Advances in Heat Transfer, Volume 34 (Advances in Heat Transfer)

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ADVANCES IN HEAT TRANSFER Volume 34

a This Page Intentionally Left Blank

Advances in

HEAT TRANSFER Serial Editors

James P. Hartnett

Thomas F. Irvine, Jr.

Energy Resources Center University of Illinois at Chicago Chicago, Illinois

Department of Mechanical Engineering State University of New York at Stony Brook Stony Brook, New York

Serial Associate Editors

Young I. Cho

George A. Greene

Department of Mechanical Engineering Drexel University Philadelphia, Pennsylvania

Department of Advanced Technology Brookhaven National Laboratory Upton, New York

Volume 34

San Diego San Francisco New York Boston London Sydney Tokyo

This book is printed on acid-free paper. Copyright 2001 by Academic Press All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. The appearance of code at the bottom of the ﬁrst page of a chapter in this book indicates the Publisher’s consent that copies of the chapter may be made for personal or internal use of speciﬁc clients. This consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc. (222 Rosewood Drive, Danvers, Massachusetts 01923), for copying beyond that permitted by Sections 107 or 108 of the U. S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for chapters are as shown on the title pages; if no fee code appears on the chapter title page, the copy fee is the same for current chapters. 0065-2717/01 $35.00 ACADEMIC PRESS 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA http://www.academicpress.com Academic Press Harcourt Place, 32 Jamestown Road, London NW1 7BY, UK International Standard Book Number: 0-12-020034-1 International Standard Serial Number: 0065-2717

Printed in the United States of America 00 01 02 03 QW 9 8 7 6 5 4 3 2 1

CONTENTS Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix xi

Transport Phenomena in Heterogeneous Media Based on Volume Averaging Theory V. S. Travkin and I. Catton I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . II. Fundamentals of Hierarchical Volume Averaging Techniques . . . . . . . . . . . . . . . . . . . . . . . .

1

A. Theoretical Veriﬁcation of Central VAT Theorem and Its Consequences .

10

III. Nonlinear and Turbulent Transport in Porous Media A. Laminar Flow with Constant Coefﬁcients . . . . . . . . . B. Nonlinear Fluid Medium Equations in Laminar Flow . . . . C. Porous Medium Turbulent VAT Equations . . . . . . . .

. . . .

. . . .

. . . .

. . . . D. Development of Turbulent Transport Models in Highly Porous Media . E. Closure Theories and Approaches for Transport in Porous Media . . . IV. Microscale Heat Transport Description Problems and VAT Approach . . . . . . . . . . . . . . . . . . . . . . . A. Traditional Descriptions of Microscale Heat Transport . . . . . . . . B. VAT-Based Two-Temperature Conservation Equations . . . . . . . . C. Subcrystalline Single Crystal Domain Wave Heat Transport Equations . D. Nonlocal Electrodynamics and Heat Transport in Superstructures . . . E. Photonic Crystals Band-Gap Problem: Conventional DMM-DNM and VAT Treatment . . . . . . . . . . . . . . . . . . . . . .

V. Radiative Heat Transport in Porous and Heterogeneous Media . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Flow Resistance Experiments and VAT-Based Data Reduction in Porous Media . . . . . . . . . . . . . . . VII. Experimental Measurements and Analysis of Internal Heat Transfer Coefﬁcients in Porous Media . . . . . . . . . . VIII. Thermal Conductivity Measurement in a Two-Phase Medium . . . . . . . . . . . . . . . . . . . . . . . . . IX. VAT-Based Compact Heat Exchanger Design and Optimization . . . . . . . . . . . . . . . . . . . . . . A. A Short Review of Current Practice in Heat Exchanger Modeling . v

4 14 17 19 21 26 32

37 38 43 45 46

.

52

. .

56

. .

66

. .

85

. .

96

. . 111 . . 112

contents

vi B. C. D. E.

New Kinds of Heat Exchanger Mathematical Models . . VAT-Based Compact Heat Exchanger Modeling . . . . Optimal Control Problems in Heat Exchanger Design . . A VAT-Based Optimization Technique for Heat Exchangers

X. New Optimization Technique for on VAT . . . . . . . . . . . . XI. Concluding Remarks . . . . . Nomenclature . . . . . . . . . References . . . . . . . . . . .

. . . . . . . . . . . . Material Design Based . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

116 117 123 124

. . . .

. . . .

. . . .

127 129 131 133

Two-Phase Flow in Microchannels S. M. Ghiaasiaan and S. I. Abdel-Khalik I. Introduction . . . . . . . . . . . . . . . . . . II. Characteristics of Microchannel Flow . . . . . III. Two-Phase Flow Regimes and Void Fraction in Microchannels. . . . . . . . . . . . . . . . . A. Deﬁnition of Major Two-Phase Flow Regimes . . . . B. Two-Phase Flow Regimes in Microchannels . . . . .

. . . . . . . 145 . . . . . . . 146

. . . . . . . . . . . . Review of Previous Experimental Studies and Their Trends . . . Flow Regime Transition Models and Correlations . . . . . . Flow Patterns in a Micro-Rod Bundle . . . . . . . . . . . Void Fraction . . . . . . . . . . . . . . . . . . . . . Two-Phase Flow in Narrow Rectangular and Annular Channels .

. . . . . . . .

. . . . . . . .

. . . C. . D. . E. . F. . G. . H. Two-Phase Flow Caused by the Release of Dissolved Noncondensables . IV. Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . A. General Remarks . . . . . . . . . . . . . . . . . . . . . . B. Frictional Pressure Drop in Two-Phase Flow . . . . . . . . . . . C. Review of Previous Experimental Studies . . . . . . . . . . . . . D. Frictional Pressure Drop in Narrow Rectangular and Annular Channels . V. Forced Flow Subcooled Boiling . . . . . . . . . . . . . . . A. General Remarks . . . . . . . . . . . . . . . . . . . . . . B. Void Fraction Regimes in Heated Channels . . . . . . . . . . . . C. Onset of Nucleate Boiling . . . . . . . . . . . . . . . . . . . D. Onset of Signiﬁcant Void and Onset of Flow Instability . . . . . . . E. Observations on Bubble Nucleation and Boiling . . . . . . . . . . VI. Critical Heat Flux in Microchannels . . . . . . . . . . . . . . . . . A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . B. Experimental Data and Their Trends . . . . . . . . . . . . . . . C. Effects of Pressure, Mass Flux, and Noncondensables . . . . . . . . D. Empirical Correlations . . . . . . . . . . . . . . . . . . . . E. Theoretical Models . . . . . . . . . . . . . . . . . . . . . . VII. Critical Flow in Cracks and Slits . . . . . . . . . . . . . . . A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . B. Experimental Critical Flow Data . . . . . . . . . . . . . . . .

147 148 150 153 161 166 169 170 178

180 180 180 184 189

191 191 192 195 198 205

209 209 210 215 216 221

224 224 225

contents

vii

C. General Remarks on Models for Two-Phase Critical Flow in Microchannels . . . . . . . . . . . . . . . . . . . . D. Integral Models . . . . . . . . . . . . . . . . . . . . E. Models Based on Numerical Solution of Differential Conservation Equations . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. VIII. Concluding Remarks . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

230 232

. 236 . 240 . 242 . 244

Turbulent Flow and Convection: The Prediction of Turbulent Flow and Convection in a Round Tube Stuart W. Churchill I. Introduction . . . . . . . . . . . . A. Turbulent Flow . . . . . . . . . . B. Turbulent Convection . . . . . . . II. The Quantitative Representation of A. Historical Highlights . . . . . . . .

. . . . . . . . . . . . . . . Turbulent . . . . .

. . . . . . . . . Flow . . .

. . . . .

. . . . .

. . . . .

. . . . .

. 256 . 257 . 259 . 260 . 260

B. New Improved Formulations and Correlating Equations . . . . . . . . .

III. The Quantitative Representation of Fully Developed Turbulent Convection . . . . . . . . . . . . . . . A. Essentially Exact Formulations . . . . . . . . . . . . B. Essentially Exact Numerical Solutions . . . . . . . . . C. Correlation for Nu . . . . . . . . . . . . . . . . . IV. Summary and Conclusions . . . . . . . . . . . . . A. Turbulent Flow . . . . . . . . . . . . . . . . . . B. Turbulent Convection . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

294

. . . . . . . .

. 304 . 305 . 323 . 335 . 348 . 348 . 353 . 356

Progress in the Numerical Analysis of Compact Heat Exchanger Surfaces R. K. Shah, M. R. Heikal, B. Thonon, and P. Tochon I. Introduction . . . . . . . . . . . . . II. Physics of Flow and Heat Transfer of A. Interrupted Flow Passages . . . . . . . B. Uninterrupted Complex Flow Passages . .

. . . . . . . CHE Surfaces . . . . . . . . . . . . . .

C. Unsteady Laminar versus Low Reynolds Number Turbulent Flow . . . . . . . . . . . . .

III. Numerical Analysis . . . . . . . . . . A. Mesh Generation . . . . . . . . . . . B. Boundary Conditions . . . . . . . . . . C. Solution Algorithm and Numerical Scheme .

. . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . .

. . . .

. . . .

. . . .

. 363 . 366 . 366 . 371

. . . . .

. . . . .

. . . . .

. . . . .

. 374 . 375 . 376 . 376 . 378

viii

contents

IV. Turbulence Models . . . . . . . . . . . . . . A. Reynolds Averaged Navier—Stokes (RANS) Equations . B. Large Eddy Simulation (LES) . . . . . . . . . . C. Direct Numerical Simulation . . . . . . . . . . . D. Concluding Remarks on Turbulence Modeling . . . . V. Numerical Results of the CHE Surfaces . . . . A. Offset Strip Fins . . . . . . . . . . . . . . . . B. Louver Fins . . . . . . . . . . . . . . . . . C. Wavy Channels . . . . . . . . . . . . . . . . D. Chevron Trough Plates . . . . . . . . . . . . . VI. Conclusions . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

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. . . . . . . . . . . . .

. . . . . . . . . . . . .

380 381 392 395 397

397 398 406 416 425

432 434 435

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . 445 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors’ contributions begin

S. I. Abdel-Khalik (145), G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332. I. Catton (1), Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, Los Angeles, California 90095. Stuart W. Churchill (255), Department of Chemical Engineering, The University of Pennsylvania, Philadelphia, Pennsylvania 19104. S. M. Ghiaasiaan (145), G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology Atlanta, Georgia 30332. M. R. Heikal (363), University of Brighton, Brighton, United Kingdom. R. K. Shah (363), Delphi Harrison Thermal Systems, Lockport, New York 14094. B. Thonon (363), CEA-Grenoble, DTP/GRETh, Grenoble, France. P. Tochon (363), CEA-Grenoble, DTP/GRETh, Grenoble, France. V. S. Travkin (1), Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, Los Angeles, California 90095.

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PREFACE For over a third of a century this serial publication, Advances in Heat Transfer, has ﬁlled the information gap between regularly published journals and university-level textbooks. The series presents review articles on special topics of current interest. Each contribution starts from widely understood principles and brings the reader up to the forefront of the topic being addressed. The favorable response by the international scientiﬁc and engineering community to the thirty-four volumes published to date is an indication of the success of our authors in fulﬁlling this purpose. In recent years, the editors have undertaken to publish topical volumes dedicated to speciﬁc ﬁelds of endeavor. Several examples of such topical volumes are Volume 22 (Bioengineering Heat Transfer), Volume 28 (Transport Phenomena in Materials Processing), and Volume 29 (Heat Transfer in Nuclear Reactor Safety). As a result of the enthusiastic response of the readers, the editors intend to continue the practice of publishing topical volumes as well as the traditional general volumes. The editorial board expresses their appreciation to the contributing authors of this volume who have maintained the high standards associated with Advances in Heat Transfer. Lastly, the editors acknowledge the efforts of the professional staff at Academic Press who have been responsible for the attractive presentation of the published volumes over the years.

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ADVANCES IN HEAT TRANSFER Volume 34

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ADVANCES IN HEAT TRANSFER, VOLUME 34

Transport Phenomena in Heterogeneous Media Based on Volume Averaging Theory

V. S. TRAVKIN and I. CATTON Department of Mechanical and Aerospace Engineering University of California, Los Angeles Los Angeles, California 90095

I. Introduction Determination of ﬂow variables and scalar transport for problems involving heterogeneous (and porous) media is difﬁcult, even when the problem is subject to simpliﬁcations allowing the speciﬁcation of medium periodicity or regularity. Linear or linearized models fail to intrinsically account for transport phenomena, requiring dynamic coefﬁcient models to correct for shortcomings in the governing models. Allowing inhomogeneities to adopt random or stochastic character further confounds the already daunting task of properly identifying pertinent transport mechanisms and predicting transport phenomena. This problem is presently treated by procedures that are mostly heuristic in nature because sufﬁciently detailed descriptions are not included in the description of the problem and consequently are not available. The ability to describe the details, and features, of a proposed material with precision will help reduce the need for a heuristic approach. Some aspects of the development of the needed theory are now well understood and have seen substantial progress in the thermal physics and in ﬂuid mechanics sciences, particularly in porous media transport phenomena. The basis for this progress is the so-called volume averaging theory (VAT), which was ﬁrst proposed in the 1960s by Anderson and Jackson [1], Slattery [2], Marle [3], Whitaker [4], and Zolotarev and Radushkevich [5]. 1

ISBN: 0-12-020034-1

ADVANCES IN HEAT TRANSFER, VOL. 34 Copyright 2001 by Academic Press. All rights of reproduction in any form reserved. 0065-2717/01 $35.00

2

v. s. travkin and i. catton

Further advances in the use of VAT are found in the work of Slattery [6], Kaviany [7], Gray et al. [8], and Whitaker [9, 10]). Many of the important details and examples of application are found in books by Kheifets and Neimark [11], Dullien [12], and Adler [13]. Publications on turbulent transport in porous media based on VAT began to appear in 1986. Primak et al. [14], Shcherban et al. [15], and later studies by Travkin and Catton [16, 18, 20, 21], etc., Travkin et al. [17, 19, 22], Gratton et al. [26, 27] and Catton and Travkin [28] present a generalized development of VAT for heterogeneous media applicable to nonlinear physical phenomena in thermal physics and ﬂuid mechanics. In most physically realistic cases, highly complex integral—differential equations result. When additional terms in the two- and three-phase statements are encountered, the level of difﬁculty in attempting to obtain closure and, hence, effective coefﬁcients, increases greatly. The largest challenge is surmounting problems associated with the consistent lack of understanding of new, advanced equations and insufﬁcient development of closure theory, especially for integral—differential equations. The ability to accurately evaluate various kinds of medium morphology irregularities results from the modeling methodology once a porous medium morphology is assigned. Further, when attempting to describe transport processes in a heterogeneous media, the correct form of the governing equations remains an area of continuously varying methods among researchers (see some discussion in Travkin and Catton [16, 21]). An important feature of VAT is being able to consider speciﬁc medium types and morphologies, lower-scale ﬂuctuations of variables, cross-effects of different variable ﬂuctuations, interface variable ﬂuctuations effects, etc. It is not possible to include all of these characteristics in current models using conventional theoretical approaches. The VAT approach has the following desirable features: 1. Effects of interfaces and grain boundaries can be included in the modeling. 2. The effect of morphology of the different phases is incorporated. The morphology decription is directly incorporated into the ﬁeld equations. 3. Separate and combined ﬁelds and their interactions are described exactly. No assumptions about effective coefﬁcients are required. 4. Effective coefﬁcients correct mathematical description — those ‘‘theories’’ presently used for that purpose are only approximate description, and often simply wrong. 5. Correct description of experiments in heterogeneous media — again, at present the homogeneous presentation of medium properties is used

volume averaging theory

3

for this purpose, and explanation of experiments is done via bulk features. Those bulk features describe the ﬁeld as by classical homogeneous medium differential equations. 6. Deliberate design and optimization of materials using hierarchical physical descriptions based on the VAT governing equations can be used to connect properties and morphological characteristics to component features. What is usually done is to carry out an experimental search by adding a third or a fourth component to the piezoelectric material, for example. This can be done in a more direct, more observable way, and with a more correct understanding of the effects of adding additional components and, of course, of the morphology of the fourth component. In this work we restrict ourselves to a brief analysis of previous work and show that the best theoretical tool is the nonlocal description of hierarchical, multiscaled processes resulting from application of VAT. Application of VAT to radiative transport in a porous medium is based on our advances in electrodynamics and microscale energy transport phenomena in twophase heterogeneous media. Some other governing conservation equations for transport in porous media can be found in Travkin and Catton [21] and the references therein. One of the aims of this work is to outline the possibilities for a method for optimizing transport in heterogeneous as well as porous structures that can be used in different engineering ﬁelds. Applications range from heat and mass exchangers and reactors in mechanical engineering design to environmental engineering usage (Travkin et al. [19]). A recent application is in urban air pollution, where optimal control of a pollutant level in a contaminated area is determined, along with the design of an optimal control point network for the control of constituent dispersion and remediation actions. Using second-order turbulent models, equation sets were obtained for turbulent ﬁltration and two-temperature or two-concentration diffusion in nonisotropic porous media and interphase exchange and microroughness. Previous work has shown that the ﬂow resistance and heat transfer over highly rough surfaces or in a rough channel or pipe can be properly predicted using the technique of averaging the transport equations over the near surface representative elementary volume (REV). Prescribing the statistical structure of the capillary or globular porous medium morphology gives the basis for transforming the integral—differential transport equations into differential equations with probability density functions governing their coefﬁcients and source terms. Several different closure models for these terms for some uniform, nonuniform, nonisotropic, and speciﬁcally random nonisotropic highly porous layers were developed. Quite

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v. s. travkin and i. catton

different situations arise when describing processes occuring in irregular or random morphology. The latest results, obtained with the help of exact closure modeling for canonical morphologies, open a new ﬁeld of possibilities for a purposeful search for optimal design of spacial heterogeneous transport structures. A way to ﬁnd and govern momentum transport through a capillary nonintersecting medium by altering its morphometrical characteristics is given as validation of the process.

II. Fundamentals of Hierarchical Volume Averaging Techniques Since the porosity in a porous medium is often anisotropic and randomly inhomogeneous, the random porosity function can be decomposed into additive components: the average value of m(x) in the REV and its ﬂuctuations in various directions, m (x ) : m (x ) ; m (x ),

. m :

The averaged equations of turbulent ﬁltration for a highly porous medium are similar to those in an anisotropic porous medium. Five types of averaging over an REV function f are deﬁned by the following averaging operators arranged in their order of seniority (Primak et al. [14]): average of f over the whole REV, f : f ; f : m f ; (1 9 m ) f , phase averages of f in each component of the medium,

1 f : m f (t, x ) d : m f 1 f : m f (t, x ) d : m f and intraphase averages,

(1)

(2) (3)

1 f : f : f (t, x ) d (4) 1 f : f : f (t, x ) d. (5) When the interface is ﬁxed in space, the averaged functions for the ﬁrst and second phase (as liquid and solid) within the REV and over the entire REV fulﬁll the conditions

volume averaging theory f ; g : f ; g

for steady-state phases and

f t

:

a : a : const

and

f

, t

except for the differentiation condition,

f g : f g

5 (6)

(7)

1 f : f ; f ds D 1U f : f 9 f, f , (8) D where S is the inner surface in the REV, and ds is the solid-phase, U s : ndS). The fourth condiinward-directed differential area in the REV (d tion implies an unchanging porous medium morphology. The three types of averaging fulﬁll all four of the preceding conditions as well as the following four consequences: f : f, f : f 9 f : 0 (9) f g : f g , f g : f g : 0 (10) Meanwhile, f and f fulﬁll neither the third of the conditions, a " a, a : m a, (11) nor all the consequences of the other averaging conditions. Futher, the differential condition becomes 1 f : f ;

f ds , (12) 1 in accordance with one of the major averaging theorems — the theorem of averaging the operator (Slattery [6]; Gray et al. [8]; Whitaker [10]). If the statistical characteristics of the REV morphology and the averaging conditions with their consequences lead to the following special ergodic hypothesis: the spacial averages, ( f , f , and f ), then this theorem converges with increases in the averaging volume to the appropriate probability (statistical) average of the function f of a random value with probability density distribution p. This hypothesis is stated mathematically as follows: f @(x ) :

\

lim f : f @.

f (x , )p , x d ? (13)

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v. s. travkin and i. catton

Quintard and Whitaker [29] expressed some concern about the connection between different scale volume averaged variables, for example, 1 T : D D

D

T d D

(14)

1 T : T d. (15) D D D D D In a truly periodic system it is known that the steady temperature in phase f can be written as T (r ) : h · r ; T (r ) ; T , (16) D D D D D D where h and T are constants and T (r ) is a periodic function of zero mean D D D over the f-phase. Applying the phase averaging operator to this D function, one ﬁnds T (x) : h r (x) ; T (r ) ; mT , D D · D D D D D D while Quintard and Whitaker [29] obtain (their Eq. (13)) T (r ) : T (r ) : h · r ; mT , D D D D D D D D D D meaning that

(17)

T (r ) : 0. (18) D D D The parameter T (r ) cannot always be equal to zero, because it D D D depends on the peculiarities of the chosen REV. In some instances, when the REV is not the volume that contains the known number of exact function periods, the averaged function T (r ) value should not be zero. If it is D D D assumed, however, that the REV volume contains the exact number of spacial periods, then T (r ) : 0. D D D Averaging the ﬂuid temperature, T , over yields the intrinsic average D D T (x) : h · r ; T : h · x ; h · y ; T , (19) D D D D D D D D because the averaging of r (see Fig. 1) results in D r : x ; y , (20) D D D D while Quintard and Whitaker [29] obtain (their Eq. (15)) T : T ( r ) : h · r ; T (21) D D D D D D D D D They note (see p. 375), ‘‘now represent r in terms of the position vector, D x, that locates the centroid of the averaging volume, and the relative

volume averaging theory

7

Fig. 1. Representative elementary averaging volume with the ‘‘virtual’’ points of representation inside of the REV (Carbonell and Whitaker [31]; Quintard and Whitaker [29]).

position vector y as indicated in Fig. 3’’ (see Fig. 1), or D r : x ; y $ r (x, y ) : x ; y (r , x) , (22) D D D D D D D D so that Eq. (21) can be written with dependence on both x and r , D T (x, r ) : T (x, y ) : h · x ; h · y (r , x) ; T , (23) D D D D D D D D D D D meaning that after averaging, T (r ) continues to be dependent on the D D position of the ‘‘virtual’’ point r , which may have changed location within D the . D To do this, they introduce a so-called ‘‘virtual REV’’ allowing the averaged value inside of the REV to be variable (see the remark on p. 354 of Quintard and Whitaker [30]: ‘‘In all our previous studies of multiphase transport phenomena, we have always assumed that averaged quantities could be treated as constants within the averaging volume and that the average of the spatial deviations was zero. We now wish to avoid these assumptions. . . .’’), and the result is a ‘‘virtual’’ averaged variable that is not

8

v. s. travkin and i. catton

constant within the ﬁxed volume of the REV. When Quintard and Whittaker derive the gradient of the average of the function (23), they use its dependence on x for the two right-hand-side terms in (23) to obtain (24)

T (x, r ) : h ; h · x y (r , x) . D D D D D D Several comments about the Quintard and Whittaker treatment that need to be considered are the following: 1. How the communication of the variables from different spaces r at the D lower scale space and x at the upper scale space is established is not meaningful. Their connection must be determined at the beginning of the averaging process and their communication is very limited. 2. One should only connect a value at a point at the higher level to the lower level REV, not only to a point within the lower level REV. When one considers an averaged variable at any point other than the representative point x for a particular REV, then

T (x) : (h · x ; T (x) ; T ), D D D D D and for the upper scale, the exact result is

T (x) : (h · x ; T ) : h. (25) D D D 3. If a function and its gradient are periodic, then the averaged function should be periodic. The VAT-based answer should be seen by determination of the averaged values, which are not averaged, only the REV being used at the lower scale. The work by Quintard and Whittaker and the improving of understanding of some basic principles of averaging has led us to state the following lemma and then point out differences from the work of Whittaker and his colleagues. Lemma. If a function , representing any continuous physical ﬁeld, is averaged over the subdomain , which is the subregion occupied by phase D f ( ﬂuid phase) of the REV in the heterogeneous two-phase medium (Fig. 1), and the averaged function (x) is assumed to have different values at D different locations x within the , then the averaged function (x) can G D D have discontinuities of the ﬁrst kind at the boundary of the REV . Proof. Consider the situation where the point y (Fig. 1; see also Fig. 3, D p. 375, in the paper by Quintard and Whitaker [29]) is located an inﬁnitely distance from the boundary of the REV within . It represents the intrinsic phase averaged value of variable averaged in the REV D D . According to Carbonell and Whitaker [31] and Quintard and Whitaker [29], its value can be different from or . D D V D

volume averaging theory

9

Fig. 2. Representative elementary averaging volumes with the ﬁxed points of representation.

Next, consider a point y located an inﬁnitely small distance outside of D the initial REV within a boundary . The point y represents the D averaged value which belongs to some REV , as shown in Fig. D D 2, with its center at the point x : (y 9 ) ; (R /2), with being an D inﬁnitely small constant. Following arguments of Carbonell and Whitaker [31], this point y is D allowed to be in at least one more REV, , which has its center x just shifted from the point x an inﬁnitely small distance, as does y from the D boundary . Further, suppose that one is approaching the boundary from both sides by points y and y . According to Carbonell and Whitaker [31] and D D Quintard and Whitaker [29], the values and can be different D D D D when is reached, which means that the averaged value experien D ces (can have) a discontinuity at each and every point of the boundary . As long as the boundary of the REV can be arbitrarily moved, changed or assigned, then the consequence of this change is that can D D have discontinuities at each point of a REV.

v. s. travkin and i. catton

10

The relationships between different scale variables and their points of representation can be found by noting the following points: 1. There is a ﬁxed relationship between the location of the point x of the upper scale ﬁeld and averaging within the REV . In other words, for each determined there is only one x that represents the value (x ) on the upper ﬁeld level (macroscale ﬁeld) if both are mapped on D D the same region (excluding close to boundary regions). 2. If there is the value (x ) , (x 9 x ) , then there is another D D " , and in it 1 1 (r ) d " (r ) d, (26) (x ) : D D D D D D D D D D where (r ) " const. D D

A. Theoretical Verification of Central VAT Theorem and Its Consequences When the coefﬁcient of thermal conductivity k is a constant value, the D ﬂuid stedy-state conduction regime is described by

k k (mT ) ; · D D D

k T ds ; D D

T · ds : 0. (27) D 1U 1U The full 1D Cartesian coordinates version of this equation, without any source, for a ﬁxed solid matrix in is

1 T m D ; x x x

1U

1 T ds ; D

T D · ds : 0, x 1U G

T m D ; MD ; MD : 0, x x

(28) (29)

where the second and third terms on the right-hand side are the so-called morphodiffusive terms, MD and MD , respectively (see also, for example, Travkin and Catton [21]), The solid-phase equation with constant k equation is of the same form, Q T

1 1 T Q Q ; Q · ds : 0, s T ds ; (30) Q x x x x 1U 1U G which can also be written in terms of the ﬂuctuating variable,

s

T 1 Q; x x

1U

1 T ds ; Q

T Q · ds : 0. x 1U G

(31)

volume averaging theory

11

Travkin and Catton [16, 18, 20] suggested that the integral heat transfer terms in Eqs. (28, (30), and (31) be closed in a natural way by a third (III) kind of heat transfer law. The second integral term reﬂects the changing averaged surface temperature along the x coordinate. Equations (28) and (30) can be treated using heat transfer correlations for the heat exchange integral term (the last term). Regular dilute arrangements of pores, spherical particles, or cylinders have been studied much more than random morphologies. Using separate element or ‘‘cell’’ modeling methods (Sangani and Acrivos [32] and Gratton et al. [26]) to ﬁnd the interface temperature ﬁeld allows one to close the second, ‘‘surface’’ diffusion integral terms in (28), (30), and (27). Many forms of the energy equation are used in the analysis of transport phenomena in porous media. The primary difference between such equations and those resulting from a more rigorous development based on VAT are certain additional terms. The best way to evaluate the need for these additional more complex terms is to obtain an exact mathematical solution and compare the results with calculations using the VAT equations. This will clearly display the need for using the more complex VAT mathematical statements. Consider a two-phase heterogeneous medium consisting of an isotropic continuous (solid or ﬂuid) matrix and an isotropic discontinuous phase (spherical particles or pores). The volume fraction of the matrix, or f-phase, is m : m : /, and the volume fraction of ﬁller, or s-phase, is D D m : 1 9 m : /, where : ; is the volume of the REV. Q Q D Q D The constant properties (phase conductivities, k and k ), stationary (timeD Q independent) heat conduction differential equations for T and T , the local D Q phase temperatures, are 9 · q : k T : 0, 9 · q : k T : 0, D D D Q Q Q with the fourth (IVth) kind interfacial ( f —s) thermal boundary conditions T : T , ds · q : ds · q . Q 1U D Q D 1U Here q : 9k T and q : 9k T are the local heat ﬂux vectors, S D D D Q Q Q U is the interfacial surface, and ds is the unit vector outward to the s-phase. No internal heat sources are present inside the composite sample, so the temperature ﬁeld is determined by the boundary conditions at the external surface of the sample. After correct formulation of these conditions, the problem is completely stated and has a unique solution. Two ways to realize a solution to this problem were compared (Travkin and Kushch [33, 34]). The ﬁrst is the conventional way of replacing the actual composite medium by an equivalent homogeneous medium with an effective thermal conductivity coefﬁcient, k : k (s, k , k ), assuming one CDD D Q

v. s. travkin and i. catton

12

knows how to obtain or calculate it. The exact effective thermal coefﬁcient was obtained using direct numerical modeling (DNM) based on the mathematical theory of globular morphology multiphase ﬁelds developed by Kushch (see, for example, [35—38]). Averaging the heat ﬂux, q, and temperature, T , over the REV yields q : k T , and for the stationary case there results CDD

· (k T ) : 0. (32) CDD The boundary conditions for this equation are formulated in the same manner as for a homogeneous medium. The second way is to solve the problem using the VAT two-equation, three-term integrodifferential equations (28) and (30). To evaluate and compare solutions to these equations with the DNM results, one needs to know the local solution characteristics, the averaged characteristics over the both phases in each cell, and, in this case, the additional morphodiffusive terms. An inﬁnite homogeneous isotropic medium containing a three-dimensional (3D) array of spherical particles is chosen for analysis. The particles are arranged so that their centers lie at the nodes of a simple cubic lattice with period a. The temperature ﬁeld in this heterogeneous medium is caused by a constant heat ﬂux Q prescribed at the sample boundaries, which, X because of the absence of heat sources, leads to the equality of averaged internal heat ﬂux q : Q . X When all the particles have the same radii, the result is the triple periodic structure used widely, beginning from Rayleigh’s work [39], to evaluate the effective conductivity of particle-reinforced composites. The composite medium model consists of the three regions shown in Fig. 3. The half-space lying above the A—A plane has a volume content of the disperse phase m : m , and for the half-space below the B—B plane Q m : m . To deﬁne the problem, let m m . The third part is the Q composite layer between the plane boundaries A—A and B—B containing N double periodic lattices of spheres (screens) with changing diameters. Solutions to the VAT equations (28) and (30) for a composite with varying volume content of disperse phase with accurate DNM closure of the micro model VAT integrodifferential terms were obtained implicitly, meaning that each term was calculated independently using the results of DNM calculations. For the one-dimensional case, Eq. (32) becomes

T k : 0, (z z z ), z CDD z

m : m (z), Q Q

where k (m ) is the effective conductivity coefﬁcient. CDD Q

volume averaging theory

13

Fig. 3. Model of two-phase medium with variable volume fraction of disperse phase.

The normalized solution of the both models (VAT and DNM) for the case of linearly changing porosity m : m ; z(m 9 m), where m (z ) : m, Q Q Q Q Q Q m (z ) : m, z : 0, z : 1, between A—A and B—B and with effective Q Q conductivity coefﬁcients of k : 0, 0.2, 1, 10, and 10,000, are presented in CDD Fig. 4. There is practically no difference (less than 10\) between the solutions, and what there is is probably because of numerical error accumulation (Travkin and Kushch [34]). Lines 1—5 represent solutions of the one-term equation, respectively, whereas the points (circles, triangles, etc.) represent the solutions of the VAT equations with accurate DNM closure of the micro model VAT integrodifferential terms MD and MD for the composite with varying volume content of disperse phase. Here the number of screens is nine, corresponding to a relatively small particle phase concentration gradient. The coincidence of the results of the exact calculation of the two-equation, three-term conductive-diffusion transport VAT model (28) and (30) with the exact DNM solution and with the one-temperature effective coefﬁcient model for heterogeneous media with nonconstant spatial morphology clearly demonstrates the need for using all the terms in the VAT equations. The need for the morphodiffusive terms in the energy equation is further demonstrated by noting that their magnitudes are all of the same order. Conﬁrmation of the fact that there is no difference in solutions between the correct one-term, one-temperature effective diffusivity equation and the

14

v. s. travkin and i. catton

Fig. 4. Comparison of VAT three-term equation particle temperature (symbols) with the exact analytical based on the effective conductance coefﬁcient obtained by exact DNM (solid lines).

three-terms, two-temperature VAT equations does not mean that it is better to take for modeling and analysis the effective diffusivity one-term, onetemperature equation (see Subsection VI, E and arguments in Sections VII and VIII). Among other issues one needs to analyze goals of modeling and to understand that the good solution of the effective diffusivity one-term one-temperature equation as it was found and described in the preceding statements means nothing less than the ground of the exact solution of the VAT problem. Also, it is important that for the exact (or accurate) solution of conventional diffusivity equation, the effective coefﬁcient needs to be found, and this means in turn that ﬁnding the solution of the two-ﬁeld problem is imperative and consequently appears to be the major problem. Meanwhile, this is the problem that was posed just at the beginning as the original one.

III. Nonlinear and Turbulent Transport in Porous Media To a great extent, the analysis of porous media linear transport phenomena are given in the numerous studies by Whitaker and coauthors; see, for

volume averaging theory

15

example, [10, 30, 31, 40—46], as well as by studies by Gray and coauthors [8, 47—50]. Our present work is mostly devoted to the description of other physical ﬁelds, along with development of their physical and mathematical models. Still, the connection to linear and partially linear problem statements needs to be outlined. The linear Stokes equations are

V : 0, (33)

0 : 9 p ; V ; g, D

and although the Stokes equation is adequate for many problems, linear as well as nonlinear processes will result in different equations and modeling features. The general averaged form of the transport equations will be developed for permeable interface boundaries between the phases. Two forms of the right-hand-side Laplacian term will be considered. First, one can have two forms of the diffusive ﬂux in gradient form that can be written V : V ; D D

V : m V ; D

V ds

(34)

V ds.

(35)

1U

or

1U

It was pointed out ﬁrst by Whitaker [42, 43] that these forms allow greater versatility in addressing particular problems. Using the two averaged forms of the velocity gradient, (34) and (35), one can obtain two averaged versions of the diffusion term in Eq. (33), namely,

1 ( V ) : · ( mV ) ; · D

1U

V ds ;

1U

V · ds, (36)

where the production term V · ds is a tensorial variable, and the version with ﬂuctuations in the second integral term

1 ( V ) : · (m V ) ; · D

1U

V ds ;

1U

V · ds, (37)

v. s. travkin and i. catton

16

Using these two forms of the momentum viscous diffusion term, one can write two versions of the averaged Stokes equations. The ﬁrst version is

1

V ; D

1U

U · ds : 0, U Y V G G

(38)

and 1 0 : 9 p 9 D ; ·

1

1U

pds ; · ( mV )

V ds ;

V · ds ; m g , D

(39) 1U 1U and the second version is found by using the following relation for the pressure gradient:

1 9 p 9 D

pds : 9m p 9

1

p ds. (40) 1U 1U Using the averaging rules developed by Primak et al. [14], Shcherban et al. [15] and Travkin and Catton [16, 18] facilitated the development of the momentum equation. By combining equations (37) and (40), one is able to write the momentum transport equations in the second form with velocity ﬂuctuations

1

V 9 V m ; D D

1U

U · ds : 0, G

(41)

obtained using

1

V : V ; V D 1 V ds : 9 V m ; D

1U and the momentum equation 0 : 9m p 9

1

1

1U

1U

U · ds, G

(42)

p ds ; · (m V )

V ds ;

V · ds ; m g. (43) D 1U 1U The third version of these equations is almost never used but can be found in [21]. ; ·

volume averaging theory

17

A. Laminar Flow with Constant Coefficients The transport equations for a ﬂuid phase with linear diffusive terms are U G:0 (44) x G U U 1 p U G ;S G;U G:9 ; (45) H x S x x x t H D H H H D;U D:D D ;S . (46) D H x D x x t H H H Here represents any scalar ﬁeld (for example, concentration C) that might be transported into either of the porous medium phases, and the last terms on the right-hand side of (45) and (46) are source terms. In the solid phase, the diffusion equation is

Q:D Q ;S . (47) Q Q x x t H H The averaged convective operator term in divergence form becomes, after phase averaging,

1 (U U ) : (U U ) : U U ; H G D H G D H G x D H

1 : [mU U ; m u u ] ; H G H G D

1U

U U · ds H G

U U · ds. (48) H G 1U Decomposition of the ﬁrst term on the right-hand side of (48) yields ﬂuctuation types of terms that need to be treated in some way. The nondivergent version of the averaged convective term in the momentum equation is

1 (U U ) : mU U ; U U ; u u ; U U · ds H G H G G H D H G D H G x D 1U H 1 : mU U 9 U U · ds H x G G H 1U H 1 ; u u ; U U · ds. (49) H G D H G 1U The divergent and nondivergent forms of the averaged convective term in

v. s. travkin and i. catton

18

the diffusion equation are 1 (CU ) : CU ; G D G D

CU · ds G

1U

1 : [mC U ; m c u ] ; G G D

1 : mU C 9C G x G

1U

1U

CU · ds G

1 U · ds ; c u ; G G D

CU · ds. G

1U

(50)

Other averaged versions of this term can be obtained using impermeable interface conditions (see also Whitaker [42] and Plumb and Whitaker [44]). For constant diffusion coefﬁcient D, the averaged diffusion term becomes

1 · (D C ) : D · (mC ) ; D · D

D

D

Cds ;

1U

C · ds,

1U

(51)

or

1 · (D C ) : D · (m C ) ; D · D

1U

c ds ;

1U

C · ds, (52)

or

1 D · (D C ) : Dm C ; D · D

1U

c ds ;

D

1U

c · ds. (53)

Other forms of Eq. (52), using the averaging operator for constant diffusion coefﬁcient, constant porosity, and absence of interface surface permeability and transmittivity, can be found in works by Whitaker [42] and Plumb and Whitaker [44], as well as by Levec and Carbonell [46]. A similar derivation can be carried out for the momentum equation to treat cases where Stokes ﬂow is invalid. Two versions of the momentum equation will result. The equation without the ﬂuctuation terms is

V 1 m ; mV · V 9 V D t :9 (mp ) 9 ; ·

1

1

1U

1U

1U

1 V · ds ; v v ; D

V V · ds 1U

pds ; · (mV )

V ds ;

1U

V · ds ; m g . D

(54)

volume averaging theory

19

with the ﬂuctuation diffusion terms it becomes

V 1 ; mV · V 9 V m D t :9m p 9

1

1

1U

1 V · ds ; v v ; D

1U

V V · ds 1U

p ds ; · (m V )

v ds ;

V · ds ; m g . (55) D 1U 1U The steady-state momentum transport equations for systems with impermeable interfaces can readily be derived from Eq. (54) and (55). They are ; ·

1 (mV · V ; v v ) : 9 (mp ) 9 D D ;

1U

1U

pds ; · (mV )

V · ds ; m g , D

(56)

or 1 (mV · V ; v v ) : 9m p ) 9 D D ;

1U

1U

p ds ; · (mV )

V · ds ; m g. D

(57)

B. Nonlinear Fluid Medium Equations in Laminar Flow To properly account for Newtonian ﬂuid ﬂow phenomena within a porous medium in a general way, modeling should begin with the Navier— Stokes equations for variable ﬂuid properties, D

V ; V · V : 9 p ; · [( V ; ( V )*)] ; g D t

(58)

: (V, C , T ), G rather than the constant viscosity Navier—Stokes equations. The following form of the momentum equation will be used in further developments:

V ; V · V : 9 p ; · (2S ) ; g D t D : (V, C , T ). G

(59)

v. s. travkin and i. catton

20

The negative stress tensor in this equation is GH N : 9 : 2( V )Q : 2S, GH GH and the symmetric tensor S is the deformation tensor

(60)

1 S : ( V )Q : ( V ; ( V )*), 2

(61)

with ( V )* being the transposed diad V. The homogeneous phase diffusion equations are

D ;S D;U D: ( (x , , V ) D H x D x D x t H H H

(62)

and

Q: Q ;S . (63) Q Q x t x H H Here and are scalar ﬁelds and nonlinear diffusion coefﬁcients for D these ﬁelds. The averaging procedures for transport equation convective terms were established earlier. The averaged nonlinear diffusion term yields

1 · (D C) : · (mD C ) ; · D D 1 ; · (D c ) ; D

1U

c ds

D C · ds. (64) 1U The other version of the diffusive terms with the full value of concentration on the interface surface is

1 · (D C) : · (D (mC )) ; · D D 1 ; · (D c ) ; D

Cds

1U

D C · ds.

(65)

1U General forms of the nonlinear transport equations can be derived for impermeable and permeable interface surfaces. The averaged momentum diffusion term is

1 (2S ) : · (2S ) : · (2S ) ; 2S · ds D D x D 1U H 2 : · 2(m S ; m S ) ; S · ds. (66) D 1U

volume averaging theory

21

The general nonlinear averaged momentum equation for a porous medium is

V 1 ; mV · V 9 V m D t : 9 (mp ) 9

1

1U

1U

1 V · ds ; v v ; D

V V · ds

1U

pds ; · 2(m S ; m S ) D

2

S · ds ; m g. (67) D 1U The steady-state momentum transport equations for systems with impermeable interfaces follows from Eq. (67), ;

(mV · V ; v v ) D D : 9 (mp ) 9

1

1U

pds ; · 2(m S ; m S ) D

S · ds ; m g. (68) D 1U The averaged nonlinear mass transport equation in porous medium follows ;

m

C C D ; mU C 9 D G D t

1U

1 U · ds ; c u ; G D G D

: · (D (mC )) ; · D 1 ; · (D c ) ; D

1

1U

C U · ds D G

Cds

1U

(69) D C · ds ; mS . AD 1U A few simpler transport equations that can be readily used while maintaining fundamental relationships in heterogeneous medium transport are given by Travkin and Catton [21].

C. Porous Medium Turbulent VAT Equations Turbulent transport processes in highly structured or porous media are of great importance because of the large variety of heat- and mass-exchange equipment used in modern technology. These include heterogeneous media for heat exchangers and grain layers, packed columns, and reactors. In all cases there occurs a jet or stalled ﬂow of ﬂuids in channels or around the

22

v. s. travkin and i. catton

obstacles. There are, however, few theoretical developments for ﬂow and heat exchange in channels of complex conﬁguration or when ﬂowing around nonhomogeneous bodies with randomly varied parameters. The advanced forms of laminar transport equations in porous media were developed in a paper by Crapiste et al. [41]. For turbulent transport in heterogeneous media, there are few modeling approaches and their theoretical basis and ﬁnal modeling equations differ. The lack of a sound theoretical basis affects the development of mathematical models for turbulent transport in the complex geometrical environments found in nuclear reactors subchannels where rod-bundle geometries are considered to be formed by subchannels. Processes in each subchannel are calculated separately (see Teyssedou et al. [51]). The equations used in this work has often been obtained from two-phase transport modeling equations [52] with heterogeneity of spacial phase distributions neglected in the bulk. Three-dimensional two-ﬂuid ﬂow equations were obtained by Ishii [52] using a statistical averaging method. In his development, he essentially neglected nonlinear phenomena and took the ﬂux forms of the diffusive terms to avoid averaging of the second power differential operators. Ishii and Mishima [53] averaged a two-ﬂuid momentum equation of the form v I I I ; · ( v v ) : 9 p ; · ( ; R ) I I I I I I I I t ; g ; v ; M 9 · , (70) I I IG I GI I G where is the local void fraction, is the mean interfacial shear stress, R I I G is the turbulent stress for the kth phase, is the averaged viscous stress for the kth phase, is the mass generation, and M is the generalized I GI interfacial drag. Using the area average in the second time averaging procedure, Ishii and Mishima [53] introduced a distribution of parameters to take into consideration the nonlinearity of convective term averaging. This approach cannot strictly take into account the stochastic character of various kinds of spatial phase distributions. The equations used by Lahey and Lopez de Bertodano [54] and Lopez de Bertodano et al. [55] are very similar, with the momentum equation being D u I HI : 9 p ; · [ u 9 (u u )] I I Dt I I I I HI I HI HI 9 g ; M 9 M 9 · ; ( p 9 p ) . (71) I I GI UI G I IG I I Here the index i denotes interfacial phenomena and M is the volumetric UI wall force on phase k. Additional terms in Eq. (70) and (71) are usually based on separate micro modeling efforts and experimental data.

volume averaging theory

23

One of the more detailed derivations of the two-phase ﬂow governing equations by Lahey and Drew [56] is based on a volume averaging methodology. Among the problems was that the authors developed their own volume averaging technique without consideration of theoretical advancements developed by Whitaker and colleagues [10, 42] and Gray et al. [8] for laminar and half-linear transport equations. The most important weaknesses are the lack of nonlinear terms (apart from the convection terms) that naturally arise and the nonexistence of interphase ﬂuctuations. Zhang and Prosperetti [57] derived averaged equations for the motion of equal-sized rigid spheres suspended in a potential ﬂow using an equation for the probability distribution. They used the small particle dilute limit approximation to ‘‘close’’ the momentum equations. After approximate resolution of the continuous phase ﬂuctuation tensor M , the vector A A (x, t), and the ﬂuctuating particle volume ﬂux tensor, M , they recog" " nized that (p. 199) ‘‘Closure of the system requires an expression for the ﬂuctuating particle volume ﬂux tensor M . . . . This missing information " cannot be supplied internally by the theory without a speciﬁcation of the initial conditions imposed on the particle probability distribution.’’ They also considered the case of ‘‘ﬁnite volume fractions for the linear problem’’ where the problem equations were formulated for inviscid and unconvectional media. The development by Zhang and Prosperetti [57] is a good example of the correct application of ensemble averaging. The equations they derive compare exactly with those derived from rigorous volume averaging theory (VAT) [24]. Transport phenomena in tube bundles of nuclear reactors and heat exchangers can be modeled by treating them as porous media [58]. The two-dimensional momentum equations for a constant porosity distribution usually have the form [59] U V ; :0 y x

(72)

U UV 1 P ; :9 ; U 9 A V LU CDD V x y x

(73)

V 1 P UV ; :9 ; V 9 A V LV, n 0, CDD W x y y

(74)

where the physical quantities are written as averaged values and the solid phase effects are included in two coefﬁcients of bulk resistance, A and A , V W and an effective eddy viscosity, , that is not equal to the turbulent eddy CDD viscosity. These kinds of equations were not designed to deal with non-

24

v. s. travkin and i. catton

linearities induced by the physics of the problem and the medium variable porosity or to take into account local inhomogeneities. Some of the more interesting applications of turbulent transport in heterogeneous media are to agrometeorology, urban planning, and air pollution. The ﬁrst signiﬁcant papers on momentum and pollutant diffusion in urban environment treated as a two-phase medium were those by Popov [60, 61]. In these investigations, an urban porosity function was deﬁned based on statistical averaging of a characteristic function (x, y, z) for the surface roughness that is equal to zero inside of buildings and other structures and equal to unity in an outdoor space. The turbulent diffusion equation for an urban roughness porous medium after ensemble averaging is m(x )C G L ; (m(x )V C ) G G L x t G C L , n : 1, 2, 3, 4 . . . , (v! )(c! ) 9 v c ; D :9 G L G L L x x x x G G G G (75)

where means porous volume ensemble averaging, and m(x ) is porosity. G Closure of the two ‘‘morphological’’ terms, the ﬁrst and the second terms on the right-hand side, was obtained using a Boussinesque analogy, C L . (v! )(c! ) ; v c : 9K G L G L GH x x x G G H

(76)

A descriptive analysis of the deviation variables (v! ), (c! ) and the effective G L diffusion coefﬁcient K was not given. In many studies of meteorology and GH agronomy, the only modeling of the increase in the volume drag resistance is by addition of a nonlinear term as done by Yamada [62], U 1 P :f V 9 ; (9uw) 9 (1 9 m )c S(z)U U I Q B t x z

(77)

V 1 P : 9f U 9 ; (9vw) 9 (1 9 m )c S(z)V V , I Q B t y z where (1 9 m ) is the fraction of the earth surface occupied by forest, m is Q Q the area porosity due to a tree volume, and f is a Coriolis parameter. I The averaging technique used by Raupach and Shaw [63] to obtain a turbulent transport equation for a two-phase medium of agro- and forest

volume averaging theory

25

cultures is a plain surface 2D averaging procedure where the averaged function is deﬁned by

1 f d, (78) f : ND ND ND with being the area within the volume occupied by air. Raupach ND N et al. [64] and Coppin et al. [65] assumed that the dispersive covariances were unimportant, u! "u! " , (79) G H ND where u! " is a ﬂuctuation value within the canopy and u! " " u . The G G G contribution of these covariances was found by Raupach et al. [64] to be small in the region just above the canopy from experiments with a regular rough morphology. This ﬁnding has been explained by Scherban et al. [15], Primak et al. [14], and Travkin and Catton [16, 20] for regular porous (roughness) morphology. Covariances are, however, the result of irregular or random two-phase media. When the surface averaging used by Raupach and Shaw [64] is used instead of volume averaging, especially in the case of nonisotropic media, the neglect of one of the dimensions in the averaging process results in an incorrect value. This result should be called a 2D averaging procedure, particularly when 3D averaging procedures are replaced by 2D for nonisotropic urban rough layer (URL) when developing averaged transport equations. Raupach et al. [64—66] later introduced a true volume averaging procedure within an air volume that yielded the averaged equation for D momentum conservation 1 U G ; U (U ) : 9 P ; 9u u ; U G HD H x G G x x t H D G H ; U · ds x G D 1U H 1 9 u! "u! " 9 P ds, i, j : 1 9 3, (80) G H D x H D D 1U where S is interfacial area. Development of this equation is based on U intrinsic averaged values of or U , whereas averages of vector ﬁeld D G variables over the entire REV are more correct (Kheifets and Neimark [11]). Raupach et al. [64] next simpliﬁed all the closure requirements by developing a bulk overall drag coefﬁcient. The second, third, and ﬁfth terms on the right-hand side of Eq. (80) are represented by a common drag resistance term. For a stationary fully developed boundary layer, they write

v. s. travkin and i. catton

26

U 1 uw u! "w! " 9 : 9 C S U , D D z 2 BC NC z

(81)

where C is an element drag coefﬁcient and S is an element area BC NC density — frontal area per unit volume. A wide range of ﬂow regimes is reported in papers by Fand et al. [67] and Dybbs and Edwards [68]. The latter work revealed that there were four regimes for regular spherical packing, and that only when the Reynolds number based on pore diameter, Re , exceeded 350 could the ﬂow regime AF be considered to be turbulent ﬂow. The Fand et al. [67] investigation of a randomnly packed porous medium made up of single size spheres showed that the fully developed turbulent regime occurs when Re 120, where Re T N is particle Reynolds number. Volume averaging procedures were used by Masuoka and Takatsu [69] to derive their volume-averaged turbulent transport equations. As in numerous other studies of multiphase transport, the major difﬁculties of averaging the terms on the right-hand side were overcome by using assumed closure models for the stress components. As a result, the averaged turbulent momentum equation, for example, has conventional additional resistance terms such as the averaged momentum equation developed by Vafai and Tien [70] for laminar regime transport in porous medium. A major assumption is the linearity of the ﬂuctuation terms obtained, for example, by neglect of additional terms in the momentum equation. A meaningful experimental study by Howle et al. [71] conﬁrmed the importance of the role of randomness in the enhancement of transport processes. The results show the very distinct patterns of ﬂow and heat transfer for two cases of regular and nonuniform 2D structured nonorthogonal porous media. Their experimental results clearly demonstrate the inﬂuence of nonuniformity of the porous structure on the enhancement of heat transfer. D. Development of Turbulent Transport Models in Highly Porous Media Fluid ﬂow in a porous layer or medium can be characterized by several modes. Let us single out from among them the three modes found in a highly porous media. The ﬁrst is ﬂow around isolated ostacle elements, or inside an isolated pore. The second is interaction of traces or a hyperturbulent mode. The third is ﬂuid ﬂow between obstacles or inside a blocked interconnected swarm of channels (ﬁltration mode). The models developed by Scherban et al. [15], Primak et al. [14], and Travkin and Catton [16—21] are primarily for nonlinear laminar ﬁltration and hyperturbulent modes in

volume averaging theory

27

nonlinear transport. Speciﬁc features of ﬂows in the channels of ﬁltered media include the following: 1. Increased drag due to microroughness on the channel boundary surfaces 2. Gravity effects 3. Free convection effects 4. The effects of secondary ﬂows of the second kind and curved streamlines 5. Large-scale vortex effects 6. The anisotropic nature of turbulent transfer and resulting anisotropy of turbulent viscosity It is well known that in spacial boundary ﬂows, an important role is played by the gradients of normal Reynolds stresses and that this is the case for ﬂows in porous medium channels as well. As a rule, ﬂow symmetry is not observed in these channels. Therefore, in channel turbulence models, the shear components of the Reynolds stress tensors have a decisive effect on the ﬂow characteristics. At present, however, turbulence models that are less than second-order can not be successfully employed for simulating such ﬂows (Rodi [72], Lumley [73], and Shvab and Bezprozvannykh [74]). Derivation of the equations of turbulent ﬂow and diffusion in a highly porous medium during the ﬁltration mode is based on the theory of averaging of the turbulent transfer equation in the liquid phase and the transfer equations in the solid phase of a heterogeneous medium (Primak et al. [14] and Scherban et al. [15]) over a speciﬁed REV. The initial turbulent transport equation set for the ﬁrst level of the hierarchy, microelement, or pore, was taken to be of the form (see, for example, Rodi [72] and Patel et al. [75])

U 1 p! U U G:9 G 9 u u ; S G;U ; (82) G H H x 3G x x x t D G H H H D: D 9 u # ; S D;U D (83) G D D H x D x t x H H H U G : 0. (84) x G Here and its ﬂuctuation represent any scalar ﬁeld that might be D transported into either of the porous medium phases, and the last terms on the right-hand side of (82) and (83) are source terms.

28

v. s. travkin and i. catton

Next we introduce free stream turbulence into the hierarchy Let us represent the turbulent values as U : U ; u : U ; U ; u ; u I P I P U : U ; u! ,

(85)

where the index k stands for the turbulent components independent of inhomogeneities of dimensions and properties of the multitude of porous medium channels (pores), and r stands for contributions due to the porous medium inhomogeneity. Being independent of the dimensions and properties of the inhomogeneities of the porous medium conﬁgurations, sections, and boundary surfaces does not mean that the distribution of values of U I and u are altogether independent of the distance to the wall, pressure I distribution, etc. Thus, the values U or u stand for the values generally I I accepted in the turbulence theory, that is, when a plane surface is referred to, these values are those of a classical turbulent boundary layer. When a round-section channel is involved, and even if the cross-section of this channel is not round, but without disturbing nonhomogeneities in the section, then the characteristics of this regular sections (and ﬂow) may be considered to be those that could be marked with index k. Hence, if a channel in a porous medium can be approximately by superposition of smooth regular (of regular shape) channels, it is possible to give such a ﬂow its characteristics and designated them with the index k, which stands for the basic (canonical) values of the turbulent quantities. Triple decomposition techniques have been used in papers by Brereton and Kodal [76] and Bisset et al. [77], among others. The latter utilized triple decomposition, conditional averaging, and double averaging to analyze the structure of large-scale organized motion over the rough plate. It should be noted that there are problems where U and u can be found I I from known theoretical or experimental expressions (correlations) where the deﬁnitions of U and u are equivalent to the solution of an independent I I problem (for example, turbulent ﬂow in a curved channel). The same thing can be said about ﬂow around a separate obstacle located on a plain surface. In this case one can write U : U : U ; U , u : u . I I P

(86)

The term u : u! appears if the ﬂow is through a nonuniform array of P obstacles. If all the obstacles are the same and ordered, then u! can be taken equal to 0. Naturally, the term u in this particular case does not equal the I ﬂuctuation vector u over a smooth, plain surface. IQ

volume averaging theory

29

The following hypothesis about the additive components is developed to correct the foregoing deﬁciencies: U : U ; u! : U ; U ; u , u : u P I I P D U : U ; U 9 U ; U , u! : u P I P I P (87) (u ) : 0, u ! : u : 0. P D I D It should be noted that solutions to the equations for the turbulent characteristics may be inﬂuenced by external parameters of the problem, namely, by the coefﬁcients and boundary conditions, which themselves can carry information about porous medium morphological features. The adoption of a hypothesis about the additive components of functions representing turbulent ﬁltration facilitates the problem of averaging the equations for the Reynolds stresses and covariations of ﬂuctuations (ﬂows) in pores over the REV. After averaging the basic initial set of turbulent transport equations over the REV and using the averaging formalism developed in the works by Primak et al. [14], Shcherban et al. [15], and Primak and Travkin [78], one obtains equations for mass conservation,

1 U ; U · ds : 0, (88) G D G x 1U G for turbulent ﬁltration (with molecular viscosity terms neglected for simplicity), U 1 G; (mU U ):9 (mp ! ) ; 9u u ; 9u! u! H G D x H G H G D t x x x H D G H H 1 1 p! ds 9 9 U U · ds 1U 1U H G D 1 9 u u · ds ; mS , i, j : 1—3, (89) H G 3G 1U and for scalar diffusion (with molecular diffusivity terms neglected),

m

m

D; (mU ) G D t x G 1 : 9u # ; 9u! #! 9 U · ds G D D x G D D G D x 1U G G 1 9 u # · ds ; mS , i : 1—3. G D D 1U

(90)

30

v. s. travkin and i. catton

Many details and possible variants of the preceding equations with tensorial terms are found in Primak et al. [14], Scherban et al. [15], and Travkin and Catton [16, 21]. Using an approximation to K-theory in an elementary channel (pore), the equation for turbulent diffusion of nth species takes the following more complex form after being averaged: m

C L ; V C : 9 v! c! D L G L D t

1 ; · (K (mC )) ; · K A L A 1 ; · (k c! ) ; A L D ;

C L

1U

1 U · ds 9 G

1U

1U

C ds L

K C · ds A L

C U · ds ; mS , L G L 1U n : 1, 2, 3, 4 . . . . (91)

In the more general case, the momentum ﬂux integrals on the right-hand sides of Eq. (89) through (91) do not equal zero, since there could be penetration through the phase transition boundary changing the boundary conditions in the microelement to allow for heat and mass exchange through the interface surface as the values of velocity, concentrations, and temperature at S do not equal zero (see also Crapiste et al. [41]). The ﬁrst term U on the right-hand side of Eq. (91) is the divergence of the REV averaged product of velocity ﬂuctuations and admixture concentration caused by random morphological properties of the medium being penetrated and is responsible for morphoconvectional dispersion of admixture in this particular porous medium. The third term on the right-hand side of Eq. (91) can be associated with the notion of morphodiffusive dispersion of a substance or heat in a randomly nonhomogeneous medium. The term with S may also L reﬂect, speciﬁcally, the impact of microroughness from the previous level of the simulation hierarchy. The importance of accounting for this roughness has been demonstrated by many studies. The remaining step is to account for the microroughness characteristics of the previous level. One-dimensional mathematical statements will be used in what follows for simplicity. Admission of speciﬁc types of medium irregularity or randomness requires that complicated additional expressions be included in the generalized governing equations. Treatment of these additional terms becomes a crucial step once the governing averaged equations are written. An attempt to implement some basic departures from a porous medium with

volume averaging theory

31

strictly regular morphology descriptions into a method for evaluation of some of the less tractable, additional terms is explained next. The 1D momentum equation with terms representing a detailed description of the medium morphology is depicted as

U m(K ; ) ; K x x x

u! K K x

;

(9u! u! ) D x

D 1 U U 9 (K ; ) · ds : m U K x x 1U G H 1 1 ; p! ds ; (mp! ) x 1U D D U 1 1 p! ds ; ; u S (x) ; (mp ! ), (92) : mU *PI U x x 1U D D H where K is the turbulent eddy viscosity, and u is the square friction *PI K velocity at the upper boundary of surface roughness layer h averaged over P interface surface S . U General statements for energy transport in a porous medium require two-temperature treatments. Travkin et al. [19, 26] showed that the proper form for the turbulent heat transfer equation in the ﬂuid phase using one-equation K-theory closure with primarily 1D convective heat transfer is

T T D: c mU m(K ; k ) D ; ND D 2 D x x x x

T D K 2 x

D ;k ) (K 2 D ;c (m 9T u! ) ; ND D x D D x

1U

T ds D

T (K ; k ) D · ds, (93) 2 D x 1U G whereas in the neighboring solid phase, the corresponding equation is ;

1

T

Q Q ; (1 9 m) K

Q2 Q x x x

K

Q2 Q ; x

T Q K Q2 x

Q

T Q · ds . K (94) Q2 x 1U 1U G The generalized longitudinal 1D mass transport equation in the ﬂuid phase, including description of potential morphoﬂuctuation inﬂuence, for a 1 T ds ; Q

v. s. travkin and i. catton

32

medium morphology with only 1D ﬂuctuations is written C D ; m(K ; D ) ! D x x x

C D K ! x

D (K ; D ) ! D ; (m 9c! u! ) ; D D x x

1U

c! ds D

C C D, (K ; D ) D · ds ; mS : mU ! D x ! x 1U G whereas the corresponding nonlinear equation for the solid phase is ;

1

C

Q Q ; (1 9 m) D

Q Q x x x D

Q Q ; x

1U

(95)

c Q D Q x

Q 1 C ds ; Q

C Q · ds . D Q x 1U G

(96)

E. Closure Theories and Approaches for Transport in Porous Media Closure theories for transport equations in heterogeneous media have been the primary measure of advancement and for measuring success in research on transport in porous media. It is believed that the only way to achieve substantial gains is to maintain the connection between porous medium morphology and the rigorous formulation of mathematical equations for transport. There are only two well-known types of porous media morphologies for which researchers have had major successes. But even for these morphologies, namely straight parallel pores and equal-size spherical inclusions, not enough evidence is available to state that the closure problems for them are ‘‘closed.’’ One of the few existing studies of closure for VAT type equations is by Hsu and Cheng [79, 80]. They used a one-temperature averaged equation [Equation (40a) in Hsu and Cheng [80]) without the morphodiffusive term

· [(k 9 k )T (9 m)] : · [(k 9 k )T ( m)]. D Q Q D The reasoning often applied to the morphoconvective term closure problem in averaged scalar and momentum transport equations is that the terms needing closure may be negligible. The basis for this reasoning is (see Kheifets and Neimark [11]) d c $ Cd , and j : D C, so c j $ D C AF , AF D D l

volume averaging theory

33

where l is the characteristic length associated with averaging volume (see, for example, the work of Lehner [81] and others) and d the mean diameter AF of pores in a REV. It is not obvious that the length scale, d , taken for the AF approximation of c follows from use of l as a scale for the second derivative. Furthermore, assuming that the variable to be averaged over the REV changes very slowly over the REV does not mean that it changes very slowly in the neighborhood of the primary REV. Various closure attempts for heterogeneous medium transport equations resulted in various ﬁnal equations. One needs to know what these equations are all about. Treatment of the one-dimensional heat conduction equation with a stochastic function for the thermal diffusivity in a paper by Fox and Barakat [82] yielded a spatially fourth-order partial differential equation to be solved. Gelhar et al. [83], after having eliminated the second-order terms in the species conservation equation for a stochastic media, were able to develop an interesting procedure for deriving a mean concentration transport equation. The equation form includes an inﬁnite series of derivatives on the right-hand side of the equation. Analysis of this equation allows the derivation of the ﬁnal form of the mass transport equation, C * C * C * C * C * ;U : (A ; a )U 9B 9 BU , * x x tx x t where the most important term is the second term on the right-hand side. In the derivation of this equation, the stochastic character of the existing assigned ﬁelds of velocity, concentration, and dispersion coefﬁcients were assumed. A simple form of the advective diffusion equation with constant diffusion coefﬁcients was developed without sorption effects by Tang et al. [84]: mC ; mV · C : D · (m C ). G t They transformed the equation with the help of ensemble averaging into a stochastic transport equation, mC * C * ; mu * C * : D · (m C *) ; m , H HI x x t H I where the tensor of the ensemble dispersion coefﬁcient is a correlation function denoted by 1 u u * H I : x · u *, HI 2 u * u * with u * being the ensemble averaged velocity. The additional term,

34

v. s. travkin and i. catton

reﬂecting the inﬂuence of the stochastic or inhomogeneous nature of the spatial velocity and concentration ﬂuctuations in the ensemble averaged stochastic equation developed by Tang et al. [84], has the dispersivity coefﬁcient fully dependent on the velocity ﬂuctuations. As can be seen by this equation, the effect of concentration ﬂuctuations was eliminated. Torquato and coauthors (see, for example, Torquato et al. [85], Miller and Torquato [86], Kim and Torquato [87]) have been developing means to characterize the various mathematical dependencies of a composite medium microstructure in a statistically homogeneous medium. Some of the quantities considered by Torquato are useful in obtaining resolution to certain closure problems for VAT developed mathematical models of globular morphologies. In particular, the different near-neighbor distance distribution density functions deserve special mention (Lu and Torquato [88], Torquato et al. [85]). Carbonell and Whitaker [89] combined the methods of volume averaging and the morphology approach to specify the dispersion tensor for the problem of convective diffusion for cases where there is no reaction or adsorption on the solid phase surface, 9D

C : 0, n

x + S , U

and considered a constant diffusion coefﬁcient and constant porosity m, which greatly simpliﬁes the closure problems. They expressed the spatial deviation function as c : f ( r ) · C , where f is a vector function of position in the ﬂuid phase. Averaged equations of convective diffusion are the same as the convective heat transfer equation given by Levec and Carbonell [46] with the exclusion of ﬂux surface integral term. The closure technique used in their paper is analogous to a turbulence theory scheme, helping them to derive the closure equation for the spatial deviation function in the form of a partial differential equation, · f : n, x + S , V ; (V ; V ) f : D f , 9n U One should note that the spatial deviation functions deﬁned for a periodic medium are periodic themselves. Nozad et al. [40] suggested that the same closure scheme be used to represent the ﬂuctuation terms T and T for a one-temperature model by D Q using

T : f T ; , T : g T ; % D Q

volume averaging theory

35

for a transient heat conduction problem with constant coefﬁcients in a two-phase system (stationary). Partial differential equations for f , g , , and % are found. They obtained excellent predictions of the effective thermal conductivity for conductivity ratios k : k /k & 100. Q D Carbonell [90] attempted to obtain an averaged convective-diffusion equation for a straight tube morphological model and obtained an equation with three different concentration variables. This demonstrates that the averaging procedures, taken too literally, can result in incorrect expressions or conclusions. A common form of the averaged governing equations for closure of multiphase laminar transport in porous media was obtained by Crapiste et al. [41]. They developed a closure approach that led to a complex integrodifferential equation for the spatial deviations of a substance in the void or ﬂuid phase volume of the macro REV. This means that solving the boundary value problem for spatial concentration ﬂuctuations, for example, requires that one obtain a solution to second-order partial differential or coupled integro differential equations in a real complex geometric volume within the porous medium. For a heterogeneous porous medium, this means that the coupled integrodifferential equation sets for the averaged spatial deviation variables must be solved for at least two scales. For averaged variables the scales are the external scale or L domain, and for the spatial deviations it is the volume of the ﬂuid phase considered at the local (pore) scale. This presents a great challenge and has not yet been resolved by a real mathematical statement. To close the reaction-diffusion problem Crapiste et al. [41] made a series of assumptions: (1) the diffusion coefﬁcient D and the ﬁrst-order reaction rate coefﬁcient k are constant; (2) diffusion is linear in the solid part of the P porous medium, (3) the spatial concentration ﬂuctuation is linearly dependent on the gradient of the intrinsic averaged concentration and the averaged concentration itself, (4) the intrinsic averaged concentration and solid surface averaged concentration are equal, (5) the restriction kd P N1 D should be satisﬁed; and (6) spatial ﬂuctuations of the intrinsic concentration and the surface concentration ﬂuctuations are equal. The fourth and sixth assumptions are equivalent to an equality of surface and intrinsic concentrations, which means that the adsorption mechanisms are taken to be volumetric phenomena. In our previous efforts we have obtained some results for both morphologies and demonstrated the strength of morphological closure procedures.

36

v. s. travkin and i. catton

A model of turbulent ﬂow and two-temperature heat transfer in a highly porous medium was evaluated numerically for a layer of regular packed particles (Travkin and Catton [16, 20]; Gratton et al. [26, 27]) with heat exchange from the side surfaces. Nonlinear two-temperature heat and momentum turbulent transport equations were developed on the basis of VAT, requiring the evaluation of transport coefﬁcient models. This approach required that the coefﬁcients in the equations, as well as the form of the equations themselves, be consistent to accurately model the processes and morphology of the porous medium. The integral terms in the equations were dropped or transformed in a rigorous fashion consistent with physical arguments regarding the porous medium structure, ﬂow and heat transfer regimes (Travkin and Catton [20]; Travkin et al. [17]). The form of the Darcy term as well as the quadratic term was shown to depend directly on the assumed version of the convective and diffusion terms. More importantly, both diffusion (Brinkman) and drag resistance terms in the ﬁnal forms of the ﬂow equations were proven to be directly connected. These relations follow naturally from the closure process. The resulting necessity for transport coefﬁcient models for forced, single phase ﬂuid convection led to their development for nonuniformly and randomly structured highly porous media. A regular morphology structure was used to determine the characteristic morphology functions, (porosity m, and speciﬁc surface S ) that were U used in the equations in the form of analytically calculable functions. A ﬁrst approximation for the coefﬁcients, for example, drag resistance or heat transfer, was obtained from experimentally determined coefﬁcient correlations. Existing models for variable morphology functions such as porosity and speciﬁc surface were used by Travkin and Catton [20] and Gratton et al. [27] to obtain comparisons with other work in a relatively high Reynolds number range. All the coefﬁcient models they used were strictly connected to assumed (or admitted) porous medium morphology models, meaning that the coefﬁcient values are determined in a manner consistent with the selected geometry. Comparison of modeling results was sometimes difﬁcult because other models utilized mathematical treatments or models that do not allow a complete description of the medium morphology; see Travkin and Catton [16]. Closures were developed for capillary and globular medium morphology models (Travkin and Catton [16, 17, 20]; Gratton et al. [26, 27]). It was shown that the approach taken to close the integral resistance terms in the momentum equation for a regular structure allows the second-order terms for the laminar and turbulent regimes to naturally occur. These terms were taken to be analogous to the Darcy or Forchheimer terms for different ﬂow velocities. Numerical evaluations of the models show distinct differences in

volume averaging theory

37

the overall drag coefﬁcent among the straight capillary and globular models for both the regular and simple cubic morphologies.

IV. Microscale Heat Transport Description Problems and VAT Approach Study of energy transport at different scales in a heterogeneous media or system emphasizes the importance of transport phenomena at subcrystalline and atomic scales. Among many works addressing subcrystalline transport phenomena (see Fushinobu et al. [91]; Caceres and Wio [92]; Tzou et al. [93]; Majumdar [94]; Peterson [95], etc.), the governing energy transport equations, whether they are of differential type or integrodifferential, are for homogeneous or homogenized matter. This idealization signiﬁcantly reduces the value of the physical description that results. VAT shows great promise as a tool for development of models for this type of phenomena because it becomes possible to include the inherent nonlinearity and heterogeneity found at the subcrystalline level and reﬂect the impact at the upper levels or scales. A heuristic approach suggested by Tzou [96] lumps all the atomic and subcrystalline scale phenomena ‘‘into the delayed response in time in the macroscopic formulation.’’ This approach was proposed by author to close the existing gap in knowledge and to help engineers develop applications. Unfortunately, the coupling between the characteristics of the subscale phenomena and delayed response time is lost. There is an ongoing search for the transport equations describing many-body systems that exhibit highly nonequilibrium behavior, including non-Markovian diffusion. The more exact the description of a physical phenomenon provided by a mathematical model, the more possibilities there are for innovative improvements in the function of a particular material or device. Our contribution to the effort is an extensive analysis of existing approaches to the development of theories for the subcrystalline and atomic scale levels. We have also made progress in the development of VAT-based tools applicable at the atomic and nanoscale level for description of transport of heat, mass, and charge in SiC and superconductive ceramics. At the subcrystalline scale, we will consider energy transport using a VAT description for effects of crystal defects and impurities on phonon—phonon scattering, which has a substantial impact on thermal conductivity. At the crystal scale, the importance of thermal resistance (different models) due to various mechanisms — lattice unharmonic resistance and crystal boundary defects — will be treated. Including these phenomena shows that they have a major impact on the transport characteristics in critical applications such as optical ceramics and superconductive ceramics.

v. s. travkin and i. catton

38

A. Traditional Descriptions of Microscale Heat Transport Kaganov et al. [97] ﬁrst developed a theory to describe energy exchange between electrons and the lattice of a solid for arbitrary temperatures using earlier advances in this ﬁeld by Ginzburg and Shabanskii [98] and by Akhiezer and Pomeranchuk [99]. In their work, they assumed that the electron gas was in an equilibrium state. After a brief summary of this early work, an analysis leading to a method for estimating the relaxation processes between the electron ﬂuid temperature T and the phonon temC perature T will be presented. J The heat balance equation for the electron temperature is T c (T ) C : 9U ; Q, C t

(97)

where Q is the heat source, c (T ) is the electronic speciﬁc heat, C

' k T , c (T ) : k n C C 2 and : (3n /8')(2' )/(2m*). C U is the heat exchange term, U:

2' m*cn (T 9 T ) (T 9 T ) Q C C J C J , 3 (T ) T J J

U:

' m*cn (T 9 T ) Q C C J , T T ; (T 9 T ) T , J " C J J 6 (T ) T J J

T T ; (T 9 T ) T J " C J J

(98) (99)

where m* is the effective electronic mass, c is the sound velocity, n is the Q C number of electrons per unit volume, (T ) is the time to traverse a mean J free path of electrons under the condition that the lattice temperature coincides with the electron temperature and is equal respectively to T . J When the lattice temperature is assumed to be much less than the temperature of the electrons (an assumption later found to be weak), then

2' m*cn Q C , (T T ; T T ); C " J C 15 (T ) C U: . ' m*c n Q C , (T T ; T T ). C " J C 6 (T ) C

(100)

volume averaging theory

39

Kaganov et al. [97] used an equation for elastic lattice vibration of the form

U U J : c U 9

((r 9 Vt). Q J r

(101)

This also allowed them to develop the heat exchange term (here U is the J displacement vector). In this equation, : M/d is the density of lattice, M is the mass of the lattice atom, V is the lattice volume, and U is the interaction constant of the electron with the lattice that appears in the expression for the time to travel the mean free path. It was nearly 20 years before needs in different physical ﬁelds (namely, intense short-timespan energy heating in laser applications) brought attention to this phenomenon and to use it for further technological advances. Anisimov et al. [100, 101] introduced a simpliﬁed two-ﬂuid temperature model for heat transport in solids, T C (T ) C : )T 9 (T 9 T ) ; f (r, t) C CN C J C C t

(102)

T J : (T 9 T ), C J t CN C J

' m*c n Q C. : (103) CN 6 (T ) C Further development of the idea of a two-ﬁeld two-temperature model for energy transport in metals by Qiu and Tien [102—104] used this model. They modiﬁed the energy exchange rate coefﬁcent (heat transfer) model in a way that uses the coefﬁcient of conductivity K in the following formula C instead of time between collisions (T ): C '(n c k ) C Q . (104) U:G: K C Tzou et al. [93] used the two-ﬂuid model with two equations for the electron—phonon transport in metals based on previous works by Anisimov et al. [101], Fujimoto et al. [105], Elsayed-Ali [106], and others. The equation for diffusion in an electron gas is a parabolic heat conduction equation with an exchange term T C : · (K T ) 9 G (T 9 T ), C C t C C CN C J

(105)

with phonon transport (phonon—electron interactions) for the metal lattice (just simpliﬁed equation) being described by T J : G (T 9 T ), C J t CN C J

(106)

40

v. s. travkin and i. catton

where K is the thermal conductivity of the electron gas. Using the C Wiedermann—Franz law for the electron—phonon interaction, Qiu and Tien [102, 103] show that the coupling factor G can be approximated by CN '(n c k ) C Q , (107) G : CN K C where c , the speed of sound in solid, is Q k (6'n )\T , (108) c : ? " Q 2'

T is the Debye temperature, is Planck’s constant, and n and n are the " C ? electronic and atomic volumetric number densities. Assuming constant thermal properties, the two equations can be combined, yielding a onetemperature equation T T 1 T 1 T J; C J : J; J, (109) x C xt t C t 2 2 2 where the thermal diffusivity of electron gas , equivalent thermal diffusivC ity , and thermal wave speed C are deﬁned by 2 2 K K K G C , C : C . : C, : (110) 2 C C C C 2 C ;C C J C J C Tzou [96] later proposed a uniﬁed two-ﬂuid model to derive the general hyperbolic equation with two relaxation times and , 2 O 1 T T ( T ) : ; O , (111)

T ; 2 t t t which he argues is the same equation derived from two-step models in metals. A more complex two-temperatures model was obtained by Gladkov [107] using parabolic equations T T T ;V :) 9 (T 9 T ) x x t

(112)

T T :) ; (T 9 T ), x t

(113)

and

It can be seen from his work that the coefﬁcients of heat transfer and are not equal. After combining the two equations into one, an equation

volume averaging theory

41

for a mobile (liquid) medium results:

T 1 T V T V T ; ; ; 1 ; t t x tx V T ) ) T T . 9 ;) :) (114) x x x There are other works (see, for example, Joseph and Preziosi [108]) treating the two-ﬂuid heat transport and obtaining the same kind of hyperbolic equation. 1. Equation of Phonon Radiative Transfer Majumdar [94] suggested an equation for phonon radiative transfer (EPRT). In three dimensions the equation is L I (T (x)) 9 I S ; (V · I ) : S S, NF S t (, T )

(115)

where I is the directional-spectral phonon intensity, V is the phonon S NF propagation speed, and I (T (x)) is the equilibrium intensity corresponding S to a blackbody intensity at temperatures below the Debye temperature. To make matters more complex, it should be noted that as stressed by Peterson [95], ‘‘However, fundamental differences exist between phonon and photon behavior in the regime where scattering and collisional processes are important . . . . Even in perfect crystals, the so-called unklapp processes that are responsible for ﬁnite thermal conductivity do not obey momentum conservation.’’ 2. Hyperbolic Heat Conduction Equations The work by Vernotte, Cattaneo, Morse, and Feshbach that led to the hyperbolic heat conduction equation was primarily heuristic in nature (without a ﬁrst principle physical basis). The ﬁnal form is often presented as a telegraph equation (see Joseph and Preziosi [108]), 1 T k T ; :

T, t (*) t

(116)

T T 1 ; : · (K T ), t t *

(117)

or

for nonconstant thermal conductivity K; * here is the heat capacity.

v. s. travkin and i. catton

42

Majumdar et al. [109] produced microphotographs of thermal images that show the grain structure, visible in the topographical image, and notes that ‘‘the grain boundaries appear hotter than within the grain. It is at present not clear why this occurs . . . . The hot electrons collide with the lattice and transfer energy by the emission of phonons.’’ The governing equations for a nonmagnetic medium they use are conservation of electrons, n ; · (nV ) : 0; C t

(118)

conservation of electron momentum,

V V e k C ; (V · )V : 9 E9

(nT ) 9 C ; (119) C C C t m* m*n K where the last term ‘‘is the collision and scattering term analogous to the Darcy term in porous media ﬂow’’; conservation of electron energy, W C ; · (W V ) : 9e(nV · E) 9 k · (nV T ) C C C C C t

(W 9 (3/2)k T ) C M ; C\M conservation of lattice optical phonon energy, ; · (k T ) 9 C C

T (W 9 (3/2)k T ) (T 9 T ) M: C M 9 C M ? ; M t M C\M M\? and conservation of acoustical energy, C

(120)

(121)

(T 9 T ) T M ? . ? : · (k T ) ; C (122) C ? ? M ? t M\? The last four equations have terms, the last term on the right-hand side, that qualitatively reﬂect the collision and scattering rates in each process. Here is the electron momentum relaxation time, is the electron optical K C\M relaxation time, is the optical acoustical relaxation time, and k is M\? Boltzmann’s constant. In those equations assumed a scalar effective mass for the electrons m*. The electric ﬁeld is determined using the Gauss law equation written in terms of electric potential (E : 9 ),

· ( ) : 9e(N 9 N 9 n ; p) : 9eNC L N NC : (N 9 N 9 n ; p), L N

(123)

volume averaging theory

43

where is the dielectric constant of Si, N is the n-doping concentration, N L N is the p-doping concentration, and p is the hole number density. B. VAT-Based Two-Temperature Conservation Equations Conservation equations derived using VAT enable one to capture all of the physics associated with transport of heat at the micro scale with more rigor than any other method. VAT allows one to avoid the ad hoc assumptions that are often required to close an equation set. The resulting equations will have sufﬁcient generality for one to begin to optimize material design from the nanoscale upward. The theoretical development is brieﬂy outlined in what follows. The nonlinear paraboic VAT-based heat conduction equation in one of the phases of the superstructure (where superstructure is to be determined as the micro- or nanoscale material’s organized morphology along with its local characteristics) is s (c ) N

T

: · [s K T ] ; · [s K T ] t ; ·

K

1

1 T ds ;

T · ds ; s S . K 2 x 1 G (124)

For constant thermal conductivity, the averaged equation for heat transfer in the ﬁrst phase can be written

T 1 : k (s T ) ; k · s (c ) N t k ;

1

T ds

T · ds ; s S . (125) 2 1 These VAT equations (124) and (125), written for the two phases, will be seen to yield the same pair of parabolic equations derived by researchers such as Gladkov [107], but with quite different arguments. Closure to Eq. (125) is needed for the second and third terms on the right-hand side. The steps to closure are 1

T 1 T k · ds : 9 k ds · n x n 1 1 G 1 : q · ds : S ( T 9 T ), 1

(126)

v. s. travkin and i. catton

44

with the heat transfer coefﬁcient, (S ), deﬁned in phase 2. This closure procedure is appropriate for description of ﬂuid—solid medium heat exchange and might be considered as the analog to solid—solid heat exchange found in many works. A more precise integration of the heat ﬂux over the interface surface, S , yields exact closure for that term in governing equations for both neighboring phases. Industry needs to lead one to attempt to estimate, or simulate by numerical calculation or other methods, the effective transport properties of heterogenous material. Among the many diverse methods used to do this, VAT presents itself as an effective tool for evaluating and bringing together different methods and is useful in providing a basis for comparative validation of techniques. To demonstrate the value of a VAT-based process, the effective thermal conductivity will be determined within the VAT framework. The averagd energy equation in phase 1 of a medium is

1 k (s T ) ; k ·

T ds : · [9q ]. 1 The right-hand-side (‘‘diffusive’’-like) ﬂux is different from that conventionally found, q

k : [9k

T ] : 9k (s T ) 9 CDD

where

k k : k (s T ) ; CDD

1

T ds ,

(127)

T ds ( T )\. (128) 1 After these transformations, the heat transfer equation for phase 1 becomes

T : · [k s (c )

T ] ; S ( T 9 T ) ; s S . CDD 2 N t (129) This is the same type of heat transport equation routinely used in two-ﬂuid models. The equation for heat transport in the second phase (if any) would be the same, and one can easily obtain the hyperbolic type two-ﬂuid temperature model. A similar VAT-based equation can be obtained for the heat transfer in phase 1 when the heat conductivity coefﬁcient is a function of the temperature or other scalar ﬁeld (nonlinear) (Eq. (124)), but the effective conductivity will have an additional term reﬂecting the mean surface temperature over

volume averaging theory

45

the interface surface inside of the REV,

K

K : K (s T ) ; s K T ; CDD

1

T ds ( T )\.

