a decoupled two-step algorithm. The analysis is exemplified on a hypothetical three-species system and a calcite dissolution test case, adopted from Saaltink et al. (2001). For computer simulations, based on the mathematical derivation, two numerical methods are described and implemented. The very popular SNIA, which is based on operator splitting, has the advantage that it can be applied much easier and that already available transport and/or geochemistry codes can be included. However its disadvantage are additional numerical errors, which stem from decoupling at the time step scale. The more complex problems DSA method is superior in that respect. It is shown how the two numerical methods can be implemented using MATLAB. In the proposed implementation, the DSA is coupled with automatic time-stepping for the solution of partial differential equations; in contrast using SNIA, as in other implementations, a fixed time step as to be specified by the modeller before the simulation starts. An advantage of the MATLAB implementation is that due to the build-in solver for partial differential equations, spatial and temporal errors of the transport solution are reduced effectively. Both methods are tested on several test cases, including both equilibrium and kinetic reactions. With 3-9 species, the selected test cases are more complex than most others which have been proposed as benchmarks for the operator splitting method. For all test runs of this study the DSA delivered better results than the SNIA. With increasing dynamics of the kinetics errors seem to appear, when a certain margin is exceeded. It can be concluded that the caution with the SNIA, which is already recommended in studies with simpler test cases, is justified. However in the test cases the SNIA for a wide range of the kinetic parameters delivered very reliable results. The advantage of the SNIA of less requirements concerning computer resources, is not relevant any more, when speciation is performed within a partial differential solver, as proposed here and implemented in MATLAB. A 2D generalized model using FEMLAB is finally presented.

ACKNOWLEDGEMENT The author is grateful to NASRI, an interdisciplinary project of the Kompetenzzentrum Berlin, funded by Veolia Water and Berliner Wasserbetriebe.

E. HOLZBECHER

337

REFERENCES Appelo, C. A. and Postma, D. (1993). Geochemistry, groundwater and pollution. Rotterdam.

Balkema,

Ashino, R., Nagase, M., and Vaillancourt, R. (2000). Behind and beyond the MATLAB ODE suite. Comp. Math. Appl. 40,491-512. Bacon, D. H. and McGrail, R B. (2003). Lessons learned from reactive transport modeling of a low-activity waste glass disposal system. Comp. Geosci. 29, 361-70. Barry, D. A., Miller, C. T., and Culligan-Hensley, R J. (1996). Temporal discretisation errors in non-iterative split-operator approaches to solving chemical reaction/groundwater transport models. J. Contaminant Hydrology 22, 1-17. Barry, D. A., Miller, C. T., CuUigan, P. J., and Bajracharya, K. (1997). Analysis of split operator methods for nonlinear and multispecies groundwater chemical transport models. Math. Comp. Simulation 43, 331-41. Barry, D. A., Prommer, H., Miller, C. T., Engesgaard, P., Brun, A., and Zheng, C. (2002). Modelling the fate of oxidisable organic contaminants in groundwater. Adv. Water Res. 25, 945-83. Bear, J. (1972). Flow through porous media. Elsevier, New York. Bethke, C. M. (1996). Geochemical reaction modelling—concepts and application. Oxford University Press. Brassard, P. and Bodurtha, P. (2000). A feasible set for chemical speciation problems. Comp. Geosci. 26,277-91. Browning, L., Murphy, W. M., Manepally, C , and Fedors, R. (2003). Reactive transport model for the ambient unsaturated hydrogeochemical system at Yucca Mountain, Nevada. Comp. Geosci. 29, 247-63. Carrayrou, J., Mose, R., and Behra, P. (2004). Operator-splitting procedures for reactive transport and comparison of mass balance errors. J. Contaminant Hydrology 68, 239-68. Delany, J. M. and Lundeen, S. R. (1990). The LLNL thermodynamical database. Lawrence Livermore National Laboratory Report UCRL-21658. Doussan, C , Poitevin, G., Ledoux, E., and Detay, M. (1997). River bank filtration: modelling of the changes in water chemistry with emphasis on nitrogen species. J. Contaminant Hydrology 25, 120-56. Eckert, P. and Appelo, C. A. J. (2002). Hydrogeochemical modelling of enhanced benzene, toluene, ethylbenzene, xylene (BTEX) remediation with nitrate. Water Resources Res. 38, 5.1-11. FEMLAB (2003). (Version 2.3.) COMSOL AB. Tegnergatan 23, SE-111 40 Stockholm, Sweden. Herzer, J. and Kinzelbach, W. (1989). Coupling of transport and chemical processes in numerical transport models. Geoderma 44, 115-27. Hiscock, K. M. and Grischek, T. (2002). Attenuation of groundwater pollution by bank filtration. J. Hydrology 266, \39-A4. Holzbecher, E. (2004). The interaction between transport and biogeochemistry. In Applications of porous media (eds A. H. Reis and A. F. Miguel), pp. 497-504. ICAPM 2004. Centro de Geofisica de Evora, Portugal. Holzbecher, E. and Homer, Ch. (2002). Modeling of redox-kinetics in sediments and groundwater. In Computational methods in water resources. Proceedings of the XIV international conference.

338

REACTIVE TRANSPORT IN POROUS MEDIA

Delft (eds S. M. Hassanizadeh, R. J. Schotting, W. G. Gray, and G. F. Finder), Vol. 1, pp. 563-70. Elsevier, Amsterdam. Holzbecher, E. and Sorek, S. (2005). Numerical models of groundwater flow and transport. Encyclop. Hydrolog. Sci. In press. Holzbecher, E., Homer, Ch., MaBmann, G., Fekdeger, A., and Merz, C. (2002). Redox-transport modelling for the Oderbruch aquifer. In Geochemical processes: conceptual models for reactive transport in soil and groundwater (eds H. D. Schulz and G. Teutsch), pp. 191-214. Wiley-VCH, Weinheim. Ingham, D. B. and Fop, I. (eds) (1998). Transport phenomena in porous media. Fergamon, Oxford. Ingham, D. B. and Fop, I. (eds) (2002). Transport phenomena in porous media. Vol. II. Fergamon, Oxford. Kallvenius, G. and Ekberg, C. (2003). TACK—a program coupling chemical kinetics with a two-dimensional transport model in geochemical systems. Comp. Geosci. 29, 511-21. Kaluarachchi, J. J. and Morshed, J. (1995a). Critical assessment of the operator-splitting technique in solving the advection-dispersion-reaction equation: 1. First order reaction. Adv. Water Res. 18, 89-100. Kaluarachchi, J. J. and Morshed, J. (1995b). Critical assessment of the operator-splitting technique in solving the advection-dispersion-reaction equation: 2. Monod kinetics and coupled transport. Adv Water Res. 18, 101-10. Kanney, J. F., Miller, C. T., and Kelley, C. T. (2003). Convergence of iterative split-operator approaches for approximating nonlinear reactive transport problems. Adv. Water Res. 26, 247-61. Keating, E. H. and Bahr, J. (1998). Reactive transport modeling of redox geochemistry: approaches to chemical disequilibrium and reaction rate estimation at a site in northern Wisconsin. Water Resources Res. 34, 3573-84. Khan, L. A. and Liu, F. L.-F. (1995). An operator-splitting algorithm for coupled one-dimensional advection-diffusion-reaction equations. Comp. Meth. Appl. Mech. Eng. 127, 181-201. Kim, S.-B. and Coracpcioglu, M. Y. (2002). Contaminant transport in riverbank filtration in the presence of dissolved organic mater and bacteria: a kinetic approach. J. Hydrology 266, 269-83. Konikow, L. F. and Bredehoeft, J. D. (1978). Computer model of two-dimensional solute transport and dispersion in ground water. In US Geological Survey techniques of water-resources investigations. Vol. 7, Chapter C2. Krauskopf, K. B. and Bird, D. K. (1995). Introduction to geochemistry. McGraw-Hill, New York. Lichtner, F. C. (1996). Continuum formulation of multicomponent-multiphase reactive transport. In Reactive transport in porous media (eds F. C. Lichtner, C. I. Steefel, and E. H. Oelkers), Reviews in mineralogy. Vol. 34, pp. 1-81. Mineralogical Society of America, Washington, DC. Lorente, S., Frizon, F., Khitab, A., and Ollivier, J. F. (2004). Ionic contamination and decontamination of porous media. In Emerging technologies and techniques in porous media (eds D. B. Ingham, A. Bejan, E. Mamut, and I. Fop), pp. 409-17. Kluwer, Dordrecht. MATLAB (2003). (Release 13.) The MathWorks, Inc. 3 Apple Hill Drive, Natrick, MA 017602098, USA. Melin, G. (2003). Riverbank filtration—the future is now. In Proceedings of the 2nd international riverbank filtration conference (RBF2). The National Water Research Institute.

E. HOLZBECHER

339

Nield, D. A. and Bejan, A. (1999). Convection in porous media (2nd edn). Springer-Verlag, New York. Parkhurst, D. L. (1995). PHREEQC: a computer program for speciation, reaction-path, advective transport, and inverse geochemical calculations. Water Resources Investigations Report 95-4227, US Geological Survey, Lakewood, CO. Pitzer, K. S. (1995). Thermodynamics. McGraw-Hill, New York. Prommer, H. (2002). PHT3D—a reactive multicomponent transport model for saturated porous media. The University of Edinburgh, UK. PHT3D user manual. Prommer, H., Davis, G. B., and Barry, D. A. (1999). PHT3D—a three-dimensional biogeochemical transport model for modelling natural and enhanced remediation. In Proceedings of the contaminants site remediation conference (ed. C. Johnston), pp. 351-8. Fremantle, Western Australia. Rau, H. and Rau, J. (1995). Chemische Gleichgewichtsthermodynamik. Vieweg, Braunschweig. Regnier, P., O'Kane, J. P., Steefel, C. I., and Vanderborght, J. P. (2002). Modeling complex multicomponent reactive-transport systems: towards a simulation environment based on the concept of a knowledge base. Appl. Math. Model. 26, 913-27. Rubin, J. (1983). Transport of reacting solutes in porous media: relation between mathematical nature of problem formulation and chemical nature of reactions. Water Resources Res. 19,1231-52. Saaltink, M. W., Ayora, C , and Carrera, J. (1998). A mathematical formulation for reactive transport that eliminates mineral concentrations. Water Resources Res. 34, 1649-56. Saaltink, M. W., Carrera, J., and Ayora, C. (2001). On the behaviour of approaches to simulate reactive transport. J. Contaminant Hydrology 48, 213-35. Sigg, L. and Stumm, W. (1989). Aquatische Chemie. VDF-Verlag, Zurich. Steefel, C. I. and MacQuarrie, K. T. B. (1996). Approaches to modeling of reactive transport in porous media. In Reactive transport in porous media (eds P. C. Lichtner, C. I. Steefel, and E. H. Oelkers), Reviews in mineralogy. Vol. 34, pp. 83-129. Mineralogical Society of America, Washington, DC. Stuyfzand, P. J. (1998). Simple models for reactive transport of pollutants and main constituents during artificial recharge and bank filtration. In Artifical recharge of groundwater. Proceedings of the third international symposium on artificial recharge of groundwater. TISAR '98, Amsterdam, 21-25 September (eds J. H. Peters et al.\ pp. 427-34. Balkema, Rotterdam. Tebes-Stevens, C. and Valocchi, A. J. (2000). Calculation of reaction parameter sensitivity coefficients in multicomponent subsurface transport models. Adv. Water Res. 23, 591-611. Tripathi, V. S. (1986). An algorithm and a fortran program (CHEMEQUIL-2) for calculation of complex equilibria. Talanta 33, 1015-20. Valocchi, A. J. and Malmstead, M. (1992). Accuracy of the operator splitting for advectiondispersion-reaction problems. Water Resources Res. 28, 1471-6. van Cappellen, P. and Wang, Y. (1996). Cycling of iron and manganese in surface sediments: a general theory for the coupled transport and reaction of carbon, oxygen, nitrogen, sulfur, iron, and manganese. Amer. J. Set 296, 197-243. van der Lee, J. (1998). Thermodynamic and mathematical concepts of CHESS. Technical Report LHM/RD/98/39, Ecole des Mines de Paris. von Gunten, H. R., Karametaxas, G., Krahenbiihl, U., Kuslys, M., Giovanoli, R., Hoehn, E., and Keil, R. (1991). Seasonal biogeochemical cycles in riverbome groundwater. Geochim. Cosmochim. Acta 55, 3597-609.

340

REACTIVE TRANSPORT IN POROUS MEDIA

Wang, S., Jaffe, P. R., Li, G., Wang, S. W., and Rabitz, H. A. (2003). Simulating bioremediation of uranium-contaminated aquifers; uncertainty assessment of model parameters. J. Contaminant Hydrology 64, 283-307. Yeh, G. T. and Tripathi, S. V. (1989). A critical evaluation of recent developments in hydrogeochemical transport models of reactive multichemical components. Water Resources Res. 25, 93-108. Zheng, C. (1990). MT3D—a modular three-dimensional transport model for simulation of advection, dispersion, and chemical reactions of contaminants in groundwater systems. Report to the Kerr Environmental Research Laboratory, US Environmental Protection Agency, Ada, OK.

13

NUMERICAL AND ANALYTICAL ANALYSIS OF THE THERMOSOLUTAL CONVECTION IN AN ANNULAR FIELD: EFFECT OF THERMODIFFUSION R. BENNACER and A. LAKHAL LEEVAM/LEEE-IUPGC Universite Cergy-Pontoise, 5 mail Gay Lussac, Neuville sur Oise, 95031, France email: Rachid.Bennacer0iupgc.u-cergy.fr and I a k h a l ® i u t c . u - c e r g y . f r

Abstract The chapter presents an analytical and numerical study of the separation of the components of a binary mixture in a vertical circular porous annulus. The motion is driven by an externally apphed constant heat flux imposed across the vertical cyhnder while the horizontal surfaces are adiabatic. All surfaces are impermeable. In the analysis, the Soret effect was taken into consideration. The first part of the study is an analytical solution valid for the Darcy model and for flows in relatively high aspect ratio enclosures. The second part of the study is a numerical investigation of the full conservation equations that validates the analytical model. The effect of the buoyancy ratio, Lewis number and the curvature, R, on the heat and mass transfer and the separation ability is found to be significant. The analysis of optimum separation conditions as a function of the Rayleigh and Lewis numbers, and the curvature parameter is presented. The analytical model is found to be in good agreement with the numerical results obtained by solving the complete system of governing equations. The particular situation where the buoyancy forces induced by the thermal and solutal effects are opposing each other and of equal intensity (A^ = —1) is considered. For this situation a purely diffusive rest state is possible. Keywords: double diffusion, natural convection, thermosolutal, porous media, Soret effect, thermodiffusion, separation

341

342

13.1

THERMOSOLUTAL CONVECTION IN AN ANNULAR FIELD

INTRODUCTION

Double-diffusive natural convection in porous media occurs in many engineering systems and in nature. It includes the disposal of waste material, drying processes, migration of moisture contained in fibrous insulation, grain storage installations, food processing, chemical transport in packed-bed reactors, contamination transport in saturated soil, the underground disposal of nuclear wastes, etc., see for instance Alavyoon et al (1991) and Montel (1994). Most of the existing studies on this topic are concerned with the case of a vertical cavity subject to a horizontal temperature and concentration gradients, i.e. 'double-diffusive convection'. Analytical and numerical results have been reported for this configuration, see Trevisan and Bejan (1985, 1986) Masuda et al. (2002), and Bennacer et al. (2003a). The second kind of double-diffusive convection result for a cavity subject to horizontal temperature and induced concentration gradients by the 'Soret effect' was first cited by Ludwig (1856) and studied by Soret (1879). Early investigations on the role of the Soret effect in natural convection of binary fluids were primarily focused on the problem of convective instability in a horizontal layer and the separation ability. A review of such studies is given in Flatten and Legros (1984). In the last decade, renewed interest of this problem has been considered theoretically, numerically and experimentally, see for example Lorenz and Emery (1959), Estebe and Schott (1970), Lhost and Flatten (1991), Ouazani and Bois (1994), Jamet et al. (1996), Marcoux and Charrier Mojtabi (1998), and Bennacer et al. (2003b). All these studies indicate the existence of an optimum separation condition for a particular value of the permeability of the porous medium. The behaviour of a binary mixture saturating a vertical porous annular and subjected to thermogravitational diffusion has been studied for vertical cylinders maintained at different and uniform temperatures by Marcoux and Charrier Mojtabi (1999) and constant heat flux by Lakhal and Bennacer (2002). The influence of curvature in the separation process, for cooperating thermal and solutal buoyancy forces, was investigated both numerically and analytically.

13.2 MATHEMATICAL MODEL Consider afluid-saturated,porous layer enclosed by two concentric cylinders as illustrated in Figure 13.1. The height of the layer is denoted by H and the inner and outer radii by r[ and r^, respectively. All boundaries of the annular region are impermeable. Both horizontal boundaries of the enclosure are thermally well insulated while the inner and outer vertical boundaries are subjected to a constant heat flux. Gravity acts in the zdirection and the mass fraction of the denser component of the mixture, 5o, is assumed to be initially uniform. The porous matrix is assumed to be rigid and in thermal equilibrium within the fluid. Thermophysical properties are assumed to be constant. The flow is assumed to be laminar, incompressible and the Boussinesq approximation is utilized. The cross contribution 'Soret effect' is taken into account where the mass flux, j , including

343

R. BENNACER AND A. LAKHAL

Figure 13.1 Schematic diagram of the geometry and the coordinate system.

the Fick and Soret contributions, is given by: j = -pDVS'

- pDrS'oil - S'o)VT',

(13.1)

where p is thefluiddensity, T' the temperature, 5 ' the concentration, D the mass diffusivity and DT the phenomenological coefficients for the Soret coefficients. The reference length is chosen as the width of porous annulus e = r'^—r[. The following dimensionless variables (primed quantities are dimensional) are employed: ,

,

ir',z')

.

.

{u',w')e

p' Pov'^je^ '

1i^

S'-SL - - ^ ^ ,

where

DT

A 5 ' = - - ^ 5 o ( l - 5o)AT',

where u' and w' are the volume-averaged velocity components in the r- and z-directions, respectively, p' the hydrodynamic pressure, a, A and u are the effective thermal diffusivity.

344

THERMOSOLUTAL CONVECTION IN AN ANNULAR FIELD

thermal conductivity and kinematic viscosity, respectively, of the fluid mixture, AT' is a characteristic temperature difference, and A 5 ' is the corresponding value for the constituent. The density p of the mixture is related to the temperature and solute concentration by a linear equation of state, see Bennacer and Gobin (1996), namely p{T', S') = po [1 - PT{T' - Ti) - PsiS' - S'o)],

(13.2)

where PT and Ps are the thermal and solutal expansion coefficients, respectively, and the subscript 0 indicates a reference state, see Bennacer and Gobin (1996). By employing the above assumptions into the conservation equations of mass, momentum, energy and species, a set of dimensionless governing equations, see Bennacer et al. (2003b), is obtained as follows: V'V

(13.3)

= 0, V = - V P -h GrT(T + NS),

(13.4)

z^V^T, (13.5) Pr {V • V ) 5 = ^{V^S - V^T). (13.6) be The above equations show that the present problem is governed by the following dimensionless parameters. {V'V)T=

• Thermal Grashof number Gr^ = gPrQiKe'^/Xi/^ or thermal Rayleigh number Ra* = Gr^^Pr, where K is the porous media permeability and g is the magnitude of the gravitational acceleration. • Buoyancy ratio N = GrJ/Gr^. = fis^S'/Pr^T' or N = {I3S/PT){DT/D)S'Q{1 • • • •

Prandtl number Pr = u/a. Schmidt number Sc = u/D or Lewis number Le = Sc/Pr = a/D. Annulus aspect ratio A — H/e. Curvature R = e/r^, the i? = 0 case corresponds to the Cartesian case.

The dimensionless boundary conditions on the vertical walls and horizontal surfaces are as follows: — - -1 — - -1 at r - dr ^ dr i? ' dT 1 dS 1 R-\-l ,,^^.

^^--RTV

^ = -R-^

'' ^ ^ ^ - '

^''-'^

oz oz The domain limit is considered as impermeable and slip dynamic boundary conditions are applied: dujdz = 0 on the horizontal surfaces and dw/dr = 0 on the vertical cylinders.

R. BENNACER AND A. LAKHAL

345

The Nusselt and Sherwood numbers, which are of interest for engineering applications, based on the average temperature difference AT between the two vertical walls, and on the average concentration difference between the two vertical surfaces, are defined as follows: QiH 1 . ni JH 1 In the conduction regime, the Nusselt and Sherwood numbers approach the asymptotic value of the curvature and are given by:

(Nu)difF = (Sh)diff = n^^^^rry •

(13.8)

13.2.1 Numerical solution The governing equations (13.3)-(13.6) were solved using the finite volume method. The computational domain is divided into rectangular control volumes with one grid point located at the centre of each control volume that forms a basic cell. The control volume formulation used in the algorithm ensures continuity of the convective and diffusive fluxes as well as the overall momentum and energy conservation. The set of conservation equations are integrated over the control volumes, leading to a balance equation for the fluxes at the interfaces. A central difference scheme is used to discretize the equations, and the false transient procedure is used in order to obtain steady-state solutions. For faster convergence, the SIMPLER algorithm, see Patankar (1980), is coupled to the SIMPLEC algorithm of Van Doormaal and Raithby (1984); for more information about the numerical scheme see Bennacer (1993). Further, non-uniform grids are used as this allows fine grid spacing near the inner and outer vertical boundaries. These fine grids are necessary to resolve the flow in the narrow gap. Numerical tests, using various mesh sizes, were performed for given conditions in order to determine the best compromise between accuracy of the results and computer time. The difference between predictions using a 99 X 35 mesh and the reference grid 399 x 141 was less than 1.0% for Tmax and C^max on the domain. The difference between the results obtained using a 199 x 71 and the reference grid was less than 0.1%. The typical grid used in the present study was 199 X 71. The convergence criterion is based on both the maximum error in the continuity equation and the average quadratic residual over the whole domain for each equation was less than a prescribed value, say ^^, which was, in general, within machine error. Note that the computational model has been used successfully to describe accurately problems involving convection and solidification. General validation of the code was undertaken by comparison to the existing numerical results of Prasad et al. (1986) and Marcoux et al. (1999) and a comparison of the heat and mass transfer, for a range of Ra* and Le numbers, is presented in Table 13.1. We observe that good agreement is obtained for a large range of values of Ra* and Le.

346

THERMOSOLUTAL CONVECTION IN AN ANNULAR FIELD Table 13.1 Comparison of our results for different curvature (a) with those of Prasad et aL (1986) for thermal convection, Ra* = 100, Le = 1, and iV = 0, and (b) with Marcoux et al. (1999) for thermosolutal convection, Ra* = 100, Le = 10, A^ = 1, and A = 10. (a) A = --2 2 3

Curvature Prasad era/. (1986) Present work

Nu

1.5

=4 2

4.05 4.09

4.74 5.71

2.88 2.77

3.22 3.23

0

1

3

10

(b)) Curvature

Marcoux..«/ (1999) Marcoux €ffl/.U^^y; ^ , , Present work

13.3

^^ ^"^^ ^'^ ^'^^ Sh 18.11 12.75 8.58 Nu 5.43 4.10 4.87 ^^ ^^^^^ ^^.83 8.62

^'^^ 5.10 2.40 5.12

ANALYTICAL SOLUTION

Figure 13.2 illustrates the temperature and concentration fields and the stream function for the borderline cases of low and strong curvature, namely R = 0 and R = 8, respectively, and of aspect ratio of unity, A = 1, or high aspect ratio, A = 8. They illustrate clearly that in spite of the loss of symmetry in the cylindrical configuration, the establishment of flow for the aspect ratio. A , raised in the vertical direction is possible. The iso-lines of the stream function become parallel and independent of z. This situation corresponds physically to a flow mainly in the vertical direction, z, and of which the velocity is a function only of the horizontal position, r. Also, it can be observed that the formation of stratification of the fields of the temperature and concentration is evidence when A increases. For these fields, it appears that the iso-lines characteristic is duplicated, i.e. translation, in the vertical direction and the parallel flow, see Figure 13.2(c,d). Because of the relatively high values of Le considered, the concentration boundary layers are observed to be thinner than the thermal boundary layers. In this section, an analytical solution is developed for the case ofA^l and for steady state flows. Tall geometrical configurations have been considered in the past by Nguyen et al. (1986) and Hasnaoui et al. (1995) for convection induced by a single source of buoyancy and by Marcoux et al. (1999) and Bahloul et al. (2003) for convection induced by two sources of buoyancy. In all these studies it has been demonstrated that an approximate analytical model can be developed using the parallel flow approximation, which leads to the following simplifications: ^(r,z) = ^ ( r ) ,

T{r,z) = CTZ-^e{r),

S{r,z) ^ Csz + (f>{r)

(13.9)

347

R. BENNACER AND A. LAKHAL

(ii) :»•- r^^, '^^/l F/V/""T~""~ |^:;> ;:•.•;:;„

•-'

- X .

y

;if

'=M ^-->^/3

(d)

iS^ ///^

^

Figure 13.2 Iso-lines of the (i) temperature, (ii) concentration, and (iii) stream function, for (a) A = 1, i? = 0, (b)A=l,R = 8, ic)A = S,R = 0, and (d)A = 8,R = 8, and for Pr = 1, Le = 10, Ra* = 50 and iV = - 1 .

on the major part of the enclosure, where CT and Cs are unknown temperature and concentration vertical gradients (stratification) parameters, respectively. Thus the equations (13.3)-(13.5) can be reduced to the following set of ordinary differential equations: w - GT*T{e + N(t)) = -P' + Gr^(CT +

NCs),

(13.10) (13.11)

348

THERMOSOLUTAL CONVECTION IN AN ANNULAR FIELD

(13.12)

wCs = ^{Ar(l)-Are), where A^ is the radial part of the cylindrical Laplacian. Equation (13.10) may be written in the form of separate variables as follows: w-GiT{e-{-N(t))

= k,

(13.13)

where fc is a constant to be determined. Using the mass conservation and the global conservative condition, i.e. pre

/

Jn

pTe

e{r)rdr = 0,

/

J ri

0 ( r ) r d r = O,

(13.14)

the integral of equation (13.13) combined with the expression (13.14), it became clear that the constant A: is zero and the final system of equations is given by

(13.15a)

Pr be

From the system of equations (13.15a), we can obtain the final vertical velocity from the governing equation Alw-u'^w = 0, (13.15b) where uo^ = Ra^[CT + A^(Le Cs + CT)] and Ra^ = Gr^Pr. The solution of the differential equation (13.15b), for UJ^ positive, is given by: w{r) = Cihioor) 4- C2Ko(a;r),

(13.16)

where In and K^ are the Bessel functions of order n and Ci and C2 are two unknown constants. For cj^ negative, equation (13.15b) does not admit a real solution. Further, the presented solution in this work, equation (13.16), namely cu^ > 0, will exhibit some non-continuity. The dynamic boundary conditions, i.e. the slip conditions on the two vertical cylinders, yield the two unknown constants in equation (13.16): Ci = C2 =

GrT(l + N) LJT

GTTJI + N)

-Ki{uri) —h{wri)

-Ki(a;re) -li{ujre)

where the group F is given by F = Ii(c(;re)Ki(wri) — Ii(a;rj)Ki(wr-e).

349

R. BENNACER AND A. LAKHAL

The horizontal thermal and solutal profiles, solution of equations (13.11) and (13.12), result from the developed flow field function: 1 e{r) = — [FvCrwir) -

NGI*T

(13.17)

{Di Inr + D2)],

0(r) = - 1 [(Pr CT + Sc Cs) w{r) + Gr^ {Di In r + 1^2)] •

(13.18)

The constants Di and D2 are identified from the energy and species conservation over a partial volume of the enclosure as follows: D2 =

Tg In Te — rf In r^

-Di

and

Di =

CsScri.

The remaining unknowns in the obtained solution, equations (13.16)-(13.18), are the values of the constant CT and Cs- With the approximations involved in equation (13.9), it is not possible to apply the boundary conditions on the horizontal walls (in the z-direction) in order to close the obtained system of equations. However, it can be easily demonstrated, see for instance Trevisan and Bejan (1986), that the heat and species transport across a transversal section, at any value of z is constant. Then, it is sufficient to use the integral conditions of equations (13.11) and (13.12), expressed as follows:

r^

CT r^

/

w{r)0{r)r dr =-—

/

rdr,

/

w{r)(t){r)r dr =—{Cs

- CT)

Jn

^'- Jn

(13.19) rdr.

By introducing the obtained solution of equations (13.16)-( 13.18) into the integral (13.19), a system of two equations with two unknown factors CT and Cs is obtained: rlwUr,)-rM{r,)

=

G^r{r'e-rl)

Di [reWi{re)\nre - riWi{ri)\nri] Sc 2

CT

N

(13.20)

- [wo{re) - ^o(^i)]

{r'e - rl) [{Cs - CT) - Le {CT + CsSc)] ^

(13.21)

where Wn is given by equation (13.16) in terms of In and K^ (Bessel functions of order n). The equations (13.20) and (13.21), for given values of G^, N, r^, rg (or R), Pr and Sc, represent a coupled set of nonlinear equations which can be resolved by the Brown algorithm, see Engeln-Mullges and Uhlig (1996), yielding the values of CT and Cs-

350

THERMOSOLUTAL CONVECTION IN AN ANNULAR FIELD

13.4 RESULTS AND DISCUSSION Computations have been performed to determine the minimum aspect ratio above which the flow can be assumed to be parallel. In the range of the parameters considered in this investigation, it was found that the numerical results can be considered independent of the aspect ratio A when 8. For this reason most of the numerical results reported here were obtained for A = 8. The number of parameters is numerous and it is difficult to investigate the effect of all these controlling parameters. Therefore it is decided to fix certain parameters, such as the geometry of the enclosure and Prandtl number to A = 8 and Pr == 1.^ The Lewis number is fixed at a moderate value, namely Le = 10 in the major part of this work. Figure 13.3 shows the effect of the buoyancy ratio, N, and the curvature effect, i?, on the values of the Nu and Sh numbers obtained analytically and numerically. The analytical solution is represented by solid lines and they are seen to be in excellent agreement with the numerical solution of the full governing equations, depicted by the symbols. The excellent agreement is for all the parameters analysed. The non-continuity of the analytical solution is, as previously stated, due to the necessary condition on cj^ > 0 for obtaining a real solution. Such criterion on u^ = Ra^ [CT 4- N (Le Cs + CT)] induce a condition on the buoyancy ratio, N, namely

\m>

1

LeCs/Cr + l

(13.22)

All the obtained results show an increase in Nu with an increase in the curvature, i?, and two distinct regimes of Sh where we observe an increase of Sh with R in the negative N domain and decreases in the positive N domain. For positive values of N, both the solutal and thermal buoyancy forces increase the convection intensity given by the clockwise circulation flow patterns, as shown by the vertical velocity component in the horizontal mid-plane, see Figure 13.4(a). For high values of N, the solutal buoyancy force principally drives the flow. Asymptotic values of Nu and Sh are reached in the mainly solutal buoyancy forces, i.e. |A^| > 1. The strength of the flow circulation is progressively annihilated in the bulk of the cavity due to the imposition of an increasingly strong stable vertical concentration gradient and the situation tends to decrease the solutal boundary layer, resulting in a mainly diffusive temperature profile as represented in Figure 13.4(b). Figure 13.4(b) shows that the weakest difference in the temperature between the two walls is obtained for a rate of separation N = 0 and this confirms that the maximum of the Nusselt number, see Figure 13.3(a), obtained in the case of the thermal convection, N = 0. In addition, the difference in temperature increases as the value of A^ increases and this is the origin of the reduction in Nu previously stated. As A^ decreases, N < 0, the solutal and thermal effect oppose each

^The presented results are still valid for different values of Pr in the Darcy regime. Just by considering the value of Ra* instead of Gr* and Le as Sc in the obtained system.

351

R. BENNACER AND A. LAKHAL

(a)

Nu 2 i

(b)

\ 1 Analytical Numerica1 1

)'] 1

lOH

J

Sh

J

o

^

R=0 R=2 R=4 R=S

/ •

/y

.

( / . A-"

:u^ < 0;

^^^

10

10' -10

1

0

'

10

Figure 13.3 Effect of buoyancy ratio A'' and curvature on (a) Nu, and (b) Sh, for Ra* = 50, Le = 10, and A = 8.

352

THERMOSOLUTAL CONVECTION IN AN ANNULAR FIELD

-0.05

Figure 13.4 Profiles of (a) the vertical velocity component, (b) the temperature, and (c) the concentration, in the horizontal mid-plane for /^ = 8, Le = 10 and Ra* = 50.

