Mechanical Engineering Series Frederick F. Ling Series Editor Advisory Board Applied Mechanics F.A. Leckie University of California, Santa Barbara Biomechanics V.C. Mow Columbia University Computational Mechanics H.T. Yang Purdue University Dynamic Systems and Control K.M. Marshek University of Texas, Austin Energetics W.A. Sirignano University of California, Irvine Mechanics of Materials I. Finnie University of California, Berkeley Processing K.K. Wang Cornell University Thermal Science A.E. Bergles Rensselaer Polytechnic Institute Tribology W.O. Winer Georgia Institute of Technology Mechanical Engineering Series Introductory Attitude Dynamies F.P. Rimrott Balancing of High-Speed Machinery M.S. Darlow Theory of Wire Rope G.A. Costello Theory of Vibration VoI. I Introduction Vol. II Discrete and Continuous Systems A.A. Sbabana Laser Machining: Theory and Practice G. Chryssolouris Underconstrained Structural Systems E.N. Kuznetsov Principles of Heat Transfer in Porous Media M. Kaviany M. Kaviany Principles of Heat Transfer in Porous Media With 158 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest M. Kaviany Department of Mechanical Engineering and Applied Mechanics The University of Michigan Ann Arbor, MI 48109-2125 Series Editor Frederick F. Ling President, Institute for Productivity Research New York, NY 10018 and Distinguished William Howard Hart Professor Emeritus Department of Mechanical Engineering, Aeronautical Engineering and Mechanics Rensselaer Polytechnic Institute 'll"oy, NY 12180-3590 USA Library of Congress Cataloging-in-Publication Data Kaviany, M. (Massoud) Principles of heat transfer in porous media 1 Massoud Kaviany, p. cm. - (Mechanical engineering series) Includes bibliographical references and indexes. ISBN-13: 978-1-4684-0414-2 e-ISBN-13: 978-1-4684-0412-8 DO!: 10.1007/978-1-4684-0412-8 1. Heat - Transmission. 2. Porous materials - Thermal properties. I. Title. II. Series: Mechanical engineering series (Berlin, Germany) TJ260.K29 1991 621.402'2-dc20 91-14658 Printed on acid-free paper. © 1991 Springer-Verlag New York Inc. Softcover reprint ofthe hardcover 1st edition 1991 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereaf ter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Camera-ready copy prepared from the author's LaTeX file. 987654321 To my parents Farideh and Morad Series Preface Mechanical engineering, an engineering discipline born of the needs of the industrial revolution, is once again asked to do its substantial share in the call for in dust rial renewal. The general call is urgent as we face profound issues of productivity and competitiveness that require engineering solu tions, among others. The Mechanical Engineering Series is a new series, featuring graduate texts and research monographs, intended to address the need for information in contemporary areas of mechanical engineering. The series is conceived as a comprehensive one that will cover a broad range of concentrations important to mechanical engineering graduate edu cation and research. We are fortunate to have a distinguished roster of consulting editors, each an expert in one of the areas of concentration. The names of the consulting editors are listed on the first page of the volume. The areas of concentration are applied mechanics, biomechanics, compu tational mechanics, dynamic systems and control, energetics, mechanics of materials, processing, thermal science, and tribology. Professor Bergles, the consulting editor for thermal science, and I are pleased to present this volume of the series: Principles of H eat Transfer in Porous Media by Professor Kaviany. The seledion of this volume under scores again the interest of the Mechanical Engineering Series to provide our readers with topical monographs as weH as graduate texts. New York, New York Frederick F. Ling Preface This monograph aims at providing, through integration of available theo retical and empirical treatments, the differential conservation equations and the associated constitutive equations required for the analysis of transport in porous media. Although the empirical treatment of fluid flow and heat transfer in porous media is over a century old, only in the last three decades has the transport in these heterogeneous systems been addressed in suffi cient detail. So far, single-phase flow and heat transfer in porous media have been treated or at least formulated satisfactorily. But the subject of two-phase flow and the related he at transfer in porous media is still in its infancy. This monograph identifies the pr in ci pIes of transport in porous media, reviews the available rigorous treatments, and whenever possible compares the available predictions, based on these theoretical treatments of various transport mechanisms, with the existing experimental results. The theoretical treatment is based on the local volume-averaging of the momentum and energy equations with the closure conditions necessary for obtaining solutions. While emphasizing abasie understanding of heat transfer in porous media, the monograph does not ignore the need for the predictive tools. Therefore, whenever a rigorous theoretical treatment of a phenomenon is not available, semiempirical and empirical treatments are given. The monograph is divided into two parts: part 1 deals with single-phase flows and part 2 covers two-phase flows. For single-phase flows, all mo des of heat transfer are examined first using a single-continuum treatment based on the assumption of a Iocal thermal equilibrium. A two-medium treatment is then made. In part 2, pore-level fluid mechanics and the thermodynamics for the simultaneous presence of both fluid phases in porous media are ad dressed. Conduction and convection he at transfer are then examined. The he at and mass transfer from surfaces bounding these porous media, which contain both liquid and gaseous phases, is also presented. Since the fluid dynamics of two-phase flow involving phase change is not