Equation (124) simpliﬁes to

(130)

T : · [K

T ] ; S ( T 9 T ) ; s S . s (c ) CDD 2 N t (131) The third term on the right-hand side of (124) plays a different role when the interface between two phases is only a mathematical surface without thickness neglecting the transport within the surface means there is no need to consider this medium separately. When this is the case, this term can be equal for the both phases, simplifying the closure problem. The problem becomes signiﬁcantly more complicated when transport within the interface must be accounted for. C. Subcrystalline Single Crystal Domain Wave Heat Transport Equations Some features of energy transport, including electrodynamics, that are above the scale of close capture quantum phenomena are considered next. Limiting the scope of the problem allows us to concentrate on the description of heat transport phenomena in the medium above the quantum scale where there are at least the three substantially different physical and spatial scales to consider. Within this scope, the heat transport equation in a single grain (crystalline) can be written in the form T 1 1 T E ; E : · (K T ) ; S . (132) E t * * 2E t Comparing this equation with the equation developed by Tzou [96] with two relaxation times, and , 2 O T T ; : T ; ( T ) ; S , (133) O t 2 t 2E t

and the parabolic equation obtained by Gladkov [107] for the model with two temperatures for constant coefﬁcients,

T 1 T V T V T ; ; ; 1 ; t t tx x V T ) ) T T 9 ;) , :) x x x

(134)

46

v. s. travkin and i. catton

one can see that all belong to the family of VAT two-temperature conduction problems with nonconstant effective coefﬁcients for the charged carriers, T A : a · [K T ] ; b ( T 9 T ) ; S , A A A A J A 2A t

(135)

and for phonon temperature transport, T J : a · [K T ] 9 b (T 9 T ) ; S . J J J J J A 2J t

(136)

This pair of equations is the wave transport equations shown in previous sections. Our current interest, however, is not to justify past assumptions made to develop appropriate scale level energy transport equations, but to develop mathematical models for heat transport and electrodynamics in multiscale microelectronics superstructures. D. Nonlocal Electrodynamics and Heat Transport in Superstructures Many microscale heterogeneous heat transport equations and some of the solutions provided elsewhere (see, for example, [110, 111, 112, 113, 109]) required substantial analysis, and many need improvement. Goodson [113], for example, directly addresses the need to model nonhomogeneous medium (diamond CVD layer) thermal transport by accounting for the presence of grains. The Peierls—Boltzmann equation for phonon transport was used along with information on grain structure. In the present work, the goal is to give some insight to situations (and those are substantial in number) where the medium cannot be considered as homogeneous even at the microscale level. For these circumstances, the governing ﬁeld equations should be based on conservation equations for a heterogeneous medium, for example, the VAT governing equations. The VAT governing equations for heterostructures will be found starting from a set of governing equations for a solid-state electron plasma ﬂuid. Phase averaging of the electron conservation equation (118) yields

n t

; · (nV ) : 0 (137) C K K where means averaging over the major phase of the material. The VAT K ﬁnal form for this equation is n 1 K ; · nV ; C K t

1KQ

nV · ds : 0, C

(138)

volume averaging theory

47

or 1 n K ; · [s n V ; m n V ] ; K C C K t

(139) nV · ds : 0, K C 1 where S is the ‘‘interface’’ (real or imaginary) of phases and scatterers. KQ It will be assumed that only immobile scatterers produce phase separation. This is not an essential restriction and is only taken to simplify the appearance of the equations and streamline the development. We recognize that defects and other scattering objects where processes are also occuring, such as nonmajor phases, occur, but we are not interested in them at this time because their volumetric fractions are very small and their importance is decreased by scattering of the ﬁelds in a major phase. The electron ﬂuid momentum transport equation can be written in two forms, and the form inﬂuences the ﬁnal appearance of the VAT equations. The ﬁrst is

V C t

K

e k ; (V · )V : 9 E 9 C C K K m* m*

1

(nT ) C n

V 9 C . K ^ K^ (140)

Using the transformation

1

(nT ) C n

K

1 : T ; T

n C Cn

: T ; T (ln n) C C K K : T ; T (Z ) , Z : ln n, C C L K L

(141)

it can be written as

V C t

e k V ; (V · )V : 9 E 9 T ; T (Z ) 9 C , C C K K m* C C L K m* ^ K^ K (142)

+^ + deﬁne the problem uncertainty in the treatment of where the brackets ^ this relaxation term. Strictly speaking, this term should not be in this form and may not exist. The same equation written in conservative form is

nV C t

e k nV C , ; · (nV V ) : 9 nE 9 (nT ) 9 C C K K m* C K m* ^ K^ K (143)

v. s. travkin and i. catton

48 Using

(V · )V : · (nV V ) 9 V ( · (nV )) C C K C C K C C K 1 (nV V ) · ds : · nV V ; C C K C C K 1KQ 9 s V · (nV ) ; V ( · (nV )) () , K K C C K C C Eq. (142) can be written in the VAT form as

V 1 C K ; · nV V ; C C K t

(144)

(nV V ) · ds C C K 1KQ () 9 s V · (nV ) ; V ( · (nV ))

K K C C K C C e k 1 :9 E 9

T ; T ds K m* C K C K m* 1KQ k 1 Z ds ; T ( (Z )) () , 9 s T Z ; L K K C L K C L K m* K 1KQ (145)

where the last term on the right-hand side of (142), the scattering and collision reﬂection term, has been replaced by a number of terms, each reﬂecting interface-speciﬁc phenomena, including scattering and collision. Some manipulation of the convection terms of the conservative form of the momentum equation has been done to combine the forms of the equations of mass and momentum. The second conservative form of the momentum equation is derived in the form V C ; (nV · )V ; n V ; · nV V n C K C t C K K t C C K 9 V

1 C

1 nV · ds ; C

(nV V ) · ds C C K

1KQ 1KQ e k 1 :9 nE 9

[s n T ; s n T ] ; K m* K C K C K m*

1KQ

nT ds , C K (146)

where a number of the integral terms are scattering and collision terms. There are other possible forms of the left-hand side of the momentum equation VAT equations that will not be pursued at this time.

volume averaging theory

49

The homogeneous volume averaged electron gas energy equation for a heterogeneous polycrystal becomes

W C t

K

; · (W V ) : 9enV · E 9 k · (nV T ) C C K C K C C K

(W 9 (3/2)k T ) C M , ; · (k T ) 9 C C K ^ ^ C\M

(147)

or W 1 C K ; · W V ; C C K t

1KQ

(W V ) · ds C C K

k : 9enV · E 9 k · (nV T ) 9 C K C C K

(nV T ) · ds C C K 1KQ ; · [ k (s T )] ; · [s k T ] C K K C K C C K k

T 1 C K C · ds . ; · T ds ; k (148) C K C x K 1KQ 1KQ G The integral terms again reﬂect scattering and collision that appear as a result of the heterogeneous medium transport description. The equation for longitudinal phonon temperature is

T M C M t

K

:9

W W C ; *- , t t A A

(149)

or

T M C M t

K

:

k

1 (nV T ) · ds ; C C K

1KQ k

C K 9 ·

1KQ

(W V ) · ds C C K

T C · ds k C x K 1KQ 1KQ G k

1 T C K ? · ds . T ds 9 k 9 · ? K ? x K 1KQ 1KQ G The equation of acoustical phonon energy is

C

T ? ? t

1 T ds 9 C K

: · (k T ) ; C ? ? K M ^ K

(T 9 T ) M ? M\? ^

(150)

(151)

v. s. travkin and i. catton

50 or

T ? C ? t

K

: · [ k (s T )] ; · [s k T ] ? K K ? K ? ? K ; ·

k

C K

1KQ

1 T ds ; ? K

T ? · ds . k ? K x 1KQ G

(152)

Describing phonon scattering and collision is an unsolved problem and as noted by Peterson [95], ‘‘The complexity of this aspect of the problem precludes the relatively simple solution used in simulating rareﬁed gas ﬂows.’’ Another kind of single phase equation for momentum transport of electronic ﬂuid results for magnetized materials:

V C t

K

; (V · )V C C K :9

e 1 k E ; V ; B 9

(nT ) C K m* n C m*

V C . 9 K ^ K^

(153)

The VAT form of this equation is V 1 C K ; · nV V ; C C K t

(nV V ) · ds C C K 1KQ 9 s V · (nV ) ; V ( · (nV )) ()

K C C K C C K e k 1 :9 (E ; V ;B ) 9

T ; K C K C K m* m* 9

k 1 s T z ; K C L K m* K

1KQ

1KQ

T ds C K

Z ds ; T ( (Z )) () . L K C L K (154)

The Maxwell equations for electromagnetic ﬁelds used to develop the VAT Maxwell equations for electromagnetic ﬁelds are

· ( E ) : , · ( H ) : 0 K K K K K B

;E : 9 K K t

;H : j ; (D ), K K t K

(155) (156) (157)

volume averaging theory

51

with constitutive relationships B : H , D : E , j : E . (158) K K K K K K K K K A full description of the derivation of the VAT nonlocal electrodynamics governing equations is given by Travkin et al. [114, 115] with only a limited number shown here. For the electric ﬁeld, the Maxwell equations, after averaging over phase (m) using , become K 1 ( E ) · ds :

· [s E ] ; · [s E ] ; K K K K K K K K K K K K 1 (159)

1

;(s E ) ; K K

ds ;E : 9 H . K K t K K

(160)

1KQ The phase averaged magnetic ﬁeld equations are 1

· (s H ) ; · [s H ] ; K K K K K K K

and

1

;(s H ) ; K K :

1KQ

1KQ

( H ) · ds : 0, K K K

(161)

ds ;H K K

E ; [s E ; s E ]. K K K K K K t K K K

(162)

These equations and some of their variations, such as the electric ﬁeld wave equations

E E K9 K: K ,

E 9 K K K t K K t K

(163)

which becomes

1

(s E ) ; · K K : K K

1KQ

1 E ds ; K K

1KQ

E · ds K K

E E 1 1 K; K ; (s ) ; K K t K K t K K

1KQ

ds , K K (164)

are the basis for modeling of electric and magnetic ﬁelds at the microscale level in heterostructures.

52

v. s. travkin and i. catton

The primary advantage of the VAT-based heterogenous media electrodynamics equations is the inclusion of terms reﬂecting phenomena on the interface surface S that can be used to precisely incorporate multiple KQ morphological effects occuring at interfaces. E. Photonic Crystals Band-Gap Problem: Conventional DMM-DNM and VAT Treatment One of the possible applications of VAT electrodynamics is the formulation of models describing electromagnetic waves in a dielectric medium of materials considered to be photonic crystals [118, 116, 117, 119, 126, 120]. The problem of photonic band-gap in composite materials has received great attention since 1987 [118, 116] because of its exciting promises. The most interesting applications appear in the purposeful design of materials exhibiting selective, at least in some wave bands, propagation of electromagnetic energy [120]. Figotin and Kuchment [122] were the ﬁrst who theoretically demonstrated the existence of band-gaps in certain morphologies. Unfortunately, this problem as presently formulated is based on the homogeneous Maxwell equations. The most common way to treat such problems has been to seek a solution by doing numerical experiments over more or less the exact morphology of interest, a method called detailed micromodeling (DMM), which is often done using direct numerical modeling (DNM) (for example, see [124]). As a result, questions arise about differences between DMMDNM and heterogeneous media modeling (HMM), which is the modeling of an averaged medium to determine its properties. How the averaging for HMM is accomplished is often not clear or not done at all. So, why cannot DMM be self-sufﬁcient in the description of heterogeneous medium transport phenomena? The answers can be primarily understood by analyzing, among others [23], the following issues: 1. A basic principal mismatch occurs at the boundaries, causing boundary condition problems. This means that for DMM and for the bulk (averaged characteristics) material ﬁelds, the boundary conditions are principally different. 2. The DMM solution must be matched to a corresponding HMM to make it meaningful at the upper scales. This can only be done for regular morphologies. Discrete continuum gap closure or mismatching will occur with DMM-DNM, precluding generalization to the next or higher levels in the hierarchy. 3. The spatial scaling of heterogeneous problems with the chosen REV (for DMM) is needed to address large or small deviations in elements

volume averaging theory

53

considered that are governed by different underlying physics. When spatial heterogeneities of the characteristics or morphology are evolving along the coordinates, DMM cannot be used. 4. Numerical experiments provided by DMM-DNM need to be translated to a form that implies that the overall spatially averaged bulk characteristics model random morphologies. It is not clear what kind of equations are to be used as the governing equations, nor what variables should be compared. In the case of the local porosity theory [128, 129], for example, the results of using real porous medium digitized images for morphological analysis to calculate the effective dielectric constant assumes that the HMM equations are applicable. 5. Interpretation of the results of DMM-DNM is always a problem. If results are presented for a heterogeneous continuum, then the previous point applies. If the results are being used as a solution for some discrete problem, then the question is how to relate that solution to the continuum problem of interest or even to a slightly different problem. If the results obtained are ﬁt into a statistical model, then the phenomena are being subjected to a statistical averaging procedure that is in most cases only correct for independent events. 6. The most sought-after characteristics in heterogeneous media transport studies are the effective transport coefﬁcients that can only be correctly evaluated from 1 9 j : * : ; ( 9 )

d, using the DMM-DNM exact solutions for a small fraction of the problems of interest. The issue is that problems of interest having inhomogeneous, nonlinear coefﬁcients and, in many transient problems, two-ﬁeld DMMDNM exact solutions are not enough to ﬁnd the effective coefﬁcients. Fractal methods are sometimes used to describe multiscale phenomena. The use of fractals is not relevant to most of the morphologies of interest and the fractal phenomenon description is generally too morphological, lacking many of the needed physical features. For example, descriptions of both phases, of the phase interchange, etc., are need to represent the physical phenomenon. For the simplest case of a superlattice or multilayer medium there can be many difﬁculties. When the boundaries are not evenly located, crossing the regular boundary cells of the medium, then the problem must be solved again and again. If the coefﬁcients are space dependent, because of the layers or grain boundaries, they will inﬂuence scattering. Grain boundaries are not perfect and are not just mathematical surfaces without thickness or physical

v. s. travkin and i. catton

54

properties. They cannot be treated as mathematical surfaces without any properties. Imperfections in the internal spacial structures must be treated as domain morphologies are not perfect at any spacial level. The insufﬁciency of a homogeneous wave propagation description of a heterogeneous medium was addressed in [125] from a pure mathematical point of view by searching for another type of governing operator that could better explain the behavior of the frequency spectrum eigenmodes via ‘‘heuristic arguments.’’ The general band-gap formulation should be treated using the HMM statements developed from the analysis of the VAT equations. A straightforward description of one of the band-gap problems is given next. Representing electromagnetic ﬁeld components with time-harmonic components, E(x, y, z)e it,

H(x, y, z)e it,

i : ((91) ,

(165)

The equations describing a dielectric medium becomes

· (E) : 0, · H : 0, : 1

(166)

;E : 9iH, ;H : i! E,

(167)

where ! is the complex dielectric ‘‘constant’’ deﬁned by ! : 9 i(/), and : 0(x ), : (x ), ! : ! (x , ). Taking the curl of the both sides of the vector equations,

;

1

;H : ;(iE ), ;( ;E) : ;(9iH ), !

(168)

yields

;

1

;H : i(9iH ) : H !

;( ;E ) : 9i(i! E) : ! E.

(169) (170)

This is the set of equations usually used when problems of photonic band-gap materials are under investigation; see the study by Figotin and Kuchment [123], p. 1564. These equations can be transformed to 9E (x ) : !(x )E (x ) for E-polarized ﬁelds and 9 · for H-polarized ﬁelds.

1

H (x ) : H (x ) ) ! (x

(171)

(172)

volume averaging theory

55

The further treatment by Figotin and Kuchment [123] reduces the mathematics to two equations,

1

f (x ) : f (x ) N ! (x ) N

(173)

9

1 f (x ) : f (x ), N ! (x ) N

(174)

9 ·

where f is the H or E polarization determined components of electric or N magnetic ﬁelds. These equations state the eigenvalue problem characterizing the spectrum of electromagnetic wave propagation in a dielectric two-phase medium, which is supposed to describe the photonic materials band-gap problem of EM propagation (see equations on p. 1568 in Figotin and Kuchment [123]) There are no spatial morphological terms or functions involved in the description, just the permittivity, which is supposed to be a space-dependent function with changes at the interface boundary. When these equations are phase averaged to represent the macroscale characteristics of wave propagation in a two-phase dielectric medium, the equations become

1

· (* (m f )) ; · * N 1 ; · (* f ) ; N

1

1

* f · ds : 9m f N N

*(x ) :

1

(m f ) ; · N

f ds N

(175)

1 !

1 f ds ; N

f · ds N

1 1 : 9[m f ; m f ]. (176) N N The three additional terms appear along with the porosity (or volume fraction) function m as a factor on the right-hand side of each of the equations. When the dielectric permittivity function is homogeneous in each of the two phases, then the VAT photonic band-gap equations can be reduced to one equation in each phase and written in a simpler form,

1

(m f ) ; · N

1

1 f ds ; N

1

f · ds : 9! m f N N (177)

v. s. travkin and i. catton

56 and

1

(m f ) ; · N

1

1 f ds ; N

1

f · ds : 9! m f . N N (178)

The equations are almost the same as equations for heat or charge conductance in a heterogeneous medium. The similarity of the equations means that the analysis of the simplest band-gap problem should also be very similar. Using DMM-DNM, Pereverzev and Uﬁmtsev [121] found that exact micromodel solutions among others features can have ‘‘medium . . . internal generation’’ that might be well characterized by the impact of the additional terms in the VAT Maxwell equations in both phases and in the combined electric ﬁeld and effective coefﬁcient equations; see Sections V and VIII. The exact closure and direct numerical modeling derived by Travkin and Kushch [33, 34] demonstrated how important and inﬂuential the additional VAT morphoterms can be (Section I). These terms do not explicitly appear in either the microscale basic mathematical statements or in microscale ﬁeld solutions. The terms appear and become very important when averaged bulk characteristics are being modeled and calculated.

V. Radiative Heat Transport in Porous and Heterogeneous Media Radiation transport problems in porous (and heterogeneous) media, including work by Tien [130], Siegel and Howell [131], Hendricks and Howell [132], Kumar et al. [133], Singh and Kaviany [134], Tien and Drolen [135], and Lee et al. [139], are primarily based on governing equations resulting from the assumption of a homogeneous medium. This implicitly implies that speciﬁc problem features due to heterogeneities can be decribed using different methods for evaluation of the interim transport coefﬁcients, as, for example, done by Al-Nimr and Arpaci [136], Kumar and Tien [137], Lee [138], Lee et al. [139], and Dombrovsky [140]. Although this kind of approach is legitimate, it presents no fundamental understanding of the processes because the governing equations suffer from the initial assumption that strictly describes only homogeneous media. Further, it is difﬁcult to represent hierarchical physical systems behavior with such models as will be touched on later. Review papers like that of Reiss [141] describe the progress in the ﬁeld of dispersed media radiative transfer. The few works on heterogeneous radiative or electromagnetic transport (see Dombrovsky [140], Adzerikho

volume averaging theory

57

et al. [142], van de Hulst [143], Bohren and Huffman [144], Lorrain and Corson [145], Lindell et al. [146], and Lakhtakia et al. [147]) approach the study of transport in disperse media with the emphasis on known scattering techniques and their improvements. The area of neutron transport and radiative transport in heterogeneous medium being developed by Pomraning [148—151] and Malvagi and Pomraning [152] treats linear transport in a two-phase (two materials) medium with stochastic coefﬁcients. This approach is the same as that which has been used to treat thermal and electrical conductivity in heterogeneous media, and to this point it has not been brought to a high enough level to include variable properties, their nonlinearities, and cross-ﬁeld (electrical and thermal or magnetic) phenomena. Research by Lee et al. [139] on attenuation of electromagnetic and radiation ﬁelds in ﬁbrous media has shown a high extinction rate for infrared radiation. The problem is treated as a scattering problem for a single two-layer cylinder by Farone and Querfeld [153], Samaddar [154], and Bohren and Huffman [144]. The process of radiative heat transport in porous media is very similar to propagation of electromagnetic waves in porous media and will also be evaluated. These two very close ﬁelds seem not to have been considered as a coherent area. Complicated problems of propagation of electromagnetic waves through the ﬁber gratings have been primarily the subject of electrodynamics. The most notable work in this area is that of Pereverzev and Uﬁmtsev [121], Figotin and Kuchment [122, 125], Figotin and Godin [124], Botten et al. [155], and McPhedran et al. [156, 157]. No effort seems to have been made to translate results obtained for polarized electromagnetic radiation to the area of heat radiative transfer. Detailed micromodeling (DMM) of electromagnetic wave scattering has been based on single particles or speciﬁc arrangements of particulate media. Direct numerical modeling (DNM) of the problem seems enables one to do a full analysis of the ﬁelds involved. As already discussed, the analysis of the results of a DNM is limited in the performance of a scaling analysis, which is the goal in most situations. Performing DNM without a proper scaling theory is like performing experiments, often very challenging and expensive; without proper data analysis, it yields a certain amount of detailed ﬁeld results, but not the needed bulk or mean media physical characteristics. Most recent work on radiative transport is based on linearized radiative transfer equations for porous media. We ﬁrst review this work to set the stage for the development that follows. This radiative transport related work extends our results in the theoretical advancement of ﬂuid mechanics, heat transport, and electrodynamics in heterogeneous media (Travkin et al. [19]; Catton and Travkin [28, 158]; Travkin and Catton [20, 159—163]; Travkin et al. [114, 115]) and provides a means for formulation of radiative

v. s. travkin and i. catton

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transport problem in porous media using the heterogeneous VAT approach and electrodynamics language. Based on our previous work, a theoretical description of radiative transport in porous media is developed along with the Maxwell equations for a heterogeneous medium. 1. L inear Radiative Transfer Equations in Porous Media The equation for radiative transport in a homogeneous medium can be written in the general form 1 I J ; · (I ) ; [, (r) ; , (r)]I J ? Q J c t 1 : , (r)I (T ) ; , (r) ? J@ 4' Q

p( · )I (r, )d (179) J L I : I (r, , t), J J with , (r) the absorption and , (r) scattering coefﬁcients, and for steady ? Q state, using the identity in the form

· (I ) : · I , J J

1 , (r) · I ; [, (r) ; , (r)]I : , (r)I (T ) ; J ? Q J ? J@ 4' Q

L

p( · )I (r, ) d. J (180)

In terms of a spectral source function S (s), the equation can be written in J a particularly simple form, 1 · I ; I : S (s), (181) J J J J where the extinction coefﬁcient (total cross section — Pomraning [150, 151] is - : , (r) ; , (r). J ? Q Linear particle (neutron, for example) transport in heterogeneous medium is assumed by Malvagi and Pomraning [152] and Pomraning [151] to be decribed by · ; -(r) : S(r, ) ;

1 4'

, (r, · )(r, ) d, (182) Q L where the quantities -(r), , (r), and S(r, ) are taken to be two-states Q discrete random variables. By assuming this, one needs to treat the porous

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59

(heterogeneous) medium as a binary medium that has two magnitudes for each of the random variables, and a particle encounters alternating segments of medium with those magnitudes while traversing the medium. When -, , , Q and S are assumed to be random variables, Eq. (182) is treated as an ensemble-averaged equation (see Malvagi and Pomraning [152] and Pomraning and Su [164]) p QC p QC ,

· ( p );p - :p S ; QG p # ; H H 9 G G , i:1, 2, j"i G G G G G G G 4' G G H G (183)

(r, )d, L where is the conditional ensemble averaged function at some point r G that is in phase i, and QC and QC are the interface ensemble-averaged ﬂuxes. H G The solution to this equation is also supposed to be ensemble-averaged. The overall averaging over the both phases is given by #:

(r, ) : p ; p , (184) where p and p are the probabilities of point r being in medium i : 1 or 2, and is the conditional ensemble averaged value of , when r is in G medium i. Ensemble averaging in this representation is obtained by averaging of medium features, including coefﬁcients, along a straight line the direction — or by nonlocal 1D line averaging in terms of the physical ﬁelds considered. Most of this kind of work is related to the Markovian statistics by alternating along the line of two phases of the medium (Pomraning [148, 151]). The ensemble averaging procedure suggested in (183) signiﬁes that the two last terms in the averaged equation reﬂect the ﬁnite correlation length (interconnection) in a single nonlinear term -(r). This kind of averaging results in very simple closure statements derived using hierarchical volume averaging theory procedures, as shown later. A major problem in using ensemble averaging techniques is that the processes and phenomena going at each separate site within separate elements of the heterogeneous medium cannot be resolved completely with the purely statistical approach of ensemble averaging. To make an ensemble averaging method workable, researchers always need to formulate the ﬁnal problem or solution in terms of spacially speciﬁc statements or in terms of the original spatial volume averaging theory (VAT). Examples of this are numerous; see the review by Buyevich and Theofanous [165].

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2. Nonlocal Volume Averaged Radiative Transfer Equations The basis for the development in this ﬁeld will be the volume averaging theory. We will present some aspects of VAT that are now becoming well understood and have seen substantial progress in thermal physics and in ﬂuid mechanics. The need for a method that enables one to develop general, physically based models of a group of physical objects (for example, molecules, atoms, crystals, phases) that can be substantiated by data (statistical or analytical) is clear. In modern physics it is usually accomplished using statistical data and theoretical methods. One of the major drawbacks of this widely used approach is that it does not give a researcher the capability to relate the spatial and morphological parameters of a group of objects to the phenomena of interest when it is described at the upper level of the hierarchy. Often the equations obtained by these methods differ from one another even when describing the same physical phenomena. The drawbacks of existing methods do not arise when the VAT mathematical approach is used. At the present time, there is an extensive literature and many books on linear, homogeneous, and layered system electromagnetic and acoustic wave propagation (Adzerikho et al. [142]; Bohren and Huffman [144]; Dombrovsky [140]; Lindell et al. [146]; Lakhtakia et al. [147]; Lorrain and Corson [145]; Siegel and Howell [131]; van de Hulst [143]). It is surprising that these phenomena are often described by almost identical mathematical statements and governing equations for both heterogeneous and homogeneous media. Major developments in the use of VAT, showing the potential for application to eletrophysical and acoustics phenomena in heterogeneous media, are found in Travkin and Catton [21], Travkin et al. [159, 114, 115], and with experimental applications to ferromagnetism in Ryvkina et al. [160, 162] and Ponomarenko et al. [161]. It has been demonstrated during the past 20 years of VAT-based modeling in the thermal physics and ﬂuid mechanics area (see Slattery [6]; Whitaker [10]; Kaviany [7]; Gray et al. [8]) that the potential of the approach is enormous. Substantial success has also been achieved in analyzing the more narrow phenomena of electromagnetic wave propagation in porous media. We consider here radiative transfer in porous media using a hierarchical approach to describe physical phenomena in a heterogeneous medium. The physical features of lowest scale of the medium are considered and their averaged characteristics are obtained using special mathematical instruments for describing hierarchical processes, namely VAT. At the next higher level of the hierarchy, physical phenomena have the physical medium pointwise characteristics resulting from averaged lower scale characteristics.

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61

The same kind of operators and averaging theorems used in preceding sections are applied to the following development, involving the rot operator, in which averaging will result because of the following averaging theorems:

1 ;f : ;f ; 1 ;f : ; f ;

1

ds ;f

(185)

ds ;f . (186) 1 Rigorous application to linear and nonlinear electrodynamics and electrostatic problems is described in Travkin et al. [114, 115]. The phase averaging the equation for linear local thermal equilibrium radiative transfer, 1 · I ; - (r)I : , (r)I (T ) ; , (r) J J J ? J@ 4' Q

p( · )I (r, ) d, (187) J L in phase 1 yields the VAT radiative equation (VARE)

1 · (I ; J

1

I ds ; - I J J J

, : , (r)I (T ) ; Q # 9 - I , i : 1 ? J@ 4' J J

(188)

p( · )I (r, ) d, J L when it is assumed that , is a constant, as done by Malvagi and QG Pomraning [152], Pomraning and Su [164], and others. The additional terms appearing in the VARE in some instances are similar, but in others they have a different interpretation in the ensemble averaged equation (183). For example, the term #:

9 - I (189) J J in (188) is the result of ﬂuctuations correlation inside of medium 1 in the REV, but it is described by p QC p QC H H 9 G G (190) H G in Malvagi and Pomraning [152], as it is an exchange of energy term between the two phases across the interface surface area S . Because ensemble averaging methodologies in Malvagi and Pomraning [152] do not

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62

treat nonlinear terms very well and incorrectly average differential operators such as , terms do not appear in equation (183) that reﬂect the interface ﬂux exchange. In VARE, Eq. (188), the interface exchange term naturally appears as a result of averaging the operator, ·

1

I ds . J

(191)

1 When the coefﬁcients in the radiative transfer equation are dependent functions, more linearized terms are observed in the corresponding VARE, · I ; - I J J J 1 : , I (T ) ; , I ; (, # ; , # ) ? J@ ? J@ 4' Q Q

1

I ds 9 - I , i : 1, (192) J J J 1 while continuing to treat the emissivity as via the Planck’s function. This equation should be accompanied by the VAT heat transfer equations in both porous medium phases (see, for example, Travkin and Catton [21]). The heat transport within solid phase 2, combining conductive and possible radiative transfer, is described by 9·

T 1 : k (s T ) ; k · s (c ) N t k ;

1 1

T · ds ; · qP ;

T ds

qP · ds .

(193)

1 1 The third and ﬁfth terms on the r.h.s. model the heat exchange rate between the phases. In an optically thick medium, for example, the radiation ﬂux term written in terms of the total blackbody radiation intensity is

4

· qP : · 9

(I ) @ 3-

4 nT

3'

, (194) where - is the total extinction coefﬁcient. An energy equation similar to Eq. (193) needs to be written for the ﬂuid-ﬁlled volume, phase 1 of the porous medium. The radiation ﬂux term would be much more complex because of the spectral characteristics of radiation in a ﬂuid. Closure is needed for the second, third, and ﬁfth terms in Eq. (193) on the r.h.s. For convective heat exchange, the last term can be written k

: · 9

T · ds : S ( T 9 T ) x 1 G

(195)

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63

by noting that

1 T T · ds : 9 ds · n k k x n 1 1 G 1 : q · ds : S ( T 9 T ). (196) 1 This type of closure procedure is appropriate for description of ﬂuid—solid media heat exchange and has been considered by many as an analog for solid—solid heat exchange. A more strict and precise integration of the heat ﬂux over the interfce surface, using the IVth kind of boundary conditions, gives the exact closure for the term in the governing equations for the neighboring phase. This would be an adequate solution for the portion of heat exchange by conduction to and from the ﬂuid phase, a conjugate problem. The radiative energy exchange across the interface surface is difﬁcult to formulate because of its spectral characteristics and the boundary conditions that must be satisﬁed. When the ﬂuid phase is assumed to be optically thin, an approximate closure expression results, 1

1

1

1 qP · ds :

1

(T 9 T ) 1 1 ; 91

· ds

((T Q ) 9 (T Q ))S , (197) 1 1 ; 91 using an interpretation of the averaged surface temperatures on opposite sides of the interface developed by Malvagi and Pomraning [152]. Another approximation is justiﬁable for an optically thick ﬂuid phase. It uses the speciﬁc blackbody surface radiation intensity I : nT to close the @ integral energy exchange term as follows: 5

1

1 qP · ds 5

(nT )ds 5 P (T Q )S .

(198) 1 1 Here, P is the total radiative hemispherical emissivity from phase 2 to phase 1 in the REV. The closure of Eqs. (183) is accomplished by assuming equality (Malvagi and Pomraning [152]; Pomraning and Su [164]) between the interface surface and ensemble (1D in this case) averaged functions,

QC : , H G

(199)

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64

as was done in heat and mass transfer porous medium problems; see, for example, Crapiste et al. [41]. 3. Radiation Transport in Heterogeneous Media Using Harmonic Field Equations Representing the electromagnetic ﬁeld components with time-harmonic components results in

· ( E) : , · ( H) : 0 (200) B K

;E : 9i H, ;H : i! E. (201) K B Here, as outlined earlier, ! is the complex dielectric function ! : B B B ), : (x ), : (x , ), ! : ! (x , ). In many 9i( /), and : (x C C K K B B C B B contemporary applications the spatial dependency of these functions is neglected. Electrophysical coefﬁcients often need to be treated as nonlinear. For example, the dielectric function can depend on E and : (x , E). The B B wave formulation of the Maxwell equations with constant phase coefﬁcients for the magnetic ﬁeld is H H

H 9 9 : 0, K C t K B t

(202)

whereas the electric ﬁeld wave equation is almost the same,

E E

E 9 9 : . K C t K B t

(203)

Another form of the equation for E appears in Cartesian coordinates when electromagnetic ﬁelds are time-harmonic functions:

E ; kE : 0,

(204)

Here, the inhomogeneous function k : is the wave number K B squared. This equation is often applicable to linear acoustics phenomena. This category of equations can be transformed to a form legitimate for application to heterogeneous media problems. The time-harmonic forms of equations for rot of electromagnetic ﬁelds are 1

;(m E ) ;

1

1

;(m H ) ;

ds ;E : 9i[m H ; m H ] K K (205)

1

ds ;H : i[m ! E ; m ! E ]. B B (206)

volume averaging theory

65

The magnetic ﬁeld wave form equation with constant coefﬁcients, when averaged over phase 1, transforms to

1

(m H ) ; ·

1

1 H ds ; : K C

1

H · ds

H H ; , K B t t

(207)

and the electric ﬁeld wave equation (203) becomes

1

(m E ) ; ·

1 E ds ;

E · ds

1 1 E 1 1 E ; (m ) ; ; : K B t K C t B B

1

ds . (208)

An analogous form of the averaged equation is obtained for the timeharmonic electrical ﬁeld:

1

(m E ) ; ·

1

1 E ds ;

1

E · ds ; m kE : 0. (209)

It is the naturally appearing feature of the heterogeneous medium electrodynamics equations as the terms reﬂecting phenomena on the interface surface S , and that fact is to be used to incorporate morphologically precise polarization phenomena as well as tunneling into heterogeneous electrodynamics, as is being done in ﬂuid mechanics and heat transport (Travkin and Catton [21]; Catton and Travkin [28]). Using the orthogonal locally calculated directional ﬁelds E and E of J P averaged electrical ﬁeld E , one can seek the Stokes parameters I, Q, U, and V, I : E E * ; E E * (210) J J R P P R Q : E E * 9 E E * (211) J J R P P R U : Re[2E E * ] (212) J P R V : Im[2E E * ], (213) J P R which characterize the intensity of polarized radiation in a porous medium. We will not here construct the general forms of equations for effective coefﬁcients, as this will be done in a succeeding section for the case of

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temperature ﬁelds; still, the same questions of multiple versions, applicability of current methods, and variance in interpretation are the present agenda. VAT-based models were developed recently while addressing the problems of modeling of electrodynamic properties of a liquid-impregnated porous ferrite medium (Ponomarenko et al. [161]), coupled electrostaticdiffusion processes in composites (Travkin et al. [159]), and to analyze heat conductivity experimental data in high-T superconductors (Travkin and Catton [166]). Powders of ferrites with NFMR frequency in the microwave range were used as the porous magnetic medium in Ponomarenko et al. [161]. The search for tunable levels of reﬂection and absorption of electromagnetic waves was conducted using a few morphologies that were arbitrarily chosen. Thus, the need for closer consideration of experiment and models presenting the data using VAT heterogeneous description tools for both became obvious.

VI. Flow Resistance Experiments and VAT-Based Data Reduction in Porous Media It is well known that existing measurements of transport coefﬁcients in porous (and heterogeneous) media must be used with care. As long as a complete description of an experiment is provided and the data analysis is carried out using correct mathematical formulations (models), the relationship between the experiment and its analysis is maintained in a comsistent, general, and useful way. Unfortunately, this is not always the case, because heuristic equations and models are often the basis for coefﬁcient matching and model tuning when heterogeneous medium experimental data is reduced to correlations. The various approaches, and even disarray, in the ﬁeld can be contributed to a lack of understanding of the general theoretical basis for transport phenomena in porous and heterogeneous media. As long as the correlations used for momentum transport comparison are generated from empirical Darcy and Reynolds—Forchheimer expressions, or effective heat and electrical conductivity and permittivity derived from homogeneous models, problems in heterogeneous media experimental validation and comparison will persist. Modeling based on volume averaging theory will be shown to provide a basis for consistency to experimental procedures and to data reduction processes by a series of analyses and examples. Many of the common correlations, and their weaknesses, are examined using a uniﬁed scaling procedure that allows them to be compared to one another. For example, momentum resistance and internal heat transfer dependencies are analyzed

volume averaging theory

67

and compared. VAT-based analysis is shown to reveal the inﬂuence of morphological characteristics of the medium; to suggest scaling parameters that allow a wide variety of different porous medium morphologies to be normalized, often eliminating the need for further experimental efforts; and to clarify the relationships between differing experimental conﬁgurations. The origin, and insufﬁciency, of electrical conductivity and momentum transport ‘‘cross-correlation’’ approaches based on analogies using mathematical models without examining the physical foundation of the phenomena will be described and explained. 1. Experimental Assessment of Flow Resistance in Porous Medium A one-term ﬂow resistance model for porous medium experimental data analysis often used is 9

dp ! :f dx

S u ! U D , m 2

(214)

where f is some coefﬁcient of hydraulic resistance. On the other hand, most two-term models used for ﬂow resistance experimental data reduction have ﬁrst-order and second-order velocity terms, the Darcy—Forchheimer ﬂow resistance models. These models were obtained primarily for direct comparison with established empirical and semiempirical Darcy and Darcy—Forchheimer type ﬂow resistance data. Thus, the momentum equation for laminar as well as the high (turbulent) ﬂow regime often used is the model by Ergun [167], dp! : m u! ; Amu! . (215) D dx k " Similarly, the model given by Vafai and Kim [168] for the middle part of a porous layer is 9

dp! F : mu ! ; m u! , (216) D dx k k " " and the Poulikakos and Renken [169] equation for the turbulent regime is 9

dp ! : u ! ; Au ! . (217) D dx k " Analysis of a simple idealized morphology where solutions are known will show that the Darcy and Darcy—Forchheimer or Ergun type model correlations are not matched consistently for any regime. Further, they are also without theoretical foundation. Thus, problems arise when studies to 9

68

v. s. travkin and i. catton

improve the description of transport use combined models for ﬂow resistance and momentum transport in a porous medium because the analysis does not start with the correct theoretical basis. Further, which of the three equations just listed should one use? A model of ideal parallel tube morphology yields the following Darcy friction coefﬁcient (see, for example, Schlichting [170]): d p f U 8 U , : F , u : U : " (218) f : * U " ( U ) 4L 8 D D p dp ! 4 f U : 9 : u : " D . (219) dx d D * d L 2 F F The morphology function S /m for a straight equal-diameter tube morU phology is S 2'R 'R S 4 U : S : U: , m : , (220) U py py m d F and an exact expression for the Darcy friction factor is U 2d p p F :f D , f : . (221) " d 2 " U L L F D The Fanning friction factor for this speciﬁc morphology is (using (220)) p f 4 U f S U D : " U D : " (222) L 4 d 2 4 m 2 F d p F f : , (223) D 2 U L D and a relationship to the Darcy friction coefﬁcient is (Travkin and Catton [16, 20])

f f : ". D 4

(224)

The friction coefﬁcient c for smooth tubes often calculated using the B Nikuradze and Blasius formulas [170] is the same as the Fanning friction factor. A model representing a porous medium with slit morphology was treated in conformity with the deﬁnition p U 2 2u 2h p h p U : *: : h : c D , c : , u : . D 2 D U U U L * L U L D D D (225)

volume averaging theory

69

The morphology ratio S /m for a porous medium morphology model of U straight equal slits is found as follows: (HL y) H (2L y) 2 : , m : : S : U ( pL y) p ( pL y) p

(226)

S 2 1 U : : , d : 4h, F m H h

(227)

yielding the Fanning friction factor, 9

dp! p 1 : :f D h dx L

U S D U :f D m 2

U D 2

(228)

H p f : (229) D U L D As one can easily see, these ﬂow resistance models are written with the second power of bulk velocity variable. The convergency of the VAT-based ﬂow resistance transport models to these classical constructions was demonstrated on several occasions by Travkin and Catton [16, 20, 21, 23] and Travkin et al. [25]. Exact ﬂow resistance results obtained on the basis of VAT governing equations by Travkin and Catton [16, 26, 23] for the random pore diameter distribution for almost the same morphology as was used by Achdou and Avellaneda [171] demonstrated the wide departure from the Darcy-lawbased treatments. That was shown even for the morphology where a single pore exists with diameter different from the all others. Meanwhile, by consistently using the VAT-based procedures (Travkin and Catton [23]), one can easily develop the needed variable, nonlinear permeability coefﬁcient for Darcy dependency, U 2

S U k : c BA B m

\ ,

(230)

where c : f is derived for this particular morphology using exact analytiB D cal (in the laminar regime) or well-established correlations for the Fanning friction factor in tubes. 2. Momentum Resistance in 1D Membrane and Porous Layer Transport The steady-state VAT-based governing equations for laminar transport in

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70

porous media (Travkin and Catton [21]) are U 1 ; u u ; (m p ) D x D x x D mU 1 pds ; ; :9 x x 1U D

mU

and

U · ds x 1U G

(231)

T mT D:k D ;c c mU (m 9T u ) ND D D ND D D D x x x x k D x

T D · ds k (232) D x 1U 1U G 1 T s T

1 Q Q ; Q · ds : 0. (233) T ds ; Q x x x x 1U 1U G The momentum equation for turbulent ﬂow of an incompressible ﬂuid in porous media based on K-theory can be written in the form (Gratton et al. [26], Travkin and Catton [20]) ;

U U (K ; ) · ds ; m(K ; ) K K x x x 1U G u! ; m K ; (m 9u! u! ) K D x x x D 1 1 9 p! ds 9 (mp! ). (234) x 1U D D By comparing these equations with conventional mathematical models and experimental correlations, one can easily see the differences. The one-dimensional momentum equation for a homogeneous, regular porous medium is

m

U U 1 ; U : t x

1 T ds ; D

1 p : D m x

m

V · ds. (235) 1U 1U Closure of the ﬂow resistance terms in the simpliﬁed VAT equation can be obtained following procedures developed by Travkin and Catton [16, 17]. The skin friction term is 9

U · ds : D x 1U* G D

1U

p ds 9

1 · ds : 9 c (x )S (x )[ U (x )], U* U* D 2 D* (236)

volume averaging theory

71

with U , u : c U (x ), : 1PI D* U* x G and closure of the form drag resistance integral term using a form drag coefﬁcient, c , is BN 1 1 pds : c S (x )[ U (x )]. (237) D 2 BN UN 1U For these equations, the speciﬁc surface has two parts. The ﬁrst part, S , is U* 1 1 S (x ) : ds, , (238) U* m 1U* where S is the laminar subregion of the interface surface element S , U* U and

1 S (x ) : UN

S ds : , ,

1 , m

(239) 1UN where S is the cross ﬂow projected area of the surface of the solid phase UN inside the REV. Substitution into the one-dimensional momentum equation yields 9

U (x) p p : (c (x)S (x) ; c S (x)) D ; ( m). D D* U* BN UN x 2m m

(240)

When the porosity is constant, the ﬂow is laminar and S : S , the U* U equation becomes

dp S S U S U UN U D U D : c ;c : c (U , M ) , (241) D BN S B m 2 2 dx m U where c is the friction factor and c the form drag, S is the cross ﬂow D BN UN form drag speciﬁc surface, and M is a set of porous medium morphological parameters or descriptive functions (see Travkin and Catton [16, 20]). The drag terms can be combined for simplicity into a single total drag coefﬁcient to model the ﬂow resistance terms in the general simpliﬁed momentum VAT equation 9

S UN . c (U , M ) : c ; c (242) B D BN S U Correlations for drag resistance can be evaluated for a homogeneous porous medium from experimental relationships for pressure drop. For

v. s. travkin and i. catton

72

example, the equation often used for packed beds is 9

S U dp ! U D . :f D m 2 dx

(243)

The complete VAT version of this equation is 1 (mp ) ; x

pds ; u S (x) D 1PI U 1U U mU : 9m U ; ; [ m 9u u ]. D x D x x D x

(244)

If the porosity function is constant (a frequent assumption), the left-hand side of Eq. (244) reduces to 9

dp S U U D . 9f D m dx 2

(245)

Setting Eq. (245) equal to zero recovers equation (243). As a result, data correlation using Eq. (243) incorporates the right-hand side of Eq. (244) implicitly into the correlation. Friction factor data presented in this way detracts from objectivity. The correlation can be written to reﬂect all the right-hand terms from Eq. (244),

S U d(mp ) UN ; F ; F ; F (S (x)) D : c ;c , (246) D BN S U 2 dx U where F , . . . , F are deduced from the following relationship: U U (F ; F ; F ) S (x) D : m U ; [m u u ] U D x D x D 2 9

9

mU . x x

(247)

In the middle part of a porous medium sample, one can assume that the porosity and ﬂow regime are constant and steady state and then neglect all terms on the right-hand side of (244). In reality, a large number of experiments are being carried out under conditions where input—output zones are present and can add signiﬁcantly to the value of the friction coefﬁcient because of the input—output pressure losses. If one wants to separate the effects of input—output pressure loss from the viscous friction and drag resistance components inside the porous medium, then taking into account the terms shown in Eq. (247) is essential. There are correlations that reﬂect a dependence on sample thickness as a result of this oversight. An

volume averaging theory

73

even more complex situation arises when the ﬂow and temperature inside the medium are transient, such as one might ﬁnd in a regenerator, and very inhomogeneous in space because of sharp gradients. The inhomogeneity in space and time precludes neglecting the four right-hand terms in Eq. (244). The inhomogeneous terms on the right-hand side of (247) may be analyzed by scaling. Some of these terms are easily interpreted. For example, the ﬁrst term on the right-hand side is the convective term

S (x) U

U U D F : m U , D x 2

(248)

and its importance can be strongly dependent on the thickness of the porous specimen. This is why many studies report an obvious correlation with specimen thickness. The remaining terms are the ‘‘morphoconvective’’ term

S (x) U

U D F : ( m u! u! ) x D D 2

(249)

and the momentum diffusion term

S (x) U

U mU D F : 9 . 2 x x

(250)

The complete momentum equation written in a proper form for experimental data reduction is 9

S U d(mp ) UN ; F ; F ; F (S (x)) D : c ;c D BN S U 2 dx U U : (c ; R )(S (x)) D , B + U 2

(251)

where S UN c :c ;c B D BN S U

(252)

and R :F ;F ;F . (253) + The features of an experiment needed to treat terms such as F , F , F are discussed later. The momentum resistance coefﬁcient for a heterogeneous porous medium can be written in the form f

NMP

:c ;R . B +

(254)

v. s. travkin and i. catton

74

This is the variable usually determined in most of porous medium ﬂow resistance experiments. Nevertheless, if this correlation value taken from an experiment is later substituted into a modeling equation (with variable porosity) of the form mU

1 mU S (x) U U D :9 (m p ) ; 9 c (x) U D " x x x m 2 x D (255)

or

1 mU U :9 (m p ) ; D x x x x D F 9 mU 9 m U , (256) D k k " " as is done by many, then the ﬂuctuation term [m 9u u ]/x is D neglected and the equation mU

U 1 mU :9 (m p ) ; 2 D x x x x D 1 U 9 pds ; · ds ; [m 9u u ] D x x 1 1U G D is being used as the problem’s model instead of 2mU

U 1 mU :9 (m p ) ; D x x x x D 1 U 9 pds ; · ds ; [m 9u u ] D x x 1U 1U G D because the model used the coefﬁcient c (x) determined from " S (x) U S (x) U D : (c (x) ; R (x)) U D c (x) U " B + m(x) 2 m(x) 2 mU

(257)

(258)

S S (x) U UN ;F ;F ;F U D : c ;c D BN S m(x) 2 U 1 U : pds9 · ds9 [m 9u u ] D x x 1U 1U G D mU U 9 ; mU , (259) x x x

volume averaging theory

75

instead of using the coefﬁcient c (x) determined from B S S (x) U S (x) U D UN ; F U D : c ;c c (x) U D BN S m(x) 2 B m(x) 2 U 1 U pds 9 : · ds ; [m u u ]. D x x 1U 1U G D (260)

The terms needed for experimental data reduction model should include all ﬁve active terms, 9

S U d(mp ) UN ; F ; F ; F (S (x)) D : c ;c D BN S U 2 dx U U : (c ; R )(S (x)) D , B + U 2

(261)

with :c ;R . (262) NMP B + The general 1D VAT laminar regime constant viscosity momentum equation has six terms, f

U 1 ; u u ; (m p ) D x D x x D U 1 mU pds ; ; · ds. (263) :9 x x x 1U 1U G D For simplicity, Eq. (263) is written in the following shorthand notation: mU

UC ; UMC ; UP : 9UMP ; UD ; UMF . (264) The two right-hand integral terms reﬂect the morphology-induced ﬂow resistance of the medium. Three ﬂow resistance models are needed to properly tie everything together. a. Flow Resistance Model 1 The ﬁrst ﬂow resistance model is for the internal frictional and form drag resistance:

S (x)U (x) U (x) 9c (U , M , x) U : (9c S (x) 9 c (x)S (x)) B BN UN D* U* 2 2 :9

1 D

1U

pds ;

U · ds. x 1U G (265)

v. s. travkin and i. catton

76

b. Flow Resistance Model 2 The second ﬂow resistance model reﬂects the addition of the ﬂuid ﬂuctuation term UMC : S (x)U (x) U 9c (U , M , x) B 2

S U U 2

S UN ; F :9 c ;c D BN S U

:9

1 D

1

pds ;

U (x) U 9 S (x) F U* 2 2

: (9c S (x) 9 c (x)S (x)) BN UN D* U*

U · ds 9 u u . D x x 1U G

(266)

c. Flow Resistance Model 3 The third ﬂow resistance model reﬂects all of the terms responsible for momentum resistance in a porous medium:

S (x)U (x) c (U , M , x) U B 2

U S U U : S (x) R ; c (U , M ) U + B 2 2 : mU ;

U mU 9 x x x

1 D

1U

pds 9

U · ds ; u u , D x x 1U G

(267)

where

U U s (x) R : (F ; F ; F ) S (x) U + U 2 2 : mU

U mU ; [m u u ] 9 . (268) D x x x x

Using the notation developed earlier for the terms in the momentum equation (264) leads to a form for each of the ﬂow resistance models that properly reﬂects their completeness, c (U , M , x) : (UMP 9 UMF ) B

S (x)U (x) U 2

(269)

volume averaging theory c (U , M , x) : (UMP 9 UMF ; UMC ) B c (U , M , x) B

77

S (x)U (x) U 2

: (UC 9 UD ; UMP 9 UMF ; UMC )

(270)

S (x)U (x) U . 2

(271)

Each of the different forms will yield a correlation of a given set of data. The problem is that the effects of the different characteristics that are manifested in the terms in the equations are lost from consideration. If predictive tools are to be developed, consideration must be given to the impact of the details that the terms reﬂect. 3. Scaling in Pressure L oss Experiments and Data Analysis Direct use of any Ergun type friction factor in a Fanning or Darcy friction factor correlation is incorrect. Ergun [167] suggested two types of friction factors, one of which is the so-called kinetic energy friction factor f , which ICP differs from the Fanning friction factor by a factor of three for the same medium:

P f d f : F : ICP . (272) D 2 u! L 3 D For the same reason, direct implementation of the correlations given by Kays and London [172] should be treated with care. For example, the correlations for friction factor (Fanning) given by Kays and London for ﬂow through an inﬁnite randomly stacked, woven-screen matrix uses surface porosity p, and speciﬁc surface [1/m] to deﬁne a hydraulic radius r , F p m Q. r : : F S U Here the speciﬁc surface S is deﬁned as the interface surface divided by the U volume of the REV. Unfortunately, the surface porosity m and volume Q porosity m are not of the same value and even if they were, the expression differs from that found earlier by a factor of 2. Bird et al. [173] used the ratio of the ‘‘volume available for ﬂow’’ to the ‘‘cross section available for ﬂow’’ in their derivation of hydraulic radius r . F@ This assumption led them to the formula md N . r : F@ 6(1 9 m)

(273)

v. s. travkin and i. catton

78

It would be double this value if a consistent deﬁnition were used for all systems, 4m 2m 4m : : d : 4r , (274) d : F@ F a (1 9 m) 3(1 9 m) N S T U where a is the ‘‘particle speciﬁc surface’’ (the total particle surface area T divided by the volume of the particle), and S : a (1 9 m). (275) U T The expression given by (274) is justiﬁed when an equal or mean particle diameter is 6 d : , N a T which is the exact equation for spherical particles and is often used as substitution for granular media particles. The value of hydraulic radius given by Bird et al. [173], (273), was chosen by Chhabra [174] and was used in determining the speciﬁc friction factor in capillary media. Media of globular morphologies can be described in terms of S , m, U and d and can generally be considered to be spherical particles with N 6(1 9 m) 2 m S : , d : d . (276) U F 3 (1 9 m) N d N This expression has the same dependency on equivalent pore diameter as found for a one-diameter capillary morphology, leading naturally to S : U

6(1 9 m) 6(1 9 m) 4m : : . d 3 (1 9 m) d N F d F 2 m

(277)

This observation leads to deﬁning a simple ‘‘universal’’ porous medium scale, 4m d :d : , (278) F NMP S U that meets the needs of both major morphologies, capillary and globular. A large amount of data exists that demonstrates the insufﬁciencies of the Ergun drag resistance correlation (287). Because it was developed for a speciﬁc morphology, a globular ‘‘granular’’ medium, application of the Ergun correlation to a medium with arbitrary relationships between porosity m, speciﬁc surface S , and pore (particle) diameter d can lead to large U F errors.

volume averaging theory

79

The particle diameter d is often used as a length scale when reducing N experimental data. Chhabra [174], for example, writes the friction factor p d N , (279) f : A@ m u! L D This friction factor can be related to the friction factor f , given by Eq. @ (6.4-1) of Bird et al. [173], to the Fanning friction fator f , and to the Ergun D kinetic energy friction factor f as follows: ICP 1 9 m 1 9 m f :2f : f :f 3 . (280) A@ @ ICP D m m

These models all use different length scales, leading to large uncertainties and confusion when a correlation must be selected for a particular application. Little attention is paid to these differences, often requiring new experimental data for a new medium conﬁguration. Only a few of the many issues important to modeling of pressure loss in porous media are addressed here. As it is known, the two-term quadratic Reynolds—Forchheimer pressure loss equation is P 1 : U m ; - U m; : . (281) D k L " By comparison with the simpliﬁed VAT (SVAT) momentum equation for constant morphological characteristics and ﬂow ﬁeld properties and only the resistance coefﬁcient c , B P S U U D , :c (282) B m L 2

a set of transfer relationships can be found to transform Ergun-type correlations and the SVAT expression. The transfer formula (Travkin and Catton [21]) is

where

c :f : ; -m B D U D : 150

2m , S U

(283)

1 9 m) (1 9 m) , - : 1.75 , d m d m N N

(284)

or A 8m c :f : ; B, A : , B D Re S U NMP

B : 2-

m , S U

(285)

v. s. travkin and i. catton

80 where

4U m . NMP S U The Ergun energy friction factor relation can be written in terms of the VAT-based formulae (Travkin and Catton [21]) as Re

:

S U p U D :f . CP m L 2

(286)

If the Ergun correlation is written using common notation, it becomes

p (1 9 m) (1 9 m) : 150 mU ; 1.75 mU , (287) D d m d m L N N and if it can be further transformed to the (SVAT) Fanning friction factor, then

A* 50(1 9 m) 3.5 , B* : : 0.583, f : N ; B* , A* : N N N CP Re 6 m N where the particle Reynolds number is Re : (U d )/, N N and

(288)

(289)

A* 100 f : AF ; B* , with A* : : 33.33, and B* : B* : 0.583, AF AF AF N CP Re 3 NMP (290) where U d 2 m U d 3(1 9 m) U d N , and Re : Re : N . Re : F : NMP N NMP 3 (1 9 m) 2m

(291)

The common scaling length just derived will allow a great deal of data to be brought to a common basis and allow greater conﬁdence in predictions. 4. Simulation Procedures A large amount of data exists that demonstrates the inadequacies of the Ergun drag resistance correlation (287). This is because the Ergun correlation is used with arbitrary relationships between porosity m, speciﬁc surface S , and pore (particle) diameter d when it was originally developed U F for granular media. How unsatisfactory it can be is shown in Fig. 5.

volume averaging theory

81

Fig. 5. Fanning friction factor f (bulk ﬂow resistance in SVAT for different medium D morphologies, materials, and scales used), reduced based on VAT scale transformations in experiments by 1, Gortyshov et al. [175]; 2, Kays and London [172]; 3, Laminar, intermediate, and turbulent laws in tube; 4, Gortyshov et al. [176]; 5, Beavers and Sparrow [177]; 6, SiC foam (UCLA, 1997); 7, Ergun [167]; 8, Souto and Moyne [181]; 9, Macdonald et al. [180]; 10, Travkin and Catton [23].

With speciﬁcally assigned morphology characteristics (primarily S ), the U Ergun drag resistance correlation will be much closer to correlations by Beavers and Sparrow [177] and Gortyshov et al. [176], as shown in Fig. 5. A similar behavior was seen between the Ergun drag resistance correlation and the drag resistance correlation by Gortyshov et al. [175]. Several other correlations are compared in Fig. 5. Gortyshov et al. [175] experimentally derived correlations for the Reynolds—Forchheimer momentum equation in the form : 6.61 · 10(d )\ m\ , F - : 5.16 · 10(d )\ m\ , F

(292) (293)

v. s. travkin and i. catton

82

where hydraulic diameter d (mm) is F d [m] . d : F F 0.001[m]

(294)

These correlations have to be used in (285) and are for highly porous (m : 0.87—0.97) foamy metallic media. A Darcy type of friction factor obtained by Gortyshov et al. [176] for very low conductivity porous porcelain with high porosity is 40 (1 ; 2.5 · 10\m\ Re ), f (Re ) : F " F Re F where Re : F

m : 0.83 9 0.92,

(295)

U d m F .

To transform this correlation, the Reynolds number must be transformed and the result divided by 4 to yield the Fanning friction factor,

40 1 f (Re ) : (1 ; 2.5 · 10\m\ Re m) , D NMP NMP 4 Re m NMP

(296)

with Re 5 Re /m (297) NMP F The correlation derived by Beavers and Sparrow [177] seems to be of little value in the original form, 1 F (R ) : ; 0.074, @Q U R U because the Reynolds number, R : U

U m(k ",

(298)

(299)

contains the permeability of the medium and is usually not known. Noting that, as pointed out by Beavers and Sparrow [177] the viscous resistance coefﬁcient : 1/k , where k is the Darcy permeability, and using the " " transformation 1 F : ; -(k , @Q R " U where

(300)

volume averaging theory

83

1 U m (k : , R : " ( U (

(301)

4 4 S U , Re : R : R ( , R : Re NMP U S (k U U NMP 4( S U U "

(302)

yields 1 F : ; -(k @Q " Um(k "

or

1 P F (R ) : , @Q U ( Um x D and when compared to

(303)

2m P f (Re ) : , D NMP U S x D U

one obtains

(304)

1 1 P (m(2m) f (R ) : · D U S (m U x U D 2(m : F (R ) . (305) @Q U S U This means that the Fanning friction factor, f , can be assessed from the D friction factor suggested by Ward [178] and Beavers and Sparrow [177], f , from @Q 2(m . (306) f (R ) : F (R ) D U @Q U S U To accomplish the transformation of F to f , the permeability k or the @Q D " viscous coefﬁcient of resistance porosity m and speciﬁc surface S must U be known. Estimates of f were obtained from measured values of F for D @Q FOAMETAL (Beavers and Sparrow sample Type C) using

k : 19.01 · 10\ [cm] : 19.01 · 10\ [m] " 1 1 : : 0.0526 · 10 k m "

(307)

84

v. s. travkin and i. catton

and Eqs (299), (298) or (300), and (306) to transform the Beavers and Sparrow [177] experimental data correlation to the Fanning friction factor correlation. With 1 ; 0.074 and F (R ) : @Q U R U

S U R : Re U NMP 4(

1 4( F (Re ) : ; 0.074, @Q NMP Re S NMP U

(308)

then

1 4( 2(m f (Re ) : ; 0.074 . (309) D NMP Re S S NMP U U Kurshin [179] has analyzed a vast amount of data using a consistent procedure he developed to embrace all three ﬂow regimes in porous media. To carry out the procedure, the following parameters must be known: (a) The viscous resistant coefﬁcient , evaluated for laminar ﬂow in a pipe from the following:

P : mU , L

1 d P : , U : . (310) k 32 x " (b) A characteristic length d evaluated by equating the preceding ex pressions:

32 32k " d : : . (311) m m (This is only justiﬁed for straight parallel capillary morphology where d : d .) F (c) Critical numbers Re and Re to distinguish the viscous, transiCP CP tional, and turbulent ﬁltration regimes. (d) Dimensionless viscous ! and inertial resistance - coefﬁcients in the turbulent regime. Unfortunately, Kurshin [179] did not present any data for foam materials and the porous metals he evaluated have low porosity in the range m & 0.5. Now one can say that by reformulating existing experimental correlations to the SVAT 1D form, P S U U D : f (Re ) , D NMP m L 2

(312)

volume averaging theory

85

the Fanning friction factor correlations can be easily compared with one another as they have a common consistent basis. A number of correlations were transformed and are in Fig. 5. The reason for the spread in the results is thought to be inadequate accounting for details of the medium. Analysis of Macdonald et al. [180] reformulated with the help of the foregoing developed procedures gives the corrected Ergun-like type of correlation 40 ; 0.6. f : D+ Re NMP

(313)

Meanwhile, Souto and Moyne [181], using the DMM-DNM solutions, came to the number of resistance curves that are separate for each morphology. One of them for rectangular rods in VAT terms appears as f

D1+

:

1 54.3 f : , Re ; 0. NMP 3 ICP Re NMP

(314)

VII. Experimental Measurements and Analysis of Internal Heat Transfer Coefﬁcients in Porous Media A VAT-based approach applied to heat transfer in a porous medium allows one to analyze and measure effective internal heat transfer coefﬁcients in a porous medium. As noted by Viskanta [182], ‘‘Convective heat and mass transfer in consolidated porous materials has received practically no theoretical research attention. This is partially due to the complexity which arises as a result of physical and chemical heterogeneity that is difﬁcult to characterize with the limited amount of data that can be obtained through experiments.’’ Viskanta [182, 183] generalized the data he analyzed for internal heat transfer coefﬁcient porous ceramic media using a correlation of the form Nu : 2.0 ; a Re@Pr, T

(315)

by assuming that the limiting Nusselt number should be 2.0 when the Re decreases to zero. This assumption is only justiﬁed for unconsolidated sparse spherical particle morphologies and is suspect for other porous medium morphologies, especially consolidated media. For this reason, some researches neglect this artiﬁcial low Re limit and correlate their ﬁndings without it. The VAT approach is applied to heat transfer in porous media to develop a more consistent correlation.

v. s. travkin and i. catton

86

1. Experimental Assessment and Modeling of Heat Exchange in Porous Media The correct form of the steady-state heat transfer equation in the ﬂuid phase of a porous media with primarily convective 1D averaged heat transfer is k T D: D c mU ND D x

1U

;c ND D

T mT D · ds ; k D D x x G k D (m 9u T ) ; D D x x

Equation (316) can be rewritten as

1U

T ds . (316) D

mT S ( T 9 T ) : 9 k ;c (m 9u T ) 2 U Q D D x ND D x D x ;

k D x

where

T ds

1U

T D, ; c mU ND D x

(317)

T D · ds : S ( T 9 T ). 2 U Q D x 1U G The right-hand side of Eq. (316) can also be written in the form k D

T D : S ( T 9 T ) ; S ( T 9 T ) ; K [9q ], 2 U Q D CDDE x 2 U Q D DV x x (318)

where the right-hand side (‘‘diffusive’’-like) ﬂux contains more terms than are conventionally considered: q

T D : 9K CDDE DV x

T k D ; c m 9u T ; D : 9 mk D x ND D D D The corresponding equation for the solid phase is

s T

1 Q Q ; x x x

1U

T ds . D

1 T Q · ds : 0. T ds ; Q x 1U 1U G The three terms are written in the following shorthand form:

T D ; T MD ; T ME : 0. Q Q Q

(319)

(320)

(321)

volume averaging theory

87

Equation (320) can also be written 0:

1 T

Q S ( T 9 T ) ; k CDDQ 2 U D Q k x x Q

: S ( T 9 T ) ; [9q ]. QV 2 U D Q x

(322)

Using the closure term for interface heat ﬂux found earlier (they are equal),

T Q · ds . x 1U G Equation (322) has a term that is usually overlooked (the second term on the right): k S ( T 9 T ) : Q 2 U D Q

sT T

1 Q; Q :9 x x

T ds . (323) Q 1U Three heat transfer coefﬁcient models are needed to properly tie everything together. The ﬁrst model incorporates only the heat transfer coefﬁcient between the phases. q

QV

: 9K CDDQ

a. Model 1 of Heat Transfer Coefﬁcient in Porous Media: Conventional Modeling If it is assumed that the porous medium heat transfer coefﬁcient is deﬁned by

T D · ds [S ( T 9 T )], U Q D x 1U G then the heat transfer equation becomes

2

:

k D

mT T D ; S ( T 9 T ), D:k c mU D x 2 U Q D ND D x x

(324)

(325)

and when the porosity is constant, the equation becomes

T T D:k D ; S ( T 9 T )/m. c U ND D x D x x 2 U Q D

(326)

Most work uses an equation of this type. The experiments carried out will reﬂect the use of Eq. (326), and the data reduction will lead to a correlation for S that is only valid for the particular medium used in the experi2 U ment. There will be no generality in the results. By redeﬁning , further 2 medium characteristics can be incorporated into the correlation. The second model incorporates velocity and temperature ﬂuctuations.

v. s. travkin and i. catton

88

b. Model 2 of Heat Transfer Coefﬁcient in Porous Media: With Nonlinear Fluctuations If we deﬁne the heat transfer coefﬁcient in a way that includes the ﬂuctuations,

2

:

k D

T D · ds ; c (m 9u T ))/[S ( T 9 T )], ND D D D U Q Q D D x x 1U G (327)

the second heat transfer model in porous media is almost the same as the ﬁrst,

T mT D:k D ; S ( T 9 T ). c mU ND D D x 2 U Q Q D D x x

(328)

The third model is obtained by using the complete energy equation for the ﬂuid phase. This is again done by redeﬁnition of the heat transfer coefﬁcient. c. Model 3 of Heat Transfer Coefﬁcient in Porous Media: Full Equation Energy Equation : 2 k D

T D · ds ; c (m 9u T ) ; ND D x D x x 1U G S ( T 9 T

U Q D

k D

1U

T ds D

(329)

The energy equation is again very similar:

T mT D:k D ; S ( T 9 T ). c mU ND D D x 2 U Q Q D D x x

(330)

Each of the models reﬂects the data obtained for a given medium. Only the coefﬁcient , however, allows for a complete representation of the par2 ameters that reﬂect the characteristics of the medium. In attempts by some researchers to improve the modeling, a more complete equation is used along with the more conventional deﬁnitions of the heat transfer coefﬁcient. The relative inaccuracy of substitution of coefﬁcient into the correct mathematical model, T D;c c mU (m u T ) ND D ND D x D x

mT k D :k ; D x x x

1U

T ds ; S ( T 9 T ), D 2 U Q D

(331)

volume averaging theory

89

can easily be seen by comparison with the deﬁnition of . The additional 2 terms are already a part of the coefﬁcient, and double accounting has occurred. The seriousness of such a mistake depends on the problem. To summarize, the heat transfer coefﬁcients and their respectively ﬂuid heat transport equations can be written in terms of the notation given by Eq. (321), : (T ME )/[S ( T 9 T )], (332) 2 D U Q Q D D T k D · ds [S ( T 9 T )], D U Q D x 1U G : (T ME ; T MC )/[S ( T 9 T )], (333) 2 D D U Q Q D D T k D · ds ; c D (m 9u T ) [S ( T 9 T )], ND D x D D U Q Q D D x 1U G : (T ME ; T MC ; T MD )/[S ( T 9 T )], (334) 2 D D D U Q D k T k D D · ds ; c D (m 9u T ) ; T ds ND D x D D D x x 1U 1U G . S ( T 9 T ) U Q D Substitution of either of the preceding effective coefﬁcients into the equation

T mT D:k D ;c c mU (m 9T u ) ND D D x ND D x D G D x x k D x

T D · ds, (335) k D x 1U 1U G T C : T D ; T MC ; T MC ; T ME , D D D D D would result having different models for experimental data reduction and even for experimental setup. ;

1 T ds ; D

2. Simulation Procedures Kar and Dybbs [184] developed several correlations for the internal heat transfer in different porous media. Their model for assessment of internal surface heat transfer coefﬁcient is based on the formula (constructed slightly differently than done by Kar and Dybbs [184] but with all the features) U S (c T 9 c T ) N D , : D AP N D 2\)" S ( T 9 T ) U D Q D

(336)

v. s. travkin and i. catton

90

which accounts for the heat exchange when T and T are the temperaD D tures of ﬂuid exiting and entering the control volume, which is taken to be equal to , through cross ﬂow surface area S [m] with mass ﬂow rate D AP M : U S [kg/s]. This deﬁnition of heat transfer coefﬁcient corresponds D AP to the continuum mathematical model of heat exchange in the porous medium formulated as m( c ) U T : S ( T 9 T ), N D G D 2\)" U Q D instead of the correct equation,

(337)

m( c ) U T : (c ) · 9T u ; k

(mT ) D D N D G D ND D G D 1 k ;k · T ds ; D

T · ds. (338) D D D 1U 1U The last term can be modeled using the heat transfer coefﬁcient given by

T · ds : S ( T 9 T ), 2 U D x 1U G which results from the closure relationship k D

(339)

T 1 T k · ds : 9 k ds · n D D x n 1U 1U G 1 : q · ds : S ( T 9 T ). (340) 2 U D 1U Kar and Dybbs measured the temperatures T and T and treated them as Q D if they were the mean (averaged) temperatures. As a result, they measured yet another heat transfer coefﬁcient, , that is deﬁned by 2 S ( T 9 T ) : S ( T 9 T ) 2 U Q D 2\)" U Q D : (c ) · 9T u ; k

(mT ) D N D D G D D 1 k ;k · T ds ; D

T · ds. (341) D D D 1U 1U The second and third terms in Eq. (341) are usually negligible. When they are, the measured heat transfer coefﬁcient reduces to the second heat transfer coefﬁcient in porous medium , 2 S ( T 9 T ) : S ( T 9 T ) 2 U Q D 2\)" U Q D k : (c ) 9T u ; D

T · ds. (342) N D D G D D 1U 1

volume averaging theory

91

Fig. 6. Internal effective heat transfer coefﬁcient in porous media, reduced based on VAT scale transformations in experiments by 1, Kar and Dybbs [184] for laminar regime; 2, Rajkumar [185]; 3, Achenbach [186]; 4, Younis and Viskanta [187]; 5, Galitseysky and Moshaev [189]; 6, Kokorev et al. [190]; 7, Gortyshov et al. [175]; 8, Kays and London [172]; 9, Heat Exchangers Design Handbook [191].

This is probably why the correlation developed by Kar and Dybbs [184] is located low among the second group of correlations in Fig. 6, where a number of correlations are presented after being rescaled using VAT. If the measured coefﬁcient is , the result will be even lower than . 2 2 As the number of terms that can be estimated increases, the value of the coefﬁcient decreases. This is probably the case with the ﬁrst group of correlations shown in Fig. 6. A large amount of the data analyzed by Viskanta [182, 183] was used to deduce consistent correlations for comparison of internal porous media heat transfer characteristics. The same scaling VAT approach used for ﬂow resistance in porous media is used for heat transfer.

v. s. travkin and i. catton

92

One of the correlations developed by Kar and Dybbs [184], correlation (11) on p. 86, is for laminar ﬂow in sintered powder metal specimens. It is h d (343) Nu : Q F : 0.004 Re Pr, F F D where both Nu and Re are based on the mean pore diameter. If a single hydraulic diameter d is F 4m , (344) d 5d : F NMP S U then 4U m Re : Re : F NMP S U

(345)

h d Nu (Re ) : Q NMP 5 Nu (Re , m, S ) : 0.004 Re Pr. (346) NMP NMP NMP F F U D This correlation is shown in Fig. 6. The correlation developed by Rajkumar [185] for hollow ceramic spheres is

h d d , (347) Nu : Q N : 1.1 Re Pr N N N L D with d : 2.5—3.5 [10\m], 18 & Re & 980, m : 0.38—0.39, Pr : 0.71, N N and u! d Re : N . N The particle Reynolds number Re can be rewritten using N 3(1 9 m) Re : Re . N NMP 2m

(348)

Nu needs to be transformed to Nu by relating the particle diameter d to N NMP N the hydraulic diameter. The result is Nu

2m 2m h d Nu (Re ) : Nu (x), 5 Q F : Nu : F 3(1 9 m) N N N NMP 3(1 9 m) D (349)

where x:

3(1 9 m) Re . NMP 2m

volume averaging theory

93

Then 2m Nu (x, Pr, d , L ) Nu : N N NMP 3(1 9 m) :

2m Nu N 3(1 9 m)

3(1 9 m) Re , Pr, d , L . NMP N 2m

(350)

Achenbach [186] developed the correlation

Re F Nu : (1.18 Re ) ; 0.23 , F F m

(351)

for Pr : 0.71, m : 0.387, and 1 & (Re /m) & 7.7;10. The Reynolds F number used by Achenbach is based on hydraulics and Re 5 Re m, F NMP and his deﬁnition of Nu is F Nu (Re ) 5 Nu (Re m). (352) NMP NMP F NMP A correlation developed for cellular consolidated ceramics by Younis and Viskanta [187, 188] is

d h d Nu : T F : 0.0098 ; 0.11 F Re Pr, (353) TF F L D where m : 0.83—0.87. The correlation yields an increasing Nu when the TF test specimen thickness is decreased. This is a clear inﬂuence of inﬂow and outﬂow boundaries on heat transfer. Transforming from a volumetric Nusselt number Nu to a conventional surﬁcial value Nu yields T Nu (Re m) NMP Nu : TF . (354) NMP 4m Viskanta [183] presents a correlation from a study of low porosity media, 0.167 & m & 0.354, by Galitseysky and Moshaev [189]: Nu : Am(1 9 m) Re Pr. TF F The coefﬁcient, A given by Viscanta [183] is

A : 37.2

d F 9 0.59 (m(1 9 m)) , L

(355)

(356)

for 0.15 & d /L & 0.23, 10 & Re & 530, Pr : 0.71. The volumetric Nusselt F F number is transformed to the surﬁcial Nusselt number with Eq. (354).

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94

A semiempirical theory was used by Kokorev et al. [190] to develop a correlation between resistance coefﬁcient and heat transfer coefﬁcient for extensive ﬂow regimes in porous media that only contains one empirical (apparently universal for the turbulent regime) constant. On the basis of this relationship, the concept of ﬂuctuation speed scale of movement is used to obtain an expression for the heat transfer coefﬁcient from the Darcy friction factor, f : 4 f : 4c : " D B h d (357) Nu : Q N : [0.14(4c Re) Pr]. B F N D Transforming their expression to the general form of the media Nusselt number yields Nu

NMP

:

2m Nu (Re m). N NMP 3(1 9 m)

(358)

The heat transfer coefﬁcient given in the Heat Exchanger Design Handbook [191] is based on a single sphere heat transfer coefﬁcient for the porous medium, h : D ( f Nu ), Nu : 2 ; (Nu ; Nu ), J 2 Q d R Q Q N

(359)

where Nu : 0.664Re Pr N J (0.037Re Pr) N , Nu : 2 (1 ; 2.443Re\ (Pr 9 1)) N for 1 & Re & 10, 0.6 & Pr & 10, and the form coefﬁcient for 0.26 N & m & 1.0 is f : 1 ; 1.5(1 9 m). R Transformation of the Nusselt number yields 2m Nu : Nu (Re ). NMP 3(1 9 m) N N

(360)

Nu values at low Reynolds number are unrealistic, leading to the NMP conclusion that the transition type expression used to treat both laminar and turbulent ﬂows is probably not adequate for heat transfer in porous media. Gortyshov et al. [175] developed a correlation for the internal heat transfer coefﬁcient for a highly porous metallic cellular (foamy) medium

volume averaging theory

95

with porosity in the range 0.87 & m & 0.97, h d Nu : T F : 0.606Pe m\ , F TF D

(361)

where U md F, (362) a D d is in millimeters (see (294)), and Nu is the volumetric internal heat F TF transfer coefﬁcient assessed using Pe : Re Pr : F F

S h q U : S . T T: U h q Q Q

(363)

Also, Nu (Re ) : NMP NMP

Nu (m Re ) TF NMP . 4m

(364)

The correlation given by Kays and London [172] is StPr : 1.4Re\ , NMP

(365)

which is transformed by NuPr : 1.4Re\ , $ Nu : 1.4Re Pr. (366) NMP NMP NMP Re Pr NMP Some useful observations can be made by comparing the heat transfer relationships shown in Fig. 6. One of the most signiﬁcant observations is that the large differences between the correlations by Kar and Dybbs [184], Younis and Viskanta [187, 188], Rajkumar [185], and others cannot be explained if one does not take into account the speciﬁc details of the medium and the experimental data treatment. Given this, the remarkable agreement, almost coincidence, of the correlations by Kays and London [172], Achenbach [186], and Kokorev et al. [190] should be noted. These correlations were developed using different techniques and basic approaches. The correlation given in the Heat Exchangers Design Handbook [191] reﬂects careful adjustment in the low Reynolds number range. The correlation is not based on a speciﬁc type of medium (for example, a globular morphology with a speciﬁc globular diameter). Rather, it was developed to summarize heat transfer coefﬁcient data in packed beds for a wide range of Reynolds numbers using an assigned globular diameter. As a result, it is not solidly based on physics, and a simple transformation from, particle to pore scale does not work properly.

96

v. s. travkin and i. catton VIII. Thermal Conductivity Measurement in a Two-Phase Medium

A majority of thermal conduction experiments are based on a constant heat ﬂux through the experimental specimen and measurement of interface temperatures. Data reduction (see, for example, Uher [192]) is accomplished using K:

QL , AT

(367)

where Q is the electrical power from heater dissipated through the specimen, L is the distance used to measure the temperature difference, and A is the uniform cross-sectional area of the sample. 1. Traditional L ocal and Piecewise Distributed Coefﬁcient Heat Conductivity Problem Formulations In DMM-DNM as, for example, for a dielectric medium, the equation usually used is

· (k(r) T (r)) : 0, r + ,

(368)

where the conductivity coefﬁcient function k is k(r) : k )(r) ; k )(r), (369) and )G is the characteristic function of phase i : 1 ^ 2 (see, for example, Cheng and Torquato [193]). Interface boundary conditions assumed for these equalities are T (r) : T (r), r + S k (n · T (r)) : k (n · T (r)), r + S .

(370) (371)

2. Effective Coefﬁcients Modeling To begin, we choose the conductivity problem and ﬁrst will be treating the example of constant phase conductivity coefﬁcient conventional equations (368) for the heterogenous medium. As shown elsewhere (see, for example, Travkin and Catton [21]), this mathematical statement is incorrect when the equation is applied to the volume containing both phases, even when coefﬁcient k(r) is taken as a random scalar or tensorial function. The reason for this is incorrect averaging over the medium, which has discontinuities.