R. BENNACER AND A. LAKHAL

353

Other and therefore induce a lower velocity and below N = —Siht flow direction became counterclockwise, see Figure 13.4(a). Figure 13.4(c) represents the concentration, 5, in the horizontal mid-plane for Ra* = 50, Le = 10, Pr = 1, and R = S. The solutal field is more sensitive than the temperature to the fluid flow due to the considered Lewis number value of 10. The obtained solutal boundary layer is thinner than that of the temperature. Figure 13.4(c) shows that the difference in the concentration between the two walls decreases as | A^| increases and this confirms the increase of the Sherwood number with |iV|, see Figure 13.3(b). It should be noted that this tendency can generate a difference in the concentration, 5, null and corresponds to a Sherwood number of infinity. This explains the exponential increase in the Sherwood number observed in Figure 13.3(b) for i? = 0 in the vicinity of N = 6. In such a situation, the Soret effect and the convective flow induce a concentration field with a significant'S' shape in the bulk cavity with no difference in the concentration on the two cylinders. Based on the non-difference concentration values in the immediate vicinity of the two vertical cylinders the domain appears to be in a homogeneous concentration as no separation occurs. Such situations are also observed in the cylindrical case, i? 7^ 0, as represented in Figure 13.5 for i? = 8, and N = 10. The figure represents the evolution of the Sherwood number with the porous Rayleigh number. The appearance of an opposite transfer (negative), i.e. Sherwood number, is noted in the vicinity of Ra* = 4.6 x 10^. As the Rayleigh number increases, the effect of the convective term on the concentration field increases. The obtained case can exceed a threshold corresponding to the zero concentration difference between the two cylinders. In such a situation, we obtain an inversion in the density gradient, and such inversion in concentration, i.e. density, 'S' shape on each horizontal level engender an unstable situation. Generally the Fick contribution opposes the Soret effect, but in the 'S' shape a particular situation is observed as represented in Figure 13.5(b). Due to the temperature field, the induced Soret mass transfer is from the outer cylinder to the inner one. The Fick law mass transfer contribution is more complicated, as it follows the concentration gradient. It remains in the two thin solutal boundary layers, from the inner cylinder towards the outer one and in the bulk domain it is from the outside concentration optimum towards the inner concentration optimum, as represented on the concentration profile in Figure 13.5(b). The Soret and Fick laws act in the same direction in the bulk and tend to reduce the unstable density situation by limiting the maximum concentration difference on a given level. The previous section clearly illustrated the ability of the analytical solution to estimate the Nusselt and Sherwood numbers, and in representing the temperature, concentration and vertical velocity component on the mid-plane. The main assumption of the analytical approach was the vertical stratification of the temperature and concentration fields. Figure 13.6(a) illustrates the iso-levels of the stream function, isotherm and isoconcentration obtained numerically for two curvatures, namely R = 0 and 8. Figure 13.6 exemplifies the effects of varying R, when i? -> 0 then the temperature field is centrosymmetric, corresponding to the case of a rectangular cavity consistent with the flow structure. On the other hand, the concentration field is characterized by thinner boundary

354

THERMOSOLUTAL CONVECTION IN AN ANNULAR FIELD

(a) 1000

500

Sh

0

y

f

Analytical Numerical 1

-500

-1000 - | 200

(b)

1

1

400

. —

1

600 Ra*

.

1

800

1

1000

0.000

0.005 H

O.OIOH

0.015 Figure 13.5 (a) The effect of Rayleigh number on the Sherwood number, and (b) the asymptotic concentration profile in the horizontal mid-plane, for Ra* = 460, N = 10 and R = S.

355

R. BENNACER AND A. LAKHAL

(a)

R=0

R=8

Cs = dS/dz Separation ability

Figure 13.6 (a) Iso-lines of (i) temperature, (ii) concentration, and (iii) stream function, and (b) the schematic separation processes.

layers with a vertical solutal stratification. The increase in curvature (R = 8) increases the non-symmetrically obtained solutions. In both situations the vertical stratification in the temperature and the concentration is clear. We focus on the solutal stratification given analytically by Cs and corresponding to the resulting efficient vertical separation ability, see Figure 13.6(b). It results from the combination of the Soret effect separation, i.e. horizontal pump, and the convective flow, i.e. vertical pump generated by gravity. The analytical value of Cs is compared to the numerical value resulting from the evaluation of the bulk concentration slope in the vertical direction. The determination of numerical value of the Cs coefficient, only valid on the domain bulk far from the horizontal surfaces in order to avoid the border effects. Figure 13.7 shows the effect of the buoyancy ratio, iV, and the curvature effect, R, on the Cs coefficient, i.e. concentration gradient, obtained analytically and numerically. It is clear that there is an excellent agreement between the numerical and the analytical results for all the parameters analysed. All the obtained results, see Figure 13.7, show a decrease of the solutal gradient in the z-direction with increasing values of the curvature, R. For positive values of N, both the solutal and thermal buoyancy forces increase the convection intensity given by the clockwise circulation and a positive concentration gradient is obtained. For moderate value of N, i.e. 0 < A/" < 5, the Cs coefficient is not significantly affected by the curvature, R.

356

THERMOSOLUTAL CONVECTION IN AN ANNULAR FIELD

0.2

J

Cs

i

0.0

-0.2

1 ^ ^ ^ *-<-^ <-..-<-i r^"°-^'---o-<:..-.e..G'']

Analytical Numerical •

-0.4

.'^ ^

-0.6 _l -10

,

A —4

i? = 8|

^

,

1

0 N

1

11 7? — n /v U n —0 /v — Z

1

1

10

Figure 13.7 Effect of the buoyancy ratio N and the curvature on the concentration stratification. In the mainly solutal buoyancy forces dominated region, i.e. |Ar| > 1, an asymptotic value of Cs coefficient is obtained corresponding to a vertical diffusion between the two obtained concentration extrema. The vertical diffusion is a consequence of the annihilated flow circulation in the bulk of the cavity. The Cs coefficient increases in the negative N domain, and this is a consequence of global flow strength decrease as |A/^| increases. The optimum separation ability is obtained in conditions of reversing flow direction from clockwise to counterclockwise, i.e. N values closer to —2. The weakest Cs value appears to be obtained for AT" = 0 and this is a consequence of the strong mixing flow ability. These flow intensities result from the fixed relative high value of the porous Rayleigh number, Ra* = 50. In the next section we discuss the effect of the Rayleigh number on the vertical separation ability. Figure 13.8 represents the variation of the concentration gradient with the porous Rayleigh number and for various values of the curvatures, JR, in the thermally-driven flow domain AT
357

R. BENNACER AND A. LAKHAL

^•"-•'7

^i

^.y

'

^-

•"'•v^

_.„-

-o

"~-o. X

,..- * ^ --"^.v 1

a'

^

0-, ^X^

0.1-

^

if

Cs

*

* • " • * • • » . . .

. . « • '

.

* • '

< .«^' ..•*'

<'••''

Analytical Numerical • •

y

G ^

^

.01-

»

'

'

'

'

1^

1^ 1

Ra*

If — C\\ K — \j\

R = l\ D

/I

A —4

i? = 8|

10

Figure 13.8 Effect of Ra* and R on Cs in the thermally-driven case N <^0. (ii) For a high Ra* number, the vertical concentration gradient is weak because of the strong flow strength inducing an important mixing ability and changing the Soret separation. (iii) For an intermediate Ra* number, the vertical concentration gradient passes through a maximum and this corresponds to a better separation ability. Such a situation corresponds to an equivalent horizontal specie pump and a vertical pump as explained in Figure 13.6. The Ra* number corresponding to the optimum separation depends on the curvature; the higher the curvature, R, then the higher is the Ra* optimum. This is due to the fact that the presence of the cylindrical geometry induces a loss of symmetry, which reduces the convection strength by reducing the globally obtained temperature difference. It is then necessary to increase the porous Rayleigh number in order to obtain the necessary flow strength (vertical pumping), allowing the optimum separation conditions to be achieved. The optimal Ra* number as a function of the curvature, R, is represented in Figure 13.9(a) and the result illustrates a relatively linear dependency over the analysed range of values. The maximum value of Cs obtained decreases with an increasing value of the curvature, R, and such an evolution is represented on Figure 13.9(b). Such a decrease with increasing R does not mean that the real dimensional physical separation ability is lower for higher curvature because of the choice of the characteristic concentration difference for defining the dimensionless form is expressed in terms of the heat flux density. The reference flux density is curvature dependent.

358

THERMOSOLUTAL CONVECTION IN AN ANNULAR FIELD

Analytical Numerical •

Figure 13.9 Effect of R on (a) Ra* ^P*^"^^^ and (b) on CJ^^^, in the thermally-driven case A^ < 0.

359

R. BENNACER AND A. LAKHAL

Figure 13.10 represents an example of the obtained concentration fields corresponding to the three Rayleigh numbers previously presented in Figure 13.8 (indicated by arrows). It illustrates the three limiting cases which represent the previously presented bell-shaped curve. The concentration fields for three values of the porous Rayleigh numbers, namely 0.5, 1.2 and 10, demonstrate how the advective term modifies the specie transfer from being diffusive, see Figure 13.10(a), to convective, see Figures 13.10(b,c). Figure 13.10(b) represents the conditions for coupling related to the convection and the thermodiffusion for which separation is optimal. For porous Rayleigh number values lower than the optimal Rayleigh number, the mainly horizontal thermodiffusion dominates, the convection being then insufficient so that the fields tends to be diffusive, as represented by vertical parallel isoconcentration lines. For porous Rayleigh number values, Ra*, higher than the optimal Rayleigh number, a strong convective contribution takes place, see Figure 13.10(c), i.e. the 'S' shape, and this induces a strong mixing and the resulting specie separation is then less. These optimum conditions, corresponding to equilibrium between the diffusion and the convection, are clearly a function of the Lewis number because they are more sensitive to the species concentrations at higher Lewis number. Figure 13.11 represents the variation of the vertical separation, Cs coefficient, as a function of the porous Rayleigh number for different values of the Lewis number, Le. We observe that the maximum separation value varies slightly with the Lewis number. It increases when the Lewis number decreases. As the specie concentration is less sensitive to the flow as the Lewis number decreases, the Rayleigh number value is higher in order to obtain the optimal condition. The evolution of the optimal Ra* as a function of the Lewis number is represented in Figure 13.12, which clearly illustrates two distinct modes according to the Lewis number. (i) For small values of the Lewis number (Le < 0.1), the optimal Rayleigh number tends to an asymptotical value and became independent of the Lewis number. This can be explained by the fact that for a small value of Le the specie became less sensitive to the flow (advective term) and from equation (13.6) we have V^5 — V^T ^ Sc( V • V ) 5 (a)

(c)

(b)

yP /

^\

-' ^,'

,^-'-'"

\ '' '' \/-^ ' 1'''.''' 1

\'^' y^y ' /

"x

/

'

X

.

.........

'[^'-"" j

^

A

;

-""

"P^ d.

/

^-<j

Figure 13.10 Concentration fields for different values of Ra*: (a) Ra* Ra* = 1.2, and (c) Ra* = 10, for AT = 0, i^ = 0.

0.5, (b)

360

THERMOSOLUTAL CONVECTION IN AN ANNULAR FIELD

0.25-

/ 0.20/ /

Cs 0.15-

/

/ /

/ / /

/

/

/

/

/ /

/

/

X

^/ \ /

/ '^ \

/

\

"^ \ \

/

\

\

\ \ \

\

i

\ \

\

/

0.10/

/

\

\

^

/ Analytical Numerical

0.05-

•

T r» — ^

a • O

T j3 1n LiQ — l U To 1 ^

'-' ,

y

Ra*

Le — 1J ,

,

,

T — I —r - 1

,

10

Figure 13.11 Effect of Ra* and Le on Cs for A^ = 0 and R = 2.

Figure 13.12

Effect of Le on Ra* °P*^"^^^ for iV = 0 and R = 2.

R. BENNACER AND A. LAKHAL

361

and then Sc -> 0 implies that V^5 — V^T -> 0. It became obvious that the solutal field tends asymptotically to be equal to the temperature field, (ii) For the values of the Lewis higher than about 0.2, the optimal Rayleigh number decreases with an increase in the Lewis number as Ra* °P*^"™^ « 1/Le. Such a result is related to the particular case of the thermally-driven flow. A particular case where the buoyancy forces are induced by the thermal and solutal effects are opposing each other and of equal intensity, namely A^ -^ - 1 , was also analysed. The obtained results are illustrated in Figure 13.13 for the normalized heat and mass transfer, Nu/Nudiff and Sh/Shdifr, as a function of the Rayleigh number, Ra^. The Nudiff is the diffusive heat transfer which is a function of curvature and given by equation (13.8). For this situation a diffusive solution prevails within the porous layer, below a critical value of Ra*. In this way the fluid remains stable, i.e. at a state of rest, up to a critical value, while for higher values of Ra* the solution bifurcates from the conductive branch (in the state of rest) to convective. An example of diffusive and convective solution is illustrated in Figure 13.14 for values of Ra* below and above the critical value. We found that the diffusive solution remains more stable with the curvature increase. This is a direct consequence of imposing a flux (temperature gradient), allowing lower temperature on the inner cylinder with the curvature increases. The lower is the temperature difference, the lower are the buoyancy forces.

Figure 13.13 Effect of Ra* on heat and mass transfer for A^ = - 1 , Le = 10, and ^ = 8.

362

THERMOSOLUTAL CONVECTION IN AN ANNULAR FIELD

Figure 13.14 Iso-lines of the (i) temperature, (ii) concentration, and (iii) stream function, for (a) RaJ^ = 5, and (b) Ra^ = 50, and for i^ = 8, Le = 10, and N = - 1 .

13.5

CONCLUSIONS

In this chapter we have analysed both analytically and numerically the contribution of the Soret effect on the thermogravitation convection both in a vertical annular porous field which is subjected to a radial constant heat flux. The influence of the thermal Rayleigh number, Ra*, buoyancy ratio, N, and Lewis number, Le, on the strength of the convection, Nusselt number, Nu, Sherwood number, Sh, and the vertical separation ability, Cs, are predicted and discussed. The main conclusions of the present analysis are as follows. (i) The numerical solution indicates that for high aspect ratio enclosures (A ^ 1), the flow is parallel in the central part of the domain while the temperature and concentration fields are linearly stratified. (ii) The developed analytical solution is based on the assumption of a parallel flow in the central area of the cavity. This has led to results which agree well with those obtained numerically when the aspect ratio of the porous layer is sufficiently high. (iii) Increase in the buoyancy ratio, N, induces a reduction in the Nusselt number, Nu, for the various analysed curvature values, R. On the other hand, the mass transfer passes through a maximum in the case of the cylindrical cavity but in the case of the

R. BENNACER AND A. LAKHAL

363

Cartesian cavity there is an inversion in the rate of separation of the species for N ranging between 5 and 6. (iv) The use of a flux boundary condition, and this, coupled with the use of an annular domain, strongly affects the separation ability, (v) The Ra* values corresponding to the optimal separation condition linearly increase as the curvature increases and decreases linearly with the Lewis number as Ra* ^P*^"^^^ ^ 1/Le. (vi) The particular case of iV = — 1 shows the existence of a critical value of Ra* above which a transition from a diffusive to a convective solution occurs. The critical value of Ra* depends more strongly with the curvature. In order to reduce the strong convective mixing, the previous studies in rectangular domains have explored the possibility of inclining the domain in order to reduce the buoyancy forces. Such a strategy is more complicated in annular geometries due to the fact that the problem becomes three-dimensional. An alternative strategy is the possible use of a horizontal partitioning in order to increase the separation by multiplying the temperature gradient, see for instance Bennacer and Mohamad (2004).

ACKNOWLEDGEMENT The authors gratefully acknowledge the fruitful discussions with Professor A. A. Mohamad.

REFERENCES Alavyoon, F , Eklund, A., Bark, F. H., and Simonsson, D. (1991). Theoretical and experimental studies of free convection and stratification of electrolyte in a lead-acid cell during recharge. I. Electrochim. Acta 14, 2153-64. Bahloul, A., Boutana, N., and Vasseur, P. (2003). Double-diffusive and Soret induced convection in a shallow horizontal porous layer. / Fluid Mech. 28, 325-52. Bennacer, R. (1993). Convection naturelle thermosolutale: simulation numerique des transferts et des structures d'ecoulement. These, Universite Paris 6, France. Bennacer, R. and Gobin, D. (1996). Cooperating thermosolutal natural convection in enclosures. I. Scale analysis and mass transfer. Int. J. Heat Mass Transfer 39, 2671-81. Bennacer, R. and Mohamad, A. A. (2004). Thermodiffusion in porous media: multi-domain constitutant separation. Int. J. Heat Fluid Flow. In press. Bennacer, R., Beji, H., and Mohamad, A. A. (2003a). Double-diffusive natural convection in a vertical multilayer saturated porous media. Int. J. Therm. Sci. 42, 141-51. Bennacer, R., Mahidjiba, A., Vasseur, P., Beji, H., and Duval, R. (2003b). The Soret effect on convection in a horizontal porous domain under cross temperature and concentration gradients. Int. J. Numer. Meth. Heat Fluid Flow 13, 199-215.

364

THERMOSOLUTAL CONVECTION IN AN ANNULAR FIELD

Engeln-Mullges, G. and Uhlig, F. (1996). Numerical algorithms with Fortran. Springer-Verlag, Beriin. Estebe, J. and Schott, J. (1970). Concentration de solutions salines et cristallisation dans un milieu poreux, par effet thermogravitationnel. CRAS Paris, Serie D 111, 805. Hasnaoui, H., Vasseur, P., Bilgen, E., and Robillard, L. (1995). Analytical and numerical study of natural convection heat transfer in a vertical porous annulus. Chem. Eng. Comm. 131, 141-59. Jamet, Ph., Farguer, D., and Costeseque, P. (1996). Determination of the effective transport coefficients for the separation of binary mixtures of organic compounds into packed thermal diffusion columns. Chem. Eng. Sci. 51, 4463-75. Lakhal, A. and Bennacer, R. (2002). Effet Soret sur le modele de Brinkman pour la convection thermosolutal dans une couche verticale annulaire poreuse. In Fourth international conference on applied mathematics and engineering sciences, Casablanca, Morocco, October, pp. 333-8. Lhost, O. and Flatten, J. K. (1991). Free convection in binary fluids: sensitivity to lateral boundary conditions. Phys. Rev. A 44, 3756-68. Lorenz, M. and Emery, A. H. (1959). The packed thermal diffusion column. Chem. Eng. Sci. 11, 16-23. Ludwig, C. (1856). Sitz. Ber. Akad. Wiss. Wien Math. -Naturw. Kl 20,539. Marcoux, M. and Charrier Mojtabi, M. C. (1998). Etude parametrique de la thermogravitation en milieux poreux. CRAS Paris, Serie lib 326, 539-46. Marcoux, M. and Charrier Mojtabi, M. C. (1999). Diffusion thermogravitationnelle dans un melange binaire saturant un espace annulaire poreux vertical. Entropie 218, 8-12. Marcoux, M., Charrier Mojtabi, M. C , and Azaiez, M. (1999). Double-diffusive convection in an annular vertical porous layer. Int. J. Heat Mass Transfer 42, 2313-25. Masuda, Y., Yoneya, M., Ikeshoji, T., Kimura, S., Alavyoon, F., Tsukada, T., and Hozawa, M. (2002). Oscillatory double-diffusive convection in a porous enclosure due to opposing heat and mass fluxes on the vertical walls. Int. J. Heat Mass Transfer 45, 1365-9. Montel, F. (1994). Importance de la thermodiffusion en exploration et prodution petrolieres. Entropie 184/185, 86-93. Nguyen, T. H., Nguyen, D. L., and Lavoie, S. (1986). Convection naturelle dans un espace annulaire poreux soumis a un flux de chaleur constant. Rapport technique EP 86-1, Ecole Polytechnique de Montreal. Ouazani, M. N. and Bois, P. A. (1994). Convective instability of a fluid mixture in a porous medium with time-dependent temperature gradient. Eur. J. Mech. B-Fluids 13, 275-8. Patankar, S. V. (1980). Numerical heat transfer and fluid flow. Hemisphere, New York. Flatten, J. K. and Legros, J. C. (1984). Convection in liquids. Springer, Berlin. Prasad, V., Kulacki, F. A., and Kulkami, A. V. (1986). Free convection in a vertical porous annulus with constant heat flux on the inner wall: experimental results. Int. J. Heat Mass Transfer 29, 713-23. Soret, Ch. (1879). Etat d'equilibre des dissolutions dont deux parties sont portees a des temperatures differentes (On the equilibrium state of solutions with two parts kept at different temperatures). Archives des Sciences Physiques Naturelles Geneve iM, 48-61. Trevisan, O. V. and Bejan, A. (1985). Natural convection with combined heat and mass transfer buoyancy effects in a porous medium. Int. J. Heat Mass Transfer 28, 1597-611.

R. BENNACER AND A. LAKHAL

365

Trevisan, O. V. and Bejan, A. (1986). Mass and heat transfer by natural convection in a slot filled with porous medium. Int. J. Heat Mass Transfer 29, 403-15. Van Doormaal, J. P. and Raithby, G. D. (1984). Enhancement of the simple method for predicting incompressible fluid flows. Numer. Heat Transfer, Part A 7, 147-63.

1 4 PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA X.R PENG and H.L.WU Laboratory of Phasechange and Interfacial Transport Phenomena, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China email: pxf~dtefimail.tsinghua.edu.cn and hailing.wufigmail.com

Abstract The conjugated transport phenomena with pore and matrix strucnire widely exist in both the natural world and practical applications. It is of critical significance to understand these phenomena accounting for the dynamical processes and structure deformation taking place in the inner pores, as one of the mo.st important topics in the area of porous media. In this chapter, a series of different experimental observations and associated theoretical investigations have been conducted to understand the transport phenomena at the pore-scale level, including the transport phenomena with/without phase change and chemical reaction, and concerning a wide range of practical applications. Keywords: pore scale, conjugated transport phenomena, boiling, bubble dynamics, two-phase flow, micro-CT, bead-packed structure, sludge, bio-tissues

14.1

INTRODUCTION

Transport phenomena in porous media commonly exist in the natural world and in various engineering applications. Classical porous media transport theory is based on a continuum approach, applying a statistical volume averaging method with the concept of 'representative elementary volume'. The associated formulations are primarily used to predict integral macroscopic characteristics of porous media, see Lin (1995), Ingham and Pop (1998) and Boer (2000). However, as is well known, the permeability and transport in porous media is in close relation with the detailed pore geometries, such as pore scale, shape, connectivity and tortuousness, see Dullien (1992) and Sahimi (1995), and can rarely be derived directly from macroscopic parameters such as porosity. The state-ofthe-art in diverse scientific and technologic fields has endowed a more abundant meaning 366

X. F. PENG AND H. L. WU

367

and implied numerous motivations for research, with a variety of applications such as compact heat exchangers, composite materials, drying processes, and environmental and biochemical applications. Although porous media cover different scales, modeling should be initiated at thefirstscale of interest, typically the pore-scale, as a basis for successively upscaling to larger scales. Recently there has been an explosion of interest in pore-scale modeling to meet the urgent demand of fundamental insight into transport mechanisms in porous media, see Blunt (2001). In many heat transfer applications, microstructure design of heat transfer enhancement is one of the key subjects in compact heat exchangers and micro fluid assemblies. The understanding of heat and mass transfer mechanism is sought to construct a tailored flow structure for optimal heat transfer. In composite casting processes, the macro physical and mechanical characteristics are controlled by the reinforcement-matrix microstructure and the morphology of the solidification interface evolution, see Michaud and Mortensen (2001). In the petroleum industry and environmental applications, the porescale displacement mechanisms for both imbibition and drainage are of great importance for predicting the percolation of oil and controlling underground pollutant diffusion. The knowledge of pore-scale transport phenomena is actually the foundation for all involved multi-physical and/or multi-scale coupling processes. Consequently, investigations on transport in porous media are no longer limited to the prediction of macroscopic parameters (permeability, porosity, etc.) of single phase or simple two-phase flow. Instead it is recognized as a comprehensive platform to adequately explore a far-ranging scope of phenomena. More and more attention is being paid to representative microstructures of the porous volume, and the pore-scale transport behavior in porous media, including capillarity, phase change,fluid-structureinteraction, activated processes, etc. As far as research work is concerned, understanding of pore-scale transport is divided into two aspects. Firstly, advanced microscopic techniques are necessary to measure and analyze the integrated three-dimensional information of pore-scale microstructures. Secondly, instead of macroscopic and simple pore geometries, predictive micro-modeling is used to develop the essence of the heat and mass transfer in complex pores in order to better understand the underlying transport mechanism in porous media, and to predict macroscopic behavior of various complicated porous media. Of course, this investigation would be highly dependent upon experimental technology, observations and measurements. In this chapter a series of different experimental observations and associated theoretical investigations are presented on the pore-scale transport phenomena concerning a wide range of practical applications with/without phase change and chemical reaction.

14.2 CONJUGATED TRANSPORT PHENOMENA WITH PORE STRUCTURE In this chapter we consider transport phenomena in porous media both with and without phase change. Chemical and biochemical reactions are usually conjugated with a pore structure variation, such as reduction of SO2 and NOx emissions. In these processes, the associated transport phenomena are highly dependent upon the pore structure. Inversely,

368

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDU

the transport processes easily alter the pore structure and further influence the transport characteristics. In this section, the typical experimental evidence is presented with a detail discussion of the conjugated transport phenomena with pore structure. 14.2J

Conjugated phenomena in sludge drying

As a typical example, the drying of sludge cakes has been investigated. The test facility, shown in Figure 14.1, consisted of an air duct, and data acquisition and image acquisition systems, with a steady air flow supplied by a fan, and a heating rate of the oncoming air controlled by an electrical heater. One thermocouple was arranged in the duct to measure the air temperature and another in the middle of the sludge cake tested to measure the interior temperature. Both the relative humidity and velocity of the air flow were measured by a humidity meter and an anemometer. The test samples were mechanically dewatered sludge cakes, with moisture content of 80-82%. Each was placed in an open container (80 mm in diameter and 5-13 mm in height), and was set on an electronic balance. Therefore the real-time weight of the sample was monitored. A CCD camera (WAT-505EX) connected to a personal computer recorded the appearance of sludge cakes for further processing by an image analyzer. The moisture distributions of the tested sludge cakes were continuously monitored during whole test by weighing and drying small pieces of samples collected from the dried cake. As a comparison, the sand cake samples were also performed under similar test conditions. The test conditions and results are illustrated in Table 14.1 and Figure 14.2, see Chen et al. (2002). Figure 14.2 presents consecutive photographs of the cracks growing on the cake surface for cases a-2 and a-3. In the first 15 minutes, the sludge cake showed no noticeable change in its appearance, except for many initial defects or small seams randomly emerging on the surface. From 15 to 60 minutes, some of these small seams gradually grew up and

CCD

Temperature and i moisture sensor |

Anemometer

3^ t

213.5g

P

Electronic balance

Figure 14.1 Experimental set-up.

369

X. F. PENG AND H. L. WU

Table 14.1

Experimental conditions and results of sludge.

CQ

Relative humidity (RH%)

Initial thickness (nmi)

2.3 2.2 2.9 3-0 3.0

30.0 40.0 41.0 41.0 40.5

26% 20% 21% 26% 14%

8 8 8 5 5

2.1 2.1 2.0 2.1 2.2

40.2 39.9 39.6 42.6 40.9

28% 23% 21% 28% 29%

<0.1 0.154^0.2 0.2--0.25 0.28<-0.35 0.355'N^0.45

Wind Speed (m/s)

Wind temperature

Sludge

a-1 a-2 a-3 a-4 a-5

Sand

S-1 S-2 S-3 S-4 S-5

Cake type

Case no.

Bound water content (kg/kg-DS*)

Constant drying rate (g/m^^ min)

2.4 2.1 1.8 1.8 1.8

26^0 32.0 35.0 33.0 34.0

0.025 0.025 0.025 0.03 0.025

23.5 24.0 26.0 24.0 22.5

tc (min) 80^ 85 95 80 80 145 145 140 170 160

*DS = dry sand

Figure 14.2 The cracks growing during the first hour for (a) the case a-2, and (b) the case a-3, at (i) t = 15 min, (ii) t = 30 min, (iii) t = 45 min, and (iv) t = 60 min.

became cracks, and extended to meet others or the edge. Finally these cracks divided the cake into several smaller piles. In the following process, each part experienced individual drying, accompanied with further shrinkage. It is interesting to note that not all small seams become cracks. Some of them were initially preponderant in the drying and expanded faster than adjacent ones. Competition mechanism of the crack growth was highly depended upon the inner pore structure and moisture content. The cracks grew more rapid where there were favorable pore structures for water evaporation. The evaporation and the crack growth were mutually stimulated. Consequently, the preponderant seams or larger pores expanded as the moisture evaporated and escaped, resulting in much easier growth and the development of cracks. In contrast, the smaller pores in the adjacent zone were suppressed and remained almost unchanged, see the circled ones in Figure 14.2.

370

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA

Zhang and Peng (2004) have investigated the inner pore structure behavior with moisture transport in a cake using a micro-CT, see Figure 14.3. They found that the pore structure of a dried cake was much altered after it had fully absorbed the water when compared with the original one. Many small inner pores were destroyed, while a few of the larger pores were extended. The pore structure of the original cake was more uniform than that of the dried cake. After drying, the solid matrix was denser and the pores became larger. Figure 14.4 depicts the drying rates, averaged over a 4 minute period, of the sludge and sand cakes. For sand cakes, although there is a large data scatter, these curves resemble a conventional drying curve composed of a constant-rate period and a reducing-rate period, and the drying rates for the constant-rate period ranged from 26-36 g/m^ min. For sludge cakes, the humidity played a more crucial role, and the constant-rate period could be regarded as being evaporated at saturated condition (relative humidity, RH = 100%). Compared with sand, the drying curvefluctuated,and the transition from a constant-rate to a reducing-rate period cannot be clearly identified. Both Figure 14.4 and Table 14.1 indicate a much higher drying rate in the constant-rate period than that of sand cakes under similar conditions. Vaxelaire et al. (2000) derived a parameter, namely the drying potential or DP* = 0.758w^'^DP, to evaluate the external drying conditions for a uniform media, and DP is the difference between the chemical potential of the saturated vapor at the wet bulb temperature (temperature at the surface of the wet material) and the vapor temperature at the dry bulb. The drying theory indicates that the constant drying rate is controlled by external drying conditions. However, the drying rate of sludge was larger than that of sand by 20'^40%, see Figure 14.5. This may be caused by the incidental cracking and associated changes of the inner structure in sludge cakes. Accounting for the extra evaporation from cracks and other important effects, Chen et al (2002) explained the phenomena and modified the correlation for calculating the drying rate. The cracks on the surface increased the drying area and the change in the pore structure near the cracks significantly affected the escape of moisture. Figure 14.5 illustrates their results for the drying rates of sand cakes, sludge cakes and sludge cakes not

(b)

Figure 14.3 Images of the inner pore structure of a cake: (a) the original cake, and (b) a dried cake after fully absorbing water.

371

X. F. PENG AND H. L. WU (a) 45 40 ^

30 —o— a-2 —A— a-3 —V— a-4 - 0 - a-5

"B) 25

I 20 £

15

Q

10 5 0 50

100

150

200

250

Time (min) (b)

30 H

^

25

t

20

Sand 1 (<0.1mm) Sand 2 (0.154-0.2 mm) Sand 3 (0.2 ~ 0.25 mm) Sand 4 (0.28 ~ 0.35 mm) Sand 5 (0.355 ~ 0.45 mm)

S 15 S. 10

40

80

120

160

200

Time (min)

Figure 14.4 The drying rate of (a) sludge cakes, and (b) sand cakes.

including cracks under the same drying potential. The drying rates of sludge cakes, not including cracks, are almost equal to those of the sand cakes. The drying of sludge cakes in the constant-rate region was still controlled by external conditions. The discrepancy in the drying rates, both with and without consideration of cracks was highly dependent upon the drying capability of the cracks on the cake surface. 14.2,2 Effect of inner evaporation on tlie pore structure Conjugated transport phenomenon clearly exists in the loose porous media, and also in relatively dense or non-deformable matrix media. Hydration and calcination is a technology to reform appropriate pore structure inside the sorbent due to explosive evaporation and

372

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA 36 1 .•a-3

34 C\J

a-4

§ 32 H

a-2

-•-•*

5 30 a*-3

Sand cakes Sludge cakes Sludge cakes not including cracks

Ic 26H

o ^

24 J S-1 280

300

320

340

360

380

DP* (kJ/Kg)

Figure 14.5 Constant drying rates as a function of the drying potential. rapid escape of the water contained in or reactively released from hydrated CaO, and consequently improve the desulfurization performance, as one of its important applications, see Fu and Peng (2004). The origin material used in this study was a kind of pure Ca(0H)2 and was calcined in a furnace at a temperature of 700 °C for 15 minutes. CaO rapidly reacted with H2O to form Ca(0H)2 again in a sample container at a constant temperature. Five slurry samples (mol ratio of H2O to CaO) were selected, namely 1.40, 2.08, 2.77, 3.60 and 6.86, to produce hydrated samples. The slurry samples were directly calcined at 560 ""C, 760 °C and 950 ''C, and were intensely decomposed, and the contained water was evaporated and quickly escaped. This powerful process made a significant inner pore reconstruction in CaO particles. Figures 14.6 and 14.7 illustrate the measured cumulative area and volume of the samples, respectively. The pore cumulative area performed a clear decreasing trend as the calcining

40 r

Mol ratio of H2O to CaO

Figure 14.6 Cumulative area of samples.

373

X. F. PENG AND H. L. WU

^ 560°C "*" 760°C -^ 950°C -K-

Ca(0H)2

Mol ratio of HgO to CaO

Figure 14.7

Cumulative volume of samples.

Origin 560°C/1.40 560°C/2.08

560°C/2.77 560°C/3.60 560°C/6.86

100

1000

Pore diameter (nm)

Origin

950°C/1.40 950°C/2.08 950°C/2.77 950°C/3.60 950°C/6.86

100 Pore diameter (nm)

Figure 14.8

Pore size distribution calcined at (a) 560 "^ C, and (b) 950 ° C.