yet fully under stood, specific phase-change processes and their peculiarities are discussed in the last chapter. The contents of each chapter are briefly reviewed here. The historical and practical aspects of heat transfer in porous media, as well as the length, time, and temperature scales encountered in this field, are reviewed in chapter one. The fluid mechanics of the single-phase flow, beginning with the Darcy law and developing along more rigorous treat ments based on the local volume averaging, are given in chapter two. In this chapter we also examine the porosity variations near the bounding x Preface solid surfaces and the hydrodynamic boundary conditions at the interface of the porous plain media. Heat conduction is treated in chapter three and deterministic, stochastic, and semiempirical treatments are examined. Variations of the effective thermal conductivity tensor ne ar the bounding surfaces are also discussed. Hydrodynamic dispersion, which always exists in convective he at transfer in porous media, and the various treatments of it are given in chapter four. These include the local volume-averaging treatment for periodic and disordered structures. Again, dispersion near the bounding surfaces is addressed. Radiation heat transfer, which is significant in low-temperature insulation applications as weIl as in high-temperature combustion applications, is discussed in chapter five. The derivation of the radiative properties from the optical properties and the various approxi mations used in the radiative heat transfer calculations are also considered. The theory of independent scattering is examined for large particles and the inclusion of dependent scattering is formulated and a solution method is presented. Mass transfer in gases, including the low-pressure, small pore size Knudsen regime, the surface diffusion, and chemical reactions are con sidered in chapter six. Part 1 ends with chapter seven, where two-medium treatment of transient heat transfer is given. The discussion aims at clar ifying the existing misunderstanding about this approach, and several ex amples are given. Part 2 begins with chapter eight, which discusses two-phase flow and the complexity of the fluid mechanics, including hysteresis. The local volume averaging and semiempirical treatments and the constitutive equations for the capillary pressures and the relative permeability are discussed. In ad dition, the coefficients for the inertial and surface-tension gradient terms in the moment um equations are presented. In chapter nine, we address the peculiarities of liquid-vapor coexistence in porous media, including the reduction in vapor pressure and capillary condensation. Conduction and convection heat transfer in two-phase flows are examined in chapter ten using both local volume averaging and semiempirical treatments. Heat and mass transfer from partially liquid-saturated permeable surfaces, including the zero Bond number asymptote, are given in chapter eleven. The phase change in the liquid-vapor systems in porous media is discussed in chapter twelve, where specific steady-state and transient processes are considered. The monograph is written as a detailed description of the fundament als of heat transfer in porous media. Familiarity with fluid mechanics and heat transfer is assumed. The concepts and physical phenomena are emphasized more than the step-by-step development. When intermediate steps in the derivations are not given, they can either be found in the references or arrived at by the material supplied in the discussion. The symbols used are defined in the nomenclature after chapter twelve. A glossary of the common terms is given following the nomenclature. Acknowledgment I would like to thank my students working in the area of heat transfer in porous media; M. Fatehi, C.-J. Kim, M. Mittal, J. Rogers, M. Sahraoui, B.P. Singh, and Y.-X. Tao. They have been a constant source of ideas and encouragement and have made this mono graph possible. I am grateful for their patience and devotion. Special thanks to C.-J. Kim for word process ing the entire monograph. The illustrations for the monograph were drawn by R. Hill and S. Erring ton of the University of Michigan. I am very thankful to them for their ex cellent professional services and to other staff members of my department who have provided assistance. My research in the area of heat transfer in porous media has been spon sored by the National Science Foundation, the Ford Motor Company, the Whirlpool Corporation, and by NASA-LeRC. I am grateful for their finan cial support and for the encouragement of their project monitors. Finally, I would like to thank my wife Mitra, who took on many extra responsibilities so that I could attend to this monograph. She remains a constant sour ce of support and strength. Contents Series Preface Vll Preface. . . . . IX Acknowledgment xi 1 Introduction . 1 1.1 Historical Background 1 1.2 Length, Time, and Temperature Scales . 5 1.3 Scope .. 8 1.4 References....... 10 Part 1: Single-Phase Flow . 13 2 Fluid Mechanics . . . . . 15 2.1 Darcy Momentum Equation 15 2.2 Porosity . . . . 18 2.3 Pore Structure ..... 22 2.4 Permeability ...... 26 2.4.1 Capillary Models 27 2.4.2 Hydraulic Radius Model 30 2.4.3 Drag Models for Periodic Structures 32 2.5 High Reynolds Number Flows . . . . 42 2.5.1 Macroscopic Models . . . . . 42 2.5.2 Microscopic Fluid Dynamics . 45 2.5.3 Turbulence . . . . . . . . . . 47 2.6 Brinkman Superposition of Bulk and Boundary Effects . 49 2.7 Local Volume-Averaging Method 50 2.7.1 Local Volume Averages 51 2.7.2 Theorems....... 52 2.7.3 Momentum Equation . 54 2.8 Homogenization Method . . . 58 2.8.1 Continuity Equation . 59 2.8.2 Momentum Equation . 60 2.9 Semiheuristic Momentum Equations 62 2.10 Significance of Macroscopic Forces . 64 2.10.1 Macroscopic Hydrodynamic Boundary Layer 65 2.10.2 Macroscopic Entrance Length . . . . . . . . . 66 2.11 Porous Plain Media Interfacial Boundary Conditions 67