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97

Conventional theories of treatment of this problem do not specify the meaning of the ﬁeld T, assuming that it is the local variable, or 9T : T (r), where at the point r the point value of potential T exists. Next, the analysis shows that the coefﬁcient k : k(r), as long as in each separate lower scale level point r there exists the local k with the value of either phase 1 or phase 2, and in each of the phases the value of k is G constant. In the DMM-DNM approaches the mathematical statement usually deals with the local ﬁelds, and as soon as the boundary conditions are taken in some way, the problem became formulated correctly and can be solved exactly, as in work by Cheng and Torquato [193]. Difﬁculties arise when the result of this solution needs to be interpreted — and this is in the majority of problem statements in heterogeneous media, in terms of nonlocal ﬁelds, but averaged in some way. The averaging procedure usually is stated as being done either by stochastic or by spatial, volumetric integration. Almost all of these averaging developments are done incorrectly because of a disregard of averaging theorems for differential operators in a heterogeneous medium. More analysis of this matter is given in work by Travkin et al. [115]. Further, a more complicated situation arises when the intention is to formulate and ﬁnd effective transport coefﬁcients in a heterogeneous medium. Let us consider the conductivity problem in a two-phase medium. According to most accepted mathematical statements this problem is given as (368)—(371). 3. Conventional Formulation of the Effective Conductivity Problem in a Two-Phase Medium One of the methods of closure of mathematical models of diffusion processes in a heterogeneous medium is the quasihomogeneous method (Travkin and Catton [21]). In this case, the transfer process is modeled as an ideal continuum with homogeneous effective transport characteristics instead of the real heterogeneous characteristics of a porous medium. This method of closure of the diffusive terms in the heat and mass diffusion equations results in certain limitations: (a) the two-phase medium components are without ﬂuctuations of the type T , c in each of the phases; and (b) the transfer coefﬁcients being constant in each of the phases (Khoroshun [194, 195]) results in reducing them to additional algebraic equations. These equations relate the unknown averaged diffusion ﬂows in each of the phases in the form j ; j : 9k* T , CDD D Q

(372)

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v. s. travkin and i. catton

when for constant (effective) coefﬁcients it is 9kD T 9 kQ T : 9k* T , CDD CDD CDD D Q

(373)

and also T : T ; T , D Q so it might be written as

(374)

(kD )\ j ; (kQ )\ j : 9 T . (375) CDD CDD D Q Here kD , kQ are the transfer coefﬁcient tensors in each of the phases, and CDD CDD k* is the effective conductivity coefﬁcient. Thus, at least in this case, the CDD problem of closure has been reduced to ﬁnding k* . CDD Applying the closure relation, for example, kD T : kQ T , CDD CDD D Q yields the effective stagnant coefﬁcient

(376)

2kD kQ CDD CDD , k* : (377) CDD (kD ; kQ ) CDD CDD which represents the lower bound of the effective stagnant conductivity for a two-phase material from the known boundaries of Hashin—Shtrikman (see, for example, [196], Kudinov and Moizhes [197]) for equal volume fraction of phases. Other closure equations for calculating the stagnant effective conductivity are found in work by Hadley [198] and by Kudinov and Moizhes [197]. The quasi homogeneous approach has several defects: (a) The basis for the quasi-homogeneous equations is in question, (b) the local ﬂuctuation values, as well as inhomogeneity and dispersivity of the medium, are neglected, and (c) the interdependence of the correlated coefﬁcients and arbitrary adjustment to ﬁt data signiﬁcantly reduce the generality of the results. 4. VAT-Based Considerations for Heterogeneous Media Heat Conductivity Experimental Data Reduction Let us consider the data reduction procedure of the heterogeneous material thermal conductivity experiment. a. Constant Heat Conductivity Coefﬁcient We treat the example of the constant coefﬁcient heat transfer equation for a heterogeneous medium and show the problem in terms of conventional experimental bulk data reduction procedures and pertinent modeling equations.

volume averaging theory

99

Consider an experiment on determining the thermal coefﬁcient of phase 1 (for example) in composite (or in material that is considered as being a pure substance, but really is composite) material. The heat transport for material phase 1 is described by

1 T : k (s T ) ; k · s (c ) N t

k T ds ;

T · ds , 1 1 which needs the closure of the second and the third r.h.s. terms. The latter is

T · ds : S ( T 9 T ), (378) x 1 G where the closure procedure is quite applicable to description of the ﬂuid—solid medium heat exchange and might be considered as the analogs for the case of solid—solid heat exchange, as done in many papers. The more strict and precise integration of the heat ﬂux over the interface surface gives the exact closure for that term in governing equations for both neighboring phases. Also considering the two terms on the r.h.s., having them as diffusion bulk terms means that k

1 k (s T ) ; k · where the ‘‘diffusive’’-like ﬂux q conventionally considered, q

T ds : · [9q ],

1 contains some more terms than are

k : 9k

(s T ) : 9k (s T ) 9 CDD

T ds ,

(379)

1 where the heat ﬂux in phase 1 is determined through the averaged temperature T . So, the effective (not homogeneous) conductivity coefﬁcient in phase 1 is

1 k : k (s T ) ; CDD 1 :k 1;

1

T ds ( (s T ))\

T ds ( (s T )) . (380) 1 There is a difference between this introduced coefﬁcient k and that CDD traditionally determined through the ﬂux in phase 1, which is q

1 : [9k T ] : 9k (s T ) ; C

1

T ds .

(381)

v. s. travkin and i. catton

100

Arising in this situation is the effective conductivity coefﬁcient determination

1 k : k (s T ) ; C

1

T ds

T

:k , (382) which is a different variable indeed and which is still the one that is not the traditional effective heterogeneous medium heat conductivity coefﬁcient (determined in all phases), : [9k T ] : 9k [ T ; T ] CDD CDD : 9k (s T ; s T ) : 9k T . (383) CDD CDD After those transformations the heat transfer equation in phase 1 becomes q

T : · [k s ( c )

(s T )] ; S ( T 9 T ). CDD N t

(384)

Repeating all of this for the steady-state heat conductivity equation

1

(s T ) ; · one obtains

1

1 T ds ;

1 k : (s T ) ; CDD for the equation

1

1

T · ds : 0,

T ds ( (s T ))\

(385)

(386)

· [k

(s T )] ; S ( T 9 T ) : 0, (387) CDD k where k does not even depend explicitly on the phase heat conductivity CDD coefﬁcient k (if the latter is taken as a constant value). Generally speaking, it depends on k implicitly through the boundary conditions and the conditions at the interface surface S . Of course, the situation changes if the heat exchange term (last term in (385)) is taken into account as the input correlation factor for conventional bulk effective heat conductivity coefﬁcient k in the equation CDD

· [k

(s T )] : 0. (388) CDD The main reason why in the present problem treatment the interphase heat exchange term is separated from the other two terms in the r.h.s. of Eq. (385) is that this logistics gives clarity in analysis and modeling of interface

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101

transport processes, which is not present in conventional composite medium modeling. Also, in the more complete and challenging physics of interface transport modeling as in the third phase, this third interphase exchange term, along with the second term, is an issue tightly connected to the closure problem and to the models of interface surface transport. b. Nonlinear Heat Conductivity of a Pure Phase Material Meanwhile, for materials such as high-temperature superconductors (HTSC), a constant heat conductivity coefﬁcient is not a justiﬁable choice, as the usual analysis of approaches has shown above. That means complications in treating the equation with a nonlinear heat conductivity coefﬁcient in phase 1, T

: · [ K (s T )] ; · [s K T ] s (c ) N t ; ·

K

1

1 T ds ;

T · ds ; s S , K 2 x 1 G (389)

where the effective conductivity model has two additional terms, one of which reﬂects the mean surface temperature over the interface surface inside of the REV, and the other of which results from nonlinearity of the ﬁelds inside subvolume ,

K : K (s T ) ; s K T

CDD ;

K

T ds ( (s T ))\,

1 which when inserted in the heat transport equation gives

(390)

T : · [K s (c )

(s T )] ; S ( T

CDD N t 9 T ) ; s S . (391) 2 Meanwhile, when an experimentalist evaluates his or her experimental data using the equation T : · [k T ] (c ) N t

(393)

with the calculation shown earlier of the thermal conductivity coefﬁcient using experimental data, he or she makes two mistakes:

v. s. travkin and i. catton

102

1. He is confusing the material’s clear homogeneous conductivity coefﬁcient k (which is the subject of his experiment) with the effective coefﬁcient k of the same phase in a composite — which is just another variable. CDD 2. Doing data reduction as for the modeling equation T : · [k

T ] (c ) CVN N t

(392)

meaning that 5k , (394) CDD and seriously believing that he measures the real homogeneous k he seeks, he drops out (but in reality he takes implicitly into account) the term reﬂecting the exchange rate, k

S ( T 9 T ), (395) in the composite material, which is experiencing at least two temperatures and usually a great inﬂuence of the internal exchange rate (see work by Travkin and Kushch [33, 34] and Travkin et al. [21]). In this way, an experimentalist makes a second mistake due to miscalculation of the inﬂuence of this additional term — yet the conductivity coefﬁcient k evaluated from experiment is not the value it is considered to be — CVN k 5 / k . CVN When the experimentalist’s goal is the measurement, not of a bulk effective coefﬁcient of a material, but of the pure material’s conductivity coefﬁcient, considerations regarding the issues of homogeneity and experimental data modeling are of primary interest. The standard deﬁnition of the effective (macroscopic) conductivity tensor is determined from j : 9k* T , GH

(396)

in which it is assumed that j : j ; j : 9k T 9 k T : 9k* T : 9k* T GH GH : 9k* [ T ; T ] : 9k* T 9 k* T , (397) GH GH GH so, for the usually assumed interface S physics, the effective coefﬁcient is determined to be k* T : [k T ; k T ] GH 1 : k (m T ) ; k (m T ) ; (k 9 k )

1

T ds (398)

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103

or

1 k* : k (m T ) ; k (m T ) ; (k 9 k ) GH

1

T ds T \, (399)

or

1 k (m T ) ; k (m T ) ; (k 9 k ) k* : GH [ T ; T ]

1

T ds

, (400)

which involves knowledge of three different functions, T , T , T S , in the volume . This formula for the steady-state effective conductivity can be shown to be equal to the known expression 1 k* T : k T ; (k 9 k ) GH

T d : k T ; (k 9 k ) T . (401) It is worth noting here that the known formulae for the effective heat conductivity (or dielectric permittivity) of the layered medium k* : . m k , i : 1, 2, C G G G for a ﬁeld applied parallel to the interface of layers, and

(402)

m \ G k* : . (403) C k G G when the heat ﬂux is perpendicular to the interface, are easily derived from the general expression (399) using assumptions that intraphase ﬁelds are equal, T : T , that interface boundary conditions are valid for averaged ﬁelds, and that adjoining surface interface temperatures are close in magnitude. The same assumptions are effectual when conventional volume averaging techniques are applied toward the derivation of formulae (402) and (403). 5. Bulk Heat Conductivity Coefﬁcients of a Composite Material The problem becomes no easier in the case when the effective conductivity coefﬁcient is meant to serve for the whole composite material. Combining

v. s. travkin and i. catton

104

both temperature equations (if only two phases are present) for the simplest case of constant coefﬁcients,

1 T : k (s T ) ; k · s (c ) N t T 1 : k (s T );k · s (c ) N t

1

k T ds ;

k T ds ;

1 into one equation by adding one to another, we obtain

T · ds

1

1

T · ds ,

T T ; s (c ) : · (k (s T ) ; k (s T )) s (c ) N t N t ; ·

k

k ;

1

1

k T ds ;

k

T · ds ;

1

1

T ds

T · ds , (404)

keeping in mind that the two-phase averaged temperature is T : s T ; s T . (405) One can write down the mixture temperature equation when summation of the equations gives (when taking into account the boundary condition of temperature ﬂuxes equality at the interface surface, (k T ) : (k T )) T T ; s (c ) s (c ) N t N t

1 : · (k (s T ) ; k (s T )) ; (k 9 k ) ·

or, written in terms of thermal diffusivities a and a , T 1 : · [a (s T ) ; a (s T )] ; (a 9 a ) · t

1

1

T ds , (406)

T ds

a k 1 k ; 19

T · ds , a : G , i : 1, 2, (407) G (c ) a k 1 N G which has the three different temperatures 9T , T , and T (S ) (here T : s T ; s T ).

volume averaging theory

105

And, assuming only a local thermal equilibrium, T : s T ; s T : T * : T : T , (408) the mixed temperature equation becomes two-temperature T *, T (S ) dependable with simpliﬁed left hand part of the equation T * : · [(k (s T *) ; k (s T *))] (s (c ) ; s (c ) ) N N t

1 ;(k 9 k ) ·

T ds . (409) 1 With the two different temperatures, the effective coefﬁcient of conductivity is equal to

k* : [(k (s T *) ; k (s T *))] CDD

1 ; (k 9 k )

T ds ( T *)\. (410) 1 This formula coincides with the effective coefﬁcient of conductivity for the steady-state effective conductivity in the medium and can be shown to be equal to the known expression

1 k* T : k T ; (k 9 k ) CDD

T d.

(411)

From this formula an important conclusion can be drawn: that the most sought-after characteristics in heterogeneous media transport, which are the effective transport coefﬁcients, can be correctly determined using the conventional deﬁnition for the effective conductivity — for example, for the steady-state problem 1 9 j : k* T : k T ; (k 9 k ) CDD

T d, (412) but only in a fraction of problems, while employing the DMM-DNM exact solution. The issue is that in a majority of problems, such as for inhomogeneous, nonlinear coefﬁcients and in many transient problems, having the two-ﬁeld DMM-DNM exact solution is not enough to ﬁnd effective coefﬁcients. As shown earlier, only the requirement of thermal equilibrium warrants the equality of steady-state and transient effective conductivities in a two-phase medium.

106

v. s. travkin and i. catton

The second form of the same equation with the surface integral of the ﬂuctuation temperature in phase 1 is T * : · [(k s ; k s ) (T *)] (s ( c ) ; s (c ) ) N N t

1 ; (k 9 k ) ·

1

T ds ,

(413)

still having the phase 1 temperature ﬂuctuation variable in one of the terms. The following equality arises while comparing the two last equations (409) and (413):

1 [(k (s T *) ; k (s T *))] ; (k 9 k )

1

1 : [(k s ; k s ) (T *)] ; (k 9 k )

T ds

T ds .

(414)

1 As can be seen, the transient effective diffusivity coefﬁcent a° in the VAT CDD nonequilibrium two-temperature equation (407) can be derived through the equality

1 a° T : a (s T ) ; a (s T ) ; (a 9 a ) CDD

a ; \ a 1 9 a

k k

1

1

T ds

T · ds

1

(415)

or

1 a° T : a (s T ) ; a (s T ) ; (a 9 a ) CDD

1

T ds ; A, (416)

where \ is the inverse operator 9 · ( \( f )) : f such that if

a

·A:a 19 a

k k

1

1

T · ds ,

(417)

then

A : \ a

a 19 a

k k

1

1

T · ds

.

(418)

volume averaging theory

107

From the preceding expression, the transient effective nonequilibrium coefﬁcient in a two-phase medium can be determined as k° : a° (s (c ) ; s (c ) ), CDD CDD N N

(419)

which looks rather inconvenient for analytical or experimental assessment or numerical calculation. The solution of this problem, which includes as an imperative part the ﬁnding of the effective bulk composite material heat conductivity (diffusivity), coefﬁcient, is equal to the solution of the exact two-phase problem. We see that the two-temperature DMM-DNM is not enough for the convenient construction of the effective coefﬁcient of conductivity. As we can compare the expressions for transient coefﬁcient (419) and thermal equilibrium coefﬁcient (410) they are of great difference in deﬁnition and in calculation. And it does not matter which kind of mathematical statement is used for the problem — the two separate heat transfer equations or the VAT statement — the problem complexity is the same. Only by using the VAT equations is the correct estimation of the transient effective coefﬁcients on the upper scale available. If we adopt the idea that phase temperature variables in each of the subvolumes and can be presented as sums of the overall tempera ture and local ﬂuctuations (Nozad et al. [40]),

T : T ; T , T : T ; T ,

(420)

which means an introduction of the two new variables T and T , then the equation for the composite averaged temperature follows (Nozad et al. [40]) in the form (s (c ) ; s (c ) ) N N

T 1 : · s k T ; T ds t 1 1 ; s k T ; T ds 1 T T ; s (c ) 9 s (c ) N t N t

9 · (s k T ; s k T )

(421) which has ﬁve variable temperatures. If the assumptions and constraints given in Nozad et al. [40] are all satisﬁed, then the ﬁnal equation with only

108

v. s. travkin and i. catton

three different temperatures resumes: (s (c ) ; s (c ) ) N N

1 T T ds : · s k T ; t 1 1 ; s k T ; T ds . 1 (422)

This means that the neglect of the global deviation T , T terms still does not remove the requirement of a two-temperature solution. a. Effective Conductivity Coefﬁcients in a Porous Medium When Phase One Is a Fluid In phase 1 the VAT equation is written for the laminar regime. In the work by Kuwahara and Nakayama [199] is given the DMM-DNM solution of the 2D problem of uniformly located quadratic rods with equal spacing in both directions. Studies were undertaken of both the Forchheimer and post-Forchheimer ﬂow regimes. This work is a good example of how DMM-DNM goals cannot be accomplished, even if the solution on the microlevel is obtained completely, if the proper VAT scaling procedures basics are not applied. The one structural unit — periodic cell in the medium — was taken for DMM-DNM. Equations were taken with constant coefﬁcients, and in phase 1 the VAT equation was written for the laminar regime as T D ; m(c ) U T : (c ) · 9T u ; k

(mT ) m(c ) ND G D ND D G D D D N D t

1 ;k · D

k T ds ; D D

T · ds. D

(423)

1U 1U Adding this equation to the VAT solid-phase (second phase) twotemperature equation gives T T D ; s (c ) ; m(c ) U T m(c ) N D t N t N D G D : (c ) · 9T u ; · (k (mT ) ; k (s T )) N D D G D D D k k ; · D T ds ; T ds D 1 1 k k ; D

T · ds ;

T · ds , (424) D 1 1

volume averaging theory

109

which reduces because of interface ﬂux equality to T T ; m(c ) U T D ; s (c ) m(c ) N t N D G D N D t : · (k (mT ) ; k (s T )) ; (c ) · 9T u D D N D D G D 1 ; (k 9 k ) · T ds , (425) D D 1 which has two averaged temperatures T and T , interface surface integrated D temperature T (S ), and two ﬁelds of ﬂuctuations T (x) and u (x), D D G assuming that the velocity ﬁeld is also computed and known. We now write the effective conductivity coefﬁcients for (425) and for the one-temperature equation when temperature equilibrium is assumed. In the ﬁrst case, for the weighted temperature,

T U : (m(c ) T ; s (c ) T )/w N D D N 2 w : m(c ) ; s (c ) : const, N 2 N D the equation can be written as w 2

(426) (427)

T U ; m(c ) U T N D G D t

: · (k (mT ) ; k (s T )) ; (c ) · 9T u D D N D D G D 1 ; (k 9 k ) · T ds , (428) D D 1 where three temperatures are unknown, T U, T , and T , plus the interface D surface temperature integral T (S ) and ﬂuctuation ﬁelds T (x) and u (x). D D G The effective coefﬁcient of conductivity can be looked for is

k° T U : (k (mT ) ; k (s T )) ; (c ) 9T u CDD D D N D D G D 1 ; (k 9 k ) T ds . (429) D D 1 In order to avoid the complicated problems with effective conductivity coefﬁcient deﬁnition in a multitemperature environment, Kuwahara and Kakayama [199], while performing DMM-DNM for the problem of laminar regime transport in a porous medium, decided to justify the local thermal equilibrium condition

T : mT ; s T : T * : T : T , D D which greatly changes the one effective temperature equation. This equation

110

v. s. travkin and i. catton

becomes simpler with only one unknown temperature T * and variable ﬁeld T and is written as D T * ; m(c ) U T * (m(c ) ; s (c ) ) N D G N D N t : · (k (mT *) ; k (s T *)) ; (c ) 9T u D N D D G D 1 T ds , ; (k 9 k ) · (430) D D 1 as the variable temperature and velocity ﬂuctuation ﬁelds T and u should D G be known, although this is a problem. As long as the deﬁnition of the effective conductivity coefﬁcient is

k* T * : k (mT *) ; k (s T *) ; (c ) 9T u CDD D ND D G D 1 ; (k 9 k ) T ds , (431) D D 1 then the effective conductivity can be calculated subject to known T *, T , D T , and u . At the same time, the important issue is that in DMM-DNM the D G assumption of thermal equilibrium has no sense at all — as long as the problem have been already calculated as the two-temperature problem. To further perform the correct estimation or calculation of effective characteristics, one needs to know what are those characteristics in terms of deﬁnition and mathematical description or model? This is the one more place where the DMM-DNM as it is performed now is in trouble if it does not comply with the same hierarchical theory derivations and conclusions as the VAT (see also the studies by Travkin et al. [115] and Travkin and Catton [114, 21]). As shown earlier, only the requirement of thermal equilibrium warrants the equality of steady-state and transient effective conductivities in a two-phase medium. Consequently, if taken correctly, the two-temperature model will introduce more trouble in treatment and even interpretation of the needed bulk, averaged temperature (as long as this problem is already known to exist and is treated in nonlinear and temperature-dependent situations) and the corresponding effective conductivity coefﬁcient (or coefﬁcients).

1. Thus, comparing the two effective conductivity coefﬁcients (429) and (431), one can assess the difference in the second term form and consequently, the value of computed coefﬁcients. Comparing the expressions for one equilibrium temperature and one effective weighted temperature, as well as for their effective conductivity coefﬁcients, one can also observe the great imbalance and inequality in their deﬁnitions and computations.

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2. Summarizing application of DMM-DNM approach by Kuwahara and Nakayama [199], it can be said that it is questionable procedure to make an assumption of equilibrium temperatures when the problem was stated and computed as via DNM for two temperatures. 3. In the calculation of the effective coefﬁcients of conductivity — stagnant thermal conductivity k ; tortuosity molecular diffusion k ; and C RMP thermal dispersion k — Kuwahara and Nakayama [199] used a BGQ questionable procedure for calculation of the two last coefﬁcients. They used one-cell (REV) computation for surface and ﬂuctuation temperatures for periodical morphology of the medium, and at the same time they used the inﬁnite REV deﬁnition for the effective temperature gradient for their calculation (assigned in the problem); see the expressions for calculation of these coefﬁcients, (21)—(24) on p. 413. That action means the mixture of two different scale variables in one expression for effective characteristics — which is incorrect by deﬁnition. If this is used consciously, the fact should be stated on that matter explicitly, because it alters the results.

IX. VAT-Based Compact Heat Exchanger Design and Optimization At the present time, compact heat exchanger (CHE) design is based primarily on utilization of known standard heat exchanger calculation procedures (see, for example, Kays and London [172]). Typical analysis of a heat exchanger design depends on the simple heat balance equations that are widely used in the process equipment industry. Analytically based models are composed for a properly constructed set of formulas for a given spatial design of heat transfer elements that allow, most of the existing heat transfer mechanisms to be accounted for. Analogies between heat transfer and friction have been shown by Churchill [200] and by Churchill and Chan [201] to be inadequate for describing many of the HE conﬁgurations of interest. This has been suspected for some time and will seriously affect the use of the ‘‘j-factor’’ in HE modeling and design. Modeling of a speciﬁc heat exchanger geometry by Tsay and Weinbaum [202] provides a useful preliminary step and a potential benchmark test case. Though the study only considered hydrodynamic effects and restricted itself to consideration of regular media and the creeping ﬂow regime, the effects of morphology-characteristic variation upon momentum transport phenomena were explored. The authors show that the overall bed drag coefﬁcient in the creep ﬂow regime increases dramatically as the innercylinder spacing approaches the order of the channel half-height.

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Analysis of processes in regular and randomly organized heterogeneous media and CHE can be performed in different ways. Some CHE structures have the characteristics of a porous medium and can be studied by application of the developments of porous media modeling. In this work, a theoretical basis for employing heat and momentum transport equations obtained from volume averaging theory (VAT) is developed for modeling and design of heat exchangers. Using different ﬂow regime transport models, equation sets are obtained for momentum transport and two- and threetemperature transfer in nonisotropic heterogeneous CHE media with accounting for interphase exchange and microroughness. The development of new optimization problems based on the VATformulated CHE models using a dual optimization approach is suggested. Dual optimization is the optimization of the morphological parameters (size, morphology of working spaces) and the thermophysical properties (characteristics) of the working solid and liquid materials to maximize heat transfer while minimizing pressure loss. This allows heat exchanger modeling and possible optimization to be based on theoretically correct ﬁeld equations rather than the usual balance equations. The problems of shape optimization traditionally have been addressed in HE design on the basis of general statements that include heat and momentum equations along with their boundary conditions stated on the assigned known volumes and surfaces; see, for example, Bejan and Morega [203]. A. A Short Review of Current Practice in Heat Exchanger Modeling Analysis of heat exchanger designs, as described by Butterworth [204], depends on the heat balance equations that are widely used in the heat design industry. The general form of the thermal design equation for heat exchangers (see, for example, Figs. 7—9) can be written (Butterworth [204]) dQ : dAT, where Q is the heat rate, and A is the transfer surface area. As outlined by Martin [205], the coupled differential equations for a cross ﬂow tube heat exchanger (Fig. 7) modeling are (for simplicity only one row is considered) d/ : / 9 /?T d0 d/ 9 :/ 9/ , d0 where / , / , and /?T are dimensionless ﬁrst and second ﬂuid temperatures 9

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Fig. 7. Three-phase tube heat exchanger unconsolidated morphology.

and the second temperature being averaged over the tube’s row width. As follows from these equations, all information about a given heat exchanger’s peculiarities and design speciﬁcs is included in the dimensionless coordinates A 0 : (Mc ) N G

z G, L G

i : 1, 2,

where is the overall heat transfer coefﬁcient and M is the mass ﬂow rate. Second-order ordinary differential equations are developed for HE as well (see, for example, Paffenbarger [206]).

Fig. 8. CHE morphology with separated subchannels for each of the ﬂuids.

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Fig. 9. Compact heat exchanger (CHE) with contracted-tube layer morphology for one of the ﬂuids.

Webb in a book [207], and in his invited talk at the 10th International Heat Transfer Conference [208], distinguishes four basic approaches to predicting the heat transfer j-factor and the Fanning friction factor f for heat exchanger design. They are (1) power-law correlations; (2) asymptotic correlations; (3) analytically based models; and (4) numerical solutions. Analytically based models are properly constructed set of formulas for a given spatial construction of heat transfer elements that allows most of the existing heat transfer mechanisms to be accounted for. Many examples are given in publications by Webb [207, 208], Bergles [209], and other researchers. The major differences between the measured characteristics of air-cooled heat exchangers with aluminum or copper ﬁnned tubes with large height, small thickness, and narrow-pitch ﬁns, and high-temperature waste heat recovery exchangers with steel ﬁnned tubes with rather low height and thickness and wide-pitched ﬁns, are given in a paper by Fukagawa et al. [210] Despite the fact that morphology of the heat exchange medium is essentially the same, the correlations predicting heat transfer and pressure drop values do not work for both HE types altogether. For this particular heat exchange morphology, a wide-ranging experiment program is needed for different ratios of the morphology parameters. There is, at present, no general approach for describing the dependencies of heat transfer effectiveness or frictional losses for a reasonably wide range of morphological properties and their ratios. The ﬁeld of compact heat exchangers has received special attention during the past several years. A wide variety of plate ﬁn heat exchangers (PFHE) has been developed for applications in heat recovery systems, seawater evaporators, condensers for heat pumps, etc. It is proposed that a theoretical

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Fig. 10. Initial optimization scheme for benchmark tube heat exchanger morphology.

basis for employing heat and momentum transport equations obtained with volume averaging theory be developed for the design of heat exchangers. An assumption of the equilibrium streams is common in HE design (see, for example, Butterworth [204]). Almost all commercial design software assume plug ﬂow with occasional simple corrections to reﬂect deviations from the plug ﬂow. CFD has applications in simpliﬁed situations, when the geometry of the channels or heat transfer surfaces can be described fairly. Butterworth [204] further noted that ‘‘the space outside tubes in heat exchangers presents an enormously complicated geometry’’ and ‘‘modeling these exchangers fully, even with simpliﬁed turbulence models just mentioned is still impracticable.’’ We do not agree with this view and propose to use techniques developed as part of our work to show that practical modeling methods exist. During the past few years considerable attention has been given to the problem of active control of ﬂuid ﬂows. This interest is motivated by a number of potential applications in areas such as control of ﬂow separation, combustion, ﬂuid—structure interaction, and supermaneuverable aircraft. In this direction, Burns et al. [211, 212] developed several computational algorithms for active control design for the Burgers equation, a simple model for convection—diffusion phenomena such as shock waves and trafﬁc ﬂows. Generally, the optimal control problems with partial differential equations (PDE), to which VAT-based HE models convert, can have detailed

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solutions of the linear quadratic regulator problem, including conditions for the convergence of modal approximation schemes. However, for more general optimal control problems involving PDEs, the main approach has been to use some method for constructing a particular ﬁnite-dimensional approximating optimal control problem and then to solve this problem by some method or other (Teo and Wu [213]). It seems that no attention has been given to the optimal control systems governed by the partial integrodifferential equations like volume averaging theory equations for HE design. B. New Kinds of Heat Exchanger Mathematical Models Our earlier work has shown that ﬂow resistance and heat transfer in HEs and CHEs can be treated as highly porous structures and that their behavior can be properly predicted by averaging the transport equations over a representative elementary volume (REV) in the region neighboring the surface. The averaging of processes in regular and randomly organized heterogeneous media and in HE can be performed in different ways. Travkin and Catton [21, 28] discussed alternate forms for the mass, momentum, and heat transport equations recently presented by various researchers. The alternate forms of the transport equations are often quite different. The differences among the transport equation forms advocated by the numerous authors demonstrate the fact that research on the basic form of the governing equations of transport processes in heterogeneous media is still an evolving ﬁeld of study. Derivation of the equations of ﬂow and heat transport for a highly porous medium during the ﬁltration mode is based on the theory of averaging by certain REV of the transfer equation in the liquid phase and transfer equations in the solid phase of the heterogeneous medium (see, for example, Whitaker [42, 10] for laminar regime developments, and Shcherban et al. [15], Primak et al. [14], and Travkin and Catton [16, 21, 23] for turbulent ﬁltration). These models account for the medium morphology characteristics. Using second-order turbulent models, equation sets are obtained for turbulent ﬁltration and two-temperature diffusion in nonisotropic porous media with interphase exchange and micro-roughness. The equations differ from those found in the literature. They were developed using an advanced averaging technique, a hierarchical modeling methodology, and fully turbulent models with Reynolds stresses and ﬂuxes in every pore space. Independent treatment of turbulent energy transport in the ﬂuid phase and energy transport in the solid phase, connected through the speciﬁc surface (the solid—ﬂuid interface in the REV), allows for more accurate modeling of the heat transfer mechanisms between rough surfaces or porous insert of HE and the ﬂuid phases.

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C. VAT-Based Compact Heat Exchanger Modeling For a pin ﬁn (PFHE), with cross-ﬂow morphology, the governing equations can be written in the following form: Momentum equation for the ﬁrst ﬂuid: U u! ; m (K ; ) K ; (9u! u! ) K K D x x x x x D U 1 U 9 · ds (K ; ) : m U x K x 1U G 1 1 p! ds ; ; (m p ! ). (432) x 1U D D momentum equation for the second ﬂuid:

W ; m (K ; ) K z z z

w! K K z

W 1 9 : m W z

D

;

(K

(9w! w! ) D z

W · ds ; ) K x G

1U 1 1 ; p! ds ; (m p! ). z 1U D D Energy equation for the ﬁrst ﬂuid:

c

T T : m U m (K ;k ) ND D x 2 x x

; ;

(433)

T m (K ;k ) 2 z z

x

T K 2 x

;

D

T K 2 z

z

;c (m 9T u! ) ND D x D ;

(K ; k ) 2 x

;

(K ; k ) 2 z

1 ;

(K

1U

T ds

T ds

1U

D

1U

T ; k ) · ds. 2 x G

(434)

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Energy equation for the solid phase:

T Q K Q2 x

T

Q Q ; s K

Q2 Q x x x T Q K Q2 x

;

z

;

K

Q2 Q z

Q

1U

;

K

Q2 Q x

1 T ds ; Q

Q

1U

T

Q Q s K

Q2 Q x z

;

T ds Q

T Q · ds : 0. K Q2 x 1U G

(435)

Energy equation for the second ﬂuid: c

T T : m (K ; k ) m W 2 x ND D z x

;

T m (K ;k ) 2 z z

;

x

T K 2 x

;

D

T K 2 z

z

;c (m 9T w! ) ND D z D ;

(K ; k ) 2 x

;

(K ; k ) 2 z

1 ;

(K

1U

T ds

T ds

1U

D

1U

T ; k ) · ds. 2 x G

(436)

The volumes for averaging in equations are , , , . D D Q A majority of the additional terms in these equations can be treated using closure procedures developed in previous work (see, for example, Travkin and Catton [16, 19]), for selected ﬁn geometries and solid matrices of a HE. Our generic interest, however, is in the theoretical applications of the VAT governing equations and possible advantages gained by introduction of irregular or random morphology into heat exchange volumes and surfaces. Cocurrent parallel ﬂow matrix type CHE morphology can be described using the next VAT-based set of governing equation.

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Momentum equation for the ﬁrst ﬂuid: 1 U 9 m U x

U 1 · ds ; ; ) p! ds x 1U 1 G D U U 1 (m p! ) ; (m (K ; ) :9 K z x x D u! ; K ; ( 9 u! u! ). (438) K x D x x D (K

K

Momentum equation for the second ﬂuid: U 1 9 m U x

U 1 · ds ; ; ) p! ds K x 1U 1U D G 1 U (m p! ) ; (m (K ; ) :9 K x x x D u! ; K ; ( 9 u! u! ). (437) K x D x x D

(K

The corresponding energy equations are like those given earlier. A simple example typiﬁes the general morphology of cocurrent and countercurrent CHEs when widths of the channels are different and the heat transfer enhancing devices are to be determined by shape optimization. For this purpose, consider two conjugate ﬂat channels of different heights that are both ﬁlled with unknown (or assigned) heat transfer elements or porous media. A set of governing equations for each of the channels were developed by Travkin and Catton ([16, 20]). A model of the momentum equation for a horizontally homogeneous stream under steady conditions has the form

U H ; m(K ; v ) KH H z z z

u! H K KH z

;

(9u w! ) H H D z

D U 1 U H · dS H · dS K ; v KH x S H x S U2H U*H G G 1 1 p! DH . 9 p! dS : H x 1UH DH DH 1 ;

(439)

This equation can be further simpliﬁed for turbulent ﬂow in a layer with a

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120

porous ﬁlling or insert that has regular morphology, U (z) H m(z)(K ; v ) ;U (U , S , K ) ; U (U , S , v ) KH H H+2 H U KH H+* H U H z z

1 (m(z)p! ) H , ;U ( p! , S ) : H+DMPK H U x DH where the three morphology-based terms are deﬁned by

(440)

U H · dS (441) KH x S U2H G U 1 H · dS v (442) U (U , S , v ) : H x H+* H U H S U*H G 1 U (p! , S ) : 9 p! dS . (443) H+DMPK H U H 1 DH UH It is obvious that the result is ‘‘controlled’’ by three morphology terms. The equation for the mean turbulent ﬂuctuation energy b(z) is written in the following simple form, which includes the effect of obstacles in the ﬂow and temperature stratiﬁcation across the layer, the z direction: 1 U (U , S , K ) : H+2 H U KH

U d H ; z dz

K

K db (z) f (c )S (z) KH ; v H ; H B UH U H H m dz @ g db(z) b (z) T H H ; 2v :C H 9 K . (444) K KH z T dz ? 2 KH Here, f (c ) is approximately the friction factor for constant and nearly B constant morphology functions, and the mean eddy viscosity is given by K (z) KH

K (z) : Cl(z)b (z), (445) H KH where l(z) is the turbulent scale function deﬁned by the assumed porous medium structure. Similarly, the equation of turbulent heat transfer in the homogeneous porous medium ﬂuid phase is c

T (x, z) T (x, z) m U (z) H : m (K ; k ) H H 2H DH NDH DH H H x z z

;T (T , S , K ) ; T (T , S , k ) H+2OGL H H+*OGL H U 2H U H 1 T H · dS ; k , (446) D x S U*H G

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with two morphology terms that ‘‘control’’ the solution being

1 T (T , S , K ) : H+2OGL H U 2H 1 T (T , S , k ) : H+*OGL H U H

S U2H

T H · dS K 2H x G

(447)

S U*H

T H · dS k . H x G

(448)

In the solid phase of CHE, the energy equation is

T (x, z) (1 9 m)K (z) Q ;T (T , S , K ) : 0, Q+OGL Q Q2 U Q2 z z

(449)

with the one ‘‘control’’ term 1 T (T , S , K ) : Q+OGL Q U Q2

K

1U

T Q · dS , Q2 x G

where dS : 9dS . If we apply the closure procedures described earlier, the equation of motion becomes U (z) H m(z)K (U , b, l ) KH z z

1 : [c (z, U )S (z) ; c (z, U )S (z) ; c (z, U )S (z)]U H H U* B H U2 BN H U. 2 D* ;

1 dp! U 1 dp! H; H D:c S H D, B U 2 dx dx D D

(450)

where K : K ; v , KH KH H and the lumped ﬂow resistance coefﬁcient c is the complex morphology B dependent function. The energy equation in the jth ﬂuid phase is c

T (x, z) T (x, z) H mU (z) H : (m(z)K (z) NDH DH H 2H z z x ; (z)S (z)(T (x, z) 9 T (x, z)), 2 U Q H

(451)

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with (x, z) + , and the energy equation in the solid phase D T (x, z) (1 9 m(z)K (z) Q 12 z z : (z)S (z)(T (x, z) 9 T (x, z))(x, z) + , 2 U Q H

(452)

with P $ 1 : K $ K c ; k , (453) P2 2H KH NDH DH DH where index j determines the ﬂuid phase number j : 1, 2 in conjugate channels 1 and 2. In Eqs. (444), (445), (450), and (452), the coefﬁcient functions and speciﬁc surface functions must be determined by assuming real or invented morphological models of the porous structure. The pressure gradient term in Eq. (450) is modeled as a constant value in the layer, or simulated by the local value of the right-hand side of the experimental correlations. The boundary conditions for these equations are z : 0 : U : 0, H

b H:0 z

T H K : v, Q : 9K K 2H z T 1 Q : 9K 12 z z:

(454)

; U b H:0 h : H : 0, 9 H z z

T T H : 0, 1 : 0, z z

(455)

where h is the half channel width. The control terms in the preceding H equations depend on temperature and velocity distributions as well as on morphological characteristics of the media. Comparing the three latest equation (450)—(452) with the equations derived by Paffenbarger [206] for practically the same structural design of HE, one will ﬁnd numerous discrepancies. For example, the energy balance equations in Paffenbarger’s [206] work have energy conservation terms that do not match each other. The VAT-based general transport equations for a single phase ﬂuid in an HE medium have more integral and differential terms than the homogenized or classical continuum mechanics equations. Various descriptions of the

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porous medium structural morphology determines the importance of these terms and the range of application of the closure schemes. Prescribing regular, assigned, or statistical structure to the capillary or globular HE medium morphology gives the basis for transforming the integrodifferential transport equations into differential equations with probability density functions governing their stochastic coefﬁcients and source terms. Several different closure models for these terms for some uniform, nonuniform, nonisotropic, and speciﬁcally random nonisotropic highly porous layers were developed in work by Travkin and Catton [16, 17, 23], etc. The natural way to close the integral terms in the transfer equations is to attempt to ﬁnd the integrals over the interphase surface, or over outlined areas of this surface. Closure models allow one to ﬁnd connections between experimental correlations for bulk processes and the simulation representation and then incorporate them into numerical procedures. D. Optimal Control Problems in Heat Exchanger Design A variety of the optimization problems that can be formulated in the area of heterogeneous medium transport involve differential equations modeling the physics of the process. Many of them have a fairly complicated form. The contemporary literature on optimal control deals with problems that are mathematically similar but consider much simpler formulations of the optimization problem with constraints in the form of differential equations. Linear optimal control systems governed by parabolic partial differential equations (PPDEs) are relatively well studied. The CHE modeling equations resulting from the VAT-based analysis are also PPDEs, but they are nonlinear and have additional integral and integrodifferential terms. The models presented and the resulting differential equations contain additional integral and integrodifferential terms not studied in the literature. The performance of a heat exchanger depends on the design criteria for optimizing the liquid ﬂow velocity, dimensions of the heat exchanger, the heat transfer area between hot side and cold side, etc. Thermal optimization of an HE requires selection of many features — for example, both the optimum ﬁn spacing and optimum ﬁn thickness, each determined to maximize total heat dissipation for a given added mass or proﬁle area. These criteria set the optimal conditions for HE operation. Theoretically, the optimal dimensions of an HE require a large number of tiny tubelets with diameters tending to zero with increasing number of tubes. This leads to a very ﬁne dispersion problem with porous medium—like behavior. Extremely compact micro heat exchangers with plate—ﬁn cross ﬂow have already been built. However, the optimization problems involving such designs are more complex than traditional designs and require new simulation techniques.

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E. A VAT-Based Optimization Technique for Heat Exchangers A variety of optimal control problems that can be formulated in the area of heterogeneous medium transport involve differential equations modeling the physics of the process. Some of them have a fairly complicated form. Meanwhile, the contemporary literature on optimal control considers too simple formulations of the optimization problems with constraints in form of differential equations. Optimal control systems governed by parabolic partial differential equations have been studied intensively. For example, Ahmed and Teo [214] give a survey on main results in this ﬁeld. Questions concerning necessary conditions for optimality and existence of optimal controls for these problems have been investigated in work by Ahmed and Teo (215—217] and Fleming [218]. Moreover, a few results by Teo et al. (1980) on the computational methods of ﬁnding optimal controls are also available in the literature (Teo and Wu [213]). However, turbulent transport equations in highly porous media were proposed by Travkin et al. [19] for optimization problems and developed in more detail in Section IV with additional ‘‘morphlogical’’ as well as integral and integrodifferential terms. Recent literature studies show optimal control problems involving PPDE either in general form or in divergence form and propose computational methods such as variational technique and gradient method (see, for example, Ahmed and Teo [214]). These studies seems to be helpful for solving various optimization problems involving integro—differential transport equations considered by Travkin et al. [19]. However, complete research has to be done for this class of equations, including analysis of necessary conditions and existence of optimal control, as well as developing computational methods for solving various optimal control problems. Optimal control for some classes of integrodifferential equations has also been considered in recent years. Da Prato and Ichikawa [219] studied the quadratic control problems for integrodifferential equations of parabolic type. A state-space representation of the system is obtained by choosing an appropriate product space. By using the standard method based on the Riccati equation, a unique optimal control over a ﬁnite horizon and under a stabilizability condition is obtained and the quadratic problem over an inﬁnite horizon is solved. Butkovski [220] was the ﬁrst to discuss the optimal control problems for distributed parameter systems. The maximum principle as a set of necessary conditions for optimal control of distributed parameter systems has been studied by many authors. Since it is well known that the maximum principle may be false for distributed parameter systems (see Balakrishnan [221]), there are many papers that give some conditions ensuring that the maximum principle

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125

remains true (see, for example, Ahmed and Teo [214]; Balakrishnan [221]; Curtain and Pritchard [222]). We note that the references just mentioned discuss the cases for distributed parameter systems or functional differential systems with no end constraints and/or with the control domain being convex; thus, they do not include Pontryagin’s original result on maximum principle as a special case. Fattorini [223] also proposed an existence theory and formulated maximum principle for relaxed inﬁnite-dimensional optimal control problems. He considered relaxed optimal control problems described by semilinear systems ODE and used relaxed controls whose values are ﬁnitely additive probability measures. Under suitable conditions, relaxed trajectories coincide with those obtained from differential inclusions. The existence theorems for relaxed controls were obtained; they are applied to distributed parameter systems described by semilinear parabolic and wave equations, as well as a version of Pontryagin’s maximum principle for relaxed optimal control problems. Optimal control problems involving equations such as (432)—(438) have control terms with the structures

(m f (x ) f (x ) ) D

(m f (x ) f (x ) ) D

# (x )

1U

1U

f (x ) · ds

[# (x ) ( f (x , f (x )))] · ds,

(456)

with controls f , f , f , f . Such statements of the control problem are hardly seen in the contemporary literature on optimal control distributed-parameter systems (see, for example, Ahmed and Teo [214]). The existence of optimal controls for equations much simpler than those here were developed only very recently; see Fattorini [223]. Thus, for linear heat- and massdiffusion problems with impulse control that is a function of magnitude or spatial locations of the impulses, Anita [224] obtained a formulation of maximum principles for both optimal problems. Ahmed and Xiang [225] proved the existence of optimal controls for clear nonlinear evolution equations on Banach spaces with the control term in the equations being represented as an additive—multiplicative term B(t)u(t). Reduction of ‘‘hererogeneous’’ terms in the corresponding momentum equation by an overall representation of diffusive and ‘‘diffusionlike’’ terms

v. s. travkin and i. catton

126 yields

A a! A : m(K ; ) ; K k KCDD x K K x x

D

; 9a! a! . D

(457)

Here, the velocity and ﬂuctuating viscosity coefﬁcient variables are taken in a form suitable for both laminar and turbulent ﬂow regimes. For problems with a constant bulk viscosity coefﬁcient (K : constant), the second term K in this relation vanishes and the whole problem essentially becomes one of evaluating the inﬂuence of dispersion by irregularities of the soil medium on the momentum. Thermal dispersion effects realized through the second derivative terms and relaxation terms and, for example, in the ﬂuid phase with constant thermal characteristics heat transport dispersion can be expressed as

T T T : m(K ; k ) ; K K 2CDD x 2 D x 2 x

9 c m T u!

ND D D D (K ; k ) D ; 2 T ds , (458) 1U where the ﬁrst and last terms resemble the effective thermal conductivity coefﬁcient for each phase, using constant coefﬁcients, found in the work by Nozad et al. [40]. By allowing the control terms to be added to the bulk transport coefﬁcients, another variation of a mathematical statement for optimal control can be found. As far as optimal control problems with PDE dynamics are concerned, one can ﬁnd a detailed solution of the linear quadratic regulator problem, including conditions for the convergence of modal approximation schemes. However, for more general optimal control problems involving PDE, the main approach has been to use some method for constructing a particular ﬁnite-dimensional approximating optimal control problem and then to solve this problem. The relationship between the solutions and stationary points of the approximating optimal control problem and those of the original optimal control problem is not established in these papers. For the models and differential equations describing HEs to be useful, the additional integral and integrodifferential terms need to be addressed in a systematic way. VAT has the unique ability to enable the combination of direct general physical and mathematical problem statement analysis with the convenience of the segmented analysis usually employed in HE design. A segmented approach is a method where overall physical processes or groups of phenomena are divided into selected subprocesses or phenomena that are interconnected to others by an adopted chain or set of depend-

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encies. A few of the obvious steps that need to be taken are the following: 1. Model what increases the heat transfer rate 2. Model what decreases of ﬂow resistance (pressure drop) 3. Combine the transport (thermal/mass transfer) analysis and structural analysis (spatial) and design 4. Find the minimum volume (the combination of parameters yielding a minimum weight HE) 5. Include nonlinear conditions and nonlinear physical characteristics into analysis and design procedures The power and convenience of this method is clear, but its credibility is greatly undermined by variability and freedom of choice in selection of subportions of the whole system or process. The greatest weakness is that the whole process of phenomena described by a voluntarily assigned set of rules for the description of each segment is sometimes done without serious consideration of the implications of such segmentation. Strict physical analysis and consideration of the consequences of segmentation is not possible without a strict formulation of the problem that the VAT-based modeling supplies. Structural optimization of a plate HE, for example, using the VAT approach might consist of the following steps: (1) optimization of the number of plates, plate spacing and ﬁn spacing; (2) optimization of the ﬁn shape; (3) simultaneous optimization of multiple mathematical statements. This approach also allows consideration and description of hydraulically and thermally developing processes by representing them through the distributed partial differential systems.

X. New Optimization Technique for Material Design Based on VAT A variety of optimal control problems that can be formulated in the area of heterogeneous medium transport involve differential equations modeling the physics of the process. Many of them have a fairly complicated form, and the contemporary literature on optimal control considers much simpler formulations of the optimization problems with constraints in form of differential equations. When the diffusion equations are written in nonlocal VAT form, there are additional terms appearing in the mathematical statements. These terms can be considered to be morphology controls involving differential and integral operators. The nonlinear diffusion equation written without source terms

v. s. travkin and i. catton

128

has three control terms,

1 C : · (D s C ) ; · D s t 1 ; · (D c ) ;

S

@

S @

C ds

D C · ds

: · (D s C ) ; F (C , D , M ) ; F (c , D , M ) ! S ! S ; F (C , D , M ), (459) ! S where the morphology characteristics set M contains many parameters, , S L such as phase fraction s and speciﬁc surface area S , M : (s , S , , , . . . ). S @ The equation for an electrostatic electrical ﬁeld in a particulate medium (polycrystalline medium) is 1

· [s E ] ; · E ;

1

( E ) · ds : ,

which becomes

· [s E ] ; F ( , E , M ) ; F ( , E , M ) : . # S # S Additional equations are 1

; (s E ) ;

(460)

ds ; E : 0

1

; (s E ) ; F (E , M ) : 0. (461) # S A temperature control equation for the solid phase with the two morphology control terms can be written T T K;T K:a (T , S , t, z) ; T (T , S , t, z), K+GL K K+OGL K K z U t

(462)

where

1 a T :a T ds , T : K K+GL K+OGL K z K K 1 K and in the void phase

T K · ds , x 1 G

T

T

:a ;T (T , t, z) ; T (T , t, z) z +GL +OGL t

(463)

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129

1 T (T , S , t, z) : a T ds (464) +GL z 1 a T (T , S , t, z) :

T · ds . (465) +OGL 1 These terms are not equal and their calculation or estimation presents a challenge. However, these are the real driving forces that will differentiate the behavior of one composite from another. Their application will lead to a direct connection between design goals and morphological solutions.

XI. Concluding Remarks Determination of the effective parameters in model equations are usually based on a medium morphology model and there are dozens of associated quasi-homogeneous and quasi-stochastic methods that claim to accomplish this. In most cases, quasi-homogeneous and quasi-stochastic methods have no well treated solutions and, most important, they are not sufﬁcient for description of the physical process features in heterogeneous media, especially when treating a multiscale processes. The hierarchical approach applied to radiative transfer in a porous medium and to the electrodynamics governing equations (Maxwell’s equations) in a heterogeneous medium yielded new volume averaged radiative transfer equations — VAREs. These equations have additional terms reﬂecting the inﬂuence of interfaces and inhomogeneities on radiation intensity in a porous medium and, when solved, will allow one to relate the lower scale parameters to the upper scale material behavior. The general nature of this result makes it applicable to any subatomic particle transport, including neutron transport, as well as radiative transport in the heterogeneous media ﬁeld. Direct closure based on theoretical and numerical developments that have been developed for thermal, momentum, and mass transport processes in a speciﬁc random porous and composite medium established a basis for closure modeling in problems in radiative and electromagnetic phenomena. In this work, transport models and equation sets were obtained for a number of different circumstances with a well substantiated mathematical theory called volume averaging theory (VAT) that included linear, nonlinear, laminar, and turbulent hierarchical transport in nonisotropic heterogeneous media, accounting for modeling level, interphase exchange, and microroughness. Models were developed, for example, for porous media using an advanced averaging technique, a hierarchical modeling methodology, and fully turbulent models with Reynolds stresses and ﬂuxes. It is worth

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v. s. travkin and i. catton

noting that nonlocal mathematical modeling is very different from homogenization modeling. The new integrodifferential transport statements in heterogeneous media and application of these nonclassical types of equations is the current issue. The theory allows one to take into consideration characteristics of multicomponent multiphase composites with perfect as well as imperfect morphologies and interphases. The transport equations obtained using VAT involved additional terms that quantify the inﬂuence of the medium morphology. Various descriptions of the porous medium structural morphology determine the importance of these terms and the range of application of closure schemes. Many mathematical models currently in use have not received a critical review because there was nothing to review them against. The more common models were compared with the more rigorous VAT-based models and found deﬁcient in many respects. This does not mean they do not serve a useful purpose. Rather, they are incomplete and suffer from lack of generality. VAT-based modeling is very powerful, allowing random morphology ﬂuctuations to be incorporated into the VAT-based transport equations by means of randomly varying morphoconvective and morphodiffusive terms. Closure of some of the resulting morphoﬂuctuation in the governing transport equations has been outlined, resulting in some well-developed closure expressions for the VAT-based transport equations in porous media. Some of them exploit the properties of available solutions to transport problems for individual morphological elements, and others are based on the natural morphological data of porous media. Statistical and numerical techniques were applied to classical irregular morphologies to treat the morphodiffusive and morphoconvective terms along with integral terms. The challenging problem in irregular and random morphologies is to produce an analytical or numerical evaluation of the deviations in scalar or vector ﬁelds. In previous work, the authors have presented a few exact closures for predetermined regular and random porous medium morphologies. The questions related to effective coefﬁcient dependencies, boundary conditions, and porous medium experiment analysis are discussed. Analysis of heat exchanger designs depends on the heat balance equations that are widely used in the heat design industry. A theoretical basis for employing heat and momentum transport equations obtained with volume averaging theory was developed for modeling and design of heat exchangers. This application of VAT results in a correct set of mathematical equations for heat exchanger modeling and optimization through implementation of general ﬁeld equations rather than the usual balance equations. The

volume averaging theory

131

performance of a heat exchanger depends on the design criteria for optimizing the liquid ﬂow velocity, dimensions of the heat exchanger, the hea‘t transfer area between the hot side and cold side, etc. However, the optimization problems involving such designs are more complex than for traditional designs and require new optimal control simulation techniques. A variety of optimal control problems that can be formulated in the area of heterogeneous medium transport involve differential equations modeling the physics of the process. Many of them have a fairly complicated form, and the contemporary literature on optimal control considers much simpler formulations of the optimization problems with constraints in the form of differential equations. Linear optimal control systems governed by parabolic partial differential equations (PDEs) are relatively well studied in the literature. The modeling CHE equations resulting from VAT-based analysis are also PDEs, but they are nonlinear and have additional integral and integrodifferential terms. It is well known that some matrix composites (often porous) represent the promise for design of a series of materials with highly desirable characteristics such as high temperature accommodation and enhanced toughness. Their performance is very dependent on the volume fraction of the constituent materials, reinforcement interface and matrix morphologies, and consolidation. Scale characteristics (nanostructural composites) give the abnormal physical properties, such as magnetic, and mechanical transport and state a great challenge in formulating the hierarchical models containing the design objectives. The importance of the physical processes taking place in a heterogeneous multiscale—multiphase—composite medium creates the need for the development of new tools to characterize such media. It leads to the development of new approaches to describing these processes. One of them (VAT) has great advantages and is the subject of this review. Acknowledgments This work was partly sponsored by the Department of Energy, Ofﬁce of Basic Energy Sciences, through the grant DE-FG03-89ER14033 A002.

Nomenclature a c B

thermal diffusivity [m/s] mean drag resistance coefﬁcient in the REV [-]

c B

mean skin friction coefﬁcient over the turbulent area of S [-] U

v. s. travkin and i. catton

132 c BN c BQNF c D* c N C

d AF d G d N ds D

D

D

F

D

Q

S U f Y f

f D f g H h

h P S U k D k Q K

D

mean form resistance coefﬁcient in the REV [-] drag resistance coefﬁcient upon single sphere [-] mean skin friction coefﬁcient over the laminar region inside of the REV [-] speciﬁc heat [J/(kg · K)] constant coefﬁcient in Kolmogorov turbulent exchange coefﬁcient correlation [-] character pore size in the cross section [m] diameter of ith pore [m] particle diameter [m] interphase differential area in porous medium [m] molecular diffusion coefﬁcient [m/s]; also tube or pore diameter [m] ﬂat channel hydraulic diameter [m] diffusion coefﬁcient in solid [m/s] internal surface in the REV [m] averaged over value f — D intrinsic averaged variable value f, averaged over din D an REV — phase averaged variable morphoﬂuctuation value of f in a D gravitational constant [1/m] width of the channel [m] averaged heat transfer coefﬁcient over S U [W/(m/K)]; half-width of the channel [m] pore scale microroughness layer thickness [m] internal surface in the REV [m] ﬂuid thermal conductivity [W/ (mK)] solid phase thermal conductivity [W/(mK)] permeability [m]

K @ K A K K K Q2 K 2 l L m m Q n n G Nu NMP p

Pe F Pe N Pr Q Re AF Re F

Re N Re NMP

S AP S U S UN

turbulent kinetic energy exchange coefﬁcient [m/s] turbulent diffusion coefﬁcient [m/s] turbulent eddy viscosity [m/s] effective thermal conductivity of solid phase [W/(mK)] turbulent eddy thermal conductivity [W/(mK)] turbulence mixing length [m] scale [m] averaged porosity [-] surface porosity [-] number of pores [-] number of pores with diameter of type i [-] h d : Q F , interface surface D Nusselt number [-] pressure [Pa]; or pitch in regular porous 2D and 3D medium [m]; or phase function [-] :Re Pr, Darcy velocity pore F scale Peclet number [-] :Re Pr, particle radius Peclet T number [-] : , Prandtl number [-] a D outward heat ﬂux [W/m] Reynolds number of pore hydraulic diameter [-] mu! d F , Darcy velocity : Reynolds number of pore hydraulic diameter [-] u! d : N , particle Reynolds number [-] u! d : NMP , Reynolds number of general scale pore hydraulic diameter [-] total cross-sectional area available to ﬂow [m] speciﬁc surface of a porous medium S / [1/m] U :S / [1/m] ,

volume averaging theory S , T T ? T Q T U T U, u u 1PI

V V " W

:S cross ﬂow projected area NP of obstacles [m] temperature [K] characteristic temperature for given temperature range [K] solid phase temperature [K] wall temperature [K] reference temperature [K] velocity in x direction [m/s] square friction velocity at the upper boundary of HR averaged over surface S U [m/s] velocity [m/s] :u ! m Darcy velocity [m/s] velocity in z direction [m/s]

k

L m r s T w

effective ﬂuid phase component of turbulent vector variable; or species or pore type component of turbulent variable that designates turbulent ‘‘microeffects’’ on a pore level laminar scale value or medium roughness solid phase turbulent wall

Superscripts

*

value in ﬂuid phase averaged over the REV

value in solid phase averaged over the REV mean turbulent quantity turbulent ﬂuctuation value equilibrium values at the assigned surface or complex conjugate variable

Greek Letters 2 D Q , B K

Subscripts e f i

133

K C

Q H ) , :, J? ? , :, JQ Q

averaged heat transfer coefﬁcient over S [W/(mK)] U representative elementary volume (REV) [m] pore volume in a REV [m] solid phase volume in a REV [m] electric permittivity [Fr/m] dynamic viscosity [kg/(ms)] or [Pas] magnetic permeability [H/m] kinematic viscosity [m/s]; also , frequency [Hz] density [kg/m]; also , electric charge density [C/m] medium speciﬁc electric conductivity [A/V/m] electric scalar potential [V] particle intensity per unit energy (frequency) ensemble-averaged value of interface ensemble-averaged value of , with phase j being to the left angular frequency [rad/s] magnetic susceptibility [-] absorption coefﬁcient [1/m] scattering coefﬁcient [1/m]

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ADVANCES IN HEAT TRANSFER, VOLUME 34

Two-Phase Flow in Microchannels

S. M. GHIAASIAAN and S. I. ABDEL-KHALIK G. W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, Georgia 30332

I. Introduction The application of miniature thermal and mechanical systems is rapidly increasing in various branches of industry. Recent technological advances have led to extremely ﬁne spatial and temporal thermal load resolutions, requiring the analysis of microscale heat transfer phenomena where the system characteristic dimension can be smaller than the mean free path of the heat carrying particles [1]. In this article the recently published research dealing with gas—liquid two-phase ﬂow in microchannels is reviewed. Only microchannels with hydraulic diameters of the order of 0.1 to 1 mm and with long length-tohydraulic diameter ratios are considered. In these systems the channel characteristic dimension is of the same order of magnitude, or smaller than, the neutral interfacial wavelengths predicted by the Taylor stability analysis. The review is also restricted to situations where the ﬂuid inertia is signiﬁcant in comparison with surface tension. Such microchannels and ﬂow conditions are encountered in miniature heat exchangers, research nuclear reactors, biotechnology systems, the cooling of high-power electronic systems, the cooling of the plasma-facing components in fusion reactors, and the heat rejection systems of spacecraft, to name a few. The ﬂow through cracks and slits when such cracks develop in vessels and piping systems containing high-pressure liquids is another application of two-phase ﬂow in microchannels of interest here. Two-phase ﬂow in capillaries with complex geometry (porous media) has been reviewed in the recent past [2—4] and will not be addressed. 145

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ADVANCES IN HEAT TRANSFER, VOL. 34 Copyright 2001 by Academic Press. All rights of reproduction in any form reserved. 0065-2717/01 $35.00

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For steady-state and developed two-phase ﬂow in a smooth pipe, and assuming incompressible phases, the variables that can affect the hydrodynamics of gas—liquid two-phase ﬂow are , , , , , D, g, 1!, , * % * % U , and U . Since the minimum number of reference dimensions in *1 %1 hydrodynamics is three (time, mass, and length), according to the Buckinham theorem eight independent dimensionless parameters can be deﬁned that in general affect the hydrodynamics of gas—liquid two-phase ﬂow. The following three dimensionless parameters are particularly important for the characterization of microchannels: Eo :

gD

(1)

U D We : *1 * *1

(2)

U D We : %1 % . %1

(3)

The remainder of the dimensionless parameters can be represented as / , * / , 1!, , and the phasic superﬁcial Reynolds numbers: % * Re : U D/ (4) *1 *1 * Re : U D/ . (5) %1 %1 % Note that the phasic Froude numbers Fr : U /gD and Fr : U /gD, %1 *1 %1 *1 and the capillary number Ca : U /, can all be derived by combining * *1 two or more of these dimensionless parameters. When Eo, We , and We *1 %1 are all less than 1, gravitation and inertia are both insigniﬁcant in comparison with surface tension. Re & 1, furthermore, implies small inertia com*1 pared with viscous forces. Flow ﬁelds in capillaries where surface tension and viscosity dominate buoyancy and inertia have important applications and have been extensively studied in the past [5—8]. In microchannels of interest here, however, Eo 1, at least one of the Weber numbers is typically of the order of 1 to 10, and Re 2 1. Thus, although surface *1 tension mostly dominates buoyancy, inertia can be signiﬁcant. Similar conditions apply to two-phase ﬂow in microgravity, resulting in important and useful similarities between the two categories of systems with respect to hydrodynamics of two-phase ﬂow. In both types of systems the predominance of the surface tension force on buoyancy leads to the insensitivity of two-phase hydrodynamics to channel orientation, and in nonseparated two-phase ﬂow patterns it leads to the suppression of velocity difference

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147

between the two phases in the absence of signiﬁcant acceleration. Suo and Grifﬁth [9] derived the following criterion for negligible buoyancy effect in two-phase pipe ﬂow: gD * & 0.88/3.

(6)

Based on an analysis of stratiﬁed to nonstratiﬁed ﬂow regime transition, Brauner and Moalem-Maron [10] derived the following criterion for the predominance of surface tension on buoyancy: Eo & (2').

(7)

Experiments with water and air ﬂowing in pipes indicate that the transition to the surface tension—dominated regime (where ﬂow patterns are not affected by channel orientation) occurs in the 1 & D & 2 mm range [11, 12]. Equation (6) agrees well with the latter observations. The channel hydraulic diameters considered here thus cover the aforementioned critical range. Another important characteristic of microchannels of interest here is that for them, D O(), C

(8)

(9)

D 0.3.

(10)

where :

. g

The Laplace length scale, , represents the order of magnitude of the wavelength of the interfacial waves in Taylor instability, and the latter instability type is known to govern important hydrodynamic processes such as bubble and droplet breakup. Equation (8) evidently implies that some Taylor instability-driven processes may be entirely irrelevant to microchannels. The criterion of Suo and Grifﬁth, Eq. (6), can approximately be recast as

III. Two-Phase Flow Regimes and Void Fraction in Microchannels The gas and liquid phases in a two-phase ﬂow system can exist in various distinct morphological conﬁgurations. Flow regimes represent the major morphological conﬁgurations of the phases and are among the most important characteristics of two-phase ﬂow systems, since they strongly inﬂuence all the hydrodynamic and transport processes, such as pressure

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drop, heat and mass transfer, and ﬂow stability. A methodology for predicting the two-phase ﬂow regimes is thus required for the design of two-phase ﬂow systems and for the speciﬁcation of appropriate closure relations for two-phase conservation equations. Flow regimes and conditions leading to their establishment have been extensively investigated in the past several decades. Methods for predicting the ﬂow regimes are often based on the ﬂow regime map concept, according to which the empirically determined ranges of occurrence of all major ﬂow patterns are speciﬁed on a two-dimensional map, with the two coordinates representing some appropriate hydrodynamic parameters [13, 14]. However, since the gas—liquid hydrodynamics are affected by a large number of independent dimensionless parameters, the two-dimensional ﬂow regime maps are often in disagreement with respect to the parameters they use as coordinates and their ranges of applicability are limited to the ranges of their databases. More recently, semianalytical methods, where the ﬂow regime transition processes are mechanistically or semianalytically modeled, have been proposed [15—17] and have undergone extension and improvement [18—20]. The existing ﬂow regime maps, as well as the aforementioned semianalytical models, however, generally do poorly when compared with experimental two-phase ﬂow regime data representing microchannels. A. Definition of Major Two-Phase Flow Regimes Experiments indicate that ﬂow regimes, which are morphologically similar to the major two-phase ﬂow regimes common in large channels, occur in microchannels as well. Therefore, a brief review of the major ﬂow regimes observed in large channels is provided in this section. Detailed explanations of these ﬂow regimes and their characteristics can be found in various textbooks and monographs [13, 14] and in more recent review articles [18]. Figure 1 schematically depicts the major ﬂow regimes in common gas—Newtonian liquid two-phase ﬂow systems in large vertical channels. Bubbly ﬂow occurs at low gas and liquid ﬂow rates and is characterized by bubbles distorted-spherical in shape and moving upward in zigzag fashion. The slug ﬂow regime is characterized by bullet-shaped Taylor bubbles that have diameters close to the diameter of the channel and are separated from the channel wall by a thin liquid ﬁlm, with lengths that can widely vary and may reach more than 15 times the channel diameter. The Taylor bubbles are separated by liquid slugs that often contain small entrained bubbles. The churn ﬂow is established following the disruption of the Taylor bubbles due to high gas ﬂow rates and is characterized by chaotic oscillations and churning. In the annular ﬂow pattern a thin liquid ﬁlm, which can be smooth or wavy depending on the gas velocity, ﬂows on the wall, while the

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Fig. 1. Major ﬂow patterns in a large vertical pipe.

gas ﬂows through the channel core. The gas often contains dispersed droplets. In the dispersed-bubbly ﬂow regime, which is established at very high liquid ﬂow rates, small spherical bubbles with little or no interaction with each other are mixed with the liquid. Unlike the common bubbly ﬂow regime, in which the bubble size is controlled by Taylor instability and aerodynamic forces, the size of the bubbles in the dispersed bubbly ﬂow regime is dictated by turbulence in the liquid. The commonly observed two-phase ﬂow patterns associated with the ﬂow of a gas and a Newtonian liquid in a horizontal large channel are depicted in Fig. 2. Bubbly ﬂow occurs at high liquid and low gas ﬂow rates and is followed by plug (elongated bubble), slug, and annular/dispersed ﬂow patterns as the gas ﬂow rate is increased. The stratiﬁed-smooth ﬂow regime occurs at low liquid and low gas ﬂow rates and is followed by stratiﬁed wavy and annular/dispersed ﬂow regimes as the gas ﬂow rate is increased. The dispersed-bubbly regime occurs at very high liquid ﬂow rates, and its characteristics are similar to those of the dispersed-bubbly ﬂow regime in vertical channels. The plug and slug ﬂow patterns are often referred to collectively as the intermittent ﬂow pattern, since the distinction between them is not always clear or important. The ﬂow patterns in Figs. 1 and 2 only display the major ﬂow regimes that are easily discernible visually and with simple photographic techniques and are commonly addressed in ﬂow regime maps and transition models.

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Fig. 2. Major ﬂow patterns in a large horizontal pipe.

Numerous subtle variations within some of the ﬂow patterns can be recognized using more sophisticated techniques, however [21]. B. Two-Phase Flow Regimes in Microchannels Early studies dealing with two-phase ﬂow in microchannels were mostly concerned with surface tension-driven ﬂows [5—8]. Two-phase ﬂow regimes in microchannels under conditions where inertia is signiﬁcant have been experimentally investigated by Suo and Grifﬁth [9], Oya [22], Barnea et al. [23], Damianides and Westwater [11], Barajas and Panton [24], Fukano and Kariyasaki [25], Mishima and Hibiki [26], and Triplett et al. [12]. Two-phase ﬂow patterns in narrow, rectangular channels, some simulating slits and cracks, have also been reported in [27—33]. Narrow et al. [34] and Ekberg et al. [35] studied the two phase ﬂow regimes in a micro-rod bundle

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and in narrow annuli, respectively. Two-phase ﬂow and transport phenomena in the slug (bubble train) regime in microchannels have also been investigated [36, 37]. The commonly observed ﬂow patterns in microchannels are depicted here using the photographs provided by Triplett et al. [12]. The major ﬂow regimes shown in these pictures are in agreement with the observation of most of the other investigators, although, as will be shown later, some ﬂow patterns have been given different names by different authors. Triplett et al. [12] conducted experiments using air and water at room temperature, in horizontal, transparent circular test sections with 1.09 and 1.45 mm diameter, and in microchannels with semitriangular (triangular with one corner smoothed) cross sections with 1.09 and 1.49 mm hydraulic diameters (see Fig. 3). They identiﬁed the ﬂow regimes using high-speed videocameras recording ﬂow details near the centers of the test sections. Figure 4 displays typical photographs of the ﬂow patterns identiﬁed in their 1.09-mmdiameter circular test section. The overall ﬂow pattern morphologies observed with the other test sections used by Triplett et al. [12] were similar to the pictures in Fig. 4.

Fig. 3. Cross-sectional geometry of the test sections of Triplett et al. [12].

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Fig. 4. Representative photographs of ﬂow patterns in the 1.1-mm-diameter test section of Triplett et al. [12]. (With permission from [12].)

two-phase flow in microchannels

153

Bubbly ﬂow (Fig. 4a) was characterized by distinct and distorted (nonspherical) bubbles, typically considerably smaller in diameter than the channel. The slug ﬂow (Fig. 4b) is characterized by elongated cylindrical bubbles. This ﬂow pattern has been referred to as slug by some investigators [9, 26] and plug by others [11, 24]. Unlike plug ﬂow in larger channels where the elongated gas bubbles typically occupy only part of the channel cross section (Fig. 2), the bubbles in slug ﬂow in microchannels appear to occupy most of the channel cross section [12, 26]. Figures 4c and 4d display the churn ﬂow pattern in the experiments of Triplett et al. [12], who assumed two processes to characterize churn ﬂow. In one process, the elongated bubbles associated with the slug ﬂow pattern become unstable as the gas ﬂow rate is increased and their trailing ends are disrupted into dispersed bubbles (Fig. 4c). This ﬂow pattern has been referred to as pseudo-slug [9], churn [26], and frothy-slug by Zhao and Rezkallah [38] in their microgravity experiments. The second process that characterizes churn ﬂow is the occurrence of churning waves that periodically disrupt an otherwise apparently wavy-annular ﬂow pattern (Fig. 4d). This ﬂow pattern is referred to as frothy slug-annular by Zhao and Rezkallah [38]. At relatively low liquid superﬁcial velocities, increasing the mixture volumetric ﬂux leads to the merging of long bubbles that characterize slug ﬂow, and to the development of the slug—annular ﬂow regime represented by Fig. 4e. In this ﬂow pattern long segments of the channel support an essentially wavy-annular ﬂow and are interrupted by largeamplitude solitary waves that do not grow sufﬁciently to block the ﬂow path. With further increase in the gas superﬁcial velocity, these large amplitude solitary waves disappear and the annular ﬂow pattern represented by Fig. 4f is established. C. Review of Previous Experimental Studies and Their Trends The important studies of microchannel two-phase ﬂow that have addressed parameter ranges of interest here are reviewed, and their experimental results are compared, in this section. Table I is a summary of the experimental investigations reviewed here. 1. General Trends The study by Suo and Grifﬁth [9] is among the earliest experimental investigations. They could observe slug—bubbly, slug, and annular ﬂow patterns. They observed no stratiﬁcation, attributed its absence to the predominane of surface tension over buoyancy, and proposed the criterion in Eq. (6). Oya [22, 39] was concerned with ﬂow patterns and pressure drop

TABLE I Summary of Experimental Data for Microchannel and Narrow Passage Two-Phase Flow Regimes 154 Author Suo and Grifﬁth [9]

Orientation Horizontal

Channel characteristics Circular, D : 1.0 and 1.4 mm

Horizontal and Glass, circular, D : 4—12.3 mm vertical Triplett et al. [12] Horizontal Pyrex, circular, D : 1.1 and 1.45 mm; semitriangular (Fig. 3), D : 1.1 and C 1.49 mm Damianides and Westwater [11] Horizontal Pyrex, circular, D : 1—5 mm; stack of ﬁns Fukano and Kariyasaki [25] Horizontal Circular, D : 1, 2.4, 4.9, 9 and 26 mm and vertical Mishima and Hibiki [26] Vertical Pyrex and aluminum, D : 1.05—4.08 mm Barajas and Panton [24] Horizontal Pyrex, polyethylene, polyurethane, ﬂuoropolymer resin Barnea et al. [23]

U %1 (m/s)

U *1 (m/s)

Water—N , heptane—N , heptane—He Water—air

Not given

Not given

0.04—60

0.002—10

Water—air

0.02—80

0.02—8.0

Water—air Water—air

0.03—100 0.1—30

0.08—10 0.02—2

Water—air Water—air

0.1—50 0.1—100

0.02—2 0.003—2

Fluids

Narrow et al. [34] Lowry and Kawaji [27]

Horizontal Vertical

Wambsganss et al. [28]

Horizontal

Ali and Kawaji [29]

Horizontal/ Vertical Horizontal/ Vertical Vertical

Ali et al. [30] Mishima et al. [31] Wilmarth and Ishii [32]

155

Fourar and Bories [33]

Horizontal/ Vertical Horizontal

Ekberg et al. [35]

Horizontal

Glass, seven-rod bundle, D : 1.46 mm C Rectangular, W : 8 cm, L : 8 cm, S : 0.5, 1, 2 mm Rectangular, W : 19.05 mm, L : 1.14 m, S : 3.18 mm Rectangular, W : 80 mm, L : 240 mm, S : 1.465 mm Rectangular, W : 80 mm, L : 240 mm, S : 0.778 and 1.465 mm Rectangular, W : 40 mm, L : 1.5 m, S : 1.07, 2.45, 5.0 mm Rectangular, L : 630 mm; W : 15 mm and S : 1 mm; W : 20 mm and S : 2 mm Rectangular glass slit, W : 0.5 m, L : 2 m, S : 1 mm; brick slit, W : 14 cm, L : 28 cm, S : 0.18, 0.4, 0.54 mm Glass annuli; D : 6.6 mm, D : 8.6 mm; and G M D : 33.2 mm and D : 35.2 mm; L : 35 cm G M for both annuli

Water—air Water—air

0.02—40 0.1—18

003—5.0 0.1—8

Water—air

0.05—30

0.2—2

Water—air

0.15—16

0.2—7.0

Water—air

0.15—16

0.15—6.0

Water—air

0.02—10

0.1—10

Water—air

0.02—8

0.07—4.0

Water—air

0.0—10

0.005—1

Water—air

0.02—57

0.1—6.1

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resulting from the conﬂuence of air—fossil liquid fuel in short vertical tubes with 2, 3, and 6 mm diameters, and L /D ratios of 20 to 25, and could identify nine distinct ﬂow patterns. Because of the predominance of entrance effects, however, Oya’s data may not be representative of fully developed ﬂow patterns. In the study by Barnea et al. [23], air—water ﬂow regimes were compared with the ﬂow regime transition models of Taitel and Dukler [15] for horizontal ﬂow and Taitel et al. [12] for vertical ﬂow, with some minor modiﬁcations. The latter models predicted their data well. Since the tube diameters were relatively large, however, their data clearly show the effects of gravity and test section orientation. The two-phase ﬂow regime data of Triplett et al. [12] are shown in Fig. 5. Regime transition lines representing a micro-rod bundle (Narrow et al. [34]) are also depicted and are discussed in Subsection E of this part. The ﬂow patterns representing the four test sections of Triplett et al. are similar, and none of the test sections supported stratiﬁed ﬂow. The depicted ﬂow patterns indicate the predominance of intermittent (slug, churn, and slug— annular) ﬂow patterns that together occupy most of the maps. The two-phase ﬂow regimes representing the ﬂow of air—water mixture in glass tubes with D : 1 mm to 2.4 mm reported by several authors are

Fig. 5. Experimental ﬂow regime maps for air—water ﬂow in microchannels (Triplett et al. [12]) and a micro-rod bundle (Narrow et al. [34]).

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157

Fig. 6. Comparison among air—water ﬂow regime maps obtained in glass tubes with D 5 1 mm. (Symbols represent the test section (a) in Fig. 3 [12]).

depicted in Fig. 6. For the air—water—Pyrex system 1! $ 34° [40], implying a partially wetting liquid. In Fig. 6, the ﬂow pattern names in capital and bold letters represent those reported by Damianides and Westwater [11] and Fukano and Kariyasaki [25], respectively; the lowercase letters are from Mishima and Hibiki [26]; and the symbols represent the data of Triplett et al. [12]. Damianides and Westwater [11] were concerned with two-phase ﬂow patterns in compact heat exchangers. The ﬂow patterns in their 1 and 2 mm diameter tubes, which are of interest here, included dispersed bubbly, bubbly, plug, slug, pseudoslug, dispersed-droplet, and annular. As noted, the ﬂow pattern identiﬁed as churn by Triplett et al. (Figs. 4c and 4d) appears to coincide with the ﬂow pattern identiﬁed as dispersed by Damianides and Westwater. Furthermore, the slug and slug—annular regimes in Triplett’s experiments (Figs. 4e and 4f ) coincide with the plug and slug ﬂow regimes in Damianides and Westwater, respectively. These differences are evidently associated with subjective identiﬁcation and naming of ﬂow patterns, and the two experimental sets are otherwise in good overall agreement.

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Fukano and Kariyasaki [25] reported no signiﬁcant effect of channel orientation on the ﬂow patterns for channel diameters 4.9 mm and smaller. Fukano and Kariyasaki identiﬁed only three ﬂow patterns: bubbly, intermittent, and annular. They compared the ranges of occurrence of the ﬂow regimes with the ﬂow regime map of Mandhane et al. [41], with poor agreement. However, their ﬂow transition lines agreed with the transition lines of Barnea et al. [23] for the latter authors’ 4 mm diameter tube tests. The ﬂow regime transition lines of Fukano and Kariyasaki representing the data obtained with their 1 mm and 2.4 mm-diameter test sections are depicted in Fig. 6. Their data are evidently in disagreement with the data of Triplett et al. [12], and Damianides and Westwater [11], except for the intermittent-to-bubbly ﬂow transition line, where all three data sets are in good agreement. In the investigation by Mishima and Hibiki [26], except for void fraction measurements, which were carried out in aluminum test sections, all experiments were performed in Pyrex test sections, and ﬂow regimes were identiﬁed using a high-speed camera. The identiﬁed ﬂow regimes were bubbly, slug, churn, annular, and annular-mist. Mishima and Hibiki compared their data representing 2.05 and 4.08 mm diameter test sections with the ﬂow regime transition models of Mishima and Ishii [42] with very good agreement and argued that the latter ﬂow regime transition models should be applicable to capillary tubes as well. The ﬂow transition lines of Mishima and Hibiki [26] are displayed in Fig. 6 for the data obtained with their 2.05 mm diameter test section, and are noted to disagree with the data of other investigators. Mishima and Hibiki have indicated that the ﬂow patterns in their 1.05 mm diameter test section were similar to the patterns for their 2.05 mm diameter test section. 2. Effect of Surface Wettability The experimetal studies just discussed all utilized materials that represented partially wetting ( & 90°) conditions. In view of the signiﬁcance of surface tension, however, the surface wettability can evidently affect the two-phase ﬂow hydrodynamics in microchannels. Barajas and Panton [24] conducted experiments with air and water, using four different channel materials. These included Pyrex ( : 34°), polyethylene ( : 61°), and polyurethane ( : 74°) as partially wetting; and the FEP ﬂuoropolymer resin ( : 106°) as a partially nonwetting combination. Figure 7 displays a summary of their ﬂow regime maps, where the data of Triplett et al. [12] representing their 1.09 mm diameter circular test section are also included for comparison. The data of Barajas and Panton [24] representing their Pyrex test section agreed well with the experimental ﬂow regime of Damian-

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159

Fig. 7. The effect of surface wettability on the air—water ﬂow regimes. (Symbols represent test section (a) in Fig. 3 [12]; ﬂow regime names are from Barajas and Panton [24]).

ides and Westwater [11] representing the latter authors’ 1 mm and 2 mm diameter test sections (which were also made of Pyrex), with the exception of the wavy stratiﬁed ﬂow pattern, which did not occur in the 1 mm test section of Damianides and Westwater. With the other partially wetting test sections, polyethylene and polyurethane, the ﬂow regimes and their ranges of occurrence were similar to those obtained with Pyrex, with the difference that with polyethylene and polyurethane the wavy ﬂow pattern was now replaced with a ﬂow regime characterized by a single rivulet. A small multirivulet region also occurred on the ﬂow regime map representing the polyurethane test section. The ﬂow regimes observed with the partially nonwetting channel FEP ﬂuoropolymer were signiﬁcantly different, however, and compared with the partially wetting tubes, the ranges of occurrence of the rivulet and multirivulet ﬂow patterns were signiﬁcantly wider. 3. Flow Regimes in Microgravity As mentioned earlier in Part II, dimensional analysis indicates that two-phase ﬂow in common large channels in microgravity has important similarities with two-phase ﬂow in terrestrial microchannels, since in both

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systems the surface tension predominates buoyancy, while inertia can be signiﬁcant. Experiments conducted aboard aircraft ﬂying parabolic trajectories that can maintain microgravity (<0.02g) for periods up to 22 seconds have been reported by several research groups. Based on the published data, empirical correlations for ﬂow regime transitions applicable over wide ranges of ﬂuid properties have been proposed by Rezkallah [43] and Jayawardena et al. [44]. Zhao and Rezkallah [38] performed experiments in 9.52 mm and 12.7 mm diameter tubes, where the regimes associated with water—air two-phase ﬂow were identiﬁed using video cameras. Four major ﬂow patterns were identiﬁed: bubbly, slug, frothy slug—annular, and annular. The bubbly, slug, and annular ﬂow regimes were morphologically similar to those described before (see Figs. 4a, 4b, and 4f ). The frothy slug—annular regime as described by Zhao and Rezkallah, however, is similar to the ﬂow pattern depicted in Fig. 4d and has been called the pseudo-slug by Suo and Grifﬁth [9], and churn by others [12, 26]. Figure 8 compares the ﬂow regimes of Triplett et al. [12] representing their test section (a) (see Fig. 3), with the experimental ﬂow regime transition lines of Zhao and Rezkallah [38]. Bousman et al. [45] utilized test sections with 12.7 mm and 25.4 mm diameters, and studied the two-phase ﬂow regime, void fraction, and liquid ﬁlm thickness in the annular regime, using air—water, air—water ; glycerin

Fig. 8. Comparison of microchannel ﬂow regime data of Triplett et al. for their 1-mm diameter circular test section (Fig. 3a) with the microgravity data of Zhao and Rezkallah [43] and Bousman et al. [45].