374

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA

temperature was increased. CaO sample calcined at 950 °C obtained a much lower pore area than the origin one, while at a relatively lower calcinging temperature effectively enhanced the sample pore area. The mol ratio of H2O to CaO had a strong effect on the cumulative volume and the pore volume was significantly increased when the slurry samples were made with a weight mol ratio greater than about 2.77. Meanwhile, the CaO sample, calcined at 950 °C and hydrated at ratio of 2.08, exhibited a possible negative effect on the pore volume. The pore distributions in the form of an incremental volume as a function of the pore diameter are presented in Figure 14.8. Those samples had three significant characteristics. First, the intensive heat and mass transfer made a considerable pore volume constriction. The dominant pore size region tended to be larger as the calcining temperature increased. Second, the small pores, diameter between 10 and 50 nm, decreased as the calcining temperature and mol ratio increased. In Figure 14.8(b) the pores of 10 to 40 nm for the original sample almost disappeared after the treatment. Third, large pores, 500 to 1000 nm, were considerably decreased. As the weight mol ratio reached 2.77, these phenomena became quite significant. Apparently, the intensive and rapid quick-evaporation and capillary effects are considered to be the important factors for the reconstruction of the micro pore configuration. When the water contained was evaporated quickly, the rapid decompounding of Ca(0H)2 also produced steam. The explosive release of steam induced an intense expansion of the micro pores. Meanwhile, the reduction of the liquid in the pores also produced a strong capillary effect, which inevitably caused the collapse of some of the larger pores. 14,3 TRANSPORT-REACTION PHENOMENA 14.3.1 Reaction in a porous solid In solid-gas transport-reaction systems, gas reactant diffuses into the solid pore holes and simultaneously reacts with the solid reactant. Generally, the reaction performs in an interfacial zone where the solid reactant and solid product co-exist. These transportreactions are expected to be associated with the pore structure. Figure 14.9 illustrates a series of SEM photos of the CaO particle reacting with SO2 at different times, see Yan (2003). The change of porous structure greatly influenced the gas diffusion and absorptive reaction, see Ishida and Wen (1968), and Wen and Ishida (1973). For a chemical reaction between a gas reactant (A) and a solid reactant (B), aA(g) + 6B(s) = cC(g) -f dD(s), Yan et al. (2003) presented a modified dual-zone model, assuming that the reaction at the second stage occurred in all the inner pores rather than reacted from the surface into the inner region, see Figure 14.10. As a result, the second-stage reaction is dependent upon the inner pore distribution. Solving the mass transfer equation of gas reactant, the dimensionless reaction time of the surface product layer formation, and corresponding conversion rate, were obtained as

375

X. F.PENG AND H.L.WU (b)

Figure 14.9 SEM photos of CaO absorbing SO2 at (a) i ^ 0 min (before reaction), (b) t = 10 min, (c) t = 30 min, and (d) t = 60 min.

Solid reactant

Reaction product

Figure 14.10 Modified dual zone reaction model: (a)firststage, formation of surface product, and (b) second stage, reaction in pores.

follows: 9c = 1 +

Th[cosh(Th)/sinh(Th)]-l Sh

and

X=

sinh(Th)

K-

<"•"

where the Thiele number Th = rQ{akyCBo/bDey^^ and the Sherwood number Sh = aoro/De. The Thiele number was explained to be the ratio of the chemical reaction to the diffusion rate. Crystallinely, the transport-reaction is controlled by the reaction resistance for Th < 1, while it is dependent on the diffusion when Th > 1. For the cases Th ~ 1,

376

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDU

both the chemical reaction and diffusion are important. This demonstrates that the pore structure plays a significant role in the transfer-reaction processes. During the second stage, the mass balance is given, see Yan (2003) and Yan et al. (2004), as follows: dl

{DyJ^)=2C^ dl J

1

\n{ry;/rpt)

A/7*-III

Ue

-1 — f^s^A )

(14.2)

where kg is the effective reaction constant, r^ and Vpt are the radii of the reacting surface and pore surface at time t, respectively, and Ds is the gas diffusive coefficient in the product layer. The rate of change in the pore radius is derived as follows: dr.p

It

_

ksCAd 2ar„

(14.3)

dpB J '

PD

where PB and po are the density of the solid product and reactant, respectively. Accounting for the pore structure, the local reaction conversion rate, or the ratio of the reacted and original solid reactant for any pore r^o in solid particle(s), is defined and derived as follows: ^i^pO^t) =

7r{rl-rlQ)MipQ

^0

a{rpo)

\rpoJ (14.4)

Accordingly, the total conversion rate is given by

;)

(14.5)

where the pore distribution parameter is defined as follows: a(rpo)

_ Virpo) _

r\pO AZ

Vtotal

Erpo^pO^^

(14.6)

Apparently, the final conversion rate and formation time of the product layer is mainly determined by the initial porosity. The pore structure, including the average pore radius and distribution, significantly affects the dynamic characteristics of the transfer and reaction processes. 14.3.2 Experimental investigation In the experiments conducted, the reaction of SO2 with solid CaO particles was investigated using a thermogravity analyzer. The pore distributions of the samples were analyzed using a mercury porosimeter. The samples investigated have average diameters of about 4 jum and cumulative special pore volume is 0.40cm^/g (sample 1) and 0.39cm^/g (sample

377

X. F. PENG AND H. L. WU

2). The reacting temperature was 800 °C and the reaction period was 80 minutes. The concentration of SO2 in the flue gas was 2000 ppm in the facility shown in Figure 14.11. Figures 14.12(a,b) illustrate the distribution of the pore characteristic parameters with the pore diameter, both having a very close final pore volume and quite different pore fraction Balance \ A/o

Flue gas

i

Weight signal Baffles

Furnace Computer

-.+ Temperature signal

"I

Figure 14.11 Diagram of the TGA system.

(a)

0.40

Cumulative special pore volume Pore fraction

10 100 Pore diameter (nm) (b)

0.40 r

Cumulative special pore volume Pore fraction

10

100

1000

Pore diameter (nm)

Figure 14.12

Characteristic pore distribution for (a) sample 1, and (b) sample 2.

378

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA

— Sample 1 - - Sample 2

20 30 40 Reaction time (min)

Figure 14.13

(a) .9 O

50

60

Desulfurization reaction conversion rate.

0.25


0.20 h

»^ c 0.15 o o c w .•g 2 55 > 0.10 h

... -—

11

t t t t

= = = =

5 min 15min 30 min 60 min

t t t t

= = = =

5 min 15 min 30 min 60 min

0.05 0.00

100 Pore diameter (nm)

1000

— — — —

100 Pore diameter (nm)

Figure 14.14

1000

Distribution of the conversion rate for (a) sample 1, and (b) sample 2.

X. F.PENG AND H.L.WU

379

distribution. Thefinalconversion rate was almost the same, while their transient dynamic conversion rates were quite different, see Figure 14.13. Clearly the final conversion was determined by the initial porosity and the dynamic reacting process was highly dependent on the pore distribution. Apparently, the coupled phenomena of the pore structure or distribution with the transport played very important roles in the reaction. Here a distributed conversion parameter is introduced to describe the pore influence on the reaction, i.e.

-fe)Vfe"<-'hfe)1)-

(14.7) This parameter means the ratio of the reacted solid reactant for a pore having an initial radius Vpo and the total reacted reactant at time t. Figures 14.14(a,b) present the distributed conversion parameters at different reaction times for the samples 1 and 2, respectively. Clearly, different pores have quite different reaction behaviors. The smaller pores exert dominant reacting areas at the early stages, while the larger pores make a greater contribution later. Yan (2003) found that this behavior greatly changed the gas transport in the pores due to the variation of the pore structure and distribution during the reaction process, and consequently the understanding of the reaction and transport phenomena at the pore scale would be very useful to explore the characteristics of the high-temperature desulfurization of thefluegases and improve the performance of the SO2 and NOx absorbents.

14.4 BOILING AND INTERFACIAL TRANSPORT The contribution of interfacial effects on boiling heat transfer is extremely distinct in porous media, where unique phenomena may arise because the bubble interface is significantly affected by the porous structure, see Peng et al. (2002). Wang et al. (2002, 2004) and Wang and Peng (2004) have conducted a series of investigations for the boiling and interfacial phenomena in bead-packed structures. 14.4.1 Experimental observations For water at ambient pressure and 18.4 °C, consecutive snapshots of a typical bubble growth-collapse cycle are presented in Figure 14.15 at 36.8 kW/m^. The bubble release frequency was very low due to high subcooling. Bubble sites in the narrow-gap comer (narrow region between a bead and surface) were more plentiful than in other regions. Small bubbles from the comers expanded and penetrated into the main cavities formed by neighboring beads, and merged to a primary bubble, see Figures 14.15(b,c). As the primary bubble grew, the pore size became the restriction for bubble growth, and the bubble distorted and elongated, see Figures 14.15(d,e). Subsequently, the bubble was truncated at the neck of the elongated bubble and quickly escaped, see Figure 14.15(f). Then, the pore space was re-occupied by the replenished liquid due to capillary effects,

380 (a)

PORESCALE TRANSPORT PHENOMENA IN POROUS MEDIA (b)

(c)

(d)

Figure 14.15 Bubble behavior in a 7 mm bead-packed structure at g" = 36.8 kW/m^ (water) at (a) e = Os, (b) t = 0.15 s, (c) t = 0.36s, (d) t = 0.82 s, (e) t = 1.12s, (f) t = 3.41 s, (g) t - 3.48 s, and (h) t = 3.72 s.

see Figure 14.15(g), and another bubble cycle followed, see Figure 14.15(h). The growth of a fully-developed bubble was mainly controlled by the evaporative heat transfer on the liquid meniscus in the narrow comer. The replenishing liquid was pulled into the comer by the capillary pressure and it maintained an excellent wetting performance. The boiling heat transfer was enhanced with this process. As the heat flux was increased, approaching to the CHF, the pore-scale phase-change behavior was highly disordered, see Figure 14.16 and Wang et aL (2002). The wall temperature was highly unsteady, and sometimes as high as 200 °C. However, dryout of the entire heated surface could not be observed, even at very high heat fluxes, and dry spots on the heated surface were observed to be rather unsteady because of the intensive *evaporation-replenishment' in the narrow-gap comers. Figure 14.16 illustrates that the bubble interface expanded horizontally. In particular, the pore space near the heated surface was vapor-filled except for the liquid menisci in the narrow-gap comers, and the vapor front on the heated surface tended to expand into the comer. The variation of the bubble interface would result in a capillary pressure gradient for the replenishing-liquid flow, and dry areas were then wetted. Clearly, the evaporation-replenishment became more intensive with increasing heat flux, and the narrow-gap comers were highly resistant

Figure 14.16 Bubble behavior in a 7 mm bead-packed structure at g" = 121.3 kW/m^ (water) at (a) < = 0 s, (b) t = 0.04 s, and (c) t = 0.12 s.

381

X. F. PENG AND H. L. WU

to dryout even at very high heat fluxes. The pore-scale bubbles on a wire in porous structures also exhibit some unique characteristics, see Figure 14.17 and Wang and Peng (2004). 14.4.2 Static description of primary bubble interface The interface of a fully-developed static bubble is divided into three parts: top, concave and base sections, see Figure 14.18 and Wang et aL (2004). The top section interface is a

Figure 14.17 Bubble behavior on a wire in a bead-packed structure (water, bead diameter 4 mm).

A-A

Bubble interface Bubble

Figure 14.18 Bubble interface model.

382

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA

spherical coronal, the base section interface is approximately a part of a hemisphere, and the concave section interface is assumed to touch the beads directly. The bubble profile can be described by several configuration parameters as follows: i?i = (i?2 -f Rp) csc(7r - 62) -Rp, J?3 = (sec^i - l)Rp ,

R2 = i?p(sec30° - 1), ] R2^Rp tan01 = Rp

(14.8)

These four relations indicate that the bubble shape is determined by only one parameter, 62, if the particle diameter is specified. For a given fluid, the departing bubble shape can be obtained through a force balance. Neglecting inertial effects, interfacial tension tends to hold the bubble, while the buoyancy acts to draw the bubble away, see Carey (1992). The departing bubble profile corresponding to the departure angle 62d is determined by the following equation, see Wang et al (2004): 27rR2a

Ru ^

'

^

^^

where Vd and Khade are the volumes of the departing and shaded part of the bubble, respectively. The predictions were found to be in very good agreement with several experimental cases. 14.4.3 Replenishment and dynamic behavior of the interface For boiling in the bead-packed structure, a downward replenishing-liquid flow is driven by the capillary pressure gradient. Peng et al, (20(X)) indicated that the liquid flow around the bubble is induced virtually by the evaporation and condensation at the bubble interface. The replenishment provides the necessary liquid supply required by the evaporation at the bubble interface in the narrow-gap comer area. With a view of the pores being a system of parallel, equilaterally-staggered capillary ducts, a preliminary analysis was conducted to determine the driving forces for the replenishing-liquid flow using Darcy*s approximation, see Figure 14.19. The following equation was obtained, see Wang et al. (2004):

where /3 is the vapor fraction in the pore, J ' the Leverett function, see Leverett (1941), and correlated by Udell (1983). Equation (14.10) can be integrated numerically to obtain the vapor fraction profile in the pore, or a dynamic description of the bubble interface. 14.4.4 Interfacial heat and mass transfer at pore level At high heatfluxes,the bubble interface profile, especially the interface in the comer, varies with the applied heat flux. To balance the replenishment flux and the evaporation flux,

X. F. PENG AND H. L. WU

•

383

Liquid phase

00 Solid particle •

Vapor phase Heated wall

Narrow-gap corner area

Figure 14.19 Replenishment and liquid-vapor interface.

an appropriate capillary pressure gradient is established for the flow by the evaporative heat transfer. The vapor fraction increases continuously from the top of the bubble to the heated surface; and correspondingly, a negative capillary pressure gradient, driving a downward liquid flow. As Ca increases, the vapor fraction increases, i.e. the bubble interface expands horizontally, which is in agreement with the experimental observation in Figure 14.16. Consequently, a larger capillary pressure gradient is established for a higher value of Ca to maintain the mass balance between the replenishment and the evaporation, as described by Wang et al. (2004). The replenishment is induced by the interfacial transport. The associated boiling heat transfer is improved by this 'evaporation-replenishment' mode. Following the analysis of Chien and Webb (1997), the pore-scale interfacial heat transfer can be idealized as the evaporation on the liquid meniscus, see Figure 14.19. The dimensionless evaporative heat transfer coefficient is defined as, see Wang et al (2004), /i* = Ca/AT^, where AT^ = (T^ - Ty)/Tv. As shown in Figure 14.20, as Ca (or heat flux) increasing, h* gradually increases, reaches a peak value, CapA:, and then falls rapidly. For Ca < Ca^A;, the increase of h* may partly be attributed to the fact that the evaporating liquid film in the comer region becomes thinner. Additionally, the heat transfer is enhanced by the violent, repeated dryout-rewetting process, which results from the unceasing interactions between the replenishment and the interfacial evaporation. This dynamical process can accelerate the collapse of the dry regions. As the heat flux increases further (Ca > Ca^jt), most regions of the heated surface are covered by vapor, and /i* begins to reduce. The comer region may be surrounded by the vapor phase and the path for the replenishing-liquid flow may be disconnected. If this trend continues, dryout of the heated surface may occur. As

384

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA

11.5

Dp = 3.5 mm

Figure 14.20 Dimensionless evaporative heat transfer coefficient as a function of the capillary number.

seen from Figure 14.21, the higher heat transfer coefficients are exhibited at the smaller pore sizes. This is due to the liquid replenishment being significantly intensified in smaller pores. However, it can be found that the Cdipk of a 3.5 mm pore structure is much lower than that of a 7 mm pore structure. Clearly, the spatial limitation becomes more serious with decreasing pore size.

Particle - Interface

Water

Step motor TL

Figure 14.21

Light source

TH

Test set-up of unidirectional freezing/thawing.

X. F. PENG AND H. L. WU

385

14.5 FREEZING AND THAWING The development of modem medicine and bioengineering has increasingly demanded a high technology of cryo-conservation of biomaterials or cells. The demand is very urgent to understand the associated heat and mass transfer phenomena during the freezing/thawing processes, which has a determinative influence on the improvement on the survival rate or conservation quality of the biomaterials, see Rubinsky (1998). The freezing/thawing dewatering of the porous media with a high moisture is also widely employed in environmental and food processing applications. Motivated by these requirements, more attention is being paid to recognize and understand the transport phenomena and the dynamical behavior at the pore scale. 14.5.1 Experimental facility Figure 14.21 illustrates the experimental facility employed, similar to the classical Bridgman unidirectional freezing set-up, see Rubinsky (1998). A linear temperature gradient was generated between the high- and low-temperature zones. A semi-conductor served as a heat sink to adjust the temperature gradient and interface advancing rate. The image acquisition system included a microscope, a CCD camera, and a high-speed frame grabber. The image was captured by the image acquisition and transferred to a computer for online observation, see Tao et al (2003,2004a, 2004b) and Wu et al (2004a, 2004b). 14.5.2 Sludge agglomerates during freezing During the sludge freezing, test slurry was placed in a shallow vessel with a 1 mm thick liquid film. Si02 powders, soil and crumb suspending in the solution were agglomerated as particles with diameter d of 0.02'>-0.5mm, see Tao et al (2004a, 2004b). As the freezing interface advanced, the loose agglomerates were repelled, engulfed or broken up by the advancing interface. There were typically two modes of the agglomerate merging. One is *low-rate merging', where agglomerates were repelled continuously, or repelled to a long distance and accumulated at the freezing interface before being trapped. The other mode is *high-rate merging', where agglomerates were entrapped immediately after being repelled to a short distance (less than the particle/agglomerate diameter), or even being engulfed without repulsion. Tao et al (2003) also observed a similar phenomena for rigid particles. Structure alternation Loose agglomerates are featured by their surface and inner structures, which are expected to result in some unique transport phenomena. Figure 14.22 illustrates the behavior of a large agglomerate (d = 300/um) at Vj = 13.8/xm/s. Some scraps were desquamated from the agglomerate surface and scattered along the interface. These scraps were engulfed immediately. From Figure 14.22, the agglomerate was reduced to about 3/4 of its original

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA

^p* Figure 14.22 Desquamation from a loose surface of an agglomerate at (a) t = 0 s, and (b)t = 70s.

size when being repelled for about 1 mm. This is expected to be the consequence of the inner structure alternation. The loose structure of agglomerates also affected the interaction between the freezing interface and the agglomerates. Rigid particle always acted as a whole part at the interface, and therefore once one point of it was frozen, the whole particle would be halted and be engulfed by the interface, see Tao et al (2003,2004a). In contrast, for loose agglomerates, once any part was frozen and engulfed, the other part could still be repelled forward. This caused the agglomerates to be stretched and thinned, or to break up and be reconstructed if the agglomerates could not sustain the increasing inner stress. Additionally, the inside water would have an important action on the inner structure during freezing/thawing. Figure 14.23 presents a soil agglomerate being thinned and finally split into two parts at Vf = 7.8/im/s. Internalflow The internal pores in loose agglomerates have a great impact on both the flow field prior to the freezing interface and the interaction between the interface and the agglomerates. Figure 14.24 illustrates the flow pattern in front of the interface around the agglomerate

Figure 14.23 Change of loose stmcture during freezing at (a) t = 150 s, and (b) t = 225 s.

X. F. PENG AND H. L. WU

387

mi 100|im

Figure 14.24 Flow around a bread crumb. (bread crumb) at Vj = 3.9/xm/s. The flow was traced with some tiny particles {d = 10/xm). Trajectories of these particles were identified from the continuous frames of experimental videos. Some particles went along the border of the agglomerate, and others went towards the agglomerate at a lower speed, indicating a weak flow through the agglomerate. Clearly, such flow penetration would reduce the repulsion force on the agglomerate, but it would exert a significant influence on the inner structure, and contribute to the disintegration of the agglomerate, such as desquamation on the surface, or breakup of the main body. These observation and investigations demonstrate that the energy transfer through the solid matrix, mass, energy and momentum transport inside the inner pores and their conjugated transport phenomena would play critical roles during the freezing of the porous media, particularly loose porous media. 14.5*3 Botanical tissues during freezing Wu et al. (2004a, 2004b) employed the facility shown in Figure 14.21, together with Micro-CT technology to observe the freezing-thawing characteristics of typical botanical tissues, and tried to understand the phenomena from the viewpoint of water morphology. The temperature gradient along the slide was generated between the two ends at a high temperature of 25 °C and a lower one of —10 °C, respectively. Interfacial advancing behavior Figure 14.25 illustrates snatched images of ice crystal growth and freezing interface in a thin pear slice. The interface front, denoted as white lines, did not advance at a uniform speed, or keep a straight and smooth line as commonly observed for a dilute solution. Instead, the interface advancing rate would alter spatially and instantaneously as it went on. Moreover, dendritic crystals formed in the vicinity of the interface, with coexistence of a few spicular or lumpish crystals. These observations just explore the heterogeneity of the transport and phase change in bio-tissues. It is likely that the freezing and advancing behavior exhibited a disparity in different regions, generally advancing faster in the region of looser matrix. Sometimes dispersed freezing occurred and ice crystals abruptly formed in isolated front areas of the interface, see Figures 14.25(c-e). These scattered crystals would expand

388

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA (a)

(b)

Figure 14.25 Ice crystal growth and interface advancing in a thin pear slice at (a) t = 1.833 s, (b) t = 1.867 s, (c) t = 3.833 s, (d) t = 4.467 s, and (e) t = 4.833 s.

rapidly and merge with the advancing interface. Therefore, the development of the palingenetic interface was greatly influenced by these dispersed ice crystals, which usually appeared where the tissue exhibited appropriate pore configuration. Such appropriate geometrical configuration and tissue matrix could enhance the associated transport processes and promote the occurrence of phase change. From these experimental observations, it is reasonable to expect that the pore structure and water morphology in tissues would be important factors affecting the transport process, and therefore the survival or damage degree of biomaterials. Inner structure The growth of ice crystals caused mechanical stresses on the microstructure of the tissues, such as squeezing, dragging or even lacerating the pericellular membrane, contributing to the damage of the tissue structure integrity. Figure 14.26 shows the images of a thin

Figure 14.26 Appearance images of a cabbage slice (a) before freezing, and (b) after thawing.

X. F. PENG AND H. L. WU

389

cabbage slice before freezing and after thawing. Apparently, the inner structure was more blurred in accordance with the tissue relaxation and softening. Figure 14.27 illustrates the reconstructed sectional images of a pear slice obtained using a Micro-CT. Before freezing, the pear tissue has a reticular microstructure and probably contains various modes of bio-water. After thawing, many fine crescent cracks appear, see Figure 14.27(b). This might suggest that the freezing has effectively altered the water morphology and separated water from biomaterials. The cracks might be caused by damage to the tissue matrix because of the ice crystal growth and interface advancing. In addition, the temperature distribution in the sample might be affected by the natural convection, which induces a slightly slower freezing rate near the top. Perhaps the gravity effect is also a significant reason for more cracks to appear in the upper part of the tissue. During the freezing-thawing process, micro bubbles were always observed, gathering on the surface of the tissue. Clearly, some non-condensable gas was dissolved in the biowater solution, and as the tissue solution is frozen, the non-condensable gas is separated from the solution and forms bubbles. The gas is more apt to separate out and accumulate in the upper part due to the relatively slower freezing rate and the gravitation effect. This gas accumulation in some pores would probably cause further damage of the tissue matrix and result in micro cracks. The effect of water morphology Generally, the modes of water in biologic tissues could be simply classified into free, absorbed and combinative water. Typically, absorptive water means water attached to the surfaces of biological macromolecules and membranes, and combinative water in the interior of the cell compartments. For living systems, the confined water would affect the water transport and metabolism in biological activities, see Vogler (1998) and Jhon et al. (2003), while the free water plays a crucial role in the formation of the ice crystal. The probability for an ice crystal to form at any temperature is a function of the volume, see Tumbull (1969), and thereby the ice will most likely form in the free water during freezing. As for high-hydrous tissues, there is an abundance of water existing in a free or absorbed mode. In the incipient freezing period, ice crystals preferentially form in regions rich of free water. Since the distributions of free water are uneven, an irregular zonal interface

Figure 14.27 CT images of a pear slice (a) before freezing, and (b) after thawing.

390

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA

is always observed in a unidirectional freezing, see Figure 14.25. It also explains the abrupt appearance and expansion of dispersed crystals in front of the interface. On the other hand, the morphology of the water near the interface is conversely varied somewhat under the action of phase change and transfer. During freezing, the concentration of solutes increases in the solution around the ice crystal, which causes mass transfer. As a result, the absorbed, or even combinative water permeates out to the surrounding solution, and are said to in the free mode. Such hydration is visibly manifested by the change of the tissue macroscopic characteristics and the emergence of its inner micro cracks. For relatively denser tissues with less free water, such as animal hepatic tissues also investigated in the present experiments, the absorbed and combinative water is better conserved in the tissue structure. Hence the dehydration is comparatively less significant during the freezing/thawing. Additionally, the whole freezing at an approximate cooling rate of about l ° C / s was conducted in a refrigerator for similar samples. Comparison of tissues undergoing unidirectional and whole freezing suggests that the latter has less appearance changes and lower dehydration rates. Ice forms firstly on the surface of the tissue, enveloping the tissue and restraining the loss of water. While in the case of unidirectional freezing, the advancing interface pushes the free water, as well as detrimentally separating the absorbed or combinative water in the tissue. Thus the dehydration performs more adequately than that in the whole freezing case. Further damages would be induced at a lower cooling rate because of the higher dehydration rate.

14.6 TWO-PHASE FLOW BEHAVIOR The available literature related to two-phase flow always treats the vapor and liquid phases separately and regard the flow as a continuous medium, and normally both the interfacial interaction between two phases and the influence caused by the non-continuity of the vapor phase are not included in these approximate methods. In this section, we focus on the visually observed vapor phase transport in liquid-vapor flows through a bead-packed structure at the pore scale. 14.6.1 Experimental observation The experimental rig employed is illustrated in Figure 14.28, consisting of a boiling section, visual section, heater, and water bath. The visual section was a quartz glass vessel, 200 x 100 x 20 mm rectangular cross-section, randomly filled with glass beads. The experiments were performed at different conditions (vapor/liquid mass ratios, saturation and bead-packed structure), see Fang et al. (2004a, 2004b). There were two types of bubbly flow, small bubbles uniformly distributed and transported with the liquid, and flows with large bubbles having interface distortion and elongation. It was observed that the average bubble size was strongly influenced by the diameter of the

X. F. PENG AND H. L. WU

391

Visual section

Figure 14,28 The experimental system. glass beads. In a large bead structure, the vapor phase was primarily transported in small bubbles, while in a small bead structure large bubbles apparently increased, see Figure 14.29, and bubbles were likely to be entrapped and turn into large bubbles. When the vapor fraction was not very high, small bubbles, with an average diameter of 0.1 mm, were the primary form of vapor transport, see Figure 14.30. These small bubbles periodically collided with the porous structure. After collision, the bubbles stopped immediately and then reaccelerated due to the liquid drag and buoyancy forces. In horizontal ducts the small bubbles finally reached the liquid velocity, while in non-horizontal ducts the final velocity of the small bubbles slightly exceeded the liquid velocity. When the vapor fraction

Figure 14.29 Bead size effect (dp mm).

Figure 14.30 Small bubble transport.

392

PORE^CALE TRANSPORT PHENOMENA IN POROUS MEDIA

increased, the average small bubble size and velocity remained nearly the same. However, the bubble number increased and the bubble distribution became denser. Compared with a non-horizontal duct, bubbles more easily tended to be entrapped by the porous structure and this led to bubbles coalescing and occasionally turning into huge bubbles covering a large area of the duct. Large bubbles, see Figure 14.31, penetrated through the bead channels with distortion and elongation. The surface area change of the bubble would cause an increase in the surface energy. If the kinetic energy of a bubble exceeded the potential increase caused by the interface distortion, the bubble could go through the pore throat directly. Due to the geometrical configuration, large bubbles with sufficient energy would follow in a zigzag path in the porous structure. The other flow modes were also observed and discussed by FangetaL (2004a, 2004b). 14.6.2

Critical diameter

From the above experimental observations, the flow modes and the transport mechanisms of bubbles were greatly influenced by the diameter of the bubble and the pore size. A critical diameter was introduced to identify the small and large bubbleflowshaving bubble size smaller and larger than the critical diameter, respectively. The critical diameter is defined as the maximum diameter at which a bubble can penetrate through the pore structure without interface deformation. Figure 14.32 depicts three typical bead arrangements and

Figure 14.31 Large bubble transport with distortion.

Figure 14.32 Critical diameter.

393

X. F.PENG AND H.L.WU

the corresponding critical diameters, where dp indicates the average diameter of the glass beads and d the critical diameter. Correspondingly, the critical diameter d is 0.154 dp to 0.414 dp. For an average bead diameter of 8 mm, the corresponding critical diameter is 1.23'^3.31mm. 14.6.3 lY-ansport of small bubbles Normally, due to the drag force a bubble accelerates from rest to the liquid velocity, and then it moves at the same velocity as the liquid until it re-collides with the porous structure. The momentum equation and corresponding solutions are given as follows, see Fang et al. (2004a, 2004b): k{uo — 5)^ = mt's,

s=

UQ

2mb {t " C2)K\

-

= u.t^l'^\ K-^J T-

S=

UQI

C2

H-C4

(14.11) for the initial condition 5(0) = 0, s(0) = 0. In equation (14.11), m?, is the bubble mass, UQ the liquid phase velocity, k the resistance coefficient, and K = 5pw'7rdl/4y/dtpw/lJ'^ C2 - ~2mb/Ky/uE, C4 = {2mb/K)'^{l/c2). Figure 14.33 depicts the transport of the bubbles having different radii. The time of acceleration is different for each size of bubble, and finally all bubbles approach the liquid velocity. The larger bubbles will take a longer time to speed up. Figure 14.34 illustrates the velocity-time relationship for bubbles at different liquid velocities. For the bubble in a high-velocity liquid, the drag is relatively high, and this corresponds to a quick acceleration process. For a bubble being transported in a non-horizontal duct, the buoyancy force F^ should be included. Accordingly, the momentum equation of a bubble is expressed as

U. 1 ^

1

1

1

1

1

01 -^ 0.08

-

.^ 0.06

8

>

0.04 0.02 n u -10 10

d^ = 0.05 ^m

'hi'

JJ' 10"®

10"^

10-"^

di^= 0.1 Jim d^ = 0.5 ^im

10-2

1

Time (s)

Figure 1433 Change of bubble velocity with time.

394

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA 10

E

1

"o = 10m/s

6

o

"n = 5m/s

L

>

r

/

2h

1

v^

/

/

_i—«

10-

10-

/ / /

^n = 1 m/s

/

"o = 0.5 m/s

-

. — — •^i — 1 » -

^0 =

0.1 m/s

^^

10-6 Time (s)

10"

10-

Figure 14.34 Velocity at various liquid velocities. foUows: k(u^ — s)^ + Fb = TUb's or

- fc(uo - s)^ -\- Ft = mt's.

(14.12)

The latter expression is for the case of the liquid exerting a resistance on the bubble. Similar to a horizontal duct, analytical solutions of equation (14.12) can be obtained, see Fang et al. (2004b). In a non-horizontal duct, the final equilibrium velocity of the bubble was greater than the liquid velocity, and the excess velocity was proportional to the bubble radius. For a non-continuous phase transport, the collision between the porous structure and the bubbles prevents bubbles from continuous movement and serves as the resistance on vapor transport. Meantime, the bubbles are propelled by the liquid phase, so the vapor phase also exerts an opposing force on the liquid. We can define the time interval, r, between two consecutive collisions as the average free time and the distance the bubble travels between two consecutive collisions as the average free distance. A, which is the order of the size of the pores. Given the average free time between two consecutive collisions, T, in which the average acceleration time is TQ, the average velocity of a bubble is u^. The average force of the liquid exerts on the bubble and the opposing force that the small bubbles exert on the liquid are derived as: _ rritUb , Fa = and Ta

rn rtsTaLAes rribUb ngLAcsmbUb Fu = = , T

Ta

r

., with

A r = —. Ub

(14.13) Finally, the additional pressure that small bubbles exert on the liquid is obtained as follows: Pls =

Fls Ae

QvLUbPg

XAe

^

.

(14.14)

395

X. F. PENG AND H. L. WU

where ^s denotes a parameter depending on the geometry. In the experiments on small bubbles in two phase flows, the pressure difference was measured for various vapor fractions 5 at a constant liquid flow rate of 1.67 x 10~^m^/s. Figure 14.35 illustrates the relationship between the pressure difference increase and the vapor fraction. When S is relatively low, the experimental data agrees well with the prediction of equation (14.14). For relatively large S, the effect of the large bubble distortion should be included in predicting the pressure difference. 14.6.4

TYansport of big bubbles

For bubbles having a size greater than the critical diameter, the shape of a bubble in its intermediate stage of cross-over is shown in Figure 14.36. Parts A and C are the portion of a sphere with radius r and part B an axis-symmetrical evolving body with a profile of the neighboring glass beads. The impetus of bubble transport comes solely from the liquid. The pressure of a cross-over bubble exerts on the liquid depends the surface energy of the distorted bubbles. This was also investigated in the work of Fang et al. (2004b). \il

CO

0)

o c 0

2

1

'

' + '

-H

y

10 8

/^ + y

6 4

Q.

T

2

-

+

-1-

1

-I-

Experimental data Prediction

-

^

•4> 1

1

0.05

0.1

1

0.15

1

0.2

1

0.25

1

0.3

0.35

Vapor fraction

Figure 14.35 Pressure difference with vapor fraction.

Figure 14.36 Bubble during crossing-over.