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161

( : 6 cP, : 63 dyn/cm), and air—water ; Zonyl FPS (a surfactant; re* sulting in : 1 cP, : 21 dyn/cm) mixtures. Bousman et al. identiﬁed four * major ﬂow regimes similar to those reported by Zhao and Rezkallah [38]. Their experimental ﬂow regime transition lines are also depicted in Fig. 8 and are noted to be in relatively poor agreement with the data of Triplett et al. [12] with respect to the bubbly—slug ﬂow regime transition, and in good agreement with respect to the transition to annular ﬂow. D. Flow Regime Transition Models and Correlations As noted earlier, theoretical arguments and experimental evidence indicate that ﬂow patterns in microchannels should be insensitive to gravitational ﬁeld, and therefore to channel orientation. Criteria for calculating the maximum channel size for which gravity is inconsequential have been proposed in [9, 10] (see Eqs. (6—8)). Equation (6) [9], which provides a criterion for the dominance of surface tension over buoyancy to the extent that for channel diameters smaller than the provided limit the two-phase ﬂow patterns are not affected by the channel orientation, agrees with the experimental data of [11, 12]. The ﬂow regime map of Mandhane et al. [41], a widely used empirical ﬂow regime map for common horizontal channels, was compared with microchannel data by some authors with poor agreement [12, 25]. Barnea et al. [23] invetigated the two-phase ﬂow regimes in small channels (4 mm D 12.3 mm) and extended the methodology of Taitel and Dukler [15] for horizontal ﬂow. In the original model of Taitel and Dukler [15], transition from stratiﬁed to intermittent ﬂow regimes is assumed to take place when small but ﬁnite amplitude disturbances that occur on the liquid surface grow. The position of the liquid surface (i.e., the liquid depth in the channel) is predicted from the solution of one-dimensional (1D) gas and liquid momentum conservation equations assuming steady-state and fully developed stratiﬁed ﬂow [15]. Barnea et al. [23] argued that in very small channels the predominance of surface tension on gravitational force, and not the interfacial wave instability, is responsible for regime transition from stratiﬁed to intermittent. Based on a simple model, Barnea et al. proposed that the latter ﬂow regime transition occurs in very small channels when the liquid depth in the channel, found from the solution of 1D momentum equations for steady-state and fully developed stratiﬁed ﬂow, satisﬁes the following equation:

. (11) D9h * (1 9 '/4) % Barnea et al. [23] also argued that when D is smaller than the right-hand

162

s. m. ghiaasiaan and s. i. abdel-khalik

side of the preceding equation, the aforementioned ﬂow regime transition occurs when

' D. h 2 19 * 4

(12)

The modiﬁed Taitel and Dukler [15] models for horizontal ﬂow and the ﬂow regime transition models of Taitel et al. well predicted the aforementioned data of Barnea et al. [23]. When applied to smaller channels, however, the aforementioned semianalytical ﬂow regime transition models appear to do poorly [11, 12], with the exception of the model of Taitel and Dukler [15] for the establishment of dispersed bubbly ﬂow. For the transition to dispersed bubbly ﬂow, Taitel and Dukler [15] derived the following relation based on a mechanistic model according to which in the latter regime spherical bubbles have diameters within the size range of inertial eddies predicted by Kolmogorov’s theory of locally isotropic turbulence, and their diameters are controlled by interation with the latter eddies:

D (/ ) g( 9 ) * * % . (13) * * This relation is valid as long as & 0.52, the latter approximately representing the upper limit of void fraction for the existence of spherical bubbles. In large horizontal channels, in addition to Eq. (13), another criterion should be met according to which turbulence dominates over buoyancy so that bubbles do not gather near the channel top [15, 20]. The latter criterion is evidently redundant in microchannels where buoyancy effect is insigniﬁcant. Although the basic assumptions for the development of Eq. (13) are usually not met in microchannels [12], it appears to predict well the transition line representing the development of bubbly ﬂow [12, 25]. Based on the drift ﬂow model, and arguing that the void fraction is a suitable parameter for correlating ﬂow regime transitions, Mishima and Ishii [42] derived expressions for ﬂow regime transition in vertical, upward ﬂow. The major ﬂow regimes considered were bubbly, slug, churn, and annular. The transition from bubbly to slug ﬂow was assumed to occur when a void fraction of 0.3 is reached, and led to U ; U : 4.0 *1 %1

3.33 0.76 g 91 U 9 . (14) %1 C 2 M * The slug-to-churn transition was assumed to occur when the liquid slugs become unstable because of the wake effect caused by the Taylor bubbles. U : *1

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163

For this transition, Mishima and Ishii derived (C 9 1)(U ; U ) ; 0.35(gD/ M *1 %1 * (15) gD gD * U ; U ; 0.75 *1 %1 * * where is predicted using the following expression provided by the drift ﬂux model [46, 47]: : 1 9 0.813

U %1 . (16) C (U ; U ) ; V M *1 %1 EH Mishima and Ishii [42] assumed that the transition from churn to annular ﬂow occurred either because of ﬂow reversal in the liquid ﬁlms separating large bubbles from the wall, or because of the disruption of liquid slugs. The ﬁrst mechanism is applicable when :

N\ g I* D , [(1 9 0.11C )/C ] M M

(17)

where

N : * * I*

\ . g

(18)

When the ﬁrst mechanism applies, the churn-to-annular transition occurs when

gD . (19) % (Note than used in the preceding equation must be larger than the right-hand side of Eq. (15)). The second mechanism causes the regime transition when U : ( 9 0.11) %1

g (20) N\ . I* % The drift ﬂux parameters C in the preceding expressions should be found from [48] U : %1

C : 1.2—0.2( / for round tubes (21) % * C : 1.35—0.35( / for rectangular ducts. (22) % * Mishima and Hibiki [26] compared their experimental data representing

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two-phase ﬂow in tubes with 1.05 mm to 4.08 mm diameters with the aforementioned ﬂow regime transition models of Mishima and Ishii [42], apparently with good agreement, despite the fact that the empirical drift ﬂux model parameters (namely C and V ) of the latter authors may be EG inappropriate for microchannels. The representation of V in the derivation EG of the foregoing transition models based on expressions valid for larger channels is evidently in disagreement with experimental data associated with microchannels, where velocity slip and the buoyancy effect are both negligible. Zhao and Rezkallah [38] and Rezkallah [43], in correlating their microgravity ﬂow regime data, argued that when the buoyancy force is negligible compared with surface tension while inertia is signiﬁcant, the phasic Weber numbers are the most appropriate dimensionless parameters for the correlation of ﬂow regime transitions. Based on their experimental data, Zhao and Rezkallah [38] correlated the bubbly-to-slug transition assuming equal phasic velocities, and that this transition occurs at a void fraction of : 0.18, and derived U : 4.56U accordingly. *1 %1 Zhao and Rezkallah empirically correlated the ﬂow regime transitions from slug to frothy slug—annular (equivalent to the slug—annular ﬂow pattern in Triplett et al. [12]; see Fig. 4e) according to We : 1 and from %1 frothy slug—annular to annular ﬂow according to We : 20. %1 The preceding correlations well predicted the experimental data of Zhao and Rezkallah [38]. When data from several other sources were also considered, the aforementioned We : constant correlations were found %1 inadequate [43]. However, correlations in the form We : constant could % well predict the entire data [43], where We : U D/, (23) % % % with U representing the gas phase velocity. The latter We : constant % % correlations are inconvenient for application, however, since the void fraction is needed for the calculation of U . Rezkallah [43], however, % showed that the data from several microgravity sources, which covered a relatively wide range of ﬂuid properties, could be well correlated in a two-dimensional map using We and We as coordinates, provided that %1 *1 the entire ﬂow regime map is divided into three regions: the surface tension region including bubbly and slug ﬂow regimes; the intermediate (transitional) region including the frothy slug—annular regime; and the inertial region representing annular ﬂow. The experimental data of Triplett et al. [12] representing all four of their test sections, and the experimental ﬂow regime transition lines of Damianides and Westwater [11] representing their 1 mm diameter test section, are depicted in a ﬂow regime map with We and We as coordinates in Fig. 9. *1 %1

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Fig. 9. Comparisn of microchannel data representing D $ 1 mm with the microgravity ﬂow regime transition lines of Rezkallah [43]. Capitals: data from (11); Lowercase: data from [43].

As noted, the depicted data agree with the empirical transition lines of Rezkallah [43] relatively well with respect to the ﬂow regime transitions among surface tension (bubbly or slug), transitional, and inertia (annular) regions. The transitional region in Fig. 9 includes the churn (Figs. 4c and 4d) and slug—annular (Fig. 4e), also referred to as pseudoslug [9] and frothy slug—annular [38] ﬂow patterns. The data of Fukano and Kariyasaki [25] are not shown, since the latter authors deﬁned only three ﬂow patterns (bubbly, intermittent, and annular). Bousman, McQuillen, and Witte [45], in correlating their microgravity ﬂow regime data, argued for the use of the void fraction as the criterion for ﬂow regime transition. The transition from bubbly to slug ﬂow was assumed to occur when : 0.4, and transition to annular ﬂow was assumed to take place when : 0.70—0.75, with the exact value depending on the liquid type. They thus divided the entire ﬂow regime map into three regions: bubbly, transitional (slug and slug/annular), and annular. They calculated the void fractions used for the derivation of the aforementioned criteria using the drift ﬂux model [46, 47], Eq. (16), assuming V : 0 because of the absence EH of velocity drift in nonseparated ﬂow patterns in microgravity. Based on their measurement of void fractions in their smaller (12.7 mm diameter) test section, they derived C : 1.21 and assumed that the same value was

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applicable to their larger (25.4 mm diameter) test section. The : 0.4 criterion for transition from bubbly to slug ﬂow patterns is in disagreement with the aforementioned : 0.18 criterion suggested by Zhao and Rezkallah [38]. It is also in disagreement with the 0.25 to 0.3 value in large terrestrial channel ﬂow data [17, 42]. Jayawarden et al. [44] considered the aforementioned data of Zhao and Rezkallah [38] and Bousman et al. [45], as well as data from several other sources covering ﬂuids with a wide range of properties. Similar to Bousman et al. [45], they divided the entire ﬂow regime map into three zones: bubbly, slug/slug—annular, and annular. They argued that the phasic superﬁcial Weber and Reynolds numbers were the primary dimensionless parameters that should be used for correlating regime transitions in microgravity. They demonstrated that the transition from bubbly to slug/slug—annular for the entire data considered could be correlated as

where

Re %1 : K Su\ Re *1

(24)

Re : U D/ (25) %1 %1 % Re : U D/ (26) *1 *1 * Re D *. (27) Su : *1 : We * *1 Equation (24) was recommended for the range 10 & Su & 10. For transition to the annular ﬂow regime, Jayawarden et al. [44] derived

for Su & 10, and

Re %1 : K Su\ Re *1

(28)

(29) Re : K Su %1 for Su 10, where K : 4,641.6, and K : 2 ; 10\. Unfortunately, the available microchannel two-phase ﬂow data represent very narrow ranges of the Su parameter, and meaningful comparison between the available data and the aforementioned ﬂow regime transition correlations of Jayawarden et al. [44] is not feasible at this time. E. Flow Patterns in a Micro-Rod Bundle Micro-tube bundles have potential applications in miniature heat exchangers. Experimental data dealing with two-phase ﬂow in micro-rod and -tube bundles are scarce, however.

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Fig. 10. Cross-section of the micro-rod bundle test section of Narrow et al. [34]. (With permission from [34].)

Narrow et al. [34] experimentally investigated the air—water two-phase ﬂow patterns and pressure drop in a horizontal, 23 cm long glass micro-rod bundle that included seven rods conﬁgured as in Fig 10. Their test section was entirely transparent, had an average hydraulic diameter of 1.29 mm, and included several short components speciﬁcally designed to reduce the entrance and exit effects. They used a high-speed digital video camera near the test section center to directly view subchannels 11 and 12 in Fig. 10, and subchannels 5 and 6, which were visible through the transparent rod in front of them. The ﬂow regime map of Narrow et al. is depicted in Fig. 5. Froth ﬂow was characterized by the absence of a discernible interfacial geometry. In the stratiﬁed—intermittent ﬂow pattern the upper subchannels in the test section were in plug or slug ﬂow patterns, while some of the bottom subchannels carried single-phase liquid. In the annular—intermittent ﬂow pattern (a ﬂow pattern not reported in the past) the inner subchannels supported an intermittent (slug or plug) or froth ﬂow pattern, while the ﬂow regime in the peripheral subchannels was predominantly annular. In the annular—wavy ﬂow pattern, liquid ﬁlms ﬂowed on all rods and on the test section wall. The ﬂow regime maps representing single channels and the micro-rod bundle depicted in Fig. 5 are in fair agreement with respect to annular and churn or froth ﬂow patterns. The range of occurrence of the slug—annular

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ﬂow pattern in single channels coincides with the annular—intermittent ﬂow pattern in the bundle. Furthermore, the plug/slug and the stratiﬁed-intermittent ﬂow patterns in the micro-rod bundle are replaced everywhere with the slug ﬂow pattern in single channels. The occurrence of the stratiﬁed— intermittent regime in the micro-rod bundle, which implies sensitivity to buoyancy and rod bundle orientation, however, is a crucial difference in comparison with single microchannels. Horizontal rod bundles are used in CANDU nuclear reactors. The conditions leading to bundle-wide or partial stratiﬁcation in the latter rod bundles may lead to dryout and critical heat ﬂux and have been experimentally studied in the past [49—51]. The micro-rod bundle ﬂow regime map of Narrow et al. did not agree with the available ﬂow regime maps for large horizontal rod bundles. Bundle-wide stratiﬁcation, which can readily occur in large horizontal rod bundles [49—51], was not observed by Narrow et al. [34]. Narrow et al. [34] developed an empirical ﬂow regime map, using void fraction and mass ﬂux as the coordinates (Fig. 11), where the void fraction was predicted everywhere using the homogeneous ﬂow assumption whereby the forthcoming Eq. (32) with S : 1 was applied. For & 0.25, transition occurred at a mass ﬂux of G : 1000 kg/ms. At higher void fractions, the transition could be represented by the following equation: ln(G) : 94.7 ; 8.1.

(30)

Fig. 11. Empiricial ﬂow rgime transition lines for the micro-rod bundle data of Narrow et al. [34]. (With permission from [34].)

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F. Void Fraction Void fractions in microchannels have been measured by Kariyasaki et al. [52], Mishima and Hibiki [26], Bao et al. [53], and Triplett et al. [54]. Fukano and Kariyasaki [25] and Mishima and Hibiki [26] also attempted to measure and correlate the velocity of large bubbles. Equation (16), which is a result of the drift ﬂux model [46, 47], has long been utilized for the correlation of void fraction in two-phase channel ﬂow. In the latter equation the two-phase distribution coefﬁcient, C , is the cross-sectional average of the product of total volumetric ﬂux and void fraction, divided by the product of the averages of the two, while the gas drift velocity, V , is a weighted mean drift velocity of the gas phase with EH respect to the mixture. The parameters C and V represent the global and EH local interfacial slip effects, respectively, and are often determined empirically. Widely used correlations for C and V , developed based on experi EH mental data for common large channels, have also been used for correlating microchannel data (Mishima et al. [31]). Correlations applied in this way include Eq. (22). An empirical correlation based on the drift ﬂux model formulation has been developed by Chexal and co-workers [55], which is based on an extensive database, includes a large number of empirically adjusted parameters, and can address channels with various conﬁgurations. The data base for this correlation does not include microchannels of interest here, however. Mishima and Hibiki [26] correlated their void fraction data for upward ﬂow in vertical channels, as well as the data of Kariyasaki et al. [52], using the drift ﬂux model [46], Eq (16), with V : 0 for bubbly and slug ﬂow EH regimes. The distribution coefﬁcient C , however, was found to be a function of channel diameter and was correlated according to [52] (31) C : 1.2 ; 0.510e\ "C. where D is in millimeters. C When the slip ratio, deﬁned as S : U /U , is known, the void fraction % * can be calculated using the fundamental void-quality relation in onedimensional two-phase ﬂow: x . (32) x ; S( / )(1 9 x) * % In homogeneous two-phase ﬂow, S : 1. A correlation for the slip ratio, proposed by the CISE group (Premoli et al. [56]) and recommended by Hewitt [57], can be represented as :

y S:1;B yB 1 ; yB

(33)

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where y : U /U (34) *1 %1 B : 1.578Re\ ( / ) (35) * * % B : 0.0273We Re\ ( / )\ (36) * * * % Re : GD/ (37) * * We : GD/( ). (38) * * Bao et al. [53] measured pressure drop and void fraction in tubes with 0.74 mm to 3.07 mm diameters, using air and water mixed with various concentrations of glycerin. The void fractions were measured using two solenoid valves located near the two ends of the test sections, which could be closed simultaneously. Bao et al [53] compared their void fraction data with predictions of several correlations, all taken from literature dealing with commonly used large channels, and based on the results they recommended the aforementioned empirical correlations for the slip ratio S proposed by the CISE group (Premoli et al. [56]). Butterworth [58] has shown that the void fraction correlations of Lockhart and Martinelli [59] and several other investigators can be represented in the generic form

19x N 19 :A ( / )O( / )P % * * % x

(39)

where A : 0.28, p : 0.64, q : 0.36, r : 0.07 for Lockhart and Martinelli [59]. Bao et al. [53] found good agreement between their measured void fractions and the predictions of the Lockhart and Martinelli [59] correlation. Triplett et al. [54] compared their void fraction data, estimated from photographs taken from their circular test sections, with predictions of the preceding correlations of Lockhart and Martinelli as presented by Butterworth [58], the aforementioned correlation due to the CISE group [56, 57], and the correlation of Chexal et al. [55]. Figure 12 depicts a typical comparison between the data of Triplett et al. and the preceding correlations. These comparisons indicated that with the exception of the annular ﬂow regime where all the tested correlations overpredicted the data, the homogeneous model provided the best agreement with experiment. G. Two-Phase Flow in Narrow Rectangular and Annular Channels Two-phase ﬂow in rectangular channels with ( O(1) mm occurs in the coolant channels of research reactors, and during critical ﬂow through cracks that may occur in vessels containing pressurized ﬂuids. Investigations

two-phase flow in microchannels

171

have been reported by [27—33]. The recent experimental investigations are sumarized in Table I. Experiments in vertical narrow channels, using air and water, with consistent overall results with respect to the two-phase ﬂow regime maps, have been reported by Kawaji and co-workers [27, 29, 30], Mishima et al. [31], and Wilmarth and Ishii [32]. Minor differences with respect to the description and identiﬁcation of the ﬂow patterns exist among these investigators, however. Lowry and Kawaji [27] used strobe ﬂash photography and could identify bubbly, slug, churn, and annular ﬂow regimes. In the bubbly ﬂow the bubbles were small and near-spherical. The slug ﬂow was characterized by large irregular and ﬂattened bubbles, while the curn ﬂow pattern contained large irregularly shaped, as well as small, bubbles. Their ﬂow transition lines for the establishment of dispersed bubbly and annular ﬂow patterns for the test sections with ( : 1 and 2 mm disagreed with the models of Taitel et al. [17]. Ali and Kawaji [29] and Ali et al. [30] performed an extensive experimental study using room-temperature and near-atmospheric air and water in rectangular narrow channels with six different conﬁgurations: vertical, cocurrent up and down ﬂow; 45° inclined, cocurrent up and downﬂow; horizontal ﬂow between horizontal plates; and horizontal ﬂow between vertical plates. Their observed ﬂow regimes and ﬂow regime maps, except for the last conﬁguration, were similar and are displayed in Fig. 13. The rivulet ﬂow pattern occurred at very low liquid superﬁcial velocities and was relatively sensitive to the orientation of their test section. The ﬂow regimes for horizontal ﬂow between vertical plates included bubbly, intermittent, and stratiﬁed—wavy, and the ﬂow regime maps for both gap sizes (( : 0.778 mm and 1.465 mm) were similar to the ﬂow regime maps observed in large pipes. Mishima et al. [31] identiﬁed four major ﬂow regimes in their experiments: bubbly ﬂow, characterized by crushed or pancake-shaped bubbles; slug ﬂow, represented by crushed slug (elongated) bubbles; churn ﬂow, in which the noses of the elongated bubbles were unstable and noticeably disturbed; and annular ﬂow. Figure 14 depicts the ﬂow regimes in their 1.07 mm gap test section. The ﬂow regime map for their 2.4 mm-gap test section was similar except for the presence of a small churn region in the latter. With a gap of ( : 5.0 mm, however, the ﬂow regime transition boundaries were displaced in comparison with Fig. 14. The predictions of the following correlation for the slug—annular ﬂow regime transition, due to Jones and Zuber [60], are also shown in Fig. 14: U : *1

19 R U 9V . %1 EH R

(40)

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s. m. ghiaasiaan and s. i. abdel-khalik

Fig. 12. Comparison of some void fraction correlations with the measured data of Triplett et al. [54] for the test section (a) depicted in Fig. 3: (a) homogeneous ﬂow model; (b) Chexal et al. [55]; (c) Lockhart—Martinelli—Butterworth, Eq. (39) [58]; (d) CISE [56]. (With permission from [54].)

where : 0.8 represents the void fraction for the slug—annular transition. R The correlation of Jones and Zuger [60] was based on air—water experiments in a vertical retangular channel with ( : 5 mm. The drift ﬂux model, Eq. (16), with C found from the aforementioned correlation of Ishii [48], Eq. (22), and V obtained from the forthcoming EH Eq. (41) [60], could well predict all the void fraction data of Mishima et al. [31] for ( : 1.07 and 2.45 mm, except for the annular ﬂow regime, for which the data and correlation deviated signiﬁcantly: V : (0.23 ; 0.13(/W )(gW /(). (41) EH Wilmarth and Ishii [32] studied the two-phase ﬂow regimes in vertical up

two-phase flow in microchannels

173

Fig. 12. (continued).

ﬂow and horizontal ﬂow between vertical plates. Their ﬂow regimes for vertical up ﬂow are compared with the aforementioned data of Mishima et al. in Fig. 14, where bubbly and ‘‘cap-bubbly’’ ﬂow patterns have been combined in the depicted bubbly ﬂow regime zone. In comparing their vertical ﬂow data with models, Wilmarth and Ishii noted relatively good agreement for the ﬂow regime transition from bubbly to slug, with the ﬂow regime transition models of Mishima and Ishii [42], Eq. (14), and Taitel et al. [17].

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s. m. ghiaasiaan and s. i. abdel-khalik

Fig. 13. Flow patterns in the experiments of Ali et al. [30]. (With permission from [30].)

For horizontal channels (i.e., ﬂow between two horizontal parallel plates), several experimental studies have been published. Differences with respect to the ﬂow regime description and identiﬁcation among various authors can be noted, however. The experimental ﬂow regimes of Ali et al. [30] were shown in Fig. 13. For the horizontal ﬂow conﬁguration, Wilmarth and Ishii [32] could identify stratiﬁed, plug, slug, dispersed bubbly, and wavy—

Fig. 14. Flow patterns in the experiments of Mishima et al. [31] and Wilmarth and Ishii [32] in test sections with 1 mm gap. (With permission from [32].)

two-phase flow in microchannels

175

Fig. 15. Flow regimes in air—water experiments in horizontal rectangular channels. (With permission from [32].)

annular ﬂow patterns. In their experiments in similarly conﬁgured narrow channels, as mentioned before, Ali et al. [30] identiﬁed bubbly, intermittent, and stratiﬁed—wavy ﬂow regimes only. The two ﬂow regime maps are compared in Fig. 15. The wavy—annular ﬂow pattern occurred at the low U and high U range, which appears to be outside the range of the *1 %1 experiments of Ali et al. [30]. The two sets of data are qualitatively in agreement with respect to the bubbly—plug/slug transition. In the study by Wambsganss et al. [28], air—water tests were conducted in transparent, horizontal rectangular test sections with two conﬁgurations, one with the 3.18 mm side oriented vertically (i.e., ﬂow between two horizontal plates), and the other with the 19.05 mm side oriented vertically (ﬂow between two vertical plates). Their ﬂow regime transition lines, furthermore, disagreed with several ﬂow regime maps that are based on large channel data, including the ﬂow regime map of Mandhane et al. [41]. Using an image processing technique, Wilmarth and Ishii [61] measured the void fraction and interfacial concentrations in their air—water experiments using vertical rectangular channels with ( : 1 and 2 mm, and calculated the drift ﬂux parameters C and V . The V values have large EH EH uncertainties. For the smaller gap, they found C : 0.81—1 for bubbly ﬂow, indicating that bubbles moved with a smaller velocity than liquid; and C : 1 for the slug and churn ﬂow patterns. For the larger channel they obtained C : 0.4—1 for bubbly ﬂow and C : 1 and 1.2 for slug and churn-turbulent ﬂow regimes, respectively.

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Fig. 16. Flow regimes of Fourar and Bories [33] for air—water ﬂow between horizontal ﬂat plates with ( : 1 mm. (With permission from [33].)

Fourar and Bories [33] conducted experiments in glass and brick slits, addressing low-liquid superﬁcial velocities. Figure 16 depicts their experimental ﬂow regime map. Bubbly ﬂow was characterized by small, isolated bubbles, whereas in the ﬁngering bubbly ﬂow large, ﬂat, and unstable bubbles were visible. The ‘‘complex’’ ﬂow pattern was chaotic, without an apparent structure (i.e., similar to froth ﬂow). In the annular regime the liquid was reported to have ﬂowed in the form of unstable ﬁlms on the walls and may refer to the rivulet ﬂow pattern identiﬁed by Ali et al. [30]. The ﬁlms were replaced by entrained droplets at very low liquid ﬂow rates. Fourar and Bories [33] also measured the average void fraction in their test section by careful measurement of its water contents. In all ﬂow regimes excluding annular, their test section void fraction closely agreed with the correlation :19

X , 1;X

(42)

with the Martinelli factor, X, to be estimated from

U * *1 . (43)* U % %1 Recently, Ekberg et al. [35] conducted experiments using two horizontal X:

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177

Fig. 17. Flow regimes in narrow horizontal annuli. Regime names in capital and small letters are for the small and large test sections of Ekberg et al. [34], respectively. Regime names in bold letters are from Osamusali and Chang [63]. (With permission from [35].)

glass annuli with 1.02 mm spacing and studied the two-phase ﬂow regimes, void fraction, and pressure drop. The two-phase ﬂow patterns in vertical and horizontal large annular channels had earlier been studied by Kelessidis and Dukler [62] and Osamusali and Chang [63], respectively. Osamusali and Chang carried out experiments in three annuli, all with D : 4.08 cm, and with D /D : 0.375, 0.5, and 0.625 (( : 4.75, 6.35 and 11.75 mm, G respectively), and noted that the ﬂow patterns and their transition lines were relatively insensitive to D /D . The experimental ﬂow regime transition lines G of Ekberg et al. [35] are displayed in Fig. 17, where they are compared with the experimental results of Osamusali and Chung [63]. These transition lines disagree with the ﬂow regime map of Mandhane et al. [41]. Stratiﬁed ﬂow occurred in the experiments of Ekberg et al. [35]. Ekberg et al. [35] compared their measured void fractions with the predictions of the homogeneous mixture model, the correlation of Lockhart and Martinelli [59] as presented by Butterworth [58], Eq. (39), the correlations of Premoli et al. [56, 57], Eqs. (33)—(38), and the drift ﬂux model, Eq. (16), with C : 1.25 and V : 0, following the results of Ali et EH al. [30] for narrow channels. The Lockhart—Martinelli—Butterworth correlation best agreed with their data.

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H. Two-Phase Flow Caused by the Release of Dissolved NONCONDENSABLES Liquid forced convection in microchannels is an effective cooling mechanism for systems with very high volumetric heating such as accelerator targets, high-power resistive magnets, and high-power microchips and electronic devices. The widely used correlations for turbulent friction factor and convection heat transfer in large channels have been shown to be inadequate for microchannels by some investigators, indicating that the turbulence characteristics of microchannels may be different from those of large channels [64—66]. An interesting issue related to the forced ﬂow of liquids in microchannels is the potential effect of noncondensables dissolved in the liquid. Dissolved noncondensables typically have a negligible effect in experiments with large channels where they undergo little desorption. In microchannels, however, because of the typically large axial pressure drops, signiﬁcant desorption of noncondensables is possible. Such desorption will lead to the development of a two-phase mixture, will increase the convection heat transfer coefﬁcient, and may be at least partially responsible for the experimental data of some investigators, which suggest that the widely used correlations representing heat and mass transfer in large channels underpredict the heat and mass transfer in microchannels rather signiﬁcantly [65, 66]. Adams et al. [67, 68] recently studied the effect of dissolved noncondensables on the hydrodynamic and heat transfer processes in a microchannel. A theoretical model [67] indicated that the release of the dissolved noncondensables can have a relatively signiﬁcant effect on the channel hydrodynamics. In [68], experiments were performed in a channel 0.76 mm in diameter with a 16 cm heated length, subject to water forced convection. The range of experimental parameters were as follows: wall heat ﬂux, 0.5 to 2.5 MW/m; liquid mean velocity at inlet, 2.07 to 8.53 m/s; channel exit pressure, 5.9 bar. The water remained subcooled in all the experiments. Typical results, depicting the Nusselt numbers at their test section exit obtained with pure (degassed) water, and with water initially saturated with air at a pressure equal to the test section exit pressure, are displayed in Fig. 18. The Nusselt numbers, deﬁned as Nu : hD/k , were calculated assuming * single-phase liquid ﬂow properties. The apparent dependence of the enhancement in Nu due to the noncondensables (air) on Re and heat ﬂux in fact represents the dependence of the results on the pressure drop and the liquid temperature rise in the test section. Higher pressure drop and higher liquid temperature rise in the test section both lead to increased desorption of dissolved noncondensables from the liquid, and therefore to the enhance-

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179

Fig. 18. The enhancement in the local convective heat transfer due to the presence of dissolved air [68]. (With permission from [68]).

ment of the local heat transfer coefﬁcient. The presence of dissolved air in water could increase the measured Nu by as much as 17%. An upper limit of voidage development resulting from the release of dissolved air from water can be obtained by assuming (a) homogeneousequilibrium two-phase ﬂow; (b) the release of dissolved air is accompanied by evaporation such that the gas—vapor mixture is everywhere saturated with vapor; and (c) the liquid and the gas—vapor mixture are everywhere at equilibrium with respect to the concentration of the noncondensable, and the latter equilibrium can be represented by Henry’s law. Using these assumptions, Adams et al. [68] showed that void fractions up to several percent were possible, implying the occurrence of the bubbly ﬂow regime in their test section. For forced convection heat transfer in developed bubbly ﬂow, the Nu augmentation factor resulting from the presence of the gas phase, which increases the ﬂow velocity, can be shown to be of the order of (1 9 )\L, with n representing the power of Re in the appropriate singlephase forced convection heat transfer correlation [69]. The augmentation in Nu in the data of Adams et al. [68] was signiﬁcantly higher, however, indicating that the observed heat transfer enhancement should be primarily due to the ﬂow ﬁeld disturbance caused by the formation and release of microbubbles on the channel walls.

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s. m. ghiaasiaan and s. i. abdel-khalik IV. Pressure Drop

A. General Remarks Pressure drop is among the most essential design parameters for piping systems. However, frictional pressure drop in microchannels, for both single and two-phase ﬂow, is not well understood. Turbulence characteristics in microchannels are known to be different from those in large channels. Thus, although the predictions of theoretial solutions for laminar ﬂow closely agree with measured friction factors in microchannels [70, 71], most of the recent experimental investigations indicate that the well-proven correlations for turbulent ﬂow in large channels fail to correctly predict frictional pressure drop in circular and rectangular microchannels [71—74]. Some investigators, on the other hand, have reported good agreement between their data and turbulent friction factor correlations for smooth pipes [26], and for pipes with carefully measured roughness characteristics [70]. Measurement uncertainties and uncertainties associated with roughness evidently contribute to the disagreement among published experimental data, while the lack of adequate understanding of turbulence in microchannels is believed to be the main reason for disagreement between existing data and the commonly used correlations for large channels.

B. Frictional Pressure Drop in Two-Phase Flow The existing methods and correlations applied in microchannel pressure drop are mostly similar to those applied to large channels. A brief review of the principles of two-phase frictional pressure drop modeling, and the predictive methods that have been applied to microchannels, is provided in this section. The simplest method for calculating the two-phase frictional pressure drop is to assume homogeneous ﬂow and apply an appropriate single-phase turbulent friction factor correlation using the homogeneous two-phase mixture properties everywhere. Thus,

P z

D2.

: 9f

G , 2. 2 D F

(44)

(45)

where

x 1 9 x \ : ; . F % *

two-phase flow in microchannels

181

Now, applying the Blasius correlation for turbulent friction factor, for example, one can write f

2.

: 0.316Re\ 2.

(46)

where : GD/ . (47) 2. 2. One of the most popular correlations for homogeneous mixture viscosity is that of McAdams [75]: Re

1 9 x \ x ; . (48) 2. * % In the preceding equations x represents the ﬂow quality. In dealing with a single-component two-phase ﬂow (i.e., the ﬂow of a liquid and its own vapor), and further assuming thermal equilibrium between the two phases (the homogeneous equilibrium mixture model, HEM), x : x , where CO x : (h 9 h )/h . (49) CO D DE The homogeneous ﬂow assumption has limited applicability, however, and the foregoing method for calculating two-phase frictional pressure drop is inaccurate in most applications. The two-phase multiplier method is the most common technique for correlating two-phase frictional pressure drop in channels [59], according to which

P z

D2.

:

: *-

P z

D*-

: %-

P z

D%-

(50)

or

P z

D2.

: *

P z

D*

P z

: %

D%

,

(51)

where

P z

and

and

P z

D*D* represent single-phase frictional pressure gradients in the channel when pure liquid at mass ﬂuxes G and G(1 9 x), respectively, ﬂows in the channel. The terms P z

D%

P z

D%

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are deﬁned similarly when pure gas at mass ﬂuxes G and Gx, respectively, ﬂows in the channel. Equations (50) and (51) evidently provide four ways for representing two-phase pressure drop, by correlating any of , , * % , or . Based on the two-phase multiplier concept, many correlations * % have been proposed in the past. These correlations are usually applicable without restriction to all ﬂow regimes. Martinelli and co-workers were the ﬁrst to correlate and , graphi* % cally in terms of the Martinelli factor X, deﬁned as X :

(P/z) D* . (P/z) D%

(52)

Lockhart and Martinelli’s graphical representation of have been utilized % for the development of algebraic correlations by some investigators [76, 77]. A widely used correlation, suggested by Chisholm and Laird [78], is : 1 ; C/X ; 1/X, *

(53)

where C may have values between 10 and 20. An alternative expression for this correlation is [79] : 1 ; CX ; X. %

(54)

In the foregoing equations the constant C depends on whether the gas and liquid phases, when ﬂowing alone, are laminar (viscous) or turbulent, and its recommended values are as follows: C : 20 for turbulent—turbulent ﬂow; C : 12 for viscous liquid and turbulent gas; C : 10 for turbulent liquid and viscous gas; and C : 5 for viscous—viscous ﬂow [78]. A correlation representing as a function of ﬂow quality, x, P /P , and n with n *D%D%representing the power of Re in the single-phase friction factor correlation, has also been proposed by Chisholm [80, 14]. Other correlations that account for the effect of phasic properties and the two-phase mixture mass ﬂux have been described in [14]. A widely used correlation, proposed by Friedel [81], is based on a vast database covering an extensive parameter range. For horizontal channels, the Friedel correlation is

% 19 % Fr\ We\ :A;3.21x (19x) * 2. 2. * % * * , (55)

two-phase flow in microchannels

183

where A : (1 9 x) ; x f ( f )\ (56) * %- % *G (57) Fr : 2. gD 2. GD We : . (58) 2. 2. According to Friedel, the single-phase friction factors should be calculated from f : 0.25[0.86859 ln Re /(1.964 ln Re 9 3.8215) ]\ (59) HH H when Re 1055, where H Re : DG/ . (60) H H When Re 1055, the appropriate laminar Fanning friction factor relation H is used. The following correlation, derived by Beattie and Whalley [82], is convenient to use because of its simplicity. In this method, homogeneous mixture ﬂow is assumed and Eqs. (44)—(47) are applied. The mixture viscosity, however, is deﬁned as : ; (1 9 )(1 ; 2.5 ) (61) 2. F % * F F where , the homogeneous void fraction, is found from Eq. (32) with S : 1. F Beattie and Whalley recommend that the single-phase friction factor be calculated, for all values of Re , from the Colebrook [83] correlation, 2. 9.35 1 : 3.48 9 4 log 2 ; (62) D Re ( f (f 2. 2. 2. where f : f /4 represents the Fanning friction factor. The preceding correlations, as mentioned earlier, do not explicitly account for ﬂow patterns and have been developed to cover various ﬂow patterns covered by their databases. The fractional pressure drop, like many other important hydrodynamic phenomena, depends on ﬂow pattern, however. Flow regime—dependent models have been derived for the stratiﬁed [84] and annular [85, 86] ﬂow regimes in the past, because of the relatively simple morphology of the latter ﬂow patterns. These models, in addition to correlating the wall friction, also account for the gas—liquid interfacial friction.

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s. m. ghiaasiaan and s. i. abdel-khalik

C. Review of Previous Experimental Studies The occurrence of ﬂashing two-phase ﬂow in refrigerant restrictors was the impetus for early studies of single- and two-phase pressure drop in long microchannels [87—90]. Two-phase pressure drop in these studies was generally modeled using the homogeneous ﬂow model. Table II is a summary of the more recent experimental investigations that are reviewed here. Koizumi and Yokohama [91] modeled their experimental data using Eqs. (44)—(48), where the liquid and vapor phases were assumed to be at equilibrium everywhere, and the two-phase viscosity was calculated from : based on the argument that the ﬂashing two-phase ﬂow in their 2. F * simulated refrigerant restrictor was predominantly bubbly. Although their calculations were in good agreement with their total measured pressure drops, they noted that the preceding model was in fact signiﬁcantly in error since it did not account for the evaporation delay in their experiments. Further investigation into the ﬂashing and two-phase ﬂow processes as associated with refrigerant restrictors was conducted more recently by Lin et al. [92]. They measured pressure drop with single-phase liquid ﬂow, and noted that their data could be well predicted using the Churchill [93] correlation, f :8 where

8 1 ; , Re (A ; B)

A : 2.457 ln

B:

37530 Re

1

7 ; 0.27 Re D

(63)

(64)

(65)

Based on an argument similar to that of Koizumi and Yokohama [91], Lin et al. [92] applied the homogeneous-equilibrium mixture model (Eqs. (44), (45), and (47)) for two-phase pressure drop calculations. For calculating the two-phase friction factor, f , they used the aforementioned Churchill 2. correlation (Eq. (63)) by replacing Re with Re , and over a quality range 2. of 0 & x & 0.25 they empirically correlated the two-phase mixture viscosity according to % * : , (66) 2. ; xL( 9 ) % * % with n : 1.4 providing the best agreement between model and data.

TABLE II Summary of Experimental Data for Microchannel and Narrow Passage Two-Phase Flow Pressure Drop Author Koizumi and Yokohama [91] Lin et al. [92] Ungar and Crowley [94] Bao et al. [53]

Bowers and Mudawar [95] Fukano and Kariyasaki [25] Mishima and Hibiki [26] 185

Triplett et al. [12] Narrow et al. [34] Lowry and Kawaji [27] Ali and Kawaji [29] Ali et al. [30] Mishima et al. [31] Fourar and Bories [33]

Yan and Lin [96, 97] Ekberg et al. [35]

Channel characteristics

Fluid(s)

Adiabatic circular channels, D : 1, 1.5 mm Adiabatic copper tubes, D : 0.66 mm ( : 2 m), 1.17 mm ( : 3.5 m) Adiabatic circular tubes, D : 1.46—3.15 mm Glass and copper tubes, D : 0.74—1.9 mm

R-12 R-12

Heated copper channels (subcooled boiling) D : 0.51, 2.54 mm Circular channels, D : 1—26 mm Pyrex and aluminum circular channels, D : 1.05—4.08 mm Pyrex circular channels, D : 1.1, 1.45 mm; semitriangular channels, D : 1.1, 1.49 mm C Glass seven-rod bundle, D : 1.46 mm C Rectangular, W : 8 cm, L : 8 cm, S : 0.5, 1, 2 mm Rectangular, W :80 mm, L :240 mm, S : 1.465 mm Rectangular, W : 80 mm, L : 240 mm, S : 0.778 and 1.465 mm Rectangular, W : 40 mm, L : 1.5 m, S : 1.07, 2.45, 5.0 mm Rectangular glass slit, W : 0.5 m, L : 1 m, S : 1 mm; brick slit, W : 14 cm, L : 28 cm S : 0.18, 0.4, 0.54 mm Circular, D : 2 mm; 28 parallel pipes with condensation and evaporation Glass annuli; D : 6.6 mm, D : 8.63 mm; and M D : 33.15 mm, D : 35.2 mm; L : 35 cm G M

Flow Range Re : 4.3;10—6.4;10 * G : 1.44;10—5.09;10 kg/ms P : 6.3—13.2 bar; T : 0—17 K GL Re 700; 450 Re 1.1;10, 0.09&x&0.98 * % 15Re 2;10; 0.05Re 4;10 % *

Ammonia Water—air, water—aqueous, glycerin solutions R-113 U *7.7 m/s *1 Water—air Water—air

0.02*U *2 m/s; 0.1*U *30 m/s *1 %1 0.02*U *2 m/s; 0.1*U *50 m/s *1 %1

Water—air

0.02*U * 8 m/s; 0.02*U *80 m/s *1 %1

Water—air Water—air Water—air Water—air

0.03*U *5 m/s; 0.02*U *40 m/s *1 %1 0.1*U *8 m/s; 0.1*U *18 m/s *1 %1 0.15*U *16 m/s; 0.2*U *7.0 m/s *1 %1 0.15*U *16 m/s; 0.15*U *6.0 m/s *1 %1

Water—air

0.1*U *10 m/s; 0.02*U *10 m/s; *1 %1 0.5*x*100 0.005*U *1 m/s; 0.0*U *10 m/s *1 %1 0.1*x*40

Water—air

R-134a Water—air

Re : 200—12,000; mean quality : 0.1—0.95 * 0.1*U *6.1 m/s; 0.02*U *57 m/s *1 %1

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s. m. ghiaasiaan and s. i. abdel-khalik

Ungar and Cornwell [94] conducted experiments at high quality (0.09 & x & 0.98); their data were therefore predominantly in the annulardispersed two-phase ﬂow regime. They calculated their experimental twophase friction factors using channel-average properties, neglecting the effect of axial variations in vapor properties and quality, and compared them with predictions of several correlations. The homogeneous-equilibrium mixture model (Eqs. (44)—(47)), when applied with McAdam’s correlation for mixture viscosity, Eq. (48), provided the best agreement with data. An empirical correlation for vertical annular ﬂow, due to Asali et al. [85], also predicted their data well. Bao et al. [53] performed an extensive experimental study using air and aqueous glycerin solutions with various concentrations and calculated the experimental friction factors using channel-average properties. They compared their data with the correlations of Lockhart and Martinelli [59], Chisholm [80], Friedel [81], and Beattie and Whalley [82], apparently without consideration for the effect of channel roughness. With the exception of the correlation of Lockhart and Martinelli [59], which did relatively well for Re & 1000, all the tested correlations failed to predict the data well. * By implementing the forthcoming simple modiﬁcation into the correlation of Beattie and Whalley [82], Eq. (61), the latter correlation predicted all their data well. The correlation of Beattie and Whalley is based on the application of the homogeneous ﬂow model, Eqs. (44), (45), and (47), and the Colebrook—White correlation (Eq. (62)) for the friction factor over the entire two-phase Reynolds number range. Bao et al. [53] modiﬁed the correlation of Beattie and Whalley [82] simply by using f : 16/Re in 2. 2. the Re & 1000 range. 2. Bowers and Mudawar [95] studied high heat ﬂux boiling in ‘‘mini’’ (D : 2.54 mm) and ‘‘micro’’ (D : 0.5 mm) channels. They applied the homogeneous-equilibrum model, Eqs. (44), (45), and (49), assuming f : 0.02 2. The homogeneous-equlibrium model could well predict the total experimental pressure drops. Because of the signiﬁcance of the acceleration pressure drop in most of their tests, the accuracy of the homogeneous-equilibrium model for the calculation of the frictional pressure drop in their experiments cannot be directly assessed. The good agreement between model-predicted and measured total pressure drops, nevertheless, may indicate that the homogeneous-equilibrium model in its entirety is adequate for similar applications. The experiments of Fukano and Kariyasaki [25] were described in Subsection B of Section III (see Table I). They compared their two-phase pressure drop data with the correlation of Chisholm [78, 79] (Eq. (53) or (54)), indicating a large discrepancy. The discrepancy was particularly signiﬁcant in the intermittent (plug and slug) ﬂow patterns. Fukano and

two-phase flow in microchannels

187

Kariyasaki [25] indicated that the pressure loss associated with the expansion of the liquid as it ﬂows from the ﬁlm region surrounding elongated bubbles into the liquid slugs is a signiﬁcant component of the total frictional pressure loss. Mishima and Hibiki [26] reported that with single-phase liquids, the Blasius correlation well predicted their turbulent friction factor data, whereas in the laminar range their data agreed with the Hagen—Poussuille relation within 2%. For calculating the two-phase frictional pressure drop, they neglected the acceleration along their test sections, and chose to modify the Chisholm correlation [78, 79], Eqs. (53) or (54), by empirically correlating the constant C according to C : 21(1 9 e\ ")

(67)

where the channel diameter, D, is in millimeters. Triplet et al. [12] measured the frictional pressure drop for air—water ﬂow in circular and semitriangular microchannels with D : 1.1 to 1.49 mm (see C Fig. 3). They compared their measured frictional pressure drops with the predictions of the homogeneous mixture method, Eqs. (44)—(48), and the correlation of Friedel [81], Eqs. (55)—(60). They noted that because of the signiﬁcant axial variation of pressure in microchannels, the gas density cannot be assumed constant. They applied a one-dimensional model, based on the numerical solution of one-dimensional mass and momentum conservation equations for the calculation of pressure drops, using the aforementioned correlations for two-phase wall friction. Overall, the homogeneous mixture model better predicted the data. Both correlations did poorly when applied to the annular ﬂow regime data, however. Yan and Lin recently measured the two-phase pressure drop and heat transfer associated with evaporation [96] and condensation [97] of Refrigerant 134a in a horizontal, 28-tube bundle consisting of tubes with 2 mm inner diameter. In calculating the frictional pressure drops for the tubes they needed to estimate the pressure losses at inlet and exit to their tube bundle, and the deceleration pressure change associated with the condensing twophase ﬂow. Following Yang and Webb [98], who investigated the twophase pressure drop associated with the adiabatic two-phase ﬂow in extruded aluminum tubes, Yan and Lin based the aforementioned estimates on the test section average quality and an average void fraction. The average void fraction was calculated using the following slip ratio correlation originally derived by Zivi [99] based on the assumption of minimum entropy generation in steady-state, annular two-phase ﬂow: S : ( / ). (68) * % Yan and Lin calculated the entrance and exit pressure losses using common-

188

s. m. ghiaasiaan and s. i. abdel-khalik

ly applied contraction and expansion models They correlated the experimental frictional pressure drops obtained in this way using a method originally suggested by Akers et al. [100], and recently applied by Yang and Webb [98], according to which

where (Akers et al. [100]):

L G COT , P : 4 f D 2. D 2 *

(69)

: G[(1 9 x ) ; x ( / ) ], (70) COT K K * % where x is the average channel quality. For condensation, Yan and Lin K [97] obtained G

f : 498.3Re\ , COT 2.

(71)

where Re : G D/ . (72) COT COT * Using their experimentally measured frictional pressure drops, Yan and Lin calculated the experimental friction factors in their evaporating twophase ﬂow tests based on the homogeneous mixture assumption, with the homogeneous density deﬁned based on channel average quality and void F friction: L G P : 4 f (73) D 2. D 2 F They, however, correlated the friction factors in terms of the aforementioned equivalent Reynolds number, deﬁned in Eq. (72), according to f : 0.11Re\ . (74) CO 2. Recently Narrow et al. [34] investigated the hydrodynamic processes associated with air—water two-phase ﬂow in a seven-rod micro-rod bundle. Their experiments were described in Section III, E. They measured the pressure drop in their experiments and compared their data with the predictions of the homogeneous mixture model (Eqs. (44)—(48)) and the correlation of Friedel [81], Eqs. (55)—(58) for two-phase frictional pressure drop. Neither correlation could satisfactorily predict the data over the entire ﬂow regime map. The correlation of Friedel could predict most of the data typically within a factor of 2, except for the data representing very low superﬁcial velocities. The homogeneous mixture model, on the other hand, consistently underpredicted the frictional pressure drop for all ﬂow patterns, ecept for the annular/intermittent and plug/slug ﬂow patterns.

two-phase flow in microchannels

189

D. Frictional Pressure Drop in Narrow Rectangular and Annular Channels For laminar, single-phase ﬂow, the measured friction factors in rectangular channels agree well with theory [101], For turbulent ﬂow, the friction factor in rectangular channels is known to depend on the Reynolds number based on hydraulic diameter, as well as on the channel aspect ratio. Jones [102] derived a simple method that allows for the application of smooth pipe laminar and turbulent friction factor correlations to rectangular channels. Accordingly, a laminar equivalent diameter, D , is deﬁned as *COT D : *D , (75) *COT C where the shape factor * is a function of the aspect ratio, W /(, and is formulated using the theoretical solution for friction factor in rectangular channels, such that for laminar ﬂow f : 64/Re ,

where the modiﬁed Reynolds number Re* is deﬁned according to Re :

GD *COT .

(76)

(77)

The function * can be found from the analytical solution of Cornish [103]. The following simple correlation, however, agrees with the aforementioned analytical solution within 2% [102]: 2 11 ( * $ ; 3 24 W

29

( . W

(78)

Using Eqs. (75)—(78), the turbulent smooth pipe ﬂow correlations can be applied to rectangular channels. Kawaji et al. [27, 29, 30], Mishima et al. [31], and Fourar and Bories [33] have reported two-phase pressure drop data dealing with narrow rectangular channels (see Table II). Lowry and Kawaji [27] indicated that the wall roughness was only 1.5 m in their test section and that the Blasius correlation for fully turbulent single-phase liquid ﬂow did extremely well for their data. This result is evidently in disagreement with the aforementioned well-known effects of the aspect ratio on the turbulent friction factor in narrow channels. Lowry and Kawaji compared their two-phase pressure drop data with the predictions of the correlation of Lockhart and Martinelli [59] and indicated that the correlation did not account for the clear dependence of the data on mass ﬂux. Their experimental two-phase multiplier, furthermore, was a weak function of U and (, and a strong function of U . The experiments of Ali *1 %1

190

s. m. ghiaasiaan and s. i. abdel-khalik

et al. [30], described earlier in Section III, G, were performed in narrow rectangular channels with ( : 0.778 mm and 1.465 mm, with the following six orientations: vertical upward and downward ﬂow; 45°-inclined upward and downward ﬂow; horizontal ﬂow between horizontal plates; and horizontal ﬂow between rectangular plates. With the exception of the last ﬂow conﬁguration, the effect of orientation on pressure drop was quite small, and the correlation of Chisholm and Laird [78], Eq. (53), agreed well with their data using C values between 10 and 20, depending on mass ﬂux. In these comparisons, for the calculation of the single-phase frictional pressure gradients, Ali et al. [30] used the following expressions for the D’Arcy friction factors, which they derived by curve ﬁtting their own experimental data. For laminar ﬂow (Re & 2300), f : 95/Re for ( : 0.778 mm, and f :94/Re for (:1.465 mm. For turbulent ﬂow (Re3500), f :0.339Re\ for ( : 0.778 mm, and f : 0.338Re\ for ( : 1.465 mm. For the transition 2300 & Re & 3500 range, they applied a linear interpolation on a logarithmic scale. For horizontal ﬂow between vertical plates, the effect of mass ﬂux on the two-phase multiplier was strong, and ﬁxed values of C could not correlate the data. For the stratiﬁed ﬂow regime in the latter conﬁguration, Ali et al. [30] derived the following expression, based on a simple separated ﬂow model that can be applied when the two phases are both either laminar or turbulent:

: [1 9 XK\]\K *

(79)

Here, m is the power of Re in the appropriate single-phase friction factor. This expression well predicted the turbulent—turbulent data of Ali et al. The experiments of Mishima et al. [31] were described in Section III, G. For single-phase ﬂow pressure drop, their results were in agreement with the recommendations of Jones [102]. They also noted good agreement between their data and a correlation proposed by Sadatomi et al. [104]. For two-phase frictional pressure drop, Mishima et al. chose the correlation of Chisholm and Laird, Eq. (53), for modiﬁcation and correlated their data according to C : 21 tanh (0.199D ) $ 21[1 9 1056 exp(90.331D )], C C

(80)

where D is in millimeters. C Fourar and Bories [33] noted that, except for the annular ﬂow regime, the frictional pressure drops could be well correlated using f:

96 , Re K

(81)

two-phase flow in microchannels

191

where the mixture Reynolds number, Re , is obtained from K 2S (U ; U ) K *1 %1 (82) Re : K K : ; (1 9 ) (83) K % * U *1 . (84) : K * U ;U *1 %1 Annular ﬂow appeared to occur at Re 4000 apparently corresponding to K the establishment of ‘‘turbulent’’ mixture ﬂow. John et al. [105] performed an extensive series of experiments dealing with critical ﬂow in cracks and slits. The work of John et al. [105] is discussed in Section VII. In view of the important effect of friction on critical mass ﬂux, John et al. measured and correlated the single-phase friction factors in their test sections. They could correlate their data according to

D \ C 9 0.866 f : 3.39 log 2

(85)

where represents the surface roughness. Ekberg et al. [35] measured pressure drops associated with air—water two-phase ﬂow in two horizontal annuli with ( : 2 and 3 mm (see Table II). When compared with predictions of a one-dimensional model based on the homogeneous ﬂow assumption, Friedel’s correlation [81] for two-phase frictional pressure drop, Eqs. (55)—(60), predicted the experimental data better than the homogeneous ﬂow wall friction model.

V. Forced Flow Subcooled Boiling A. General Remarks Cooling by the ﬂow of a highly subcooled liquid, and subcooled boiling, are the heat transfer regimes of choice in numerous applications, because of the extremely high heat ﬂuxes they can sustain at relatively low heated surface temperatures. The extensive past studies have been reviewed in a number of textbooks and monographs, including [14, 106—108]. The work by Tuckerman and Peasa [109], and that pursued by many other investigators, has shown that the inclusion of networks of microchannels cooled by subcooled liquids in circuit boards, in particular, can provide effective cooling for extremely high thermal loads. A good review of the past research dealing with single-phase ﬂow heat transfer in microchannels can be found in [110]. An extensive summary is provided by Duncan and Peterson [111].

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s. m. ghiaasiaan and s. i. abdel-khalik

These experimental studies have also shown that the hydrodynamic and heat transfer characteristics of microchannels are different from those of the commonly applied large channels. These differences, the causes of which are not fully understood, include the Reynolds number range for laminar-toturbulent ﬂow pattern transition, and the wall friction factor and forced convection heat transfer in the transition and fully turbulent regimes. Despite its evident signiﬁcance, forced-ﬂow boiling in microchannels with De < 0.1 to 1 mm has attracted little investigation in the past, although heat transfer in small channels representative of modern compact heat exchangers (with D values typically in the few-millimeters range) has been C investigated by several investigators recently [112—115]. The limited available data for microchannels, nevertheless, indicate major differences between small and commonly used large channels, with respect to the basic bubble ebullition phenomenology. The experimental data of Wambsganss et al. [114] and Tran et al. [115], for example, indicate that, unlike in large channels, in small channels used in compact heat exchangers the nucleation process, and not the forced convection process, is the dominant boiling heat transfer mechanism at high ﬂow qualities. In the forthcoming sections, the subcooled boiling phenomena, for which recent investigations have led to reasonably consistent results, are discussed. Table III is a summary of the recently published experimental studies dealing with subcooled boiling phenomena in microchannels. B. Void Fraction Regimes in Heated Channels The voidage development in heated channels with subcooled inlet conditions has been studied extensively in the past, and is described in textbooks [107, 116]. Figure 19 is a schematic of the axial void fraction distribution along a uniformly-heated channels with subcooled liquid inlet conditions. This schematic is consistent with experimental observations with water at high pressures in commonly used large channels [117]. The ﬂow ﬁeld upstream of point A is single-phase liquid, and bubbles attached to the wall can be seen at and beyond point A, referred to as the onset of nucleate boiling (ONB) point. Bubbles remain predominantly attached to the wall upstream of point B. Beyond the latter point bubbles can be seen detached from the wall. Beyond point C, referred to as the onset of signiﬁcant void (OSV), or the point of net vapor generation (NVG), the detached bubbles can survive condensation and a rapid increase in the gradient of the void fraction curve is observed. Recent low-pressure experiments with water by Bibeau and Salcudean [118—120], also carried out in test sections representative of common large channels, have shown that the phenomenology implied in the schematic of

TABLE III Summary of Recent Experimental Data Dealing with Subcooled Boiling in Small and Microchannels Author

Channel characteristics

Inasaka et al. [130] D : 1, 3 mm; L : 1, 3, 5, 10 cm; vertical

193

Vandervort et al. [131] Kennedy et al. [132] Roach et al. [135] Blasick et al. [139] Peng and Wang [153]

Fluid Water

D : 0.3—2.6 mm; L :2.5—66 mm; vertical

Water

D : 1.17, 1.45 mm, L : 22 cm; L : 16 cm; horizontal & D : 1.17, 1.45 mm, circular; D : 1.13 mm, semitriangular; & L : 22 cm; L : 16 cm; horizontal & Annuli with r : 6.4 mm and G ( : 0.724—1.0 mm; L : 17.4—19.7 cm; & horizontal Rectangular, width : 0.2—0.8 mm, depth : 0.7 mm; L : 4.5 cm; horizontal

Water

Peng and Wang [110]

Rectangular, width : 0.6 mm; depth : 0.7 mm; L : 6.0 cm

Hosaka et al. [155]

D : 0.5, 1, 3 mm; L /D : 50, vertical

Water

Water

Parameter range G : 7000, 13,000, 20,000 kg/ms; T : 20,60°C; P $ 1 bar GL G : 8400—42,700 kg/ms; P : 1—22 bar G : 800—4500 kg/ms; P : 3.44—10.34 bar G : 220—790 kg/ms; P : 2.4—9.33 bar

ONB, OSV, OFI

G : 85—1,428 kg/ms; P : 3.44—10.34 bar

OFI

Water, U : 0.2—2.1 m/s, T : 65—90°C *GL QS@GL methanol for water; U : 0.2—1.5 m/s, *GL T : 45—50°C for methanol; QS@GL P $ 1 bar Water U : 1.5—4.0 m/s; T : 30—60°C; * *GL P $ 1 bar R-113

Phenomena studied

G : 9300—32,000 kg/ms; T : 50—80°C; P : 11—24 bar QS@GL

ONB ONB, OFI OFI

Liquid single-phase and subcooled boiling heat transfer Liquid single-phase and subcooled boiling heat transfer Subcooled boiling heat transfer and CHF

194

s. m. ghiaasiaan and s. i. abdel-khalik

Fig. 19. Void fraction variation along a uniformly heated channel.

Fig. 19 is not accurate, at least for low-pressure subcooled boiling of water, where the region of attached voidage (the region beween points A and B in Fig. 19, where bubbles are presumed to grow and collapse while attached to the wall) is essentially nonexistent, and bubbles that are detached always slide on the wall before being injected into the liquid core. The phenomenology depicted in Fig. 19, nevertheless, has been the basis of successful correlations [121, 122] and analytical models [123—125] in the past. Figure 20 schematically depicts the pressure drop-ﬂow rate characteristics (the demand curve) of a heated channel subject to a constant heat load (constant heat ﬂux for a uniformly heated channel). The demand curve can be used for the analysis of static instabilities [126, 127]. When the channel is part of a forced or natural circulatory loop, the segment of the heated channel demand curve with negative slope (between points OFI and S in Fig. 20) can be unstable, and the onset of ﬂow instability (OFI) point is deﬁned as the relative minimum point on the demand curve. The occurrence of OFI is due to the increase in the channel pressure drop which results from

two-phase flow in microchannels

195

Fig. 20. Pressure drop-mass ﬂow rate characteristic curve of a uniformly heated channel.

subcooled boiling voidage, and in experiments with steady heat ﬂux (or steady mass ﬂow rate), OFI is known to occur at a ﬂow rate slightly lower (or a heat ﬂux slightly higher) than the ﬂow rate (or the heat ﬂux) that leads to OSV. The OSV point can thus be considered as a conservative estimate of OFI. C. Onset of Nucleate Boiling The onset of nucleate boiling (ONB), or boiling incipience, has been modeled by several authors. A good compilation of the existing correlations can be found in Marsh and Mudawar [128]. Most of the models are based on the assumption that at boiling incipience stationary and stable bubbles, attached to wall crevices, exist, and that the steady-state liquid temperature proﬁle is tangent to the temperature proﬁle predicted by the Clapeyron bubble superheat proﬁle. According to the model of Bergles and Rohsenow [129], one of the most widely used models of this kind, bubbles attached to the wall at the ONB point are hemispherical, and the heat ﬂux and wall

196

s. m. ghiaasiaan and s. i. abdel-khalik

temperature at ONB are related to each other according to T 9 T (r ) * q" : k U U * r

(86)

q" : h (T 9 T ), (87) *- U * where T (r ) is the local liquid temperature at a distance r away from the * heated surface. The aforementioned tangency criterion, furthermore, requires that T (r ) : T (88) * T T * : , (89) y r WP where T , the bubble temperature, accounts for the superheat required for mechanical equilibrium between the bubble and its surroundings based on Clapeyron’s relation:

R 2 T :T ;T T ln 1 ; (90) Q?R Q?R Mh r P DE An empirical curve ﬁt to the predictions of the above equations for water in the 1 bar P 136 bar, was derived by Bergles and Rohsenow [129] as q" : 5.30P [1.8(T 9 T ) ]L, n : 2.41/P (91) U-, 5 Q?R -, where q" is in W/m, P is in kPa, and temperatures are in K. U-, The preceding correlation of Bergles and Rohsenow has been compared with microchannel data with water as the working ﬂuid by Inasaka et al. [130], Vandervort et al. [131], and Kennedy et al. [132]. Inasaka et al. [130] performed experiments in heated tubes with D : 1 and 3 mm, with G : 7000 to 20,000 kg/ms. The correlation of Bergles and Rohsenow, Eq. (91), predicted the data of Inasaka et al. relatively well. Signiﬁcant scatter can be noted in their comparison results, however, with maximum dicrepancies of about < 50% in the prediction of q" . U-, Vandervort et al. [131] carried out an experimental investigation of subcooled boiling phenomena in microchannels with diameters in the D : 0.3 to 2.6 mm range, subject to high heat ﬂuxes, with water as the working ﬂuid. They reported that the correlation of Bergles and Rohsenow predicted their ONB data well. They also applied the model of Bergles and Rohsenow (Eqs. (87)—(91)), for the estimation of the released bubble diameters, and showed that the released bubbles were typically only a few micrometers in diameter. Using the estimated bubble sizes, they calculated the order of magnitude of various forces that act on the bubbles, and

two-phase flow in microchannels

197

Fig. 21. Comparison between the ONB data of Kennedy [132] and the correlation of Bergles and Rohsenow [129]. (D : 1.17 mm). (With permission from [132].)

showed that the thermocapillary (Marangoni) force and the lift force resulting from the ambient liquid velocity radient are signiﬁcant forces that must be considered in modeling of bubble ebullition phenomena in microbubbles. A more detailed discussion of these forces is presented in the forthcoming Subsection E of this section. Kennedy et al. [132] studied the ONB and OFI phenomena in heated microchannels with D : 1.17 and 1.45 mm, using water as the working ﬂuid. Figure 21 displays the comparison between the ONB data for their 1.17 mm test section and the correlation of Bergles and Rohsenow. The correlation of Bergles and Rohsenow systematically overpredicted the experimental data, typically by a factor of 2. The correlation, however, agreed reasonably well (with a slight systematic overprediction) with the data of Kennedy et al. representing their 1.45 mm diameter test section. Inasaka et al. [130] and Kennedy et al. [132] utilized the pressure drop characteristic curves of their test sections (similar to Fig. 20) for specifying the ONB conditions. Inasaka et al. identiﬁed the heat ﬂux that led to the occurrence of ONB at the exit of their test sections, for given (constant) inlet temperature and mass ﬂux, as the minimum point on the P/P versus *q" curve, with P representing the measured pressure drop in the experi5 ment with heated wall, and P representing the channel pressure drop *with adiabatic single-phase liquid ﬂow only.

198

s. m. ghiaasiaan and s. i. abdel-khalik

Kennedy et al. [132] also identiﬁed the ONB point on each pressure drop versus mass ﬂow rate (demand) curve (Fig. 20) by comparing the experimental demand curves with pressure drop—ﬂow rate characteristics representing single-phase liquid ﬂow. The ONB occurs at the point where the slopes of the two curves deviate. They could also recognize an easily audible whistlelike sound from their test section, before the onset of ﬂow instability (OFI) occurred, which was evidently due to the appearance of vapor bubbles at the test section exit and could be attributed to the occurrence of ONB. Kennedy et al. [132] compared the conditions leading to ONB predicted by the aforementioned method, with the conditions where the whistlelike sound was heard, and noted good agreement between the two methods. Several other correlations for ONB have been proposed in the past. A good review of these correlations can be found in Marsh and Mudawar [128]. Most of the correlations, however, are based on data with water only. Yin and Abdelmessih [133] and Hino and Ueda [134] have proposed correlations that are based on data with Freon 11 and Freon 113, respectively. With the exception of the correlation of Bergles and Rohsenow [129], however, these correlations have not been systematically compared with microchannel experimental data.

D. Onset of Significant Void and Onset of Flow Instability The onset of signiﬁcant void (OSV) point is usually identiﬁed in experiments with commonly applied large channels by measuring the void fraction proﬁle along the channel, and deﬁning the OSV as the point downstream of which the slope of the void fraction proﬁle is signiﬁcantly high (see Fig. 19). Since OSV occurs only slightly before the onset of ﬂow instability (OFI), however, the conditions leading to OFI can be used for estimating the OSV conditions. The latter approach has been used by Inasaka et al. [130], Kennedy et al. [132], and Roach et al. [135] in their experiments with microchannels. The OSV phenomenon has been studied extensively in the past, and several empirical correlations and mechanistic models have been proposed for its prediction [121—125]. Saha and Zuber [121] have proposed the following widely used correlation, which has been successful in predicting a wide range of experimental data dealing with commonly applied channels: St : 455/Pe for Pe & 70,000

(92)

St : 0.0065 for Pe 70,000,

(93)

two-phase flow in microchannels

199

where St :

q" U U C (T 9 T ) * * .* Q?R *

(94)

Pe :

GDC .* k *

(95)

Equation (92) represents OSV in the thermally controlled regime, where, based on the experimental observations [117], bubbles generated on the wall roll next to the wall, and are ejected into the bulk ﬂow at the point where Eq. (92) is satisﬁed. Equation (93), on the other hand, represents OSV in the hydrodynamically controlled regime, where bubbles attached to the wall act as surface roughness, and when the roughness height reaches a characteristic height the bubbles are detached because of the hydrodynamic effects. Inasaka et al. [130] compared their OFI data (deﬁned similar to Fig. 20) with the foregoing correlation of Saha and Zuber. For their 3 mm diameter test section the correlation well predicted the data. For their 1 mm diameter test section the correlation of Saha and Zuber agreed with the data reasonably well for G : 7000 kg/ms (corresponding to the thermally controlled Pe $ 4.8;10). A similar comparison, between the OFI data and the correlation of Saha and Zuber for OSV [121], was carried out by Kennedy et al. [132]. Figure 22 depicts the results of Kennedy et al. As noted, consistent with the results

Fig. 22. Comparison between the OFI data of Kennedy et al. [132] on the correlation of Saha and Zuber [121] for OSV. (With permission from [132].)

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s. m. ghiaasiaan and s. i. abdel-khalik

of Inasaka et al. [130], within the experimental parameter range, the correlation of Saha and Zuber agrees with the data reasonably well in the 20,000 Pe range. The apparent large uncertainty bands in the experiments represent only <2% uncertainty in heat ﬂux, and are a result of the integral nature of the experiments of Kennedy et al. [132]. Successful analytical models for OSV, based on the bubble detachment mechanism, have been proposed by Levy [123], Staub [124], and Rogers et al. [125, 136]. These models, which are very similar in their basic approach, assume that OSV occurs when the largest bubbles that can be thermally sustained in the steady-state thermal boundary layer that forms on the heated surface are detached from the heated surface by the hydrodynamic and buoyancy forces parallel to the heated surface. The models of Levy [123] and Staub [124] assume a high liquid mass ﬂux and neglect the effect of buoyancy force on bubble departure, and are known to do well when applied to high-pressure data for water. The model of Rogers et al. [125] considers relatively low mass ﬂux and pressure conditions, takes account of the buoyancy effect, explicitly accounts for the effect of advancing and receding contact angles (surface wettability), and is based on a model for bubble detachment due to Al-Hayes and Winterton [137]. Rogers and Li [136] recently modiﬁed the aforementioned model of Rogers et al. [125], thereby extending its parameter range of applicability. Recent experimental studies by Bibeau and Salcudean [118—120] have cast doubt on the bubble detachment phenomenon as the process responsible for OSV. Careful visual observations in the latter studies have shown that bubble ejection normal to the wall, and not bubble detachment and motion parallel to the wall, is responsible for OSV. The aforementioned detachment models [123—125, 136] do not address bubble ejection at all. Notwithstanding, these analytical models have been relatively successful in predicting experimental data. The model of Levy, brieﬂy described next, has been applied to microchannel data by Inasaka et al. [130]. According to Levy’s model [123], in a fully turbulent subcooled ﬂow ﬁeld in a heated channel bubble departure occurs when the drag force on the bubble (itself obtained from wall frictional stress) becomes equal to the resistive surface tension force, leading to y> : y

(D U* C *, :C * *

(96)

where y is the distance from the heated wall to the tip of the bubble, and U* : ( / . U *

(97)

two-phase flow in microchannels

201

The coefﬁcient C : 0.015 is an empirically adjusted parameter, and is U obtained from

The friction factor, f

*-

G f . : *U 4 2 * , is found from

(98)

10 ; , (99) *D Re C *where /D : 10\ is assumed to represent the surface roughness caused by C the presence of bubbles at the departure. Levy [123] assumed that bubbles are at saturation temperature with respect to the local ambient pressure, and can be sustained if the local liquid temperature, T ( y ), is at least at saturation. The liquid temperature * distribution furthermore, was assumed to follow the fully developed, steadystate turbulent boundary layer temperature proﬁle [138], according to which : 0.022 1 ; 2x10

f

T 9 T ( y>) : Q f ( y>, Pr ), (100) U * * where y> : yU*/ is the dimensionless distance from the wall, and * q" U Q: (101) C U* * .* Pr y>, O y> 5 * y> f ( y>, Pr ) : 5 Pr ; ln 1 ; Pr 91 , 5 & y> 30. (102) * * * 5 5 Pr ; ln[1 ; 5Pr ] ; 0.5 ln(y>/30) 30 & y> * * In fully developed and steady-state,

T 9 T : q" /h . (103) U *U * Utilizing Eqs. (100) and (102) and requiring that T ( y*) : T at OSV, one * Q?R gets [123] q" 9T ) : 5-14 9 Q f ( y> , Pr ). (104) * -14 * h *Accordingly, based on the model of Levy [123], Eqs. (97) and (104) provide the relationship between q" and the bulk liquid subcooling at OSV. U-14 Inasaka et al. [130] compared the predictions of the model of Levy [123] for OSV with their OFI data. (See Table III for the characteristics of their experiments.) The comparison results are depicted in Fig. 23. As noted, the (T

Q?R

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s. m. ghiaasiaan and s. i. abdel-khalik

Fig. 23. Comparison between the OFI experimental data of Inasaka et al. [130] and the predictions of the OSV model of Levy [123].

model agrees with the data associated with the larger test section of Inasaka et al. (D : 3 mm) reasonably well, and systematically underpredicts q" U-14 for their smaller (D : 1 mm) test section. The model of Levy [123], as well as the aforementioned models of Staub [124] and Rogers et al. [125, 136], all assume that at OSV the bubble temperature, and liquid temperature at the bubble tip, must be equal to T , Q?R the saturation temperature corresponding to the local ambient pressure. This assumption evidently neglects the bubble superheat resulting from surface tension and is appropriate for commonly used large channels where the predicted size of the bubbles is typically large enough to render the effect of surface tension on bubble temperature negligibly small. In microchannels, however, the bubbles are small (see the discussion in the forthcoming Subsection E).

two-phase flow in microchannels

203

The model of Levy [123] can be corrected for the effect of surface tension on bubble temperature by requiring that T : T ( y>) : T ; T (105) * Q?R N where, assuming that the bubble diameter is approximately equal to Y , and using Clapeyron’s relation,

1 1 2T Q?R 9 . T : N Y * hfg J 2

(106)

Results of the modiﬁed Levy model are also depicted in Fig. 23 and are noted to agree better with the entire data of Inasaka et al. [130]. An extensive experimental study of the OFI phenomenon in microchannels cooled with water was recently carried out at the Georgia Institute of Technology, for the purpose of generating the data bases needed for the design of the proposed Accelerator Production of Tritium (APT) system [132, 135, 139]. A summary of the parameter ranges of these experiments is included in Table III. The OFI data of Kennedy et al. are compared with the correlation of Saha and Zuber [121] for OSV in Fig. 22. The experimental data of Roach et al. [135] deal with OFI at very low ﬂow rates, in channels with the cross-sectional geometries displayed in Fig. 24. Test sections (a) and (b) were uniformly heated circular channels, and

Fig. 24. Cross-sectional geometries of the test sections of Roach et al. [135]. (With permission from [135].)

204

s. m. ghiaasiaan and s. i. abdel-khalik

test sections (d) and (e) were meant to represent the ﬂow channels in a micro-rod bundle with triangular array. Test section (d) was uniformly heated over its entire surface, while the test section (e) was heated over the surfaces of the surrounding rods. Roach et al. also examined the effect of dissolved noncondensables on OFI by performing similar experiments with fully degassed water and with water saturated with air with respect to the test section inlet temperature and exit pressure. The bulk of the data indicated that OFI occurred when the coolant at channel exit had a positive equilibrium quality, indicating that, unlike in large channels and microchannels subject to high heat ﬂuxes and high coolant ﬂow rates, subcooled voidage was insigniﬁcant in these experiments. Figure 25 displays typical data. These results show that the commonly used models for OFI, which emphasize subcooled voidage, or use the onset of signiﬁcant void (OSV) as an indicator for the eminence of OFI, may be inapplicable for microchannels under low ﬂow conditions. In comparison with tests with degassed water, the total channel pressure drops in tests with air-saturated water were consistently and rather signiﬁcantly larger, indicating strong desorption of the noncondensables, which contributed to channel voidage and therefore

Fig. 25. OFI equilibrium qualities in the experiments of Roach et al. [135]. (With permission from [135].)

two-phase flow in microchannels

205

increased the total channel pressure drop. The impact of the noncondensables on the conditions leading to OFI was small, however. With all parameters including heat ﬂux unchanged, the mass ﬂuxes leading to OFI in air-saturated water experiments were different than in degassed water experiments, typically by a few percent. Blasick et al. [139] investigated the OFI phenomenon in uniformly heated horizontal annuli, using six different test sections, all with an inner radius of 6.4 mm, and gap widths in the 0.72—1.00 mm range. Among the parametric effects they examined was the impact of the inner-to-outer surface heat ﬂux ratio (varied in the 0—- range), which was found to be negligible. Kennedy et al. [132], Roach et al. [135], and Blasick et al. [139] developed simple and purely empirical correlations for their OFI data by comparing the ﬂow and boundary conditions that lead to OFI with those leading to saturation at the exit of their test sections. E. Observations on Bubble Nucleation and Boiling Heterogeneous bubble nucleation and ebullition phenomena in commonly applied large channels, as the basis of nucleate boiling heat transfer mechanism, have been qualitatively well understood for decades [106—108]. The bubble formation and release period from wall crevices is generally divided into waiting and growth periods. The departure of a bubble from a wall crevice disrupts the local thermal boundary layer, and the waiting period represents the time during which a fresh thermal boundary layer capable of initiating bubble growth on the crevice forms. The bubble growth during the growth period is primarily due to the evaporation of a liquid microlayer that separates the bubble from the heated surface, and the bubble is detached from the solid surface when the buoyancy and hydrodynamic forces that attempt to displace the bubble overcome the resistive forces, mainly the surface tension force. Models based on the aforementioned phenomenology have been published, among others, by [140, 141]. The bubble ebullition process in reality is highly stochastic, however, and accordingly semiempirical correlations have been proposed for the nucleation site size number and distribution [142, 143], and bubble maximum size and frequency [144—146]. The applicability of the aforementioned models and correlations to microchannels is questionable, however. In extremely small channels very high wall temperatures are required for the generation of bubbles. Lin et al. [147], for example, could produce bubbles in water, methanol, and FC 43 liquids in 75 m deep microchannels by raising the channel wall temperature to the proximity of the liquid critical temperature. In microchannels of interest to this article, furthermore, the velocity and temperature gradients near the wall can be extremely large,

206

s. m. ghiaasiaan and s. i. abdel-khalik

leading to the formation and release of extremely small microbubbles. Because of the occurrence of very large temperature and velocity gradients and the small bubble size, furthermore, forces such as the thermocapillary (Marangoni) force and the lift force arising from the velocity gradient become important [131]. These forces are generally neglected in the modeling of bubble ablution phenomena in large channels. Vandervort et al. [131] investigated the heat transfer associated with the ﬂow of highly subcooled water at high velocity in microchannels with 0.3—2.5 mm diameter. At the relatively low mass ﬂux of G : 5000 kg/ms, with a wall heat ﬂux that was about 70% of the heat ﬂux that would lead to CHF at the test section exit, the ﬂow ﬁeld at the test section exit was foggy, indicating the presence of large numbers of micro bubbles too small to be discernible individually. The occurrence of fogging required higher heat ﬂuxes as the mass ﬂux was increased, and no fogging was visible at G : 25,000 kg/ms. Vandervort et al. [131] analytically estimated the size of the bubbles released from the wall crevices, and the magnitude of forces acting on them, for the following typical test conditions: D : 1.07 mm, L /D : 25, P : 1.2 MPa, G : 25,000 kg/ms, and T : 100°C. QS@ The diameter of the released bubble, as predicted by the model of Levy [123] in the latter author’s analysis of the onset of signiﬁcant void (OSV), was only 2.7 m. The estimated magnitudes of other forces acting on such a bubble, while it is still attached to the heated surface, are depicted in Fig. 26, where F , F , and F represent the forces due to surface tension, drag, Q B @ and buoyancy, respectively. The forces F and F are due to the generated TR JK vapor thrust and the inertia of the liquid set in motion by the growing bubble, respectively. All the latter forces are generally accounted for (and some are neglected because of their relatively small magnitudes) in bubble ebullition analysis for common large channels. The Marangoini force F , K which is small for large bubbles and is therefore usually not considered in bubble ebullition models for large channels, can be estimated from [148]

D F :' K 2 T

T . y

(107)

Vandervort et al. [131] also estimated the magnitudes of forces that act on the aforementioned bubble, once it is detached from the solid surface, as depicted in Fig. 27, where F is the lift force that results from the local liquid J velocity gradient and can be obtained from [149, 150] 'D U *. F :C (U 9 U ) J 4 r 6 * *

(108)

two-phase flow in microchannels

207

Fig. 26. Magnitudes of various forces acting on microbubble formed at ONB conditions in a microchannel with D : 1.07 mm [131]. (With permission from [131].)

Based on air—water experimental data in a 57.1 mm diameter test section, Wang et al. [150] correlated the coefﬁcient C in the preceding expression as C : 0.01 ;

0.490 log % ; 9.3168 cot\ , ' 0.1963

(109)

where D dU D 1 U * % , (110) U 9 U dr D Re U % * where D and D are the bubble and channel diameters, respectively, and % : e\?

Re : D U 9 U / (111) % * * U : 1.18(g / ) . (112) * The force F is a near-wall force that opposes the contact between the 5 bubble and the wall and arises because of the hydrodynamic resistance associated with the drainage of the liquid ﬁlm between the bubble and the surface when the bubble approaches the surface. A similar force opposes the coalescence of bubbles, and bubble—particle coalescence, in ﬂotation [151].

208

s. m. ghiaasiaan and s. i. abdel-khalik

Fig. 27. Magnitudes of various forces acting on a microbubble detached from the wall in a microchannel with D : 1.07 mm [131]. (With permission from [131].)

Based on a two-dimensional analysis, Antal et al. [152] derived

'D 2 (U 9 U ) D * * J F : C ;C 5 5 5 2y 6 D

,

(113)

where: : 90.104 9 0.06(U 9 U ) (114) 5 % * C : 0.147, (115) 5 where y is the distance from the wall. Observations consistent with those reported by Vandervort et al., have also been reported by Peng and Wang [110, 153]. The latter authors have studied the forced convective boiling and bubble nucleation associated with the ﬂow of subcooled deionized water and methanol in microchannels with C

two-phase flow in microchannels

209

rectangular cross sections, 0.2—0.8 mm wide, and 0.7 mm deep, with nearatmospheric test section exit pressure. The microchannels were heated on one side through a metallic cover and were covered by a transparent cover on the other side. The experimental boiling curves of Peng and Wang [110] indicated essentially no partial boiling in their microchannels, and in the portions of the boiling curves that indicated fully developed boiling the effects of liquid velocity and subcooling were small. No visible bubbles occurred in the heated channel, however, even under conditions clearly representing fully developed boiling, and instead a string of bubbles could be seen immediately beyond the exit of each test section. Peng et al. [154] have hypothesized that true boiling and bubble formation are possible if the microchannel is large enough to provide an ‘‘evaporating space’’; otherwise a ‘‘ﬁctitious boiling’’ heat transfer regime is encountered where the wellknown fully developed boiling heat transfer characteristics (e.g., lack of sensitivity of the heat transfer coefﬁcient to the bulk liquid velocity and subcooling) occur without visible bubbles. Hosaka et al. [155] have argued that careful experiments are needed to determine whether passage dimensions and length sales peculiar for each ﬂuid affect the boiling phenomena in microchannels. Evidently, experiments aimed at careful elucidation of the bubble ebullition and other phenomena associated with boiling in microchannels are needed.

VI. Critical Heat Flux in Microchannels A. Introduction Forced convection subcooled boiling in small channels is among the most efﬁcient known engineering methods for heat removal and is the cooling mechanism of choice for ultrahigh heat ﬂux (HHF) applications, such as the cooling of fusion reactor ﬁrst walls and plasma limiters where heat ﬂuxes as high as 60 MW/m may need to be handled. Critical heat ﬂux represents the upper limit for the safe operation of cooling systems that depend on boiling heat transfer, and adequate knowledge of its magnitude is thus indispensable for the design and operation of such systems. Critical heat ﬂux has been investigated extensively for several decades. The majority of the investigations in the past three decades have dealt with the safety of the cooling systems of nuclear reactors, however. Some recent reviews include [106, 156—159.] The complexity of the CHF process, the lack of adequate understanding of the phenomenology leading to CHF, and the urgent need for predictive methods have led to more than 500 empirical

210

s. m. ghiaasiaan and s. i. abdel-khalik

correlations in the past [157]. The available experimental data are extensive and cover a wide parameter range. Most of the data, nevertheless, deal with water only, and until recently data have been scarce for certain parameter ranges, such as low-ﬂow, low-pressure CHF in small channels. Some CHF experimental investigations in the past have included channels with D * 0 (1 mm) [95, 131, 155, 160—175], and microchannel CHF data & have been included in the databases of some of the widely used CHF correlations [176—178], apparently without consideration of the important differences between micro- and large channels. Systematic investigation of CHF and other boiling/two-phase ﬂow processes in microchannels, however, have been performed only recently. Most of the reent studies were concerned with the aforementioned cooling systems of fusion reactors, where channels with D $ 1—3 mm and with large L /D ratios carry highly subcooled water with high mass ﬂuxes and are subjected to large heat ﬂuxes [155, 160—173]. A few experimental investigations have also addressed CHF in microchannels under low mass ﬂux and low wall heat ﬂux conditions [95, 174]. In the forthcoming sections, the recently published data, models, and correlations relevant to CHF in microchannels are reviewed. In Subsections B and C the existing CHF data obtained with channels with D * 0 (1 mm) & and their important trends are discussed. The empirical correlations that have been recently applied to CHF in microchannels are discussed in Section D. In Section E the relevant theoretical models are discussed. B. Experimental Data and Their Trends Table IV provides a summary of recent experimental investigations and includes some older microchannel data previously reviewed by Boyd [156] and utilized for model validation by Celata et al. [175]. Among the investigations listed, only the data of Bowers and Mudawar [95] and Roach et al. [174] were obtained in horizontal heated channels, and all other experiments dealt with ﬂow in vertical channels The depicted list does not include experiments where enhancement techniques such as internal ﬁns and swirl ﬂows were utilized. Boyd [157] carried out a detailed assessment of the important parametric trends based on the data associated with subcooled ﬂow CHF available in 1983, and identiﬁed parameter ranges in need of further experimental research. With respect to the fusion reactor applications, Boyd [156, 157] recommended experiments with large L /D. The scarcity of data at low pressure is also evident in Table IV. Experiments with large L /D and at low pressure were subsequently performed by Boyd in channels with D : 3 and 10.2 mm diameters [167, 168]. The CHF experimental investigations in

TABLE IV Summary of Experimental Data Dealing with CHF in Small Channels

Source

211

Ornatskiy [160]? Ornatskiy and Kichigan [161]? Ornatsky and Vinyarskiy [162]? Loomsmore and Skinner [163]? Daleas and Bergles [164]@ Subbotin et al. [165] Katto and Yokoya [166] Boyd [167] Nariai et al. [169, 170]

Channel characteristics

Mass ﬂux (Mg/ms)

Inlet conditions

Critical heat ﬂux (MW/m)

D : 0.5 mm, L : 14 cm, vertical D : 2 mm, L : 56 mm, vertical

Water Water

1.0—3.2 1.0—2.5

20—90 5.0—30.0

T : 1.5—154°C GL T : 2.7 9 204.5°C GL

41.9—224.5 6.4—64.6

D : 0.4—2.0 mm, L : 11.2—56 mm, vertical

Water

1.1—3.2

10.0—90.0

T : 6.7—155.6°C GL

27.9—227.9

D : 0.6—2.4 mm, L : 6.3—150 mm, vertical

Water

0.1—0.7

3.0—25.0

T : 3.2—130.9°C GL

6.7—44.8

D : 1.2—2.4 mm, L /D : 14.9—26, vertical

Water

0.2

1.52—3.0

D : 1.63 mm, L : 180 mm, vertical

Helium

0.1—0.2

0.08—0.32

D : 1 mm, L /D : 25—200, vertical

Liquid He

0.199

11—10

Water Water

0.77 at exit 0.1

4.6—40.6 6.7—20.9

Water R-113 Water Water

0.3—1.1 1.1—2.4 0.6—2.6 0.1—2.3

4.3—30 9.3—32.0 10.1—40.0 8.4—42.7

R-113

0.138 at inlet

Water

0.344—1.043 at exit

0.031—0.15 for D : 2.54 mm; 0.12—0.48 for D : 0.5 mm 0.25—1.0

D : 3 mm, L /D : 96.9, horizontal D : 1, 2, 3 mm; L : 1.0—100 mm, vertical Inasaka and Nariai [171] D : 3 mm, L :100 mm, vertical Hosaka et al. [155] D : 0.5, 1, 3 mm; L /D : 50, vertical Celata et al. [172] D : 2.5 mm, L : 100 mm, vertical Vandervort et al. D : 0.3—2.6 mm, L : 2.5—66 mm, vertical [131, 173] Bowers and Mudawar D : 0.51, 2.54 mm, L : 10 mm, horizontal [95] Roach et al [174]

Fluid

Pressure (MPa)

D : 1.17, 1.45 mm, circular; D : 1.13 mm, & semitriangular; L : 160 mm; horizontal

?From Celata, Cumo, and Mariani [175]. @From Boyd [156].

0.31—3.1 x 2 90.25 GL h 9 h : 93.5 to D GL ;7.0 kJ/kg T : 20°C GL T : 15.4—64°C GL T : 25—78°C GL T : 50—80°C T QS@GL : 29.8—70.5°C GL T : 6.4—84.9°C GL T

QS@GL

6.25—41.58 4.6—70 7.3—44.5 12.1—60.6 18.7—123.8

: 10—32°C

T : 49—72.5°C

0.86—3.7

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s. m. ghiaasiaan and s. i. abdel-khalik

[169, 170], as noted, are primarily focused on low pressure and high mass ﬂux. The experiments of Boyd [167] all represented CHF under high heat ﬂux and subcooled bulk liquid conditions; the local subcooling at their test section exit varied in the 30—74°C range. The CHF varied linearly with G in Boyd’s experiments and was correlated accordingly [167]. Boyd [168] examined the effect of L /D on CHF in a 1.0 cm-diameter channel. The experimental data of Nariai et al. [169, 170] include subcooled as well as saturated (two-phase) CHF data. The dependence of CHF on channel diameter was found to vary with the local quality. When CHF occurred in subcooled bulk liquid, CHF monotonically increased as D decreased. With CHF occurring under x 0 conditions, however, the trend was reversed and CHF decreased with decreasing D. The aforementioned trend, i.e., increasing CHF in subcooled forced ﬂow as D is decreased, had been noted earlier by Bergles [179], who suggested that three mechanisms, all of which deal with the vapor bubbles as they grow and are released from wall crevices, lead to increasing CHF as D becomes smaller. As D is decreased, (a) the vapor bubble terminal diameter (the diameter of bubbles detaching from the wall) decreases as a result of larger liquid velocity gradient; (b) the bubble velocity relative to the liquid is increased; and (c) condensation at the tip of bubbles is stronger due to the large temperature gradient in the liquid. Nariai et al. [169, 170] thus explained the aforementioned trend of increasing CHF with decreasing D in subcooled liquids by arguing that smaller bubbles imply a thinner bubble layer and a smaller void fraction, and lead to a higher CHF. A recent systematic assessment of the effect of channel diameter on CHF in subcooled ﬂow by Celata et al. [180], based on experimental data from several sources, has conﬁrmed the aforementioned mechanism. Hosaka et al. [155], in their experiments with R-113, observed a similar trend and attributed the increase in CHF associated with decreasing D to the decreasing bubble terminal size. The trends of the available data, however, indicate that a threshold diameter exists beyond which the effect of channel diameter on subcooled ﬂow CHF is negligible. Figure 28 depicts the results of Vandervort et al. [173]. Below a threshold diameter (about 2 mm for the depicted data), CHF in subcooled ﬂow increases with decreasing D, whereas for larger diameters the inﬂuence of variations in D on CHF is small. CHF is more sensitive to D at higher values of G. Similar trends have been noted by some other investigators [180]. The magnitude of the aforementioned threshold diameter, which is likely to depend on geometric as well as thermal—hydraulic parameters, may not be speciﬁed with precision at the present time because of the limited available data. It should be mentioned that the foregoing trend (i.e., increasing CHF with decreasing D) applies when CHF occurs in subcooled bulk ﬂow. An

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213

Fig. 28. Parametric dependencies in the CHF data of Vandervort et al. [131]. (With permission from [131].)

opposite trend was reported by Nariai et al. [169, 170] for CHF occurring when x 0. CO The experiments of Celata et al. [172] covered the intermediate and low pressure range of 0.6—2.6 MPa, and were all carried out in the relatively large 2.5 mm diameter test section. They are, however, part of a database utilized by Celata et al. [175, 181] for the validation of various models and correlations, as well as the identiﬁcation of some important trends in the CHF data. Vandevort et al. [173] systematically examined the effects of inlet subcooling, channel diameter, pressure, and length-to-diameter ratio, dissolved noncondensable gas, and heated wall material on CHF of subcooled water ﬂow. Their data, along with data from several other sources, were used for parametric trend identiﬁcation by Celata [181]. Vandervort et al. [173] noted the frequent occurrence of premature burnout in their tests, which they deﬁned as any thermal failure not directly attributable to CHF or other obvious failure mechanisms. Premature failure occurred following boiling

214

s. m. ghiaasiaan and s. i. abdel-khalik

incipience, which was typically accompanied by a ‘‘boiling song’’ in the test section. Each test section that failed had some region experiencing incipient boiling. The majority of the premature failures occurred in two types of tubing (1.9 and 2.4 mm diameter stainless steel capillary tubing) and otherwise showed no discernible dependence on the primary variables, P, G, and subcooling. Although the cause of, and the conditions leading to, these premature failures could not be identiﬁed with certainty, the evidence indicated that they resulted from some thermal—hydraulic phenomenon subsequent to incipient boiling. The development of a metastable superheated liquid because of the scarcity of wall crevices, which can lead to sudden and explosive boiling, was mentioned as a possible cause, and it was argued that channel wall roughness may thus be a stabilizing factor that reduces the possibility of premature burnup. The experiments of Bowers and Mudawar [95] and Roach et al. [174] addressed CHF under very low mass ﬂux conditions. Bowers and Mudawar [95] compared the characteristics of a ‘‘mini’’ (D : 2.54 mm) and a ‘‘micro’’ (D : 0.51 mm) channel. CHF occurred when the channel exit equilibrum quality was quite high, typically at x ) 0.5 for the larger channel and CO x ) 1 for the smaller channel. At very low mass ﬂuxes, furthermore, CO superheated vapor exited from the test section and the CHF results were insensitive to the inlet subcooling. Figure 29 displays the test section exit qualities measured by Bowers and Mudawar, where the Weber number is deﬁned as We : GL /( ), and L is the heated length. For the smaller & D &

Fig. 29. Equilibrium quality at CHF in the experiments of Bowers and Mudawar [95].

two-phase flow in microchannels

215

(micro) test section (D : 0.51 mm) the equilibrium quality at CHF was evidently quite high and could approach 1.5. These high equilibrium qualities imply that a metastable superheated liquid ﬂow occurred upstream of the CHF point in the channel. The potential occurrence of metastable superheated liquid also seems to be consistent with the observations of Peng and Wang [110, 153] in their experiments dealing with boiling in microchannels. The latter authors studied boiling heat transfer in a channel with a 0.6 mm;0.7 mm rectangular channel, and did not observe visible bubbles in their test section even under conditions that implied fully developed boiling. Instead, strings of bubbles could be seen at the exit of their test section. The experiments of Roach et al. [174] dealt with CHF in subcooled water at low mass ﬂuxes in heated microchannels. The results were consistent with the aforementioned observations of Bowers and Mudawar [95]. CHF occurred when x ) 0.36 at the exit of their test sections, suggesting the CO occurrence of dryout; and x ) 1 was noted in many of their tests, CO suggesting the potential occurrence of metastable superheated liquid ﬂow. C. Effects of Pressure, Mass Flux, and Noncondensables CHF is affected by more than 20 parameters, which include subcooling, pressure, channel diameter, length, surface conditions and orientation, heat ﬂux distribution, dissolved noncondensables, and various thermophysical properties [157]. Although the available microchannel data are limited and do not allow for a systematic assessment of all dependencies, the existing database associated with CHF in subcooled water ﬂow is sufﬁcient for the identiﬁcation of some important trends applicable to high mass ﬂux CHF, where CHF occurs under high local subcooling conditions. A useful systematic study of various parametric effects associated with CHF in microchannels was performed by Vandervort et al. [173]. Utilitizing the experimental data of several authors, Celata [181] assessed several important parametric dependencies. The dependence of CHF on pressure is in general monotonic. At pressures well below the critical pressure, P* , CHF is expected to increase with AP increasing pressure [156]. The available data relevant to subcooled CHF in microchannels (which virtually all represent P & P* ) indicate that CHF is AP insensitive to pressure [173, 181]. A slight decreasing trend in CHF with respect to increasing pressure has been noted by Vandervort et al. [173] and Hosaka et al. [155], however. CHF monotonically inceases with increasing mass ﬂux; it increases monotonically, and approximately linearly, with increasing local subcooling. Figure 30 depicts the effect of subcooling on CHF in the experiments of Vandervort et al. [131, 173].

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Fig. 30. The effect of exit subcooling on CHF in the experiments of Vandervort et al. [173]. (With permission from [173].)

Vandervort et al. [173] and Roach et al. [174] attempted to measure the effect of dissolved air in subcooled water on CHF. The impact of the release of the dissolved noncondensables on liquid forced-convection in microchannels were discussed in Section III, H, and the impact of dissolved air on critical (choked) ﬂow in cracks and slits is discussed in Section VII, D. The experimental results of both Vandervort et al. [173] and Roach et al. [174] indicated a negligibly small effect of dissolved air on CHF. Since the solubility of air in water is very low, and in view of the fact that considerable evaporation due to boiling occurs in CHF, the insigniﬁcant contribution of dissolved air to CHF is expected. It should be noted, however, that for other ﬂuid—noncondensable pairs for which the solubility of the noncondensable in the liquid is high, the impact of the dissolved noncondensable may not be negligible. D. Empirical Correlations Most of the more than 500 models and correlations proposed in the past for CHF are applicable over limited parameter ranges and are often in disagreement with one another. Good reviews can be found in [157—159]. In this section, only empirical correlations that have recently been applied to, and have been successful in predicting, some microchannel CHF data are discussed.

two-phase flow in microchannels

217

The following simple empirical correlation for subcooled ﬂow CHF was proposed by Tong [182]: G q" D . !&$ : C (116) D h DE Tong [182] correlated the coefﬁcient C in terms of the local equilibrium quality, x , according to CO C : 1.76 9 7.433x ; 12.222x . (117) CO CO Nairai et al. [169, 170] applied the correlation of Tong [182] to their experimental data and noted that in order to achieve agreement they needed to correlate the parameter C separately for low and high heat ﬂux conditions. More recently, Celata et al. [175] noted that the data used by Nariai et al. [169, 170] were limited to relatively low heat ﬂux and low pressure, and their modiﬁcation of the correlation of Tong was inadequate. Celata et al. [175] correlated the parameter C according to C : (0.216 ; 4.74;10\P)3

(118)

3 : 0.825 ; 0.987x , for 90.1 & x & 0 (119) CO CO 3 : 1 for x & 90.1 (120) CO (121) 3 : 1/(2 ; 30x ) for x 0, CO CO where P must be in MPa in Eq. (118). This modiﬁed Tong correlation evidently should apply to saturated exit conditions as well. Celata et al. [175] indicated that the preceding correlation could predict 98.1% of their compiled data points (which covered 0.1 & P & 9.4 MPa, 0.3 & D & 25.4 mm, 0.1 & L & 0.61 m, 2 & G & 90 Mg/ms, and 90 & T & 230 K) QS@ within <50%. The agreement of the correlation with the microchannel data included in the database of Celata et al., furthermore, appeared to be satisfactory. Celata et al. [172, 175] also compared their compiled database with the predictions of several other empirical correlations, generally with poor agreement in comparison with the aforementioned modiﬁed-Tong correlation. Hall and Mudawar [183] have recently assessed the validity of previously published CHF experimental data and have compiled a qualiﬁed CHF database (referred to as the PU-BTPFL CHF Database). This database includes experiments representing 0.3 D 45 mm, 10 G 2484 kg/ms, and 92.25 x 1.0, in vertical, upﬂow tubes, with x representing the CO CO local equilibrium quality at the CHF (i.e., the end of the heated segment of test sections) point. They compared 25 widely referenced correlations dealing with CHF in vertical, upﬂow channels with their database and

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showed that the empirical correlations of Caira et al. [184] provided the most accurate predictions. The correlation of Bowring [178] also showed relatively good agreement with data. Although the PU-BTPFL Database and the aforementioned correlations generally address vertical channels with upﬂow, the data included in the database that represent small channels (D * 1 mm) may represent horizontal microchannels as well because of the small inﬂuence of channel orientation with respect to gravity on two-phase ﬂow in such microchannels. The correlation of Bowring [178] can be expressed as A 9 DGh x /4 DE CO C

(122)

2.317(h DG/4)F DE 1 ; 0.0143F DG

(123)

q" : !&$ where q" is in W/m, and !&$ A: C:

0.077F DG G L 1 ; 0.347F 1356

n : 2.0 9 0.5P 0 P : 0.145P. 0

(124)

(125) (126)

P in Eq. (126) is in MPa, and F : P exp[20.891(1 9 P )] ; 0.917 /1.917 0 0

(127)

F : 1.309F / P exp[2.444(1 9 P )] ; 0.309

0 0

(128)

F : P exp[16.658(1 9 P )] ; 0.667 /1.667 0 0

(129)

F : F P . 0

(130)

The preceding correlation could predict the low and high mass velocity data of the PU-BTPFL database with mean absolute errors of 21.9 and 53.5%, respectively. The correlation of Bowring systematically underpredicted the data of Roach et al. [174], on the average by 36%. The correlation of Caira et al. [184] can be represented as q" : !&$

; [0.25(h 9 h) ]W1! GLJCR D 1 ; LW

(131)

two-phase flow in microchannels

219

where

: y DWGW M 1! : y DWGW

: y DWGW. All parameters are in SI units, and

(132) (133) (134)

y : 10,829.55 y : 90.0547 y : 0.713 y : 0.978 y : 0.188 y : 0.486 y : 0.462 y : 0.188 y : 1.2 y : 0.36 y : 0.911. The preceding correlation agreed with the low and high mass velocity data in the PU-BTPFL Database with mean absolute errors of 16.5 and 22.6%, respectively. Caira’s correlation agreed with the data of Roach et al. [174], with an average overprediction of the data by only 18%. Vandervort et al. [173] developed a statistical correlation based on their subcooled ﬂow water CHF data. Their correlation, however, includes more than 20 constants. The experimental data of Bowers and Mudawar [95], as noted in the previous section, represented low-ﬂow CHF, where CHF at high equilibrium qualities occurred. Noting the insensitivity of their data to inlet subcooling, Bowers and Mudawar developed the following empirical correlation:

GL \ q !&$ : 0.16 (L /D)\ . (135) G D FDE This correlation is remarkable for the implied recognition of the importance of surface tension. The correlation, however, has not been validated

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against data from other sources and is unphysical in its monotonic dependence on D. Shah [185] has developed an empirical CHF correlation based on a vast pool of data representing upﬂow in vertical channels with parameter ranges 0.315 D 37.5 mm, 1.3 L /D 940, 4 G 29041 kg/ms, 0.0014 Pr 0.96, and inlet qualities covering 92.6 to 1.0. Shah’s database includes a wide variety of ﬂuids, and his empirical correlation is dimensionless and utilizes all important thermophysical properties. Shah’s correlation has two versions: the upstream condition correlation (UCC), and the local condition correlation (LCC). The UCC vesion can be expressed as q" /G : 0.124(D/L ) (10/Y )(1 9 x ) (136) !&$ FDE # G# where x and L are the effective inlet equilibrium quality and effective tube G# # length, respectively, and n is an empirical exponent. When x 0, L is the G # axial distance from the channel inlet and x : x ; when x 0 L is equal G# G G # to the boiling length (i.e., the axial distance from the point where equilibrium quality is equal to zero) and x : 0. Distinction is made between G# helium and other ﬂuids. For helium n : (D/L ) , and for other ﬂuids # n : (D/L ) , for Y 10 (137) # 0.12 , for Y 10. (138) n: (1 9 x ) G# The parameter Y (Shah’s correlating parameter) is deﬁned as GDC N* ( gD/G)\ ( / ) . * * % k * The LCC correlation of Shah can be expressed as Y:

(139)

q" /Gh : F F Bo , (140) !&$ DE # V M where F , the entrance effect factor, is the smaller of 1 and [1.54 9 0.032(L / # A D)], with L representing the axial distance from entrance. Parameters Bo A M and F are functions of the local quality, reduced pressure, P* , and the P V parameter Y. Shah recommends that the UCC correlation be used when Y 10 or L 160/P1 ; otherwise, the correlation version predicting a P # lower q" should be chosen. Hosaka et al. [155] compared the predictions !&$ of Shah’s correlation with their data. On the average, the correlation overpredicted the data only slightly. The correlation, however, has not been adequately compared with other recent microchannel CHF data. Katto [159] has indicated that the strong dependence of the parameter Y in Shah’s CHF correlation on g for high mass ﬂux forced ﬂow may be physically questionable.

two-phase flow in microchannels

221

E. Theoretical Models Theoretical modeling of CHF has undergone great advances in the recent past. Most of the developed models, however, may be inapplicable to microchannels. Weisman [186] has summarized the status of theoretical models for CHF and indicated that the phenomenology of CHF depends on the two-phase ﬂow regime. In annular (high quality) ﬂow, ﬁlm dryout leads to CHF. In the slug/plug ﬂow regime, CHF appears to occur if the time period associated with the passage of a gas plug is long enough to allow for the complete evaporation of the liquid ﬁlm that is left behind on the wall following passage of a liquid slug. In highly subcooled ﬂow, CHF is triggered by the thermal and hydrodynamic processes adjacent to the heated wall. Two modeling approaches have been pursued for CHF in highly subcooled ﬂow. Weisman and co-workers [187, 188] suggested that the coalescence of microbubbles that form on the wall and the occurrence of a critical void fraction in the bubble layer lead to CHF. For subcooled or low-quality CHF, based on careful ﬂow visualization, Lee and Mudawar [189] proposed that CHF in the aforementioned regime occurs when the liquid sublayer that separated vapor blankets or slugs from the wall is disrupted. They developed a mechanistic model accordingly. More recent ﬂow visualization studies further support the latter liquid sublayer dryout model [190], and models based on this mechanism, and following the essential elements of the model by Lee and Mudawar [189], have recently been compared with data including small channel data by Katto [191, 192] and Celata et al. [193]. The outline of the model, as elaborated by Katto [191], is now presented. Figure 31 is a schematic of the ﬂow ﬁeld [191], where vapor slugs, formed as a result of the coalescence of smaller bubbles, are separated from the heated surface by a liquid sublayer. The vapor blankets are assumed to remain thin because of condensation, and their velocity is assumed to be closely related to the local ambient ﬂuid velocity. CHF is assumed to occur when the residence time of a vapor slug over the liquid sublayer is sufﬁciently long to allow for its complete evaporation and breakdown. Katto assumed that the local ﬂow quality can be obtained from the quality proﬁle ﬁt of Ahmad [194], according to which

x CO 9 1 x COMQT for x &x , (141) x: CO, CO x CO 19x exp 91 COMQT x COMQT where the equilibrium quality at the point of onset of signiﬁcant void, x , COMQT x 9x exp CO COMQT

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s. m. ghiaasiaan and s. i. abdel-khalik

Fig. 31. Schematic of ﬂow ﬁeld near the CHF conditions with high subcooling and high mass ﬂux. (With permission from [191].)

is obtained from the empirical correlation of Saha and Zuber [121]. The sublayer initial ﬁlm thickness at the front end of the vapor slug is found based on an empirical correlation due to Haramura and Katto [195];

h J DE , ( : 1.705;10\' J 1; J (142) DGJK q" @ * * J where the boiling heat ﬂux, q" , is obtained by assuming that the wall heat @ ﬂux is the summation of convective and boiling terms, thereby q" : q" 9 h (T 9 T ), (143) @ U $! U * where h is obtained from the well-known correlation of Dittus and $! Boelter, assuming purely liquid ﬂow, and T 9 T is obtained from U * ( 9 1)(T 9 T ) ; q" /h U $! Q?R * T 9T : M (144) U * M : 230(q" /Gh ) . (145) U M DE CHF is assumed to occur when

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223

q" : ( h /t , (146) U * DGJK DE PCQ where t , the residence time of the vapor slug over the ﬁlm underneath it, PCQ can be found from 2'( ; ) L * J . : (147) t : PCQ U U * J Here, the length of the vapor slug has been assumed to be equal to the neutral wave length according to the Helmholtz stability theory. The velocity of the vapor slug relative to the ﬁlm is assumed to follow U : ,U , (148) *B where U is the velocity at the distance ( from the wall, as predicted by *B DGJK the Karman’s universal turbulent boundary layer velocity proﬁle, based on a wall frictional shear stress found from : ( f /4)G/2 . (149) U F The homogeneous density, , is found using the ﬂow quality obtained from F Eq. (45), and the D’Arcy friction factor f is found from the Prandtl— Karman correlation: 1/( f : 2.0 log (Re( f ) 9 0.8 (150) Re : GD/[ ; (1 9 )(1 ; 2.5)]. (151) J * Katto empirically correlated the parameter , in Eq. (148) in terms of Re( / ) and [191, 192]. J * The model of Katto [191, 192] just described has been shown to predict experimental data well for channels with D 2 1 mm, for a variety of ﬂuids and a pressure range of 0.1 to 2.0 MPa. More recently, Celata et al. [193] pointed out that Katto’s model is unable to calculate the CHF when the void fraction is larger than 70%. Celata et al. further modiﬁed the aforementioned phenomenological model of Lee and Mudawar [189] and Katto [191, 192]. They assumed that the thickness of the vapor blanket (slug) is equal to the bubble departure diameter, and the vapor blanket is always surrounded with saturated liquid. Celata et al. [193] obtained the vapor blanket velocity, U , from the balance between drag and buoyancy forces, assuming a vertical, upﬂow conﬁguration. Other essential model elements were similar to [189, 191]. Celata et al. [193] compared their model with experimental data covering the following range of parameters, with good agreement between model and data: 0.2 & D & 25.4 mm, 25 T GQS@ 255 K, 10 & G & 9;10 kg/ms, and 0.1 & P & 8.4 MPa. The application of the aforementioned force balance on vapor blankets in the model of

224

s. m. ghiaasiaan and s. i. abdel-khalik

Celata, however, implies dependence on channel orientation, which may not apply to microchannels. CHF under high-quality conditions is primarily due to the ﬁlm dryout phenomenon in the annular ﬂow regime. The ﬁlm dryout models are based on the solution of two-phase mass, momentum, and energy conservation equations, whereby the ﬂow rate of the liquid ﬁlm on the heated wall and the ﬁlm thickness are calculated. CHF is assumed to occur when the liquid ﬁlm is depleted [159]. Annular-dispersed ﬂow is the ﬂow regime that occurs during ﬁlm dryout CHF in commonly used large channels, where entrained liquid droplets are mixed with the vapor phase and the two-phase ﬂow is accompanied by continuous entrainment of new droplets from the liquid ﬁlm and deposition of droplets on the ﬁlm. The variation of the ﬁlm thickness is evidently affected by evaporation as well as droplet entrainment and deposition. Dryout CHF models utilize empirical correlations for droplet entrainment and deposition [196, 197]. A model by Sugawara et al. [198] also accounts for the inhibition of droplet deposition due to the counter ﬂow of vapor resulting from ﬁlm evaporation. Film dryout models have not been systematically applied to microchannels, and the current models [196, 198] employ constitutive relations associated with interfacial transfer processes and droplet entrainment and deposition that may not be applicable to microchannels. The recent lowﬂow and high-quality CHF data [95, 174] indeed suggest that the dryout phenomenology in microchannels may be signiﬁcantly different than in larger channels.

VII. Critical Flow in Cracks and Slits A. Introduction Critical or choked ﬂow represents the maximum discharge rate of a ﬂuid through an opening connecting a pressurized vessel to a low-pressure environment. When choking happens, the ﬂow conditions downstream of a location where critical conditions occur do not affect the ﬂow rate, implying that the hydrodynamic signals originating downstream are unable to pass through the critical location. Critical ﬂow of a compressible ﬂuid can be well predicted by assuming that the one-dimensional ﬂuid velocity at the critical location is equal to the local isentropic speed of sound, and knowledge of ﬂuid stagnation properties is sufﬁcient for calculating the conditions at the critical cross-section. Critical two-phase ﬂow is considerably more complicated, however, because of the development of thermal and mechanical nonequilibria between the

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225

phases. The ﬂuid stagnation properties are thus insufﬁcient for uniquely determining the conditions at the critical cross-section in two-phase ﬂow. Critical ﬂow can be used for ﬂow control and plays an important role in a number of hypothetical nuclear reactor accidents. Critical two-phase ﬂow has been extensively studied in the past. Good reviews can be found in [106, 199, 200]. The critical ﬂow of initially highly subcooled liquids through narrow cracks and slits is of great interest in relation to the safety of nuclear and chemical reactors. When cracks occur in high-pressure piping systems, they often support critical ﬂow and, in accordance with the leak-before-break concept, their detection and correct characterization are necessary for prediction and prevention of major leaks. Extensive research effort has been devoted to the critical ﬂow in cracks and slits in the past 15 years. Most of the studies have focused on critical ﬂow in slits or simulated cracks with simple and regular cross-sections. Photomicrographs of typical intergranular stress corrosion-induced cracks in type 304 stainless steel [201], however, show that such cracks often have highly tortuous and irregular ﬂow passages. Cracks often have hydraulic diameters in the D & 1 mm range and have C large length-to-diameter ratios; their nucleation, boiling heat transfer, and two-phase ﬂow characteristics are different from those of large channels. Consequently, the correlations and models that have been developed in the past for critical ﬂow in commonly applied large channels do not apply to cracks without modiﬁcation. In the forthcoming discussion, previous experimental studies dealing with critical ﬂow of initially subcooled liquids in cracks are reviewed in Section B. The theoretical models are discussed in Sections C—E. B. Experimental Critical Flow Data Table V is a summary of the recent experimental data. Collier et al. [202, 203] performed two sets of experiments. In Phase I of their experiments, they measured critical ﬂow in artiﬁcial slits. The geometric and test parameters associated with their Phase I experiments are summarized in Table V. The test sections were produced by splitting ﬂanges and machining crack faces in the center of each half, then reassembling the two ﬂange halves. The simulated crack surfaces were roughened by shot blasting them, and the crack width, (, was adjusted with spacer blocks placed between the two ﬂange halves. They measured the temperatures and pressures at three locations along their simulated cracks. In phase II of their experiments, Collier et al. [202, 203] used a stainless steel pipe with 32.4 cm outside

TABLE V Summary of Experimental Data for Critical Flow in Cracks and Microchannels

226

Author

Fluid

Maximum inlet pressure (MPa)

Collier et al. [202, 203]

Water

11.5

33—118

Collier et al. [202, 203]

Water

11.5

0—72

Amos and Schrock [204] Kefer et al.? [205]

Water Water

16.2 16.0

0—65 0—60

John et al. [105]

Water

14

2—60

Rectangular, W : 80 mm; S : 0.2—0.6 mm

Nabarayashi et al. [206, 207]

Water

7

&30

Real cracks in pipe wall, Dp : 114.3, 216.3 mm, W : 60—160 mm, S0.5 mm; rectangular cracks, W : 30, 60 mm, S : 0.07—1 mm Circular, D : 0.78 mm

Ghiaasiaan et al. [208] ?Cited in John et al. [105].

Water

7.24

Inlet subcooling (K)

34—258

Cross-section Rectangular, W : 57.2 mm; S : 0.2—1.12 mm Rectangular, W : 0.74—27.9 mm; S : 0.0183—0.247 mm Rectangular, W : 20.4 mm; S : 0.127—0.381 mm Rectangular, W : 19—108 mm; S : 0.097—0.325 mm

Length (mm)

Roughness (m)

63.5

0.3—10.2; simulated crack Real crack

20 60—75 10—33

8.6, 12.7 mm

Smooth 20—40; simulated and real cracks 2—150; real cracks with 240 5.4—12.1

10, 20, 36 mm

3.6—12

0.78 mm

Not measured

46

two-phase flow in microchannels

227

diameter that contained a girth butt weld at midlength with a full circumferential stress-corrosion crack near the welds. The initial circumferential cracks were deep about 90% of the wall thickness, and through cracks of varying lengths were obtained by machining away the pipe surface in the vicinity of the crack. Amos and Schrock [204] used two stainless steel blocks, with their surfaces ground ﬂat, to provide ﬂow channels with smooth walls. Various slit widths were created by adjusting the distance between the stainless steel blocks. They measured axial pressure variations using several pressure taps. The experimental data of Kefer et al. [205] are referred to by John et al. [105]. John et al. [105] used two blocks of stainless steel to produce slits with adjustable distances between the steel blocks in steps of 0.1 mm. Using several pressure taps they measured pressure distribution along their simulated cracks. To examine the effect of slit surface roughness, they varied the surface roughness of the steel blocks by shot blasting them with sand and steel grit. For tests with a real crack, they used blocks of real reactor pipe steel (20 Mn Mo In 55), with a crack that was produced by cyclic bending. They measured the width of the cracks at inlet and exit of their test section after mounting them. In some cases the slit width at exit was slightly larger than at its inlet. Nabarayashi et al. [206] and Matsumoto et al. [207] were interested in fatigue cracks in carbon steel and stainles steel pipes and performed two sets of experiments. In the ‘‘fundamental’’ tests [206], critical ﬂow of saturated water and steam in artiﬁcial slits with W : 30 and 60 mm, L : 10, 20, and 36 mm, S : 0.07—1 mm was studied. They also conducted experiments in through-wall cracks [207], initiated by electric discharge machining and popagated by bending. They performed a careful measurement of surface roughnesses in the artiﬁcial and fatigue-induced cracks. The ranges of average roughnesses are given in Table V. The maximum surface roughnesses varied in the 20—55 m range for their artiﬁcial cracks and in the 16—60 m range in the fatigue-induced cracks. The surface roughnesses, furthermore, did not vary noticeably after tests. They empirically correlated their data for the effect of bending undulation on the crack pressure loss. The experimental investigation of Ghiaasiaan et al [208] was concerned with critical ﬂow of highly subcooled water through a very short capillary. Some important trends in critical ﬂow data are now described. The experimental data indicate the occurrence of metastable liquid ﬂow near the entrance of cracks, due to delayed nucleation (pressure undershoot). Figure 32 is an example pressure proﬁle [204] that shows the effect of delayed ﬂashing. The solid line represents the liquid single-phase pressure drop, the dashed line represents the measured pressure proﬁle, and the horizontal line is the saturation pressure corresponding to the inlet temperature. Flashing

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Fig. 32. Pressure proﬁle in a critical ﬂow experiment [204]. (With permission from [204].)

occurs when the slope of the dashed line abruptly deviates from the slope of the solid line. This type of delay in ﬂashing is also common in large channels and is typically 1—2 K for water [209]. Amos and Schrock [204] noted that the degree of subcooling before the inception of ﬂashing increased with increasing the stagnation subcooling. An empirical correlation

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for pressure undershoot has been developed by Alamgir and Lienhard [210] and has been applied by some investigators in mechanistic modeling of critical ﬂow in large channels. Amos and Schrock [204] noted that the majority of depressurization rates in their experiments were outside the range of the latter correlation. With respect to the important trends, the available critical ﬂow experimental data indicate that G (a) increases with increasing inlet subcooling; AP (b) increases with increasing stagnation pressure; (c) decreases with increasing L /D; and (d) is very sensitive to frictional pressure drop and consequently decreases with increasing the channel surface roughness. Figures 33 and 34 depict typical parametric trends in the experiments of Amos and Schrock [204] and John et al. [105], respectively. The depicted model predictions are discussed later. In Fig. 35 typical data of Ghiaasiaan et al. [208] are depicted and are compared with some relevant data points from Collier and Norris [202]. The ﬁgure demonstrates that critical ﬂow in short capillaries may be

Fig. 33. Some parametric trends in the experiments of Amos and Schrock [204]. (With permission from [204].)

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s. m. ghiaasiaan and s. i. abdel-khalik

Fig. 34. Some parametric trends in the experiments of John et al. [105]. (With permission from [105].)

signiﬁcantly different from that in cracks and slits with large L /D. The measured G in the experiments of [208] are evidently much larger than AP those measured by Collier and Norris [202] under similar conditions and demonstrate the signiﬁcance of channel length. The data of [208], furthermore, are relatively insensitive to the inlet subcooling. The latter data, however, represent extremely high inlet subcooling, and their apparent insensitivity to small variations in inlet temperature may not be representative of crack critical ﬂow in their typical applications. C. General Remarks on Models for Two-Phase Critical Flow in Microchannels Many semianalytical and mechanistic models have been developed for two-phase critical ﬂow in common large passages. These models can be

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231

Fig. 35. Comparison of the data of Ghiaasiaan et al. [208] with the data of Collier and Norris [202], the predictions of the correlations of the RETRAN-03 code [219], and the correlation of Leung and Grolmes [220]. (With permission from [208].)

divided into two broad groups: integral models and models based on numerical solution of differential conservation equations. The two groups of models follow similar principles: They all search for conditions that lead to the maximum possible mass ﬂux without violating the ﬁrst and second laws of thermodynamics. In the integral models, the conservation equations are analytically integrated along the ﬂow passage in order to derive a closedform solution for the mass ﬂux. The resulting closed-form solutions, of course, may need iterative numerical solution. Major simplifying assumptions are evidently needed to make closed-form solutions possible. Reviews of the most widely used integral models can be found in [106, 200].

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Mechanistic critical ﬂow models based on the numerical solution of the one-dimensional differential conservation equation can rigorously account for the thermal and mechanical nonequilibria and the system integral effects on local ﬂow properties. In these models, the 1D conservation equations are numerically solved and, by iteratively varying the passage discharge rate, the ﬂow rate leading to critical conditions (an inﬁnitely large pressure gradient, a vanishing determinant of the coefﬁcient matrix for the system of onedimensional quasi-linear conservation equations, etc.) is speciﬁed. Early contributions to this method include [211, 212], and a discussion of mathematical bases can be found in [213]. This technique is particularly appropriate for cracks and slits with large L /D where the 1D ﬂow assumption is adequate. Models based on direct numerical solution of differential conservation equations for cracks and slits have been developed by Amos and Schrock [204], Schrock and co-workers [201, 214, 215], and Feburie et al. [216]. More recent contributions include [217, 218], where the effect of noncondensables was modeled. D. Integral Models Integral models dealing with two-phase critical ﬂow in large passages are many and are reviewed among others in [106, 200]. These models are mostly based on assumptions and/or simple models that allow for the calculation of the velocity slip and the magnitude of thermodynamic nonequilibrium at the critical cross-section. Most of the widely used methods for the prediction of two-phase critical ﬂow neglect the frictional pressure losses, which are often small in commonly applied large passages. These models overpredict the critical ﬂow rates in small passages and cracks where such pressure losses are signiﬁcant. The isentropic homogeneous-equilibrium model is the simplest among the models that neglect irreversible losses and leads to G : [2 (h 9 h )] AP M AP AP

(152)

where

s 9s M D (153) s DE AP h : (h ; xh ) (154) AP D DE AP v : \ : (v ; xv ) . (155) AP AP D DE AP The critical mass ﬂux is obtained by applying (dG/dP) : 0, which leads to AP x : AP

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v 2[h 9 h (s , P )] M M AP P

; v(s , P ) : 0. (156) M AP AP Note that (v/P) also corresponds to the thermodynamic state speciﬁed with (s , P ). Equation (156) along with Eqs. (153)—(155) are thus iterativeM AP ly solved for P , as well as other thermodynamic properties at the critical AP cross-section, whereupon G is obtained from Eq. (152). AP Since the aforementioned iterative solution is cumbersome, simpler correlations for water have been developed [219, 220]. The correlation in [219] is a simple polynomial-type curve ﬁt to the predictions of Eqs. (152)—(156) for water, and the correlation can be found in [200]. Utilizing an approximate equation of state for water, Leung and Grolmes [220] derived the following correlation, based on the isentropic homogeneous-equilibrium ﬂow assumption for water:

2w G : AP w91

19 19

1 2w 9 1 2w Q

x

P M DM Q .

(157)

Here, : P (T )/P (158) Q Q?R M MGL C T P (T ) DEM , : DM M Q?R M (159) h DM DEM where T is the subcooled liquid stagnation temperature and P is the M MGL stagnation pressure at channel inlet when the effect of channel entrance irreversible pressure loss is accounted for:

A G AP . AP :P 9K (160) M CLR A 2 GL M Predictions of the aforementioned approximate correlations are compared with typical experimental data of Ghiaasiaan et al. [208] in Fig. 35. The ﬂow passage in the latter data is very short (L /D $ 1) and their frictional losses are negligible. Model predictions are of course relatively sensitive to the entrance pressure loss coefﬁcient, K . For sharp-edged CLR sudden contractions with very large contraction ratio, K : 0.5 [221], and CLR with smooth or conical entrances K will be lower. The isentropic homoCLR geneous-equilibrium model well predicted the latter data, with standard deviations of about 10% with K : 0.5 and about 13% for K : 0.3. CLR CLR The isentropic homogeneous-equilibrium model is not appropriate for application to cracks and slits with larger L /D, however, and overpredicts the data because of the signiﬁcance of frictional pressure losses in cracks and slits. P

MGL

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The critical ﬂow of initially saturated liquid of a liquid—vapor mixture in a channel, with frictional pressure loss, has been modeled by Moody [222]. This model assumes thermal equilibrium between the vapor and liquid phases and accounts for velocity slip by assuming a slip ratio everywhere equal to the expression derived by Zivi [99], Eq. (68), for minimum entropy production in a steady-state liquid—vapor ﬂow. In an earlier, and widely used, model for critical, isentropic ﬂow, Moody derived the same expression for the slip ratio at the channel throat. The model of Moody [222] is based on the integration of the one-dimensional two-phase momentum conservation equation along a constant-area channel, d G : [xU ; (1 9 x)U ]G , (161) 9dP/dz 9 f L /D *- 2 % * *dz * where f is an average D’Arcy channel friction factor for liquid ﬂow. The *one-dimensional energy conservation also gives

1 (1 9 x) x h : h ; h ; G ; . (162) M * DE 2 (1 9 ) * % Note that and x are related by the slip ratio, according to Eq. (32). Since the ﬂuid mixture is saturated everywhere, the right-hand side of Eq. (161) can be expanded by replacing d/dz with d/dP(dP/dz), obtaining d/dP of all properties using appropriate thermodynamics data, and factoring out (dP/ dz) in all the terms that include it. Equation (162) is also utilized in the manipulation of Equation (161), noting that dh /dz : 0. The ﬁnal result can M be represented as

. L , (163) (P, h , G)dP : f M *- D . where P is the pressure at inlet to the channel (i.e., the point beyond which irreversible pressure losses occur), P is the pressure at the location of the throat, and is a function that involves the pressure, slip ratio, and thermodynamic properties and their partial derivatives. The critical mass ﬂux is obtained by iteratively changing P and solving Eq. (163) for G, until dG/dP : 0, at which point P : P and G : G, Moody [222] suggested AP AP the following expression for the two-phase multiplier:

19x : . *19

(164)

Nabarayashi et al. [206, 207] compared their measured ﬂow rates with predictions of the aforementioned model of Moody [222], and noted that Moody’s model well predicted their data when pressure losses due to the

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235

entrance effect and wall friction, and ﬂow area reduction due to the wall roughness, were correctly accounted for. The aforementioned model of Moody [222] well predicted the measured ﬂow rates in their fatigue-induced cracks as well, once the entrance and frictional pressure losses, ﬂow area blockage by wall roughness, and pressure losses due to bending undulation were all included in the calculations. Based on the experimental data of Collier et al. [202, 203], a homogeneousnonequilibrium model was developed at Battelle-Columbus Laboratory, which is brieﬂy described by Abdollahian et al. [223]. The model was based on the widely used homogeneous, nonequilibrium model of Henry [224]. In the model, which was coded into the computer program LEAK, the ﬂow was assumed to remain single-phase liquid between the entrance and a point located at a distance of z : 12D downstream from the entrance; the acceleration and frictional pressure drops between the channel inlet and the critical cross-section were obtained using channel averaged properties and assuming homogeneous ﬂow everywhere; the expansion of the vapor phase was assumed to be isentropic; and quality was assumed to vary linearly between z : 12D and the throat. Abdollahian et al. [223] and Chexal et al. [225] modiﬁed and improved the LEAK model and developed the LEAK 01 model by using an isenthalpic ﬂow assumption for pressure drop calculations, assuming a linear ﬂow area variation along the channel, and improving the two-phase frictional pressure drop calculation. Noting that for subcooled liquid inlet conditions the models based on the homogeneous-equilibrium ﬂow in cracks appear to generally predict saturated liquid conditions at the critical cross-section. Abdollahian et al. [223] also proposed the following simple expression for the critical mass ﬂux in long racks and slits:

2[P 9 P (T )] M Q?R M . (165) G : AP (1 ; f L /D ) ; K K & CLR M Here, is the average homogeneous two-phase mixture speciﬁc volume: K : ! ; x! ! . (166) K D DE The average properties, including ! and ! , are calculated at D DE P : [P ; P (T )]/2. (167) M Q?R M The average quality is deﬁned as x! : [h 9 h (P )]/h . (168) M D DE Abdollahian et al. used the unrealistically high K : 2.7, however, and CLR obtained f from Karman’s correlation for rough-walled channels. The two

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models of Abdollahian et al. (i.e., LEAK-01 and the aforementioned simpliﬁed model) well predicted the data of Collier et al. [201, 203], but were shown by John et al. [105] to do relatively poorly when applied to the latter authors’ data, as well as the data of Amos and Schrock [204].

E. Models Based on Numerical Solution of Differential Conservation Equations Most of the models dealing with two-phase critical ﬂow in large channels include constitutive relations that may not apply to cracks and slits. The models of [209, 217, 226, 227] all apply the two-ﬂuid technique and include interfacial transfer models of doubtful applicability to microchannels and cracks where interfacial slip is likely to be suppressed by surface tension. Equal phasic velocities are assumed in [228, 229]. The latter references, however, use models for bubble nucleation and relaxation-type delayed evaporation that may not apply to microchannels. Several models that speciﬁcally address cracks and slits have been developed and published. These models are now brieﬂy reviewed. Amos and Schrock [204] and Lee and Schrock [214] modeled critical two-phase ﬂow in cracks assuming a homogeneous (equal phasic velocities) nonequilibrium ﬂow, by solving numerically the one-dimensional mixture mass, momentum, and energy conservation equations. They used empirically adjusted closure relations, however, to obtain agreement with experimental data. Flashing was assumed to occur when a pressure undershoot (ie., local pressure lower than the saturated pressure corresponding to the local pressure) is obtained, given by P : SP , S S*

(169)

where P is the pressure undershoot associated with bubble nucleation, S* as correlated empirically by Alamgir and Lienhard [210]: P : C S*

0.252 (T /T * ) (1 ; 144 ) AP (k T * ) (1 9 / ) Q AP % *

(170)

In this equation, k is Boltzmann’s constant, and T * is the thermodynamic AP Q critical temperature. Except for 4, the depressurization rate, which must be in Matm/s, the remaining dimensions are consistent. The correction parameter, C, was empirically correlated by Amos and Schrock in terms of local velocity [204], and as a function of Reynolds and Jacobs numbers by Lee and Schrock [214], to obtain agreement between model predictions and

two-phase flow in microchannels

237

data. The evaporation of the metastable liquid phase (the relaxation of the liquid phase superheat) was assumed to follow an exponential form, with a time constant that was also empirically adjusted to ﬁt their data. Critical ﬂow was assumed to occur when the mixture velocity at the critical cross-section was equal to the local velocity of sound, and the latter was represented by an expression by Kroeger [230] representing zero interfacial slip. Lee and Schrock [214] compared the predictions of their model with data from several sources. Most of the data represented critical ﬂow in large channels, however. The real intergranular cracks have complicated and tortuous conﬁgurations. Schrock and Revankar [215] argued that for critical ﬂow in real intergranular corrosion cracks, the homogeneous-equilibrium model is appropriate, since signiﬁcant ﬂow separation and thermal nonequilibrium are unlikely. They developed the fast-running homogeneous-equilibrium code SOURCE, in which the mixture mass, momentum, and energy conservation equations were numerically solved using the ﬁnite-difference technique, and the discharge rate was iteratively varied until the mixture velocity at the critical cross-section was equal to the velocity of sound in homogeneousequilibrium two-phase ﬂow. They assumed an equivalent friction factor that, in addition to wall friction, accounts for the irreversible pressure losses associated with ﬂow disturbances and tortuosity. Shrock et al. [215] compared the predictions of SOURCE with 61 out of the 82 BCL data that were found to be acceptable. An optimum friction factor was found that best ﬁtted the data associated with each test section, and thereby they developed a methodology for the prediction of the equivalent friction factor for real cracks. Since SOURCE predictions showed a systematic dependence on inlet subcooling, furthermore, to obtain agreement between model and data, Schrock and Revankar also developed an inlet subcooling correction factor that must be multiplied by the mass ﬂux predicted by the code in order to calculate the correct mass ﬂux. Feburie et al. [216] developed a nonequilibrium model assuming the existence of three phases: saturated vapor, saturated liquid, and metastable superheated liquid. The pressure undershoot leading to the initiation of evaporation was modeled according to [231] P :k , P (T ) Q?R *+

(171)

where T is the temperature of the superheated liquid and k : 0.95 to *+ 0.97. Following the initiation of evaporation, the three phases are assumed to move at the same velocity (homogeneous ﬂow). Their one-dimensional steady-state conservation equations for the mixture mass, momentum, and

s. m. ghiaasiaan and s. i. abdel-khalik

238 energy are

d (Au/ ) : 0 K dz

(172)

dP dU p ; U :9 U U K dz dz A

(173)

ds T T dy K : GAC Q?R ; *+ 9 1 ln .* dz T T dz *+ Q?R p q" p q" ; p q" F% U% ; &*+ U*+ ; F*1 U*1 T T Q?R *+ p ;p p U% U% , ; U U*+ U*+ ; U*1 U*1 (174) T T Q?R *+ where subscripts L M, L S, and G represent the metastable superheated liquid, saturated liquid, and saturated vapor, respectively, and (1 9 y) represents the metastable liquid mass fraction in the mixture. The last two terms on the right-hand side of Eq. (174) are the entropy sources due to wall heat transfer and friction. The parameter p represents the wall-super&*+ heated liquid perimeter through which the heat ﬂux q" is transferred, and U*+ p and represent the wall-superheated liquid perimeter and its U*+ U*+ associated shear stress, respectively. For simplicity, T : constant is as*+ sumed once bubble nucleation starts. Other parameters are deﬁned similarly. The mixture entropy is deﬁned as GA

s : (1 9 y)s ; xys ; (1 9 x)ys , (175) K *+ E *1 where xy and (1 9 x)y are the mass fractions of saturated vapor and saturated liquid in the mixture, respectively. Other mixture properties, including , are deﬁned similarly. Two additional equations are provided K by the equation of state, : (P, s , y), K K K and a closure relation in the form [232]

(176)

P (T ) 9 P dy Q?R *+ , (177) : k(1 9 y) P 9 P (T ) dz CVGR Q?R *+ where k is a constant assumed to depend on geometry according to p k:k U. A

(178)

two-phase flow in microchannels

239

Using the Moby-Dick experimental data, k : 0.02 m. Equations (172)— (174), (176) (upon differentiation), and (177) constitute ﬁve coupled equations, which can be represented in the form of a quasi-linear set of coupled ordinary differential equations: A

dY :B dz

(179)

Y : (P, U, , s , y)2. (180) K K Note that if and s are known, x and T can be calculated. These K K *+ equations can be numerically integrated, and by iteration the critical mass ﬂow rate, which leads to the vanishing of the determinant of the coefﬁcient matrix A, can be speciﬁed. For wall friction and heat transfer, Feburie et al. used various correlations. Feburie et al. compared preditions of their model for G with 70 data AP points from John et al. [105], with agreement within <12%, and with 14 data points from Amos and Schrock [204], with agreement within about <5%. Sensitivity analysis indicated that the model predictions were relatively sensitive to the magnitude of the constant k in Eq. (171), the wall friction, and entrance pressure loss, and were insensitive to the magnitude of the constant k . Geng and Ghiaasiaan [218] recently developed a homogeneous-equilibrium model for the critical ﬂow of an initially subcooled liquid through cracks and slits, where the effect of a dissolved noncondensable in the inlet subcooled liquid was accounted for. They assumed that: (a) the solubility of the noncondensable in the liquid is low; (b) the liquid and vapor gas phases are everywhere at thermodynamic equilibrium and at equilibrium with respect to the concentration of the noncondensable; and (c) the vapor— noncondensable gas mixture is everywhere saturated with respect to vapor. For a channel with uniform cross-sectional area that is inclined with respect to the horizontal plane by the angle 1!, the mixture mass, momentum, and energy conservations are

d ( U) : 0 dz F

(181)

dU dP p U : 9 9 g sin 1! 9 U U F dz F dz A

(182)

d 1 U (1 9 )h ; h ; U * D E E 2 F dz

: 9 Ug sin 1! ; p q" /A, F U U (183)

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240

where, assuming that the noncondensable is an ideal gas,

P 9 P (T ) Q?R , (184) : (1 9 ) ; ; E F * (R/M )T L where T is the local mixture temperature. Utilizing Henry’s law [233] for the equilibrium between the gas and liquid phases with respect to the concentration of the noncondensable, Geng and Ghiaasiaan derived

d M * U (1 9 )M ; : 0 (185) D * % (P/C ) ; [1 9 (Mg/M )]M dz &C L * d M P 9 P (T ) L Q?R M 9 : 0. (186) * dz M C E &C Geng and Ghiaasiaan expanded Eqs. (181)—(183), (185), and (186) using thermodynamic relation, and cast the aforementioned equations in the form represented by Eq. (179), with

Y : (U, P, , h , M )2. (187) D * Geng and Ghiaasiaan applied the correlation of John et al. [105], Eq. (85), for single-phase wall friction and applied the homogeneous mixture model, along with McAdams’ correlation (Eqs. (44)—(48) for two-phase pressure drop. They numerically integrated the foregoing differential conservation equations and, by iteratively varying the channel mass ﬂow rate, speciﬁed G as the mass ﬂux leading to det A : 0 at the critical crossAP section. Geng and Ghiaasiaan compared the predictions of their model with the experimental data of Amos and Schrock [204] and John et al. [105] with satisfactory agreement. To examine the effect of noncondensables, they chose the data of Amos and Schrock as the basis for parametric calculations. Figure 36 represents typical results, where model predictions with pure water and water saturated with nitrogen at inlet are both presented. The results indicate that desorption of nitrogen (or air) initially dissolved in subcooled water can reduce the critical mass ﬂux by several percent.

VIII. Concluding Remarks The recent developments related to gas—liquid two-phase ﬂow, forcedﬂow subcooled boiling, and the critical (choked) ﬂow of initially subcooled liquids, in channels with hydraulic diameters of the order of 0.1 to 1 mm, were reviewed in this article. The hydrodynamic phenomena reviewed

two-phase flow in microchannels

241

Fig. 36. Comparison between the predictions of the model of Geng and Ghiaasiaan [218] and the experimental data of Amos and Schrock [204] and the effect of noncondensable gas on model predictions. (With permission from [218].)

included the two-phase ﬂow regimes, void fraction, and frictional pressure drop in narrow rectangular and annular passages, a micro-rod bundle, and microchannels under conditions where surface tension and inertial forces are both signiﬁcant. The boiling phenomena reviewed included the onset of nucleate boiling (ONB), onset of signiﬁcant void (OSV), onset of ﬂow instability (OFI), and critical heat ﬂux. The critical (choked) ﬂow of initially subcooled liquids in capillaries, cracks, and slits was also addressed. The observed major two-phase ﬂow regimes in microchannels are morphologically similar to the ﬂow regimes in large channels. However, they can be insensitive to channel orientation and are inﬂuenced by surface wettability. The commonly applied predictive methods for the ﬂow regime transitions in large channels overall fail to predict the microchannel data well. Some empirical correlations that have been developed based on the microgravity experiments, however, appear to agree with microchannel data satisfactorily. The bulk of the existing microchannel data have been obtained with air and water, and more experimental data examining the effects of ﬂuid properties, in particular the surface tension, surface wettability, and liquid viscosity, are needed. The existing predictive methods for void fraction and two-phase frictional pressure drop are also generally inadequate for microchannels, in particular for the annular ﬂow regime.

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With respect to boiling, the fundamental bubble ebullition phenomena in microchannels are likely to be different from the qualitatively well-understood bubble phenomena associated with boiling in large channels, because of the occurrence of very large velocity and temperature gradients in the former. Bubble ebullition in microchannels needs to be investigated. The critical heat ﬂux data associated with moderate and high mass ﬂuxes in microchannels are predicted satisfactorily by some of the recent and widely used empirical correlations, because of the presence of relevant data in the data bases of these correlations. The available data associated with low-ﬂow critical heat ﬂux in microchannels, however, are unlike similar data in large channels and suggest the occurrence of metastable superheated liquid prior to dryout. Investigations aimed at the elucidation of the hydrodyamic and evaporation phenomena associated with annular ﬂow regime in microchannels are thus needed. The critical (choked) ﬂow of an initially subcooled liquid in a capillary, crack, or slit is strongly inﬂuenced by pressure drop and can be predicted using models based on the homogeneous ﬂow assumption.

Nomenclature A A B C Ca C Cp D D C D,D G M Eo F F ,F , @ B F ,F ,F, JK K Q F ,F TR U f f Fr G g h

coefﬁcient matrix ﬂow area; dimensionless coefﬁcient column vector constant capillary number two-phase distribution coefﬁcient speciﬁc heat diameter hydraulic diameter inner, outer diameters Eotvos number force term forces acting on a bubble (Figs. 26 and 27)

D’Arcy friction factor Fanning friction factor Froude number mass ﬂux gravitational constant convection heat transfer coefﬁcient

h * h h DE k k Q K CLR L M M N I P P S p ,p & 5 Pe Pr P* AP P* P Q q" U R r Re S s

liquid depth speciﬁc enthalpy speciﬁc heat of vaporization thermal conductivity Boltzmann’s constant entrance loss coefﬁcient length molar mass mass fraction viscosity number pressure pressure undershoot heated and wetted perimeters Peclet number Prandtl number thermodynamic critical pressure reduced pressure dimensionless heat ﬂux wall heat ﬂux universal gas constant radius Reynolds number slip ratio; gap distance speciﬁc enthalpy

two-phase flow in microchannels s DE St T t T* AP U U* U v

speciﬁc enthalpy of vaporization Stanton number temperature time thermodynamic critical temperature velocity friction velocity rise velocity speciﬁc volume

V EG W We X x Y y

z

243

gas drift velocity width Weber number Martinelli parameter quality Shah’s correlation parameter distance from wall; the combined mass fraction of vapor and saturated liquid axial coordinate

Greek Symbols ( 1! ,

Void fraction Volumetric quality Gap distance; ﬁlm thickness Function in Moody’s critical ﬂow model Roughness Pressure ratio function deﬁned in Eq. (133); angle with respect to the horizontal plane dimensionless coefﬁcient Laplace length scale dynamic viscosity

% *

3

density surface tension shear stress kinematic viscosity parameter deﬁned in Eq. (110) two-phase multiplier shape factor deﬁned in Eq. (78) function deﬁned in Eq. (132); contact angle function deﬁned in Eqs. (119)—(121) function deﬁned in Eq. (145) function deﬁned in Eq. (134) function deﬁned in Eq. (159)

Subscripts B b cr E eq eqv f FC G g GS G0 H h i

bubble, vapor blanket boiling critical (choked) effective equilibrium equivalent frictional; saturated liquid forced convection gas saturated vapor superﬁcial gas all-gas heated homogeneous inlet

Superscripts mean

L LM LS L0 m mod n p sat TP v w 0 (

liquid metastable liquid superﬁcial liquid; saturated liquid all liquid average modiﬁed noncondensable pipe saturation two-phase vapor wall stagnation ﬁlm edge

s. m. ghiaasiaan and s. i. abdel-khalik

244 Abbreviations CHF HHF NVG

critical heat ﬂux high heat ﬂux net vapor generation

OFI ONB OSV

onset of ﬂow instability onset of nucleate boiling onset of signiﬁcant void

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132. Kennedy, J. E., Roach, G. M., Jr., Dowling, M. F., Abdel-Khalik, S. I., Ghiaasiaan, S. M., Jeter, S. M., and Qureshi, Z. H. (2000). The onset of ﬂow instability in uniformly heated horizontal microchannels. J. Heat Transfer 122, 118—125. 133. Yin. S. T., and Abdelmessih, A. H. (1974). Prediction of incipient ﬂow boiling from a uniformly heated surface. AIChE Symp. Ser. 164, 236—243. 134. Hino, R., and Ueda, T. (1985). Studies on heat transfer and ﬂow characteristics in subcooled ﬂow boiling — Part I. Boiling characteristics. Int. J. Heat Mass Transfer 11, 269—281. 135. Roch, G. M., Jr., Abdel-Khalik, S. I., Ghiaasiaan, S. M., and Jeter, S. M. (1999). Low-ﬂow onset of ﬂow instability in heated microchannels. Nucl. Sci Eng. 133, 106—117. 136. Rogers, J. T., and Li, J.-H. (1992). Prediction of the onset of signiﬁcant void in ﬂow boiling of water. ASME, Fundamentals of Subcooled Flow Boiling, HTD-Vol. 217, 41—52. 137. Al-Hayes, R. A. M., and Winterton, R. H. S. (1981). Bubble diameter on detachment in ﬂowing liquids. Int. J. Heat Mass Transfer 24, 223—230. 138. Martinelli, R. C. (1947). Heat transfer to molten metals. Trans. ASME 69, 947—951. 139. Blasick, A. M., Dowling, M. F., Abdel-Khalik, S. I. Ghiaasiaan, S. M., and Jeter, S. M. (2000). Onset of ﬂow instability in uniformly-heated thin horizontal annuli. Proc. 8th Int. Conf. Nucl. Eng. (ICONE-8), April 2—6, Baltimore, MD. 140. Zijl, W., Ramakers, F. J. M., and Van Stralen, S. J. D. (1979). Global numerical solutions of growth and departure of a vapor bubble at a horizontal superheated wall in a pure liquid and a binary mixture. Int. J. Heat Mass Transfer 22, 401—420. 141. Lee, R. C., and Nydahl, J. E. (1989). Numerical calculation of bubble growth in nucleate boiling from inception through departure. J Heat Transfer 111, 474—479. 142. Kocamustafaogullari, G., and Ishii, M. (1983). Interfacial area and nucleation site density in boiling systems. Int. J. Heat Mass Transfer 26, 1377—1387. 143. Yang, S. R., and Kim, P. H. (1988). A mathematical model of the pool boiling nucleation site density in terms of the surface characteristics, Int. J. Heat Mass Transfer 31, 1127—1135. 144. Unal, H. C. (1976). Maximum bubble diameter, maximum bubble-growth time and bubble-growth rate during the subcooled nucleate ﬂow boiling of water up to 17.7 MN/m. Int. J. Heat Mass Transfer 19, 643—649. 145. Shin, T. S. and Jones, O. C. (1993). Nucleation and ﬂashing in nozzles — 1: A distributed nucleation model. Int. J. Multiphase Flow 19, 943—964. 146. Klausner, J. F., Mei, R. and Zeng, L. Z. (1997). Predicting stochastic features of vapor bubble detachment in ﬂow boiling. Int. J. Heat Mass Transfer 40, 3547—3552. 147. Lin, L., Udell, K. S. and Pisano, A. P. (1993). Vapor bubble formation on a micro heater in conﬁned and unconﬁned micro channels. ASME, Heat Transfer on the Microscale, HTD Vol. 253, 85—93. 148. Lahey, R. T., Jr., and Drew, D. A. (1988). The three-dimensional time and volume averaged conservation equations of two-phase ﬂow. In Advances in Nuclear Science and Technology (J. Lewis and M. Becker, eds.), Vol. 20. pp. 1—69, Plenum Press, New York. 149. Drew, D. A., and Lahey, R. T., Jr. (1987). The virtual mass and lift force on a sphere in rotating and straining inviscid ﬂow. Int. J. Multiphase Flow 13, 113—121. 150. Wang, S. K., Lee, S. J. Jones, O. C., Jr., and Lahey, R. T., Jr. (1987). 3-D turbulence structure and phase distribution measurements in bubbly two-phase ﬂows. Int. J Multiphase Flow 13, 327—343. 151. Shultze, H. D. (1984). Physico-chemical Elementary Processes in Flotation, pp. 123—129. Elsevier, Amsterdam. 152. Antal, S. P., Lahey, R. T., Jr., and Flaherty, J. E. (1991). Analysis of phase distribution in fully developed laminar bubbly two-phase ﬂow. Int. J. Multiphase Flow 17, 635—652.

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196. Hewitt, G. F., and Govan, A. H. (1990). Phenomena and prediction in annular two-phase ﬂow. ASME Advances in Gas—L iquid Flows, FED Vol. 99, pp. 41—56. ASME, New York. 197. Sugawara, S. (1990). Droplet deposition and entrainment modeling based on the threeﬂuid model. Nucl. Eng. Design 122, 67—84. 198. Sugawara, S. (1990). Analytical prediction of CHF by FIDAS code based on three-ﬂuid and ﬁlm dryout model. J. Nucl. Sci. Technol. 27, 12—29. 199. Abdollahian, D., Healzer, J. Janssen, E., and Amos, C. (1982). Critical ﬂow data review and analysis. Electric Power Research Institute Report EPRI NP-2192, Palo Alto, CA. 200. Elias, E., and Lelluche, G. S. (1994). Two-phase critical ﬂow. Int. J. Multiphase Flow 20, Suppl. 91—168. 201. Schrock, V. E., Revankar, S. T., and Lee, S. Y. (1988). Critical ﬂow through pipe cracks. In Particulate Phenomena and Multiphase Transport (N. T. Veziraglu, ed.). Hemisphere, New York. 202. Collier, R. P., and Norris, D. M. (1983). Two-phase ﬂow experiments through intergranular stress corrosion cracks. Proc. CSNI Specialist Meeting on L eak-Before Break in Nuclear Reactor Piping, U.S. Nucl. Reg. Comm. Report NUREG/CP-d005, pp. 273—299. 203. Collier, R. P. Stuben, F. B., Mayﬁeld, M. E., Pope, D. B., and Scott, P. M. (1984). Two-phase ﬂow through intergranular stress corrosion cracks. Electric Power Research Institute Report EPRI-NP-3540-LD, Palo Alto, CA. 204. Amos, C. N., and Schrock, V. E. (1984). Two-phase critical ﬂow in slits. Nucl. Sci. Eng. 88, 261—274. 205. Kefer, V., Kastner, W., and Kra¨tzer, W. (1986). Leckraten bei unterkritischen Rohrleitung srissen. Jahrestagung, Kerntechnik, Aachen, Germany. 206. Nabarayashi, T., Ishiyama, T., Fujii, M., Matsumoto, K., Harimizu, Y., and Tanaka, Y. (1989). Study on coolant leak rates through pipe cracks: Part I — Fundamental tests. Proc. ASME Pressure Vessels and Piping Conf., JSME Co-sponsorship, ASME PVP Vol. 165, pp. 121—127. ASME, New York. 207. Matsumoto, K., Nakamura, S., Gotoh, N., Nabarayashi, T., Tanaka, Y. and Horimizu, Y. (1989). Study on coolant leak rates through pipe cracks: Part 2 — Pipe test. Proc. ASME Pressure Vessels and Piping Conf., JSME Co-sponsorship, ASME PVP Vol. 165, pp. 113—120. ASME, New York. 208. Ghiaasiaan, S. M., Muller, J. R., Sadowski, D. L., and Abdel-Khalik, S. I. (1997). Critical ﬂow of initially highly subcooled water through a short capillary. Nucl. Sci. Eng. 126, 229—238. 209. Richter, H. J. (1983). Separated two-phase ﬂow model: Application to critical two-phase ﬂow. Int. J. Multiphase Flow 9, 511—530. 210. Alamgir, M. D., and Lienhard, J. H. (1981). Correlation of pressure undershoot during hot-water depressurization. J. Heat Transfer 103, 52—55. 211. Giot, M., and Fritz, A. (1972). Two-phase two- and one-component critical ﬂow with the variable slip model. Prog. Heat Transfer 6, 651—670. 212. Ardron, K. H. (1978). A two-ﬂuid model for critical vapor—liquid ﬂow. Int. J. Multiphase Flow 4, 323—327. 213. Boure´, J. A. (1997). The critical ﬂow phenomena with reference to two-phase ﬂow and nuclear reactor systems. Proc. ASME Symp. T hermal-Hydraulic Aspects of Nuclear Reactor Safety, pp. 195—216. ASME, New York. 214. Lee, S. Y., and Schrock, V. E. (1988). Homogeneous non-equilibrium critical ﬂow model for liquid stagnation states. Proc. National Heat Transfer Conf., 7th, HTD Vol. 96, pp. 507—513. ASME, New York. 215. Schrock, V. E., Revankar, S. T., and Lee, S. Y. (1988). Critical ﬂow through pipe cracks. In Particulate Phenomena and Multiphase Transport (N. T. Veziroglu, ed.), Vol. 1, pp. 3—17. Hemisphere, Washington, D.C.

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216. Feburie, V., Giot, M. Granger, S., and Seynhaever, J. M. (1993). A model for choked ﬂow through cracks with inlet subcooling. Int. J. Multiphase Flow 19, 541—562. 217. Ghiaasiaan, S. M., and Geng, H. (1997). Mechanistic non-equilibrium modeling of critical ﬂashing ﬂow of subcooled liquids containing dissolved noncondensables. Num. Heat Transfer: B 32, 435—457. 218. Geng, H., and Ghiaasiaan, S. M. (1998). Mechanistic modeling of critical ﬂow of initially subcooled liquid containing dissolved noncondensables through cracks and slits based on the homogeneous-equilibrium mixture method. Nucl. Sci. Eng. 129, 294—304. 219. McFadden, J. H., Paulsen, M. P., Gose, G. C., Peterson, McClure, J. A., Jensen, P. J., and Westacott, J. L. (1992). RETRAN-03. A program for transient thermal-hydraulic analysis of complex ﬂuid systems. Electric Power Research Institute Report EPRI NP-7450, Vol. 1, Palo Alto, CA. 220. Leung, J. C., and Grolmes, M. A. (1988). A generalized correlation for ﬂashing choked ﬂow of initially subcooled liquid. AIChE J. 34, 688—691. 221. Idelchik, I. E. (1994). Handbook of Hydraulic Resistances, 3rd ed., CRC Press, London. 222. Moody, F. J. (1966). Maximum two-phase vessel blowdown from pipes. J. Heat Transfer 88, 285—295. 223. Abdolahian, D., Chexal, B., and Norris, D. M. (1983). Prediction of leak rates through intergranular stress corrosion cracks. Proc. CSNI L eak-Before-Break Conf., Nucl. Reg. Comm. Report NUREG/CP-00151, pp. 300—326. 224. Henry, R. E. (1970). The two-phase critical discharge of initially saturated or subcooled liquid. Nucl. Sci. Eng. 41, 336. 225. Chexal, B., Abdollahian, D., and Norris, D. (1984). Analytical prediction of single-phase and two-phase ﬂow through cracks in pipes and tubes. AIChE Symp. Ser. 80(236), pp. 19—23. 226. Schwellnus, C. F., and Shoukri, M. (1991). A two-ﬂuid model for non-equilibrium two-phase critical discharge. Can. J. Chem. Eng. 69, 187—197. 227. Dagan, R., Elias, E., Wacholder, E., and Olek, S. (1993). A two-ﬂuid model for critical ﬂashing ﬂows in pipes. Int. J. Multiphase Flow 19, 15—25. 228. Blinkov, V. N., Jones, O. C., Jr., and Nigmatulin, B. I. (1993). Nucleation and ﬂashing in nozzles — 2. Comparison with experiments using a ﬁve-equation model for vapor void development. Int. J. Multiphase Flow 19, 965—986. 229. Downar-Zapolski, Z., Bilicki, Z., Bolle, L., and Franco, J. (1996). The non-equilibrium relaxation model for one-dimensional ﬂashing liquid ﬂow. Int. J. Multiphase Flow 22, 473—483. 230. Kroeger, P. G. (1978). Application of a non-equilibrium drift-ﬂux model to two-phase blowdown experiments. Paper presented at OECD/NEA Specialists’ Meeting on Transient Two-Phase Flow, Toronto, Canada, August 1998. 231. Lackme, C. (1979). Incompleteness of the ﬂashing of supersaturated liquid and sonic ejection of the produced phases. Intl. J. Multiphase Flow 5, 131—141. 232. Hardy, Ph., and Mali, P. (1983). Validation and development of a model describing subcooled critical ﬂow through long tubes. Energie Primaire 18, 5—23. 233. Garrels, R. M., and Christ, C. L. (1965). Solutions, Minerals and Equilibria. Harper and Row, New York.

ADVANCES IN HEAT TRANSFER, VOLUME 34

Turbulent Flow and Convection: The Prediction of Turbulent Flow and Convection in a Round Tube

STUART W. CHURCHILL Department of Chemical Engineering The University of Pennsylvania Philadelphia, Pennsylvania 19104

The quantitative prediction of turbulent ﬂow and convection in channels extends at least from Boussinesq, who in 1877 proposed the eddyviscosity model, to Papavassiliou and Hanratty, who in 1997 computed the rate of transport of molecular species by the turbulent ﬂuctuations using Lagrangian direct numerical simulation. The history and current state of the art of such predictions are examined herein. It is concluded that the eddydiffusivity, mixing-length, and ,— models should now be completely abandoned in favor of generalized correlating equations for the turbulent shear stress and the turbulent heat ﬂux density based on semitheoretical asymptotic expressions, even though those for the latter quantity are yet uncertain and incomplete. Correlating equations for the time-averaged velocity distribution, the friction factor, the time-averaged temperature distribution, and the heat transfer coefﬁcient may serve as conveniences, but such expressions are unessential since these four quantities may be determined numerically with comparable accuracy from simple, single integrals of the two more elementary quantities. Such direct numerical evaluations reveal that all of the classical algebraic analogies between heat and momentum transfer are in signiﬁcant error functionally as well as numerically, in large part because of inaccurate representations of the radial variation of the total heat ﬂux density. In the interests of simplicity and clarity, this presentation is primarily limited to fully developed heat transfer in a uniformly heated round tube, but the methodologies and formulations may readily be adapted 255

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ADVANCES IN HEAT TRANSFER, VOL. 34 Copyright 2001 by Academic Press. All rights of reproduction in any form reserved. 0065-2717/01 $35.00

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or extended for other one-dimensional ﬂows and other thermal boundary conditions.

I. Introduction Progress in engineering and science occurs by discarding old concepts and correlations in favor of new or improved ones. Turbulent ﬂow and convection have been somewhat resistant to this process of renewal; many early concepts and correlations remain enshrined in our current textbooks and handbooks and in the software for design calculations even though they long ago became obsolete in terms of accuracy or were shown to be untenable in a theoretical sense. We should of course recognize and honor the pioneers of our ﬁeld and preserve their contributions in a historical context, but at the same time be alert and aggressive about identifying and incorporating experimental and theoretical improvements. Turbulent forced convection is analogous to turbulent ﬂow in many respects, but is far more complex and demanding analytically, experimentally, and in practice. Under many circumstances, ﬂow may be studied independently from heat transfer. On the other hand, the detailed analysis of forced convection requires a quantitative description of the details of the ﬂow. Also, turbulent convection invokes the Prandtl number as a parameter as well as several other complexities that are not encountered in the description of the ﬂow. For these reasons, ﬂow is examined ﬁrst and convection thereafter. That division and order is observed in this Introduction as well as in the presentation as a whole. In the interests of simplicity and clarity, the detailed descriptions that follow are limited in the main to fully developed turbulent ﬂow inside a straight smooth round tube and to fully developed forced convection from a uniformly or isothermally heated wall. However, the adaptation or extension of these methodologies and formulations for other one-dimensional ﬂows is examined brieﬂy and shown to be rather straightforward. Attention is also limited to single-phase Newtonian ﬂuids with invariant physical properties, including the speciﬁc density. The latter restriction excludes natural convection. Physical property variations are of course often signiﬁcant in magnitude and may greatly inﬂuence the ﬂow and convection. However, since the variations of the viscosity and thermal conductivity are different for every ﬂuid and that of the density for every liquid, accounting for these effects would eliminate almost all of the generality that is a primary feature of the new and old developments described herein. At the present time the best compensation for this oversimpliﬁcation is to incorporate empirical corrections in the ﬁnal idealized results for invariant physical

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properties. As computational tools and techniques advance, it may become feasible to incorporate such variations in direct numerical simulations of the time-dependent differential equations of conservation or numerical integrations of their time-averaged counterparts. For convenience, a single standardized notation is utilized throughout rather than necessarily those that were originally employed in particular contributions. All symbols are deﬁned when they appear ﬁrst or in a new context. A. Turbulent Flow Lamb [1], in his classical treatise Hydrodynamics, ﬁrst published in 1879 but periodically revised by himself through 1932, a span of over half a century, begins his rather brief treatment of turbulent motion in even the latest of those editions with the statement, ‘‘It remains to call attention to the chief outstanding difﬁculty of our subject.’’ He then proceeds to explain the reasons for that difﬁculty, noting that ‘‘the motion becomes wildly irregular and the tube appears to be ﬁlled with interlacing and constantly varying streams, crossing and recrossing the pipe.’’

Fig. 1. Self-portrait of Leonardo da Vinci observing turbulent vortices behind a disturbance in a river. (from Richter [2], Plate XXV).

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Leonardo da Vinci in 1515, at age 63, sketched in his notebook, as shown in Fig. 1 (Plate XXV from Richter [2]), a self-portrait with a river ﬂowing past obstructions. As indicated by the accompanying text, he was interested in the pattern of ﬂow in a scientiﬁc as well as an artistic sense. His description has been translated (Richter [2], p. 200) as ‘‘Observe the motion of the surface of the water which resembles that of hair, and has two motions, of which one goes on with the ﬂow of the surface, the other forms the lines of the eddies; thus the water forms eddying whirlpools one part of which are due to the impetus of the principal current and the other to the incidental motion and return ﬂow.’’ Even such a universal genius and perceptive observer as Leonardo may sometimes err; he sketched symmetrical pairs of vortices rather than the alternating ones that actually occur. The later renowned physicists and applied mathematicians who have written on the subject of turbulent ﬂow include Subrahmanyan Chandrasekhar [3], Albert Einstein [4], Werner Heisenberg [5], Pyotr Kapitsa [6], Lev Landau [7], Hendrik Lorentz [8], Isaac Newton [9], Lord Rayleigh [10], Arnold Sommerfeld [11], George Uhlenbeck [12], Richard von Mises [13], C. R. von Weizsa¨cker [14], and Yakob Zel’dovich [15]. It is intimidating and humbling for anyone who undertakes the study of turbulent ﬂow to realize that even these great scientists made few signiﬁcant contributions to this subject outside the special topic of stability. The focus herein on shear ﬂows and in particular on fully developed ﬂow in a round tube avoids the necessity of reviewing the statistical developments that have found applicability primarily in the idealized domains of isotropic and homogeneous tubulence. Another quotation is appropriate in this regard. Schlichting [16], in the Author’s Preface of the ﬁrst German edition of Boundary L ayer T heory, wrote (in translation) ‘‘No account of the statistical theories of turbulence has been included because they have not attained any practical signiﬁcance for engineers.’’ In a later edition he writes begrudgingly and deﬁantly in apparent response to criticisms of that statement that ‘‘This (the statistical theory of turbulence) admittedly has contributed to our understanding of turbulent ﬂows but it has not yet acquired any importance to engineers.’’ Although statistical representations of turbulence have recently been utilized in the development of approximate expressions for the kinetic energy of turbulence, ,, and the rate of dissipation of the energy of turbulence, , in connection with the ,— and related models, the details of that usage are outside the chain of development herein and only merit this brief mention. The general partial-differential equations for the conservation of mass and momentum in time-dependent form are generally presumed to describe turbulent ﬂow insofar as the ﬂuid may be treated as a continuum. Because of their complexity, and in particular their nonlinearity, they resisted

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numerical as well as analytical methods of solution until roughly the last decade. Accordingly, as an alternative approach, Sir Osborne Reynolds [17] in 1895 reduced these expressions to manageable proportions and a tractable form by space-averaging. This great simpliﬁcation has inspired a century-long development of semitheoretical models and approximate solutions for the reduced equations. In particular, Ludwig Prandtl and his associates and contemporaries developed a very useful structure for the prediction and correlation of turbulent shear ﬂows by postulating mechanistic models for the unknown term(s) in the time-averaged (rather than space-averaged) equations of conservation in differential form. This procedure proved to be so successful in an applied sense that only sporadic and limited improvements were made over the ensuing half-century. Finally, in the 1980s and 1990s, a signiﬁcant breakout occurred in the form of essentially exact solutions of the general time-dependent equations of conservation by direct numerical simulation (DNS). Although the results obtained by this technique are yet very restricted in scope, both intrinsically and because of their computational demands, this development has invigorated the ﬁelds of turbulent ﬂow and convection and has played a key role in the development of the new formulations that constitute the principal contribution of this article. Before the presentation of these improved formulations, a retrospective assessment of the historical development and validity of the quantitative descriptions of turbulent shear ﬂow in the classical and current literature is provided. This survey is limited to representative contributions with longlasting consequences, both positive and negative, rather than being exhaustive, since the primary objective is to provide a framework and perspective for the new improved expressions. This portion of the presentation is organized chronologically on the mean, but also topically, in part in the interests of clarity, continuity, and the avoidance of repetition. B. Turbulent Convection Although the path of development of a structure for the correlation and prediction of convection in turbulent ﬂow might have been expected to follow the path of development for the ﬂow itself because of the similar structure of the equation of conservation for energy to those for momentum, this is not found to be the case. Turbulent convection is much more complicated because of (1) the coupling of the equations for the conservations of energy and momentum, (2) the possibility of different boundary conditions, and (3) the appearance of additional parameters. Reynolds [18] in 1874, and thus 21 years prior to his great contribution to the development of a simpliﬁed structure for turbulent ﬂow by means of

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space averaging, derived a simple algebraic equation, now known as the Reynolds analogy, by postulating equal mass rates of transport of energy and momentum from the bulk of the ﬂuid stream all the way to a conﬁning surface by the turbulent eddies. This expression, which is free of any explicit empiricism and independent of geometry, is only of ﬁrst-order accuracy at best, but it lives on after 125 years by virtue of its implicit or explicit presence within many predictive correlations for heat and mass transfer. An alternative approach to the prediction of the rate of heat transfer involves the adaptation of the mechanistic models of Prandtl and others to represent the primary unknown term (the turbulent heat ﬂux density) in the time-averaged differential equation for the conservation of energy. Such modeling has, however, evolved more slowly and less successfully for convection than for ﬂow because of the inherently more complex behavior mentioned earlier. Direct numerical simulation has also been utilized for convection, but even more severe limitations are encountered than for ﬂow and, as a consequence, less success has been achieved. In contrast with ﬂow, the historical development of a structure for turbulent convection is reviewed retrospectively in the light of the new and improved formulations that prompted this article. Finally, some representative numerical results obtained by means of these new formulations are presented and generalized.