396

14.7

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA

CONCLUSION

The conjugated transport phenomena with a pore and matrix structure widely exist in both the natural world and practical applications. It is of critical significance to understand these phenomena accounting for the dynamical processes and structure deformation taking place in the inner pores. In these processes, the associated transport phenomena are highly dependent upon the pore structure. Inversely, the transport processes easily alter the pore structure and further influence the transport characteristics. In this chapter, a series of different experimental observations and associated theoretical investigations have been conducted to understand the transport phenomena at the pore-scale level, including the transport phenomena with/without phase change and chemical reaction, and concerning a wide range of practical applications. Several typical processes, namely drying, transportreaction, boiling and two phase flow in porous media, freezing and thawing treatment of loose and bio materials, were visually observed and theoretically described at the porescale level. These investigations provide some new understanding and insights into the nature of transport phenomena in porous media from different points of views. Without doubt, it is quite a new and challenging area to investigate the transport phenomena in porous media at the pore-scale level, and the present investigations are very preliminary and ongoing. In particular, more attempts should be addressed on the understanding and theoretical descriptions of the fundamentals. Of course, this kind of investigation would be highly dependent upon the experimental technology, observations, measurements and innovatory theoretical description. These will be the emphases in future investigations.

ACKNOWLEDGEMENT This research was supported by the National Natural Science Foundation of China (Contract No. 50136020 and 50306009).

REFERENCES Blunt, M. J. (2001). Flow in porous media—pore-network models and multiphaseflow.Current Opinion Colloid Interface Sci, 6, 197-207. Boer, R. (2000). Theory of porous media: highlights in historical development and current state. Springer, Berlin. Carey, V. P. (1992). Liquid-vapor phase change phenomena. Hemisphere, Washington, DC. Chen, J. B., Feng, X. F., and Lee, D. J. (2002). Convective drying of sludge cake. /, Therm. Sci. 11, 277-82. Chien, L. H. and Webb, R. L. (1997). A nucleate boihng model for stmctured enhanced surfaces. Int. J. Heat Mass Transfer 41, 2183-95.

X. F. PENG AND H. L. WU

397

Dullien, F. A. L. (1992). Porous media: fluid transport and pore structure. Academic Press, New York. Fang, C , Peng, X. F., and Yang, Z. (2004a). Vapor phase transport of two-phase flow in porous media. In Applications of porous media (eds A. H. Reis and A. F. Miguel), pp. 259-65. ICAPM 2004. Centro de Geofisica de Evora, Portugal. Fang, C , Peng, X. E, and Yang, Z. (2004b). Visualization of liquid-vapor two-phase flow in porous media. In Heat transfer science and technology 2004, Proceedings of 6th international symposium on heat transfer, pp. 424-9. Zhongke Multimedia Electronic Press, Beijing. Fu, G. G. and Peng, X. F. (2004). Pore behavior of hydrated and calcined CaO and effect on flue gas desulfurization. In Heat transfer science and technology 2004, Proceedings of 6th international symposium on heat transfer, pp. 403-7. Zhongke Multimedia Electronic Press, Beijing. Ingham, D. B. and Pop, I. (eds) (1998). Transport phenomena in porous media, Pergamon, Oxford. Ishida, M. and Wen, C. Y. (1968). Comparison of kinetic and diffusion model for solid-gas reactions. A/C/iE 7.14,311-17. Jhon, Y. I., Kim, H. G., and Jhon, M. S. (2003). The structure of water near platinum and its significance in water-adsorbent system: molecular dynamics study. J. Colloid Interface ScL 260,9-18. Leverett, M. C. (1941). Capillary behavior in porous solids. AIME Trans. 142,152-69. Lin, R. T. (1995). Introduction of heat and mass transfer in porous media. Science Press, Beijing. Michaud, V. and Mortensen, A. (2001). Infiltration processing of fibre reinforced composites: governing phenomena. Compos, Part A 32,981-96. Peng, X. F , Wang, B. X., and Christopher, D. M. (2000). Some fundamentals of boiling in microgravity. In Heat transfer science and technology 2000, Proceedings of the 5th international symposium on heat transfer, pp. 60-76. High Education Press, Beijing. Peng, X. F , Wang, Z., and Lee, D. J. (2002). Dynamic behavior of bubble interface during boiling. J. Therm. Sci. 11, 308-19. Rubinsky, B. (1998). Microscale heat transfer in biological systems at low temperatures. In Microscale energy transport (eds C.-L. Tien, A. Majumdar, and F. M. Gemer), pp. 344-68. Taylor & Francis, Washington, DC. Sahimi, M. (1995). Flow and transport in porous media andfractured rock: from classical methods to modem approaches. Weinheim, VCH, New York. Tao, T, Peng, X. F., Chu, C. P., and Lee, D. J. (2003). Interaction of ice-water interface with agglomerates of activated sludge during freezing. In Proceedings of the ASME summer heat transfer conference 2003, Las Vegas, NV, pp. 541-5. Tao, T., Peng, X. E, Chu, C. P., and Lee, D. J. (2004a). Rigid or loose particles interacting with ice front during unidirectional freezing. In Applications of porous media (eds A. H. Reis and A. F. Miguel), pp. 479-84. ICAPM 2004. Centro de Geofisica de Evora, Portugal. Tao, T., Peng, X. F , and Lee, D. J. (2004b). Unidirectional freezing of loose agglomerates. In Heat transfer science and technology 2004,*^ Proceedings of 6th international symposium on heat transfer, pp. 155-8. Zhongke Multimedia Electronic Press, Beijing. TumbuU, D. (1969). Under what conditions can a glass be formed? Contemp. Phys. 10, 473-88. Udell, K.S. (1983). Heat transfer in porous media heated from above with evaporation, condensation, and capillary effects. J. Heat Transfer 105,485-92.

398

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA

Vaxelaire, J., Mousques, P., Bongiovanni, J. M„ and Puiggali, J. R. (2000). Desorption isotherms of domestic activated sludge. Environ. Tech. 21,327-35. Vogler, YE. A. (1998). Structure and reactivity of water at biomaterial surfaces. Adv. Colloid Interface ScL 74,69-117. Wang, Z. and Peng, X. F. (2004). Pool boiling on a wire porous media: effect of pore size on bubble nucleation and dynamics. In Applications of porous media (eds A. H. Reis and A. F. Miguel), pp. 279-84. ICAPM 2004. Centro de Geofisica defivora,Portugal. Wang, Z., Peng, X. F, and Liu, T. (2002). Visualization of boiling phenomena in a bead-packed structure. Experim. Heat Transfer 15, 177-89. Wang, Z., Peng, X. F, and Ochterbeck, J. M. (2004). Dynamic bubble behavior during boiling in bead-packed structures. Int. J. Heat Mass Transfer 47,4771-83. Wen, C. Y. and Ishida, M. (1973). Reaction rate of sulphur dioxide with particles containing calcium oxide. Environ. Sci. Tech. 7,703-8. Wu, H. L., Ma, Y., and Peng, X. F. (2004a). Freezing-thawing characteristics of botanical tissues and influence of water morphology. Chin. Phys. Lett. 21,345-7. Wu, H. L., Ma, Y, and Peng, X. F. (2004b). Freezing-thawing characteristics of some bio-material tissues. In Applications of porous media (eds A. H. Reis and A. F. Miguel), pp. 485-90. ICAPM 2004, Centro de Geofisica de Evora, Portugal. Yan, Y (2003). Transfer-reaction characteristics inner pores and simulation of desulfurization in fiuidized beds. Ph.D. thesis. Tsinghua University, Beijing 100084, China. Yan, Y, Peng, X. F, and Lee, D. J. (2003). Transport and reaction characteristics in flue gas desulfurization. Int. J. Therm. Sci. 42, 943-9. Yan, Y, Peng, X. F, Wang, B. X., and Lee, D. J. (2004). Dual-stage model for the transport-reaction in porous particles. Chin. J. Eng. Thermophys. IS, 133-5. Zhang, Y and Peng, X. F (2004). Inner structure analysis of porous media using micro-CT. In Proceedings of 2004 Chinese national heat transfer conferencej Vol. 2, pp. 1047-51.

15

DYNAMIC SOLIDIFICATION IN A WATER-SATURATED POROUS MEDIUM COOLED FROM ABOVE S. KIMURA Institute of Nature and Environmental Technology, Kanazawa University, 2-40-20 Kodatsuno, Kanazawa, 920-8667, Japan email: skimuraQt.kanazawa-u.ac.jp

Abstract This chapter describes a fundamental study on an ice-layer formation and the melting along a cooling surface, positioned at the top boundary of a rectangular space filled with watersaturated porous medium. The temperature of the cooling surface is varied periodically with time, while the temperature at the bottom is kept constant, with all side walls being thermally insulated. In such conditions natural convection has a significant impact on the heat balance at the solid-liquid boundary that develops into the unfrozen layer The goal of this study is to develop a one-dimensional model which is capable of predicting the transient response of the ice-layer to a prescribed cooling temperature variation. In the course of the formulation, the Stefan number, the Rayleigh number and the cooling temperature conditions are identified as the important parameters of the problem. The one-dimensional theory predicts that a higher cooling temperature frequency generally 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 tested against two-dimensional numerical simulations for the dynamic response of the interface movement. An excellent agreement between the one-dimensional model and two-dimensional simulations has been obtained, and further experiments have been conducted in order to verify the numerical models. In our experiments, spherical glass beads saturated with distilled water is used as a porous medium. The lower boundary is kept at 20 ° C, while the upper plate is initially set at an arbitrary subzero temperature. After a steady state has been reached, the growth and decay of the ice-layer due to the oscillating cooling temperature is recorded. Again the one-dimensional analysis is found to be good enough to predict the dynamic response of the spatially-averaged interface position. Keywords: porous medium, heat transfer, solidification, natural convection, convecrive heat transfer, transient process, periodic cooling temperature, steadily oscillating ice-layer, numerical analysis

399

400

15.1

SOLIDIFICATION IN A WATER-SATURATED POROUS MEDIUM

INTRODUCTION

Solidification is one of the most commonly observed phenomena around us. In engineering applications, solidification processes are encountered in a variety of material productions and processes, where technologies to control the solidifying process are crucial for manufacturing high quality products as well as for new material development. For example, the solidifying speed of alloys is an important factor that determines the morphology of the resulting solid phase, see Flemings (1974) and Kurz and Fisher (1992). In geophysics and planetary science, solidification is the driving mechanism that forms the crust of the Earth and determines the deeper structures of the planetary interior, see Turcotte and Schubert (1982). It is a well-established fact that the amount of ice in the arctic regions has a significant impact on the global climate change, see Maykud and Untersteiner (1971), Hibler III and Walsh (1982) and Wettlaufer et al. (1997). Glaciers and permafrost observed in the arctic and sub-arctic regions are other geophysical phenomena that also interact with the oceans and the atmosphere, see Fowler (1997). Therefore, it is important to know how fast the ice sheets will respond to a general rise in Earth temperatures. Studies on solidification in liquid-saturated porous medium have attracted much attention, especially for the understanding of the transport processes in soil freezing. Most of them have been conducted with boundary conditions of constant cooling temperatures, as summarized by Nield and Bejan (1999). In the 1970s, an analysis using the complex variable theory has been developed by Goldstein and Reid (1978) in order to analyze the phase change of a water saturated porous medium in the presence of a seepage flow. The finite element method was employed by Hashemi and Sliepcevich (1973) and Frivic and Comini (1982) to investigate the freezing around a row of pipes when the ground water flow normal to the pipe axis is present. Okada (1981) developed an approximate analysis of the freezing around cooled pipes in Darcy flow. O'Neill and Albert (1984) and Oosthuizen (1988) have studied solidification in porous media in the presence of natural convection, but they were not specifically concerned with water freezing. Water freezing in porous media has been investigated relatively recently. For example, Chellaiah and Viskanta (1988), Sasaki et al. (1990) and Sasaki and Aiba (1992) investigated a solidifying process in a rectangular cavity cooled from the side, emphasizing the influence of natural convection in the unfrozen space. Chellaiah and Viskanta (1989) performed numerical simulations using an enthalpy method. Sugawara et al. (1988) made an experimental study on the freezing in a horizontal porous layer. They were particularly concerned with natural convection heat transfer under the presence of a density extreme. However, solidification often involves a condition of unsteady cooling rates, e.g. the cooling temperature is no longer constant but time-dependent. Probably there are tvC^o different approaches to this class of problem. One is to consider the cooling temperature as given conditions and to predict the response of the solid layer, i.e. a time-history of the solid layer thickness. The other is to determine the cooling temperatures that are required for achieving a prescribed solidification speed. The latter will be an interesting problem in materials development, while the former is more related to geophysical and environmental phenomena.

401

S. KIMURA

In this work we focus on thefirstproblem, and investigate the freezing processes in a watersaturated porous layer, when cooling takes place at the upper boundary of a rectangular space. The lower boundary temperature is kept constant above the solidifying point. Therefore, a liquid layer is always present at steady state, and convection is expected to develop in it. We use time-dependent cooling temperatures with periodical changes that clearly distinguish our approach from other related studies. The main focus of our work is on the dynamic solid layer response to such periodic temperature variations, see Kimura and Vynnycky (1999) and Kimura et al (2003,2004a, 2004b). It is also worth mentioning that Echigo et al (2002,2003) have recently studied growth and decay of permafrost with a time-dependent cooling temperature.

15.2 MATHEMATICAL FORMULATION 15.2.1 Two-dimensional model A schematic description of the freezing problem in a liquid-saturated porous medium is shown in Figure 15.1. The rectangular space is initially filled with a water-saturated porous medium at the same temperature as the bottom boundary. At t = 0 the temperature on the upper boundary is instantly dropped below the solidifying point of water, i.e. 0 °C and an ice-layer starts growing along the cooled horizontal wall. Since the lower boundary temperature is kept unaltered, there is always an unfrozen region below the frozen layer where convecting water may be present. In developing the mathematical formulations we have made the following assumptions, see Bejan and Klaus (2003). (i) The porous matrix and the saturating water are in a thermal equilibrium.

Adiabatic

Figure 15.1 Two-dimensional model.

402

SOLIDIFICATION IN A WATER-SATURATED POROUS MEDIUM

(ii) The thermophysical properties are constant and independent of the temperature, except for the density appearing in the buoyancy term, (iii) Darcy's law is valid for flows through the porous matrix. One critical issue in the numerical solution for the solidification is the ever-moving solid-liquid interface. In the present study, a coordinate transformation technique is employed, so that in the computational space the interface becomes fixed, see, for example, Saitoh (1978). Introducing the streamfunction and taking the curl of the two-dimensional Darcy's equation, the momentum equation is reduced to a single Poisson equation for the streamfunction. This formulation is routinely employed for analyzing two-dimensional natural convection in enclosures. Many such examples are found in a monograph edited by Ingham and Pop (1998), and a book by Nield and Bejan (1999). With the above coordinate transformation, as shown in the work by Sasaki et al. (1990), the dimensionless momentum equation is given by:

where ^ is the nondimensional streamfunction, and i^ is a nondimensional quantity measuring the magnitude of buoyant force and defined by (15.2)

R = ^ ^ ^ .

Here K, g, H, a^ and z/ denote the permeability of the porous matrix, the gravitational acceleration, a characteristic length (cavity height), the thermal diffusivity of the porous medium and the kinematic viscosity of the water, respectively. All thermophysical properties in equation (15.2) are evaluated at f = {TH + To)/2, the average temperature in the liquid region. We assume that the density of the water is in general a nonlinear function of temperature, p is the density of the water evaluated at the average temperature in the liquid region, and po is the maximum density of the water at T = 3.98 °C. When density is a linear function of the temperature, R could be interpreted as the conventional Rayleigh number. However, in our case the properties of R in the linear range differ from those of the conventional Rayleigh number. This is due to the fact that the Rayleigh number is based on the maximum density change in the system, while R in equation (15.2) is based on the average density of the system. For better consistency with the conventional Rayleigh number, the following definition is employed: KgH

(15.3)

Therefore, Ra and R are related by the identity, R8i = R— 1 Po

P

(15.4)

Po

The dimensionless energy equation in the liquid region, see Sasaki et al (1990), is given by:

403

S. KIMURA

dr

dC dr

d0L dX

dCdY

dOL d<: dC dX

(15.5) V^^L

dC, dY

dX "*" dC dX where the differential operator V^ is defined as follows:

d\

V2 =

dY

d

dC "^ dx dxdc + dX'^dC +

92

dX

(15.6)

where C(^) is the vertical coordinate in the transformed computational space for the liquid region, and defined by C(-'^) = Y/F{X). Here F{X) is the dimensionless vertical position of the solid-liquid interface, defined by F{X) = f{X)/H. The apparent complexity of the differential operators in equations (15.5) and (15.6) is the result of the coordinate transformation. According to Sasaki et al. (1990), the dimensionless energy equation in the solid region becomes

where the differential operator V^ is similarly defined as follows:

de ^

dX dXdi

-h

+

92

(15.8)

where £,{X) is the vertical coordinate in the transformed space for the solid region, defined by ^(X) = {Y - F{X))I{1 - F{X)). Due to these coordinate transformations, in the computational domain, the solid-liquid interface can be treated as a fixed horizontal boundary. At steady state, convection heat flux arriving at the interface from below must balance with the upward conduction cooling in the solid layer. Any imbalance between them causes growth or melting of the solid layer and the associated release or absorption of latent heat. Therefore, the energy balance at the solid-liquid interface, see also Sasaki et ah (1990), yields the following condition: Ste

- To 86L dC To - Tc dC dY

TH

keL d^ dY

1-

dx

dF_ dr

(15.9)

where Ste is the Stefan number defined by Ste = {PC)L{TO - Tc)/(l>psL. In the above equation the left-hand side expresses the imbalance between the convection heat flux from below and the upward conduction cooling, while the right-hand side indicates the release of latent heat due to the advance of the interface. Equation (15.9) also stipulates that a thermodynamic equilibrium persists on the solid-liquid interface. After a new temperature field is generated from equations (15.1) and (15.5), the equation (15.9) must be solved for F , providing a new location of the solid-liquid interface at a new time level.

404

SOLIDIFICATION IN A WATER-SATURATED POROUS MEDIUM

The boundary conditions for equations (15.1), (15.5) and (15.7) are as follows: at

C= 0

(2/ = 0),

6L = 0

at

C= i

{y = f),

es = o es = -i

at

^= 0

(y = / ) ,

at

i=l

iy = H)

* = 0,

OL

* = 0,

* = 0,

= OH

dOL

dx

dOs

at

dx

(15.10)

X = 0,1.

Initially the system has the same temperature as the bottom boundary. The time integration is then initiated with an instantaneous temperature drop at the top (cooling) boundary to a value below 0 °C. Since our computation assumes the presence of a finite thickness solid layer at all time, a very thin uniform solid layer and the temperature profile there are provided analytically by the Neumann solution of the Stefan problem at r = 0.02. Further reduction of the initial thickness does not cause a significant difference in the subsequent computational results. After a steady state has been reached for a fixed cooling temperature, the temperature condition at the top boundary, ^ = 1, is replaced with a periodic function of time, typically with a square wave shape. The nondimensional quantities used in the above equations are defined in the following manner: * -

V QeL

_

.

>

_ n-To

^L -

I.

keL

" ' ' ' ~ {pC)L '

_ TS-TQ

7f 7f- . J-H-J-C

_

^5 -

I.

keS

" ' ^ " ipC)eS '

P

7^; TfT- , J-O-J-C

. _

^ -

PO

/ A

aeS[P'~^)eL

~ a^LipCh

—,

(15.11) '

The Roman letters k, C, / and L denote the thermal conductivity, specific heat, interface position in the ^/-direction and the latent heat of solidification. The Greek letters 6, d and (j) stand for the nondimensional temperature, density ratio and porosity. The subscripts e, L and 5 indicate the effective value, liquid and solid, respectively. In the case where the saturating fluid is water, the density variation in the buoyancy term can be computed by the following nonlinear equation of temperature, see Seki et al (1976): ^^ 1 + a'T -f 6'r2 + c'T^ + d'T4 '

(15.12)

where T is the temperature deviation from the value T = 3.98 °C, at which the maximum density of water occurs, and the coefficients have the following values:

S. KIMURA

405

po = 999.8396 k g / m ^ a' ^ -0.678964520 x 10"^ (1/°C), h' = 0.907294338 x 10"^ (l/^C)^ ,

(15.13)

c' = -0.964568125 x 10"^ (1/°C)^, d' = 0.873702983 x 10"^ (1/°C)'*. In order to solve the set of equations (15.1), (15.5), (15.7) and (15.9), first we discretized them with a central finite-difference of the fourth-order accuracy in space, and then used the Crank-Nicolson method for the time integration. Solutions were generated for different values of R (or Ra), Ste, TH and TQ. 15.2.2 A reduced one-dimensional model One-dimensional models are often favored over more complex two- and three-dimensional models, mainly because of their ease of use and ability to produce a clear picture of the physical processes for seemingly complex problems. In addition, for many geophysical processes, the solid layer formation is restricted to a small distance from the cooling surface. Frozen soils, known as permafrost, and ice-layer formations over lakes in cold regions are some good examples of thin solid layer formations relative to their horizontal extents. In such cases, the average ice-layer thickness over a large area and its timedependent behavior due to, for instance, seasonal climate change are of major concern for geophysicists and construction engineers. If a horizontal cooling surface of infinite extent is assumed, the thickness of solid layer at steady state can be determined from the energy balance between the heat arriving from below by either convection or conduction and the conductive cooling across the solid layer. Therefore, the imbalance between the two heat fluxes is the sole driving force inducing the growth and melt of the solid layer. However, in many engineering and geophysical situations, the solid layer thickness is much smaller than the liquid layer below, so that the heat flux by convection or conduction from below becomes insensitive to the solid layer growth. Taking the above into consideration, we show in Figure 15.2 a sketch of the onedimensional model and its coordinate system. This is essentially a reduced version of the earlier two-dimensional model. The convective or conductive heat flux from the unfrozen region is assumed to be constant q", and independent of the solid layer thickness. Since the value of q^' determines the frozen layer thickness at steady state for a given specific cooling temperature, it is natural to take the following characteristic length scale in the model:

^^ks{To-Tc) According to Kimura et al. (2003), the one-dimensional energy equation in the solid layer and the energy balance at the solid-liquid interface are written as follows:

406

SOLIDIFICATION IN A WATER-SATURATED POROUS MEDIUM

Tc{t)

x= m

Temperature profile

XJ

Figure 15.2 One-dimensional model.

(15.15)

dr dr

= Ste ('-^\

-1

(15.16)

and the corresponding boundary conditions to equations (15.15) are given by: es = Oc{T)

at

X = 0,

(9s = 0

at X = F.

(15.17)

The second condition indicates that the solidification takes place at the liquidus temperature without any undercooling. Accordingly, the nondimensionalization from the previous two-dimensional case is modified, using the new length scale (., i.e. Y - -

TP-L

^-££

(15.18)

and it should be noted that / now indicates the solid layer thickness as shown in Figure 15.2. A similar coordinate transformation for the vertical coordinate, ^(X,r) = X/F{T), is employed in order to deal with the moving boundary problem. The following equations, corresponding to equations (15.7) and (15.9) in the two-dimensional model, are derived as a result: dds dr

f-'^

d^dOs dr d^

dx ac^ '

(15.19)

-1

(15.20)

c=i/

407

S. KIMURA

The boundary conditions of equation (15.17) are also transformed to those at ^ = 0 (cooling boundary) and ^ = 1 (interface). The initial condition is the same as that for the two-dimensional model. Similar numerical schemes are used for solving equations (15.19) and (15.20) simultaneously, i.e. a fourth-order, finite difference in space and the second-order Adams-Bashforth method for the time integration.

15.3 NUMERICAL RESULTS 15.3.1 Development of a solid layer and convecting flow It is assumed that the water-saturated porous medium is initially ai 6H- At r = 0 the temperature of the upper boundary is lowered instantly to a subzero value, and henceforth the ice-layer growth is computed numerically. Just to demonstrate the solidification processes in the presence of convection, we show in Figure 15.3 ice-layer growths for three different values of R. It is evident from Figure 15.3 that the ice-layer becomes thinner as the value of R increases. This is due to the stronger convective heat transfer that develops at higher values of R. In Figure 15.4 the streamlines for the two different values, R — 500 and R = 3000, are plotted. In Figure 15.4(a), for R = 500, convection is still weak and the solid-liquid interface is horizontal. The convective motion appears restricted to the lower half region, while the upper half is filled with an almost stagnant fluid. This is clearly reflecting the existence of a stably-stratified layer in the upper portion. In Figure 15.4(b), for R = 3000, however, the strong current in the liquid region takes over the stable layer, and a deformed solid-liquid interface develops. An amount of volumetric water driven by the buoyancy force ai R = 3000 is about four times greater than that a.tR = 500. Two convecting cells are clearly seen for the both cases. However, whether or not this is the only realizable convecting pattern is still open to question. For one-dimensional calculations, the speed to reach a steady state as time progresses is determined by a single parameter, namely the Stefan number. The convective heat flux

0.5 R = 500 0.4 R = 2000 0.3 0.2

jf[y^

R = 3000

0.1 0.0,0

Figure 15.3 Growth of the ice-layer over time for different Rayleigh numbers. (Kimura etai, 2003.)

408

SOLIDIFICATION IN A WATER-SATURATED POROUS MEDIUM

Figure 15.4 Streamlines of the convective motion and the resulting solid-liquid interface shape (A^ = 1): (a) R = 500, and (b) R = 3000. (Kimura et ai, 2003.)

from below must be provided as a prescribed condition, which effectively determines the ice-layer thickness at steady state. 15.3.2 Amplitude and phase lag of the oscillating solid-liquid interface Once a steady state is reached, the cooling temperature is subject to periodic variations in time. In our experiment we use square shaped cooling temperature oscillations and investigate the resulting ice-layer responses. This subsection explains how our onedimensional model can be used to predict the time-dependent behavior of the horizontallyaveraged ice-layer. For the two-dimensional calculation, the value of Ra is set to about 100, while the Stefan number, which reflects the cooling rate at the top boundary, is set to about 0.1. The values of A^/Ste, where A^ is the temperature oscillation amplitude, fall to be in the range of 2 to 10. Numerical results from the two different models are plotted together in Figure 15.5, and clearly show a good agreement. In Figure 15.5(a), the phase lag of the interface to the oscillating cooling temperature is plotted as a function of the cooling temperature period. As expected, slow temperature oscillations produce a smaller phase lag. It is also observed that the magnitude of the temperature amplitude does not have much effect on the phase delay. In Figure 15.5(b), the interface amplitude is plotted as a function of the temperature amplitude. A greater temperature amplitude, in general, produces a larger interface oscillation. It should be especially noted that the smaller period of the temperature oscillation significantly reduces the interface amplitude. This result can be explained from the temperature amplitude at an arbitrary distance X for the periodic heating at the boundary of a semi-infinite half-space, see, for example, Carslaw and Jaegar (1959):

Ae{x)

^-V^x

^^-x/^

(15.21)

where a;, P and X are the dimensionless angular frequency, period of cooling temperature oscillation, and distance from the cooling surface, respectively.

409

S. KIMURA

(a)

^

A^/Ste A6>/Ste = 5 A^/Ste = 10 - A6>/Ste = 2 - A^/Ste = 5 - Al9/Ste = 10

•

7r/2

• •

?

i>

i 7r/4' r \

^^!^ n

^

^

"

•

ID ID ID 2D 2D 2D

^:^

^

0.1

(b)

0.16 y^i

0.12

i

k

1

• A

AF

0.08 0.04

A

^^--^1 1 %-^^^^ '

P P P P P P

= = = = = =

ID ID ID 2D 2D 2D

0.1 0.2 l 0.1 0.2 l

• 4

•

A^/Ste

10

15

Figure 15.5 Comparison between the two numerical results: (a) the phase lag of the interface as a function of the period of the cooling temperature, and (b) the oscillation amplitude of the solid-liquid interface as a function of the temperature amplitude. (Kimura ^r^/.,2003.)

15.4 EXPERIMENTAL RESULTS 15.4.1 Experimental apparatus and procedure In Figure 15.6 we show a schematic diagram of the experimental apparatus. The box is nearly cubic, 18 cm in size and is built of 1.5 cm thick acrylic plate. Copper plates are used as the top and bottom boundaries, ensuring isothermal conditions. The internal space is packed with glass beads with average diameters of 2 mm or 5 mm, and is saturated with distilled water. Temperatures are measured using thermocouples installed at the top and bottom boundaries. Eighteen additional thermocouples are positioned along the sidewall for acquiring the vertical temperature variation inside the vessel. The thickness of the ice-layer formed is estimated by measuring the amount of water expelled from the vessel. The mass conservation principle enables us to calculate the average thickness of

410

SOLIDIFICATION IN A WATER-SATURATED POROUS MEDIUM

Figure 15.6 A picture of the experimental apparatus. Glass beads and distilled water are removed so that eighteen thermocouples on the left wall are clearly seen. The lower one of the two tubes seen on therightis connected to a burette to measure the expelled volume of water. (Kimura et ai, 2004a.)

ice produced in the vessel using the following expression, see Sugawara et ai (1988):

V =

Sf(t>(l-

PL

(15.22)

where V, S and / are the volume of expelled water, the horizontal cross sectional area of the vessel and the average thickness of the ice-layer, respectively. Temperature-conditioned coolants from three different cryostats are used in order to keep the bottom temperature constant, and to modulate the cooling temperature at the top periodically. In the present experiments, the bottom boundary is kept at 20 °C. The Rayleigh numbers defined by equation (15.3) are in the range from 15 to 100, and the values of-R in equations (15.1) and (15.2) are much higher and have orders of 10^ to 10^. 15.4.2 Ice-layer thickness at steady state In Figure 15.7, we show the ice-layer thickness at steady state against the cooling temperatures. The steady state thickness is important because it can be used to evaluate the convective heat flux from the liquid region. Two different diameters of glass beads, d = 2 mm and d — b mm, are used for varying the porous medium permeability and four different cooling temperatures, from - 6 °C to - 3 °C, are imposed on the top boundary. At the lower the cooling temperature, naturally the thicker ice-layer is formed. Since the porous matrix formed by the large glass beads causes less resistance to the water flow, vigorous convection can be accordingly expected. The ice-layer thickness for cJ = 2 mm

411

S. KIMURA

ou-

•

B 50-

• d = 2mm • d = 5mm

•

"

M

•

| 4 0 o Z 30-

s

•

a

• •

'Z 2 0 -

•

•

>%

1 10C/0

0-

-7

1

\

1

1

-6 -5 -4 -3 Cooling temperature T^ [°C]

-2

Figure 15.7 Ice-layer thickness at steady state with different cooling temperatures. (Kimuraera/., 2004a.)

is two times greater than that for d = 5 mm, indicating that the average heat flux for d = 2 mm is a half of that for d = 5 mm. It is difficult, however, to speculate this trend from the ordinary Nu ~ Ra correlation, because the water density in the liquid regime changes nonlinearly and goes through a maximum at about 4 °C, see Blake et al. (1984). 15.4.3 Average Nusselt number and vertical temperature variation at steady state The average Nusselt numbers at steady state were computed from the ice-layer thickness, see Figure 15.8. At the solid-liquid interface, the heat conduction through the ice-layer must balance with the convection heat flux from the liquid region. The Nusselt numbers increase as the cooling temperature decreases. A lower cooling temperature produces a thicker ice-layer, henceforth the liquid layer becomes thinner. This observation somewhat contradicts the fact that the thinner liquid layer generally reduces the convective motion. We speculate that the thickness of the stably stratified water layer, ranging from 0 °C to 4 °C, plays a critical role in determining the overall vertical heat flux. The average Nusselt number variation with respect to the cooling temperature is still visible. When the cooling temperature drops from - 3 ° C to - 6 ° C , the heat flux increases by about 10% for the large glass beads. This is probably due to the increase in the heat penetration from the environment. Figure 15.9 shows vertical temperature profiles for four different cooling temperatures and two different diameters of glass beads. The thermocouples are positioned about 4 cm from the side wall. Therefore, those profiles reflect the temperatures along a vertical line at a particular position on the horizontal plane. For d = 2 mm, the temperature profiles are nearly linear, indicating that the heat transfer mechanism in the liquid region is conduction dominated. However, a close examination proves that those profiles slightly deviate from

412

SOLIDIFICATION IN A WATER-SATURATED POROUS MEDIUM

•?

J.J

^

o d=5mm 1 • ^=2mm A Kimura et al. (2004b) (7^ = 8 °C)|

-,

3.02.5-

o

o

2.0Nu

1.5-

*

1.00.50.0-7

1

1

1

1

-6 -5 -4 -3 Cooling temperature T^ [°C]

-2

Figure 15.8 Average Nusselt numbers measured from the ice-layer thickness at steady state. (Kimura era/., 2004a.)

6 Temperature [°C]

12

Figure 15.9 Vertical temperature profiles at steady state measured from thermocouples imbedded in the experimental vessel. (Kimura et al, 2004a.)

S.KIMURA

413

linear lines, showing the evidence of weak convection. On the contrary, for d = 5 mm, the temperature profiles in the liquid regime suggest the presence of strong convection. The permeability of the latter case is about six times greater than the former. This implies that R and Ra values for d = 5 mm are also six times greater than those for d = 2 mm. 15.4.4

Oscillating cooling temperature and the response of ice-layer

In this subsection we illustrate how the interface responds when the cooling temperature oscillates periodically with a square wave as shown in Figure 15.10. The average cooling temperature is fixed at Tc = - 6 °C, and the oscillating amplitude is also fixed at 1 °C. However, the oscillating period is varied from 1 to 5 hours. Since the average cooling temperature is unaltered in these cases, the corresponding averaged ice-layer thickness stays at about 57 mm. For the one hour period, the oscillation amplitude of the interface is extremely small (about 0.2 mm). However, the 5 hours period produces a much greater amplitude (about 1.5 mm). This can be explained by the fact that the temperature oscillation amplitude decays exponentially, whose power is a product of the square root of the frequency and the distance from the boundary, as in equation (15.20). Also it can be noticed that the phase lags relative to the temperature oscillation of a square wave are greater when the temperature period is small. For instance, the five-hour period produces only a very sHght delay, while the one-hour period produces nearly a quarter period delay. In the same figure, we plot the numerical results from the one-dimensional model. The convective heat flux in equation (15.13) is computed from the ice-layer thickness at steady state, and this value is kept constant thereafter during the time integration. The experimental and the numerical results agree very well, and as plotted, are almost indistinguishable. 15.4.5

Amplitude and phase lag against oscillating cooling temperature

As we demonstrated for the one-dimensional and two-dimensional models in the earlier section, the one-dimensional numerical model is compared with the experimental results for both the phase lag and amplitude in Figure 15.11. The phase lag of the interface against the cooling temperature is shown for three different periods in Figure 15.11(a). The period of the cooling temperature has a strong effect on the phase delay. In general the smaller the period, the greater is the phase delay. The effect of the oscillating period is even more pronounced for thicker ice-layers. The agreement between the two results is good. In Figure 15.11(b) the amplitude of the interface is plotted as a function of the ice-layer thickness. As the ice-layer thickness increases, the amplitude of the interface becomes smaller. The oscillation period gives an equally significant effect. The agreement between the experiment and the one-dimertsional numerical model is again good. Although the results are not presented in these figures, it is worth mentioning that we have also conducted two-dimensional numerical calculations for all the cases. As speculated from the good agreement between the one-dimensional and two-dimensional models.

414

SOLIDIFICATION IN A WATER-SATURATED POROUS MEDIUM W 60

10

o Experimental -*- Numerical

59 ? 58H

u 4 ^^

/

(U

s

S3 56

2

I 55

0

t: 54

-? S

1^ 53 I

H^

hn

-6

-8

51 50 0

1 Time t [hour] 2

b) ca ' 6059-

3

^.A^7 S .

^ ^ '.V ^v, ^ - ^ \

f—y

I6 u

^^^^

. ^^^^/^^ . ^ ^

o o U

-10

o Experimental IT 10 -^ Numerical |h 8

i^ 58< "^^

^. S 56-

ex

-4 c

r"^

52

^ 57-

CJ

4 ^Xc^pP^ ^si^^V^

K"


L

2

G

0

C3

1^ 55-

h0

r
[- - 7 s

43

>> 53^« -2 — 0) J^ 5251^0-

— J

n

r^

T<,r 1

1—

1

1

1

Ou

ton

-4 c -6

1-

o o

u

-10

^ Time/[hour] ^^ Figure 15.10 Oscillating ice-layer with periodic cooling temperatures. Comparisons between the experimental results (d = 2 mm) and the one-dimensional numerical model: (a) Tc = - 6 ° C , P = Ihour, (b) Tc = - 6 ° C , P = 2 hours, and (c) Tc = - 6 ° C , P = 3 hours. (Kimuraer^/., 2004a.)

415

S. KIMURA

10 (b)

20

30 40 50 Ice-layer thickness/[mm]

60

20

30 40 50 Ice-layer thickness/[mm]

60

12 10

I 4 2i

10

Figure 15.11 Comparison between experimental results and the one-dimensional numerical model for three different periods P: (a) phase lag as a function of the icelayer thickness, and (b) amplitude of oscillating interface as a function of the ice-layer thickness. (Kimura et al, 2004a.)

which is demonstrated in Figure 15.5, two-dimensional numerical results follow closely the one-dimensional numerical and experimental results.

15.5

CONCLUSION

Solidification in a box of water-saturated porous medium cooled from the top and with a bottom boundary kept at a constant temperature above the solidification point has been studied numerically and experimentally. A particular interest is placed on the periodic

416

SOLIDIFICATION IN A WATER-SATURATED POROUS MEDIUM

growth and shrinking of the ice-layer, when the cooUng temperature oscillates. Both onedimensional and two-dimensional numerical models are developed, and it is demonstrated that the two numerical models agree well. They are also tested against experimental results. The following concluding remarks can be drawn. (i) The one-dimensional numerical model with a constant heat flux from the unfrozen region is capable of predicting two-dimensional numerical results. Agreement between the numerical and experimental results is good. (ii) The amplitude of the interface oscillation becomes greater as the period of the cooling temperature increases. (iii) The phase delay behind the cooling temperature oscillation in time becomes larger as the period is reduced.

REFERENCES Bejan, A. and Klaus, A. D. (eds) (2003). Heat transfer handbook. Wiley, Hoboken, NJ. Blake, K. R., Bejan, A., and Poulikakos, D. (1984). Natural convection near 4°C in a water saturated porous layer heated from below. Int. J. Heat Mass Transfer 27, 2355-64. Carslaw, H. S. and Jaegar, J. C. (1959). Conduction of heat in solids (2nd edn). Oxford University Press. Chellaiah, S. and Viskanta, R. (1988). Freezing of saturated and superheated liquid in porous media. Int. J. Heat Mass Transfer 31, 321-30. Chellaiah, S. and Viskanta, R. (1989). Freezing of water-saturated porous media in the presence of natural convection: experiments and analysis. Trans. ASME J. Heat Transfer 111, 425-32. Echigo, R., Hara, T, and Hirata, M. (2002). An analysis on frozen-melting of a system in surface periodic temperature change (thermal behavior in permafrost area). Trans. JSME,Ser. 568,1531-6. Echigo, R., Hara, T., and Hirata, M. (2003). An analysis on frozen-melting of a system in surface periodic temperature change (dual periodic temperature changes with daily and annual cycles). Trans. JSME, Sen 5 69, 1673-8. Flemings, M. C. (1974). Solidification processing. McGraw-Hill, New York. Fowler, A. C. (1997). Mathematical models in applied sciences. Cambridge University Press. Frivic, R E. and Comini, G. (1982). Seepage and heat flow in soil freezing. Trans. ASME J. Heat Transfer 104, 323-8. Goldstein, M. E. and Reid, R. L. (1978). Effect of fluid flow on freezing and thawing of saturated porous media. Proc. Roy. Soc. Lond., Series A 364, 45-73. Hashemi, H. T. and Sliepcevich, C. M. (1973). Effect of seepage stream and artificial soil freezing. ASCEMech. Found Div. 99,267-89. Hibler III, W. D. and Walsh, J. E. (1982). On modeling seasonal interannual fluctuations of arctic sea ice. J. Phys. Oceanog. 12, 1514-23. Ingham, D. B. and Pop, I. (eds) (1998). Transport phenomena in porous media. Pergamon, Oxford.

S.KIMURA

417

Kimura, S. and Vynnycky, M. (1999). Time history of ice-layer formation at the cooled top boundary and its dynamic response to the time-varying cooling temperature. In Moving boundaries V (eds B. Sarler, C. Brebbia, and H. Power), pp. 47-56. WIT Press, Southampton. Kimura, S., Okajima, A., Kiwata, T., and Fusaoka, T. (2003). Time history of ice-layer thickness in a saturated porous medium due to time-varying cooling temperature. In Proceedings of the 6th ASME/JSME joint thermal engineering conference. Paper TED-AJ03-584. Kimura, S., Okajima, A., Kiwata, T., and Fusaoka, T. (2004a). Solidification in a water-saturated porous medium when vigorous convection is present (response of solid-liquid interface to the cooling temperature fluctuation). Trans. JSME, Sen B. In press. Kimura, S., Okajima, A., Kiwata, T., and Nakamura, T. (2004b). Characteristics of solidification and melting in the water-saturated porous medium cooled from the top (response of solid-liquid interface due to time-varying cooling temperature). Heat Transfer—Asian Res. 33, 3 3 0 ^ 1 . Kurz, W. and Fisher, D. J. (1992). Fundamentals of solidification. Trans Tech Publications, Switzerland. Maykud, G. A. and Untersteiner, N. (1971). Some results from a time-dependent thermodynamic model of sea ice. /. Geophys. Res. 76, 1550-75. Nield, D. A. and Bejan, A. (1999). Convection in porous media (2nd edn). Springer-Verlag, New York. Okada, M. (1981). Approximate analysis of freezing around two cooled pipes in Darcy flow. Refrigeration 56, 3-13. O'Neill, K. and Albert, M. R. (1984). Computation of porous media natural convection and phase change. In Finite elements in water resources (eds J. P. Laible, C. A. Brebbia, W. Gray, and G. Pinder), pp. 213-29. Springer, Berlin. Oosthuizen, P. H. (1988). The effect of free convection on steady state freezing in a porous medium-filled cavity. ASME HTD 96, 321-7. Saitoh, T. (1978). Numerical method for multi-dimensional freezing problems in arbitrary domains. ASME J. Heat Transfer 100, 294-9. Sasaki, A. and Aiba, S. (1992). Freezing heat transfer in water-saturated porous media in a vertical rectangular vessel. Wdrme- und StojfUbertr. 27, 289-98. Sasaki, A., Aiba, S., and Fukusako, S. (1990). Numerical study on freezing heat transfer in water-saturated porous media. Numen Heat Transfer, Part A 18, 17-32. Seki, N., Fukusako, S., and Nakaoka, M. (1976). An analysis of free convective heat transfer with density inversion of water between two horizontal concentric cylinders. Trans. ASME J. Heat Transfer 98, 670-2. Sugawara, M., Inaba, H., and Seki, N. (1988). Effect of maximum density of water on freezing of a water-saturated horizontal porous layer. Trans. ASME J. Heat Transfer 110, 155-9. Turcotte, D. L. and Schubert, G. (1982). Geodynamics. Wiley, New York. Wettlaufer, J. S., Worster, M. G., and Huppert, H. E. (1997). Natural convection during solidification of an alloy from above with application to the evolution of sea ice. /. Fluid Mech. 344, 291-316.

16

APPLICATION OF FLUID FLOWS THROUGH POROUS MEDIA IN FUEL CELLS L. MA*, D. B. INGHAMt and M. C. FGURKASHANIAN^ * Centre for Computational Ruid Dynamics, University of Leeds, Leeds, LSI 9JT, UK email: [email protected]

^Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, UK email: aint6dbi©maths. leeds .ac.uk * Energy Resources Research Institute, University of Leeds, Leeds, LS2 9JT, UK email: f ue6mtz@leeds .ac.uk

Abstract A fuel cell is a multicomponent power generating device which relies on chemistry, rather than combustion, to convert chemical energy into electricity. The key components of the fuel cell are made of porous materials through which fuel and oxidant are delivered to the active site of the cell where electrochemical reactions take place to generate power, heat and water Fuel cell technology presents a huge economical and environmental potential in the future power markets, 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 fuel cell design. 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 where appropriate. Keywords: fuel cell, CFD model, SOFC, PEMFC, porous media, electrochemical reactions, catalysts, power generation

418

L.MAETAL.

16.1

419

INTRODUCTION

Fuel cells offer a unique combination of high efficiency and ultra-low emissions for converting hydrocarbon fuels into electricity. At present, a conversion efficiency of about 26% may be achieved by an internal combustion engine and about 30% in a CHP system using traditional power generation techniques. However, fuel cells are expected to achieve an energy efficiency of over 40% and a remarkably higher efficiency of about 80% in a fuel cell turbine hybrid CHP system, see, for example, George (2000) and Veyo et ah (2002). In addition, fuel cells have virtually no emissions if pure hydrogen is used as the fuel. This is because the by-products of the fuel cells consist primarily of water and steam, see, for example, Hoogers (2003). Unfortunately, today's fuel cells are still too expensive for general commercial use and have many technique barriers to overcome before they could compete with the well developed traditional power system. In recent years, significant research and development in the fuel cell technology have been carried out in terms of reducing production cost and increasing power density of the fuel cell products in order to make them viable for commercial applications and thus benefit from their advantages. It is anticipated that with the further development and improvement in fuel cell materials and cell designs, fuel cells will be able to compete with combustion engines in transportation applications and for stationary power supply at locations off the electrical grid, even when they can also compete head-on with grid power with an advanced CHP hybrid system.

16.2 OPERATION PRINCIPLES OF FUEL CELLS The concept of the fuel cell was invented in 1839, see, for example, Singhal and Kendall (2003). The distinct difference from a combustion engine is that the operation of the fuel cell relies on the chemistry, instead of combustion, to combine fuel and oxygen to create electricity and therefore it is more efficient than the combustion engine. At present, there exist several different types of fuel cells and all have similar geometrical structures and share the same basic operating principles. A fuel cell typically consists of two electrodes, namely the anode and the cathode, and an electrolyte which is sandwiched in between the two electrodes, see the schematic diagram in Figure 16.1. The fuel and oxidant are fed at the side of the electrodes. The electrodes are made of porous materials so that the fuel and oxidant can penetrate to the location near to the surface of the electrolyte where the electrochemical reactions take place in the presence of the catalysts. The fuel, usually hydrogen or hydrogen rich hydrocarbons, transports through the porous gas diffusion layer to the catalyst layer on the anode side of the electrolyte where electrons are stripped from the fuel and form ions and electrons. In the case of a PEMFC, the electrons make their way from the anode to the cathode through an external circuit to drive a load and the remaining ions travel through the ionic conducting electrolyte to the cathode. At the same time, the oxygen is fed from the cathode gas channel, penetrates the cathode gas diffusion layer and reaches the catalyst layer in the cathode side next to the electrolyte. The oxygen combines with the emerging hydrogen

420

APPLICATION OF FLUID FLOWS THROUGH POROUS MEDIA IN FUEL CELLS

Gas channel

Anode

Electrolyte

Cathode

Gas channel

Figure 16.1 A schematic diagram of a hydrogen-oxygen fuel cell.

ions and the free electrons coming form the anode to from the by-products, typically heat and water. Figure 16.2 shows an electron micrograph of the cross-section of an SOFC developed by Siemens Westinghouse, see Ormerod (2003), where the distinguished structures of the electrodes and the electrolyte can be clearly observed. The SOFC uses an oxygen ion conducting electrolyte which works at a relatively high temperature. Therefore, in the case of the SOFC, it is the oxygen ions in the cathode side that is conducted to the anode side and the main reduction reaction occurs at the anode side to form water. Although detailed chemistry may vary in different types of fuel cells, see, for example, Larminie and Andrew (2000), the fundamental overall reaction is the same, i.e. the

Fuel electrode ^ f

J^ . .

C , ^ /

^.^N*^*-^'J Electrolyte

Air electrode Figure 16.2 An electron micrograph of the cross-section of an SOFC developed by Siemens Westinghouse, see Ormerod (2003).

421

L.MAETAL.

exothermic hydrogen-oxygen reaction, and it may be expressed as follows: (16.1)

2H2 + 0 2 - ^ 2 H 2 0 + A E ,

where AE" denotes the heat produced in the reaction. The maximum electrochemical work that a fuel cell can do may be determined using the Nemst equation as follows:

'^=^-5'° n^r

(16.2)

where E is the electric potential that is produced by the electrochemical reaction, E^ is the potential under a standard reference pressure, usually p^ = 1 atm, and may be calculated from the changes in the free Gibbs energy of the reaction, R is the gas constant, T is the temperature, pk is the partial pressure of the reactant k in the mixture, m is a constant related to the stoichiometric coefficient of the species, Ue is the number of electrical charges (electrons or protons) transferred in the reaction and F is the charge carried by a mole of electrons (or protons), known as Faraday's constant. The Nemst equation (16.2) represents the ideal maximum voltage potential that can be achieved in a fuel cell. However, in reality, there are certain irreversible losses that reduce the magnitude of the maximum potential that a fuel cell can produce. Typically, there are three major losses in the system, namely the activation losses due to the kinetic resistance, the ohmic losses due to the material ohmic resistance and the concentration losses due to the limitations of the species diffusion through the porous electrodes, see Figure 16.3. Thus the voltage produced by a fuel cell is always lower than the ideal voltage predicted

1.25 Irreversible ideal voltage • Open-circuit losses Porosity increase

Concentration losses

0.25

0.25

Figure 16.3

0.5 0.75 Current density [A/cm^j

1.25

Schematic diagram of the potential polarisation in a fuel cell.

422

APPLICATION OF FLUID FLOWS THROUGH POROUS MEDIA IN FUEL CELLS

by the Nemst equation and it is typically expressed as follows: Kell — E - r/act - ^ohmic " ^con ,

(16.3)

where Keii is the cell voltage, E is the reversible cell potential given by the Nemst equation, ^ohmic is the ohmic overpotential, r/act is the kinetic overpotential at the anode and the cathode, and r/con denotes the concentrate overpotential. The kinetic or the activation losses are the major losses when the fuel cell operates at a low current density whilst the other two losses are small. For a well-designed fuel cell working at the designed operational condition, additional losses in cell potential are mainly due to the ohmic resistance of electrical conducting material and it is proportional to the current density in the fuel cell. However, at high current density, when the speed of the consumption of the reactants is faster than the speed of the delivery of the fuel and/or oxidant by the system of species diffusion, then the limiting effect of the concentration of the reactants on the performance of the fuel cell will appear. It is clear that the performance of the fuel cell ultimately relies on the effective delivery of the fuel and oxidant through the porous electrodes to the active catalytic reacting site where the electrochemical reactions take place to produce electricity, heat and the by-product, water. If a fuel cell is poorly designed, in particular if when the reactants could not be efficiently delivered to the reacting site then the speed of the diffusion process, at which the fuel and oxygen species are transported through the diffusion layer to the reaction surface of the catalyst layer, will control the electrochemical processes. Figure 16.3 schematically shows the effect of the species transport in the electrodes on the performance of the fuel cell. Usually, the limitations of the reactant transport to the cell power output are not noticeable until a large operating current density occurs. However, if the permeability of the porous media of the electrodes is too small, or the pores of the electrodes are blocked by water, then an insufficient supply of oxidant to the catalyst site will appear. This results in an earlier deterioration of the performance of the fuel cell. In the worst situation, if no reactant is able to reach the active site, such as when the pores of the electrodes are flooded by the presence of an excessive amount of water, then the fuel cell will lose its function. Therefore, the investigations on the fluid flows and heat and mass transfer in the porous electrodes are of vital importance in the design and the optimization of fuel cells. There are a number of distinctive differences in the nature of the fluid flow in the porous electrodes of the fuel cells over the traditional fluid flow through porous materials, such as the fluid flows through packed bed materials or in underground oil/water flows. Firstly, the length scales of the porous media in the fuel cells are extremely small and they vary considerably within different components of the fuel cell. Secondly, due to the nature of the manufacturing process and the material that is used to make up the fuel cell, the porous media used in some fuel cells, such as the PEMFC, shows significant anisotropy and/or spatial inhomogeneity. This significantly influences the behaviour of thefluidflow in the porous electrodes. Thirdly, species transport in fuel cells typically involves more than two species and therefore there exists a multicomponent species transport where the concentration of one species may significantly influence the transport of the others and this only adds to the complexity of the species transport process.

L. MA ET AL.

423

16.3 GOVERNING EQUATIONS FOR THE FLUID FLOWS IN POROUS ELECTRODES From a mathematical modelling point of view, the fluid flow and the heat and mass transfer in the fuel cell can be described by the typical conservation equations for the thermal reacting fluid flows, such as the Navier-Stokes, energy, mass conservation and the associated chemical reaction equations. For the electric current flow that is generated in the fuel cell, the conservation equation of charge may be employed and it governs the current density distributions and the overall potentials in the fuel cell. All the transport processes occurring in fuel cells are ultimately determined by the operating conditions of the fuel cell, i.e. the electric current that goes through the fuel cell and this is usually defined as a boundary condition to the governing set of the transport equation for the fuel cells. 16.3.1 Equations for the fluid flow and mass transfer in fuel ceUs Mathematically, the fluid flow in the porous electrodes may be expressed by the well known Navier-Stokes and the mass conservation equations. In writing these equations, as well as doing mathematical calculations, two representative fluid velocities are in use, namely the superficial fluid velocity, or the Darcy averaged velocity, and the physical fluid velocity, or the real velocity. The superficial fluid velocity is defined as if there is no solid matrix present. Thus the relation between the superficial fluid velocity and the physical fluid velocity may be given as follows: ^superficial — T^physical j

(lo.4)

where 7 is the ratio of the void area to the whole cross-sectional area. In a homogenous matrix, 7 is equal to the porosity of the matrix. Employing different representative velocities produces slight differences in the form of the equations. In this chapter, the superficial fluid velocity is employed. Written in tensor notation in the Cartesian coordinate system, the Navier-Stokes and the continuity equations for the fluid flow through an isotropic porous medium may be expressed as follows: 1 djpu') e dt

1 djpu'u^) _ e^ dx^

dp dx^

d dx^

7 \dx^^'^^)

+ Su,

(16.5)

where x^ is the component of the Cartesian coordinate system, t is the time, u\p, p and p are the fluid velocity components, pressure, density and molecular viscosity of the fluid flow, respectively, and e is the porosity of matrix. These equations are also valid for the fluid flows through gas channels of the fuel cells where we have e == 1.

424

APPLICATION OF FLUID FLOWS THROUGH POROUS MEDIA IN FUEL CELLS

The first term on the left-hand side of equation (16.5) represents the rate of change of momentum per unit volume with time and the second term represents the change of momentum resulting from the convective motion. On the right-hand side of equation (16.5), the first term is the force resulting from the pressure differences in the fluid flow and the second term is the viscous shear forces resulting from the motion of the fluid. Further, Su denotes the extra resistance force on the fluid flow due to the presence of the porous matrix. This momentum sink contributes to the pressure gradient in the porous media, creating a pressure drop that is usually proportional to the fluid velocity in the application of the fuel cells. The source Sm on the right-hand side of equation (16.6) is the mass added to the continuous phase from, for example, the source of the chemical species, or the change in mass due to the condensation/vaporization of a second phase such as water. Depending on the strategies employed in a mathematical model to connect different regions of thefluidflowsin a fuel cell, Sm may be used to reflect the effect of the electrochemical reactions and/or mass transfer across the boundaries of each component. Since the porosity and the permeability of the porous part of the electrodes, such as the gas diffusion and the catalyst layers, are usually very small, this produces a very large resistance to the fluid flow and results in an extremely low fluid velocity. Thus the convective acceleration and the diffusion terms appearing in the Navier-Stokes equation are relatively small and thus can often be ignored. Then the momentum equation (16.5) for the fluid flow through the porous matrix of the electrodes reduces to the Darcy law as follows:

g = -5„^-|uS

(16.7)

where K is the permeability of the matrix. Whilst the Navier-Stokes equation and Darcy's equation are primarily responsible for the convective transport of the reactants, the diffusion processes, which are significant in the transport of chemicals in a gradient species concentration field in fuel cells, are modelled by the species transport equation. The transport of reacting and non-reacting species throughout the fuel cell follows the conservation law of chemical species which can be written as follows:

djpYk) Idjpu'Yk) d f dYk\ -dr^-e—d^ = d^i[f^''-d^)+^'^

fc

= l,2,...,Ar-l,

(16.8)

where Yk and Sk denote the mass fraction and the production/consumption of the species k, respectively, Dk is the diffusivity of the species k, and A^ denotes the total number of chemical species present in the system and is dependent on the types of the fuel and oxidant that feed the fuel cell. For a simple hydrogen-oxygen fuel cell, the species in the system typically include as least hydrogen, oxygen and water, and in this situation N = S. The production/consumption of the fuel and oxidant due to electrochemical reactions are represented in the species and mass conservation equations by the source/sink terms. These terms are proportional to the electronic current that is produced by the electrochemical reaction. For hydrogen, oxygen and water species the source/sink terms may be expressed.

425

L. MA ET AL.

in a volumetric basis, respectively, as follows:

2

Q

J

*-^H2 — ~7^2F 5

C

'^02 — ~T^

•'

C

'

*^H20 —

2F

(16.9)

where j is known as the electronic charge transfer current density representing the charges generated by the electrochemical reactions. The electrochemical charge transfer process in an electrode is an activation-controlled process and the charge transfer current density it generates is determined by the effectiveness of the catalysts and the concentrations of the reactants that are available at the reaction sites. The kinetics of the catalytic electrochemical reaction in the fuel cell is usually described by the Butler-Volmer equation. Taking into consideration the effect of the concentrate polarisations, the Butler-Volmer equation for the anode and the cathode of a hydrogen-oxygen fuel cell may be expressed, respectively, as follows, see, for example. Bard and Faulkner (1980) and Beming et ai (2002): 1/2

. _ .ref / £Hi

Ja — Jo,a I ^ef

. _.reff£0^'

Jc - Jo,c I ref

exp exp

agF ^act,a RT

agF RT ^act,c

exp

- exp

OcF - RT ^ a c t , i

(16.10) (16.11)

where c denotes the concentration of the species at the reaction site, a is known as the anodic transfer coefficient (0 < a < 1) and it usually takes the value of 0.5,77act is the active overpotential representing the voltage loss due to electrochemical reaction, and the subscripts a and c denote the anode and cathode of the fuel cell, respectively. The value of the reference current density JQ^^ is catalyst material and operational-condition dependent and it usually has to be obtained experimentally. In order to calculate the transfer current density produced in an electrochemical reaction, the magnitude of the active overpotential of the reaction must be known which varies with the catalyst that is used and the operational condition of the fuel cell. At the moment, information on the overpotential for a particular catalyst has to be obtained experimentally. However, it can be estimated from the data of the open circuit voltage of the fuel cell as follows, for the anode and the cathode, respectively, see, for example, Meng and Wang (2004): (16.12) ^act,a = ^ e - ^ i ^act,c = ^ e - ^io K, where $e and $ion are the electrical potential and the ionic potential, respectively, at the interface between the solid catalyst layer and the membrane, Foe is the cell open circuit potential that has to be obtained experimentally. 16.3.2 Heat generation and transfer in fuel cells Heat generation, transfer and balance are important issues in both low and high temperature fuel cells. A low temperature PEMFC could not withstand a high temperature as the

426

APPLICATION OF FLUID FLOWS THROUGH POROUS MEDIA IN FUEL CELLS

polymer electrolyte relies on liquid water to keep it active. Heat balancing in high temperature SOFC becomes even crucial as severe nonuniform or sudden changes in the temperature may cause crack or even a structure failure to the whole fuel cell system. The heat transfer in the fuel cell is a combination of the heat transfer through the fluid flow and the heat transfer through the solid matrix and it is governed by the energy conservation equation which may take the following form:

(16.13) where p, Cp and k are the density, specific heat and the heat conductivity of the fluid, respectively, ps, Cp^s and kg are the density, specific heat and the heat conductivity of the solid matrix, respectively, and SE is the energy source that is generated within the fuel cell. In the gas channel, the convective heat transfer will be the dominate heat transfer process. However, in the solid electrical conducting elements, such as the solid current collector, the conduction is the sole heat transfer process. In the porous electrodes and/or membrane, both conduction and convection will be important. Heat generations in a fuel cell may come from a number of mechanisms. Most of the heat comes from the exothermal chemical reactions, which combine hydrogen and oxygen to form water, and the electrochemical activation (kinetic) losses that split species into electrons and ions in the catalyst layer. These heat sources are also major sources of the non-uniformity of the temperature distribution across the cell. In addition, the ohmic losses in the electronic conducting elements, due to ohmic resistance of the solid materials, also produce a significant amount of heat, particularly when the cell is operating at a high current density. Clearly, all these heat generation mechanisms are linked to the current density of the fuel cell in operation. The heat generation due to the chemical reactions may be estimated by the change in the entropy, A 5 , of the system and the current density as follows, see, for example, Lampinen and Fomino (1993) and Beming et al. (2002):

The heat source due to the activation losses is given by Ssict = m c t ,

(16.15)

where a^^ is the effective electronic conductivity of the cell, and the ohmic losses are directly proportional to the current density and are given by Soh-^,

(16.16)

where i is the local current density. In the situation where a phase change occurs, such as those that exist in low temperature fuel cells where water may experience condensation and/or evaporation, then additional

L. MA ET AL.

427

heat source/sink terms may be required to represent the heat release and absorption during the process of phase changing. 16.3.3 The electric field in fuel cells The ions and electrons generated during the electrochemical reactions in the fuel cell produce an ionic potential field within the ionic conducting elements and an electrical potential field within the electronic conducting elements. The transport of the ions and electrons within the fuel cell are governed by the conservative law of ions and charges. Thus the ionic potential, $ion, and the electric potential, $e»should follow the conservation equations which may be written in the form, respectively, as follows: _d_

dx^

' dx^

+ S<,,on-0,

(16.17)

+ 5c^, - 0 ,

(16.18)

where af^^ and cr|^ are the effective ionic and electrical conductivity, respectively, and 5«|.i^„ and 5$^ are the sources of the ionic and electric potentials, respectively. Since the only source of ions and electrons production is the electrochemical reaction taking place within the catalyst layers. Therefore, the magnitude of the sources of potentials within the catalyst layer should be equal to the local transfer current density, j , which may be determined using the Butler-Volmer equation (16.10) or (16.11). In other regions of the fuel cell, the source terms should be equal to zero. Therefore, we have the following expressions for the catalyst layer, see, for example, Um et ah (2000) and Meng and Wang (2004): 5 $ i o n = i , S^^=-j, (16.19) and in other regions of the cell we have •5<^ion-0,

5c^, = 0 .

(16.20)

Once the electrical potential has been obtained then the local current density in the electric conduction components of the fuel cell may be determined using the following equation: i = - e r f V$e •

(16.21)

16.4 MULTICOMPONENT GAS TRANSPORT IN POROUS ELECTRODES Species transport in the fuel cell is a combination of convection and diffusion processes. The convective transport governs the motion of the mixture of different species through the porous medium where the mixture is treated in the same manner as a pure gas. This is usually mathematically modelled by using Darcy's law and its various modifications.

428

APPLICATION OF FLUID FLOWS THROUGH POROUS MEDIA IN FUEL CELLS

The diffusive transport is on the other hand responsible for the diffusion of the gas species relative to the convective transport and Pick's diffusion law is often employed for a binary species diffusion. For the more complex diffusive process where more than two species are involved then the Maxwell-Stefan equations may be employed for the multicomponent diffusion. However, in the situation of the fuel cell, the species transport through various components cover a broad range of length scales stretching from microscale pores of a complex porous electrode to macroscale gas channels. The thickness of a typical electrode is about 10~^ to 10~^ m with the size of the pore of 10~^ m, whilst the length of the gas channel may be in the range of centimetres. This vast range in length scales presents a challenge to the modelling since thefluidflowsand the mechanism of the species transport in these different length scales are very different and the interfacial conditions between the different components have to be considered with special care. 16.4.1

Convective transport

A typical approach of modelling fluid flows through porous media is to use Darcy's law, such as equation (16.7). However, the use of Darcy's law has its limitations when modeUing fluid flows in fuel cells and this is because the fuel cell consists of a number of distinct layers of different porosities. The Darcy law assumes no effect of boundaries and the fluid velocity in Darcy's equation is determined by the permeability of the matrix. Therefore at the interface between the regions of different porosity in the fuel cell, particularly between the free fluid flow region such as a gas flow channel and a permeable medium, then a discontinuity in the fluid velocity and/or the shear stress could emerge. At a solid wall boundary, the fluid velocity will not reduce to the no-slip condition when the Darcy law is enforced. Therefore, in this situation, the Brinkman equation may be employed, which is an extension of the Darcy law and facilitates the matching of boundary conditions between the different regions of different levels of permeability. The Brinkman equation has the following form:

where /ig is a parameter that is used to match the shear stress boundary conditions across the interface between such as a free fluid and a porous medium, i.e. du

du

(16.23)

where y = 0'^ and y = 0~ represent the free fluid and the matrix sides of the interface, respectively. In the sense that Darcy law is only valid in the portion of the fluid flow where the fluid velocity is so small that the inertial terms can be neglect, the Brinkman equation may be used in the region with a relatively larger porosity where the inertial term is not so small. It is worth mentioning that the accurate modelling of the fluid flow and the mass transfer in the vicinity of the interface between a gas channel and a porous matrix is

L.MAETAL.

429

Still a very changing area of research and requires further investigations, see, for example, Alazmi and Vafai (2001). In addition, elements of the fuel cells that are made of fibrous porous material, such as those used in PEMFC, are highly anisotropic in permeability and the characteristics of the fibrous materials used are very different from the packed bed, carbon filters and/or the soil in underground flow where the porous media are typically composed of more or less uniform grains. Due to the process of making up the gas diffusion layers of the electrodes of a PEMFC, the fibres are usually oriented so that their axes lie primarily in the plane of the paper. Since the fluid flows along the axis of the fibers is easier than that across the fibers, thus the in-plane permeability can be significantly greater than that in the through-plane directions and they may differ by up to a factor of 20, see, for example, Stockie et al. (2003). Therefore, in this situation the permeability K in equation (16.7) should be taken as a tensor quantity to facility the anisotropy of the matrix. 16.4.2 Diffusive transport The diffusive transport in the porous electrodes is a much more complex process than the convective transport. The electrochemical reactions, taking place in the so called three boundary regions within the catalyst layer, consume the reactants to generate the end products. This leads to the gradients in the component concentrations within the catalyst layer and across the gas diffusion layer. The species diffusive transport is primarily driven by the concentration gradients. Species diffusion is frequently modelled using Pick's law which states that the species diffusive flux is directly proportional to its concentration gradient and can be expressed as follows:

4 =-^.|^-

(16.24)

The species diffusivity Dk itself is an intrinsic property of the gas mixture. However, in order to take into consideration the effect of the porous matrix on the diffusive transport, it is usually replaced by the effective diffusivity, D^^, that may include the effect of the porosity, e, and the tortuosity, r, of the matrix, as follows: i^eff = -Dk . (16.25) r In addition, the Bruggeman correction is also frequently used to estimate the effective diffusivity in a porous medium which is defined as follows: i^eff = e^-^Dk .