II. The Quantitative Representation of Turbulent Flow Structures for the prediction and correlation of the characteristics of turbulent ﬂow that are important in the subsequent prediction and correlation of rates of heat transfer are ﬁrst examined from a historical point of view. New and improved formulations are then described and compared with experimental data and numerical predictions. The derivations and ﬁnal expressions are presented herein primarily in the context of a round tube, even though they may have been formulated originally for ﬂow between parallel plates or for unconﬁned ﬂow along a ﬂat plate. The adaptation or extension of these new expressions for other geometries is ﬁnally examined brieﬂy. A. Historical Highlights 1. The Exact Structure The partial differential equations for the conservation of momentum in the ﬂow of a single-phase Newtonian ﬂuid with invariant viscosity and

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density may be expressed in cylindrical coordinates as follows: Radial component:

u u u u u u P; F P9 F ;u P P;u P X r r 1! r z t

:9

(1)

(2)

p 1 1 u 2 u u P9 F; P ; g ; (ru ) ; P P r r 1! r 1! z r r r

Angular component:

u u u u uu u F;u F; F F; P F;u F P r X z t r 1! r

:9

1 p 1 1 u 2 u u F; P; F ; g ; (ru ) ; F F r 1! r 1! r 1! z r r r

Axial component:

u u u u u X;u X; F X;u X P r X z t r 1! :9

p 1 u 1 u u X; X : g . ; r X ; X z r r 1! z r r

(3)

Here, t represents time, r, z and 1! the radial, axial and angular coordinates, g , g , and g the corresponding components of the gravitational vector, u , P X F P u , and u those of the instantaneous velocity vector, p the instantaneous F X thermodynamic pressure, the speciﬁc density, and the dynamic viscosity. Equations (1)—(3) are generally known as the Navier—Stokes equations in recognition of their derivation by Navier [19] in 1822 and their reﬁnement by Stokes [20] in 1845. After repeated attempts to derive or justify these expressions on the basis of statistical mechanics, Uhlenbeck [12] noted that ‘‘Quantitatively, some of the predictions from these equations surely deviate from experiment, but the very remarkable fact remains that qualitatively the Navier—Stokes equations always describe physical phenomena sensibly . . . . The mathematical reason for this virtue of the Navier—Stokes equations is completely mysterious to me.’’ The greatest advance ever in the analysis of turbulent ﬂow was made in 1895 by Reynolds [17], who not only conceived of the practical advantages of space averaging but also derived in detail the required mathematical procedures for this averaging. He then applied this process to Eqs. (1)—(3). Time averaging, which is generally presumed to be equivalent to space averaging, followed by specialization for steady fully developed ﬂow, reduces

stuart w. churchill

262 these equations to 9

(u u ) P 1 d F F :0 9 (ru u ) ; P P r dr r r 1 d u u (ru u ) 9 P F : 0 P F r dr r

(4) (5)

and 9

P 1 d du ; r X 9 (ru u : 0, P X z r dr dr

(6)

where here, as contrasted with Eqs. (1)—(3), u denotes the time-averaged X velocity, and P the time-averaged dynamic pressure that arises from changes in velocity only; u , u and u the instantaneous ﬂuctuations in velocity P X F about the time-averaged values; and the superbars the time-averaged values of products of these ﬂuctuations. The validity of the equations obtained by space or time averaging has been questioned, but no speciﬁc failures have been demonstrated for conditions such that treatment of the ﬂuid as a continuum is a valid approximation. Barenblatt and Goldenfeld [21] have questioned the concept of full development for turbulent shear ﬂows (the attainment of a velocity ﬁeld and pressure gradient independent from z), but such behavior is certainly attained for all practical purposes (see, for example, Abbrecht and Churchill [22]), and such a postulate has resulted in many apparently valid asymptotic predictions. Equations (4)—(6), even if exact, are, in contrast with Eqs. (1)—(3), an incomplete description of the ﬂuid motion. The terms u u , u u , u u , and P P F F P F u u , which are known quite appropriately as the Reynolds stresses, repreP X sent the information lost by time averaging since they are indeterminate from these equations alone. Most of the modeling of turbulent ﬂow has involved the postulate of empirical expressions for these time-averaged products of the ﬂuctuating components of the velocity. On the other hand, the structural gain from time averaging is quite evident, not only from the relative simplicity of Eqs. (4)—(6) as compared to Eqs. (1)—(3), but also from their susceptibility to analytical and formal integration with respect to r to obtain P : P 9 u u 9 P P U

?

P u u : 0 P F

and

dr (u u 9 u u ) F F P P r

r P du 9 : 9 X ; u u . P X 2 z dr

(7) (8)

(9)

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263

Equation (7) expresses the radial variation in pressure wholly in terms of the ﬂuctuations in u and u ; Eq. (8) indicates that the Coriolis force is zero F P at all radii. Here P is the pressure on the wall of the pipe at r : a, where U a is the radius of the pipe. Since u u and u u are not functions of z in fully F F P P developed ﬂow, it follows that P/z is also independent of z and hence a constant that may be expressed as dP/dz. From a force balance over a central cylindrical segment of the ﬂuid it may be shown that :

(10)

(11)

dP r 9 , dz 2

where is the total shear stress in the z-direction imposed on the outer ﬂuid at any radius by this inner segment. It follows that at r : a, a dP : 9 . U 2 dz

Here is the shear stress imposed on the wall in the z-direction. From the U ratio of Eqs. (10) and (11), r : . (12) a U Now for convenience and simplicity, letting u : u, u : 9v, u : u, and P X X a 9 r : y allows Eq. (9) to be reexpressed as

y du 1 9 : 9 uv. U a dy

(13)

From Eq. (13), which is the starting point for all subsequent modeling herein for ﬂow in a round tube, it is evident that the contribution of the turbulent ﬂuctuations to the time-averaged velocity is wholly represented by uv. Similarly, it is evident from Eq. (7) that the radial variation of the dynamic pressure is wholly a consequence of u u and u u . P P F F 2. Dimensional, Asymptotic, and Speculative Analyses The most useful technique for the development of functional relationships for the characteristics of turbulent ﬂow has proven to be a combination of dimensional, asymptotic, and speculative analyses. Here speculation refers to a tentative postulate whose consequences are ultimately to be tested with experimental or exact theoretical results. It is unfortunate that the uncertainty implied by this terminology has often discouraged its formal usage. The techniques and results of dimensional, asymptotic, and speculative analysis have evolved independently in many different contexts and there is

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no general agreement on the priority of the various contributions. The attributions herein are conceded to be somewhat arbitrary. Fourier [23] in 1822 established the fundamental basis for dimensional analysis by noting that all added and equated terms in a complete relationship between the variables must have the same net dimensions. Rayleigh [24, 25] in 1892 illustrated the expression of functional relationships in terms of dimensionless groups, and in 1915 proposed a general mechanistic process for the determination of an appropriate minimal set of dimensionless groupings to describe the behavior deﬁned by a listing of dependent variables, independent variables, physical properties, and any other relevant parameters. Asymptotic dimensional analysis, as used herein, refers to the reduction of such a listing, and hence of the number of dimensionless groupings, for limiting conditions or locations or times. Speculative dimensional analysis, as deﬁned by Churchill [26], refers to the tentative elimination of individual variables or parameters and thereby reduction of the number of dimensionless groupings without necessarily any justiﬁcation or rationale in advance. This latter procedure may be characterized by the question, ‘‘What if . . .?’’ Insofar as the chosen set of variables and parameters is sufﬁcient and self-consistent, the results obtained by ordinary and speculative dimensional analyses are exact. The results from an asymptotic dimensional analysis may additionally depend on arbitrary constraints. Since the original choice of a set of variables, whether from a mathematical model or a heuristic listing, is always a possible source of error, Churchill [27] has proposed that all processes of dimensional analysis be considered to be speculative and thereby tentative until conﬁrmed by experimental data or exact theoretical results. Since neither analytical nor numerical solutions of the general timedependent equations of conservation for conditions resulting in turbulent shear ﬂow have been accomplished until very recently, these several processes of dimensional analysis have proven to be invaluable in terms of suggesting forms for the efﬁcient correlation of experimental data. Such forms may be expected to serve the same role for the currently emerging exact but discrete numerical results. The ﬁrst application of dimensional analysis for turbulent ﬂow was by Reynolds [28, 17], who in 1883 determined the conditions for the onset of turbulence in ﬂow through a long pipe and then in 1895 surmised that the transition was characterized by the dimensionless grouping Du /, now K known as the Reynolds number. Here D :2a is the diameter of the pipe and u the space- and time-mean velocity. This result and its direct counterparts K for the friction factor and the time-mean velocity distribution are all that may be inferred from the variables of Eq. (13) by simple dimensional

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analysis. On the other hand, Prandtl and his associates and contempories utilized asymptotic and speculative dimensional analysis with great insight and success to choose forms of these types for the correlation of experimental data starting from either Eqs. (1)—(3) or (13). Some such analyses follow. It may be speculated on purely physical grounds, inferred from Eqs. (1)—(3), or inferred from Eq. (13) that the local time-mean velocity in steady, fully developed turbulent ﬂow in a round tube with a radius a may be expressed in general as u :# y, , a, , , (14) U where the notation # x designates an unknown function of any independent variable x. [The inference of Eq. (14) from Eq. (13) implies that uv is a function of the same variables as u.] It follows from the application of ordinary dimensional analysis to Eq. (14) that one possible set of dimensionless groupings is

y( ) y U :# , . (15) a U Prandtl [29] introduced the notation u> :u(/ ) and y> :y( )/, U U which is still used almost universally today, to reexpress Eq. (15) as u

u> : # y>,

y y> # y>, : . a a>

(16)

Equation (13) may be reexpressed in this notation as 19

y> du> : ; (uv)>, a> dy>

(17)

where, as may be inferred, (uv)>Y9uv/ . U Prandtl next speculated that near the wall u might be essentially independent of a, thereby reducing Eq. (16) to u> : # y> ,

(18)

which is now known as the universal law of the wall. The limitation of Eq. (18) to y> a> explains the terminology law of the wall, and the dependence only on y> suggests its possible applicability to all geometries, and thereby the term universal. Very, very near the wall, the contribution of the turbulent ﬂuctuations in the velocity to the local shear stress might be expected to be negligibly small relative to the viscous stress, permitting Eq. (17) to be integrated to obtain u> : y> 9

(y>) ; y>. 2a>

(19)

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The limiting form of Eq. (19) may be noted to conform to Eq. (18), and both the general and the limiting form to apply to laminar ﬂow. The corresponding speculation that near the centerline the viscous stress might be negligible with respect to that due to the turbulent ﬂuctuations in the velocity implies independence of du/dy from . From the same process of dimensional analysis for du/dy as carried out for u in Eqs. (14)—(18), it may be inferred that

y d( y/a ) du> : : , a d(y/a) d(y/a)

(20)

where and designate arbitrary functions of y/a. Formal integration of Eq. (20) from u :u , the velocity at the centerline at y :a, leads to A y y u> 9 u> : 1 9

:# . (21) A a a

The term u> 9 u>, which characterizes the behavior near the centerline in A the same general sense that u> does near the wall, is called the velocity defect (or deﬁciency), while Eq. (21) is called the law of the center. Millikan [30], with great imagination and insight, speculated that Eqs. (18) and (21) might have some region of overlap, far from the wall and far from the centerline, where both were applicable, at least as an approximation. Accordingly, he reexpressed Eq. (18) in terms of the velocity defect, that is, as u> 9 u> : # a> 9 # y> , (22) A and noted that the only functional expression for the velocity defect satisfying both Eqs. (21) and (22) is

a , u> 9 u> : B ln A y

(23)

where B is an arbitrary dimensionless coefﬁcient. The necessary counterpart for the velocity itself is u> : A ; B ln y> ,

(24)

where A is an arbitrary dimensionless constant. Although Eqs. (23) and (24) conform to both the law of the center and the law of the wall, they would be expected to have only a narrow identical region of validity far from both the centerline and the wall. Von Ka´rma´n [31] postulated that despite this restriction, Eq. (23) might provide an adequate approximation for the entire cross-section insofar as integration to determine u> is concerned. The result K

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267

of such an integration is 3B u> 9 u> : , A K 2

(25)

which may be combined with Eq. (24), as specialized for y> : a>, to obtain

2 3B : u> : A 9 ; B ln a> . K f 2

(26)

Here f Y 2 /u is the Fanning friction factor. Equation (26) may of course U K also be derived directly by integrating u> from Eq. (24) over the crosssection. The derivation of Eq. (26) by von Ka´rma´n is one of the most fateful in the history of turbulent ﬂow, in that it has remained to this day the most common correlating equation for the friction factor, with its fundamental shortcomings compensated for and disguised by differing and varying values of A 9 (3B/2) and B. For a pipe with a roughness e, the same type of analysis that led to Eq. (16) results in

y e u> : # y>, , a a

(27)

or the equivalent. The speculation that the velocity is independent of the radius and dependent primarily on the roughness rather than on the viscosity leads to the following modiﬁed law of the wall: u> : #

y . e

(28)

Equation (21) remains applicable for the region near the centerline for roughened as well as smooth pipe. The equivalent of the speculation of Millikan results again in Eq. (23) for the velocity defect in the possible region of overlap but the following different expression for the velocity distribution itself in that region: u> : C ; B ln

y . e

(29)

Here, C is a dimensionless arbitrary constant and B is implied to have the same value as in Eqs. (23) and (24). Integration of Eq. (29) over the cross-section results in

2 3B a : u> : C 9 ; B ln , K f 2 e

(30)

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268

but Eq. (25) remains applicable. It may be inferred that the effect of roughness is simply to decrease u> for a given value of y> by the quantity B ln e> ; A 9 C and to decrease u> by the same amount for a given value K of a>. Here e> Y e( )/ in conformity to the deﬁnition of y>. U Murphree [32] and several others used a variety of methods of asymptotic expansion to derive the following relationship for the time average of the product of the ﬂuctuating components of the velocity and thereby the turbulent shear stress very near a wall: 9 uv : y ; -y ; . . . .

(31)

Here, ,-, . . . are arbitrary dimensional coefﬁcients. Equation (31) may be reexpressed in terms of the previously deﬁned dimensionless variables as (uv)> : (y>) ; -(y>) . . . ,

(32)

where and - are dimensionless coefﬁcients. Substitution of (uv)> from Eq. (32) in Eq. (17), followed by integration from u> : 0 at y>: 0, leads to a corresponding expression for u>, namely, (y>) u> : y> 9 (y>) 9 ( y>) 9 ;···. 4 5 2a>

(33)

Equation (33) without the term in (y>)/2a>, which is negligible for typical values of a>, has also been derived directly by asymptotic expansion. The recognition on physical grounds that the fraction of the shear stress due to turbulence, namely 9uv/, is necessarily ﬁnite, positive, and less than unity at the centerline requires, by virtue of Eq. (17), that

y> , u> 9 u> : E 1 9 A a>

(34)

and therefore that

(uv)> ; 1 9

2E a>

19

y> , a>

(35)

where E is an arbitrary dimensionless coefﬁcient. Equations (34) and (35) are the counterparts for the region near the centerline of Eqs. (33) and (32), respectively, for the region near the wall. The range of validity, if any, of each of the foregoing speculative expressions, namely Eqs. (18)—(35), is subsequently evaluated on the basis of experimental data and direct numerical simulations.

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3. Empirical Models An alternative and supplementary approach to dimensional, speculative, and asymptotic analyses is the postulate of mechanistic empirical models for the turbulent shear stress and thereby for the prediction of the local time-mean velocity distribution and its space mean. a. The Eddy Viscosity Boussinesq [33] in 1877, and thus before the identiﬁcation by Reynolds in 1895 of the relationship between the turbulent shear stress and the ﬂuctuating components of the velocity, proposed by analogy to Newton’s law for the viscous shear stress the following expression for the total shear stress in a shear ﬂow: du : ( ; ) . R dy

(36)

Here is the eddy viscosity, an empirical quantity that is a function of local R conditions rather than a physical property such as . This expression may be recognized as equivalent to the following differential model for the principal Reynolds stress: du 9 uv : . R dy

(37)

Equation (17), with the turbulent shear stress represented by Eq. (37), may be rewritten as 19

y> du> : 1; R . a> dy>

(38)

b. The Mixing Length Another historically important model for the turbulent shear stress was proposed by Prandtl [34] in 1925 on the basis of a postulated analogy between the chaotic motion of the eddies and that of the molecules of a gas. This model may be expressed as 9uv : l

du du dy dy

(39)

where l is a mixing length for eddies corresponding to the mean free path of molecules as deﬁned by the kinetic theory of gases. Although this analogy, as noted by Bird et al. [35, p. 160], has little physical justiﬁcation, the mixing-length model has generally been accorded more respect by analysts than the eddy viscosity model, apparently because of its mechanistic rationale, however questionable that may be. Von Ka´rma´n [31] speculated on dimensionless grounds that near the wall, l might be proportional to the distance from the wall; that is, he

stuart w. churchill

270 proposed the expression

l : ky,

(40)

where k is a dimensionless factor that is now generally called the von Ka´rma´n constant. Prandtl [29] substituted l from Eq. (40) in Eq. (39) and in turn the resulting expression for 9uv in Eq. (13) to obtain

y du du , 1 9 : ; ky U a dy dy

(41)

which may be reexpressed in the canonical dimensionless form as follows: 19

y> du> : ; k(y>) a> dy>

du> . dy>

(42)

Prandtl [34], starting from Eq. (42), neglected the variation in the total shear stress with y>/a>, neglected the viscous shear stress, took the square root of the resulting expression, and integrated indeﬁnitely to derive 1 u> : A ; ln y> . k

(43)

Equation (43) is seen to be equivalent to Eq. (24) with B : 1/k. Because of the two idealizations made in the reduction of Eq. (42), the resulting expression would be expected to be invalid near the wall where the viscous shear stress is controlling and near the centerline where the variation of the total shear stress is important. Even within the remaining region, Eq. (43) is subject to the two postulates represented by Eqs. (39) and (40). The existence of a region of overlap, which was postulated by Millikan in deriving Eq. (24), may be inferred to be equivalent to these two empirical postulates of Prandtl. It is worthy of note that despite the postulate of negligible viscous shear in its derivation, Eq. (43) incorporates, when rewritten in terms of dimensional variables, a dependence on the viscosity insofar as A is a constant independent of the Reynolds number and hence of the viscosity. The subsequently demonstrated success of Eq. (43) and (24) with empirical values for A and B : 1/k in representing experimental data is a testament to the insight and ingenuity of both Prandtl and Millikan in following two different and tortuous paths in their derivations. Analytical solutions of Eq. (42) in closed form are actually possible if one or the other of the simpliﬁcations made by Prandtl in reducing Eq. (42) in order to derive Eq. (43) is avoided. Furthermore, a solution in integral form may be derived without making either simpliﬁcation. For example, if the viscous shear stress is taken into account, the resulting quadratic equation

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271

in du>/dy> may be solved and then integrated from u> : 0 at y> : 0 to obtain u> :

1 9 [1 ; (2ky>)] 1 ; ln 2ky> ; [1 ; ky>)] . k 2ky>

(44)

Because of the imposition of the boundary condition at the wall, this expression, which was apparently ﬁrst derived by Rotta [36], is free of the arbitrary constant A of Eq. (43) and provides a smooth if erroneous transition from the limiting form of Eq. (19) for y> ; 0 to Eq. (43), with an effective value of A ; (1/k)(ln 4k 9 1) as y> ; -. The correct limiting behavior for y> ; 0 and the smooth transition to Eq. (43) are a consequence of accounting for the viscous shear stress. The failure of the predicted transitional behavior to conform functionally to Eq. (33) is clearly attributable to the shortcomings of Eqs. (39) and (40), but the reason for the prediction of highly erroneous (negative) values for the equivalent of A at large values of y> for a representative value of k is more difﬁcult to assign. Conversely, accounting for the linear variation of the total shear stress with y> but neglecting the viscous shear stress permits derivation of the following solution for the so-reduced form of Eq. (42) by a process similar to that used to obtain Eqs. (43) and (44): u> :

y> y> 1 2 19 92 19 a> a> k

; ln

y> a> y> 1; 19 a> 19 19

y> 1; 19 a> y> 19 19 a>

.

(45)

Here u> : 0 at y> : y> was invoked as an arbitrary boundary condition. The choice of y> : exp 9Ak results in matching the predictions of Eqs. (45) and (43) at that location. Equation (45) shares the limitation of applicability of Eq. (43) to the turbulent core near the wall and is of interest only as a measure of the effect of neglecting the variation of the total shear stress in that regime. For larger values of y>/a> it is in serious error because of its incorporation of Eq. (40). Taking into account both the viscous shear stress and the variation of the total shear stress, that is, starting from Eq. (42), solving this quadratic equation in du>/dy>, and integrating formally from u> : 0 at y> : 0 results

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in the following integral expression:

y> dy> a> . u> : 2 y> 1; 1; 19 (2ky>) a>

19

W>

(46)

Equation (46) coincides with Eq. (44) for y> a> and represents an improvement on Eq. (45) for larger values of y> at the expense of numerical integration, but fails for y> ; a> owing to the inapplicability of Eq. (40) for that regime. Comparison of the predictions of Eqs. (44)—(46) with Eq. (43) for representative values of A, k, and y> conﬁrms the good judgment of Prandtl in making the simpliﬁcations leading to Eq. (43) since Eq. (40), which all of these ‘‘improved’’ expressions incorporate, is valid even as an approximation only in the turbulent core near the wall. Prandtl [34], and in more detail in [37], speculated that near the centerline the mixing length might be nearly invariant, i.e., l5l , (47) ? where l is the limiting value for y> : a>. He then substituted l for l in ? ? Eq. (39) and the resulting expression for 9uv in Eq. (13), neglected the viscous term, and integrated from u : u at y : a to obtain, in dimensionless A form,

2 a> y> . (48) u> 9 u> : 19 A 3 l> a> ? Equation (48) correctly predicts du/dy : 0 at y> : a>, but has a different power dependence on 1 9 (y>/a>) than does Eq. (34). In order to improve upon Eqs. (40) and (45) and thereby on Eqs. (43) and (48), von Ka´rma´n [31] postulated that l : k*

du/dy du/dy

(49)

where k* is an arbitrary dimensionless coefﬁcient similar to k. He once explained, in response to an oral inquiry from the author of this article, that Eq. (49) was chosen because it was the simplest dimensionally correct expression for l involving only derivatives of the velocity. Substituting l from Eq. (49) in Eq. (39) and following the same procedure as used to obtain Eq. (48), but with two integrations and the equivocal boundary condition du/dy ; - at y : 0, results in 1 > 9 > : 9 A k*

19

y> y> ; ln 1 9 1 9 a> a>

.

(50)

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273

If the variation of the total shear stress is neglected as well, the procedure used to derive Eq. (50), except for an indeﬁnite limit for the second integration, leads to Eq. (43) with k replaced by k*, which suggests but does not prove their identity in general. As y> ; a>, Eq. (50) may be approximated by

(51)

l : ky(1 9 exp 9*y> )

(52)

y> 1 19 , u> 9 u> : A a> 2k*

which not only has a different functional dependence on 1 9 (y>/a>) than Eq. (34) but in addition fails to predict du/dy : 0 at y> : a>. Van Driest [38] attempted to improve upon Eq. (40), the mixing length model of Prandtl for the region near the wall, by including a term for viscous damping similar to the one that holds for the laminar motion of a ﬂuid subjected to the harmonic oscillation of a plate. That is, he let where * is an empirical dimensionless coefﬁcient whose numerical value is usually taken to be 1/26, in rough correspondence to the furthest limit of the buffer layer from the wall. Introducing l from Eq. (52) in Eq. (39) and in turn 9uv in Eq. (13), neglecting the variation in the total shear stress, solving the resulting quadratic equation for du>/dy>, and integrating formally from u> : 0 at y> : 0 results in u> : 2

W>

dy> . 1 ; (1 ; [2ky>(1 9 exp 9*y> ])

(53)

For y> ; 0, Eq. (53) reduces to Eq. (29), but unfortunately with : 0 and - : k*/5. For large values of y> it reduces to Eq. (44), and thereby has the merits and shortcomings already noted for that expression. It may be inferred from Eq. (46) that the variation of the total shear stress, which van Driest neglected, may be taken into account to obtain

u> : 2

W>

19

1; 1; 19

y> a>

y> a>

dy> . (54)

(2ky>)(1 9 exp 9*y> )]

Just as noted with respect to Eq. (46), the result is only a slight improvement on Eq. (53), since Eq. (52) is not applicable in the region where the terms in 1 9 (y>/a>) have a signiﬁcant role. c. Other Models Kolmogorov [39] and Prandtl [40] independently conjectured on dimensional grounds (local similarity) that : c ,l*, R

(55)

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where here , is the kinetic energy of the turbulence and l* is an unknown length scale. Batchelor [41] subsequently conjectured that l* : c ,/, (56) where is the rate of dissipation of turbulence. Combination of Eqs. (53) and (56) results in : c c ,/. (57) R Launder and Spalding [42] proposed calculating , and numerically from differential transport equations formulated as moments of that for momentum, such as Eq. (9), and then in turn calculating from Eq. (57). R Unfortunately, the approximate expressions for the terms in these moments of the momentum balance that have been suggested by various investigators are somewhat arbitrary, and in any event introduce a number of empirical coefﬁcients in addition to c c of Eq. (57). Although some success has been achieved with the ,— model in the few simple ﬂows for which an extensive set of data exists from experimental measurements and/or direct numerical simulations and therefore for which the model is not needed, the predictions for more complex ﬂows have been disappointing in accuracy or are precluded by singularities in and/or l. The ,——uv or Reynolds-stress R model, which adds a transport equation for uv similar to those for , and , appears to be free of singularities even in a concentric circular annulus (see Hanjalic´ and Launder [43]) but is essentially a correlative rather than a predictive model for the important region near the wall. The large eddy simulation (LES) method starts from the time-dependent equations of conservation but introduces arbitrary terms such as those of the ,— and Reynolds-stress models as well as utilizing the ,— model or additional empirical terms for the region near the wall. For an illustration of the applicability of this model, again for an annulus, see Satake and Kawamura [44]. 4. T he Experimental Data of Nikuradse Nikuradse [45—47] in 1930, 1932, and 1933 obtained extensive and precise sets of experimental data for the time-mean velocity distribution in the turbulent core and for the axial pressure gradient for the fully developed ﬂow of water in smooth round tubes for 600 & Re & 3.24;10 and in round tubes with a uniform artiﬁcial roughness e corresponding to 15(a/a)507 for 600 & Re & 10. Furthermore, he presented his data in tabular as well as graphical forms, thereby making it readily available in full numerical detail to subsequent investigators. For more than 60 years these data have

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been generally accepted as the primary standard for the development of models and correlating equations, for the evaluation of arbitrary constants therein, and for evaluation of the data of subsequent investigators. However, Miller [48] in 1949 identiﬁed an apparent discrepancy in the tabulated values of y> in Table 3 of Nikuradse [46]. By means of an inquiry addressed to Prandtl, he learned that Nikuradse had added 7.0 to each value of y> (the dimensionless distance of each point of measurement from the wall) in order to force the measured values of u> in his smooth pipes to approach the limiting form of Eq. (19) as y> ; 0. This discovery by Miller may readily be conﬁrmed by comparing the values of u> y> plotted in Fig. 15 of Nikuradse [45] with those in Fig. 24 of Nikuradse [46]. This ‘‘adjustment’’ has an insigniﬁcant effect for the large values of y> but precludes the use of the tabulated values for the small values. Robertson et al. [49] in 1968 conjectured that Nikuradse [47] might also have ‘‘adjusted’’ his experimental values for the velocity distribution near the centerline of the artiﬁcally roughened pipes in order to force comformity of the values of u> —u> to 3B/2 : 3.75, and Lynn [50] in 1959 discreetly noted ‘‘the A K extraordinarily low scatter’’ in the experimental values used by Nikuradse [46] to infer (incorrectly) that the eddy viscosity approaches zero at the centerline. However, on the whole the measurements by Nikuradse of the velocity distribution in the turbulent core as well as those of the axial pressure gradient for both smooth and rough pipe have stood the test of time, and these ‘‘adjustments,’’ except possibly those implied by Lynn, have not had any serious consequences in either ﬂuid mechanics or heat transfer. Nikuradse [46] used his experimental data ﬁrst of all to test the law of the wall, Eq. (18). He found conformity for the smooth pipes for all ﬂows, all diameters, and all locations, including even the region near the centerline where it might not have been expected to hold. Next, he found that Eq. (24) with A : 5.5 and B : 2.5 represented these values well for all y> 50, again even for the region near the centerline. He also found Eq. (29) with C : 8.5 and B : 2.5 to be successful for representation of the measured velocity distribution for the artiﬁcially roughened pipes at the larger values of a>. However, his experimental values for the friction factor in the smooth pipes were found to be represented better by

2 : 2.00 ; 2.46 ln a>

f

(58)

than by Eq. (26) with A : 5.5 and B : 2.5, which yields A 9 (3B/2) : 1.75 and the same value of the coefﬁcient B as for the velocity distribution. The discrepancy in the constant (2.0 as compared to 1.75) may be attributed to

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Fig. 2. Experimental velocity distribution in fully developed turbulent ﬂow of water in a 127-mm Plexiglas tube (R : Re). (Reprinted with permission from Lindgren and Chao [51], Figure 1. Copyright 1969 American Institute of Physics.)

the neglect of the boundary layer near the wall, as represented in the limit by Eq. (33), and of the wake, as represented in the limit by Eq. (34). Both of these deviations from Eq. (24) are well illustrated by the much later data of Lindgren and Chao [51] in Fig. 2. On the other hand, the discrepancy in the coefﬁcient (2.46 as compared to 2.50) is unacceptable on theoretical grounds. The overly simpliﬁed and incongruent expressions for the velocity distribution and the friction factor that appear unexplained in most of our current textbooks and handbooks are a legacy of the failure of Nikuradse to obtain sufﬁciently precise and accurate values for the velocity distribution near the wall and near the centerline and to develop correlating equations encompassing these regions. Nikuradse is, of course, not responsible for the failure of subsequent investigators and writers to explain and provide a rational correction for these anomalies. A similar discrepancy exists for the correlating equations of Nikuradse [47] for artiﬁcially roughened pipe, for which he correlated his experimental

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data for the axial pressure gradient for asymptotically large values of the Reynolds number with the expression

2 a . : 4.92 ; 2.46 ln f e

(59)

It may be noted that Eq. (30) with C : 8.5 and B : 2.5 predicts a value of 4.75 rather than 4.92 for the constant and a value of 2.5 rather than 2.46 for the coefﬁcient. For the reasons just cited, the experimental measurements of the velocity distribution by Nikuradse do not provide a test of Eqs. (19) and (33) or allow evaluation of the coefﬁcients of the latter. Although he apparently did not recognize the existence of a wake, his experimental values of u> 9 u> A conform crudely to Eq. (34) even for y/a as low as 0.2. On the whole, they suggest a value of E between 6.7 and 7.5. The failure of the values very near the centerline to conform to this relationship may be due to the ‘‘adjustments’’ implied by Lynn [50] as well as to the very small differences in the measured velocities at closely adjacent locations in that region. Nikuradse [45] determined the values of the mixing length plotted in Fig. 3 from the slope of plots of the velocity distribution. The values of the mixing length thus determined appear to be independent of the Reynolds number and of the roughness ratio for sufﬁciently large values of the Reynolds number, implying a great generality for the relationship between l/a and y/a. Nikuradse [46] subsequently proposed representation of all of these values by the empirical expression

l y y : 0.14 9 0.08 1 9 9 0.06 1 9 , a a a

(60)

which he attributed to Prandtl and interpreted as an ‘‘interpolation formula’’ between Eq. (40) with k : 0.4 for y> ; 0 and Eq. (47) with l : 0.14 ? for y> ; a>. This value of l results in a net numerical coefﬁcient of 4.76 in ? Eq. (48). From these same slopes of the velocity distribution he determined the values of the eddy viscosity plotted in Fig. 4 and concluded erroneously that it approaches zero at the centerline. The experimental data of Nikuradse for fully developed ﬂow in a round tube and his own correlations for u>, , l, and f based on these data have R been described and analyzed here in some detail because they have had great inﬂuence on the predictions and correlations for convective heat transfer. In addition to the caveats noted earlier, some subsequent investigators have questioned the numerical values of the constants and coefﬁcients determined by Nikuradse. In particular, Hinze [52] and other have asserted that the constant A and the coefﬁcient B : 1/k of Eq. (24) are Reynolds-

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Fig. 3. Experimental mixing lengths in smooth and artiﬁcially roughened round tubes. (From Nikuradse [45], Figures 9 and 12.)

number dependent. Such uncertainties and variations have not been explored herein since none of the numerical values determined from the data of Nikuradse appear in the ﬁnal expressions for either ﬂow or heat transfer. 5. Power-L aw Models Power-law models for the velocity distribution and the friction factor might not have merited attention herein had not a recent attempt been made

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Fig. 4. Experimental eddy viscosities in smooth tubes. (From Nikuradse [46], Figure 27.)

to resuscitate them. Furthermore, they might logically have been included in Section II, A, 3. The deferral to this point is because the experimental data of Nikuradse, as described in Section II, A, 4, are essential to their interpretation and evaluation. Blasius [53] in 1913 plotted the available experimental data for the friction factor for round tubes, which then extended only up to Re : 10, versus Re in logarithmic coordinates and found that a satisfactory represen-

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tation could be achieved with a straight line equivalent to f:

0.0791 . Re

(61)

C. Freeman [54] in 1941, in a Foreword to a compilation of the extensive set of experimental data obtained in 1892 by his father, J. R. Freeman, but not published until 49 years later, speculated that had Blasius had access to these values, which extend up to Re : 9; 10, he might have developed a more general correlating equation and thereby changed the course of history in applied ﬂuid mechanics. This assertion not only was justiﬁed when it was written, but has proven prophetic. Prandtl [55] in 1921, and therefore prior to his development of the mixing-length model, recognized from dimensional considerations that Eq. (61) implies

u y , (62) : a u A which by virtue of the numerical coefﬁcient of 0.0791 may also be expressed as u> : 8.562(y>).

(63)

Nikuradse [45—47] tested Eq. (62) with his experimental data for the velocity distribution and found that it provided a good representation only for the turbulent core, only for smooth pipes, and only for Re & 10. Accordingly, he generalized Eq. (62) as

y ? u : , u a A

(64)

u> : -( y>)?.

(65)

which corresponds to

Here is an arbitrary dimensionless exponent and - is an arbitrary dimensionless coefﬁcient. From integration of Eq. (65) over the cross-section it follows that -Y

(1 ; )(1 ; 2)u> K. 2(a>)?

(66)

He determined numerical values of and - as functions of Re and e/a from his experimental velocity distributions, but abandoned this mode of correlation as inferior to Eqs. (24), (26), (29), and (30).

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Nunner [56] in 1956 somewhat revived the power-law model by discovering that the empirical relationship : 2 f

(67)

provides a good approximation for both smooth and roughened pipes. However, a separate correlation is required for the friction factor as a function of the Reynolds number and roughness ratio. Thirty-seven years later, Barenblatt [54], apparently unaware of the work of Nunner, rationalized the form of the power law for the velocity distribution using scaling arguments, and proposed, on the basis of the data of Nikuradse [46] for smooth pipe, empirical expressions for and - in Eq. (65) as functions of Re. These expressions are not reproduced herein, since that for is inferior to Eq. (67) and that for - is equivalent but inferior to most other correlating equations for the friction factor. Equation (65) with from Eq. (67) and - from Eq. (66), and with u> from K Eq. (58) or (59), whichever one is appropriate, is slightly superior to Eq. (24) for 30 & y> & 0.1a>. However, it is seriously in error for larger as well as smaller values of y>. These errors might have been anticipated from the predictions by Eq. (65) of an unbounded velocity gradient at the wall and a ﬁnite velocity gradient at the centerline. Equation (66) may be reexpressed as f:

(1 ; )(1 ; 2) ?>? , -2?\?Re?

(68)

which implies that a ﬁxed-power dependence of the velocity on the distance from the wall over the entire cross-section is required to obtain a power-law dependence of the friction factor on the Reynolds number. It may therefore be inferred from the previously cited failures of the power-law model for the velocity near the wall and near the centerline, and more importantly from the observed dependence of on Re, that a pure power-law model for the friction factor cannot have any real range of validity with respect to Re. The semilogarithmic dependence of the square root of the reciprocal of the friction factor on the Reynolds number has already been noted to be subject to a related but numerically less severe defect. Churchill [58] has recently compared the predictions of the power-law models for the velocity distribution and the friction factor with experimental data and other correlating equations. These comparisons support the preceding qualitative conclusions. The failure of the power-law models might have been anticipated on the basis of dimensional analysis. Rayleigh [25] used a power-series expansion

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as a mechanical means of identifying a minimal set of dimensionless groups from a listing of the variables, physical properties, and parameters. He fully recognized, as demonstrated by his own illustrations of this technique, that the derivation of the ﬁrst term in the expansion in the form of a product of arbitrary powers of the independent dimensionless groups did not imply either powers or products for the unknown functional relationship. Unfortunately, such misinterpretations plague us to this day. Indeed, relationships in the form of powers other than the unity ordinarily occur only in asymptotic expressions, such as the limiting form of Eq. (32), or in special cases, such as with the friction factor (but not the velocity distribution) for fully developed laminar ﬂow in a round tube. 6. T he Analogy of MacL eod Before examining some important recent work it is convenient if not essential to describe a little-known conjecture that suggests a means of obtaining congruence of the turbulent shear stress and the velocity distribution for round tubes with their counterparts for ﬂow between parallel plates of inﬁnite extent. Rothfus and Monrad [59] showed that complete congruence of the velocity proﬁle in fully developed laminar ﬂow in a round tube with that between parallel plates may be achieved by specifying : U0 U. and a : b, where the subscripts R and P designate round tubes and parallel plates, respectively, and b is the half-spacing of the plates. This requirement is excessive; a sufﬁcient condition for u> y>, a> : u> y>, b> is simply 0 . that a> : b>. MacLeod [60] subsequently speculated that this latter relationship might also hold for fully developed turbulent ﬂow. His speculation is beautifully conﬁrmed in Fig. 5, in which the experimental values of Whan and Rothfus [61] for u> in ﬂow between parallel plates are compared A with a curve representing the corresponding experimental values of Senecal and Rothfus [62] for a round tube. The velocities at the central plane and the centerline were chosen for this comparison because as extreme values they provide the most severe test of the analogy. It is evident that the analogy does not hold for the regime of transition from fully developed laminar to fully developed turbulent ﬂow. This discrepancy was to be anticipated since the onset of transitional ﬂow has long been known to occur at differing values of a> and b>. It may be inferred from Eqs. (17), (38), and (39) that the analogy of MacLeod applies directly to (uv)>, /, R and l> insofar as it is valid for u>. The special importance of the analogy of MacLeod is that it provides a formal justiﬁcation for the use of experimental data as well as values from direct numerical simulations for u> and (uv)> in ﬂow between parallel

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Fig. 5. Experimental conﬁrmation of the analogy of MacLeod for central velocities. (From Whan and Rothfus [61], Figure 3.)

plates in the development of correlating equations for round tubes. A critical assessment of the analogy of MacLeod as applied to (uv)> would appear to be of crucial importance in both ﬂow and heat transfer, but that requires values of greater precision and reliability for both geometries than are currently available from either experimental measurements or direct numerical simulations. The results from direct numerical simulations for round tubes are currently less extensive and reliable than those for parallel plates because of the computational complexities associated with curvature. On the other hand, the experimental measurements for ﬂow in round tubes are more extensive and reliable than those for parallel plates because of the difﬁculty of aligning and supporting plates of sufﬁcient extent and small enough spacing to minimize side-wall effects. Entrance effects also appear to be more serious. The law of the wall of Prandtl [Eq. (18)] may be noted to be a special case of the analogy of MacLeod for the region near the wall, but, on the other hand, it is presumed to be applicable for all shear ﬂows, not just those for round tubes and parallel plates.

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7. T he Colebrook Equation for the Friction Factor in Naturally Rough Piping The contribution of Colebrook [63] to the prediction of turbulent ﬂow in piping is perhaps second only to that of Nikuradse in practical importance. Although the results of his work appear implicitly in almost every plot of the friction factor in our handbooks and textbooks, he is seldom cited as the primary source. Nikuradse [47] chose uniform artiﬁcial roughness for his experimental investigation for the obvious reasons of reproducibility of the measurements of the ﬂow and of simple quantitative characterization of the roughness. Such measurements would not be expected to be representative for the naturally occurring roughness of commercial piping, which is characterized by the highly variable and perhaps chaotic amplitude and spacing associated with particular materials of construction (such as glass and concrete) and different methods of manufacture (such as extruding and casting), as well as with aging, corrosion, erosion, fouling, and different methods of linkage (such as welding and threading). The measurements of Colebrook for a variety of natural materials and conditions revealed that the pressure drop not only depends on the magnitude of the roughness but, even more importantly, has a completely different functional dependence on the Reynolds number: The friction factor decreases to an asymptotic value, as contrasted with an increase to an asymptotic value for uniform artiﬁcial roughness and an unending decrease for smooth piping. In order to represent this behavior, he arbitrarily deﬁned and designated by e a nominal roughness for each material and condition A that would result in the same asymptotic value of f for the friction factor A for very large Reynolds numbers as the value of the uniform artiﬁcial roughness of Nikuradse. Thus, on the basis of Eq. (50),

4.92 9 (2/ f ) e A A Y exp . 2.46 a

(69)

This ingenious concept of correlation avoids the necessity and difﬁculty of measuring and characterizing the roughness statistically, and instead focuses directly on the behavior of primary interest, namely the shear stress on the wall. Colebrook also found that his measured pressure drops at less than asymptotically large values of the Reynolds number could be represented closely by

2 e 1 A ; : 92.46 ln . f 7.39a 2.25a>

(70)

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He justiﬁed the form of Eq. (70) simply by asserting without any rationalization that the two terms in the argument of the logarithm must be additive. Churchill [64] subsequently reinterpreted Eq. (70) in terms of the canonical correlating equation of Churchill and Usagi [65], namely, y x S : y x S ; y x S, (71) where y x and y x are asymptotic values or expressions for small and large values of x, respectively, and n is an arbitrary exponent. By trial and error, exp (2/ f )/2.46 was found to be a better choice for y x in Eq. (71) than (2/ f ) or f . Then, from Eq. (58) rearranged as exp

(2/ f ) : 2.255a>, 2.46

(72)

y x : 2.255a>, and from Eq. (59) rearranged as (2/ f ) a exp : 7.389 , (73 ) e 2.46 A y x : 7.389a/e . A value of n :91 was chosen on the basis of the A experimental data of Colebrook. The result of this procedure may be expressed as

2 a> : 2.00 ; 2.46 ln , (74) f 1 ; 0.304(e /a)a> A which is exactly equivalent to Eq. (70). Most of the graphical representations of the friction factor in the turbulent regime in the current literature are simply plots of Eq. (74) or its near equivalent, most often in the form of f or f \ versus Re : a>(8/ f ) or Re f : 8a> with e /a or e as a A A parameter, accompanied by a table of values of e for various materials and A conditions. Most of the values of e that appear in the standard tabulations A were determined many decades ago by Colebrook and his contemporaries and may not be representative of modern materials and modern methods of manufacture and joining. A redetermination and recompilation of values of e would appear to be worthwhile. A The plot in Fig. 6 of the experimental data of Nikuradse and Colebrook for uniformly and naturally roughened pipe, respectively, in the form of

2 2a 9 2.46 ln f e

as a function of (e>) :

e( /) e U : (a>) a

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Fig. 6. The transitional behavior for uniformly and naturally roughened round tubes: (★) Data of Colebrook [63], natural roughness; all other points are from Nikuradse [47], uniform roughness.

demonstrates the fundamentally different paths of transition for uniformly roughened and naturally roughened pipe from ﬂow in a smooth pipe, as represented by the linear oblique asymptote, to ﬂow controlled wholly by roughness, as represented by the horizontal asymptote. On the basis of Eqs. (24), (26), (29), and (30), the velocity distribution in ‘‘the turbulent core near the wall’’ of naturally rough pipe may be predicted speculatively by dividing y> in the argument of the logarithm by 1 ; 0.304(e /a)a>. A 8. Experimental and Computed Values of uv Near the Wall Difﬁculty has been encountered in the past in determining in Eq. (32) or (33), since measurements of very small values of uv or u very near the wall are required. However, the uncertainty in has been greatly reduced in the past decade by virtue of direct numerical simulations. This computational method, which was pioneered by Orszag and Kells [66], is essentially free from empiricism except for the choice of a wave form, but it is sensitive to the number of grid points used in all three coordinate directions. The

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implementation of direct numerical simulations for channels is yet limited by computational demands, with only a few exceptions, to ﬂow between parallel plates of unlimited extent and even then to values of b> just above the minimal value of about 145 for fully developed turbulence. The computed values of uv by Kim et al. [67], Lyons et al. [68], and Rutledge and Sleicher [69] are in fair agreement with one another and with the best experimental measurements, such as those of Eckelmann [70], for the intrinsically important region very near the wall where the behavior is presumed to be independent of or at least negligibly dependent on b>. This combination of computational and experimental results for parallel plates conﬁrms beyond question the form of Eq. (32) and indicates a value 57;10\ for . 9. Experimental Values of uv and u Near the Centerline The coefﬁcient E of Eqs. (34) and (35) is the analog of for the region near the wall. It may in principle be determined from Eq. (34) by virtue of experimental values of u>, from Eq. (35) by virtue of experimental or computed values of (uv)>, or from the derivative of Eq. (34) by virtue of experimental values of du>/dy>. Unfortunately, all three of these methods require experimental values of greater accuracy and precision than are currently available. Even the values of (uv)>> computed by direct numerical simulation are marginal in this respect. It follows from Eqs. (38) and (34) that for y> ; a>

1 dy> y> 1 1 1 R : 19 91 ; 9 5 . a> a> du> a> 2E a> 2E

(75)

Equation (75) predicts that near the centerline, /a> approaches an R asymptotic value independent of y> and essentially independent of a>. Such behavior has been conﬁrmed (see, for example, Figure 5 of Churchill and Chan [71]). Groenhof [72] examined and compared experimental determinations of /a> by ﬁve sets of investigators for round tubes and one R investigator for parallel plates. These values range from 0.062 to 0.08, corresponding to values of E from 8.06 to 6.25.

10. T he Experimental Data of Zagarola Most of the correlating equations mentioned earlier, including the empirical constants, are based on the experimental data of Nikuradse [46, 47] despite their age and indicated limitations. Recently a new, comprehensive

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set of experimental data has been obtained for fully developed turbulent ﬂow in a round tube that challenges the dominant role of the measurements of Nikuradse. Zagarola [73] in 1996, using modern instrumentation and carefully controlled conditions, measured the time-averaged velocity and axial pressure gradient in air ﬂowing through a 129-mm tube with a highly polished surface. His ﬂows extended from Re : 3.55;10 to 3.526;10 and thus to higher values than those of Nikuradse, but not to as low ones. Zagarola conceded that his own measurements of the time-mean velocity were excessively high for y/a & 0.0155 for all Re. This requires discarding all of his values in the viscous sublayer (0&y>&10) and all but a few in the buffer layer (10&y>&30). Also, the slight displacement of many of the maximum measured values of the velocity from the centerline suggests that their accuracy is marginal for purposes of differential analysis in that region. Despite great effort to attain an aerodynamically smooth surface, the directly measured roughness ratio of e/a : 2.4;10\ is, according to Eq. (74), of sufﬁcient magnitude to have a signiﬁcant effect on both the friction factor and the velocity distribution at the higher values of Re. The effective roughness ratio, e /a, was concluded by Churchill [58] to have the someA what lower value of 7;10\, which, however, is still aerodynamically signiﬁcant for the largest values of Re studied by Zagarola. In spite of these caveats, the tabulated values of Zagarola for the velocity and the friction factor represent a very signiﬁcant contribution to ﬂuid mechanics and may be considered to supplant the tabulated experimental data of Nikuradse within the range of conditions for which they overlap, namely 3.158;10 & Re & 3.24;10. They justify reﬁnement or replacement of most of the algebraic and graphical correlations for the velocity distribution and the friction factor in the current literature. Zagarola found, as illustrated in Figs. 7 and 8, a signiﬁcant variation of the coefﬁcient k : 1/B of Eqs. (23), (24), and (26) with Re, but concluded that a value of 0.436 provided an adequate representation for the semilogarithmic regimes in all three instances. The constant A of Eq. (24), as determined on the basis of k : 0.436, was also found to vary somewhat with Re, as illustrated in Fig. 9, but a value of 6.13 was concluded by Zagarola to provide an adequate representation for all values of y> and Re for which this equation is applicable. The plots of u> 9 u> versus ln a/y , A such as illustrated in Fig. 10 for 3.1;10 & Re & 2.5;10, which he used as one method of evaluating k, indicated to him that the following expression was preferable to Eq. (23) for 0.01 & y/a & 0.1:

a 1 ln ; 1.51. u> 9 u> : A y 0.436

(76)

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Fig. 7. Variation with Re of the coefﬁcient k : 1/B in Eqs. (24) and (43) for 50 y> 0.1a>. (From Zagarola [73], Figure 4.38.)

Fig. 8. Variation of the coefﬁcient k : 1/B in Eq. (23) for u> and Eq. (26) for u> for various A K sets of increasing values of Re. (From Zagarola [73], Figure 4.30.)

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Fig. 9. Variation with Re of the constant A of Eqs. (24) and (43) for 50 y> 0.1a> with the coefﬁcient k : 1/B ﬁxed at 0.436 and free. (From Zagarola [73], Figure 4.40.)

Fig. 10. Determination of the deviation in the velocity at the centerline due to the wake. (From Zagarola [73], Figure 4.53.)

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Fig. 11. Comparison of semilogarithmic and power-law representations of the velocity distribution for 31;10 Re 4.4;10. (From Zagarola [73], Figure 4.44.)

Although Zagarola did not present a correlating equation for the region of the wake, Eq. (24) with A : 6.13 and B : 1/0.436 may be combined with Eq. (76) to obtain the following expression for the velocity at the centerline itself: 1 u> : 7.64 ; ln a> . (77) A 0.436 Zagarola also tested Eq. (65) and found, as illustrated in Fig. 11 for 3.1;10 & Re & 4.4;10 and 001 & y/a & 0.1, that the expression u> : 8.7(y>) ,

(78)

where 8.7 and 0.137 are purely empirical, represents the measured velocities better for 30 & y> & 500 but much more poorly for y> 500 than u> : 6.13 ;

1 ln y> . 0.436

(79).

Both representations are seen to fail for y> & 50. Similar representations and misrepresentations were found to be provided by these two expressions for other ranges of Re.

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Zagarola represented his experimental data for the friction factor with the following expression:

2 138.5 1 : 3.30 9 ; ln a> . f (a>) 0.436

(80)

He proposed the term 138.5/(a>) as a correction for the deviation of the velocity distribution in the boundary layer from the semilogarithmic regime on the basis of a subsequently described expression of Spalding [74] for the velocity distribution for all y> & 0.1a>. Zagarola actually proposed several different leading coefﬁcients for Eq. (80). The value of 3.30 was chosen here for consistency with Eqs. (76) and (79). 11. Overall Correlating Equations for the Velocity Distribution With only a few exceptions, to be noted here, the correlating equations of the past, as well as those of Zagarola, are for a single regime, primarily ‘‘the turbulent core near the wall,’’ although such expressions have often been implied to be applicable for the entire turbulent core. Equations (44)—(46), as well as Eqs. (53) and (54), purport to encompass the boundary layer, but the ﬁrst three do not include the higher-order terms of Eq. (33) and the latter two imply erroneously that : 0. Despite the presence of a> in several of these expressions, none of them approaches Eq. (34) as y> ; a>. Churchill and Choi [75] combined the limiting form of Eq. (19) for y> ; 0 and Eq. (24) with A : 5.5 and B : 2.5 in the form of Eq. (71) and chose a value of 92 for n on the basis of the experimental data of Abbrecht and Churchill [22] to obtain u> :

y> . y> 1; 2.5 ln 9.025y>

(81)

Here 5.5 ; 2.5 ln y> is expressed as 2.5 ln 9.025y> for compactness and to emphasize the presence of a singularity at y> : 1/9.025 : 0.1108. This singularity may be avoided without a signiﬁcant effect on the predictions of u> for any value of y> by simply adding unity to the argument of the logarithm to obtain u> :

y> . y> 1; 2.5 ln 1 ; 9.025y>

(82)

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Fig. 12. Representation of the velocity distribution in a smooth round tube by Eq. (82). (From Churchill and Choi [75], Figure 2.)

Equation (82) may be seen in Fig. 12 to represent the data upon which it is based very well for all but the largest and smallest values of y>. The deviations for y> & 2 are probably due to experimental error but those for the largest y> are deﬁnitely a consequence of failing to account for the wake. The previously mentioned expression of Spalding [74] for the viscous sublayer, buffer layer, and turbulent core near the wall is

y> : u> ; 0.1108 e0.4u> 9 1 9 (0.4u>) 9

(0.4u>) (0.4u>) 9 . (83) 2 3

Equation (83) approaches Eq. (24) with A : 5.5 and B : 2.5 for large values of y>, just as does Eq. (82), but approaches Eq. (33) as y> ; 0, albeit with a somewhat low value of 4.13;10\ for . It is thereby superior to Eq. (82) functionally but is not necessarily more accurate numerically. The inverse form of Eq. (83) is inconvenient functionally for a speciﬁed value of y>, but numerical values of u> may then readily be obtained by iteration. As indicated by the absence of a>, Eq. (83) also fails to account for the wake.

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Even earlier, Reichardt [76] proposed the following expression for the entire cross-section, including the region of the wake:

u> : 7.8 1 9 e9y>/11 9

y> 11

; 2.5 ln

e90.33y>

3a>(1 ; 0.4y>)(2a> 9 y>) . 2(3a> 9 4a>y> 9 2(y>))

(84 )

Equation (84) conforms to Eq. (34) with E : 7.5 for y> ; a> and to Eq. (24) with A : 5.5 and B : 2.5 for intermediate values of y>. Reichardt apparently intended it to conform to the ﬁrst two terms on the right-hand side of Eq. (33) for y> ; 0, but it fails in that respect since the coefﬁcients of (y>) and (y>) that result from the expansion of the logarithmic and exponential terms in series are not quite zero. Furthermore, the corresponding coefﬁcient of the term in (y>) has the very excessive value of 1.548;10\ as compared to 7;10\/4 : 1.75;10\ from the direct numerical simulations. (Values of 18.738 and 0.5071 in place of 11 and 0.33, respectively, would eliminate the terms in (y>) and (y>), but would decrease the coefﬁcient of (y>) only slightly to 1.371;10\ and therefore insufﬁciently.) Despite these discrepancies, the numerical predictions of Eq. (84) do not differ greatly from those of Eqs. (82) and (83) for y> & 0.1a> while it is more accurate functionally as well as numerically for 0.1a> & y> a>. B. New Improved Formulations and Correlating Equations The models just described all appear to have a defect or shortcoming and the correlating equations to be limited in scope or generality. The objective of the work described in this section has been to develop formulations and correlating equations that avoid these defects and limitations. 1. Model-Free Formulations The mixing-length model, as expressed by Eq. (39), was proposed by Prandtl to facilitate the prediction of the turbulent shear stress in the time-averaged equation for the conservation of momentum on the premise that algebraic or differential correlating equations for the mixing length, such as Eqs. (40), (47), and (49), would be simpler or more general than a correlating equation for the shear stress itself. The eddy diffusivity model of Boussinesq, as represented by Eq. (37), may be interpreted to have an equivalent objective.

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In contradistinction, Churchill and Chan [71, 77, 78] and Churchill [79] investigated the direct use of the turbulent shear stress itself as a variable for integration and correlation, thereby avoiding the need for heuristic variables such as the mixing length and the eddy viscosity altogether. The consequences are generally favorable and in some respects quite surprising. They started from Eq. (17), which is expressed in terms of (uv)> Y 9uv/ , U namely the local turbulent shear stress as a fraction of the shear stress on the wall. The negative sign was used in this deﬁnition since uv is negative over the entire radius of a round tube. Churchill [80] subsequently proposed as slightly advantageous the use of a new, alternative dimensionless quantity (uv)>> Y 9uv/, which may be recognized as the local fraction of the total shear stress due to turbulence. Equation (13) then becomes

19

y> du>/dy> [1 9 (uv)>>] : . a>

(85)

From physical considerations (uv)>> must be positive, less than unity, and greater than zero at all locations within the ﬂuid. This latter characteristic gives (uv)>> a signiﬁcant advantage in terms of correlation over (uv)>, which is zero at the centerline. Eliminating du>/dy> between Eqs. (38) and (85) reveals that (uv)>> R: 1 9 (uv)>>

(86)

The eddy viscosity is thus seen to be related algebraically to (uv)>>, which is a physically well-deﬁned and unambiguous quantity, and thereby to be independent of its heuristic diffusional origin. (Boussinesq was either very intuitive or just lucky.) It further follows from Eq. (85) and the indicated behavior of (uv)>> that / is also ﬁnite and positive at all locations R within the ﬂuid, including the centerline. It similarly follows from Eqs. (39) and (85) that (l>) :

(uv)>>

y> 19 a>

(87)

[1 9 (uv)>>]

The mixing length is thus also independent of its mechanistic and heuristic origin but is unbounded at the centerline. How did such an anomaly, which has a counterpart in all geometries and which refutes the very concept of a mixing length for practical purposes, escape attention for more than 70 years? One reason is the uncritical extension of respect for Prandtl to all details of his work. A second reason is the false mindset created by Fig. 3. A third is the requirement of even more precise data for the velocity

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distribution near the centerline than is available even today, although in retrospect the singular behavior of the mixing length at the centerline is apparent, at least qualitatively, from most sets of data. Although the anomalous behavior of the mixing length was apparently ﬁrst recognized by Churchill [80] as a consequence of his derivation of Eq. (85) and therefore because of his introduction of (uv)>> as a variable, it could have been identiﬁed much earlier merely by the substitution of du>/dy> from Eq. (34), which goes back at least to Reichardt [76] in 1951, in the combination of Eqs. (17) and (39) or the equivalent. Inference of this singularity from Eq. (17) requires consideration of Eq. (35). In any event, the continued use of the mixing length does not appear to have any justiﬁcation under any circumstance. Now reconsider Eq. (85), which represents the momentum balance in a round tube in terms of (uv)>>, which, as noted, is well behaved and constrained between zero and unity for all values of y>. Formal integration results in the following expressions for the time-averaged velocity distribution: u> :

W>

19

y> [1 9 (uv)>>]dy> a>

(y>) W> y> 9 19 (uv)>>dy> 2a> a> These may be expressed more compactly as : y> 9

a> (1 9 R) 9 (uv)>>dR (89) 2 0 0 The leftmost forms of Eqs. (88) and (89) are more convenient for numerical integration because the rightmost ones involve small differences of large numbers, but the latter have the advantage of demonstrating that the effect of the turbulence is simply to provide a deduction from the well-known expressions for purely laminar ﬂow at the same value at a>. Equation (89) may in turn be integrated formally over the cross-sectional area to obtain the following expression for the space- or mixed-mean velocity and thereby the friction factor: u> :

a> 2

(88)

[1 9 (uv)>>]dR :

2 a> : u> : [1 9 (uv)>>]dR dR. K f 2 0 Equation (90) may be reduced by integration by parts to obtain

2 a> : u> : K f 4

[1 9 (uv)>>]dR :

a> a> 9 4 4

(90 )

(uv)>>dR, (91)

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which involves only a single integral. The rightmost form of Eq. (91) reveals that the effect of the turbulence on the mixed-mean velocity is also simply a deduction from the well-known expression (Poiseuille’s law) for purely laminar ﬂow. This deductibility of an integral term from the expressions for the time-mean velocity distribution and the mixed-mean velocity in purely laminar ﬂow may seem to be obvious in retrospect, but such a structure is not so evident in the analogs of Eqs. (89) and (91) in terms of the eddy viscosity and the mixing length, and does not appear ever to have been mentioned in the literature. Also, although it is evident in retrospect that the double integral of the analogs of Eq. (90) in terms of the eddy viscosity and the mixing length may be reduced to a single integral by means of integration by parts, that simpliﬁcation was apparently never recognized or implemented because the more complex forms obscure this possibility. It may be noted that (uv)>> has no advantage over (uv)> in this respect, that is, both the deductibility of the effects of turbulence and the possibility of integration by parts are quite evident when starting from Eq. (17) rather than Eq. (85). Equations (88)—(91) are exact insofar as Eq. (13) is valid, but some empiricism is necessarily invoked in the required correlating equation for (uv)>>. Before turning to such expressions several partial precedents for Eqs. (89) and (91) should be acknowledged. Kampe´ de Fe´riet [81] derived the equivalent of Eq. (89) and the analog of Eq. (91) for parallel plates in terms of (uv)> but did not implement these expressions; while Bird et al. [35, p. 175], note that the use of a correlating equation for (uv)> rather than one for the eddy viscosity or the mixing length might lead to a simpler integration for the velocity distribution. 2. Correlating Equations for the L ocal Turbulent Shear Stress Despite the advantages of the dimensionless turbulent shear stress in predicting the velocity distribution and the mixed-mean velocity, as described in the immediately preceding paragraphs, the general failure to recognize those advantages has resulted in a dearth of correlating equations. One exception is due to Pai [82], who was inspired by the aforementioned formulations of Kampe´ de Fe´riet to represent the experimental data of Nikuradse [46] for the velocity distribution at Re : 3.24;10 (his highest rate of ﬂow) by a polynomial in R and to substitute the derivative of that expression in Eq. (17) to obtain (uv)> : 0.9835R(1 9 R).

(92 )

Equation (92) is reasonably accurate numerically and functionally for y> ; a>(R ; 0), but fails badly for both small and intermediate values of

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y>. Integration of either Eq. (92) or the velocity distribution from which it was derived leads to an obviously invalid expression for the friction factor. Churchill and Chan [71] constructed a more comprehensive and general expression for (uv)> that may be reexpressed for simplicity in terms of (uv)>> as follows: (uv)>> :

0.7

y> \ 2.5 2.5 4y> \ \ ; exp 9 . 9 1; 10 y> a> a> (93 )

The construction of Eq. (93) will be described in detail since almost all subsequent expressions herein for the velocity distribution, the friction factor, and the heat transfer coefﬁcient are based on this expression with only slight numerical modiﬁcations. Equation (93) has the form of Eq. (71) with n :98/7,

y> (uv)>> : 0.7 10

(94 )

and

2.5 2.5 4y> (uv)>> : 1 9 9 1; . y> a> a>

(95)

Equation (94) has the limiting form of Eq. (32) with the previously discussed value of 7;10\ for . Equation (95), on the other hand, is based on the following expression of Churchill [83] for the velocity distribution across the entire turbulent core: u> : 5.5 ; 2.5 ln y> ;

15 y> 10 y> 9 . 3 a> 4 a>

(96 )

The terms in ( y>/a>) and (y>/a>) were added to the correlating equation of Nikuradse [46] to encompass the wake. The coefﬁcients 15/4 and 9(10/3) were chosen to force du>/dy> ; 0 and u> 9 u> ; 7.5(1 9 y>/a>) as A y> ; a>. The coefﬁcient of 7.5 is based on Eq. (84) of Reichardt [76] and therefore indirectly on the experimental data of Reichardt himself for parallel plates as well as that of Nikuradse for round tubes. Substituting for du>/dy> in Eq. (85) from the derivative of Eq. (96) and then simplifying algebraically results in Eq. (95). The absolute value sign and the approximation of 1 9 (2.5/y>) by exp 92.5/y> in Eq. (93) are merely mathematical contrivances to avoid singularities in ranges of y> in which these terms have an otherwise negligible role. Equation (95) was constructed from the correlating equation for the velocity because the supporting data are more

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299

Fig. 13. Representation of experimental data and directly simulated values of (uv)>> at small values of b>. (From Churchill [80], Figure 1.)

extensive and reliable than those for the turbulent shear stress itself. However, the value of 98/7 for the arbitrary combining exponent is based on the experimental data of Wei and Willmarth [84] for uv in ﬂow between parallel plates, which appear to be the most accurate ones over a wide range of values of both y> and b>. Hence, the validity of the analogy of MacLeod is implied in the value of this exponent as well as in the value of 7;10\ for . Equation (93) is compared with experimental data and values determined by direct numerical simulations for small values of y> and b> in Fig. 13 and with the experimental data of Wei and Willmarth for moderate and large values of y> and b> in Fig. 14. The agreement appears to be within the bands of uncertainty of the experimental and computed values. The small oscillations in the curves in Fig. 13 are an artifact of the structure of Eq. (93) rather than an error in plotting. Because the nominal lower and upper limits of y> : 30 and y> : 0.1a>, respectively, for Eq. (24) with A : 5.5 and B : 2.5, coincide at a> : 300, a semilogarithmic regime presumably does not exist for any lesser value of a>. Equation (93), which implies the existence of a semilogarithmic regime for

stuart w. churchill

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Fig. 14. Representation of experimental data of Wei and Willmarth for (uv)>> by Eq. (93). (From Churchill and Chan [71], Figure 3.)

the velocity for all values of a>, is therefore of questionable functionality for intermediate values of y> for a> & 300 despite the reasonable representation in Figures 13 and 14. The recent experimental data of Zagarola [73] for the time-averaged velocity distribution as discussed in Section II, A, 10 suggest updating Eq. (96) as u> : 6.13 ;

1 y y ln y> ; 6.824 9 5.314 . 0.436 a a

(97)

Equation (97) may be seen to be in agreement with Eq. (79) for y> a>

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and with Eq. (77) for y> : a>. For y> ; a>, Eq. (97) leads to Eq. (34) with E : 10.264. Substituting for du>/dy> in Eq. (85) from the derivative of Eq. (97), and then simplifying, results in the following updated version of Eq. (95):

1 1 6.95y> 9 1; . (uv)>> : 1 9 0.436y> 0.436a> a>

(98 )

Combination of Eq. (98) with Eq. (94), again with a combining exponent of 98/7 and the same mathematical contrivances, results in the ﬁnal correlating equation for (uv)>>, namely (uv)>> :

0.7

; exp

y> \ 10

91 1 6.95y> \ \ 9 1; . 0.436a> a> 0.436y>

(99)

As may be inferred from the detailed description of the formulation of Eq. (93), the form of each of the three principal terms of Eq. (99) is speculative and the values of the three numerical coefﬁcients are subject to some uncertainty. The most uncertain elements of Eqs. (93) and (99) are, however, the form of Eq. (71) and the value of 98/7 for the combining exponent. On the other hand, a virtue of correlating equations with this form is the numerical insensitivity of their predictions to the value of the arbitrary exponent. Equation (99) differs from all prior expressions for the turbulent shear stress, except for Eq. (93), by virtue of its presumed generality for all values of y> and all values of a> 300 (and perhaps even for a> 145), and its incorporation of all of the known theoretical structure, namely, Eq. (32) for y> ; 0, Eq. (35) for y> ; a>, and 1 9 B/y> for the regime of overlap. It supersedes all existing correlating equations for the eddy viscosity and the mixing length. Although a correlating equation with the same generality may be constituted for the eddy viscosity by the combination of Eqs. (99) and Eq. (86), and for the mixing length by combination of Eqs. (99) and (87), such expressions would not appear to serve any useful purpose. 3. New Correlating Equations for the Velocity Distribution and the Friction Factor Although numerical values for the velocity distribution and the friction factor may be determined simply by evaluating the integrals of Eqs. (89) and

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(91), respectively, using values of (uv)>> from Eq. (99), generalized correlating equations for these two quantities may be constructed for convenience. Churchill and Chan [71, 77] developed such correlating equations using Eq. (93) for (uv)>. Their expression is not reproduced here, since these same forms have recently been updated by Churchill [58] using Eq. (99) and the experimental data of Zagarola [73]. The resulting ﬁnal expressions for u> and u> are K u> : ;

and

(y>) \ 1 ; y> 9 exp 91.75(y>/10)

1 1 ; 14.48y> y> y> \ \ ln ; 6.824 9 5.314 0.436 1 ; 0.301(e/a)a> a> a>

(100)

227 50 1 a> u> : 3.30 9 ; ; ln . K a> a> 0.436 1 ; 0.301(e/a)a>

(101)

Equation (100) has the form of Eq. (71) with u> adapted from the limiting form of Eq. (33) with : 7;10\ and - : 0, and u> adapted from Eq. (97). The modiﬁed form of u> and the added value of unity in the argument of the logarithm of u> are simply mathematical contrivances to avoid singul arities in ranges of y> for which these terms do not contribute signiﬁcantly. The coefﬁcient of 14.48 corresponds to 6.13 in Eq. (79), that is, (1/ 0.436) ln 14.48 : 6.13. The combining exponent of 93 was chosen by Churchill and Chan [71] on the basis of various early sets of experimental data (see their Fig. 1). The term 1;0.301(e/a)a> was included in the argument of the logarithmic term to extend the applicability of Eq. (100) to commercial (naturally rough) piping, at least for y> e>. The coefﬁcient of 0.301 as compared to 0.304 in Eq. (74) represents the slightly reﬁned expressions for the components for smooth and rough pipes. The leading coefﬁcient of 3.30 for Eq. (101) is adopted from Eq. (79) of Zagarola, but the terms in (a>)\ and (a>)\ are a necessary consequence of u> ; y> near the wall, although such corrections have generally been overlooked. The coefﬁcients 227 and 50 are based on the numerical integration of Eq. (99) since the experimental data of Zagarola do not encompass the regime of a> for which these terms are the most signiﬁcant. The term in (a>)\ in Eq. (80) is a purely empirical approximation for this behavior, and as was noted in Section II, A, 10, is based on integration of Eq. (83) of Spalding. Again, the term 1;0.301(e/a)a> was incorporated in the argument of the logarithmic term to extend the applicability of Eq. (101) to encompass commercial (naturally rough) piping.

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Numerical integration of Eq. (90) using (uv)>> from Eq. (99) predicts values of the ‘‘constant’’ in Eq. (97) that vary from 5.39 to 5.76 with a>, and integration of Eq. (92) predicts values of the constant in Eq. (101) that vary only slightly about 2.71. In this instance, the experimentally based values of 6.13 and 3.30 were given preference. The discrepancy between the predicted and the experimentally determined leading constants is presumed to be due to a slight underprediction of Eq. (99) in the regime of interpolation. Churchill [58] compared the prediction of u> by Eq. (100) with the A experimental values of Zagarola and found an average absolute deviation of only 0.17% and a maximum deviation of 0.39%. The deviations for small and intermediate values of y> were even less on the whole. The prediction of u> by Eq. (101) was found to have an average absolute deviation of only K 0.22% and a maximum deviation of 0.5%. In both instances these deviations are less than those corresponding to the correlating equations of Zagarola himself.

4. New Formulations and Correlating Equations for Other Geometries Equation (85) with / substituted for 1 9 (y>/a>) is applicable for all U one-dimensional fully developed turbulent ﬂows. However, the variation of / with distance from a wall is known a priori only for forced ﬂow in a U round tube and between identical parallel plates (both smooth or both equally rough) and in planar Couette ﬂow (induced by the movement of one plate parallel and uniformly with respect to an identical one). On the basis of the analogy of MacLeod, Eqs. (99) and (100) are presumed to be directly applicable for forced ﬂow between identical parallel plates if b> is simply substituted for a>, while the friction factor may be represented by

2 155 1 b> : u> : 4.615 9 ; ln . K f b> 0.436 1 ; 0.301(e/b)b>

(102)

The value of 4.615 for the constant term as well as that of 9155 for the coefﬁcient are based on computations by Danov et al. [85] using Eq. (99), since experimental values of u> for parallel plates of accuracy and modernK ity comparable to those of Zagarola for round tubes do not appear to exist. Churchill [83], Chapter 3, constructed a theoretically based correlating equation for u> in planar Couette ﬂow, for which : at all locations U within the ﬂuid. A corresponding correlating equation for (uv)>> might be postulated, but experimental data to test such an expression critically over a wide range of conditions do not appear to be available.

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Churchill [86] also constructed a generalized empirical expression from which / but not u> or (uv)>> may be estimated for forced ﬂow through U a circular concentric annulus. The construction of such expressions for combined forced and induced ﬂow between parallel plates and for rotational annular ﬂow is not yet feasible because of the lack of appropriate data for uv and/or u. 5. Recapitulation The primary purposes in deriving Eqs. (99)—(102) was to provide the basis for the development of the corresponding expressions for forced convection in a round tube and between parallel plates. In this regard, the path of their derivation is as relevant as their ﬁnal form. However, over and above this objective, these four expressions are presumed to be the most accurate and comprehensive ones in the literature for turbulent ﬂow, at least for a> and b> 300. Indeed, the ﬁrst of these, for the turbulent shear stress, has no counterpart in the current literature, while Eq. (101) for the friction factor in a round tube may be considered to be an improvement upon as well as a replacement for all current graphical and algebraic correlations. Although the structure of Eqs. (99)—(102) represents the current state of the art, and the constants, coefﬁcients, and exponents therein are based on the best available experimental data and computed values, these expressions and values should all be considered to be subject to improvement on the basis of future contributions, both theoretical and experimental.

III. The Quantitative Representation of Fully Developed Turbulent Convection The history and present state of predictive and correlative expressions for the turbulent forced convection of energy in a round tube differ greatly from those described previously for ﬂow. One reason is the greater difﬁculty in characterizing the process of thermal convection experimentally. For example, (1) the thermal conductivity and the viscosity both vary signiﬁcantly with the primary dependent variable, the temperature of the ﬂuid, forcing the use of small overall temperature differences, whereas the viscosity does not vary with the velocity for ordinary ﬂuids; (2) the mixed-mean temperature must be determined by integrating the product of the temperature and the velocity over the cross-section at a series of axial distances, whereas the mixed-mean velocity is invariant with axial length and may be determined externally and directly with a ﬂowmeter; (3) the heat ﬂux density, which is difﬁcult to measure accurately, and/or the temperature

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varies along the wall, whereas the velocity is zero at the wall and the shear stress at the wall, which is ordinarily invariant with distance, may readily be determined from a single measurement of the axial pressure drop; (4) the heat ﬂux density within the ﬂuid, which is almost impossible to measure accurately, varies complexly with the distance from the wall and depends on both the Reynolds number and the thermal boundary condition, whereas the shear stress within the ﬂuid is known a priori to vary linearly with radius; (5) the extent of the deviation from fully developed convection is more difﬁcult to determine than that for fully developed ﬂow because the latter condition is deﬁned simply as a negligible change in the velocity distribution and/or the axial pressure gradient with axial distance, whereas the former is ordinarily deﬁned as a negligible change in T 9T j T 9T U U U or and/or h : T 9T T 9T T 9T U A U K U K while T r , T , T , and j or T are still varying individually; and ﬁnally (6) A K U U the experiments must be repeated for a series of different ﬂuids encompassing a wide range of values of the Prandtl number. Corresponding complexities arise in modeling, as revealed in the sections that immediately follow. These complexities and uncertainties appear to have inspired rather than discouraged the development of purely empirical and semiempirical expressions for heat transfer since their number and variety far exceed those for ﬂow. Another difference is the focus of the work in ﬂow and convection. In many applications of ﬂow, the velocity distribution is of equal or greater interest than the friction factor, whereas in most applications of forced convection interest in the heat transfer coefﬁcient greatly exceeds that in the temperature distribution. A somewhat different order and scheme of presentation is followed for turbulent convection than that for turbulent ﬂow. The essentially exact structure is ﬁrst examined in order to provide a framework and standard for evaluation of the early work. Thereafter new developments are considered within this same framework as well as in terms of historical precedents.

A. Essentially Exact Formulations 1. New Differential Formulations The general differential equation for the conservation of energy in a moving ﬂuid with constant density, viscosity, and thermal conductivity may

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be expressed in cylindrical coordinates as follows: c N

T u T T T 1 T T T 1 ;u ; F ;u :k r ; ; P r X z t r 1! r r r r 1! z

;2

u 1 u P ; F;u P r r 1! ;

;

u X ; z

u 1 u F; X z r 1!

u u 1 u v X; P ; P;r F . z r r r r 1!

(103)

The only new variables as compared to Eqs. (1)9(3) are the temperature T, the thermal conductivity k, and the speciﬁc heat capacity c . V iscous N dissipation, as represented by the terms with the viscosity as a coefﬁcient, is signiﬁcant only for very high velocities and for very viscous ﬂuids. Such conditions and ﬂuids will not be considered herein. Hence these terms with as a coefﬁcient will be dropped. The time- averaged form of the remaining terms of Eq. (103) for steady, fully developed ﬂow and fully developed thermal convection may then be expressed as

T 1 T c u : kr ; c rT v , N z N r r r

(104)

where for consistancy with Eq. (13), the substitutions u : u and v :9u X P have been made. Equation (104) may be integrated formally to obtain

c P T T N u rdr :9 k ; c T v. (105) N y r z The terms 9k(dT /dy) and c T v represent the heat ﬂux densities in the N y-direction (negative-r-direction) due to thermal conduction and the turbulent ﬂuctuations, respectively. The integral term on the left-hand side of Eq. (105) represents the axial heat ﬂux in the central core of ﬂuid with a radius r. It may also be interpreted, by virtue of Eq. (105) itself, as the total heat ﬂux density j at r in the negative-r-direction. Equation (105) may also be expressed in the dimensionless form j dT > [1 9 (T v)>>] : , j dy> U where T > Y k(T 9 T )( )/j U U U (T v)>> Y c T v/j N

(106)

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307

and

1 0 (T /x) u> j : dR. (107) (T /x) u> R j K K U This deﬁnition of T > was chosen in order to achieve the same form for Eq. (106) as for Eq. (85) and to result in T > : 0 at y> : 0 in analogy to u> : 0 at y> : 0. The term (T v)>> is also analogous to (uv)>> in the sense that it is the local fraction of the heat ﬂux density due to the turbulent ﬂuctuations. Equation (107) was constructed by noting that, according to Eq. (105), the total heat ﬂux density at the wall may be expressed as c j : N U a

?

T z

rdr :

c a N 2

T z

u

dR Y

c au T N K K . 2 z

(108) Equation (108) may also be considered, as indicated, to deﬁne the velocityweighted (mixed) mean of the longitudinal temperature gradient. The only explicit difference between Eqs. (85) and (106) is that j/j is given U by Eq. (107) in the latter whereas / is simply equal to 1 9 (y>/a>) : R U in the former. This difference is, however, a source of great complexity in the expressions for convection. Another implicit difference is that although the velocity is ordinarily postulated to be zero at the wall, a temperature varying along the wall or a uniform or varying heat ﬂux density along the wall may be speciﬁed. Although T > remains zero at the wall, T itself may vary. The U thermal boundary condition thus becomes a parameter. An implicit difference of even greater signiﬁcance is the dependence of (T v)>> on a parameter Pr : c /k, called the Prandtl number, as well as on y> and a>. N It follows that T > depends on Pr, and in general so does j/j . This U parametric dependence is not avoidable in general simply by some other choice of dimensionless variable, although it may vanish in certain narrow regimes. The measured values of T v or T and j that are required to evaluate (T v)>> are too limited in scope and accuracy to support the construction of a generalized correlating equation in terms of y>, a>, and Pr comparable to Eq. (98) for (uv)>>. This deﬁciency may be alleviated somewhat by reexpressing Eq. (106) as u

j Pr dT > [1 9 (uv)>>] 2 : j Pr dy> U

(109)

Pr 1 9 (T v)>> 2Y 1 9 (uv)>> Pr

(110)

where

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Since (uv)>> is implied to be known in advance, the net effect of this substitution is to replace (T v)>> by Pr /Pr as an unknown. The quantity 2 Pr /Pr, as deﬁned by Eq. (110), may be recognized in physical terms as the 2 ratio of the local fractions of the transport of energy and momentum by molecular motion. This quantity suffers from the same uncertainties as (T v)>> as well from the lesser ones associated with (uv)>>, but has, as will be demonstrated, a more constrained behavior for small and moderate values of Pr. Therein lies its principal merit as a characteristic quantity. The following alternative to both Eqs. (106) and (109) also has some advantages:

j Pr (uv)>> : 1; Pr 1 9 (uv)>> j R U

dT > . dy>

(111)

Here

Pr (uv)>> 1 9 (T v)>> RY . Pr (T v)>> 1 9 (uv)>>

(112)

The quantity Pr /Pr, as deﬁned by Eq. (112), may be recognized in physical R terms as the ratio of the transport of momentum by molecular and eddy motions, divided by the equivalent ratio for the transport of energy. Although Eq. (112) appears to be more complex than Eq. (110), and Eq. (111) more complex than either Eq. (106) or (109), Pr proves to be R essentially constant for large Pr, which results in a signiﬁcant simpliﬁcation. Elimination of (T v)>> between Eqs. (110) and (112) or of j/j (dy>/dT >) U between Eqs. (109) and (111) results in 1 (uv)>> 1 9 (uv)>> : ; . (113) Pr Pr Pr R 2 This relationship between Pr and Pr will subsequently prove very useful. 2 R Since j/j differs only moderately from / : R, it is convenient to U U introduce the variable *, deﬁned by 1;*Y

(114)

(115)

j j/j j/j U : U: U . j R 1 9 (y>/a>) U Substituting for j/j in Eq. (109) from Eq. (114) results in U y> Pr dT > (1 ; *) 1 9 [1 9 (uv)>> ] 2 : . a> Pr dy>

Comparison of Eqs. (115) and (85) indicates more explicitly the complications associated with convection than does comparison of Eqs. (106) and

turbulent flow and convection

309

(85). Substitution for j/j from Eq. (111) in Eq. (114) results in U y> Pr (uv)>> dT > : 1; . (116) (1 ; *) 1 9 dy> a> Pr 1 9 (uv)>> R Equations (115) and (116) are the starting points for the subsequent exact formulations for fully developed thermal convection. Reichardt [87] in 1951 was apparently the ﬁrst to propose * as a correlative quantity. Rohsenow and Choi [88] in 1961 subsequently suggested the use of M : 1 ; * as an alternative quantity for correlation. Although the effects represented by * and M are generally signiﬁcant, as will be shown on the basis of experimental and computed values, they have been overlooked or ignored in many analyses of convection.