(16.26)

In the modelling of the species diffusion in an anisotropic medium, the diffusivity coefficient -Deff should be taken as a tensor quantity with different values of diffusivity in each direction.

430

APPLICATION OF FLUID FLOWS THROUGH POROUS MEDIA IN FUEL CELLS

Strictly speaking, Pick's law is valid only for binary mixtures. However, in fuel cells usually at least three species exist in the system and therefore the species transport is a multicomponent diffusive process in which the flux of one component is influenced by the concentration gradient of other components. Therefore, in this situation the MaxwellStefan equations may be used to account for the cross-coupling between multi-species components, see, for example, Tayler and Krishma (1993) and Beming et al. (2002): dXk

^

XiX

a;r = E^;;7K-v),

06.27)

where X is the mole fraction, v denotes the component of the diffusive velocity of the species and Dim is the binary diffusivity of any two species / and m. It is noted from equation (16.27) that the Maxwell-Stefan equation gives the concentration gradient of each species in terms of the diffusion of other spices. However, the mass conservative equation (16.8) is usually written in the form of Pick's law expressing the species diffusion as a function of the gradient of the species concentration. Therefore, the Pick diffusivities usually have to be calculated from the Maxwell-Stefan expression and this can be at a great computational expense, see, for example, Stockie et al. (2003). Another issue that may add to the complexity of the modelling of the species diffusion in fuel cells is the microscale pores existing in the porous electrode. The length scale of the pores is usually of the same order to that of the mean free path of the molecules in the mixture. In this situation, the diffusing molecules collide with the walls of the pore, known as the Knudsen diffusion, with about the same frequency as they collide with other molecules. Therefore, the effect of Knudsen diffusion on the species diffusion may need to be considered. This may be achieved by using the Bosanquet formulae which is in the Pick form as follows, see, for example, Thomson (2000): P — —

J_ Dk

1

^Knudsen

"•§.

where I^Knudsen is the Knudsen diffusion coefficient which is proportional to the pore size and the mean molecular velocity. However, due to the complexity of the geometry of most porous media and the pore structures and geometry have a significant impact on the Knudsen diffusion. Therefore, the Knudsen diffusion coefficient usually have to be obtained experimentally or using empirical data. It should be noted that, in general, issues concerning the diffusion of species in the microscale pores of the porous electrodes in fuel cells are still under intensive investigation and the values of diffusivity and other material related constants for the CPD models usually have to be obtained by experiment and/or using empirical values. In particular, the effect of anisotropy and inhomogeneity of the porous material that makes up the electrodes on the diffusivity of a gas should be carefully investigated, see Stockie et al. (2003). In addition, accurate information about the permeability of the matrix is not always easy to obtain. Even at a given porosity of the matrix, the permeability is usually a function of the fluid flow, temperature and the material of the matrix and the chemical compositions

L.MAETAL.

431

of the fluid at the microscale-nanoscale pores. In general, simulating the fluid flow in porous media is at present still very scientifically challenging and much of the detailed mechanisms of the flow in the pores of the matrix is still not fully understood, see, for example, Ingham and Pop (1998, 2002).

16.5

CFD MODEL PREDICTIONS OF FUEL CELLS

In the CFD modelling of thefluidflowsthough the porous medium in fuel cells, the system of governing partial differential equations that has been described in the previous sections may be solved by using the general CFD numerical solution techniques, such as the control volume method or finite element method. Most general purpose CFD software packages have these numerical facilities built in. Thus only the submodels and user defined transport quantities that are unique to the physical, chemical and electrochemical processes of fuel cells have to be specially developed. These submodels calculate the necessary source/sink terms, boundary conditions and the fluid flow parameters that are required by the general conservative governing equations. Since the mathematical model of the fuel cells include a complex system of equations describing the fluid flow, heat and mass transfer, chemical and electrochemical reactions and the transport of the electric current, a multi-loop iteration process is often required in order to acquire a stable numerical solution. A general numerical approach for solving the complex fuel cell problems may be schematically described in Figure 16.4. The solution process begins with specifying the appropriate boundary conditions and an initial guess to the solution. Then the governing partial differential equations are solved based on the given boundary conditions and initial settings. The solutions of the partial differential equations form the necessary input for the fuel cell specific submodels for the electrochemical reactions, ionic/electric field, overpotentials, heat and species generations/consumptions, etc. in the fuel cell. These submodels provide update to the source terms, as well as the relevant parameters and physical properties for the governing partial differential equations. The convergence of the iteration is checked, if the residue of the solution is larger than a prescribed tolerance then the calculation should continue with the updated source terms and physical properties until the convergence criteria is satisfied. The fluid flows in fuel cells are inherently three-dimensional. While a one and/or two dimensional analysis is useful in guiding the general design of the cell, see, for example, Costamagna and Honegger (1998), Gurau et ai (2000), Mann et al. (2000) and Li and Chyu (2003), many fuel cell modelling activities are now performing 3D simulations in order to obtain more accurate information on the fluid flows and electrochemical reactions within the fuel cell. Figure 16.5 shows the results of one of such calculations where the velocity vectors of the fluid flow are presented on a cross section perpendicular to the gas channel of the PEMFC investigated by Dutta et al. (2001). It can be seen from Figure 16.5 that a very complex three-dimensional flow structure may exist in the fuel cell with strong secondary flows in both the gas channel and the diffusion layer. Similar complex flow patterns have also been observed in a simulation performed for a composed SOFC

432

APPLICATION OF FLUID FLOWS THROUGH POROUS MEDIA IN FUEL CELLS

Boundary conditions and initial guess

I

Solve transport equations for species, momentum, temperature and ionic/electrical potential field

I

Update parameters and source/sink terms

Run fuel cell submodels to calculate electrochemical reactions, overall potentials and heat and species sources, water managements, etc.

Figure 16.4 CFD modelling strategy for fuel cells.

channel, see Yuan et al. (2003). Figure 16.5 also shows the coupling between the adjacent flow channels, and between the anode and cathode that occurs in the case of PEMFC due to the water transport through the membrane. Typically, the fluid density increases on the anode side when it travels from the inlet to the outlet of the cell and decreases on the cathode side as a result of the water transport. The presence of water is not a particular problem for high temperature fuel cells, such as SOFC, although it may cause corrosion to the electrodes due to chromium poisoning, see, for example, Singhal and Kendall (2003). However, for a low temperature PEMFC, a hydrated polymer membrane is used as the electrolyte and it must be in a highly hydrated state to facility proton transport since the conductivity of the electrolyte relies on its water contents which should be kept at an optimum level. If there is not enough water, the membrane becomes dehydrated and the proton conductivity decreases sharply. On the other hand, if too much water is present, the pores of the gas diffuser may be flooded by liquid water and it will block the transport of the reactants to the reaction sites. Therefore, water management in the PEMFC is a vital and difficult task. Water transport within the polymer electrolyte is controlled by two, usually opposite processes, namely the electro-osmotic drag and the back diffusion. When the PEMFC is in operation, hydrogen ions moving through the electrolyte from the anode to the cathode pull the water molecules within the electrolyte with them. This is known as the electro-

433

L. MA ET AL. (a) 0.003 i

0.0025 U

0.0021 0.0165

0.017

0.0175 0.018 Channel width [m]

0.0185 Reference vector (0.01ms-')

(b) ± t z : t ^ b^

-:nrs

E 0.0015

±^^^

g 0.001 i

0.0005

0.0165

0.017

0.0175 0.018 Channel width [ml

I ""I

r ' t

0.0185

•!

Figure 16.5 Typical velocity vectors and mixture density contours in the gas channel and the diffusion layers of the PEMFC in (a) the anode channel, and (b) the cathode channel, see Dutta et al. (2001).

osmotic drag. The larger the number of ions moving to the cathode then the more water v^ill be dragged along with them to the cathode. Figure 16.6 is a schematic diagram showing the basic concept of the water transport in the PEMFC fuel cell. In order to provide water to keep the membrane in a best hydrated condition, usually damped hydrogen is fed into the anode. Due to the effect of the electroosmotic drag, water contents decrease in the anode gas channel and increase in the cathode. At the same time, additional water is formed in the cathode because of the electrochemical reactions. Therefore water contents in the cathode side increase constantly. This may eventually produce such a large gradient in water concentrations between the cathode and the anode that the back diffusion of the water may overwhelm the forward electro-osmotic drag resulting in the water flowing back to the anode. Therefore, an increase in the water contents is often observed downstream of the anode gas channel. Water transport in a polymer electrolyte is very complex and is not a well known area of the fuel cell technology. Since the size of the pores of the polymers is often in the

434

APPLICATION OF FLUID FLOWS THROUGH POROUS MEDIA IN FUEL CELLS Water concentration decreasing

Water concentration increasing

Water concentration increasing Figure 16.6

Basic concept of the water transport in the fuel cell.

microscale, even nanoscale, which is about the same level of length scale of the free path of the gas molecules. Thus the interaction between the protons, water molecules and the pore walls strongly affect the behaviour of the water transport and this still an area requiring further investigations. At present, most existing water transport models for membranes still rely on empirical correlations often within a one-dimensional approximation, such as that proposed by Springer et al. (1991). Based on their own experimental investigation. Springer et (2/. (1991) suggested that the net water transport through the membrane can be represented as the electro-osmotic drag minuses the back diffusion of the water molecules as follows, see, for example, Springer et al. (1991) and Dutta et al. (2001): a{x,y) = nd{x,y) - 77—^Di^(x,t/)

i{x,y)

dz

(16.29)

where rid is the number of water molecules carried per proton due to thS electro-osmotic drag, which is linked with the water vapour concentrations and the saturated pressure, Cw is the water concentration across the membrane and D^ is the diffusivity of the water in the membrane. It should be noted that different forms of formulae for the water transport coefficient are in use, see, for example, Gurau et al. (2000) and Berg et al. (2004). However, in general, advanced water management model is an area that requires further investigation. It is noted that most fuel cell models that are described in the open literature assume a single phase approach so that the complexity of dealing with the phase changing and two-phase flow modelling is avoid. However, when the partial pressure of the water vapour exceeds the saturated vapour pressure, which usually occurs in the cathode, then liquid water will form and there will be a two-phase flow in the fuel cell. Figure 16.7 schematically shows the typical liquid water formations in the cathode gas channel and the gas diffuser where moisturised air is fed from the left end of the gas channel and discharged from the right, see, for example, Wang et al. (2001) and You and Liu (2002). The water mass fraction increases along the channel due to the production of water vapour by the reduction reactions in the catalyst layer. When the water partial pressure is higher than the saturated vapour pressure at the local temperature, then liquid water forms with a

L.MAETAL.

'•'

Moisturised air

^X>*- ' ^ t . ' ^ ^ , ^ ' * '^

'"'^^-0

^^^^

^IflMiS;^^

435

:*'-^'^'*'w-v";'

Condensatioir'''"'\ Saturation s^x^ increasing front

Gas channel

Figure 16.7 Typical water saturations in the cathode gas channel and gas diffuser of a co-flow PEMFC.

condensation front. The liquid saturation is zero in the single phase region and it increases along the flow direction after the condensation front and towards the catalyst layer. The existence of the liquid water in the catalyst and the gas diffusion layers blocks the effective reacting surface and the pores through which oxidant diffuses to the active surface. The immediate consequence of this is the deterioration in the cell performance, which usually shows up at the mid to high levels of current density when a high rate of water production is expected, see Li and Becker (2004). In this situation, employing a single-phase flow model is clearly inaccurate when describing a two-phase flow. There are two basic approaches of two-phase flow modelling that have been used in the fuel cell modelling. One is the multi-component mixture model in which the gas and liquid phases are regarded as constituents of a mass-averaged mixture of the two phases concerned. The flow of the mixture is described by a single set of governing equations, including conservation equations for the momentum, mass and species and the other necessary constitutional equations, and the liquid saturation is calculated from an equilibrium calculation, see, for example, Wang et al (2001) and You and Liu (2002). In the other approach, the formation and transportation of the water liquid and the water vapour are traced separately by using two transport equations and the liquid saturation is modelled via a transport equation with a water formation source term, see, for example. Nam and Kaviany (2003) and Li and Becker (2004). In addition to the bulkfluidflow,spatial distributions of the fuel and oxidant across the fuel cell is of predominant importance to the electrochemical reaction and the electrical potential over the fuel cell. Figure 16.8 shows the numerically obtained typical distributions of hydrogen and oxygen mass fractions in a co-flow PEMFC fuel cell. The electrochemical reactions occurring in the catalyst layer consume the reactants generating concentration gradients of the fuel and oxidant across the electrodes. In the flow direction, the contents of both fuel and oxygen reduce as a result of the reactions. The local concentrations of the fuel and oxygen are closely linked to the local current densities generated. The high current density is usually located in the area of relatively higher fuel and oxygen concentrations and at the same time showing a higher gradient in fuel/oxygen concentrations. For low temperature fuel cells, the variation of the temperature in the fuel cell are relatively small and therefore most CFD simulations of PEMFC employ an isothermal calculation. However, for high temperature fuel cells, such as SOFC, where large temperature dif-

436

APPLICATION OF FLUID FLOWS THROUGH POROUS MEDLV IN FUEL CELLS O2

•

0.23

SBHM^^^^^^^Bfc^^^;^^

|il|||^^

Anode

B 0.184 H 0.161

Membrane

™ 0.138

^BHlR'f'iA \i^ ^^^ Cathode

**-'"-^* 0.115 0.092 0.069 0 046 0.023

0

Figure 16.8 Typical mass fraction distributions of H2 and O2 in a co-flow PEM fuel cells.

ferences may occur which may lead to a structure failure of the whole fuel cell system. Therefore, predicting a temperature distribution within a SOFC fuel cell is very important for the cell design. Chemical reactions are one of the main heat sources in the fuel cell. Intensive reactions usually occur at locations where high concentrations of fuel and oxidant exist. This inevitably produces a high local temperature at high current density. Therefore, some measures of cooling have to be introduced and one of them is using the air cooling with an increased air flow rate. The overall temperature distribution in the fuel cell is a result of the combined effects of the heat generation and the cooling facility. Figure 16.9 shows a CFD predicted temperature distribution in the active areas of a co-flow SOFC, see Recknagle et al. (2003), in which the fuel and the air are fed from the left leaving at the right hand side of the cell. It can be seen that air flowing past the fuel cell is most effective at cooling the region near the air inlet and it carries the heat generated by various

Air and fuel

Figure 16.9 Temperature distribution C C) in the active area of a cross-flow planar SOFC stack, see Recknagle et al. (2003).

437

L. MA ET AL.

heat-generation processes in the fuel cell towards the air outlet. Therefore, the temperature usually increases along the airflow direction, reaching a maximum near to the exit. The electrolyte of the SOFC is usually made of ceramic material which is strong but fragile and it can crack very easily under an unbalanced temperature. Therefore pursuing a uniform temperature distribution is one of the key tasks in the SOFC designs. The performance of the fuel cell is typically represented by the cell polarization curve which shows the relationship between the cell voltage and the operational electric current. With the knowledge of thefluidflowand species transport in the fuel cell, the polarizations of the fuel cell may be predicted numerically by solving the transport equations for the electrical potential, i.e. equation (16.18). Figure 16.10 shows a typical numerically predicted performance of a co-flow PEMFC in terms of its potential polarization and power density, see Beming et al. (2002). As we can observe, there is a good agreement in general between the CFD model predictions and the experimental data. Some discrepancies exist at the high current density and this may suggest some limitations in the model predictions at a high rate of mass transport in the porous electrodes. However, it should be noted that predicting the performance of fuel cells is still a very scientifically challenging area of research and most existing CFD models involving many tuning parameters which are related to the processes of fluid flow, heat and mass transfer and electrochemical reactions, as well as the properties of the materials of which the fuel cells are made, and many of which are not well understood at present.

1.2 1.0

0.6 Ticianelli etal (1988) O 3D model polarization curve ^.««^r**.. 3J) rnodel power density curve

0.3

0.6 0.9 Current density [A/cm^]

1.2

0.5

1.5

Figure 16.10 A comparison between the CFD model predictions and the experimental data for a co-flow PEMFC fuel cell, see Beming et al (2002).

438

APPLICATION OF FLUID FLOWS THROUGH POROUS MEDIA IN FUEL CELLS

16.6

CONCLUDING REMARKS

Most components of the fuel cell are made of porous materials through which fuel and oxidant are distributed to the active surface in the catalyst layer where electrochemical reaction takes place to generate electricity and heat. Therefore, fluid flowing through porous electrodes through a combination of the convective and diffusive processes is a main character of the fuel cell. One of the main tasks in the fuel cell design is to distribute the fuel and oxidant uniformly to the reaction surface and remove the by-products, such as the heat and the water, effectively out from the fuel cell. It is noted that fluid flow in pours media in general is still an area of significant challenge, see Ingham and Pop (1998, 2002) and Pop and Ingham (2001), and the majority of the investigations are on the porous media in the engineering and/or environmental applications which often have a much larger length scale than those used in fuel cells. The disparity in length scales and matrix density can lead to very significant differences in the mass transport behaviour through the porous media. One distinct feature in fuel cells is its small but diverse length scales of the components ranging from microns to centimetres and possessing different levels of permeability. In some fuel cells large anisotropy exits in the electrode and the electrolyte and they have significant effects on the mass transport in the fuel cell. Therefore, the species transport in porous matrixes of the fuel cells is a very complex process. The modelling of the convection transport of the fluid flow in the electrodes of the fuel cell is often achieved using Darcy's law and the Navier-Stokes equation for thefluidflow in the gas channel. Special attentions are needed to balance in the interface of different layers of different permeability where discontinuity of velocity and/or shear stress may occur when Darcy's law is employed. As the transporting reactant species in the dense porous electrodes is mainly through diffusion, correctly evaluating the mass diffusion is critical to obtaining correct fuel/oxygen concentrations at the active catalyst surface. Since the number of the chemical species in the system is often more then two, therefore it is a multicomponent diffusive process in the fuel cell and the more sophistic Stefan-Maxwell equations may be used, although Pick's law is frequently employed since it is simple in formulation and economic in computation. However, an appropriate evaluation of the constants/parameters in association with the species diffusivities and the effect of the mironscles matrix is still an area in which further investigations are required. In particular, a correct modelling of the water management in the PEMFC requires the understanding of the fundamentals of water transport in the polymer membrane, where interactions of the water and the ions within the nanoscale pores of the membrane are the key factors need to take into account. In general, it is very scientifically challenging to mathematically model in detail such a complex system as fuel cells where innovative mathematical and numerical techniques are required. At present, whilst the general characteristic of the performance of the fuel cell may be mathematically reproduced, problems and difficulties associated with the detailed and accurate process simulations remain and a substantial amount of further research work is required. There are a number of fundamental issues, such as the species diffusion in fine porous materials of which the electrodes are made, modelling of the catalytic

L. MA ET AL.

439

electrochemical reactions, as well as water management in hydrolyte polymer membrane, fuel internal reforming, transient processes, etc. are yet to be further investigated.

REFERENCES Alazmi, B. and Vafai, K. (2001). Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer. Int. J. Heat Mass Transfer 44, 1735-49. Bard, A. J. and Faulkner, L. R. (1980). Electrochemical methods. Wiley, New York. Berg, P., Promislow, K., Pierre, J. S., Stumper, J., and Wetton, B. (2004). Water management in PEM fuel cells. 7. Electrochem. Soc. A 151, 341-53. Beming, T., Lu, D. M., and Djilali, N. (2002). Three-dimensional computational analysis of transport phenomena in a PEM fuel cell. / Power Sources 106, 284-94. Costamagna, P. and Honegger, K. (1998). Modelling at solid oxide heat exchanger integrated stacks and simulation at high fuel utilization. J. Electrochem. Soc. 145, 3995-4007. Dutta, S., Shimpalee, S., and Van Zee, J. W. (2001). Numerical prediction of mass-exchange between cathode and anode channels in a PEM fuel cell. Int. J. Heat Mass Transfer 44, 2029-42. George, R. A. (2000). Status oftubularSOFC field unit demonstrations. J. Power Sources S6,134-9. Gurau, V., Barbir, E, and Liu, H. (2000). An analytical solution of a half-cell model for PEM fuel cells. /. Electrochem. Soc. 147, 2468-77. Hoogers, G. (2003). Fuel cell technology handbook. CRC Press, Boca Raton, FL. Ingham, D. B. and Pop, I. (eds) (1998). Transport phenomena in porous media. Pergamon, Oxford. Ingham, D. B. and Pop, I. (eds) (2002). Transport phenomena in porous media, Vol. II. Pergamon, Oxford. Lampinen, M. J. and Fomino, M. (1993). Analysis of free energy and entropy changes for half-cell reactions. J. Electrochem. Soc. 140, 3537-46. Larminie, J. and Andrew, D. (2000). Fuel cell systems explained. Wiley, Chichester. Li, P. W. and Chyu, M. K. (2003). Simulation of the chemical/electrochemical reactions and heat/mass transfer for a tubular SOFC in a stack. /. Power Sources 124, 487-98. Li, S. and Becker, U. (2004). A three-dimensional CFD model for PEMFC. In Proceedings of 2nd ASMEfuel cell conference, Rochester, NY, 14-16 June. Mann, R. F , Amphlett, J. C , Hooper, M. A. I., Jensen, H. M., Peppley, B. A., and Roberge, P. R. (2000). Development and application of a generalised steady-state electrochemical model for a PEM fuel cell. J. Power Sources 86, 173-80. Meng,H.andWang,C.Y.(2004). Electron transport in PEFCs. J. Electrochem. Soc. A151,358-67. Nam, J. H. and Kaviany, M. (2003). Effective diffusivity and water-saturation distribution in singleand two-layer PEMFC diffusion medium. Int. J. Heat Mass Transfer 46, 4595-611. Ormerod, R. M. (2003). Solid oxide fuel cells. Chem. Soc. Rev. 32, 17-28. Pop, I. and Ingham, D. B. (2001). Convective heat transfer: mathematical and computational modelling of viscous fluids and porous media. Pergamon, Oxford.

440

APPLICATION OF FLUID FLOWS THROUGH POROUS MEDIA IN FUEL CELLS

Recknagle, K. P., Williford, R. E., Chick, L. A., Rector, D. R., and Khaleel, M. A. (2003). Threedimensional thermo-fluid electrochemical modelling of planar SOFC stacks. J. Power Sources 113, 109-14. Singhal, S. C. and Kendall, K. (2003). High temperature solid oxide fuel cells: fundamentals, design and applications, Elsevier, Oxford. Springer, T. E., Zawodzinski, T. A., and Gottesfeld, S. (1991). Polymer electrolyte fuel cell model. 7. Electrochem. Soc. 138, 2334-42. Stockie, J. M., Promislow, K., and Wetton, B. R. (2003). Afinitevolume method for multicomponent gas transport in a porous fuel cell electrode. Int. J. Numen Meth. Fluids 41, 577-99. Tayler, R. and Krishma, R. (1993). Multicomponent mass transfer. Wiley, New York. Thomson, W. J. (2000). Introduction to transport phenomena. Prentice Hall, Upper Saddle River, NJ. Ticianelli, E. A., Derouin, C. R., Redondo, A., and Srinivasan, S. (1988). Methods to advance technology of proton exchange membrane fuel cells. J. Electrochem. Soc. 135, 2209-14. Um, S., Wang, C. Y., and Chenb, K. S. (2000). Computational fluid dynamics modeling of proton exchange membrane fuel cells. J. Electrochem. Soc. 147, 4485-93. Veyo, S. E., Shockling, L. A., Dederer, J. T., Gillett, J. E., and Lundberg, W. Y. (2002). Tubular solid oxide fuel cell-gas turbine hybrid cycle power systems: status. J. Eng. Gas Turbines Power 124, 845-9. Wang, Z. H., Wang, C. Y, and Chen, K. S. (2001). Two-phase flow and transport in the air cathode of proton exchange membrane fuel cells. J. Power Sources 94, 40-50. You, L. and Liu, H. (2002). A two-phase flow and transport model for the cathode of PEM fuel cells. Int. J. Heat Mass Transfer AS, 2277-87. Yuan, J., Rokni, M., and Sunden, B. (2003). Three-dimensional computational analysis of gas and heat transport phenomena in ducts relevant for anode-supported solid oxide fuel cells. Int. J. Heat Mass Transfer 46, 809-21.

17

MODELLING THE EFFECTS OF FAULTS AND FRACTURES ON FLUID FLOW IN PETROLEUM RESERVOIRS S. D. HARRIS*, Q. J. FISHERt, M. KARIMI-FARD^ A. Z. VASZI* and K.WU+ *Rock Deformation Research Limited, University of Leeds, Leeds, LS2 9JT, UK email: sdhQrdr.leeds.ac.uk and attilaQrdr.leeds.ac.uk

^Rock Deformation Research Limited / School of Earth and Environment, University of Leeds, Leeds, LS2 9JT, UK email: quent inQrdr. leeds .ac.uk

^Department of Petroleum Engineering, Stanford, CA, USA email: [email protected] "^Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh, EH 14 4AS, UK email: [email protected]

Abstract Fault and fracture zones are often highly-complex heterogeneities that can have a significant effect on fluid flow within petroleum reservoirs on length scales of less than 1 /xm to more than 10 km. It is therefore important to incorporate their properties in production simulation models. Here we describe some of the numerical techniques currently being used to model the effects of faults and fractures on fluid flow. We begin by describing some of the techniques that we are currently developing to model flow around faults acting as barriers to fluid flow. At the millimetre scale, the Markov chain Monte Carlo simulation can be used to construct the 3D pore structure of fault rocks, which may then be combined with the lattice Boltzmann method to determine their permeability. A discrete fault flow model is presented for modelling fluid flow around highly-complex fault zones at the kilometre scale. The results from these models can then be incorporated directly into industry-standard production simulation models to allow

441

442

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS better decision-making. To model reservoirs where faults and fractures act as conduits for fluid flow we present a discrete fracture model.

Keywords: faults, fractures, petroleum reservoirs, multiphase flow, Markov chain Monte Carlo, control volumes, lattice Boltzmann, pore scale 17.1

INTRODUCTION

The development of petroleum reservoirs is extremely expensive, particularly in offshore, deep-water environments where it often costs more than £1 billion to build production platforms and over £20 million to drill individual wells. To reduce the financial risks associated with such operations, by allowing both better day-to-day decision-making and long-term planning, it has become standard practice within the petroleum industry to attempt to predict subsurface fluid flow patterns using production simulation models, see, for example, Mattax and Dalton (1990). The most commonly used production simulation models, e.g. the Eclipse^^ software by Schlumberger, use control volume finite-difference methods. A key problem with such models is that they are computationally very expensive because, as well as solving the fluid flow equations, the software also needs to model the phase changes that occur during petroleum production. Consequently, to allow simulation models to be run in a reasonable length of time the reservoir has to be represented by a relatively small number of grid blocks, i.e. usually less than 10^, each grid block often representing a volume of order 10 000 m^. Thus effective properties of the reservoir need to be calculated for each grid block. This process of upscaling is difficult for several reasons. (i) Reservoir rocks are very heterogeneous, even on the sub-grid-block scale. (ii) Petroleum reservoirs contain two or three immiscible fluids (water, gas and/or oil), all of whose movement often need to be modelled. (iii) The main information available on the distribution of the flow properties of rocks within petroleum reservoirs comes from taking seismic surveys or drilling wells. These are both low resolution and provide only a very sparse 'picture' of the subsurface. (iv) Features that affect fluid flow have a huge range of length scales, from pore throats with sizes down to less than 10 nm to large-scale faults that can be tens of kilometres in length. Faults and fractures are common heterogeneities found within petroleum reservoirs that often have a significant influence on fluid flow. Faults and fractures are localised zones of deformation, formed in response to shear and tensile stresses, respectively, that ultimately arose from tectonic movements within the Earth's subsurface. Faults and fractures may act as high-permeability conduits for fluid flow that enhance the rate of petroleum production. In many reservoirs, faults can also restrict fluid movement; such faults are commonly referred to as sealing. The presence of sealing faults can reduce the profitability of petroleum extraction because more wells are often needed to drain the reservoir. Faults

S. D. HARRIS ET AL.

443

and fractures are particularly problematic to incorporate into production simulation models for the following reasons. (i) They are often very thin (less than 1 mm to 1 m) compared to the size of the grid block, (ii) They often have large permeability contrasts (sometimes more than six orders of magnitude) compared to the surrounding reservoir, (iii) They have a complex structure, (iv) Their fluid flow properties are not well understood. To illustrate the problem further we begin by focusing on fault-zone structure. Figure 17.1(a) shows an aerial photograph of outcrops near the Moab fault zone, Utah, with the faults marked having an offset of more than 5 m being those that would possibly be imaged by a seismic survey. Detailed examination of these faults shows, however, that they are not single planes but instead are composed of highly-complex arrays of smaller-scale faults, see Figure 17.1(b-d) surrounding the principle fault plane; this clustered array of smallscale faults is often referred to as the damage zone to the main fault. The individual faults themselves are composed of a gouge containing undeformed grains mixed with angular fragments formed by the crushing of sand grains, see Figure 17.1(e). The grain size of the gouge is therefore reduced and the grain-sorting is poorer than the host sediment, see Figure 17.1(f). The problem that faces petroleum engineers is how to model fluid flow around and across these complex structures. We review some techniques currently being used to model the effects of faults, acting to restrict fluid flow, and fractures, acting as high-permeability conduits, on fluid flow over a variety of length scales. It is the purpose of this chapter to provide an overview of the numerical methods currently being used to model fluid flow within petroleum reservoirs. It is hoped that our review will stimulate those not directly involved in this area of research to apply their expertise to generate novel methods for tackling these complex problems; a significant amount of progress is required before practical solutions are found that can be used on a day-to-day basis by non-specialists working on individual petroleum reservoirs.

17.2 SEVGLE AND MULTIPHASE FLOW Sedimentary rocks, which form most of the world's petroleum reservoirs, are either deposited in water or become saturated with water soon after sediment deposition. In these circumstances, the rate of fluid flow may be calculated using Darcy's law: ^ kAAp Q=—XT^

(17.1)

where Q is the flow rate, Ap is the pressure differential, Al is the length of theflowpath, fi is the fluid viscosity, A is the cross-sectional area and A: is an empirical constant, with units of length^, that varies for each rock type. A rock has a 1 Darcy permeability if it allows a

444

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

Figure 17.1 (a) An areal photograph of part of the Moab fault zone in the Bartlett Wash, Mill Canyon and the area around Courthouse Springs, Utah, USA. Faults with a shear offset of greater than 5 m have been marked, (b) A map of all of the faults present within the boxed area shown in (c), the photograph of the outcrop at the end of Mill Canyon. (d) The complexity of these smaller-scale features is shown in a hand specimen. Backscattered electron micrographs show the contrast in pore structure and grain size between (e) the fault rock and (f) the undeformed sediment.

fluid with a viscosity of 1 cp to flow at a rate of 1 cm^/s, through a cross-sectional area of 1 cm^, under a pressure gradient of 1 atm/cm. Although Darcy's law was initially based on empirical relationships, it has since been derived theoretically from the Navier-Stokes equations for the motion of a viscous fluid, see King Hubbert (1956).

S. D. HARRIS ET AL.

445

The heating of organic-rich source rocks such as black shales and coals generates petroleum. Petroleum is less dense than water so, when expelled into water-saturated pore space, oil and gas will tend to migrate vertically due to their buoyancy. The buoyancy force, usually referred to as the capillary pressure, pc, increases upwards through a static petroleum column and can be related to the density of the petroleum, pp, and the aqueous phase, p^u, the vertical height of the petroleum column, H, and the acceleration due to gravity, g, by the following equation: Pc = {pw — Pp) gH,

or Pc = 0.433 {pw — Pp) H in field units.

(17.2)

In the latter expression, 0.433 is a conversion constant which takes into account g, pc is measured in psi, p^j and pp are measured in g/cm^, and H, in feet, see, for example. Watts (1987). The capillary pressure is essentially the amount by which the pressure in the petroleum phase, Pp, exceeds that of the water phase, p^, at the same depth, namely Pc = Pp — Pw The relationships Pp = Po — PpdH and Pw = Po — PwOH express the decrease in pressure within the petroleum and water, respectively, with reference to the common pressure po at the base of the petroleum column. The rise of petroleum through a rock under the influence of the buoyancy force does not go unhindered. Instead, work is required to move petroleum through pore spaces. The force required for petroleum to flow into water-saturated pore space is known as the threshold pressure, pt. The magnitude of pt is a function of the interfacial tension, cr, between the petroleum and the water, the contact angle, 6, between the fluids and rock, and the pore throat radius, r^ and can be described by pt = 2a cos O/TC, see Purcell (1949). Rocks can be considered to contain a 3D array of pores connected by pore throats; most rocks contain a range of pore and pore-throat sizes. As the capillary pressure increases, petroleum can enter ever-decreasing pore-throat sizes within an individual rock. Consequently, increases in capillary pressure cause the petroleum saturation, Sp, to increase and the water saturation, Sw, to decrease. Graphs relating capillary pressure to phase saturation, see Figure 17.2(a), are measured routinely during reservoir characterisation. The shape of these capillary pressure curves is controlled by the pore-throat and pore-space distribution within the rock, which is itself dependent upon its grain size and sorting. The rate of fluid flow through rocks containing immiscible fluids is usually estimated using a modified version of Darcy's law: Q.=

'-^^,

(17.3)

where Qx and //^ are the rate of flow and viscosity of phase x (water, gas or petroleum), respectively, and krx = kx/k is known as the relative permeability of phase x, where kx is the permeability of phase x and k is the absolute permeability of the rock. The value of krx generally lies between 0 and 1 and relationships between relative permeability and phase saturation, often referred to as relative permeability curves, see Figure 17.2(b), are usually determined experimentally. Capillary pressure and relative permeability curves

446

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

Water saturation [%]

(c) Saturation scale 100%. 0% petroleum! •water

Water saturation [%] Fault zone saturation distribution controlled by phase relative permeability and capillary pressure functions

Transition zone in fault

Host rock phase distributions controlled by relative permeability and capillary pressure functions of the host rocks

0%LJ100% petroleum water

Figure 17.2 Examples of (a) a capillary pressure curve, and (b) relative permeability curves for water and petroleum relative permeabilities {krw and krp, respectively). Each of the curves is plotted as a function of the water saturation and described in relation to height above the free-water level (FWL). (c) A schematic illustration offlowacross a fault zone and the key positions referred to in (a,b).

are the most commonly used forms of data incorporated into production simulation models to enable the rate of fluid flow of the specific phases to be predicted.