2. New Integral Formulations Equation (115) may be reexpressed in terms of R and then integrated formally to obtain the following exact expression for the temperature distribution: a> 2

(1 ; *) [1 9 (uv)>>]

Pr 2 dR. Pr

(117) 0 Integration of this expression for T >, weighted by u>/u> , over the crossK section of the pipe then gives the following expression for the mixed-mean temperature T >:

a> Pr 2 dR T>: (1 ; *)[1 9 (uv)>>] K 2 Pr 0 from which it follows that

u> dR, u> K

(118)

j D 2a> U Nu :Y : k(T 9 T ) T> K U K 4 : . (119) Pr u> 2 dR (1 ; *)[1 9 (uv)>>] dR Pr u> K 0 The quantity Nu, called the Nusselt number, may be interpreted as the dimensionless rate of convective heat transfer. The corresponding expressions for T > and Nu in terms of Pr rather than Pr are R 2 a> (1 ; *)dR T >: (120) 2 Pr (uv)>> 0 1 ; Pr 1 9 (uv)>> R

stuart w. churchill

310 and

2T > 4 : K: a> Nu

(1 ; *)dR Pr (uv)>> 0 1 ; Pr 1 9 (uv)>> R

u> dR. u> K

(121)

Equations (117)—(121) are the ﬁnal, general integral formulations herein for the temperature distribution, mixed-mean temperature, and Nusselt number. However, some reductions are possible for particular boundary conditions and particular values of the Prandtl number as described in the immediately following sections. It may be inferred that the analytical or numerical evaluation of T > by means of Eqs. (117) or (120) requires (uv)>> as a function of y> and a>, and * and Pr or Pr as a function of Pr and the thermal boundary condition 2 R as well as of y> and a>. The evaluation of T> by means of Eqs. (118) or K (120), and Nu by means of Eqs. (119) or (121), may further be inferred to require a relationship for u> as a function of y> and a> and u> as a function K of a>. However, these requirements may be relaxed somewhat. First, u> and u> are given exactly as integral functions of (uv)>> by Eqs. (89) and (91), K respectively, and approximately but probably with sufﬁcient accuracy for all practical purposes by Eqs. (100) and (101), respectively. Yahkot et al. [89] assert, although they do not prove, that Pr is a universal function of / 2 2 and Pr for all geometries and boundary conditions. By virtue of Eq. (86), this generality, if valid, must extend to Pr as a function of (uv)>> and Pr. R Although this assertion of Yahkot et al. has been implied in a number of analyses, Abbrecht and Churchill [22] appear to have provided the only experimental conﬁrmation. They found Pr to be invariant with axial R distance in developing thermal convection following a step in wall temperature in fully developed turbulent ﬂow — a severe test of independence from the thermal boundary condition. Their results for a round tube were also found to agree closely with those of Page et al. [90] for heat transfer from a plate at one uniform temperature to a parallel one at a different uniform temperature for ﬂows at the same values of a> and b> — a severe test of independence from geometry as well as from the thermal boundary condition. Thus, the evaluation of T >, T > , and Nu only requires (uv)>> as a K function of y> and a>, Pr or Pr as a function of (uv)>> and Pr, and * as R 2 a function of y>, a>, Pr and the thermal boundary condition. The dependence of * on Pr vanishes under some circumstances. Finally, as will be shown, the relationship for *, although quite complex, is known exactly, whereas those for Pr and Pr are highly uncertain both theoretically and R 2 experimentally.

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311

a. A Uniform Heat Flux Density from the Wall As noted in the ﬁrst paragraph of Section III, fully developed convection is ordinarily deﬁned as the attainment of essentially unchanging values of T 9T T 9T U U or and/or of h : j /(T 9 T ) U U K T 9T T 9T U K U A with axial distance. Although the exact point of this attainment is ill deﬁned, the concept is a useful one in both an analytical and an applied sense. The majority of the theoretical semitheoretical solutions and correlations in the literature for the Nusselt number in turbulent ﬂow are for this regime, which prevails or is closely approached over most of the length of ordinary industrial heat exchangers. A uniform heat ﬂux density from the wall to the ﬂuid may be attained approximately by passing an electrical current axially through the metal wall of a heat exchanger, which thereby functions as an electrical resistance. Small deviations from uniform heating of the ﬂuid may then occur because of end effects, for example, thermal conduction along the tube wall or nonuniform heat losses to the surroundings. A uniform heat ﬂux density from the wall to the ﬂuid may also be closely approached in the inner pipe of a concentric circular double-pipe heat exchanger operated in equal countercurrent ﬂow. In this case small deviations may be expected due to variations in the local overall heat transfer coefﬁcent as a consequence of entrance effects in ﬂow and the variation of the physical properties of the two ﬂuids with temperature. If the heat transfer coefﬁcent h : j /(T 9 T ) U U K approaches an asymptotic value with axial distance for a uniform heat ﬂux density, T 9 T must as well. Then, if (T 9 T )/(T 9 T ) approaches an U K U U K asymptotic value,

1 T T T 9T U U9 ; ;0 (T 9 T ) z z z T 9 T U K U K

(122)

T T (T 9 T ) ; U 9 K ; 0. K z U z z

(123)

and

Hence, for fully developed convection with uniform heating, T T T 5 U5 K. z z z

(124)

Then from Eqs. (107) and (114) 1;*:

1 R

0 u> dR. u> K

(125)

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By virtue of Eq. (125), Eq. (118) may be integrated by parts to obtain

Pr 4a> 8 2 dR. (1 ; *)[1 9 (uv)>>] : : (126) Pr Nu T > K The analog of Eq. (126) in terms of Pr is readily shown to be R 8 4T > (1 ; *)dR : K: . (127) a> (uv)>> Nu Pr 1; Pr 1 9 (uv)>> R The evaluation of Nu from Eqs. (126) or (127) appears to involve only a single integration. However, the quantity * must be evaluated by integration for each value of a>, as indicated by Eq. (125). This latter relationship may be expressed directly in terms of (uv)>> by substituting for u> and u> from K Eqs. (89) and (91), respectively, integrating by parts, and simplifying to obtain

*:

19R R

0

[19(uv)>>]dR;

1 9 R R

0 [1 9 (uv)>>]dR

[1 9 (uv)>>]dR .

(128)

The behavior of * for two special cases is worthy of note. From Eq. (125) it is apparent that * is zero for all values of y> only for the hypothetical case of plug ﬂow. On the other hand, it is apparent from Eqs. (125), (89), and (91) that for R ; 0, for which u> approaches a nearly constant value,

[1 9 (uv)>>]dR u> 1;*; A : . (129) u> K [1 9 (uv)>>]RdR Equation (129), which may also be derived directly but by a considerably longer path from Eq. (128), deﬁnes the maximum value of * for each value of a> and thereby characterizes the magnitude of the deviation of j/j from U / : 1/R. U Equations (117), (125), and (126), together with Eq. (128), constitute the ﬁnal exact and completely general formulations herein for fully developed turbulent convection in a uniformly heated round tube. Their numerical evaluation requires only an expression such as Eq. (99) for (uv)>> and one for Pr or Pr , presumably as a function only of (uv)>> and Pr. [Actually, 2 R Eqs. (117), (126), (127), and (128) are also applicable for fully developed

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313

laminar convection with uniform heating as well. For this case, (uv)>> : 0, Eq. (113) gives Pr/Pr : 1, Eq. (128) gives * : 1 9 R, and both Eqs. (126) 2 and (127) give Nu : 48/11.] In the limit of Pr ; 0, Eq. (113) reduces to Pr : 1 9 (uv)>>. Pr 2

(130)

Substitution of this expression in Eq. (117) gives T > Pr : 0 :

a> 2

(1 ; *)dR :

0

a> (1 9 R)(1 ; *mR), 2

(131)

where * is seen to be the integrated-mean value from R to 1. Since * is K0 ﬁnite and positive for all R " 1 for both laminar and turbulent ﬂow, the Pr term * is ﬁnite and positive as well. Similarly, substitution of from K0 Pr 2 Eq. (130) in Eq. (126) gives

8 4T > Pr : 0

: K : (1 ; *)dR : (1 ; *) , K0 a> Nu Pr : 0

(132)

where (1 ; *) is seen to be the integrated mean of (1 ; *) over R from K0 0 to 1.0. Equation (131) provides an upper bound for T >/a> and Eq. (132) a lower bound for Nu for all Pr as a function of a>. In all of these expressions, 1 ; * represents the effect of the deviation of j/j from / U U (which has often been neglected) while (1 ; *) includes the effect of the velocity distribution as well. Insofar as Pr approaches a ﬁnite value as y> ; 0, the corresponding R asymptotic solution may be derived for Pr ; -. For this case, the entire temperature development takes place within the viscous boundary layer where y>/a> may be neglected, * ; 0, and (uv)>> may be approximated by the ﬁrst term on the right-hand side of Eq. (32). Equation (116) thereby reduces to dT > : dy>

1 9 (y>) . Pr 1; 9 1 (y>) Pr R

(133)

The function on the right-hand side of Eq. (133) may be integrated analytically if Pr is postulated to be invariant with respect to y>. With the R

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314

boundary condition u> : 0 at y> : 0, the result is

T>: 3

Pr Pr 1 (1 ; z) R ln 1 9 z ; z Pr 2 91 Pr R 3' y> 2z 9 1 ; 3 tan\ ; 9 , 6 Pr 3 91 Pr R

(134)

where

Pr 91 y>. Pr R Near the wall for very large values of Pr, the last term on the right-hand side of Eq. (134) may be dropped. Finally, letting z ; - gives the following expression for the fully developed temperature, which differs negligibly from the mixed-mean temperature, and thereby: z :

2a> T >5T>: : K Nu

3

For : 7;10\, it follows that

Nu:0.07343 1 9

2'

Pr Pr R . Pr 91 Pr R

(135)

Pr Pr f Pr f R Re ; 0.07343 Re Pr Pr 2 Pr 2 R R (136)

The more general form of Eq. (136) was apparently ﬁrst derived by Churchill [91], but the equivalent of the limiting form, usually with Pr R postulated to be unity, was derived much earlier by Petukhov [92] and others. The utility of the term

19

Pr R Pr

is in providing a ﬁrst-order correction for the effect of a ﬁnite value of Pr and conversely of deﬁning the lower limit of applicability of this limiting form with respect to ﬁnite values of Pr. It may be inferred from the absence of a and * that Eqs. (134)—(136) are applicable for fully developed turbulent

turbulent flow and convection

315

convection in any fully developed shear ﬂow and for any thermal boundary condition, not just for a uniformly heated round tube. As shown subsequently, the one speculative element in the derivation of Eqs. (134) and (135), namely the attainment of a ﬁnite asymptotic value for Pr as y> ; 0 and Pr R increases, is supported by some sets of experimental data and direct numerical simulations but is contradicted by others. The postulate that Pr : Pr requires, by virtue of Eq. (113), that Pr : Pr R 2 as well. Insofar as Pr : Pr for all y>, Eq. (117) reduces to 2 a> (1 ; *)[1 9 (uv)>>]dR, (137) T > Pr : Pr : Pr : 2 R 2 0 which, by virtue of Eq. (89), may be expressed as

T > Pr : Pr : Pr : u>(1 ; * ) (138) 2 R UK0 where * is the integrated mean of *, weighted with respect to 19(uv)>> UK0 over R from R to 1.0. The deviation of the T > y>, a> from u> y>, a>

for Pr : Pr is seen to be wholly a consequence of the factor 1 ; * and thus 2 wholly due to the deviation of j/j from / . The similarity of the U U distribution of T > to that of u>, as represented by Eq. (138), is one reason for the arbitrary deﬁnition of T > herein. The postulate of Pr : Pr : Pr 2 R for all y> allows the reduction of both Eqs. (126) and (127) to

8 4 T> Pr :Pr :Pr

2 R : K : (1 ; *)[19(uv)>>]dR. a> Nu Pr : Pr : Pr

2 R (139) Comparison of Eqs. (139) and (91) reveals the following similarity for T > K and u>: K (140) T > Pr : Pr : Pr : u>(1 ; *) K K UK0 2 R Here (1 ; *) is the integrated mean of (1 ; *), weighted with respect UK0 to 1 9 (uv)>>, over R from 0 to 1.0. It follows that 2a> Re( f /2) Nu Pr : Pr : Pr : : . (141) 2 R u>(1 ; *) (1 ; *) K UK0 UK0 Equation (141) is a surprising and remarkable result. It has the same explicit functional dependence on ﬂow as the famous analogy of Reynolds [18], namely Nu : Re

f Pr, 2

(142)

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but occurs at Pr : Pr : Pr instead of Pr : 1 and differs by the factor 2 R (1 ; *) . The speculative element upon which the derivation of Eqs. UK0 (137)—(141) is based, namely the invariance of Pr with y> for the particular 2 value of Pr : Pr : Pr, has some experimental and semitheoretical support 2 R for Pr 5 0.87, in particular over the turbulent core. The observed behavior of Pr and Pr in the viscous sublayer and the buffer layer is not necessarily 2 R contradictory, just uncertain. Despite the indicated uncertainties with regard to Eqs. (136) and (141), these two expressions, together with Eq. (132), prove to be invaluable in evaluating approximate and speculative formulations and solutions and in constructing generalized correlating equations. b. A Uniform Wall Temperature Next to uniform heating, the most frequently postulated thermal boundary condition in analytical formulations for convective heat transfer in a round tube is a uniform temperature on the wall, higher (or lower) than that of the entering ﬂuid. This boundary condition is closely approximated in real exchangers cooled or heated on the outer surface of the tubes(s) by a boiling liquid or condensing vapor, respectively. Deviations from a uniform wall temperature may occur as a result of a ﬁnite value of the outer heat transfer coefﬁcient and of end effects. For a uniform wall temperature, fully developed convection may be characterized by T T K 9 (T 9 T ) U z z T 9T U : ; ; 0, T 9T (T 9 T ) z T 9 T U K U K U K which implies that

(143)

T /z T 9T T> : U : . (144) T /z T 9 T T> K K U K Substituting this expression in Eq. (107) and then that result in Eq. (114) leads to

0 T > u> 1 0 dR : T >u>dR. (145) T > u> RT >u> K K K K Seban and Shimazaki [93] were apparently the ﬁrst to identify the equivalent of Eqs. (143) and (144) as characterizing convection with a uniform wall temperature. It is apparent from Eq. (145) that * for a uniform wall temperature, as contrasted with a uniform heat ﬂux density from the wall, is ﬁnite even for the hypothetical case of plug ﬂow. Furthermore, the maximum value of 1;*:

1 R

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317

1 ; * may be inferred from Eq. (145) to be equal to (T >/T >)(u>/u>) and A K A K therefore greater than for uniform heating by the factor T > /T >. It follows A K that the error due to neglecting * is greater for a uniform wall temperature. Equations (117)—(121) are directly applicable for a uniform wall temperature, but because of the dependence of * on T >, as expressed by Eq. (145), an iterative process of solution is required. For example, for a speciﬁed value of Pr, a correlating equation for Pr /Pr and an arbitrary postulated 2 expression * y> for * y> , T > y> may be calculated from Eq. (117), T > K from Eq. (118), and then * y> from Eq. (145). These calculations are repeated, starting with * y> , and continued until convergence is achieved. 1 Now consider the three special cases of Pr ; 0, Pr ; -, and Pr : Pr : Pr for a uniform wall temperature. For Pr ; -, Eqs. (133)— 2 R (136), which are independent of *, remain directly applicable. For Pr ; 0, * must be determined iteratively from Eq. (145) using T > from Eq. (131) and T > from the following reduced form of Eq. (118): K

a> T > Pr : 0 : K 2

(1 ; *)dR

(146)

0 Similarly, for Pr : Pr : Pr, * must be determined iteratively from Eq. 2 R (144) using T > from Eq. (136) and T > from the following reduced form of K Eq. (118): a> T > Pr : Pr : Pr : K 2 R 2

u> dR. u> K

0

(1 ; *)[1 9 (uv)>>]dR

u> dR. u> K (147)

c. Generalized Expressions An alternative form of expression for T > and K Nu is useful for interpretation if not for numerical evaluations. Setting R : 0 in the lower limit of the integral of Eq. (117) results in the following expression for the temperature at the centerline: a> T>: A 2

[1 ; *][1 9 (uv)>>]

Pr 2 dR. Pr

(148)

From this it follows that Nu :

2a> 2a> T > 4(T >/T >) A : A K : T> T> T> K A K [1 ; *][1 9 (uv)>>]

Pr 2 Pr

. (149) dR

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For Pr ; 0, Eq. (149) reduces to 4(T >/T >) 4(T >/T >) 2a> A K A K , (150) : : (1 ; *) T > Pr : 0

K K0 [1 ; *]dR where here (1 ; *) is the integrated mean of 1 ; * over R from 0 to 1, KU0 whereas for Pr : Pr : Pr Eq. (150) reduces to 2 R f T > u> K Re A 2 T > u> 4(T >/T >) A A K : K , Nu Pr : Pr : Pr : 2 R (1 ; *) UK0 (1 ; *)[1 9 (uv)>>]dR (151) Nu Pr : 0 :

where (1 ; *) is the integrated mean of 1;*, weighted by [19(uv)>>], UK0 over R from 0 to 1.0. The factors T >/T > and u>/u> may be expected to A K K A compensate for each other to some extent, although T >/T > is always larger. A K Equations (149)—(151) do not have any merit relative to Eqs. (119), (121), (126), (127), (132), (139), and (141) as far as numerical calculations are concerned because of the presence of T >/T >. However, these formulations A K will be shown subsequently to be invaluable in terms of constructing a theoretically based correlating equation. As mentioned previously, the factor (1 ; *) represents in all cases the effect of the deviation of the heat ﬂux density ratio from the shear stress ratio, while the factor (1 ; *) represents the effect of the velocity distribution as well. Equations (148)—(151) are applicable for both uniform heating and uniform wall temperature. This approach does not result in an alternative expression for Nu Pr ; - since the postulate that T > : T > is A K inherent for that limiting case. It may be inferred that the effects of these two thermal boundary conditions are exerted wholly through (T >/T >)/ A K (1 ; *) for Pr : 0 and KU0 T >/T> A K (1 ; *) UK0 u>/u> A K for Pr : Pr : Pr. 2 R

3. Parallel Plates and Other Geometries Insofar as the analogy of MacLeod is applicable for (uv)>> and u>, all of the previous expressions in Section III for T > and dT >/dy> are directly applicable for fully developed convection from parallel plates heated equally

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319

on both surfaces (either uniformly or isothermally) if a> is simply replaced by b> and R by Z : 1 9 (y>/b>) wherever they appear. The expressions for * and for T > are, however, different because they invoke integrations over K a planar rather than a circular area. As an example, for equal uniform heating on both plates, the expression analogous to Eq. (126) is

3T > Pr 12 2 dZ. : K : (1 ; *)[1 9 (uv)>>] b> Pr Nu @ where here, as contrasted with Eq. (128),

*:

19Z Z

8

[1 9 (uv)>>]dZ ;

19Z Z

[1 9 (uv)>>]dZ

8 [1 9 (uv)>>]dZ

(152)

.

(153)

Again, as for a round tube, * increases monotonically from 0 at Z : 0 to u> /u> at Z : 1. A K For Pr : 0, Eq. (152) reduces to Nu Pr : 0 : @

12

:

(1 ; *)dZ

12 , (1 ; *) K8

(154)

where (1 ; *) is the integrated-mean value over Z. On the other hand, K8 for Pr : Pr : Pr, Eq. (152) reduces, by analogy with Eq. (141), to R f Re @ 2 12 : Nu Pr : Pr : Pr : @ R 2 (1 ; *) , UK8 (1 ; *)[1 9 (uv)>>]dZ (155)

where (1 ; *) is the integrated-mean value, weighted by [1 9 uv>>], K8 over Z. Equation (136) for Pr ; - is directly applicable in terms of Nu @ and Re . @ For parallel plates at different uniform temperatures, j is uniform across the channel and * : 0. It follows that T>:

W>

[1 9 (uv)>>]

Pr 2 dZ Pr

(156)

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320 and that

b> 1 j b U : : . Nu Y @ k(T 9 T ) T > Pr @ 2 U @ [1 9 (uv)>>] dZ Pr For Pr : 0, Eq. (157), by virtue of Eq. (130), reduces to simply

(157)

Nu Pr : 0 : 1. (158) @ The half-spacing b was chosen as the characteristic dimension in order to achieve this particular, obvious result. For Pr : Pr : Pr , Eq. (157) reR 2 duces to Nu : @

1 [1 9 (uv)>>]dZ

:

1 , 1 9 (uv)>>] K8

(159)

where [1 9 (uv)]>> is the integrated-mean value over the channel. For Pr ; -, Eq. (136) is directly applicable in terms of Nu and Re . @ @ Expressions for equal uniform wall-temperatures may readily be formulated by analogy to those for an isothermal round tube in Section III, A, 2, b. but are not included here in the interests of brevity. Expressions for parallel-plate channels analogous to Eqs. (149)—(151) are also omitted, even though they are referred to subsequently, since their form is readily inferred. Equivalent formulations for fully developed convection are possible for all one-dimensional ﬂows, but their implementation is dependent upon individual expressions for (uv)>>, as discussed in II, B, 4, and in turn for *. 4. Alternative Models and Formulations None of the differential models that have been proposed in the past for the heat transfer in turbulent ﬂow appear to provide any improvement over the dimensionless turbulent heat ﬂux density, (T v)>>, or its exact equivalents in terms of the dimensionless turbulent shear stress, (uv)>>, and Pr R or Pr . However, the eddy conductivity, k , and its implementation are 2 R described here in some detail because of the widespread use of this quantity or its exact equivalent, : k /c , the thermal eddy diffusivity, for correlaR R N tion and prediction in the past and present literature. The eddy conductivity itself may be deﬁned by dT 9k : c (T v) R dy N

(160)

turbulent flow and convection

321

and incorporated in the elementary differential energy balance to obtain, in dimensionless form,

k dT > j : 1; R . k dy> j U Elimination of dT >/dy> between Eqs. (161) and (106) reveals that k (T v)>> R: . k 1 9 (T v)>>

(161)

(162)

Equations (160)—(162) are directly analogous to Eqs. (37), (38), and (86), respectively, for momentum transfer. Since, from physical considerations, (T v)>> must be greater than zero and less than unity at all locations within the ﬂuid in a round tube, k may be inferred to be positive, bounded, and R interchangeable with (T v)>> in this geometry. Equation (161) has often been expanded as

j : j U

dT > c k R R 1; N : k c dy> N R

1;

Pr R Pr R

dT > dy>

(163)

or as

j c k;k ; dT > Pr dT > R R 2 : N : . (164) j k c ( ; ) dy> Pr dy> U N R 2 where here Pr Y c /k , Pr Y c ( ; )/(k ; k ), Y ; , and R N R R 2 N R R 2 R k Y k ; k . These deﬁnitions of Pr and Pr are consistent in every respect 2 R R 2 with those of Eqs. (112) and (110), respectively. Such transformations were of course initially made with the expectation that Pr /Pr or Pr /Pr would R 2 be more constrained in its behavior than k /k. R Equation (163) may be integrated formally to obtain

W>

dy> , (165) Pr R 1; Pr R and then T > from Eq. (165), weighted by u>/u>, may be integrated formally K over the cross-section of the round tube to obtain T>:

j j U

2a> :T>: K Nu

W>

j j U

dy> u> dR. (166) Pr u R K 1; Pr R For uniform heating it has been the custom, when utilizing the eddy conductivity ratio k /k or its equivalent such as (Pr/Pr )( /), to substitute R R R

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322

from Eq. (125) for j/j , thereby transforming Eq. (165) to U 0 u> dR T> : dR u> Pr a> K 0 R R 1; Pr R and Eq. (166) to

2 T> : K: Nu a>

0 u> dR u> K R

0

(167)

dR Pr 1; Pr R

R

u> dR. u> K (168)

Lyon [94] in 1951 recognized that changing the order of integration allows reduction of Eq. (168) to Nu :

2

, (169) dR dR Pr R R 1; Pr R which requires far less computation for numerical evaluations than does the triple integral of Eq. (168). The analogous reduction of Eq. (119) to (126) was much simpler and more obvious because of the use of (uv)>> rather than both u> and / as variables. Lyon further recognized that setting R Pr : 0 in Eq. (169) gives an expression for the lower limiting value of Nu that varies with a> (or Re) only by virtue of the variation in the velocity distribution. He further inferred (incorrectly, as will be shown) that this limiting value is approached asymptotically as RePr approaches zero. For a uniform wall temperature, Eqs. (165) and (166) become, by virtue of Eq. (145),

0

T> : a> 0

u> u> K

0 T > u> dR T > u> K K R

and Nu :

dR Pr 1; Pr R

R

2

0

0 T > T> K

u> u> K

dR

R

dR Pr 1; Pr R

R

(170)

. u> dR u> K (171)

turbulent flow and convection

323

Equations (165)—(171) may be expressed in terms of Pr rather than Pr 2 R simply by replacing

Pr Pr R 2. by Pr Pr R 2 The preceding expressions in terms of , u>, and Pr or Pr are exact but R R 2 more cumbersome than the corresponding ones in terms of (uv)>>, *, and Pr or Pr . An expression for / could be derived from a correlating R 2 R equation for u> by virtue of Eq. (38), but in most applications, for some unexplained reason, separate, incongruent correlating equations have been used for u> and . Despite appearances to the contrary, the use of (uv)>> R rather than / does not decrease the number of integrations to determine R values of T > and Nu; the integration or integrations of u>/u> are simply K performed separately in the process of evaluating *. The ,— and uv—T v models do not appear to have a useful role for convection in round tubes or parallel-plate channels, but the latter one has promise for circular annuli despite the considerable empiricism involved in the implementation of the supplementary equations of transport. 1;

B. Essentially Exact Numerical Solutions The integral formulations of Section III, A are exact, except possibly the reduced ones for the special case of Pr : Pr : Pr , which incorporate the R 2 postulate of invariance of Pr and Pr with y>. In addition, the closed-form R 2 solution for Pr ; - is subject to the asymptotic attainment of a ﬁnite value for Pr as y> ; 0. Some uncertainty arises in the numerical evaluation of the R integral expressions for Nu for all ﬁnite values of Pr by virtue of the empirical expression, such as Eq. (99), that is used for (uv)>>, but the net effect is presumed to be completely negligible. On the other hand, the uncertainty introduced by the expressions utilized for Pr or its equivalent R [Pr , k , k or (T v)>>] is potentially very signiﬁcant. The uncertainty 2 R 2 associated with expressions for Pr or its equivalent extends to all prior R numerical results for turbulent convection, other than those from direct numerical simulations, as well to those subsequently presented herein. The estimation of values of Pr or its equivalent is therefore given ﬁrst attention R in this section. 1. Expressions for the Turbulent or Total Prandtl Number As noted heretofore, Pr is presumed on the basis of theoretical conjecR tures, as well as experimental evaluations, to be the same unique function of

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324

(uv)>> (or /) and Pr for all geometries and all thermal boundary R conditions. This presumption, which has been overlooked or denied implicitly by many prior investigators, greatly simpliﬁes the task of correlation for Pr as well as the integrations for T > and Nu. R The high degree of uncertainty of the various expressions for Pr arises on R the one hand from the very severe requirements for precision in the measurements of either T v or dT/dy, and on the other hand from the lack of a universally accepted theoretical model. A vast but generally disappointing body of literature exists on this subject (see, for example, Reynolds [95] and Kays [95a]). Only a few directly relevant contributions will be noted here. Jischa and Rieke [96] and others have successfully correlated the extensive data for the turbulent core for ﬂuids with Pr 2 0.7 by means of a simple algebraic expression, such as 0.015 Pr : 0.85 ; . R Pr

(172)

Over the purported range of validity of Eq. (172), the turbulent Prandtl number is predicted to vary only from 0.85 to 0.87 and to be independent of y> and a> [or (uv)>>]. Such constrained behavior, insofar as this prediction is valid, appears to justify the use of Pr rather than (T v)>> or R k /k or even Pr as a variable for correlation. The nominal restriction of Eq. R 2 (172) to the turbulent core may be attributed in part to the widespread scatter of the experimental data for Pr in the viscous sublayer and the buffer R layer rather than wholly to its inapplicability in those regimes. Kays [95a] proposed the extension of Eq. (172) for small values of Pr by replacing the constant 0.85 by A / with a value for A of 0.7 based on direct numerical R simulations or 2.0 based on experimental data. Because of the excessive values of Pr predicted by this modiﬁcation of Eq. (172) very near the wall, R he proposed to set Pr : 1 in that region. R A more complex empirical expression is that of Notter and Sleicher [97], which may be rewritten in terms of (uv)>> rather than / as follows: R Pr : R

10 1; (uv)>> 35 ; 19(uv)>>

uv)>> 1 9 (uv)>> . uv)>> uv)>> 0.025 Pr ;90Pr 19(uv)>> 19(uv)>>

1 ; 90Pr

(173)

turbulent flow and convection

325

Equation (173) predicts an asymptotic value of Pr : : 0.778 as y> ; 0 R for large values of Pr, which supports the critical postulate in the derivation of Eq. (136). However, it predicts higher values than Eq. (172) for the turbulent core and values of Pr : Pr that depend slightly on (uv)>> even R in the turbulent core, which is not in accord with the critical postulate in the derivation of Eq. (141). Yahkot et al. [89] used renormalization group theory to derive

1 Pr 2 1 1.1793 9 Pr

1.1793 9

1 Pr 2 : 1 9 (uv)>> 1 2.1793 ; Pr

2.1793 ;

(174)

Equation (174) is implied by the authors to be applicable for all geometries, all thermal boundary conditions, and all values of Pr and (uv)>>. Furthermore, they assert that this expression is free of any ‘‘experimentally adjusted parameters.’’ However, they undermine its credibility somewhat by suggesting, in a footnote ‘‘added in proof,’’ the change of a theoretical index in their derivation from 7 to 4, which appears to have signiﬁcant numerical consequences. The dependence of Pr on the rate of ﬂow and on location 2 within the ﬂuid stream may be inferred from Eq. (174) itself to be characterized wholly by (uv)>>. The dependence of Pr on Pr and (uv)>> only, 2 if valid, must extend to Pr by virtue of Eq. (113). The limited experimental R support for these various presumptions has already been discussed in the paragraph following Eq. (121). In any event, Eq. (174) is attractive in terms of simplicity and purported generality, and its predictions appear to be qualitatively if not quantitatively correct. For example, it predicts Pr : Pr : Pr over the entire cross-section of ﬂow for Pr : 0.848, but on R 2 the other hand an obviously low value of Pr : 0.39 at the wall for R asymptotically large values of Pr. Elperin et al. [97a] showed that only one of the constants and exponents of Eq. (174) is independent, and determined an ‘‘improved’’ value thereof. However, the latter value does not eliminate the indicated shortcomings. The ﬁnal recent contribution to be examined here is that of Papavassiliou and Hanratty [98], who used both Lagrangian and Eulerian direct numerical simulations to predict Pr for heat transfer between parallel plates for a R series of values of Pr from 0.05 to 2400, but only for b> : 150, which is just above the minimum value for fully developed turbulence. The title of the previously cited paper by Einstein [4] on Brownian motion does not suggest any relevance to turbulent ﬂow and convection, but its statistical development serves as the origin of the Lagrangian DNS methodology of Papavassiliou and Hanratty. Their predictions appear to be in fair qualitative and

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quantitative agreement with Eqs. (172)—(174) for most conditions, but for Pr 100 they indicate an increase without limit in Pr as y> ; 0. This latter R prediction, as represented (in thermal terms) by 1.71 Pr R: , Pr (y>)

(175)

is in accord with the measurements by Shaw and Hanratty [99] of the rate of electrochemical mass transfer. This result would appear to refute the validity of Eq. (136), but since convective turbulent heat transfer is usually limited to ﬂuids with Pr & 100, Eq. (136) may remain applicable as an asymptotic element of a correlating equation as long as the latter is not utilized for higher values. In view of the contradictions among these representative results, the principal challenge in turbulent convection appears to be the resolution of the uncertainties in the qualitative and quantitative dependence of Pr on R (uv)>> (or y> and a>) and Pr. In the following section the effects of this uncertainty in Pr are avoided insofar as possible by choosing particular R conditions for which its impact is minimal. Substitution of Pr from Eq. (172) or (173) and of (uv)>> from Eq. (99) R in Eq. (112) would yield a direct analog of (uv)>> for convection, that is, an algebraic expression for (T v)>> as a function of y>, a>, and Pr. Substitution of Pr from Eq. (110) and again of (uv)>> from Eq. (99) in 2 Eq. (174) or its alternatives would yield much more complex algebraic expressions for (T v)>>, again as a function of y>, a>, and Pr. Such expressions for (T v)>> have not been presented herein because they would obviously be less convenient to apply than those for Pr and Pr . It is R 2 obvious that such expressions for (T v)>> would incorporate the considerable uncertainties of the generating expressions for Pr and Pr as well as R 2 the lesser ones attributable to Eq. (99) and its components. 2. Numerical Results for Nu for a Uniformly Heated Tube a. Solutions for Particular Conditions Consideration is ﬁrst given to those conditions for which the dependence of Nu on the uncertainty of the values of Pr is absent or minimal. Heng et al. [100] were the ﬁrst to use the new R formulations herein for numerical evaluations of Nu but the later evaluations of Yu et al. [100a] are presented herein since they are presumed to be slightly more accurate. For Pr : 0, the Nusselt number, as given by Eq. (132), is independent of both Pr and Pr . Values of Nu, and T > /T > for a> : 500, 1000, 2000, 5000, A K R and 10,000, as computed using Eq. (99) for (uv)>> and in turn Eq. (128) for *, are listed in Table I along with values for laminar and plug ﬂow. The

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327

TABLE I Computed Thermal Characteristics of Fully Developed Turbulent Convection in a Uniformly Heated Round Tube for Pr : 0 Nu a>

T > /T > A K

(YOC)

(KL)

(NS)

500 1000 2000 5000 10,000 20,000 50,000

1.862 1.884 1.889 1.911 1.918 1.924 1.930

6.480 6.675 6.808 6.932 7.004 7.063 7.130

6.490 6.695 6.845 6.895 6.995

6.82 6.935 7.03 7.175 7.30

YOC, Yu et al. [100a]; KI, Kays and Leung [101], interpolated with respect to Re ; NS, Notter and ? Sleicher [97], interpolated with respect to Re . ?

values of Re were obtained from 2a>u> and hence may be used to recover, K if desired, the computed values of u> : (2/ f ). Values of Nu were obtained K from 2a>/T > and hence can be used to recover the computed values of T > K K and in turn those of T >. The values of Nu attributed to Kays and Leung A [101] and Notter and Sleicher [97] were obtained by theoretically based interpolation of their actual computed values with respect to Re. In all cases, the new values lie between the older ones. The models and procedures used to obtain these earlier computed values are discussed subsequently. The computed values of u> , u>/u> , Re : 2a>u> , f : 2/(u> ) and Re( f /2) K A K K K : 2a>/u> corresponding to the computed values of Nu, etc. in Table I are K listed in Table II. TABLE II Computed Characteristics of Fully Developed Turbulent Flow through a Round Tube (from Yu et al. [100a]) a>

u> K

Re;10\

u>/u> A K

f ;10

Re( f /2)

500 1000 2000 5000 10,000 20,000 50,000

17.0 18.815 20.518 22.69 24.295 25.90 28.01

17.0 37.63 82.07 226.9 485.9 1036 2801

1.2696 1.2375 1.2148 1.1926 1.1793 1.1680 1.1552

6.920 5.650 4.751 3.885 3.388 2.981 2.549

58.82 106.3 194.96 440.72 823.21 1544.4 3570.15

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328

TABLE III Computed Thermal Characteristics of Fully Developed Turbulent Convection in a Uniformly Heated Round Tube for Pr : Pr : 0.8673 Based on Eq. (172) R Nu a>

T > /T > A K

(YOC)

(KL)

(NS)

500 1000 2000 5000 10,000 20,000 50,000

1.242 1.222 1.206 1.189 1.177 1.167 1.155

53.63 99.45 185.3 424.1 796.7 1502 3487

55.12 102.0 189.5 433.0 812.4

52.7 97.1 177.7 401.7 749.8

See footnotes in Table I and values of Re and ? u>/u> in Table II. Values of KL and NS were A K interpolated for both Re and Pr. ?

Equation (172) implies that Pr : Pr : Pr for Pr : 0.8673. Values of Nu R 2 computed for this condition using Eq. (139), and again Eq. (99) for (uv)>> and Eq. (128) for *, are listed in Table III. The corresponding values of T >/T > are also provided. Individual values of T > and T > may be A K K A determined from the tabulated values of Nu and T >/T >. The indicated A K values of Re( f /2) : 2a>/u> may be used to determine values of (1 ; *) . K UK0 The values of Nu attributed to Kays and Leung [101] and Notter and Sleicher [97] were in this case obtained by interpolating their computed values with respect to both Pr and Re. Again the new values are intermediate to the older ones. b. Solutions for General Values of Pr For Pr 0.867, Eq. (127) may be rearranged and approximated by Nu :

(1 ; *)

8 Pr uv)>> 19 Pr 1 9 (uv)>> R

,

(176)

dR

and hence by 8 Pr uv)>> 1; , (177) (1 ; *) Pr 1 9 (uv)>> K0 UK0 R where the weighting factor for the integrated-mean value of the term in Nu :

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329

TABLE IV Predicted Nusselt Numbers for Fully Developed Turbulent Convection in a Uniformly Heated Round Tube with Pr based on Eq. (172) (from Yu et al. [100a]) R Small Pr a>

10\

10\

0.01

0.1

0.7

Nu 500 1000 2000 5000 10,000 20,000 50,000

6.481 6.675 6.809 6.933 7.006 7.068 7.141

6.489 6.694 6.848 7.035 7.210 7.476 8.154

7.073 7.927 9.327 12.95 18.25 27.63 51.84

16.67 26.70 44.39 90.62 159.1 283.3 618.6

48.17 88.53 163.7 371.5 694.0 1302 3006

Large Pr a>

1

10

100

1000

10,000

-

Nu/(0.07343(Pr/Pr )Re( f /2)) R 500 1000 2000 5000 10,000 20,000 50,000

0.7462 0.6957 0.6510 0.5993 0.5650 0.5341 0.4979

0.9227 0.9066 0.8903 0.8688 0.8529 0.8374 0.8176

0.9794 0.9763 0.9726 0.9673 0.9631 0.9588 0.9532

0.9935 0.9934 0.9928 0.9918 0.9909 0.9899 0.9886

0.9950 0.9953 0.9953 0.9952 0.9948 0.9945 0.9936

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

square brackets is (1 ; *). Since Eqs. (172)—(174) all predict increasing values of Pr as Pr decreases, it may be inferred from Eq. (177) that the effect R of any error in the values of Pr used to compute Nu from Eq. (127) for small R values of Pr will be very limited and will continually decrease as Pr decreases below a value of 0.867. Insofar as Eq. (172) is valid, Pr only varies R slightly for Pr 0.867, at least in the turbulent core. On the basis of these considerations for small and large values of Pr, Eq. (172) might, despite its obvious shortcomings, be expected to result in a reasonable approximation for Nu for all values of Pr. Values so computed are listed in Table IV. As a quantitative test, the computations leading to the values in Table IV were repeated using Eq. (173) rather than Eq. (172) for Pr . The results differ R signiﬁcantly only for very large a> with Pr : 0.001, 0.01, 0.1 and 10 or greater and therefore are not reproduced herein.

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c. Prior Computed Values of Nu Many prior analytical and numerical solutions for turbulent convection have neglected the variation in the total heat ﬂux density with radius or postulated the same linear variation as for the total shear stress. Many have postulated Pr : 1 or some other ﬁxed R value for all conditions, and a number have incorporated the postulate that uv is proportional to y near the wall. Only the two numerical solutions that avoid all of these gross idealizations will be examined here, namely the previously mentioned ones of Kays and Leung [101] and Notter and Sleicher [97]. The solutions of Kays and Leung are ostensibly for circular annuli but include a uniformly heated round tube and parallel plates as limiting cases. They carried out numerical integrations of the partial differential energy balance using separate, incongruent correlating equations for the velocity and eddy viscosity as well as an expression of unknown accuracy for the turbulent Prandtl number. Their expressions for and u are unquestionably R less accurate than the equivalent values used by Yu et al. [100a]. For Pr ; 0 the errors due to their expressions for the eddy viscosity and the turbulent Prandtl number phase out, and the slight discrepancies between their values of Nu and those of Yu et al. must be due to the inaccuracy of their values of u as compared to those of Yu et al. for (uv)>>. The slightly greater discrepancies in Table III presumably stem only from the values that Kays and Leung used for and u since the dependence on Pr is effectively R R eliminated. Notter and Sleicher [97, 111] developed Graetz-type series solutions for Nu in developing as well as fully developed convection. The correlating equations that they utilized for u>, /, and Pr [Eq. (173)] are almost R R certainly more accurate than those used by Kays and Leung, but less accurate, with respect to u> and /, than the equivalent values used by Yu R et al. The remarks concerning the discrepancies of the values of Kays and Leung for Nu in Tables I and III are applicable at least qualitatively to those of Notter and Sleicher. All in all, the values of Yu et al. for Nu in Tables I, III and IV are presumed to be more accurate than any previously computed values, primarily because of the greater accuracy associated with Eq. (99) for (uv)>>. Neither the absolute nor the relative error in Nu associated with the values used for Pr in the numerical predictions of Yu et al., Kays and R Leung, and Notter and Sleicher can be evaluated with certainty at this time. Fortunately, the error associated with Pr is reduced in Nu in all cases by R the integration or summation involved in the evaluation of the latter. Comparison of the numerically computed values of Nu with experimental data is deferred until after their representation by correlating equations.

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331

TABLE V The Thermal Characteristics of Fully Developed Turbulent Convection in a Round Tube with Uniform Wall-Temperature for Pr : 0 (from Yu et al. [100a]) a>

Nu

T > /T > A K

500 1000 2000 5000 10,000 20,000 50,000

4.959 5.055 5.122 5.187 5.225 5.258 5.295

2.124 2.152 2.171 2.188 2.198 2.205 2.214

3. Numerical Results for an Isothermal Wall-Temperature Yu et al. [100a] carried out numerical calculations for fully developed turbulent convection in a round tube following a discrete step in wall temperature for the same conditions as those of Tables I, III, and IV. Owing to the presence of T > in Eq. (145) for * and of * in Eq. (120) for T >, an iterative method of solution is required. They concluded that stepwise solution of the differential equivalents of these two equations for trial values of T > was more efﬁcient computationally than iterative evaluation of the K integrals by quadrature. The results obtained using Eq. (172) for Pr are R summarized in Tables V, VI, and VII. The computed values of Nu for TABLE VI The Thermal Characteristics of Fully Developed Turbulent Convection in a Round Tube with Uniform Wall-Temperature for Pr : Pr : 0.8673 Based on Eq. (172) R (from Yu et al. [100a]) a>

Nu

T > /T > A K

500 1000 2000 5000 10,000 20,000 50,000

52.07 97.08 181.5 416.9 784.8 1482 3447

1.276 1.251 1.232 1.210 1.196 1.184 1.169

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332

TABLE VII Predicted Nusselt Numbers for Fully Developed Turbulent Convection in a Round Tube with Uniform Wall-Temperature with Pr based on Eq. (172) X (from Yu et al. [100a]) Small Pr 10\

10\

0.01

a>

0.1

0.7

14.61 23.89 40.53 84.57 150.4 270.6 596.8

46.50 86.01 159.7 364.1 682.0 1282 2966

Nu 500 1000 2000 5000 10,000 20,000 50,000

4.959 5.055 5.123 5.188 5.227 5.262 5.304

4.967 5.072 5.157 5.275 5.402 5.611 6.173

5.488 6.155 7.328 10.50 15.27 23.89 46.54

Large Pr a>

1

a> 150 500 1000 2000 5000 10,000 20,000 50,000

10

100

1000

10,000

-

Nu/(0.07343(Pr/Pr )Re( f /2)) R 0.8216 0.7270 0.6811 0.6394 0.5904 0.5575 0.5278 0.4928

0.9440 0.9200 0.9047 0.8887 0.8675 0.8518 0.8363 0.8166

0.9812 0.9791 0.9761 0.9725 0.9672 0.9630 0.9588 0.9531

0.9914 0.9935 0.9934 0.9928 0.9918 0.9909 0.9899 0.9886

0.9930 0.9950 0.9953 0.9953 0.9952 0.9948 0.9945 0.9936

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

uniform wall temperature are observed to be signiﬁcantly less than those for uniform heating only for Pr 1. The values of Nu in Table VIII for a> : 5000 only were, on the other hand, calculated using Eq. (173) in order to provide a direct comparison with the values computed by Notter et al. [97] who used the same expression. The differences are therefore presumed, if numerical errors in calculation are negligible, to be wholly a consequence of using (uv)>> from Eq. (99) rather than less accurate and incongruent expressions for R and u>.

turbulent flow and convection

333

TABLE VIII Comparison of the Computed Values of Nu for Fully Developed Turbulent Convection with a Uniform WallTemperature at a> : 5000 with Pr Based on Eq. (173) R Nu Pr 0 10\ 10\ 0.7 1.0 8 10 10 10

Yu et al. [100a] 5.187 9.350 84.96 360.2 454.0 1306C 3572 7915 17,119

Notter and Sleicher* [97] 5.29 11.86 65.7 332 443 1220 3436 7747 16,730

*Interpolated semitheoretically with respect to a>. C Interpolated semitheoretically with respect to Pr.

4. Numerical Results for Nu for Parallel-Plate Channels a. Equal Uniform Heating on Both Plates Danov et al. [85] utilized the integral formulations of Eqs. (152)—(155) together with (uv)>> from Eq. (99), * from Eq. (153), and Pr from Eq. (172) to compute numerical R solutions for fully developed convection in turbulent ﬂow between two parallel plates heated uniformly and equally. Their results are summarized in Table IX. The values of Re : 4b>u> in Table IX correspond to values K @ of u> determined by integrating (uv)>> as given by Eq. (99) with b> K substituted for a>. The corresponding values of Re : 4u>b>, u>/u> , K A K @ f : 2/(u> ) and Re ( f /2) : 4b>/u> are also listed in Table X. The values K K @ of Re ( f /2) were determined from 4b>/u> and those of * from Re ( f / K KU8 @ @ 2)/Nu Pr : 0.867 . Their computed values of Nu are compared in Table @ @ XI with the earlier ones of Kays and Leung [101] for the other limiting case of a concentric circular annulus. Interpolation was avoided by utilizing values of b> corresponding to the values of Re chosen by Kays and Leung. @ The comments on the discrepancies between the new and prior results in Tables I and III are presumed to be directly applicable here. b. Different Uniform Temperatures on the Two Plates Danov et al. [85] also carried out numerical integrations for this boundary condition using the integral formulations of Eqs. (157) and (159). Their results are

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TABLE IX Predicted Nusselt Numbers for Fully Developed Convection in Turbulent Flow between Uniformly and Equally Heated Parallel Plates with Pr based on Eq. (172) R (from Danov et al. [85]) Nu for small values of Pr @ Pr b>

0

0.001

0.01

0.10

500 1000 5000 10,000 50,000

10.43 10.61 10.85 10.93 11.07

10.45 10.64 11.03 11.27 12.78

11.46 12.76 21.15 30.31 89.95

28.93 46.66 162.4 288.0 1142

0.70 90.40 166.3 701.6 1314 5732

0.867 101.3 187.9 804.7 1515 6668

Nu for large values of Pr @ Pr b>

1.0

10

100

1000

10,000

25,000

500 1000 5000 10,000 50,000

107.0 198.4 876.2 1617 7176

280.6 547.8 2672 5148 25,051

701.8 1392 6927 13,815 68,715

1535 3062 15,271 30,466 152,087

3335 6667 33,334 66,656 333,209

4531 9061 45,288 90,568 452,471

TABLE X Computed Characteristics of Fully Developed Turbulent Flow between Parallel Plates per Danov et al. [85] a>

u> K

u>/u> A K

Re ;10\ @

f ;10

Re ( f /2) @

500 1000 5000 10,000 50,000

18.558 20.312 24.132 25.738 29.428

1.1605 1.1437 1.1189 1.1113 1.0974

37.116 81.249 482.64 1029.5 5885.5

5.81 4.85 3.43 3.02 2.31

107.8 196.9 828.8 1554 6796

turbulent flow and convection

335

TABLE XI Comparison of Predicted Values of Nu for Fully Developed Turbulent Convection @ from Two Uniformly and Equally Heated Plates with Pr based on Eq. (172) R Nu @ Pr : 0

Nu /Pr @ Pr : 0.867

Pr : 1000

b>

Re · 10\ @

K&L

D, A & C

K&L

D, A & C

K&L

D, A & C

415 1204 3244 9737

30 100 300 1000

10.41 10.66 10.74 10.90

10.40 10.66 10.81 10.93

84.90 212.3 514.2 1392

85.90 222.0 543.1 1478

99.90 288.6 766.5 2305

127.5 368.6 991.9 2967

K & L, Kays and Leung [101]; D, A & C, Danov, Arai and Churchill [85]; b>, is based on speciﬁed values of Re and Eq. (102).

summarized in Table XII. No appropriate prior results were identiﬁed for comparative purposes. C. Correlation for Nu Although direct numerical integrations such as those of Eqs. (89) for u> and (91) for u> using Eq. (99) for (uv)>> are perhaps feasible on demand for each K particular condition of interest, those required for Nu for each value of a> and Pr are somewhat more demanding because of the added dependence on * and Pr . Correlating equations are therefore convenient for computed values as R well as for experimental data for convection. By deﬁnition, correlating equations for computed values necessarily incorporate some empiricism. That empiricism may, however, often be minimized by the appropriate use of exact or nearly exact asymptotic expressions within the structure of the correlating equation. Such theoretically structured expressions are more reliable functionally and usually more accurate and general than purely empirical ones. 1. Dimensional Analysis Dimensional and speculative analysis proved to be very helpful in constructing the ﬁnal correlating equations for turbulent ﬂow. It is, however, much less helpful in turbulent convection, again because of the added dependence on Pr, Pr , and *. R The heat transfer coefﬁcient for fully developed convection in a smooth round tube and for invariant physical properties might, quite justiﬁably, be postulated to be a function only of D, u , , , , k, and c . However, since K U N

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336

TABLE XII Predicted Nusselt Numbers for Fully Developed in Turbulent Convection between Plates at Unequal Uniform Temperatures with Pr based on Eq. (172) R (from Danov et al. [85]) Nu for small values of Pr @ Pr b>

0

0.001

0.01

0.10

0.70

0.867

500 1000 5000 10,000 50,000

1.0 1.0 1.0 1.0 1.0

1.002 1.003 1.018 1.036 1.175

1.113 1.230 2.126 3.186 10.83

3.337 5.609 21.86 40.54 175.3

14.16 26.74 118.8 226.9 1028

16.43 31.20 139.8 267.6 1218

Nu for large values of Pr @ Pr b>

1.0

10

100

1000

10,000

25,000

500 1000 5000 10,000 50,000

18.11 34.49 155.3 180.0 1360

66.58 31.07 630.5 1240 5993

169.7 338.1 1671 3380 16,656

382.1 761.3 3797 7588 37,915

833.3 1665 8319 16,624 82,855

1132 2264 11,307 92,617 113,154

/u is known to be a unique function of D( )/, one of the ﬁve U K U variables in these latter two groupings is redundant in the listing for h. For example, eliminating , u , and individually allows the following three U K different dimensionless groupings to be derived:

Du c hD K , N :# k k

D( ) c hD U :# , N k k

or Nu : # Re, Pr

or Nu : # Re

f , Pr 2

(178)

(179)

and

hD D c f U, N :# or Nu : # Re , Pr . (180) k u k 2 K These three expressions are functionally equivalent to one another by virtue of the relationship between /u and D( )/, but on speculative K U

turbulent flow and convection

337

reduction they lead in some but not all cases to fundamentally different results. For example, the further speculation of independence of h from D leads, respectively, to Nu : Re# Pr

(181)

Nu : Re

(182)

Nu : Re

(183)

f # Pr

2

and f # Pr . 2

Equation (182) has been shown [see Eq. (136)] to be a valid asymptote for Pr ; -, whereas Eq. (183) provides a ﬁrst-order expression for Pr : Pr : Pr (neglecting the dependence on *). On the other hand, Eq. R 2 (181) does not appear to be valid for any condition. The speculation of independence from c in Eqs. (178)—(180) leads to the limiting solutions for N Pr : 0, but elimination of , k, , and individually does not appear to U lead to valid expressions. These limited results are to be contrasted with the extensive set of asymptotes obtained for ﬂow by speculative analysis. Nusselt [102] in 1909 was apparently the ﬁrst to apply dimensional analysis to turbulent convection in a round tube. Unfortunately, he postulated a power dependence of h on each of the dependent variables and thereby obtained the equivalent of Nu : AReLPrK

(184)

rather than simply Eq. (178). On the basis of the various exact integral expressions of Section III, A, Nu does not appear to be a ﬁxed power of Re for any condition. Since the friction factor, f, was found in Section II to be a non-power-function of Re, the proportionality of Nu to Re( f /2) or Re( f /2) does not constitute a power-dependence on Re. Similarly, Nu was found to be a ﬁxed power of Pr only in the limit of Pr ; -. The use of Eq. (184) for correlation of experimental data has actually impeded the representation, understanding, and prediction of turbulent convection, as illustrated in the following paragraphs. 2. Purely Empirical Correlating Equations Nusselt [102] ﬁtted the constants of Eq. (184) using his own experimental data for turbulent convection in gases in a round tube and determined an exponent n : 0.786 for Re and inexplicably the same value for the exponent of Pr. Based on experimental data for Re 10 and various gas and liquids

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with 0.7 Pr 100, Dittus and Boelter [103] in 1930 recommended A : 0.0265 and m : 0.3 for cooling, and A : 0.0243 and m : 0.4 for heating. Sherwood and Petrie [104] in 1932 plotted experimental values of Nu for Re 5 10 versus Pr in logarithmic coordinates, as shown in Fig. 15, and determined a power dependence on Pr of 0.4. The straight line in Fig. 16 represents the following expression: Nu : 0.024Re Pr .

(185)

A later correlation of this type from Coulson and Richardson [105] is shown in Fig. 16. The data appear to be well represented on the mean for large Re by Eq. (184) with A : 0.023, m : 0.4, and n : 0.8, but the scatter is suppressed visually by the logarithmic coordinates and furthermore is undoubtedly due in part to the oversimpliﬁed form of correlation as well as to experimental inaccuracy. Colburn [106] in 1933 noted the similarity of Eq. (185) to the following empirical expression for the friction factor: f : 0.023Re\ . 2

(186)

Fig. 15. Determination of power dependence of Nu on Pr for Re : 10. (From Sherwood and Petrie [104], Figure 1.)

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Fig. 16. Test of Eq. (185) with experimental data. (From Coulson and Richardson [105], p. 166.)

He thereby inferred that Nu f : . RePr 2

(187)

He chose the exponent of for Pr, not on theoretical grounds but simply as a compromise for the values of Dittus and Boelter and others ranging from 0.3 to 0.4. Equation (187) predicts the wrong functional dependence for Nu on Re except as a ﬁrst-order approximation for Pr : O 1 and on Pr except in the asymptotic limit of Pr ; -. 3. Numerical Predictions for L ow-Prandtl-Number Fluids Equations (185) and (187) failed utterly to predict the convective behavior of liquid metals in nuclear reactors in the 1950s, thereby stimulating the new

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theoretical analyses described in the following section. It was soon recognized that the distinctive thermal characteristic of liquid metals was their relatively high thermal conductivity (or low Prandtl number), which results in a signiﬁcant contribution by thermal conduction even in the turbulent core. Martinelli [107], Lyon [94], and others derived numerical solutions for turbulent convection in round tubes, using the eddy conductivity and accounting for thermal conduction over the entire cross section. These solutions predicted a lower limiting value for Nu as Pr ; 0 and an improved representation for liquid—metal heat transfer. They also had the effect of establishing the credibility of theoretical predictions as compared to purely empirical correlations of experimental data. However, Lyon conjectured that his computed values of Nu would be a function only of RePr, that is, to be independent of the viscosity, thereby leading to miscorrelations such as that of Lubarsky and Kaufman [108], as shown in Fig. 17. The dashed and dotted lines represent Eq. (185) for Pr : 0.006 and Pr : 0.03, respectively. The other two curves represent purely empirical correlating equations. Sleicher and Tribus [109] carried out numerical calculations for Nu for a wide range of values of Re and Pr, developing as well as fully developed convection, and a number of thermal boundary conditions, using a Graetztype series expansion and more accurate values of u>, , Pr, and Pr than R R those of prior investigators. They concluded from their results that the scatter in Fig. 17 and similar plots was due in part to a parametric dependence on Pr beyond that of RePr, as well as to incomplete thermal development. Notter and Sleicher [97, 110, 111] subsequently improved somewhat upon these solutions. Their numerical results are probably the most reliable in the literature other than those of Yu et al. [100a], which are based on the equivalent of more accurate values of u> and . R Churchill [112] correlated all of the computed values of Nu of Notter and Sleicher for fully developed convection as well as culled experimental data with the expression

Nu : Nu ;

f Pr 2 [1 ; Pr]

0.079Re

(188)

with Nu : 4.8 and 6.3, respectively, for a uniform wall temperature and uniform heating. He also extended this expression to encompass laminar and transitional ﬂow as follows:

Nu:Nu; J

exp (22009Re)/366

0.079Re( f /2)Pr \ \ ; Nu ; . Nu [1 ; Pr\] J (189)

Fig. 17. Representation of experimental values of Nu for liquid metals as a function of Pe´ : RePr by Eq. (185): (- - -) Pr : 0.006; ( · · · ) Pr : 0.03. (From Lubarsky and Kaufman [108], Figure 42.)

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Fig. 18. Representation of culled experimental data and predicted values for Nu and Sh by Eq. (189). (From Churchill [112], Figure 1.)

Here Nu , the solution for laminar ﬂow, equals 3.657 for a uniform wall J temperature and 4.364 for uniform heating. Equation (189) is seen in Fig. 18 to represent the computed values of Notter and Sleicher as well as the experimental data very well. 4. Mechanistic Analogies Equation (187) is commonly known as the Colburn analogy because it was constructed by postulating the same empirical dependence for Nu/RePr

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and f /2 on the Reynolds number. Many other analogies for turbulent convection have been devised by postulating similar mechanisms of transport for momentum and energy. Although Eqs. (136) and (141) and their counterparts relate Nu and f, those relationships are simply a consequence of the dependence of the rate of heat transfer on the velocity ﬁeld rather than an explicit analogy. Several of these mechanistic analogies are described brieﬂy because of the understanding they convey, and one in detail because it proves remarkably useful for correlation. Reynolds [18] in 1874, as mentioned in the Introduction, made a significant contribution to turbulent convection by postulating equal rates of transport of momentum and energy from the bulk of the ﬂuid to the wall by means of the ﬂuctuating eddies. His result, in modern notation, takes the form of Eq. (142). Prandtl [113] in 1910 attempted to improve upon the Reynolds analogy, by postulating that the transport of momentum and energy by the eddies extends only to the edge of a boundary layer and that the completion of the transport to the wall occurs by linear molecular diffusion. His result may be expressed as Nu :

Re( f /2)Pr , 1 ; (>(Pr 9 1)( f /2)

(190)

where (> : (( )/. The major contribution of the Prandtl analogy, Eq. U (190), is the prediction that the dependence of Nu on Re shifts from proportionality to Re( f /2) to proportionality to Re( f /2) as Pr increases from unity to very large values. It implies that the Reynolds analogy is valid only for Pr : 1. As contrasted with the Reynolds analogy, which is free of explicit empiricism, the Prandtl analogy contains an empirical factor (>. Thomas and Fan [114] attempted to improve upon one other deﬁciency of the Reynolds analogy by applying the penetration and surface renewal model of Higbie [115] and Danckwerts [116] to account for transport from the eddies to the wall when they reach it. Their result is Nu : Re

f Pr. 2

(191)

Comparison of Eqs. (142), (190), and (191) with Eqs. (132)—(136) and (141) reveals that three analogies all incorporate the postulates of * : 0 and Pr : 1. They all fail outright for small values of Pr because of the failure to account for conduction in the turbulent core. For large values of Pr, the Reynolds analogy fails because of the neglect of the boundary layer, the Prandtl analogy fails because of the neglect of turbulent transport within the

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boundary layer, and the Thomas and Fan analogy fails because periodic transient conduction is simply a very poor model for the combination of turbulent and molecular transport in the boundary layer. Perhaps the greatest lasting value of these analogies is the understanding provided by analysis of the reasons for their failure.

5. A Useful Differential Analogy Reichardt [87] in 1951 derived an analogy based on the differential momentum and energy balances in time-averaged form. He utilized the eddy viscosity model for turbulent transport, but his derivation will be outlined here in terms of (uv)>>. Taking the ratio of Eqs. (115) and (116), respectively, with Eq. (85) gives dT > Pr 1;* : (1 ; *) 2 : . du> Pr 1 9 (uv)>> ; (Pr/Pr )(uv)>> R

(192)

Integrating the rightmost form from the centerline to the wall results in T >: A

u> A

(1 ; *) Pr 1 9 (uv)>> ; Pr R

(193)

du>.

(uv)>>

His ingenious expansion of the equivalent integrand in terms of may be R rephrased in terms of (uv)>> as follows:

T>: A

u> A

*

1 9 (uv)>> ;

Pr Pr R

Pr 19 R Pr Pr ; R; du>. Pr Pr uv>> (uv)>> 1; Pr 19uv>> R (194)

In order to obtain a solution in closed form, Reichardt suggested that for moderate and large values of Pr the leftmost term of the integrand be approximated by *(Pr /Pr) since *;0 for small values of u> while (uv)>> R 5 1 for u> ; u>. He also concluded that the rightmost term was negligible A except very near the wall where du> 5 dy>. He further postulated Pr /Pr R to be invariant over the cross-section. Had he utilized the limiting form of Eq. (93), that is, Eq. (194) for (uv)>> in the rightmost term, he would have

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obtained

2a> 2a> T > A : T> T> T> K A K 1 : , (195) Pr Pr R (1 ; *) > ; 19 R KS Pr Pr ; u> T > f 3 T > Pr f K A Re A Re u> T > 2 2' T > Pr 2 A K K R where (1 ; *) > is the integrated-mean value over u>. By virtue of Eq. (88), KS this latter term may also be interpreted as the integrated mean, weighted by 1 9 (uv)>>, over R. Equation (195) is, by virtue of the limits of integration and several of the approximations, applicable for a uniform wall temperature as well as for uniform heating. Nu :

6. T heoretically Based, Generalized Correlating Equations On the basis of the asymptotic expressions for Pr : Pr and Pr ; - for R uniform wall temperature, namely, Eqs. (151) and (136), Eq. (195) may be interpreted as Nu :

1

(196) 1 1 Pr ; 19 R , Nu Pr Nu where Nu signiﬁes Nu Pr : Pr and Nu signiﬁes Nu Pr ; - . Accord R ingly, Eq. (196) may be postulated to be applicable for uniform heating with Nu and Nu from Eqs. (141) and (136), respectively. The analogy of Prandtl [Eq. (190)] may be noted to have the form of Eq. (196) with, however, the implicit postulates of Pr : 1 and * : 0, and a missing R dependence on Pr for Pr ; -. Equation (195) may be interpreted on the basis of the Prandtl analogy as the consequence of the resistances for Pr : Pr and Pr ; - in series, or alternatively as an application of Eq. (71) R with n :91 and limiting solutions of (Pr/Pr ) Nu and Nu /(1 9 Pr /Pr). R R Equations (195) and (196) are limited to Pr 2 Pr , which according to the R expressions of Yahkot et al. [89] and Jischa and Rieke [96] means to Pr 0.848 and 0.867, respectively. By analogy to Eq. (196), rearranged as

Pr R Pr

Nu 9 Nu : Nu 9 Nu 1;

1 Pr R Pr 9 Pr R

Nu Nu

(197)

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Churchill et al. [117] speculated that Nu 9 Nu : Nu 9 Nu 1;

1 (198) Pr Nu Pr 9 Pr Nu R might be applicable as a correlating equation for Pr Pr . However, Eq. R (198) was not found to be sufﬁciently accurate and in addition to result in a discontinuity in the derivative of Nu with respect to Pr/Pr at Pr : Pr . R R Accordingly, they introduced an arbitrary coefﬁcient as a multiplier of (Pr/Pr 9 Pr)(Nu /Nu ) and evaluated it functionally to provide a continuR ous derivative. The resulting expression Nu 9 Nu : Nu 9 Nu 1;

1 (199) Pr Nu Nu 9 Nu Pr 9 Pr Nu Nu 9 Nu R where Nu : Nu Pr : Pr : 0.07343Re( f /2), has proven as successful R as Eq. (196). Although Eq. (199) lacks the theoretical basis of Eq. (196) it is free of any explicit empiricism. Because of the generality of their structure and components, Eqs. (196) and (199) might be speculated to be applicable for all thermal boundary conditions and for all channels. As will be shown, this conjecture is conﬁrmed for all of the yet available numerical results. Although Eq. (136) is presumed to be universally applicable for Nu , different expressions are required for Nu and Nu in Eqs. (196) and (199) for each case, as discussed next.

a. Uniformly Heated Round Tubes In order to utilize Eqs. (196) and (199) for values of a> intermediate to those of Tables I—IV, it is necessary to have supplementary correlating equations for Nu and Nu . The following purely empirical expressions, together with u> from Eq. (102) (with a modiﬁed K leading constant of 3.2) reproduce the values of Nu in Tables I and III almost exactly: 8

(200) 7.7 1; (u> ) K Re( f /2) 2a>/u> K Nu : : (201) 185 185 1; 1; (u> ) (u> ) K K The choice of u> rather than u> , u> /u> , a> or Re as the independent K A A K Nu :

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variable in Eqs. (200) and (201) is arbitrary since they all bear a one-to-one correspondence. The leading constant of 3.3 in Eq. (102) was chosen on the basis of the experimental data of Zagarola [73], while the recommended value here of 3.2 corresponds more closely to the computed values of u> in K Table II and is thereby self-consistent with the computed values of Nu. b. Isothermally Heated Round Tubes Separate correlating equations might have been devised for T >/T > and u>/u> as well as for (1 ; *) and A K A K K0 (1 ; *) in Eqs. (150) and (151). However, in the interests of simplicity, UK0 the following overall expressions were derived: 8 (202) 1.538 1; (u> ) K Re( f /2) 2a>/u> K Nu : : (203) 148 148 1; 1; (u> ) (u> ) K K Equations (202) and (203) reproduce the values of Nu in Tables V and VI, respectively, almost exactly. Nu :

c. Uniform and Equally Heated Parallel Plates The corresponding expressions are Nu :

12 5.71 1; (u> ) K

(204)

and Re( f /2) 4b>/u> K : (205) 90 90 1; 1; (u> ) (u> ) K K Here, Nu and Re are based on a characteristic length of 4b and Eq. (102) is to be used for u> . K d. Convection between Isothermal Plates at Different Temperatures In this case, b is chosen as the characteristic length in order that Nu : 1. The corresponding expression is Nu :

f 2 b>/u> K Nu : : 11.707 11.707 1; 1; u> u> K K Re

(206)

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Equation (206) reproduces the values in Table XII for Pr : 0.867 very closely. e. Test of the Correlating Equations Figure 19 provides a test of Eqs. (196) and (199) for the computed values of Nu for a uniformly heated round tube and for parallel plates, both uniformly and equally heated and at different uniform temperatures in terms of Pr/Pr and Figure 20 for an isothermal R tube in terms of Pr with Pr estimated from Eq. (172). The agreement is very R good. The slight discrepancy for Pr : 0.01 and a> : 50,000 is presumed to result from the simpliﬁcations made by Reichardt [87] in deriving the equivalent of Eq. (195). f. Interpretation of Correlating Equations Equation (199) predicts a rapid increase in Nu as Pr increases followed by a point of inﬂection and a decreasing rate of increase as Pr ; Pr . Equation (196) similarly predicts a R more rapid increase beyond Pr : Pr followed by a second point of R inﬂection and a decreased rate of increase approaching a one-third-power dependence. The changes for Pr Pr are smaller than those for Pr & Pr R R and indeed almost indistinguishable in the scale of Figs. 19 and 20. Such behavior, which is presumed to be real, is far more complex than could ever be deduced from experimental or even precise computed values and is an illustration of the value of theoretically structured equations for correlation. Equations (196) and (199) together with Eqs. (136), (172) and (200)—(206) are presumed to predict more accurate values of Nu than any prior correlating equations. They are subject to signiﬁcant improvement primarily with respect to Eq. (172). A more accurate expression for Pr not only affects R the predictions of Eqs. (196) and (199) but also the numerically computed values upon which Eqs. (136) and (200)—(206) are based.

IV. Summary and Conclusions A. Turbulent Flow 1. A New Model for the Turbulent Shear Stress The new and improved representatives proposed in Section I for fully developed turbulent ﬂow in a channel are a direct consequence of the observation by Churchill and Chan [77] that the local, dimensionless, time-averaged shear stress, namely (uv)> :9uv/ , constitutes a better U variable for this purpose than traditional mechanistic and heuristic quantities such as the mixing length and the eddy viscosity. Churchill [80]

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Fig. 19. Representation of numerically predicted values of Nu by Yu et al. [100a] and Danov et al. [85] with Eqs. (196) and (199) for a> and b> : 5000. [x, equally and uniformly heated parallel plates (Nu : 4bu /v); *, uniformly heated round tube (Nu : 2au /v); ;, K K parallel plates at different uniform temperature (Nu : bu /v)]. K

subsequently noted that the local fraction of the shear stress due to turbulence, namely (uv)>> :9uv/, is an even better choice. The presentation of new integral formulations and algebraic correlations for fully turbulent ﬂow based on the time-averaged partial differential equations of conservation might appear to be atavistic in view of the recent, presumably exact, solutions of these equations in their unreduced timedependent form. However, the use of integral and algebraic structures based on the time-averaged equations may be expected to persist into the

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Fig. 20. Representation of numerically predicted values by Yu et al. [100a] of Nu for a uniform wall temperature by Eqs. (196), (199), and (172).

foreseeable future for two reasons. First, the exact numerical solutions, which have been attained only by direct numerical simulation, are currently very limited in scope by their computational requirements and perhaps their inherent structure. Second, even if these limitations are eventually eliminated or at least eased by improved computer hardware and software as well as better inherent representations, or even if the DNS calculations are replaced or supplemented by some other methodology, the results will be in the form of discrete instantaneous or time-averaged values of u, v, uv, and u for a particular condition and therefore not directly useful for applications such as the design of hydrodynamic piping. Correlating equations will accordingly continue to be useful if not essential to summarize and generalize the vast quantity of information that is generated. Theoretically based algebraic structures will likewise continue to be useful in constructing forms for these correlating equations. 2. Integral Formulations in Terms of the Turbulent Shear Stress An unexpected result from the use of (uv)>> as a variable was the realization that, by virtue of integration by parts, u> as well as u> may be K expressed as simple, single integrals of this quantity. The possibility of such a simpliﬁcation by means of integration by parts was apparently ﬁrst discovered by Kampe´ de Fe´riet [81] in the context of uv and a parallel-

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plate channel. This suggestion was ﬁrst implemented by Pai [82] for both parallel-plate channels and round tubes, but with very poor representations for uv. The advantage of using uv rather than u as a primary variable was noted by Bird et al. [35], p. 175, but only in connection with the cited work of Pai, and even then incorrectly. The major contribution of Churchill and Chan [77] in this context was the recognition that an accurate and generalized correlating equation for (uv)> was the key to successful implementation of the integral formulation. An inherent advantage of correlating equations for (uv)> or (uv)>> over those for u> apart from simplicity, is that integration is a ‘‘smoothing’’ process and somewhat dampens any minor error in the integrand. Hence, the predictions of Eqs. (89) and (90), using Eq. (99) for (uv)>>, are inherently more accurate than those of Eqs. (100) and (101). 3. T he Development of Correlating Equations for (uv)>>, u>, and u> K The structure of an almost exact correlating equation for (uv)>>, namely Eq. (99), was developed by Churchill and Chan [71] from a number of asymptotic and speculative expressions for the time-averaged velocity as well as for the time-averaged turbulent shear stress. The empirical coefﬁcient of the asymptotic solution for y> ; 0 was evaluated using the several sets of results obtained by DNS while those for intermediate values of y> and a> were evaluated from the experimental time-averaged velocity distributions measured by Nikuradse [46]. These latter constants were subsequently reevaluated by Churchill [80] using the recent improved measurements of the time-averaged velocity distribution by Zagarola [73]. The incorporation of the equivalent of the semilogarithmic expression for the time-averaged velocity nominally restricts this correlating equation to a> 300, but it provides a very good approximation for y> a> even down to a> : 145, the lower limit of fully turbulent ﬂow. The calculation of u> and u> from the integral formulations using Eq. K (99) for (uv)>> is feasible even with a handheld calculator. Hence, separate correlating equations for u> and u> are not really required. However, such K expressions were constructed in the name of convenience and tradition. Equations (100) and (101) are presumed to be the most accurate expressions in the literature for u> and u>, respectively, for both smooth and naturally K rough pipe. Since u> Y (2/ f ), the correlating equation for u> serves as K K one for the Fanning friction factor as well. Equations (99), (100), and (101) are subject to reﬁnement, at least in terms of the coefﬁcients, constants and combining exponents, upon the appearance of better values for (uv)>, u>, u> , and the roughness e from either K A experimentation or numerical simulations. The tabulated values of e in the A

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current literature are very old and are almost certainly not representative for modern piping. The reevaluation of these roughnesses for representative materials and conditions would appear to have a high priority. 4. T he Analogy of MacL eod The analogy attributed to MacLeod [60] was crucial to the development of the just-mentioned correlations for (uv)>> and u> in that it allowed experimental data and computed values for round tubes and parallel plates to be used interchangeably. This little-known analogy appears to be validated within the accuracy of the experimental values for uv and u, but has no theoretical rationale. A critical test may be beyond the accuracy of present experimental means, but should be possible, at least for a> and b> & 300, by DNS. Such a resolution would appear to have great merit. 5. Obsolete and L imited Models Science and engineering progress by discarding obsolete models as well as by new discoveries. One of the ﬁrst discoveries resulting from the use of (uv)>> as the primary dependent variable was that the mixing length is unbounded at one location in the ﬂuid in all channels and in addition is negative over a ﬁnite adjacent region in all channels other than round tubes and parallel plates. Although the eddy viscosity is well behaved in round tubes and parallel-plate channels, it shares the failure of the mixing length in all other channels. How did these anomalies completely escape attention for 75 years? The explanation has three elements. First, the initial numerical evaluation of the mixing length by Nikuradse [45, 46] not only was based on experimental data of insufﬁcient precision but also was conditioned by a preconceived notion concerning the behavior. Second, the subsequent acceptance of the mixing length by later investigators is simply inexcusable, since the aforementioned failures of this concept are readily apparent from a critical examination of most of their own sets of measurements of the velocity distribution as well as from the predictions thereof. Third, the anomalies are much more apparent in terms of (uv)>> than in terms of earlier formulations. When it was ﬁrst introduced by Launder and Spalding [42], the ,— model appeared to have great promise for the predictions of turbulent ﬂows, but it has ultimately proven to have no real utility. For round tubes and parallel-plate channels the ,— model, in all of its manifestations, not only invokes a great deal of empiricism and approximation, but is unneeded. In all other channels it shares the failure of the eddy-viscosity model, to which

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it is directly linked. The ,— 9 uv model avoids this linkage and thereby has a possible role, despite its high degree of empiricism, for geometries, such as circular annuli, in which the variation of the total shear stress is not known a priori. The large eddy simulation (LES) methodology avoids the need for time-averaging, at least in the turbulent core, and has a wider range of applicability than the DNS methodology, but at the price, at least at the present time, of a considerable degree of empiricism and approximation for the region near a surface. Barenblatt [57] and co-workers have recently attempted to resuscitate the power-law correlation of Nikuradse [46] and Nunner [56] for the timeaveraged velocity, and Zagarola [73] has demonstrated that it is more accurate than the semilogarithmic model for a narrow range of values of y>. This ‘‘improvement’’ is accomplished at the price of considerable empiricism, functionally as well as numerically, and very poor behavior outside that narrow range. Hence it does not appear to have any utility as an element of overall correlating equations for (uv)>> and u>. B. Turbulent Convection 1. Initial Perspectives As a result of the great success described earlier in developing simple formulations and improved correlating equations for fully developed turbulent ﬂow, the development of analogous expressions for fully developed turbulent convection was undertaken with consideration conﬁdence and great expectations. Unfortunately, it soon became apparent that turbulent convection is much more complex than turbulent ﬂow even in the simplest of contexts, and that the data base, both experimental and computational, and the known asymptotic structure are much more limited. Turbulent convection would be expected to be responsive to the same new numerical methodologies used for ﬂow, such as DNS, but so far the greater inherent complexity of the behavior has limited the scope and accuracy of such results. 2. New Differential Models Time-averaging of the partial differential energy balance, followed by one integration and expression in terms of dimensionless variables, results in Eq. (106), in which (T v)>> : c T v/j, the fraction of the heat ﬂux density due N to turbulence, is a new variable analogous to (uv)>>. However, the heat ﬂux density ratio, j/j , is a dependent variable, given in general by Eq. (107) U as contrasted with / : 1 9 y>/a> for ﬂow. Furthermore, very few data U

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have been obtained for T v or correlated in terms of (T v)>>. Accordingly, Eq. (106) was reexpressed as Eq. (109) with the expectation that the behavior of Pr /Pr Y 1 9 (T v)>>/1 9 (uv)>> would be more constrained 2 than that of (T v)>>. The terms Pr /Pr and j/j represent the complications 2 U associated with turbulent convection as compared with turbulent ﬂow. 3. T he Heat Flux Density Ratio For a uniform heat ﬂux from the wall, the heat ﬂux density ratio is a function only of the time-averaged velocity distribution, and Eq. (109) may be reduced to Eq. (115) with * given by Eq. (125), which may also be expressed in terms of (uv)>> [see Eq. (128)]. Owing to the accuracy and generality of Eq. (99), the uncertainty associated with Eq. (207) and thereby with the prediction of Nu herein is essentially conﬁned to Pr /Pr. Many past 2 semitheoretical expressions for Nu have, however, also been in error to an unknown degree because of the implicit postulate of * : 0. 4. T he Turbulent Prandtl Number One of the initial objectives of the investigation of turbulent convection that culminated in this article was to eliminate Pr or its equivalent, Pr [see 2 R Eq. (113)]. However, an important discovery resulting from the use of (uv)>> and (T v)>> as primary variables is that Pr and Pr bear a 2 R one-to-one correspondence to (uv)>> and (T v)>> and are therefore independent of their heuristic diffusional origin. It has generally been postulated that Pr and Pr are functions only of 2 R (uv)>> (or /) and Pr and thus independent of the thermal boundary R condition. For example, this postulate is inherent in the solutions of Notter and Sleicher [97, 110, 111] for developing thermal convection with both uniform heating and a uniform wall temperature. It is implied by Eq. (172) of Jischa and Rieke [96], Eq. (173) of Sleicher and Notter [97], and Eq. (174) of Yahkot et al. [89]. The last imply that their expression is also independent of geometry. The only direct experimental conﬁrmation of either postulate appears to be that of Abbrecht and Churchill [22], who found the eddy conductivity to be independent of length in developing thermal convection in an isothermal round tube as well as identical to that of Page et al. [90] for fully developed heat transfer across a parallel-plate channel at equal values of a> and b>, respectively. The publication of this result was greeted with two contradictory responses: one that it was obvious, and the other that it was obviously wrong. Despite the great simpliﬁcation provided by these generalizations, neither a completely satisfactory correlating equation nor a universally accepted

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theoretical expression for Pr or Pr appears to exist. This is the principal 2 R unresolved problem of turbulent convection, at least in round tubes and parallel-plate channels, and is worthy of renewed effort, experimentally, theoretically, and computationally. Thermal calculations by DNS, LES, and ,——uv—T v generate values of (T v)>> or the equivalent and therefore do not require a separate expression for Pr . However, with the exception R of the predictions of Papavassiliou and Hanratty [98], which are limited to b> : 150, these methodologies have not yet produced reliable values of (T v)>> or Pr for a broad range of Pr and (uv)>> or y> and a>. R 5. Integral Formulations for Nu Because of the great simpliﬁcation in the expression for the heat ﬂux density ratio that is possible for uniform heating, most theoretical solutions, including the present ones, have been restricted to this condition. By virtue of Eq. (125), Nu may be represented by the single integral of Eq. (126) in terms of Pr and of Eq. (127) in terms of Pr . Such a simpliﬁcation has 2 R apparently not been achieved before because of the greater complexity of the formulations in terms of / and u> as compared to these in terms of R (uv)>> only. Because of the uncertainty in the various expressions for Pr and Pr , 2 R particular attention has been given herein to three cases for which that uncertainty is eliminated or greatly reduced, namely Pr : 0, Pr:Pr :Pr , R 2 and Pr ; - while y> ; 0. The second of these conditions is implied by Eq. (172) to occur for Pr : 0.8673, by Eq. (174) for Pr : 0.848, and by Eq. (173) for values of Pr varying from 0.8 to 0.9, depending upon the value of (uv)>>. Equation (172) implies a limiting value of Pr : 0.85 for Pr ; - and Eq. R (173) a limiting value of Pr : 0.78 for y> ; 0 for large Pr, but the R calculations of Papavassiliou and Hanratty [98] using DNS suggest that such a ﬁnite limiting value is attained only for Pr & 100. The postulate that Pr ; 0.85 as y> ; 0 and Pr ; - allows the derivation of an analytical R solution in closed form, as represented by Eq. (136), which, however, may be valid only for large values of Pr but less than 100. 6. Numerical Solutions for Nu Numerical solutions for Nu have been carried out by Heng et al. [100] for a uniformly heated round tube and by Yu et al. [100a] for both a uniformly heated and an isothermal tube, in both instances for a complete range of values of Pr and a wide range of values of a> using Eq. (172) for Pr . The results of Yu et al., including the three limiting cases described in R

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the preceding section are summarized in Tables I—VIII. Similar results for parallel-plate channels, as obtained by Danov et al. [85], are summarized in Tables IX—XII. Despite the uncertainty associated with the use of Eq. (172) for Pr , these values of Nu are presumed to be more accurate than any prior R ones because of the essentially exact representation in every other respect. They are of course subject to improvement and should be updated when more accurate values or expressions for Pr or Pr become available. 2 R 7. Final Correlating Equations Because of a lack of data of proven reliability and broad scope, a new correlating equation was not devised for (T v)>> or Pr . For the same R reason, new correlating equations were not constructed for T >. Instead attention was focused directly on Nu. The integral formulations for Pr : 0 and Pr : Pr : Pr and the anaR 2 lytical solution for Pr ; - imply that all prior correlating equations for Nu, including the Colburn analogy, are in signiﬁcant error functionally as well as numerically even over their own purported range of validity. A new simple but very general correlating equation for Nu for Pr 2 0.867 was devised on this basis of the analogy of Reichardt. This expression, Eq. (196), represents all of the computed values of Yu et al. and Danov et al. for Pr 2 0.867 quite accurately and is presumed to be applicable for other conditions as well. A supplementary empirical correlating equation was devised for Pr 0.867. This expression, Eq. (199), also represents all of the computed values very well. The overall success of Eqs. (196) and (199) is displayed in Fig. 19. in terms of Pr/Pr , and in Fig. 20. The close represenR tation of the computed values of Nu in Fig. 20 does not constitute a critical test of the absolute values of Nu because Eq. (172) was used for Pr in both R cases. However, the accuracy of the predictions of Nu by Eqs. (199) and (198) is presumed to be independent of the expression used for Pr . R References 1. Lamb, H. (1945). Hydrodynamics, 1st American ed. Dover Publications, New York, p. 663. 2. Richter, J. P., ed. (1970). T he Notebooks of L eonardo da V inci, Vol. 1. Dover Publications, New York. 3. Chandrasekhar, S. (1949). On Heisenberg’s elementary theory of turbulence. Proc. Roy. Soc. (London) A 200, 20—33. 4. Einstein, A. (1905). [Engl. transl. On the motion required by the molecular kinetic theory of heat of particles suspended in ﬂuids at rest.] Ann. Phys. 17, 549—560. 5. Heisenberg, W. (1924). Uber Stabilita¨t und Turbulenz von Flu¨ssigkeitsstro¨men. Ann. Phys. 74, 577—627.