17.3 MODELLING FLOW IN PETROLEUM RESERVOIRS WHERE FAULTS ACT AS BARRIERS Once a petroleum engineer has produced a production simulation model, the first step is to refine the various input parameters (e.g. permeability and porosity distribution, relative permeability and capillary pressure curves, etc.) so that a simulation run can reproduce previous production data (flow rates, reservoir pressures, etc.)—a process known as history matching. In reservoirs where faults are suspected of being barriers tofluidflowit has been common practice to alter the transmissibility multipliers between grid blocks adjacent to faults to achieve this history match. (The 'multiplier' here reflects the influence of the fault

S. D. HARRIS ET AL.

447

and enablesfluxesto be calculated as a function of the transmissibility between pairs of grid blocks; see the discussion of Figure 17.9.) Traditionally, this is done in an ad hoc manner without establishing whether the values used are physically realistic. History matches are inevitably non-unique and therefore incorporating unrealistic properties into a simulation model can produce a convincing history match but the model may not necessarily be of use when making further development decisions, see Dake (2001). More recently, there has been a push to be more scientifically rigorous about assigning transmissibility multipliers to grid blocks adjacent to faults, see, for example, Knai and Knipe (1998), Manzocchi et al (1999, 2002), and Al-Busafi et al. (2005). In particular, attempts have been made to calculate thefluidflowproperties of faults based on estimates of their thickness and permeability. The permeability of small-scale faults found within cores retrieved from petroleum reservoirs has been measured in the laboratory, see, for example, Fisher and Knipe (1998, 2001). Laboratory measurement of the fluid flow properties of fault rocks, particularly their multiphaseflowproperties, is expensive, time-consuming and inaccurate when the faults are extremely thin and have significant permeability anisotropy. As an alternative, attempts are now being made to model the permeability of fault rocks based on images of their microstructure. Fault rock thickness is often estimated based on empirical relationships between fault throw and cumulative fault rock thickness, see, for example, Manzocchi et al. (1999). The estimates of fault thickness are also problematic because, as shown in Figure 17.1, faults are extremely complex and there is a possibility that fluids will move through a tortuous pathway in preference to flowing through the cumulative thickness of fault rock present. In this section we describe recent attempts to model both the permeability of fault rocks and flow through complex fault zones. We then show how results from such models can be incorporated into production simulation models. Finally, we describe some of the issues that still require further research. In essence, this represents a problem in the field of multiscale-multiphysics simulation in which the results of numerical models conducted at a variety of length scales need to be combined, sometimes in an iterative manner, to provide accurate forecasts. 17.3.1 Numerical modelling of the permeability of fault rocks We describe a methodology to allow the fluid flow properties of fault rocks to be modelled from 2D images of their pore structure. The first stage is to construct the 3D pore structure from orthogonal 2D images obtained by thin-section light, or back-scattered electron, microscopy. The second stage involves using the lattice Boltzmann method (LBM) to model the permeability of the fault rock from the 3D pore system. The fundamental idea of the LBM is to construct simplified kinetic models that incorporate the essential physics of the microscopic or mesoscopic processes and for which the macroscopic-averaged properties obey the desired macroscopic equations. The quantitative characterisation of the microstructure of the porous medium The fluid flow within faulted/fractured rocks is determined by the microstructure of the porous medium, which is characterised by the void space geometry and connectivity, and

448

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

the solid surface/fluid chemistry. The prediction of the fluid flow properties of porous media from their microscopic origins involves two major steps: (i) a quantitative characterisation of the microstructure, and (ii) exact or approximate solutions of the equations of motion that govern the transport phenomena of interest. Fluid flow and related processes at the pore scale essentially occur in 3D. Although direct measurements of a 3D microstructure are now available via synchrotron X-ray computed microtomography, see Dunsmoir et al (1991), it is often difficult and expensive to obtain reliable 'images' of the 3D pore structure. In practice, however, information about the microstructure of porous materials is often limited to 2D thin-section images. Several techniques have been proposed to statistically generate representative 3D pore structures from spatial information derived through 2D thin-section images, see Adler et al. (1990) and Manswart and Hilfer (1998), and more recently a multiple-point method has been proposed, see Okabe and Blunt (2003). These methods consist of measuring statistical properties, such as porosity, correlation, and lineal path functions, on 2D thin-section images and generating random 3D models in such a manner that they match the measured statistical properties. Recent quantitative comparisons of these models with tomographic images of sedimentary rocks have shown that statistical reconstructions may differ significantly from the original sample in their geometric connectivity, see Manswart et al. (2000). Meanwhile, Bakke and 0ren (1997) also developed a process-based reconstruction procedure that incorporates grain-size distribution and other petrographical data, but their procedure is computationally expensive. The statistical methods discussed above have difficulties quantifying the geometric connectivity of the original sample, and so the resulting predictions of thefluidflowproperties are not satisfactory. We present a new class of models to characterise heterogeneous porous media, based on either 2D or 3D sample information, and using a Markov chain Monte Carlo (MCMC) simulation. This is a stochastic method which utilises the heterogeneous spatial structure information of voids and solids obtained from the sample, and in which the geometrical and topological features can be modelled and simulated effectively. The approach, which has its origins in image modelling, differs from published two-point correlation methods; instead, it involves a complex voxel interaction system or a very high-order neighbourhood system to quantify the morphology and topology of the porous media. Markov random field model for a heterogeneous porous medium A widely used method for image modelling is the Markov random field (MRF) model, see Geman and Geman (1984) and Cressie (1993). There are two principal reasons for using MRPs in this specific problem. First, this methodology was developed from Markov random field theory, which allows the use of a small number of local conditions to predict global features. In other words, it considers local neighbour interactions and other geometrical descriptors to generalise the overall morphological features of the target images. Second, the Markov chain constructed in our approach will eventually converge.

S. D. HARRIS ET AL.

449

and this makes the predictions more reliable. The procedure allows a quantitative transition from microscopic to macroscopic scales in porous media. In order to simulate real porous media, such as a heterogeneous rock structure, the porosity and the local dependence functions of the pore space are measured, and the extent of the neighbourhood system must be sufficiently large; here in 3D we take 15 neighbours into consideration. Having observed the rock structure image, we would like to make inferences about the probability mechanism underlining these counts by exploring their spatial dependence. We provide a new technique that accounts for nearby-location dependence and interactions among a small number of neighbouring sites which exhibit spatial conditional dependence. As a high-level neighbourhood system is involved, a rapid convergence of the chain is vital for its application so that computational costs can be minimised, particularly for large-scale images in 3D modelling. We extend the scan scheme of Wu et al. (2004) from 2D images to the 3D simulation of heterogeneous porous media structure. To describe the 3D Markov chain model we first introduce some notation. Suppose we have a finite rectangular array of n voxels in a 3D sample space 5. In particular, the voxels are labelled by integers z G 5 = { 1 , . . . , z , . . . , n}. Let x = ( x i , . . . , X i , . . . , Xn) be the *state/colour' of the voxel set, e.g. void or solid. Each state/colour value belongs to a finite set A = { 1 , . . . , m} and this determines the complete colouring of the array or constituents distribution. Further, if we describe x as the outcome of a corresponding random vector X, then we use F{X = x) = p{x) to denote the probability distribution of X. Our objective is to specify a probabihty distribution, p(x), for X. The Markov models have been widely chosen for such probability models and are based on the assumption that the label on any site depends on the other labels only through a small number of neighbouring sites. In our application of the technique, we let VLMN = {Vijk : 0 ^ i ^ L rows, 0 ^ j ^ M columns, 0 ^ k ^ N layers} define a finite integer lattice, with Vijk being a voxel at the intersection of row i, column j and layer k, see Figure 17.3(a). We also let Xijk denote the array of colours vectors (void or solid) associated with that voxel. Since a single-pass scan scheme will be used in our algorithm, instead of an intensive iteration method, we only define our neighbourhood system within past voxels and the 'past' voxel set for the voxel at (z, j , k) is the set {{l,m,n):l
\Xi-ij,kAi,j-i,kjXij,k-i]' (17.4) we have the following joint probability function, see Qian and

450

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

(b)

Figure 17.3 (a) The definition of Vijk and 'past' voxels in a 3D Markov chain model, (b) An illustration of a 15-neighbourhood system, i.e. 7 pixels in the previous layer and 8 in the current layer.

—l,m,n> '^Z,m—1,71?

A/,m,n-l]

(17.5)

/=0 m=0 n=0

and P [^ijk I {Azmn : {l,m,n) ^ {ij,k)}]

= P [Xijk \ {Ximn ' {l,m,n) G J^ijk}], (17.6)

where Jiiju is the set of neighbours of Vijk- We also use an algorithm which generates two voxels simultaneously, namely the right voxel at (i, j + l,fc) as well as the voxel at (z, J, k). The method was first introduced in Wu et al. (2004). In comparison to a mono-site updating scheme, this approach offers both an improvement in the efficiency and effectiveness (increased quality) of the chain. Thus, we define a l.^-neighbourhood system in 3D, in which 13 voxels are *past' voxels and 2 voxels are to be determined, i.e.

Kpast

= {{i-2J,k-l),{i-2J + l,k-l), {i-lj,k-l),{i-l,j + l,k-l),{ij-l,k-l),{ij,k-l),{ij + l,k-l), (i - 2J,k), {i - 2J + l,fc), (z - 1, j - 1, A;), (z - lj,k), (i - l , j + 1, A;), {ij - l,k) ,{ij,k) ,{i,j + l,k)}, (17.7) which is the direct analogue of the 6-neighbourhood system of Wu et al (2004) and is illustrated in Figure 17.3(b). The next problem in constructing the pore architecture models is to estimate the physical parameters. Parameter estimation is a very difficult area, because the spatial association intrinsic to these models leads to the application of standard statistical approaches, such as maximum likelihood estimation, being computationally prohibitive. These approaches must be replaced by MCMC methods. We have devised a novel method in which the transition probability matrix is obtained from three orthogonal images that are assumed to provide representative directional information on the 3D properties of the sample. In ordinary Monte Carlo calculations it is required to draw a perfect random sample from the

S.D. HARRIS ETAL.

451

target distribution. We now assume that this is impractical (because the sample space S is too large for the normalisation constant to be calculated directly), and instead we construct an ergodic (i.e. regular in the finite case) Markov transition probability matrix. The chain is run for a sufficiently long time, for the discrete spatial case and using the scan scheme of Wu et al. (2004). Here a 'sufficient' run time must ensure that the simulated structure of the chain can be used as a basis for summarising features of the probability distribution P{x) for the real material. In order to test the robustness of the model, a wide variety of rocks were selected and simulated in our study, from highly-porous sandstone to low-porosity mudrocks, their corresponding permeabilities ranging over six orders of magnitude. Having measured their hydraulic conductivities, thin sections were cut and binary images depicting grain and pore spaces were produced. From the three mutually perpendicular thin-section images and having measured the Markov transition probability from these images, we can implement the MCMC method to reconstruct the 3D pore architecture models (PAMs) using the 15-neighbourhood model. The size of the simulated 3D structure can range from the centimetre to metre scale, depending on the voxel scale and computer space. In our real images for training, the pixel size ranged from 0.1 /xm (for fine materials) to tens of microns (for coarse sandstone), with corresponding frame sizes of tens of cubic microns to a few cubic centimetres. The chain constructed in this method converges after implementing 200 x 200 x 200 steps in our test, which means that the minimum size of the simulated representative structure should at least consist of 200 x 200 x 200 voxels. The simulations are targeted at reproducing the topological features of real structures, namely porosity, pore size, pore connectivity, and the associated hydrological processes. When the simulated 3D architecture became stationary the resulting porosity was close to the average porosity of the training images. The novelty in this algorithm is that the chain only requires one pass, utilising just a few minutes on common PCs, without any iterations needed in standard MCMC methods. The size of the chain depends on the homogeneity/heterogeneity of the real image targeted. Some examples of simulation results are presented in Table 17.1. Simulation ofpermeability—lattice Boltzmann method When the porous media structure is available, the second task is to determine its macroscopic transport properties. The permeability of the simulated structure was computed using the LBM which, unlike conventional numerical schemes based on discretisations of macroscopic continuum equations, is based on microscopic models and mesoscopic kinetic equations. This feature gives the LBM an advantage when studying non-equilibrium dynamics, especially in fluid flow applications involving interfacial dynamics and complex boundaries. The basic premise for using these simplified kinetic-type methods for macroscopic fluid flows is that the macroscopic dynamics of a fluid is the result of the collective behaviour of many microscopic particles in the system, and that the macroscopic dynamics is not sensitive to the underlying details in the microscopic physics, see Kadanoff (1986). Since its appearance, the LBM has been successfully applied to the study of a variety of flow and transport phenomena, including flow in porous media, see

452

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

Table 17.1 Comparisons of various properties derived experimentally and from numerical simulations. Properties

Sample 1 Mudstone

Sample 2 Siltstone

Sample 3 Sandstone

Sample 4 Soil

Two-dimensional thin-section image

Three-dimensional simulation image

Experiment Simulation Experiment Permeability [mD] Simulation Porosity [%]

Succi et al. (1989), multiphase and multicomponent flows, see Gunstensen et al. (1991), and heat transfer, see Qian (1993). The LBM can be derived in several different ways; a detailed review on the LBM and its applications in various fields can be found in Chen and Doolen (1998). Here we use the popular single-time relaxation Bhatnagar-Gross-Krook (see Bhatnagar et al, 1954) model (the so-called BGK model) proposed by Chen et al (1991) and Qian et al. (1992), in which the particle distribution function is described by

fi {x + eut + 1) = fi {x,t) - - [fi {x,t) - n T

{xM,

(17.8)

where fi {x, t) is the particle distribution for site x at time t and in direction i. In particular, we use a 19-velocity model in the 3D case (typically written as a D3Q19 model), which defines a 3D grid of nodes, each of which maintains a 19-member particle distribution (also called speeds). Each node is allowed to exchange mass with 18 of its neighbours (6 nearest nodes and 12 next-nearest nodes) and a rest speed (the 19th member) is also maintained. The particle distribution fi{x, t) (where i = 0 , . . . , 18 for D3Q19) is updated by 'collision' and 'streaming'. The second term on the right-hand side of equation (17.8) is the collision operator, in which r is a relaxation parameter related to the viscosity of the fluid. The local equilibrium distribution for each node, p^{x,t), is parameterised by the local conserved quantities, and is easily calculated from the particle density and momentum at each node. The left-hand side of equation (17.8) is the streaming operator that exchanges the particle distribution at i with adjacent lattice nodes, whose locations are specified by the relative positions ei of neighbouring nodes. Special rules are applied to preserve mass and momentum at fluid-solid interfaces, see Zou and He (1997). Fluid flow

S.D. HARRIS ETAL.

453

can be induced by specifying densities or velocities at open boundaries of the simulated domain, or, as used in this study, by applying a 'body force' to all nodes, see Succi (2001). The interface of the solid and pore space is treated as a non-slip boundary for water movement, i.e. zero fluid velocity is imposed. This treatment is the same as that proposed by Filippova and Hanel (1998) and is second-order accurate. Two types of external forces, namely a body force or a pressure gradient, can be used as a driving force. Singh and Mohanty (2000) pointed out that the use of the body force might give incorrect results, and our calculations of permeability using a body force were found to be unrealistic, especially in cases of low porosity. Instead, we have used a pressure difference as the driving force in all of the simulations presented here, and we follow a simple approach based on a mirror-image algorithm to maintain a specified pressure at a boundary, see Zou and He (1997) and Zhang et al. (2002). From a practical point of view, the most important aspect of simulation is to reproduce the associated physical and fluid flow processes, and it is only in the 3D case that the connectivity of pores can be practically explored and related to laboratory measurements. To test the flow properties of reconstructed materials based on our PAMs, the intrinsic permeability of the reconstructed porous media was calculated using the LBM and the results compared with measured permeabilities from core samples in the laboratory. We first cut the cores of rocks obtained from drilled oil wells, measuring their absolute permeability in the laboratory. The methods described here enabled us to take images from thin sections of the cores and reconstruct their 3D pore structure. A variety of rocks were used to test the robustness of this model, from low porosity to high porosity, homogeneous to anisotropic, fine-grained material to coarse sandstone, and their associated permeabilities range across six orders of magnitude. Some of these results are presented in Table 17.1. Samples 1 and 2 are fine-grained materials, mudstone and siltstone, respectively. Their training images each have 1000 times magnification and each pixel has dimension 0.1 /xm. Sample 3 is a slightly-deformed sandstone, and the magnification of its thin-section image is 50 times, with one pixel equivalent to 20 //m. Sample 4 is a soil sample, the image magnification of which is about 30 times, and consequently one pixel equals about 30 /xm. It should be noted that sample 4 has a lower porosity compared to sample 2; nevertheless, its permeability is three orders of magnitude higher than that of sample 2. It is clear that a variety of heterogeneous materials can be reconstructed using our 15-neighbourhood models (i.e. PAMs), and their permeability values agree well with laboratory measurements. This implies that the simulated reservoir rocks derived from our 3D Markov chain model are indeed functionally similar to real rocks. 17.3.2 Modelling flow in complex damage zones In recent years the rapid advance in the quality and quantity of seismic data has provided increasingly detailed information on 3D fault networks, but the imaging limitations mean that many minor faults in fault damage zones are still below the limit of resolution. Thus, information from fault damage zones is available at best in 2D, in the form of maps derived from outcrops or, more commonly, as ID line samples provided by well logs and cores.

454

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

and the issue of predicting 3D fault and fracture network characteristics is an important one. In this section, we describe a stochastic model of a 3D fault damage zone which incorporates statistical properties and spatial clustering observed in outcrop, see Harris et al. (2003). We also briefly describe the permeability upscaling techniques developed in Odling et al. (2004) for fault damage zones in 2D and describe the recent development of these for 3D fault networks. Fault damage zone characteristics Several studies in the last ten years have focused on the architecture of fault damage zones and their potential influence on fluid flow, see, for example, Antonellini and Aydin (1995). In siliclastic sedimentary rocks, faults take the form of deformation bands, along which grain size and porosity are reduced, to form a partial barrier to fluid flow, see, for example, Fisher and Knipe (1998) and Shipton et al (2002). In such rocks, fault zones are composed of a major slip zone, along which the majority of the displacement occurs, surrounded by a damage zone comprising a complex network of low-throw faults. The Moab fault in Utah, USA, see Foxford et al (1998), and the Ninety Fathom fault in NE England, see Knott et al (1996), provide well-exposed examples of normal faults and their damage zones in sandstones. The damage zone of the Moab fault is composed of an anastomosing network of deformation bands, which, in the well-exposed canyon of Bartlett Wash, has an inner zone of well-connected, high-density deformation bands and an outer damage zone of lower-density faults, see Figure 17.1. The orientation distribution of faults has a strong influence on the connectivity of a fault system. Deformation bands generally have a trend similar to that of the main slip plane, see Antonellini and Aydin (1995) and Shipton and Cowie (2001), but also show sufficient variation in orientation to generate good connectivity in the sub-horizontal plane. Subvertical sections of these damage zones show that the dip of deformation bands can be either unimodal or bimodal, with an angle of 20-30° between the two major dips. We model the variation in orientation of small faults around larger structures through Gaussian distributions for fault strike and dip, following observations from the Moab fault. Measurements on outcrop surfaces at the Moab fault reveal a wide range in fault throw to length ratios, but a value of 1 : 100 is most likely to be representative when measurement issues are accounted for. The thickness of fault rock is also related to the fault length, displacement and the lithology. Field observations provided in Manzocchi et al (1999) suggest that a ratio of thickness to displacement of 1:100 may be appropriate for minor faults and deformation bands. The displacement on isolated faults increases from the tip line, which defines the outer edge of the fault, towards its centre, see, for example, Nicol et al (1996). We assume this to be a linear relationship. Observations from seismic surveys indicate that isolated normal faults are planar, with an approximately elliptical shape and a sub-horizontal long axis, see, for example, Nicol et al (1996), and show an average aspect ratio of around 2. We model each fault as a

S. D. HARRIS ET AL.

455

simple elliptical surface with a horizontal long axis that follows a Gaussian distribution with mean 2 and standard deviation 0.05. The recent review of scaling in fault systems by Bonnet et al (2001) shows that power law exponents for fault systems range from —0.8 to —1.3 (for the cumulative frequency distribution). In both the Moab and the Ninety Fathom faults, the nature of the outcrop does not allow the determination of the fault length distribution and there are no studies of fault length distributions within fault damage zones. However, the Ninety Fathom fault damage zone shows a power law throw distribution with an exponent of around —1.6. For a power law relationship, the number of faults with lengths greater than or equal to /, i.e. the cumulative frequency of faults, F{1), with length at least /, is F{1) oc l~^^. The 3D power law exponent D3 controls the proportion of large to small faults within the population, so that small faults become increasingly dominant in the population as i^s increases. In both the Moab and the Ninety Fathom faults, deformation band density generally increases towards the major slip zone, ranging from 1 to over 100 faults per metre. Fault frequency profiles across the damage zone of both faults show a general increase towards the major slip plane, with significant localised variations in frequency on the scale of tens of metres. Due to difficulties in quantifying the spatial distribution of fault and fracture systems, approaches to incorporating them in simulation models rely on geometrical rules for placing faults, after which the simulated patterns can be tested against natural patterns. The hierarchical approach to simulating the fault spatial distribution is described in Harris et al. (2003), and this is the approach that we follow here. In this approach every 'child' fault is chosen to be 'clustered' around an existing 'parent' fault, whose location and orientation control the corresponding parameters for the 'child' fault. The process begins with an original main fault at the largest scale, and in the examples presented here its length is 3 km. Vertical cross-sections through 3D fault populations obtained for different power law exponents are shown in Figure 17.4. The sub-clustering is apparent on both the large and small scale. The main effect of Z^s on the fault population is the proportion of small to large faults. Thus, whilst for i^s = 1.6 the fault traces have a high degree of connectedness, for Ds = 2 the fault clusters can be more clearly identified and also regions of very low fault density appear. Finally, for Ds = 2.4 the fault clustering is even more apparent, and an easier passage may be expected for flow across the region unless the fault density is increased significantly. Discrete fault flow model The discrete fault flow model (DFFM), see Odling et al. (2004), approach employs simple finite-volume techniques on regular orthogonal grids to enable large numbers of faults (up to tens of thousands) to be represented. Combined with the stochastic fault damage zone model, the DFFM provides a tool for assessing the impact of complex fault networks on flow and determining upscaled properties of grid cells for large-scale reservoir simulation packages.

456

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

Sub-clusters

A:[m]

jc[m]

x[m]

Figure 17.4 Vertical cross-sections (perpendicular to the main fault) through 3D fault models for (a) D3 = 1.6, (b) D3 = 2, and (c) D3 = 2.4.

The upscaling of the permeability field in model-size blocks is carried out using the classical approach, i.e. a numericalflowmodel is used to determine the equivalent permeability of the block as a whole in the direction of the applied pressure gradient. Both the (partially sealing) faults and the rock matrix are discretised onto a regular square grid and each element is assigned a corresponding permeability. For the flow of an incompressible fluid, the mass balance equation in the steady-state situation, for the elementary control volume around the node (i, j ,fc),see Figure 17.5(a), is written as follows: (17.9) where _

Qx =

kxAyAz dp ju: : —9 x^ ,'

_

kyAxAz dp

^ = —— fi . —dy^ ^, qy

qz =

kz Ax Ay dp

/i — — dz

r^^^f^^

(iv.io)

457

S. D. HARRIS ET AL.

(a)

(b)

Pij*U

I 1 Li-

(U+hk)

ii-hj,k) I

Pi-\j,kf-j^

Mjji

fPij,k

(4Aa. I ('+i,y,ft)

OJ-1,^) Figure 17.5 A 2D (layer k) illustration of (a) the control volume around the node (i, j , A;), and (b) the associated permeability grid.

are the fluid fluxes in the x-, y- and z-directions, respectively, for a regular 3D cuboidal network of nodes covering i = 0 , . . . , n^, i = 0 , . . . , riy and k — 0 , . . . , n^. Here p is the fluid pressure, // is the fluid viscosity, Aa;, A^/ and Az are the spatial increments in the x-, y- and 2;-directions, respectively, and k^, ky and kz denote the corresponding local permeabilities along grid elements. The network of nodes in Figure 17.5(a) is referred to as the primary grid. The indices (2 — 1/2, j , k) and (i -h 1/2, j , k) represent the interfaces in the x-direction for the control volume around the node {i,j,k)y see Figure 17.5(a), and similarly for the y- and ^-directions. At these interfaces, the fluxes (17.10) in the x-direction are approximated using central differences as follows:

- -h

^y^zPi-\-lJ.k

-PiJ,k

(17.11)

and similarly for the fluxes in the y- and z-directions. The permeabilities k^i-l/2,j,k and ^Xi+i/2,j,k ^^ defined on a permeability grid, the 2D representation of which is shown in Figure 17.5(b). When a fault is crossed by a primary grid element, then, depending on the thickness of the fault at the crossing location, a certain thickness value is assigned to this element, and hence a corresponding fault volume is assigned. The permeability of each block centred around these primary grid elements is then calculated by taking the harmonic average of the permeability of the host rock, kh, and fault rock, kf, within the block.

4^8

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

Combining equations (17.9) and (17.11), together with the corresponding equations for the fluxes in the y- and z-directions, we obtain one equation at each grid node:

+ Ax Az

[(A:y. ^._i/2,fc + ^yi,j+i/2,k)

PiJ,k - ^yij-i/2,kPi,3-^.k

- ^2/i,j+i/2,fcPiJ+l,A;J

= 0 (17.12) and the resulting equation system can be solved subject to the boundary conditions defining a uniform applied pressure gradient in the x-direction and no-flow conditions on the remaining four sides of the domain. This equation system for the pressure field can be solved using iterative methods, and the conjugate gradient method proved to be very efficient in obtaining a solution. Once the pressurefieldhas been determined, theflowfieldcan be resolved and the upscaled permeability in the x-direction, Kx, is defined using the average fluid flow through either the upstream or downstream face. The efficiency of the fault zone, see Odling et al (2004), can now be characterised relative to A'*, the upscaled permeability that would be obtained for a single fault of uniform thickness, comprising all of the fault rock volume V within the block, spanning the whole region, and perpendicular to the flow direction. The value of Kl is defined using the harmonic average of the fault rock and matrix permeabilities: VIcP

d • VI dp kh

-^a-

1

or

Kilkn

(17.13)

where d is the length of the sample region in the flow direction and r — kf/kh- In this configuration the fault rock is being utilised in its most efficient way as a barrier to flow. However, equation (17.13) underestimates the bulk permeability as it assumes unidirectionalflow,whilst in reality theflowparticles follow a tortuous pathway, and therefore we use the 'efficiency' to quantify this difference. The single-fault system for which equation (17.13) applies represents 100% efficiency, while realistic fault arrangements will always result in lower levels of efficiency, a, where

[ ^

a = .Kxikh

^

[K^lkh

ll / 1

1/

1

[Kllkh

-1

-0]

(17.14)

Flow characterisation of fault damage zones for different model parameters The flow pathway lengths across a sample domain are plotted in Figure 17.6 for different values of Ds and different permeability contrasts, r. The mean and range of the results have been indicated in each case. For all three values of Ds, the flow pathway lengths increase with decreasing r, and the extent of this rise is more accentuated with increasing i?3, as observed from the mean of the results. Thus the largest path lengths are found for

459

S.D. HARRIS ETAL. 110

_

1

I

1

_ 1)3 = 2.4' r:

100 Lh ^3=2T);

-^ 80 -

I

70

60

;

'd

-

+ ;

50 h -ij. 40 ~ 1

10-

i33=1.6i

:

lii

ill

fi: -

•"•

•'•

- _

i

1

1

1

10^ 10-^ Permeability contrast

10"

Figure 17.6 Observed statistical data (maximum, mean and minimum) on the flow pathway lengths for different fault to matrix permeability contrasts.

Ds = 2.4 and r = 10"^, and this is explained by the larger proportion of small faults in the population compared to D3 = 1.6 so that theflowpathways can follow a more tortuous route. Increasing the proportion of small faults in the population leads to near-straight flow paths for relatively large r (e.g. 10"^) and very long flow paths for relatively small r (e.g. 10-^). Results for the bulk permeability (normalised by the matrix permeability, and when r = 10"'^) are presented in Figure 17.7(a) for three different domains taken at locations immediately adjacent to the major fault. The size of each domain is 25 m x 50 m in the X- and ^/-directions, whilst the region size in the z-direction is either 0.1m, I m ,

(a)

(b)

0.10

-r<^-4

0.09 !i 0.08

0.65

•^

S 0.06 10.05 ^ - '_ - -o- ^ , 0.04 0.1

^-^-i \

1 20 50 Region height (z-direction) [m]

'

X- - -..

g 0.60 ?X

g 0.07 fCO

0.70

~—_____^

^

^ x .

~"~~~^

^v

"• -

^

'.

;

•

-.

'^'\^ ^\^ \ \ " --^

S0.55 0.50 : 0.45 0.1 -J

^^s^^^X 1

I

I

1 20 50 Region height (z-direction) [m]

Figure 17.7 Results for (a) the bulk permeability, and (b) the fault zone efficiency, in three different domain realisations for D3 = 2.4 as a function of the domain height.

460

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

x[m] Figure 17.8 (a) A 2D vertical cross-section of a fault damage zone with D3 = 2, and (b) 3D illustrations of the faults and aflowpathway.

20 m or 50 m. As the domain height increases from 0.1 m to 50 m, the bulk permeability generally increases, which is expected due to the increased spatial freedom for crossing the domain and the likelihood offindingflowpaths along which less fault rock is encountered. However, as the fault rock distribution within the domain is of a statistical nature, it is possible for the overall fault rock density to increase with increasing domain height, and this could lead to a lower bulk permeability. Corresponding results for the efficiency of the fault zone are presented in Figure 17.7(b). Again, the efficiency generally decreases with increasing domain height as the fault zone forms a less effective barrier to the flow. From these results and the results for other values of D3, we can conclude that considering 3D rather than 2D properties becomes of increased importance as Ds increases. A 30 X 30 m, 2D vertical cross-section of a fault damage zone when D3 = 2 is shown in Figure 17.8(a). A thin 3D section of the fault damage zone at the same location is shown in Figure 17.8(b) to illustrate the effect of the faults on the flow pathway tortuosity. 17.3.3 Incorporation of fault properties into production simulation models Throughout this chapter we have attempted to highlight the complex nature of faults and fault rocks over length scales of less than 1 /im to over 1 km. The difficulty facing petroleum engineers is how to calculate effective flow properties of simulation grid blocks, which are typically 100 m wide areally and have a thickness of around 10 m, containing such faults and their complexity. The first attempts at quantitative fault seal analysis for production simulation modelling simply applied transmissibility multipliers to the faces of grid blocks to take into account the presence of fault rocks, see, for example. Yielding et al (1997), Knai and Knipe (1998), and Manzocchi et al (1999). For example. Figure 17.9(a) shows two grid blocks, labelled i and j , with permeabilities of ki and kj, and lengths Li and i j , respectively. The transmissibility, Trans^j, between two grid blocks straddled by a fault, see Figure 17.9(b), is Transii

2T L/if Ki ~r Juj/ kj

(17.15)

461

S. D. HARRIS ET AL.

1 ;"; k;v^>«-«*^

" '- : ,/,' V.'-t>r'iiv^^'f"'

^'

Figure 17.9 (a) The representation of a fault straddling two grid blocks as a 'multiplier' to the inter-block transmissibility Transij. (b) The modelled fault of thickness tf physically occupies a thickness tf/2 of each neighbouring grid block.

where T is the transmissibility multiplier that is used to account for the presence of the fault. For the case where the grid blocks have the same size and are not offset from each other, T can be calculated using T =

^ \

^^f2/kf-l/ki-l/kj\ L/if Ki + L/j/Kj J

-1

(17.16)

where tf is the thickness of the fault rock. In many cases, the fault permeability can be obtained directly by conducting flow experiments on small-scale faults present within core from petroleum reservoirs. However, a combination of the LBM and the MCMC method presented in Section 17.3.1 provides a way to estimate the permeability of fault rocks where flow experiments are not possible. Previously, fault rock thickness has been estimated based on correlations with the fault displacement, see, for example, Manzocchi et al. (1999). However, the use of the DFFM model described in Section 17.3.2 provides a method to more accurately estimate effective fault rock thickness based on measurements of the fault population statistics in core or outcrop. 17.3.4 Knowledge gaps and future directions A key problem with the methodology outlined above is that it does not take into account the multiphaseflowproperties of faults, see Fisher and Knipe (2001), Manzocchi et al. (2002), and Al-Busafi et al. (2005). In terms of multiphase flow, it might be expected that the buoyancy force in the petroleum column would not be sufficient to overcome the threshold pressure of the fault rock near the free-water level (FWL). Here the relative permeability to petroleum would be expected to be zero, see Figure 17.2(c), which would not be taken into account in conventional methods used to calculate transmissibility multipliers. On

462

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

the other hand, at greater distances above the FWL the buoyancy force generated by the petroleum column may be sufficient to overcome the threshold pressure of the fault rock. In such a situation the fault rock would have a finite permeability to petroleum, see Fisher era/. (2001). We see three particular areas on which research work should focus in terms of multiphase flow. Firstly, the multiphase flow properties of fault rocks have never been measured directly in the laboratory. Indeed, it is likely that such experiments will be extremely timeconsuming and expensive. We therefore aim to extend the LBM described above to model multiphase flow in fault rocks. Secondly, we have used the results from the DFFM model to suggest that the effective fault rock thickness of a damage zone is approximately 50% that of the cumulative fault rock thickness, see Odling et al. (2004). This approach seems reasonably robust for single-phase flow, but further work is needed to verify if such an approach is justified for multiphase flow. A third key area for research is identifying better methodologies to incorporate the multiphase flow properties of faults into production simulation models. It has been proposed that a possible approach is to use dynamic pseudo-functions, see Fisher and Knipe (2001) and Manzocchi et al. (2002). Essentially, dynamic pseudo-functions are generated by conducting high-resolution fluid flow models at the scale of the reservoir simulation grid block, using flow rates similar to those likely to be encountered within the reservoir. In these high-resolution models, both the fault and reservoir rock are given their own capillary pressure and relative permeability curves. The results of the simulations are then used to create relative permeability and capillary pressure curves that are then incorporated into the upscaled production simulation model to account for the presence of both the fault and undeformed reservoir. There are many ways to calculate such pseudo-functions; the reader is referred to Barker and Thibeau (1996) and Barker and Dupouy (1999) for comprehensive reviews of the subject. Although research has only just begun into the best methods for incorporating the multiphase flow properties of faults into production simulation models using pseudo-functions, it is clear that several significant problems need to be overcome. One of the most significant of these problems regards the large number of pseudo-functions that would need to be generated due to the following factors: (i) both the faults themselves and the undeformed reservoir are extremely heterogeneous, and each grid block containing a fault may therefore generate a different pseudofunction; (ii) the pseudo-functions will vary as a function of boundary conditions (such as flow rate); and (iii) the pseudo-functions will vary depending upon whether the rock is undergoing drainage or imbibition. The problem of boundary conditions can be overcome by dynamically updating the simulation model. In this approach, after several time steps the rate of cross-fault flow is recalculated and a set of pseudo-functions are determined based on these new flow rates. It is also possible to group pseudo-functions to reduce the number that are needed for the simulation, see Christie (1996). Several methods for grouping have been suggested.

S. D. HARRIS ET AL.

463

The latest version of the TransGen software (Fault Analysis Group, University College, Dublin) allows the user to choose the number of groups and the ranking of the groups. For example, the user could decide that pseudo-functions would be calculated for five different fault permeability values, four different flow rates, three fault rock thicknesses, and for both drainage and imbibition, thereby requiring 120 pseudo-functions to be calculated. An alternative would be to calculate pseudo-functions for every grid block containing a fault and then group similar tables based on physical quantities, see Christie (1996), such as the end-point and cross-over point of the relative permeability curves. At present there are only a small number of groups scattered throughout academia and industry that are conducting work on the incorporation of multiphase flow properties of faults into production simulation models. However, it is likely the interest in this area of research will increase as more examples of its importance are identified.

17.4 MODELLING FLOW IN RESERVOIRS WHERE FAULTS AND FRACTURES ACT AS CONDUITS Flow simulation through a fractured porous medium is a challenging problem. The heterogeneity of the porous medium and the connectivity of the fractures have a significant effect on flow, especially in the presence of capillary pressure and gravity effects. Flow through fractured porous media is typically simulated using dual-porosity models. These approaches are based on a continuum representation of the fractures, and this assumption requires a highly-fractured porous medium with good connectivity. The advantages of these methods are their simplicity and their applicability to large models as the local complexity of the fracture network is not represented explicitly. The use of dual-porosity models requires the knowledge of a parameter called the transfer function, which defines theflowexchange between the matrix and the fractures, and the evaluation of this parameter is the main difficulty in such models. The most accurate and also the most computationally expensive methodology for studying flow through fractured porous media is the discrete fracture model (DFM). This approach is a direct numerical simulation of flow through fractured porous media where the matrix and fractures are represented explicitly and Darcy's equations are solved locally. There is increased interest in DFMs because of the availability of more detailed geological models and the increase in computational power. Although the recent DFMs are becoming more and more efficient, the application of these methods at the reservoir scale is not realistic. They should not be seen as a replacement to existing dual-porosity models, but rather additional tools available for providing a better understanding of the flow processes at small scales through detailed information on the pressure and fluid distribution. This information can be used to improve existing dual-porosity models or to investigate new modelling techniques for fractured reservoirs. DFMs can also be used stand-alone or in combination with other techniques like dual-porosity to represent explicitly and accurately the main faults and fractures at the reservoir scale.

464

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

Another technique of interest is the discrete fracture network (DFN) model. The DFN can be seen as a special case of a DFM in which the contribution of the matrix is completely neglected and only the flow through the fracture network is considered. When applicable, this approach can handle much larger models compared to the D I ^ , since the elimination of all matrix blocks leads to a considerable reduction in model size. An improvement to the DFN is possible by applying the idea behind the dual-porosity model to account for the matrix contribution. This is achieved by defining a transfer function to model the flow between a 'Active' matrix pore volume and the fracture network. This will extend the applicability of the DFN but add the difficulty of evaluating the transfer function and defining the matrix pore volume. 17.4.1

Overview of existing discrete fracture models

The existing DFMs require a computational grid of the geological model to perform the flow simulation. To reduce the complexity of the grid generation step the majority of existing techniques use a lower-dimensional geometric object to represent the fractures. Typically, in a 2D problem the matrix block is represented by 2D polygons and the fractures are represented using segments (edges of polygons). In a similar way, in a 3D problem the matrix is represented by polyhedra and the fractures by polygons (faces of polyhedra). Figure 17.10 depicts a 2D example of a fractured porous medium. The initial problem is defined in the physical domain, which is then discretised using polygons for the matrix part and segments for the fractures. The fracture thickness is not represented in the grid domain, and is only considered when discretising the flow equations. This simplifies considerably the gridding of fractured media, because there is no need to explicitly resolve the fractures or to calculate fracture intersections, and the consequences of this simplification are particularly important in 3D systems. The complexity of fractured porous media raises some technical issues for grid generation. These issues are not addressed here. More detail can be found in Koudina et al. (1998), where Delaunay triangulation of 3D fracture networks is discussed. This approach has

Physical domain

Grid domain

Figure 17.10 A typical representation of a 2D fractured porous medium in a DFM: (a) the physical problem, and (b) a geometrical discretisation. The thick segments in the grid domain represent the fractures.

S. D. HARRIS ET AL.

465

been extended to a fully-3D model (including fractures and matrix) by Bogdanov et al. (2003 a), where the fracture network is first triangulated as described by Koudina et al. (1998) and then the space between the fractures is 'paved' by an unstructured boundaryconstrained tetrahedral mesh using an advancing front technique. Other approaches of interest for the gridding of complex geological models are the 'atomic meshing' technique of Hale (2002) and the 'growing region' technique of Karimi-Fard (2004). These methods provide some possible approaches, though the gridding of complex geological models remains a challenge. Once a geometrical discretisation of the fractured porous medium is available the flow equations can be discretised. In the case of unstructured discretisations there are two main approaches—finite-element and finite-volume methods. Baca et al. (1984) were among the first authors to propose a 2D finite-element model for single-phase flow with heat and solute transport. In a more recent paper, Juanes et al. (2002) presented a general finiteelement formulation for 2D and 3D single-phaseflowin fractured porous media. There has been some work on the extension of the finite-element method to handle multiphase flow. For example, Kim and Deo (2(X)0) and Karimi-Fard and Firoozabadi (2003) presented extensions of the work of Baca et al. (1984) for two-phase flow. They modelled the fractures and the matrix in a 2D configuration with the effects of capillary pressure included. The two media (matrix and fractures) were coupled using a superposition approach. This entails discretising the matrix and fractures separately and then adding their contributions to obtain the overall flow equations. The existing approaches based on finite-element procedures are successful in the case of single-phase flow and heat transfer, but in the case of multiphase flow in highly heterogeneous reservoirs they do not ensure local mass conservation. Finite-element formulations based on mixed or discontinuous Galerkin methods can eliminate this difficulty, though these methods are generally more expensive than standard finite-volume procedures. The research on DFMs using finite-volume approaches on an unstructured grid is quite recent. In the context of a finite-volume formulation, it is possible to use either a vertexbased or cell-based approach. In a vertex-based approach (called also control-volume finite-element) the unknowns are evaluated at each vertex. A new set of control volumes is first constructed around each vertex. The local mass conservation is then achieved on this dual grid. Among recent work using a vertex-based approach there is the extension of the DFN developed by Koudina et al. (1998) to a fully-3D model (including matrix and fractures) with application to incompressible single-phase flow, see Bogdanov et al. (2003a), slightly-compressible single-phase flow, see Bogdanov et al. (2003b), and incompressible two-phase flow, see Bogdanov et al. (2003c). A similar approach for 2D and 3D fractured porous media is presented by Monteagudo and Firoozabadi (2004). Cell-based approaches, in which control volumes can be readily aligned with the discontinuities of the permeability field, are probably more appropriate for reservoir simulation applications. In this configuration the unknowns are evaluated inside each control volume. Previous work on cell-based approaches is mainly for 2D systems discretised on triangular meshes. For example, Caillabet et al. (2000, 2001) used a two-equation model for single-phase problems, and a similar approach was employed in a single-porosity model for single-phase flow by Granet et al. (1998) and then extended to two-phase flow by

466

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

Granet et al (2001). More recently, Karimi-Fard et al. (2004) presented a cell-based DFM for 2D and 3D systems using unstructured grids. In the following section this last technique is presented in more detail. 17.4.2

Technical description of the methodology

We consider a simple and effective DFM which is applied on an unstructured polygonal (2D) or polyhedral (3D) grid to provide a better representation of the heterogeneity. The unknowns are evaluated at the centroid of each control volume. To determine the flow equations, the mass balance is applied for each control volume, which requires knowledge of neighbouring control volumes (a connectivity list) and the flow rate associated with each connection. An accurate representation of theflowrate on a flexible grid with matrix heterogeneity requires a multipoint flux approximation, see Aavatsmark et al, (1998a, 1998b) and Lee et al (2002). In the present methodology a two-point flux approximation is used for its simplicity and robustness. This approximation is acceptable because it is generally more important to capture the effects of fracture geometry and connectivity (which is accurately represented in this approach) than it is to accurately resolve the detailed effects of matrix anisotropy. However, the existing multipoint flux approximation techniques are compatible with this formulation and can be used if necessary. The flow rate between two adjacent control volumes is expressed as Qi2 = Ti2A(p2~Pi),

(17.17)

where pi is the fluid pressure in cell i, Qu is the flow rate from cell 1 to cell 2, Tu is the geometric part of the transmissibility, and A represents the fluid mobility using upstream information. In the case of multiphase flow, different flow rates, pressures and mobilities are applicable for each phase. The geometric part of the transmissibility is, however, the same for each phase and is given by: Ti2 =

with Qi = -77-rii • fi. (17.18) a i -f- a2 Di Here, Ai is the area of the interface between two control volumes (using information from CVi, where CV^ designates the ith control volume), ki is the permeability of CVj, Di is the distance between the centroid of the interface and the centroid of CVi, n^ is the unit normal to the interface inside CV^, and fi is the unit vector along the direction of the line joining the CV centroid to the centroid of the interface. These quantities are illustrated in Figure 17.11 for a 2D configuration. This procedure for computing T12 is essentially a generalisation of the transmissibility calculation for comer-point systems, see Ponting (1989). For two adjacent CVs only three different configurations are possible: both matrix CVs; one matrix CV and one fracture CV; or two fracture CVs. For two matrix CVs equation (17.18) can be applied directly to evaluate theflowrate, but for cases involving fracture CVs a special treatment is necessary. One issue is that the fracture thickness is not represented in the grid domain but it is required for flux evaluation. This issue is addressed by

467

S. D. HARRIS ET AL.

Ai = Ao /

\

D2 = IIC0C2II

Figure 17.11 Geometrical representation of two adjacent control volumes and the definition of different parameters involved in a. The shape of the control volumes (dashed lines) does not enter this calculation.

Grid domain

Computational domain

•

•

2D

Fractures

1

/

3D

Figure 17.12 Connection between matrix and fracture in 2D and 3D configurations. using a computational domain where the thickness of the fracture is locally considered. It is important to note that the computational domain is never explicitly constructed. Figure 17.12 shows the case when a fracture and a matrix CV are connected. Equation (17.18) can be applied directly on the computational domain to evaluate the flow rate between the matrix and the fracture. The connection between two or more fracture CVs requires additional consideration since, typically, two connected fracture CVs do not have the same thickness. This problem is addressed by introducing an intermediate CV in the computational domain to account for flow redirection and thickness variation. Figure 17.13 illustrates this technique for two and three connected fractures. Since the intermediate CV will introduce numerical problems due to its relatively small size, we account for it implicitly and write the transmissibility between the fracture CVs directly. For the case of two connected fractures, a harmonic average of the transmissibilities can be used to directly write the transmissibility between the fractures as follows: Tl2 =

TioTc 10^02 Tio + Tc02

(17.19)

468

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

Grid domain

Computational domain

Figure 17.13 A 2D example of fracture intersection in the grid and computational domains. Each fracture can have different thickness and properties.

where Tij denotes the transmissibility between CVi and CVj. Moreover, as the intermediate CV is generally very small compared to the surrounding fracture CVs, it is possible to introduce the additional simplifications Tio « a i and T02 « 0:2, and write the transmissibilities between fracture CVs directly: U2

Q^iQ2

(17.20)

Oil + Oi2

In the case of three connected fractures it is possible to eliminate the intermediate CV using the analogy between flow through porous media and conductance through a network of resistors. Figure 17.14 depicts the 'star-delta' transformation for three connected resistors. The use of this transformation combined with the previous simplification gives the three new transmissibilities J-ij —

TjoTjo TiQ -h T2Q -\- Tso for

OLjOLj

a i -f 0:2 + as z =: 1, J = 2,

= 1, j = 3 and

2,j=3. (17.21)

Figure 17.14 Star-delta transformation for an equivalent network of resistors.

S. D. HARRIS ET AL.

469

In fact, it is possible to write the transformation for n connected fractures as follows: Tij^-^^^^.

(17.22)

It is important to note that this transformation is applied using only information contained in the connectivity list. It can be applied to 2D as well as 3D problems and it is used to eliminate all intersection CVs. The * star-delta' transformation is exact in the case of singlephase incompressible flow since we know the transmissibilities precisely, but it is only an approximation for general multiphase flow problems. In general, the transmissibilities depend on saturations and pressure and are therefore time dependent, and so, because the transformation is applied only to the geometric part of the transmissibility, a local error is introduced into the calculations. These errors are typically small and the transformation provides good accuracy for multiphase flow problems. Table 17.2 summarises the advantages of this simplification for 2D and 3D flow problems. For 2D flow through a fractured porous medium, three different objects are needed in standard implementations. Specifically, the matrix CVs are represented by 2D objects, the fracture CVs by ID objects and the fracture intersections by OD objects. Using our simplified DFM we can eliminate the OD objects in both 2D and 3D. The ID objects can additionally be eliminated in 3D systems. This results in improved computational efficiency relative to standard DFMs. As the thickness of fractures is not represented explicitly and only considered when evaluating flow rates and computing pore volumes, there is a resulting slight mismatch between the total fracture volume in the grid and computational domains. This extra volume is negligible if only a few fractures are considered but it may become significant globally as the number of fractures increases. The volumes of the matrix blocks surrounding the fractures are therefore modified to maintain the correct pore volume. This is accomplished by removing pore volume from matrix control volumes that connect with fractures. The amount of pore volume removed depends on the number and size of fractures to which the matrix control volume is connected. 17.4.3 An example of flow simulation in a fractured reservoir To illustrate the applicability of this methodology to complex fractured porous media a 2D example from Karimi-Fard et al (2004) is presented. Figure 17.15(a) represents an

Table 17.2 Summary of objects for 2D and 3Dflowusing the standard and simplified DFMs. Standard DFM Simplified DFM 2D flow 2D, ID, OD 2D, ID 3D flow 3D, 2D, ID, OD 3D, 2D

470

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

(c)

Figure 17.15 (a) Detailed map of a fault zone obtained from outcrop measurements (adapted from Myers, 1999 and Jourde et al, 2002). (b) Simplified fault zone, and (c) the corresponding discretisation. The discrete model contains 3138 triangular and linear control volumes.

areal map of a portion of a strike-slip fault (with about 14 m of slip) in the Valley of Fire State Park in Nevada, USA. This map is approximately 1 m square and was obtained from detailed outcrop measurements, as discussed in Myers (1999) and Jourde et al. (2002). In the figure, white represents the host rock, which has a permeability of 200 mD and a porosity of 22%. The gray filled region in the centre of the model represents fault core or gouge, having a very low permeability (0.1 mD) and porosity (15%) relative to the host rock. The gray linear features in the damage zone outside of the fault core are sheared joints and deformation bands, which display reduced porosity and permeability similar to that of the fault core. The heavier black features, both inside the fault core and in the damage zone, are joints and slip surfaces (open fractures), which display permeabilities of about 10^ mD and porosities of 100%. These features provide the main pathways for flow through the fault zone. Faults can have a significant impact on large-scale flow in reservoirs. However, they are typically represented using very simple techniques, such as cross-fault transmissibility multipliers. There is therefore a clear need to enhance fault models in reservoir simulators.

S.D. HARRIS ETAL.

471

Water saturation Figure 17.16 Water saturation profiles at (a) 0.1, (b) 0.3 and (c) 0.5 pore volume of water injected.

Here a DFM is used to simulate detailed flow through a fault zone. This type of model could be used directly in a simulator or as a pre-processing calculation to determine effective fault zone properties, which could then provide input to the simulator. Due to limitations of the available gridding tools, a simplified model of this fault zone is considered, see Figure 17.15(b), although the essential features of Figure 17.15(a) are retained. A triangulated mesh of this model is also shown in Figure 17.15(c). A waterflood simulation is performed by injecting 0.01 PV (pore volume) per day of water in the bottom-left comer of the model, whilst producing liquid from the top-right comer. The system is initially saturated with oil. Figure 17.16 depicts the water saturation profiles after 0.1, 0.3 and 0.5 PV of water injection. As can be seen, the high-permeability slip surfaces provide conduits for flow through the fault core. The proper modelling of this effect is essential for the accurate representation of transport through fault zones of this type. The DFM is clearly well suited to this calculation.

17.5 DISCUSSION AND CONCLUSIONS Faults and fractures are often highly-complex stmctures that retard or enhance fluid flow on length scales of less than 1 /im to more than 1 km. Production simulation models, used to predict the behaviour of petroleum reservoirs, are computationally very expensive and cannot directly incorporate all of the small-scale heterogeneity. We have therefore developed numerical techniques that allow us to model the effects of faults on fluid flow at a variety of scales to provide effective properties that can be incorporated into production simulation models. Clearly, there has been a considerable amount of work conducted in this direction and much progress has been made. There are, however, huge questions that remain regarding the extent to which such work is currently leading to better decisionmaking within the petroleum industry. A key issue here is that there have been very few

472

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

post-mortem analyses in which the predictions made by such modelling techniques have been compared with production data. Indeed, even if post-mortems were conducted it may be difficult to assess whether the way in which faults or fractures have been incorporated into the model is correct due to the large number of other uncertainties that exist within the model. In other words, there is a real possibility that we are developing numerical methods that cannot be accurately tested against reality. A second key issue is whether some of the mathematical methods employed are a massive 'overkill'. There has been a tradition in petroleum engineering to treat complex problems in the most simple manner possible, see Dake (2001), and we are still open-minded to the suggestion that many of the problems described in this paper could be addressed just as well but in a more simple manner. We are acutely aware that petroleum resources are running out rather rapidly and a sense of urgency needs to be generated to advance some of these predictive techniques while there are still reserves to be produced. We would therefore urge future research not just into the complex methods employed here but also into more simple methods that might have more impact on future petroleum production.

ACKNOWLEDGEMENTS We would like to thank our numerous sponsors who have partially funded this work. Special mention should go to our industrial sponsors Anadarko, BG, BP, ConocoPhilips, Encana, Kerr KcGee, Maersk, Petrobras, Shell, Statoil, and Total. NERC are also thanked for providing financial support as part of their /i2M thematic research program. We are grateful to Dr Xiaoxian Zhang for the use of his LBM code.

REFERENCES Aavatsmark, I., Barkve, T., B0e, 0., and Mannseth, T. (1998a). Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: Derivation of the methods. SIAM / ScL Comp. 19,1700-16. Aavatsmark, I., Barkve, T., B0e, 0., and Mannseth, T. (1998b). Discretization on unstructured grids for inhomogeneous, anisotropic media. Part 11: Discussion and numerical results. SIAM J. ScL Comp. 19, 1717-36. Adler, P. M., Jacquin, C. G., and Quiblier, J. A. (1990). Flow in simulated porous media. Int. J. Multiphase Flow 16, 691-712. Al-Busafi, B., Fisher, Q. J., and Harris, S. D. (2005). The importance of incorporating the multiphase flow properties and thickness of fault rocks into production simulation models. Marine Petroleum Geol. In press. Antonellini, M. and Aydin, A. (1995). Effect of faulting on fluid flow in porous sandstones: geometry and spatial distribution. Amer. Assoc. Petroleum Geol. Bull. 79, 642-71. Baca, R. G., Amett, R. C, and Langford, D. W. (1984). Modellingfluidflowin fractured-porous rock masses byfinite-elementtechniques. Int. J. Numer Meth. Fluids 4, 337-48.

S. D. HARRIS ET AL.

473

Bakke, S. and 0ren, P. E. (1997). 3-D pore-scale modelling of sandstones and flow simulations in the pore networks. Soc. Petroleum Eng. J. 2, 136-49. Barker, J. W. and Dupouy, P. (1999). An analysis of dynamic pseudo-relative permeability methods for oil-water flows. Petroleum Geosci. 5, 385-94. Barker, J. W. and Thibeau, S. (1996). A critical review of the use of pseudo-relative permeabilities for upscaling. SPE 35491. Bhatnagar, P., Gross, E. P., and Krook, M. K. (1954). A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component system. Phys. Rev. 94, 511-25. Bogdanov, 1.1., Mourzenko, V. V., Thovert, J.-F., and Adler, P. M. (2003a). Effective permeability of fractured porous media in steady state flow. Water Resources Res, 39, 1023. Bogdanov, 1.1., Mourzenko, V. V., Thovert, J.-E, and Adler, P. M. (2003b). Pressure drawdown well tests in fractured porous media. Water Resources Res. 39, 1021. Bogdanov, 1.1., Mourzenko, V. V., Thovert, J.-F., and Adler, P. M. (2003c). Two-phase flow through fractured porous media. Phys. Rev. £68,026703. Bonnet, E., Bour, O., Odling, N. E., Davy, P., Main, I., Cowie, P., and Berkowitz, B. (2001). Scaling of fracture systems in geological media. Rev. Geophys. 39, 347-83. Caillabet, Y., Fabric, P., Landereau, P., Noetinger, B., and Quintard, M. (2000). Implementation of finite-volume method for the determination of effective parameters in fissured porous medium. Numer. Meth. Partial Diff. Eqns 16, 237-63. Caillabet, Y., Fabric, P., Lasseux, D., and Quintard, M. (2001). Computation of large-scale parameters for dispersion in fissured porous medium using finite-volume method. Computat. Geosci. 5, 121-50. Chen, S. and Doolen, G. D. (1998). Lattice Boltzmann method for fluid flows. Ann. Rev. Fluid Mech. 30, 329-64. Chen, S., Chen, H., Martinez, D., and Matthaeus, W. (1991). Lattice Boltzmann model for simulation of magnetohydrodynamics. Phys. Rev. Lett. 67, 3776-9. Christie, M. A. (1996). Upscaling for reservoir simulation. J. Petroleum Tech. 48, 1004-10. Cressie, N. A. C. (1993). Statistics for spatial data. Wiley, New York. Dake, L. P. (2001). The practice of reservoir engineering (revised edn). Elsevier, Amsterdam. Dunsmoir, J. H., Ferguson, S. R., D'Amico, K. L., and Stokes, J. P. (1991). X-ray microtomography: a new tool for the characterization of porous media. In Proceedings of the 1991 SPE annual technical conference and exhibition, Dallas, 6-9 October. Filippova, O. and Hanel, D. (1998). Grid refinement for lattice-BGK models. /. Computat. Phys. 147, 219-28. Fisher, Q. J. and Knipe, R. J. (1998). Fault sealing processes in siliciclastic sediments. In Faulting, fault sealing and fluid flow in hydrocarbon reservoirs (eds G. Jones, Q. J. Fisher, and R. J. Knipe), pp. 117-34. Geological Society, London, Special publications 147. Fisher, Q. J. and Knipe, R. J. (2001). The permeability of faults within siliclastic petroleum reservoirs of the north sea and Norwegian continental shelf. Marine Petroleum Geol. 18,1063-81. Fisher, Q. J., Harris, S. D., McAllister, E., Knipe, R. J., and Bolton, A. J. (2001). Hydrocarbon flow across sealing faults: theoretical constraints. Marine Petroleum Geol. 18, 251-7. Foxford, K. A., Walsh, J. J., Watterson, J., Garden, I. R., Guscott, S. C , and Burley, S. D. (1998). Structure and content of the Moab fault zone, Utah, USA, and its implications for fault seal

474

FAULTS ANfD FRACTURES IN PETROLEUM RESERVOIRS

prediction. In Faulting, fault sealing and fluid flow in hydrocarbon reservoirs (eds G. Jones, Q. J. Fisher, and R. J. Knipe), pp. 87-103. Geological Society, London, Special publications 147. Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Machine InteII. 6,721-41. Granet, S., Fabric, P., Lemonnier, P., and Quintard, M. (1998). A single-phase flow simulation of fractured reservoir using a discrete representation of fractures. In 6th European conference on the mathematics of oil recovery, Peebles, Scotland, 8-11 September. Granet, S., Fabric, P., Lemonnier, P., and Quintard, M. (2001). A two-phase flow simulation of a fractured reservoir using a new fissure element method. J. Petroleum Sci. Eng. 32, 35-52. Gunstensen, A. K., Rothman, D. H., Zaleski, S., and Zanetti, G. (1991). Lattice Boltzmann model of immiscible fluids. Phys. Rev. A 43, 4320-7. Hale, D. (2002). Atomic meshes: from seismic imaging to reservoir simulation. In 8th European conference on the mathematics of oil recovery, Freiberg, Germany, 3-6 September. Harris, S. D., McAllister, E., Knipe, R. J., and Odling, N. E. (2003). Predicting the three-dimensional population characteristics of fault damage zones: a study using stochastic models. J. Struct. Geol. 25, 1281-99. Jourde, H., Rodin, E. A., Aydin, A., Durlofsky, L. J., and Wen, X.-H. (2002). Computing permeability of fault zones in eolian sandstone from outcrop measurements. Amer. Assoc. Petroleum Geol Bull. 86,1187-200. Juanes, R., Samper, J., and Molinero, J. (2002). A general and efficient formulation of fractures and boundary conditions in the finite element method. Int. J. Numer. Meth. Eng. 54, 1751-74. Kadanoff, L. (1986). On two levels. Phys. Today 39, 7-9. Karimi-Fard, M. (2004). Growing region technique applied to grid generation of complex fractured porous media. In 9th European conference on the mathematics of oil recovery, Cannes, France, 30 August-2 September. Karimi-Fard, M. and Firoozabadi, A. (2003). Numerical simulation of water injection in fractured media using discrete-fracture model and the Galerkin method. Soc. Petroleum Eng. Reservoir Eval. Eng. 6,117-26. Karimi-Fard, M., Durlofsky, L. J., and Aziz, K. (2004). An efficient discrete-fracture model applicable for general-purpose reservoir simulators. Soc. Petroleum Eng. J. 9, 227-36. Kim, J.-G. and Deo, M. D. (2000). Finite element, discrete-fracture model for multiphase flow in porous media. AIChEJ. 46, 1120-30. King Hubbert, M. (1956). Darcy's law and the field equations of the flow of underground fluids. Trans. AIME 2^, 222-39. Knai, T. A. and Knipe, R. J. (1998). The impact of faults on fluid flow in the Heidrun field. In Faulting, fault sealing andfluidflowin hydrocarbon reservoirs (eds G. Jones, Q. J. Fisher, and R. J. Knipe), pp. 269-82. Geological Society, London, Special publications 147. Knott, S. D., Beach, A., Brockbank, P. J., Lawson Brown, J., McCallum, J. E., and Welbon, A. I. (1996). Spatial and mechanical controls on normal fault populations. /. Struct. Geol. 18, 359-72. Koudina, N., Garcia, R. G., Thovert, J.-F., and Adler, P. M. (1998). Permeability of threedimensional fracture networks. Phys. Rev. E SI, 4466-79. Lee, S. H., Tchelepi, H. A., Jenny, P., and De Chant, L. J. (2002). Implementation of a fluxcontinuous finite-difference method for stratigraphic, hexahedron grids. Soc. Petroleum Eng. J. 7, 267-77.

S. D. HARRIS ET AL.

475

Manswart, C. and Hilfer, R. (1998). Reconstruction of random media using Monte Carlo methods. Phys. Rev. £59,5596-9. Manswart, C , Torquato, S., and Hilfer, R. (2000). Stochastic reconstruction of sandstones. Phys. Rev. £62,893-9. Manzocchi, T., Heath, A. E., Walsh, J. J., and Childs, C. (2002). The representation of two-phase fault-rock properties in flow simulation models. Petroleum GeoscL 8, 119-32. Manzocchi, T., Walsh, J. J., Nell, R, and Yielding, G. (1999). Fault transmissibility multipliers for flow simulations models. Petroleum Geosci. 5, 53-63. Mattax, C. C. and Dalton, R. L. (1990). Reservoir simulation. SPE Monograph Series 13, Richardson, TX. Monteagudo, J. E. R and Firoozabadi, A. (2004). Control-volume method for numerical simulation of two-phase immiscible flow in two- and three-dimensional discrete-fractured media. Water Resources Res. 40, W07405. Myers, R. (1999). Mechanism and permeability of brittle faults in sandstone. Ph.D. thesis. Stanford University. Nicol, A., Watterson, J., Walsh, J. J., and Childs, C. (1996). The shapes, major axis orientations and displacement patterns of fault surfaces. J. Struct. Geol. 18, 235-48. Odling, N. E., Harris, S. D., and Knipe, R. J. (2004). Permeability scaling properties of fault damage zones in siliclastic rocks. /. Struct. Geol. 26, 1727-47. Okabe, H. and Blunt, M. J. (2003). Multiple-point statistics to generate geologically realistic porespace representations. In Proceedings of the society of core analysts annual meeting, September. Ponting, D. K. (1989). Comer point geometry in reservoir simulation. In 1st European conference on the mathematics of oil recovery, Cambridge, England, 25-27 July. Purcell, W. R. (1949). Capillary pressures—their measurement using mercury and the calculation of permeability therefrom. Amen Inst. Mining, MetalL, Petroleum Eng., Petroleum Trans. 186,39-48. Qian, W. and Titterington, D. M. (1991). Pixel labelling for three-dimensional sense based on Markov mesh models. Signal Proc. 22, 313-28. Qian, Y. H. (1993). Simulating thermohydrodynamics with lattice BGK models. /. Sci. Comp. 8,231-41. Qian, Y. H., d'Humieres, D., and Lallemand, P. (1992). Lattice BGK models for Navier-Stokes equation. Europhys. Lett. 17, 479-84. Shipton, Z. K. and Cowie, P. A. (2001). Damage zone and slip-surface evolution over m to km scales in high-porosity Navajo sandstone, Utah. J. Struct. Geol. 23, 1825-44. Shipton, Z. K., Evans, J. P., Robeson, K. R., Forster, C. B., and Snelgrove, S. (2002). Structural heterogeneity and permeability in faulted eolian sandstone: implications for subsurface modelling of faults. Amen Assoc. Petroleum Geol Bull. 86, 863-83. Singh, M. and Mohanty, K. K. (2000). Permeability of spatially correlated porous media. Chem. Eng. Sci. 55, 5393-403. Succi, S. (2001). The lattice Boltzmann equation for fluid dynamics and beyond. Oxford University Press. Succi, S., Foti, E., and Higuera, F. (1989). Three-dimensional flows in complex geometries with the lattice Boltzmann method. Europhys. Lett. 10, 433-8.

476

FAULTS AND FRACTURES IN PETROLEUM RESERVOIRS

Watts, N. L. (1987). Theoretical aspects of cap-rock and fault seals for single- and two-phase hydrocarbon columns. Marine Petroleum Geol. 4, 274-307. Wu, K., Nunan, N., Ritz, K., Young, I., and Crawford, J. (2004). An efficient Markov chain model for the simulation of heterogeneous soil structure. Soil Sci, Soc. Amer J. 68, 346-51. Yielding, G., Freeman, B., and Needham, D. T. (1997). Quantitative fault seal prediction. Amer. Assoc. Petroleum Geol. Bull. 81, 897-917. Zhang, X. X., Bengough, A. G., Crawford, J. W., and Young, I. M. (2002). A lattice BGK model for advection and anisotropic dispersion equation. Adv. Water Res. 25, 1-8. Zou, Q. and He, X. (1997). On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Phys. Fluids 9, 1591-8.