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6. Kapitsa, P. L. (1947). Theoretical and empirical formulas for heat transfer in twodimensional turbulent ﬂow. Doklady AN SSSR 55, 595—602. 7. Landau, L. D. (1944). Turbulence. Doklady AN SSSR 44, 339—342. 8. Lorentz, H. A. (1907). Uber die Entstehung turbulenter Flu¨ssigkeitsbewegungen und u¨ber den Einﬂuss dieser Bewegungen bei der Stro¨mung durch Rohren. Abh. T heor. Phys., L eipzig 1, 43—71. 9. Newton, I. (1701). Scala graduum Caloris. Philos. Trans. Roy. Soc. (London) 22, 824—829. 10. Rayleigh, Lord (J. W. Strutt) (1880). On the stability or instability of certain ﬂuid motions. Proc. London Math. Soc. 11, 51—70. 11. Sommerfeld A. (1909). A Contribution to the Hydrodynamical Explanation of Turbulent Fluid Motions. Atti del IV. Cong. intern. dei matematici, Roma, pp. 116—124. 12. Uhlenbeck, G. (1980). Some notes on the relation between ﬂuid mechanics and statistical physics. Ann. Rev. Fluid Mech. 12, 1—9. 13. Mises, R. von (1941). Some remarks on the laws of turbulent motion in channels and circular tubes. In Th. von Kármán Anniversary Volume, Calif. Inst. Techn. Press, Pasadena, CA, pp. 317—327. 14. Weizsa¨cker, C. F. von (1948). Das Spectrum der Turbulenz bei grossen Reynolds’schen Zahlen, Z. Phys. 124, 614—627. 15. Zel’dovich, Ya. B. (1937). Limiting laws for turbulent ﬂows in free convection. Zh. Eksp. Teor. Fiz. 7, 1463—1465. 16. Schlichting, H. (1979). Boundary Layer Theory, 7th ed., translated by J. Kestin. McGrawHill Book Co., New York, p. xxii. 17. Reynolds, O. (1895). On the dynamical theory of incompressible viscous ﬂuids and the determination of the criterion. Philos. Trans. Roy. Soc. (London) 186, 123—161. 18. Reynolds, O. (1874). On the extent and action of the heating surface of steam boilers. Proc. Lit. Soc. Manchester 14, 7—12. 19. Navier, C.-L. M. N. (1822). Me´moire sur les lois du mouvement des ﬂuides. Mém. Acad. Roy. Sci. 6, 389—416. 20. Stokes, G. G. (1845). On the theories of internal friction of ﬂuids in motion, and of the equilibrium and motion of elastic solids. Trans. Cambridge Philos. Soc. 8, 287—319. 21. Barenblatt, G. I., and Goldenfeld, N. (1995). Does fully developed turbulence exist? Reynolds number independence versus asymptotic covariance. Phys. Fluids 7, 3078—3082. 22. Abbrecht, P. H., and Churchill, S.W. (1960). The thermal entrance region in fully developed turbulent ﬂow. AIChE J. 6, 268—273. 23. Fourier, J. B. (1822). Théorie analytique de la chaleur, Gauthie´r-Villais, Paris. 24. Rayleigh, Lord (J. W. Strutt) (1915). The principle of similitude. Nature 95, 66—68. 25. Rayleigh, Lord (J. W. Strutt) (1892). On the question of the stability of the ﬂow of ﬂuids. Philos. Mag. 34, 59—70. 26. Churchill, S. W. (1981). The use of speculation and analysis in the construction of correlations. Chem. Eng. Commun. 9, 19—38. 27. Churchill, S. W. (1997). A new approach to teaching dimensional analysis. Chem. Eng. Educ. 30, 158—165. 28. Reynolds, O. (1883). An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and the law of resistance in parallel channels. Philos. Trans. Roy. Soc. (London) 174, 935—982. 29. Prandtl, L. (1926). Über die ausgebildete Turbulenz, Verhdl. 2. Kong. Techn. Mechanik, Zurich, p. 62. 30. Millikan, C. B. (1938). A critical discussion of turbulent ﬂows in channels and circular tubes. Proc. Fifth Intern. Congr. Appl. Mech., Cambridge, MA, pp. 386—392.

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31. Ka´rma´n, Th. von (1930). Mechanische Ahnlichten und Turbulenz. Proc. Third Intern. Congr. Appl. Mech., Stockholm, Part 1, pp. 85—93. 32. Murphree, E. V. (1932). Relation between heat transfer and ﬂuid friction. Ind. Eng. Chem. 24, 726—736. 33. Boussinesq, J. (1877). Essai sur la the´orie des eaux courantes. Mém. présents divers savants Acad. Sci. Inst. Fr. 23, 1—680. 34. Prandtl, L. (1925). Bericht u¨ber Untersuchungen zur ausgebildeten Turbulenz. Zeit. angew. Math. Mech. 6, 136—139. 35. Bird, R. B., Stewart, W. E. and Lightfoot, E. N. (1960) Transport Phenomena, John Wiley & Sons, New York. 36. Rotta, J. C. (1950). Das in Wandnahe gultige Geschwindigkeitsgesetz turbulenter Stro¨mungen. Ing. Arch. 18, 277—280. 37. Prandtl, L. (1933). Neuere Ergebnisse der Turbulenzforschung. Zeit. Vereines Deutscher Ingenieure 77, 105—114. 38. van Driest, E. R. (1956). On turbulent ﬂow near a wall. J. Aeronaut. Sci. 13, 1007—1011. 39. Kolmogorov, A. N. (1941). The local structure of turbulence in an incompressible viscous ﬂuid at very large Reynolds numbers. Doklady AN SSSR 30, 301—304. 40. Prandtl, L. (1945). Uber ein neues Formelsystem fu¨r die ausgebildete Turbulenz. Nachr. Ges. Wiss. Göttingen Math.-Phys. Klasse, pp. 6—19. 41. Batchelor, G. K. (1953). The Theory of Homogeneous Turbulence. Cambridge Univ. Press. 42. Launder, B. E. and Spalding, D. B. (1972). Mathematical Models of Turbulence, Academic Press, London. 43. Hanjalic´, K., and Launder, B. E. (1972). A Reynolds stress model of turbulence and its application to thin shear ﬂows. J. Fluid Mech. 52, 609—638. 44. Satake, S., and Kawamura, H. (1995). Large eddy simulation of turbulent ﬂow in concentric annuli with a thin inner rod. In Turbulent Shear Flows 9, (Ed. F. Durst, N. Kasagi, B. E. Launder, F. W. Schmidt, K. Suzuki, and J. N. Whitelaw). Springer, Berlin, pp. 259—281. 45. Nikuradse, J. (1930). Widerstandgesetz und Geschwindigkeitsverteilung von turbulenten Wasserstro¨mungen in glatten und rauhen Rohren. Proc. Third Intern. Congr. Appl. Mech., Stockholm, Part 1, pp. 239—248. 46. Nikuradse, J. (1932). Gesetzma¨ssigkeiten der turbulenten Stro¨mung in glatten Rohren, Ver. Deutsch. Ing. Forschungsheft, 356. 47. Nikuradse, J. (1933). Stro¨mungs gesetze in rauhen Rohren. Ver. Deutsch. Ing. Forschungsheft, 361. 48. Miller, B. (1949). The laminar ﬁlm hypothesis. Trans. ASME 71, 357—367. 49. Robertson, J. M., Martin, J. D., and Burkhart, T. H. (1968). Turbulent ﬂow in rough pipes. Ind. Eng. Chem. Fundam. 7, 253—265. 50. Lynn, S. (1959). Center-line value of the eddy viscosity. AIChE J. 5, 566—567. 51. Lindgren, E. R., and Chao, J. (1969). Average velocity distribution of turbulent pipe ﬂow with emphasis on the viscous sublayer. Phys. Fluids 12, 1364—1371. 52. Hinze, J. O. (1963). Turbulent pipe ﬂow. In Mécanique de la turbulence. Editions CNRS, Paris, pp. 130—165. 53. Blasius, H. (1913). Das Ahnlichkeitsgesetz bei Reibungsvorga¨ngen in Flu¨ssigkeiten. Ver. Deutsch. Ing. Forsch-arb. Ing.-wes., No. 131. 54. Freeman, J. R. (1941). Experiments upon the Flow of Water in Pipes and Fittings, ASME, New York. 55. Prandtl, L. (1929). Uber den Reibungswiderstand Stro¨mender Luft. Ergebn. Aerodyn. Versanst., Göttingen 3, 1—5. (First mentioned in I. Lieferung (1921), p. 136.)

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56. Nunner, W. (1956). Wa¨rmeu¨bertragung und Druckabfall in rauhen Rohren. Ver. Deutsch. Ing. Forschungsheft, 455. 57. Barenblatt, G. I. (1993). Scaling laws for fully developed turbulent shear ﬂows. Part 1. Basic hypotheses and analysis. J. Fluid Mech. 248, 513—520. 58. Churchill, S. W. (2000). An appraisal of new experimental data and predictive equations for fully developed turbulent ﬂow in round tubes, in review. 59. Rothfus, R. R., and Monrad, C. C. (1955). Correlation of turbulent velocities for tubes and parallel plates. Ind. Eng. Chem. 47, 1144—1149. 60. MacLeod, A. L. (1951). Liquid turbulence in a gas—liquid absorption system. Ph.D. Thesis, Carnegie Institute of Technology, Pittsburgh, PA. 61. Whan, G. A., and Rothfus, R. R. (1959). Characteristics of transition ﬂow between parallel plates. AIChE J. 5, 204—208. 62. Senecal, V. E., and Rothfus, R. R. (1955). Transition ﬂow of ﬂuids in smooth tubes. Chem. Eng. Progr. 49, 533—538. 63. Colebrook, C. F. (1938—1939). Turbulent ﬂow in pipes with particular reference to the transition region between smooth and rough pipe laws. J. Inst. Civ. Eng. 11, 133—156. 64. Churchill, S. W. (1973). Empirical expressions for the shear stress in turbulent ﬂow in commercial pipe. AIChE J. 19, 375—376. 65. Churchill, S. W. and Usagi, R. (1972). A general expression for the correlation of rates of transfer and other phenomena. AIChE J. 18, 1121—1128. 66. Orszag, S. D. and Kells, L. C. (1980). Transition to turbulence in plane Poiseuille and plane couette ﬂow. J. Fluid Mech. 96, 159—205. 67. Kim, J., Moin, P., and Moser, R. (1987). Turbulence statistics in fully developed channel ﬂow at low Reynolds numbers. J. Fluid Mech. 177, 133—166. 68. Lyons, S. L., Hanratty, T. J., and McLaughlin, J. B. (1991). Large-scale computer simulation of fully developed turbulent channel ﬂow with heat transfer. Int. J. Num. Methods Fluids 13, 999—1028. 69. Rutledge, J., and Sleicher, C. A. (1993). Direct simulation of turbulent ﬂow and heat transfer in a channel. Part I. Smooth walls. Int. J. Num. Methods Fluids 16, 1051—1078. 70. Eckelmann, H. (1974). The structure of the viscous sublayer and the adjacent wall region in turbulent channel ﬂow. J. Fluid Mech. 65, 439—459. 71. Churchill, S. W., and Chan, C. (1995). Theoretically based correlating equations for the local characteristics of fully turbulent ﬂow in round tubes and between parallel plates. Ind. Eng. Chem. Res. 34, 1332—1341. 72. Groenhof, H. (1970). Eddy diffusion in the central region of turbulent ﬂow in pipes and between parallel plates. Chem. Eng. Sci. 25, 1005—1014. 73. Zagarola, M. V. (1966). Mean-ﬂow scaling of turbulent pipe ﬂow. Ph.D. Thesis, Princeton University, Princeton, NJ. 74. Spalding, D. B. (1961). A single formula for the ‘‘Law of the Wall.’’ J. Appl. Mech. 28E, 455—458. 75. Churchill, S. W. and Choi, B. (1973). A simpliﬁed expression for the velocity distribution in turbulent ﬂow in smooth pipes. AIChE J. 19, 196—197. 76. Reichardt, H. (1951). Vollsta¨ndige Darstellung der turbulenten Geschwindigkeitsverteilung in glatten Leitungen. Zeit. angew. Math. Mech. 31, 201—219. 77. Churchill, S. W., and Chan, C. (1995). Turbulent ﬂow in channels in terms of local turbulent shear and normal stresses. AIChE. J. 41, 2513—2525. 78. Churchill, S. W., and Chan, C. (1994). Improved correlating equations for the friction factor for fully developed turbulent ﬂow in round tubes and between identical parallel plates, both smooth and rough. Ind. Eng. Chem Res. 33, 2016—2019.

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79. Churchill, S. W. (1997). New simpliﬁed models and formulations for turbulent ﬂow and convection. AIChE J. 42, 1125—1140. 80. Churchill S.W. (1996). A critique of predictive and correlative models for turbulent ﬂow and convection. Ind. Eng. Chem. Res. 35, 3122—3140. 81. Kampe´ de Fe´riet, J. (1948). Sur l’e´coulement d’un ﬂuide visqueux incompressible entre deux plaques paralle´lel indeﬁnies. La Houille Blanche 3, 509—517. 82. Pai, S. I. (1953) On turbulent ﬂow between parallel plates. J. Appl. Mech. 20, 109—114; (1953). On turbulent ﬂow in circular pipe. J. Franklin Inst. 236, 337—352. 83. Churchill, S. W. (1994). Turbulent Flows. The Practical Use of Theory. Notes, The University of Pennsylvania, Philadelphia, Chapter 5. 84. Wei, T., and Willmarth, W. W. (1980). Reynolds-number effects on the structure of a turbulent channel ﬂow. J. Fluid Mech. 204, 57—95. 85. Danov, S. N., Arai, N., and Churchill, S. W. (2000). Exact formulations and nearly exact solutions for convection in turbulent ﬂow between parallel plates. Int. J. Heat Mass Transfer 43, 2767—2777. 86. Churchill, S. W. (1997). New wine in new bottles; unexpected ﬁndings in heat transfer. Part II. A critical examination of turbulent ﬂow and heat transfer in circular annuli. Therm. Sci. Eng. 5, 1—12. 87. Reichardt, H. (1951). Die Grundlagen des turbulenten Wa¨rmeu¨berganges. Arch. ges. Wärmetechn. 2, 129—142. 88. Rohsenow, W. M., and Choi, H. Y. (1961). Heat, Mass, and Momentum Transfer, Prentice-Hall, Englewood Cliffs, NJ, p. 183. 89. Yahkot, V., Orszag, S. A., and Yahkot, A. (1987). Heat transfer in turbulent ﬂuids — 1. Pipe ﬂow. Int. J. Heat Mass Transfer 30, 15—22. 90. Page, F., Jr., Schlinger, W. G., Breaux, D. K., and Sage, B. H. (1952). Point values of eddy conductivity and eddy viscosity in uniform ﬂow between parallel plates. Ind. Eng. Chem. 44, 424—430. 91. Churchill, S. W. (1997). New wine in new bottles; unexpected ﬁndings in heat transfer. Part III. The prediction of turbulent convection with minimal explicit empiricism. Therm. Sci. Eng. 5, 13—30. 92. Petukhov, B. S. (1970). Heat transfer and friction in turbulent pipe ﬂow with variable physical properties. Adv. Heat Transfer 6, 503—562. 93. Seban, R. A., and Shimazaki, T. T. (1951). Heat transfer to a ﬂuid ﬂowing turbulently in a smooth pipe with walls at constant temperature. Trans. ASME 73, 803—808. 94. Lyon, R. N. (1951). Liquid metal heat-transfer coefﬁcients. Chem. Eng. Progr. 47, 75—79. 95. Reynolds, A. J. (1975). The prediction of turbulent Prandtl and Schmidt numbers. Int. J. Heat Mass Transfer 18, 1055—1069. 95a. Kays, W. M. (1994). Turbulent Prandtl number — Where are we? J. Heat Transfer, Trans. ASME 116, 284—295. 96. Jischa, M., and Rieke, H. B. (1979). About the prediction of turbulent Prandtl and Schmidt numbers from modiﬁed transport equations. Int. J. Heat Mass Transfer 22, 1547—1555. 97. Notter, R. H., and Sleicher, C. A. (1972). A solution to the turbulent Graetz problem — III. Fully developed and entry region heat transfer rates. Chem. Eng. Sci. 27, 2073—2093. 97a. Elperin, T., Kleeorin, N., and Rogachevskii, I. (1996). Isotropic and anisotropic spectra of passive sealer ﬂuctuations in turbulent ﬁlm ﬂow. Phys. Rev. E 53, 3431—3441. 98. Papavassiliou, D. V., and Hanratty, T. J. (1997). Transport of a passive scalar in a turbulent channel ﬂow. Int. J. Heat Mass Tranfer 40, 1303—1311. 99. Shaw, D. A., and Hanratty, T. J. (1978). Turbulent mass transfer to a wall for large Schmidt numbers. AIChE J. 23, 28—37.

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100. Heng, L., Chan, C. and Churchill, S.W. (1998). Essentially exact characteristics of turbulent convection in a round tube. Chem. Eng. J. 71, 163—173. 100a. Yu, B., Ozoe, H., and Churchill, S. W. (2000). The characteristics of fully developed turbulent convection in a round tube, in review. 101. Kays, W. M., and Leung, R. Y. (1963). Heat transfer in annular passages — Hydrodynamically developed turbulent ﬂow with arbitrarily prescribed heat ﬂux. Int. J. Heat Mass Transfer 6, 537—557. 102. Nusselt, W. (1909). Der Wa¨rmeu¨bergang in Rohrleitungen. Mitt. Forsch.-Arb. Ing.-wes., 89, 1750—1787. 103. Dittus, R. W., and Boelter, L. M. K., (1930). Heat Transfer in Automobile Radiators of the Tubular Type, University of California Publications in Engineering 2, 443—461. 104. Sherwood, T. K., and Petrie, J. M. (1932). Heat transmission to liquids ﬂowing in pipes. Ind. Eng. Chem. 24, 736—748. 105. Coulson, J. M., and Richardson, J. F. (1954). Chemical Engineering, Vol. 1. McGraw-Hill Book Co., New York, p. 166. 106. Colburn, A. P. (1933). A method for correlating forced convection heat transfer data and a comparison with ﬂuid friction. Trans. AIChE 29, 174—210. 107. Martinelli, R. C. (1947). Heat transfer to molten metals. Trans. ASME 69, 947—959. 108. Lubarsky, B., and Kaufman, S. J. (1956). Review of experimental investigations of liquid-metal heat transfer. Nat. Advisory Comm. Aeronaut. Report 1270, Washington, D.C. 109. Sleicher, C. A., and Tribus, M. (1957). Heat transfer in a pipe with turbulent ﬂow and arbitrary wall-temperature distribution. Trans. ASME 79, 789—797. 110. Notter, R. H., and Sleicher, C. A. (1971). The eddy diffusivity in the turbulent boundary layer near a wall. Chem. Eng. Sci. 26, 161—172. 111. Notter, R. H., and Sleicher, C. A. (1971). A Solution to the turbulent Graetz problem by matched asymptotic expansions — II. The case of uniform heat ﬂux. Chem. Eng. Sci. 26, 559—565. 112. Churchill, S.W. (1977). Comprehensive correlating equations for heat, mass, and momentum transfer in fully developed ﬂow in smooth tubes. Ind. Eng. Chem. Fundam. 16, 109—116. 113. Prandtl, L. (1910). Eine Beziehung zwischen Wa¨rmeaustausch and Stro¨mungswiderstand der Flu¨ssigkeiten. Phys. Z. 11, 1072—1078. 114. Thomas, L. C., and Fan, L. T. (1971). Heat and momentum analogy for incompressible boundary layer ﬂow. Int. J. Heat Mass Transfer 14, 715—717. 115. Higbie, R. (1935). The rate of pure gas into a still liquid during short periods of exposure. Trans. Amer. Inst. Chem. Engrs. 31, 365—389. 116. Danckwerts, P. V. (1957). Signiﬁcance of liquid ﬁlm coefﬁcients in gas absorption. Ind. Eng. Chem. 43, 1460—1467. 117. Churchill, S. W., Shinoda, M., and Arai, N. (2000). A new concept of correlation for turbulent convection. T hermal Sci. Eng., in press.

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ADVANCES IN HEAT TRANSFER, VOLUME 34

Progress in the Numerical Analysis of Compact Heat Exchanger Surfaces

R. K. SHAH Delphi Harrison Thermal Systems Lockport, New York 14094

M. R. HEIKAL University of Brighton Brighton, United Kingdom

B. THONON AND P. TOCHON CEA-Grenoble DTP/GRETh 38054 Grenoble, France

I. Introduction Compact heat exchangers (CHEs) are characterized by a large heat transfer surface area per unit volume of the exchanger, resulting in reduced space, weight, support structure and footprint, energy requirements, and cost, as well as improved process design, plant layout, and processing conditions, together with low ﬂuid inventory compared to conventional designs such as shell-and-tube heat exchangers. Somewhat arbitrarily, a gas-to-ﬂuid exchanger is referred to as a compact heat exchanger if it incorporates a heat transfer surface with area density above about 700 m/m (213 ft/ft) or the hydraulic diameter D 6 mm (1/4 in.) for operating in a gas stream and above about 400 m/m (122 ft/ft) for operating in a liquid or phase-change stream. In contrast, a typical process industry shell-and-tube exchanger has a surface area density of less than 100 m/m on one ﬂuid side with plain tubes, and two to three 363

ISBN: 0-12-020034-1

ADVANCES IN HEAT TRANSFER, VOL. 34 Copyright 2001 by Academic Press. All rights of reproduction in any form reserved. 0065-2717/01 $35.00

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Fig. 1. Plate-ﬁn geometries: (a) offset strip ﬁn, and (b) louver ﬁn.

times that with high-ﬁn-density low-ﬁnned tubing. A typical plate heat exchanger has about two times the heat transfer coefﬁcient h or the overall heat transfer coefﬁcient U compared to that for a shell-and-tube exchanger for water/water applications. For phase-change applications, even higher heat transfer coefﬁcients are achieved compared to a shell-and-tube exchanger. A laminar ﬂow heat exchanger (also referred to as a meso heat exchanger) has a surface area density on one ﬂuid side greater than about 3000 m/m (914 ft/ft) or 100 m D 1 mm. A heat exchanger is referred to as a micro heat exchanger if the surface area density on one ﬂuid side is greater than about 15,000 m/m (4570 ft/ft) or 1 m D 100 m. A compact heat exchanger is not necessarily of small bulk and mass. However, if it did not incorporate a surface of high area density, it would be much more bulky and massive. Plate-ﬁn, tube-ﬁn, and rotary regenerators are examples of compact heat exchangers for gas ﬂow on one or both sides; whereas gasketed, welded, brazed plate, and printed circuit heat exchangers are examples of compact heat exchangers for liquid ﬂows. Typical ﬁn geometries used in plate-ﬁn and tube-ﬁn exchangers are shown in Figs. 1 and 2, and plate geometries used in plate heat exchangers (PHEs) are shown in Fig. 3. The most commonly used ﬁn geometries for plate-ﬁn exchangers are offset strip ﬁns

Fig. 2. Tube-ﬁn geometries: (a) wavy ﬁn on round tubes, (b) louver ﬁn on round tubes, and (c) louver ﬁn on elliptical tubes.

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Fig. 3. Plate heat exchanger plate geometries: (a) washboard, (b) zigzag, and (c) chevron.

and louver ﬁns (referred to as multilouver ﬁns in the automobile industry). A considerable amount of experimental results are available in the literature for ﬂow and heat transfer phenomena in complex ﬂow passages of compact heat exchanger surfaces. Starting with the description of some of the complex ﬂows in compact heat exchanger surfaces, it is explained that ﬂows in compact heat exchanger surfaces are dominated by swirl and vortices in uninterrupted ﬂow passages, and by boundary layer ﬂows and wake regions (separation, recirculation, and reattachment) for interrupted ﬂow passages. Although unsteady laminar ﬂows are relatively easy to analyze, swirl and low Reynolds number turbulent ﬂows are difﬁcult to solve numerically because of the lack of appropriate turbulence models. This is the reason for the very slow progress in the numerical analysis of compact heat exchanger surfaces. A comprehensive experimental study of the performance of CHE surfaces is very expensive because of the high cost of the tools needed to produce a wide range of geometric variations. Numerical modeling, on the other hand, has the potential of offering a ﬂexible and cost-effective means for such a parametric investigation, with the added advantage of reproducing ideal geometries and boundary conditions, and exploring the performance behavior in speciﬁc and critical areas of ﬂow geometry. Thus, the objective of this work is to provide a comprehensive state-ofthe-art review on numerical studies of single-phase velocity and temperature ﬁelds, and heat transfer and ﬂow friction characteristics of compact heat exchanger surfaces, as well as to provide speciﬁc comparisons to evaluate the accuracy of numerical work where experimental data are available. The surfaces include offset strip and louver ﬁns used in plate-ﬁn exchangers, wavy ﬁns/channels used in tube-ﬁn exchangers and plate heat exchangers,

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and chevron (stamped) plates used in plate heat exchangers. First, a description of some of the complex ﬂows in such surfaces is presented. Next, some highlights are presented for the numerical analysis of compact heat exchanger surfaces. Since separation, recirculation, and reattachment as well as large eddies and small-scale turbulence generation are common features in CHE surfaces, a comprehensive but concise overview of turbulence models/methods is presented next to illustrate the current capabilities and limitations of these models. The rest of the paper covers numerical work reported in the literature on the following CHE surfaces: offset strip ﬁns, louver ﬁns, wavy ﬁns/channels, and chevron trough plates. For each surface, the numerical analysis is described in sufﬁcient detail, and comparisons are presented with experimental measurements where available. Thus, based on the insight gained from numerical and experimental results, the performance (ﬂuid ﬂow and heat transfer) behavior of these CHE surfaces is discussed and summarized. Also, brieﬂy mentioned is the proposed mode of research that combines numerical analysis, sophisticated experimentation on the small sample ﬁn geometries, and performance testing of actual heat exchanger cores.

II. Physics of Flow and Heat Transfer of CHE Surfaces In this section, the current understanding of the physics of ﬂow and heat transfer in compact heat exchanger surfaces is described in order to set the stage for the task of numerical analysis. The description is divided into interrupted and uninterrupted complex ﬂow passages, followed by characterization into laminar unsteady and low Reynolds number turbulent ﬂows.

A. Interrupted Flow Passages The two most common interrupted ﬁn geometries are the offset strip ﬁn and louver ﬁn geometries as shown in Fig. 1. Here the ﬁn surface is broken into a number of small sections. For each section, a new leading edge is encountered, and thus a new boundary layer development begins, and is then abruptly disrupted at the end of the ﬁn offset length l. The objective for such ﬂow passages is not to allow the boundary layers to thicken, thus resulting in the high heat transfer coefﬁcients associated with thin boundary layers. However, the interruptions create the wake region, and self-sustained ﬂow unsteadiness (see Figs. 4 and 5). As a result, the models based on the boundary layer development are not adequate and do not accurately predict

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Fig. 4. Flow phenomena in an offset strip ﬁn geometry.

Fig. 5. Flow phenomena in louver ﬁn geometry: (a) conventional louvers (section AA of Fig. 1b but not the same number of louvers); (b) and (c) CFD results of typical ﬂow path in a louver ﬁn array at Rel : 10 and Rel : 1600, respectively [113]; (d, see color insert) Flow visualization in a louver ﬁn geometry (courtesy of Hitachi Mechanical Engineering Lab).

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the heat transfer coefﬁcients (Nusselt number Nu or Colburn factor j ) and friction factors. Separation, recirculation, and reattachment are important ﬂow features in most interrupted heat exchanger geometries [1]. Consider, for example, the ﬂow at the leading edge of a ﬁn of ﬁnite thickness. The ﬂow typically encounters such a leading edge at the heat exchanger inlet or at the start of new ﬁns, offset strips, or louvers. For most Reynolds numbers, a geometric ﬂow separation will occur at the leading edge because the ﬂow cannot turn the sharp corner of the ﬁn as shown in Fig. 4. Downstream from the leading edge, the ﬂow reattaches to the ﬁn. The ﬂuid between the separating streamline (see Fig. 4) and the ﬁn surface is recirculating. This region is called a separation bubble or recirculation zone. Within the recirculation zone, a relatively slow-moving ﬂuid ﬂows in a large eddy pattern. The boundary between the separation bubble and the separated ﬂow (along the separation streamline) consists of a free-shear layer. Since free shear layers are highly unstable, velocity ﬂuctuations develop in the free shear layer downstream from the separation point. These perturbations are advected downstream to the reattachment region, and there they result in an increased heat transfer. The ﬁn surface in contact with the recirculation zone is subject to lower heat transfer because of the lower ﬂuid velocities and the thermal isolation associated with the recirculation eddy. The separation bubble increases the form drag, and thus usually represents an increase in pumping power with no corresponding gain in heat transfer. If the ﬂow does not reattach to the surface from which it separates, a wake results. A free shear layer is also manifested in the wake region at the trailing edge of a ﬁn element. Depending on the Reynolds number and geometry, the wake from the upstream ﬁns can have a profound impact on the downstream ﬁn elements. The highly unstable wake can promote strong mixing that destroys the boundary layers from the upstream ﬁns, causing downstream heat transfer enhancement. However, at low Reynolds numbers, or for very close streamwise spacing of ﬁn elements, the shear layers might not be destroyed or the next ﬁn element might be embedded in the wake of an upstream ﬁn element. In such cases, the low velocity and near-ﬁn temperature of the wake will have a detrimental effect on the downstream heat transfer. Wake management in complex heat exchanger passages poses a difﬁcult challenge, especially at moderate and high Reynolds numbers where numerical simulation is difﬁcult to perform. Nevertheless, wake management appears to be the key to further progress in improved heat exchanger surface design. 1. Offset Strip Fins The ﬂow phenomenon for the offset strip ﬁn geometry is described by Jacobi and Shah [1] as follows. The ﬂow unsteadiness begins at relatively

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low Reynolds numbers (Re : 100) as waviness in the wake of the ﬁn elements. As the Reynolds number increases, oscillating ﬂow develops in the wake region. At higher Reynolds numbers, individual strips shed vortices at regular intervals. These vortices are transverse to the main ﬂow, and as they are carried through the ﬁn array, they refresh the boundary layer to produce a time-averaged thinner boundary layer. For deep arrays, vortex shedding begins at the downstream ﬁns and moves upstream as the ﬂow rate is increased (see Fig. 6 of [1]). At low Reynolds numbers (less than 400), ﬂow through the offset strip ﬁn geometry is laminar and nearly steady, and the boundary layer effects dominate the heat transfer and friction. For intermediate Reynolds numbers (roughly 400 & Re & 1000), the ﬂow remains laminar, but unsteadiness and vortex shedding become important. For example, at Re : 850, boundary layer restarting causes roughly a 40% increase in heat transfer over the plain channel with vortex shedding causing an additional 40% increase. Unfortunately, there is a commensurate increase in the pressure drop due to boundary layer restarting and vortex shedding. For Reynolds numbers greater than 1000, the ﬂow becomes turbulent in the array, and chaotic advection may be important in the low Reynolds number turbulent regime. A factor of 2 or 3 increase in heat transfer and pressure drop over plain ﬁns can be obtained as a result of the turbulent mixing. The important variables affecting the wake region identiﬁed are the strip length l, the ﬁn spacing s, and the ﬁn thickness (. The ﬁn spacing s and the strip length l are responsible for the boundary layer interactions and wake dissipation; the ﬁn thickness ( introduces form drag and also affects the heat transfer performance. Higher aspect ratios (s/b or b/s), shorter strip lengths l, and thinner ﬁns (() are found to provide higher heat transfer coefﬁcients (Nu or j ) and friction factors f. 2. L ouver Fins Flow through louver ﬁn geometries is similar to the ﬂow through offset strip ﬁn geometries, with boundary layer interruption and vortex shedding playing potentially important roles. However, another important aspect of louver ﬁn performance is the degree to which the ﬂuid follows the louvers. At low Reynolds numbers (Re & 200), boundary layer growth between neighboring louvers becomes pronounced, and a signiﬁcant blockage effect can result. Thus, at very low Reynolds numbers, the ﬂuid tends to ﬂow mostly between the ﬁns forming the channel without following the louvers. This ﬂow is referred to as the duct ﬂow (see Fig. 5a). At intermediate Reynolds numbers, when the boundary layers are thinner, the ﬂow tends to more closely follow the louvers. This ﬂow is referred to as the louver ﬂow (see Fig. 5a). At high Reynolds numbers (5000), the louvers act as a ‘‘rough’’ surface, and the duct ﬂow oscillates after the ﬁrst bank of louvers

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in a ﬁn geometry. Effectively, at all Reynolds numbers, both the duct ﬂow and louver ﬂow components exist, but the relative amount depends on the Reynolds number. Sketches of possible ﬂow patterns in louver ﬁns are shown in Figs. 5b and 5c, and a ﬂow visualization picture of ﬂow through louvers is shown in Fig. 5d (see color insert). To effectively exploit high heat transfer associated with short ﬂow lengths (louvers act as short ﬂat plates), it is important that the ﬂuid follow the louver (louver ﬂow) rather than passing between two ﬁns (duct ﬂow) to obtain high Nu (and the resultant high f factors). The degree of the ﬂow deﬂection by the louvers is determined by the relative hydraulic resistance to the ﬂow for the louver ﬂow vs duct ﬂow. This is dependent upon the ﬁn geometry and the ﬂow Reynolds number. The degree to which the ﬂuid follows the louvers is sometimes called the ﬂow efﬁciency, which can be deﬁned as the mean angle of the ﬂow divided by the louver angle. The behavior of the ﬂow efﬁciency and its relation to heat transfer has been examined by Cowell et al. [2]. For louver angles from 15 to 35° and ﬁn pitch-to-louver length ratios (p /l ) from 1 to 2.5, the ﬂow efﬁciency drops dramatically for Rel & 100. See typical results presented later in Fig. 14. Flow efﬁciency is nearly at its maximum by Rel : 200 and is almost independent of the Reynolds number for higher ﬂow rates. The ﬂow does not align with the louver array at the louver inlet and it takes a few louvers to turn the ﬂow. The heat transfer and pumping power performance is strongly dependent on this ﬂow-directing properties of the louver array. Surfaces that cause the ﬂow to follow the louvers, i.e., those with high ﬂow efﬁciency, generally perform better than those in which the ﬂow does not follow the louvers. However, the exact heat transfer performance of these surfaces is less well understood. Although the qualitative effect of the degree of ﬂow alignment on heat transfer is accepted, more accurate quantiﬁcation of these effects is needed. The degree of ﬂattening of the Stanton number curve at low Reynolds numbers [2] should be examined further with a view to determining, more accurately, the critical Reynolds number at which this ﬂattening starts and the effect of the different geometrical parameters on this phenomenon. The experimental work of Chang and Wang [3] demonstrates clearly that general correlating equations for predicting the heat transfer and pressure drop performance of these surfaces are far from being achieved. This is mainly due to the fact that the performance of these surfaces is a function of a large number of geometric parameters and that a number of variants of the ﬁn geometry are in use. Additionally, the manufacturing tolerances in the production of the ﬁns and variations in the test conditions also play a part in producing different performances for supposedly similar ﬁns at the same ﬂow conditions.

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A novel approach for the optimization of the heat transfer performance of ﬁns with variable louver angles was presented by Cox et al. [4] as an alternative to numerical modeling. Their method utilizes the Reynolds analogy to obtain heat transfer performance characteristics from measurement of the forces acting on the louvers in a 20 : 1 large-scale model of a typical matrix. The model allows the angle of individual louver rows to be driven automatically to speciﬁc angles. Force data logging and angle control were performed automatically under computer control implementing optimization strategies for the maximization of heat transfer performance as a function of louver angles. Results for ﬁxed angle arrays based on known geometries showed good agreement with previously obtained experimental data based on thermal experiments on full-size matrix sections. Although the model used had too few ﬁns to be representative of an inﬁnite array, the authors demonstrated the viability of such method for the optimization of variable louver ﬁns. B. Uninterrupted Complex Flow Passages In this case, the heat transfer surface (the ﬁn or the prime surface) is not cut, but convoluted such that the ﬂow passage geometry does not allow the boundary layer growth. For a plate-ﬁn geometry (Fig. 6a), two ﬂow passages are possible: wavy corrugated and wavy furrowed cross section (of Fig. 6a) as shown in Fig. 6c and 6d. For plate heat exchangers, the cross section of plates having intermating troughs (washboard design) is shown in Fig. 7a and those of plates having chevron troughs are shown in Figs. 7b and 7c. The physics of ﬂow of these surfaces is discussed next. The Reynolds number for the plate heat exchanger is commonly deﬁned with one of two characteristic lengths: the hydraulic diameter (D : four times the channel volume divided by the total heat transfer surface area) or the equivalent diameter (D : twice the plate spacing). The ratio D /D characterizes the surface extension ratio (Adeveloped /Aprojected ), and it ranges from 1.1 up to 1.4 for industrial plates.

Fig. 6. (a) Plate-ﬁn exchanger, (b) tube-ﬁn exchanger with ﬂat ﬁns. At section AA: (c) wavy corrugated passage, and (d) wavy furrowed passage.

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Fig. 7. Cross-section of two neighboring plates: (a) intermating troughs, (b) and (c) chevron troughs.

1. Wavy Corrugated and Furrowed Channels Corrugated and furrowed channels, as shown in Fig. 6, differ from plain channels of constant cross-section. Wavy geometries provide little advantage at low Reynolds numbers, and maximum advantage at transitional Reynolds numbers. However, at higher Reynolds numbers, periodic shedding of transverse vortices increases the Nusselt number with a considerable increase in the friction factor. The following are important ﬂow mechanisms associated with wavy ﬁns [1]: At low Re (&200), steady recirculation zones form in the troughs of the wavy passages and heat transfer is not enhanced. For higher Reynolds numbers, the free shear layer becomes unstable; vortices roll up and are advected downstream, thus enhancing the heat transfer. Transition to turbulence occurs at Re : 1200, depending on the geometry. It appears that chaotic advection may contribute to the heat transfer in the transitional Reynolds number range. For Reynolds numbers of 4000 and over, the ﬂow is fully turbulent with very high pressure drops. Thus, wavy channels provide higher heat transfer rates than plain channels, but with higher pressure drops. Ali and Ramadhyani [5] and Gschwind et al. [6] found that a streamwise — Go¨rtler-like — vortex system forms in the transitional Reynolds number range. Although the impact on heat transfer and pressure drop is not completely clear, such a vortex system is known to increase heat transfer. Wavy passages clearly offer heat transfer enhancement over plain channel passages; however, they do not offer a performance advantage (heat transfer relative to the pressure drop) over interrupted passages.

2. Intermating and Chevron Trough Plates The high heat transfer coefﬁcients obtained in plate heat exchangers are a direct result of the corrugated plate patterns. A cross-section of one corrugation along the ﬂow length of an intermating trough design is shown

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in Fig. 7a. The ﬂuid ﬂows through wavy passages in which, depending upon the Reynolds number Re, ﬂow will separate in hills and valleys where Taylor—Go¨rtler vortices are generated. The ﬂow separation and vortices are responsible for the high performance of these surfaces. The increases in j, f, and j/ f are generally higher than those for ﬂow over a plate having a dimple surface. In chevron plate (see Fig. 3c with - deﬁned there) design, the ﬂow geometry is 3D and quite complex. The typical cross-sectional geometries for chevron plates at - : 90° are shown in Figs. 7b and 7c. In other geometries, the furrows in the bottom plate have continuous path at angle - and the mating top plate has furrows at an angle 180° 9 -; thus, the ﬂuid moves in different directions in the ﬂow passages of mating plates. Because of the criss-crossing (three-dimensional, 3D) nature of corrugations of the mating plates, the secondary ﬂows induced are swirl ﬂows, which are generally superior in terms of an increase in heat transfer over friction. Hence, the relative performance of chevron plates is superior to all other corrugation patterns and thus it is now most commonly used heat transfer surface in plate heat exchangers. A much better understanding of the ﬂow patterns in chevron plates and subsequent enhancement is now available [7—9]. Flow visualization by Focke and Knibbe [10] and Hugonnot [9] in a larger-scale channel clearly shows recirculation areas downstream of the corrugation edges. These areas are large at low Reynolds numbers, but the transition to turbulent ﬂow (which occurs at Re $ 200) reduces the size of these areas. The recirculation area induces degradation of the kinetic energy of mean ﬂow and reduces heat transfer. Experimental information on local heat transfer coefﬁcient distribution has been obtained by Gaiser and Kottke [11]. They indicate that the pitch-to-hydraulic diameter ratio and the angle of corrugation have some inﬂuence. They observed that the heat transfer coefﬁcient distribution is more homogeneous at high corrugation angles, but there are still some weak areas. Local measurements by laser pulse and thermographic analysis in a two-dimensional (2D) channel [12] show poor heat transfer coefﬁcients downstream of the corrugation. It can also be seen that the local Nusselt number tends to be more homogeneous while increasing the Reynolds number. Be´reiziat et al. [13] have measured the wall shear rate and observed some similar recirculating areas in a 3D channel ( : 60°). These areas of low heat transfer coefﬁcient could be limited by a proper design of the corrugation shape. More recently, Stasiek et al. [14] have performed measurements of the local heat transfer coefﬁcients by applying a thermochromic method. Their results (corrugations angles & 30°) show that the heat transfer coefﬁcients are linked to the ﬂow pattern and to the mixing intensity in the channel with variations of <50% where measured.

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C. Unsteady Laminar versus Low Reynolds Number Turbulent Flow In the numerical analysis as well as in the heat exchanger design, it is essential to characterize whether the ﬂow is unsteady laminar or low Reynolds number turbulent ﬂow. In case of numerical analysis, the unsteady laminar ﬂow is much easier and accurate to analyze compared to the low Reynolds number turbulent ﬂow. For the latter case, the large eddy simulation model and the direct numerical simulation (refer to Section IV for turbulence models) could be used for analyzing the ﬂow. For heat exchanger design, the pressure drop associated with the unsteady laminar ﬂow is lower than that for the low Re turbulent ﬂow. As we understand today, the following is the characterization of these ﬂows. Unsteady laminar ﬂow may be seen in the wake of a bluff body, or a streamlined body inclined to a ﬂow, when the Reynolds number is sufﬁciently low. The unsteady motion is orderly and contains structures (vortices) that are generated continuously at a characteristic frequency and are of a size closely related to the width of the body projected normal to the ﬂow. In contrast, low Reynolds number turbulent ﬂow contains haphazard or chaotic motions in addition to the underlying steady (or unsteady) ﬂow. These chaotic motions encompass a range of length and time scales, the range being related to the Reynolds number, and so, unlike unsteady laminar ﬂow behind a bluff body, cannot be characterized by a single size and frequency. In DNS studies, we have access to time and space values; the statistical analysis of ﬂuctuations can provide information on the ﬂow structure. At the moment, these statistical calculations can only give information for a speciﬁc case, and the criteria found for the change in the regime cannot be extended to other geometries. But such analysis could be generalized to experiments and DNS calculations to ﬁnd objective criteria based on time and space ﬂuctuations. As far as we know, only a few authors have tried to apply these methods to numerical modeling, but for experimental work this has already been done for two-phase ﬂow structures. It is essential that sophisticated experimentation and careful numerical analysis be conducted to characterize what type of ﬂows occur at low Re in interrupted ﬁn geometries, wavy ﬁns/ channels, and chevron plates for its far-reaching impact on compact heat exchanger design. Classical theories of turbulence require very high Reynolds number as a prerequisite for the occurrence of the phenomenon. However, recent observations from compact heat exchanger studies (Jacobi and Shah [1] or Focke and Knibbe [10]) seem to suggest that the Reynolds number need not be very high to have turbulence. This observation, which we will refer to as low

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Reynolds number turbulence, certainly deserves more studies in the context of the classical notions of turbulence. This is particularly necessary since some of the standard turbulence modeling procedures are based on the assumption of an inﬁnitely high Reynolds number.

III. Numerical Analysis Numerical analysis of compact heat exchanger surfaces started about 20 years ago with signiﬁcant progress in this time frame. However, the problems analyzed numerically are simpler models of the real complex ﬂows within the CHE surfaces. As a result, many of the phenomena observed in CHE surfaces by ﬂow visualization and partial/full-scale testing have not yet been duplicated by the numerical analysis. In spite of this, CFD (computational ﬂuid dynamics) codes allow some modeling of 2D or 3D ﬂows for better understanding of the basic mechanisms of heat transfer and pressure drop in CHE surfaces. These methods can be used as guidelines for a parametric design study and the study of new geometries. The most common numerical approaches used for the analysis of CHE surfaces are the ﬁnite difference and ﬁnite volume methods, and only in some cases, ﬁnite element. The algorithm used for the solution of the partial differential equations is the pressure-based method because of the low Mach numbers in CHEs. Also, in most analyses, structured grid is used for the analysis. Unstructured, adaptive, and composite grids have been rarely used in analyzing compact heat exchanger surfaces. Refer to Heikal et al. [15] for the governing equations, solution algorithm, 2D and 3D models for numerical meshes, boundary conditions, and the determination of performance parameters (such as Nu, St, Re, h, f ) for multilouver ﬁn geometries. As mentioned earlier, the ﬂow and heat transfer performance of CHE surfaces is mainly dictated by the boundary layer behavior over the interruptions or in complex ﬂow passages, and ﬂow separation, recirculation, reattachment, and vortices in the wake region. Careful consideration must therefore be given to the grid used. Adequate grid reﬁnement is needed to capture the boundary layer growth and separation, and this is not always possible with moderate computing resources. One must also determine whether steady-state solutions are adequate or a time-dependent model is needed to capture the correct ﬂow behavior. The decision to perform 2D or 3D modeling must also be made and assessed against grid size requirement and speed of computation. However, the accurate prediction of local and overall Nu (or j ) and f factors for CHE surfaces will only be possible by analyzing time-dependent 3D ﬂows.

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A. Mesh Generation The numerical solution of the governing CFD equations in arbitrarily shaped regions requires the generation of numerical grids. A grid is a discrete representation of the continuous ﬁeld phenomena being modeled. It is the structure on which the numerical solution is built (Thompson et al. [16]) and should therefore accurately represent the geometrical conﬁguration of the domain and the physics of the problem. The mesh density and structure have a signiﬁcant inﬂuence on the accuracy and stability of the solution. The optimum mesh should be ﬁne enough to reduce the discretization error and resolve ﬂow and heat transfer details, especially in the areas of sharp gradients. It is important to keep the grid as orthogonal as possible, and to avoid cell aspect ratios signiﬁcantly larger or smaller than unity. The ﬁnite-difference CFD algorithms for complex geometries require grid generation techniques that transform a curvilinear nonuniform grid into a uniform rectangular one in the computational space (structured grids). The boundary conditions can be accurately represented, in this case, as some coordinate line (or surface in 3D) coincides with a boundary of the physical region (body-ﬁtted coordinates). Although the body-ﬁtted structured grids are widely used for both ﬁnite-difference and ﬁnite-volume algorithms, the meshing of the complex geometries found in most compact heat exchanger surfaces can consume considerable time and effort. Finite-volume methods enable the use of unstructured grids, which allow more meshing ﬂexibility. Also, Cartesian grids with boundary cells aligned to the surface are among current trends in grid generation (Anderson [17], Melton [18]). The grid generation strategy is determined according to the size and location of ﬂow features such as shear layers, separated regions, boundary layers, and mixing zones. For wall-bounded ﬂows, the grid size at the wall can affect the accuracy of the computed shear stress and heat transfer coefﬁcient. One must address the speciﬁc requirements of the wall functions used (see Section IV). For example, using a classical k— model, the grid point closer to the wall must be inside the buffer zone of the boundary layer. Because of the strong interaction of the mean ﬂow and turbulence, the numerical results for turbulent ﬂows tend to be more susceptible to grid dependency than those for laminar ﬂows.

B. Boundary Conditions The boundary conditions are very important for computational ﬂuid dynamic techniques as they govern the solutions. Usually, inlet conditions are uniform bulk velocity (based on the speciﬁed ﬂow rate) and temperature or ﬁxed velocity/temperature distribution, although time-dependent condi-

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tions are becoming more common. No-slip velocity conditions are used at wall as a ﬂow condition, whereas uniform temperature or heat ﬂux at wall is speciﬁed as a thermal one. For the outlet, a zero spatial derivative in a direction normal to the boundary is speciﬁed (Shaw [19]). As the pressure is obtained by the solution of the Navier-Stokes equation, a uniform arbitrary pressure is usually ﬁxed at the outlet of the computational domain. However, this condition is sometimes unsuitable when the reattachment point of a separated ﬂow is near the outlet or when an eddy structure exists through it. For these cases, special conditions are used: uniform streamwise pressure gradient (Mercier and Tochon [20]) or Sommerfeld radiative conditions (Orlanski [21]) in the outer part of the domain. For the lateral part, two conditions could be used: periodicity or symmetry (free slip condition). The former is based on a direct pressure coupling between the two lateral sides and is well suited for deviated ﬂows (louver ﬁns, for example). The latter is usually used for spatially developed ﬂows inside a symmetrical geometry (offset strip ﬁns, for example). To simulate fully developed ﬂows, a cyclic condition could be used. In this case, the velocity and temperature proﬁles at the outlet of the domain are placed at the inlet at each time step. With the use of (u/x) : 0 and v : 0 for the outﬂow condition, a longer wake region downstream of the surface is required to get reasonable results, especially for unsteady ﬂow. For example, to simulate ﬂow past a cylinder at Re : 300, the ﬂow becomes unsteady with vortices in the downstream region. If we set the length of the downstream wake region as smaller than 20D (D is the cylinder diameter), the simulation may diverge or the results for ﬂow performance (such as Strouhal number, ﬂow friction, or pressure drop) may differ from what they should be by using the preceding outﬂow condition. This is because the actual ﬂow cannot meet the condition of (u/x) : 0 and v : 0 at the boundary. Thus, if we use (u/x : 0 and v : 0) for the downstream boundary, the downstream wake region would be longer (for example, 30D) for more accurate results. If we use the boundary layer approximations for the outﬂow condition, the downstream wake region in the foregoing example can be reduced to lower than 20D for accurate results, thus reducing both the number of grid cells and the computation time. The 3D boundary layer momentum and energy equations for the outﬂow condition are

P u u (u) ; (u) ; (vu) ; (wu) : 9 ; ; x y z x y y z z t v v (v) ; (uv) ; (v) ; (wu) : ; t x y z y y z z

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w w (w) ; (uw) ; (vw) ; (w) : ; x y z y y z z t (c T ) ; (c uT ) ; (c vT ) ; (c wT ) N N N N t x y z :

T T ; . y y z z

(1)

These equations are solved in the last cell near the boundary, and hence the following terms are not present in the preceding equations (they are present when solving the boundary layer equations in the interior domain): u v w P P T : 0, : 0, : 0, : 0, : 0, : 0. x x y z x x

(2)

The pressure at the outﬂow boundary is assumed uniform and used to compute the pressure correction P at all interior grid points. Then the corrected P at the last node before the outﬂow boundary is used to solve Eq. (3), the ﬁnite difference form of the foregoing boundary layer equations, for the last cell near the boundary to get a better value of u at the outﬂow boundary node. This iteration between correction P and reﬁned value of u continues until the convergence, yielding the correct values of u and pressure ﬁeld. Kieda et al. [22] implemented the boundary layer approximation for the velocity components, and Xi [23] extended the concept by the computation of the pressure ﬁeld. The foregoing boundary conditions and the computational domain employed by Xi and Shah [90] are shown as an example in Fig. 8 for a 3D analysis of the offset strip ﬁn geometry. C. Solution Algorithm and Numerical Scheme The ﬁdelity of the results from computational ﬂuid dynamics techniques for turbulent ﬂows is largely determined by the solution algorithm and the numerical scheme. This is true for Reynolds averaged numerical simulation (RANS), large eddy simulation (LES), and direct numerical simulation (DNS). LES methods need to solve accurately motions over a wide range of scales (although not wide as for DNS) and require spatial and temporal discretization schemes that are at least second-order accurate. Generally, the time discretization schemes used are the Adams—Bashford, Runge—Kutta, or Leap Frog schemes. Mostly explicit schemes are used except viscous terms, which are treated implicitly when they require smaller time steps. For ﬂows of practical interest, mostly ﬁnite-volume methods are used, since

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Fig. 8. Boundary conditions and the computational domain for an offset strip ﬁn geometry analyzed by Xi and Shah [90].

ﬁnite-element applications need a higher cost per node in terms of computer memory and CPU time are requirements. Moreover, the sub-grid-scalestress (SGS) effect is essentially dissipative. So, mostly second-order central differencing schemes are used for the discretization of convection terms, since the numerical error is dispersive. Third-order (such as the Quick scheme described by Leonard [24]) or ﬁfth-order upwind differencing schemes are used too, but Moin [25] ﬁnds them too dissipative in ﬂow-pastcylinder calculations. The DNS calculations are usually based on the ﬁnite difference or spectral element scheme for the calculation of ﬂows in CHEs, since their geometry

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is very complex. In particular, the pseudospectral method, which is commonly used for the DNS of homogeneous ﬂows, cannot be used for CHE calculations because of the presence of solid walls. And, as for LES models, explicit schemes or part implicit schemes are used for the DNS methods.

IV. Turbulence Models In the design of compact heat exchangers, complex geometries are often used to promote high heat transfer rates. These geometries involve nonstraight channels or ducts as described in Section II earlier. The complex ﬂow phenomena in such systems have profound inﬂuence on the heat transfer and ﬂow friction. Indeed, even though the mean ﬂow is represented by a Reynolds number based on the mean velocity and hydraulic diameter, Lane and Loehrke [26] and Ota et al. [27] deﬁned a Reynolds number based on half the height of the surface roughness proﬁle to describe the separated ﬂow. According to these authors, when this speciﬁc Reynolds number is greater than 300, for discrete rib roughness, the ﬂow separates at the leading edge of the ﬁns in a laminar manner but reattaches in a turbulent way. This phenomenon creates coherent eddy structures, which increase the local heat transfer. Thus, because of the complex geometries of compact heat exchangers, even for ﬂows at low Reynolds numbers based on the hydraulic diameter (at so-called classical laminar or transitional ﬂows), some turbulent phenomena can appear. Although the ‘‘global turbulence’’ that is observed in a pipe ﬂow for Re 2300 is also found in many heat exchangers, the quasi-coherent structures of the type of von Ka´rma´n streets are more important in the CHE surfaces. The selection of an appropriate models to calculate low-Re turbulent ﬂow, transition ﬂow, and turbulent ﬂows is one of the key factors in obtaining reliable prediction of ﬂow friction and heat transfer in CHEs. For engineering applications, obtaining empirical data for heat transfer and ﬂuid ﬂow is quite cumbersome and costly; as a result, the scope of evaluating various geometries and operating parameters is limited. Therefore, the development of reliable computational techniques is required to evaluate heat transfer rates and pressure drops for the CHE surfaces. In this section, the most commonly used turbulence models/methods for computational ﬂuid dynamics analyses are described. Although not all turbulence models are commonly used in CHE analysis, it is imperative that we provide a comprehensive but concise review to challenge the CFD and CHE researchers to advance the CFD technology for CHE applications. The methods for calculating turbulence can be divided into the following three broad categories.

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( Reynolds averaged Navier—Stokes (RANS) models of turbulence such as the k— model or Reynolds stress closure model (RSM), which consists of the second moment turbulence modeling ( L arge eddy simulation (L ES) techniques ( Direct numerical simulation (DNS) techniques For more information on turbulence models, refer to Rodi [28], Halbaeck et al. [29], and Wilcox [30]. We introduce the equations governing the total (mean plus ﬂuctuating) ﬂow and the heat transfer, then describe the turbulence models/methods. A. Reynolds Averaged Navier—Stokes (RANS) Equations In this method [31], the instantaneous solution variables in the governing equations (Navier—Stokes equations, continuity, and energy) are decomposed into their mean and ﬂuctuating components. For an incompressible ﬂuid, the instantaneous velocity obeys the following equation (in Cartesian tensor form):

1 p u u u G; H . G; (u u ) : 9 ; (3) x G H x x x x t H H G G H The velocity components and scalar quantities such as pressure are decomposed as u : U ; u G G G p : P ; p,

(4) (5)

where U and P are the mean components and u and p the ﬂuctuating G G components. By substituting u and p of Eqs. (4) and (5) into Eq. (3) and G time (or ensemble) averaging, the mean velocity equations can be written as

DU 1 P U U 2 U G:9 G; H9 ( I ; ; (9u u ). (6) G H GH Dt x x x x 3 x x G H H H G I The mean continuity equation for an incompressible ﬂuid can be written as (U ) : 0. (7) G x G Equations (6) and (7) are called the Reynolds-averaged Navier—Stokes equations. They have the same form as the laminar Navier—Stokes equations with the velocities and other variables representing time-averaged (or ensemble-averaged) values. However, an additional term appears in Eq. (6), which represents the effect of turbulence and is called the Reynolds stress

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tensor: (9u u ). This term needs to be modeled in order to close the system G H of equations. Several approaches already exist for this purpose: (1) eddy viscosity models (EVMs), (2) algebraic stress models (ASMs), and (3) Reynolds stress transport models (RSMs). These are now brieﬂy described. All these approaches require a special treatment of turbulent ﬂows near the wall, and some of the models for wall effects are summarized in Subsection 4 of this section. 1. Eddy V iscosity Models (EV M) This is the most common way to model the Reynolds stresses. It is based on the Boussinesq hypothesis, which assumes that the Reynolds stresses are related to the mean velocity gradients by the empirical formula

2 U U G; H u u : k( 9 G H 3 GH R x x G H

where

(8)

0 for i " j ( : , GH 1 for i : j

(9)

and k is the turbulent kinetic energy. Equation (8) is valid for incompressible ﬂuid only; Eq. (7) is already incorporated in Eq. (8). In this approach, a turbulent viscosity v is introduced and needs to be determined. The R advantage of this method is the relatively low computational cost associated with the calculation of the turbulent viscosity using one of the following three methods. The ﬁrst method does not need an additional equation, whereas the other two need one and two additional equations, respectively. a. Zero-Equation Models In these models, no additional differential equations are needed to obtain the turbulent viscosity v , which is deﬁned as a function of the mean ﬂow. The Baldwin—L omax model [32] is one such model based on the Prandtl mixing-length model. In this model, two different expressions are given for the turbulent viscosity. ( For the inner zone (0 y y ), A : l where the mixing length l and the vorticity are given by

l : Ky 1 9 exp 9

y> A>

(11)

U U U U U U G9 H ; H9 I ; I9 G x x x x x x H G I H G I with A> : 26, and the von Ka´rma´n constant K : 0.42. :

(10)

(12)

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( For the outer region (y y (), A (13) : F((U ; U ; U) G H I where y is the distance normal to the wall, y the inner zone (viscous A sublayer) thickness, and ( the boundary layer thickness. More complex models for the outer region are also published. The main drawback of these models is that the mixing length is only computed inside the boundary layer of the ﬂow. It is therefore difﬁcult to generalize the model for the complex geometries found in CHEs. So, this model is rarely used for CHE applications. b. One-Equation Models In this method, v of Eq. (10) is given by an additional transport equation. Usually, it is the transport equation for the turbulent kinetic energy k. In this case, v : kl, and an ad hoc speciﬁca tion for l is still needed. Spalart and Allmaras [33] have developed an example of such models. This model was designed speciﬁcally for aerospace applications involving wall-bounded ﬂows and has been shown to give good results for boundary layers subjected to adverse pressure gradients. In its original form, the Spalart—Allmaras model is a low Reynolds number model, requiring the viscous-affected region of the boundary layer to be properly solved. When the mesh resolution is not sufﬁciently ﬁne, some commercial CFD software uses speciﬁc wall functions. However, one-equation models are often criticized for their inability to accommodate rapid changes in the length scale, such as might be necessary when the ﬂow changes abruptly from a wall-bounded to a free shear ﬂow, a phenomenon that is encountered frequently in CHEs. For this reason, this model is rarely used in CHE applications. c. Two-Equations Models In these models, two separate transport equations determine independently the turbulent velocity and length scales. These models are usually implemented as k—, k—, or k—l. In the k— models, the two transport equations are for the turbulent kinetic energy k and its dissipation rate . The k— models are based on the Boussinesq hypothesis and assume that the turbulence is isotropic. As a result, they are expected to perform poorly in curved geometries and ﬂows with directional inﬂuence. For example, they cannot calculate the ﬂows shown later in Figs. 15 and 16 because of the rotational body forces. The ‘‘standard’’ k— model is used for practical engineering ﬂow calculations as proposed by Launder and Spalding [34]. This is an economic and numerically robust model, which gives reasonably accurate results for a wide range of turbulent ﬂows that do not involve too much rotational ﬂow. Nevertheless, it is commonly

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used in complex ﬂows of industrial applications and heat transfer simulations that have rotational ﬂows. It is a semiempirical model whose strengths and weaknesses have become known [35]. Several improvements have been made to obtain better performance with this model. T he Standard k— Model. The standard k— model proposed by Launder and Spalding [34] determines the turbulent kinetic energy k and its dissipation rate from the following transport equations: k Dk : ; ;G ;G 9 (14) x Dt x G G D : ; ;C (G ; C G ) 9 C . (15) C C C Dt x x k k G C G In these equations, G represents the generation of turbulent kinetic energy by the mean velocity gradients, G is the generation of turbulent kinetic energy by buoyancy, C , C , C are constants, and , are the turbulent C C C I C Prandtl numbers for k and , respectively. The turbulent viscosity is given by

k :C R

(16)

where C is a constant often equal to 0.09 in practical applications. The standard k— model is commonly used for analyzing ﬂow inside most CHE geometries; for example, corrugated wavy channels (Hugonnot [9] and Ergin et al. [130]), louver ﬁns (Achaichia et al. [109]), offset strip ﬁns (Michallon [87]), or chevron trough plates (Fodemsky [151]), Ciofalo et al. [152], or Hessami [153]). T he RNG k— Model. The RNG-based k— turbulence model is derived from the instantaneous Navier—Stokes equations using a rigorous statistical technique called renormalization group theory. The model equations are

Dk k : ;G ;G 9 Dt x x G G D R : ;C (G ; C G ) 9 C 9 R. C k C C k Dt x x G C G Here the term R is given by

where is given by

C (1 9 / ) R: , (1 ; - ) k :

k

U U U G; H G . x x x H G H

(17) (18)

(19)

(20)

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The quantities - and are constants having values as 0.012 and 4.38, respectively. For the RNG k— model, the eddy viscosity expression is

: 1;

C k .

(21)

The RNG theory and its application to turbulence are described by Yakhot and Orszag [37]. The scale elimination procedure in the RNG theory results in a differential equation for turbulent viscosity, which is integrated to obtain an accurate description of how the effective turbulent transport varies with the effective Reynolds number and near-wall ﬂows. In the high Reynolds number limit, the expression of turbulent viscosity is the same as in the standard k— model. The RNG model is similar in form to the standard k— model, but includes the following improvements: ( The RNG model has an additional model term R in the equation that signiﬁcantly improves the accuracy for rapidly strained ﬂows ( The effect of swirl on turbulence is included in the RNG model, enhancing accuracy for swirl ﬂows ( The RNG theory provides an analytical formula for turbulent Prandtl numbers, whereas the standard k— model uses constant values ( Whereas the standard k— model is a high Reynolds number model, the RNG theory provides an analytically derived differential formula for effective viscosity that takes into account low Reynolds number effects Thus, the RNG model is more accurate and reliable for a wider class of ﬂows than the standard k— model. It is well suited for corrugated ﬁn surfaces where the hydraulic diameters are small and the Reynolds numbers are low. Because this kind of modeling is relatively recent, few computations on CHE geometries have been performed. Sunde´n [157] used this model for chevron trough plates geometry and obtained more accurate results than those with the standard k— model. However, the accuracy for predicting the turbulent ﬂows using the RNG model is reported as poorer than that for two other k— models for vortex shedding behind the bluff objects, such as square rods and circular tubes in a heat exchanger. In these cases, the separated ﬂows are not well predicted as in chevron or corrugated plate geometry. Such vortex shedding has low frequency modulations. Saha et al. [38] compared three turbulence models to capture the essence of time-averaged ﬂow quantities in a vortex shedding

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dominated ﬂow ﬁeld through the turbulence models in two dimensions. They used the Launder and Spalding [34] standard k— model, the Kato—Launder k— model [39], and the RNG k— model of Yakhot et al. [40]. In terms of the parameters such as the Strouhal number and lift and drag coefﬁcients, the predictions due to the Kato—Launder and the standard k— models were close to each other, and reasonably close to experiments of Lyn et al. [41]. However, the predictions due to RNG k— models were not close to the experimental values. A detailed comparison of velocity proﬁles revealed the Kato—Launder model to have the closest agreement with the experiments. A comparison between the computations and the experiment were also made for the time averaged kinetic energy variation along the centerline of the domain of interest. The Kato—Launder model predicted the peak value of turbulent kinetic energy in good agreement with the experiments. The peak value of the turbulent kinetic energy due to the RNG k— model showed a signiﬁcant departure from the experimental value. Thus, the comparison of these turbulence models indicates that the accuracy of the models may depend upon the geometry investigated; a more thorough investigation is needed for establishing the utility of speciﬁc models for speciﬁc geometries. T he Realizable k— Model. The realizable k— model (Shih et al. [42]) is a recent development that satisﬁes certain mathematical constraints on the normal stress consistent with the physics of turbulent ﬂows contrary to the standard and RNG k— models. In practice, the principle differences between the realizable model and the other k— model are that the former contains a different formulation for the turbulent viscosity and that another transport equation has been used for the dissipation rate. This equation has been derived from an exact equation for the transport of the mean-square vorticity ﬂuctuation. The transport equations for the turbulent kinetic energy k and for the dissipation rate are

Dk : Dt x G

D : Dt x G

k ; R ;G ;G 9 x G I

(22)

; R ;C C G 9C ; C S (23) C C C x k k ; ( C G

where S is a scalar value of the strain tensor. The turbulent viscosity v is calculated from Eq. (16), but C is no longer a constant. It is given by J C :

1 A ;A

U*k

(24)

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where A : 4.04, and A and U* [U* deﬁned in Eq. (29)] are functions of both the mean strain and rotation rates, the angular velocity of the system rotation, and the turbulence ﬁeld (k and ) [42]. This model is more accurate for predicting in the spreading rate of planar or round jets than the standard k— model. It is likely to provide superior performance for ﬂows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation. At present, this model still requires validation for industrial applications and remains a ﬁeld of research. T he k — Model for L ow Reynolds Number. Hwang and Lin [43] F F proposed an improved low Reynolds number k — turbulence model to F F describe thermal ﬁeld. By adopting the Boussinesq approximation, the turbulent heat ﬂux is approximated as: T

9 u T : H x H

(25)

where is the thermal diffusivity and T is the mean temperature.

In the standard k— model, is adopted to be proportional to the ratio of the turbulent viscosity and the Prandtl number. Most calculations of CHE have used constant turbulent Prandtl number Pr ; it might be more appropriate to use a variable one for CHE. See Kays and Crawford [44] for variable Pr . In the k — model, the thermal diffusivity is expressed as a F F function of the velocity scale and of the thermal and mechanical time scales to take into account the variations of the turbulent Prandtl number and the fact that the thermal diffusivity is not necessarily related to the eddy diffusivity. Moreover, to give a correct asymptotic behavior in the vicinity of the wall, the dissipation rate is decomposed into two parts in this model (Jones and Launder [36]): : ;

(26)

Here is the dependent variable in the dissipation rate equation (23) and : 2((k/x ). With this decomposition, reaches zero at the wall and H equals for y> 15. This model has been validated on experimental and DNS data for duct ﬂows. For CHE applications, the preceding linear eddy viscosity model by Launder and Sharma [45] simpliﬁes the wall boundary condition of the dissipation equation. However, one of its main limitations is that it gives far too large near-wall length scales in impinging or recirculation ﬂows. As a remedy to this, Yap [46] introduced an extra source term into the dissi-

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pation equation. With Yap’s correction, near-wall turbulent length scale can be reduced in a separated ﬂow, particularly near the ﬂow reattachment point around where the maximum heat transfer occurs. Some nonlinear eddy viscosity (k—) models have been developed to essentially capture the nonisotropic behavior of the ﬂows as encountered in shear ﬂow, recirculation ﬂow, or swirling ﬂow [47]. For CHE applications, low Reynolds number models such as the k — F F model have been used for internal ﬂows inside chevron trough plates (Ciofalo et al. [152], Hessami [153], or Sunde´n [157]) and corrugated wavy channels (Yang et al. [126] and Ergin et al. [132]). However, it still requires validation for shear ﬂows, which are encountered in offset strip ﬁn and louver ﬁn geometries. Other Common Eddy V iscosity Models. Many other models, based on the eddy viscosity concept, exist such as the k— model and the shear stress transport (SST) model. For more details, refer to Wilcox [30] or Menter [48].

2. Algebraic Stress Models (ASM) or Nonlinear Eddy V iscosity Models (NL EV Ms) The ASM or nonlinear eddy viscosity model (NLEVM) [49—51] is an intermediary model between the EVM and the RSM. In this model, Reynolds stresses are represented as a tensor polynomial expansion in terms of the mean strain rate and rotation rate tensors. The expansion coefﬁcients are determined from the simpliﬁed differential Reynolds stress transport equation. The ASM is less sensitive to rotation than the EVM and need less computational cost than the RSM, but this kind of model is still not commonly used in industrial applications because they require too many empirical parameters, which have to be adjusted for each application.

3. Reynolds Stress Models (RSM) In this method, the Reynolds stress is determined by solving the differential transport equations for each components of Reynolds stresses (Launder et al. [52]; Gibson and Launder [53] ; Launder [54]). Taking moments from the exact momentum equations may derive the exact form of the Reynolds stress transport equation. The momentum equations are multiplied by a ﬂuctuating property and then are time-averaged. But the Reynolds stress transport equation contains several unknown terms that need to be modeled

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in order to close the equations: (u u ) ; (U u u ) :9 I G H x x t G H I I (1)

;

(2)

x I

(3)

U U H ; u u G (u u ) 9 u u G H G I x H I x x I I I (4)

P 9-(g u 1! ; g u 1!) ; G H H G (6) 92

P u u u ; (( u ; ( u ) G H I IH G GI H

(5)

u u G; H x x G H

(7)

u u H 9 2 (u u ; u u ) G G K HIK . I G K GIK x x I I (9)

(27)

(8)

Here, - the volumetric expansion coefﬁcient, 1! is the ﬂuctuating ﬂuid temperature, and the rotation vector. In the preceding equation, the following terms do not require any models: (1) local time derivative; (2) convection; (4) molecular diffusion; (5) stress production; and (9) production by system rotation. However, in order to close the equation set, the following terms need to be modeled: (3) turbulent diffusion due to triple correlations and pressure ﬂuctuations; (6) buoyant production; (7) pressure strain; and (8) dissipation. Since the RSM takes into account the effects of streamline curvature, swirl, rotation, and rapid changes in the strain rate in a more rigorous manner than other RANS models, it supposedly gives more accurate results for complex ﬂows. But the RSM predictions are still limited by the closure assumptions used for various terms of Eq. (27), especially the pressure-strain and the dissipation-rate terms, designated as (7) and (8). The RSM does not always yield results superior to the simpler models in all classes of ﬂows. Compared with the k— models, the RSM requires additional memory and CPU time because of a large number of transport equations computed. Furthermore the RSM could need more iterations than k— models because of the strong coupling between the Reynolds stresses and the mean ﬂow. The use of the RSM is interesting when the ﬂow features studied are the results of strong anisotropy in the Reynolds stresses (cyclone ﬂows, rotating ﬂows, etc.). At present, these models are still not used for industrial

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applications and remain a ﬁeld of research. Indeed, they are based on too many adjustable parameters (unknown quantities) that could be determined for simple geometries but that are not available for complex ones. Also, it lacks the generality of the model assumptions, which researchers have tried to overcome by including higher-order terms in the model equations. 4. Models for Wall Effects Turbulent ﬂows are signiﬁcantly affected by the presence of walls as the mean velocity ﬁeld is affected through the no-slip condition at the wall. Numerous experiments have shown that the near-wall region can be largely subdivided into three layers: ( The viscous sublayer, where the ﬂow has laminar properties and the viscosity has a dominant role in the momentum and heat transfer ( The buffer region, where the effects of molecular viscosity and turbulence are equally important ( The fully turbulent layer, where the effects of turbulence are dominant There are two standard methods to take into account wall effects in numerical simulations: wall-function modeling and the use of low Reynolds number turbulence models. a. Wall-Function Models Wall functions are a collection of semiempirical formulas and functions that link the solution variables at the near-wall cells and the corresponding parameters on the wall. They are composed of laws of the wall for mean velocity and temperature, and formulas for near-wall turbulent quantities. For industrial ﬂows, Launder and Spalding [35] wall functions can be used. Therefore, the law of the wall for the mean velocity is U* :

1 ln (Ey*), K

(28)

where UCk yC k , y* : , K : 0.42, E : 9.81, (29) / U and y is the normal distance from the wall. The value of C is nearly a constant, often equal to 0.09 in practical applications. Nevertheless, for the RNG model, Eq. (24) gives the correct value. This logarithmic law for the mean velocity is known to be valid for y* 30—60. For lower y* values, one can apply the laminar stress—strain relationship that can be written as U* : y*. U* :

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Similar to this law of the wall for the mean velocity, the law of the wall for temperature can comprise two equations: a linear law for the thermal conduction sublayer where conduction is important, and a logarithmic law for the turbulent region where effects of turbulence dominate conduction. In high Reynolds number ﬂows, the wall function approach substantially saves computational resources, as the viscosity affected near-wall region does not need to be solved. The wall-function approach is economical, robust, and reasonably accurate. It is a practical option for the near-wall treatments for industrial ﬂow simulations. Some variations of the wall functions based on the same concept are used in all CFD codes. b. Turbulence Models for Low Reynolds Number Flows When low Reynolds number effects are important in the ﬂow domain, the hypothesis underlying the wall functions cease to be valid. Therefore, models such as the two-layer model (Iacovides and Launder [55], Rodi [56]) can be used. Similar to Rodi’s model [56], Chen and Patel’s model [57] resolves the near wall region by the transport equation for k only while the energy dissipation and the eddy viscosity are prescribed in an algebraic manner. In these models, the k— model is combined with one equation model near the wall so that the dissipation rate and the turbulence viscosity near the wall are calculated with the prescribed length scales l and l as T C : C (kl , R

:

k , l C

(30)

where the length scales l and l contain the damping effects in the near wall C region and are calculated from K l : y[1 9 exp(90.236y*)] C C

(31)

K y[1 9 exp(90.016y*)], l : C

(32)

where y* : y(k/v is the dimensionless distance [a deﬁnition different from that in Eq. (29)] and y is the normal distance from the wall. C : 0.09 and K : 0.42, respectively. The Chen and Patel model [57] is observed to be robust from the point of view of numerical stability and also capable of predicting separated ﬂows and ﬂows over rough surfaces. Another commonly used low Reynolds number turbulence model is by Lam and Bremhorst [58].

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5. Assessment of RANS Models ( Among the RANS models, the k— model is the standard model used for practical engineering ﬂow calculations as it gives reasonably accurate results for a wide range of turbulent ﬂows without demanding excessive CPU time and memory. ( Other eddy viscosity models do not provide accurate enough results for turbulent ﬂows, even though the Spalart—Allmaras [33] model or the k— model [30, 48] yields good results for boundary layers subjected to adverse pressure gradients and is well adapted for aerospace applications. ( Moreover, comparing with the k— models, the RSM requires signiﬁcant additional memory and CPU time. The RSM does not always give results superior to those of the k— models because of the large number of adjustable parameters (unknown quantities). The use of RSM is interesting when the ﬂow features studied are the results of strong anisotropy in the Reynolds stresses (cyclone ﬂows, rotating ﬂows, etc.). ASMs are less sensitive to rotation than EVMs and need less computational cost than the RSM, but this kind of model is still not commonly used for industrial applications. Again the reason is that there are too many parameters that need to be adjusted for each application. B. Large Eddy Simulation (LES) The main idea of this model is to compute the large-scale turbulence and to model the smaller scales. Here lies a profound similarity between RANS and LES models. The only difference between them lies in the deﬁnition of a small scale. In RANS models, the effect of all eddies is simulated by the turbulence model, whereas in LES models, the large scales are simulated and the scales smaller than the grid size or the ﬁlter width are modeled [59]. All large-scale structures in both RANS and LES models are determined by solving the governing equations. Numerical models are applied to subscale structures. As a result, a set of ﬁltered equations with subscale correlations is obtained. The subscale structures have relatively low energy and their structure is expected to be rather universal. In the case when the grid size is less than the Kolmogorov scale (:lRe\, where the Reynolds number J is based on the mixing length l ), the ﬂuid transport properties are controlled exclusively by molecular processes, the need for modeling of subscale processes disappears and LES models turn into DNS models, when sufﬁciently small time steps are used. The major interest in the LES models is the cases when RANS models perform poorly but DNS models are

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prohibitively computer intensive. For such ﬂows, which are encountered in CHEs, the k— models plus wall functions cannot provide accurate predictions of heat transfer to meet the CHE designer’s requirements. In addition, the more advanced nonlinear second moment models and near-wall sublayer models usually result in unsatisfactory numerical stability in many engineering applications. In some CHE applications where turbulence mixing can be an important factor inﬂuencing the performance, LES can be used, as it provides not only mean ﬂow mixing characteristics but also subscale mixing, which can be very valuable. However, LES suffers from its high computational cost. The choice therefore between RANS (EVM and ASM) and LES is really dependent on the balance between accuracy and computational cost (both memory and speed requirements). At present, RANS is the only engineering tool for design of industrial CHEs. However, from the point of view that higher and higher accuracy will be required for CHE modeling and more and more powerful computers will be available to users in the near future, it might be wise for researchers to investigate the suitability of these methods for the prediction of the performance of heat exchanger surfaces. As an example, Ciofalo et al. [152] used the LES method for a chevron trough plate geometry and obtained the best prediction of friction and heat transfer coefﬁcients compared with other classical models. However, this kind of approach is still not widespread. LES methods have also been used with success for the modeling of turbulent ﬂows in complex geometries (Rodi et al. [60], Moin [61]). LES requires some averaging of the variables (e.g., velocity) to obtain the solved quantities. For this, various approaches are used and summarized next. a. Schumann’s Approach [62] In this approach, instantaneous quantities (velocities, temperature, etc.) are averaged over a control volume deﬁned by the numerical grid leading to a piecewise constant distribution of the velocity. This method is implicitly used in ﬁnite volume methods, but the averaged velocity results from the discretization and changes with it. b. Filter Approach (Leonard [63]) This approach applies a low-pass ﬁlter (top-hat ﬁlter or Gaussian ﬁlter) to the solved quantities. It leads to an averaged velocity that is now a continuous function of the position and is independent of the physical scale separation and discretization. However, in practice, the ﬁlter does not appear explicitly in many LES codes. In fact, it is implicit since scales smaller than the grids are automatically disregarded. So, it is recommended to use a ﬁlter width larger than the mesh size in order to remove the link between the subgrid scale length and the grid size because the solutions become grid-independent.

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The ﬁltered averaged equations for constant ﬂuid density are u G:0 x G p! u G; (u u ) : ; (2S 9 ! ), GH GH x G H x x t H G H

(33) (34)

where

u 1 u G; H . (35) S : GH 2 x x H G The subgrid scale stress ! : u u 9 u! u! represents the effect of unresolved GH G H G H (subgrid scale) motion on the resolved one and is modeled using subgrid scale (SGS) models. The task is to determine the subgrid-scale stress and hence to simulate the effect of the unresolved motion on the resolved motion. This effect is mainly dissipative, i.e., energy transfer is globally from large to small scales, but locally and instantaneously transfer could be in the other direction (backscatter). Eddy V iscosity Models. Most SGS models presently use the eddyviscosity (Boussinesq) concept: 1 9 ( : 92 S . GH GH 3 GH II

(36)

The task of SGS is now to determine turbulent viscosity . From a R dimensional analysis, . lq , (37) R where l is the length scale of unresolved motion (and not the mixing length here), and q the velocity scale of unresolved motion. Most active unresol ved scales are those closest to the cutoff point, so the natural l in LES is usually the grid size. For determining q , there are various approaches, similar to the RANS modeling, as follows. Smagorinsky Model (Smagorinsky [64], Lilly [65]). Similar to the Prandtl mixing model, the Smagorinsky model is related to the gradient of the space average velocity: (38) q : (S S . GH GH More recently, the subgrid viscosity is directly related to the strain tensor [59] as : C S , GH

(39)

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where C is the Smagorinsky constant (C is often taken as 0.1 in practical applications) and is the ﬁlter width or subgrid scale. Renormalization Group (Yakhot et al. [66]). The renormalization group theory can be used to derive a model for the subgrid-scale eddy viscosity, which results in an effective subgrid viscosity given by

(40)

x x0 . 0 x0

(41)

0.12! 9C () : 1 ; H (2')

where ! $ 2 ()S , GH

H(x) :

Here, H is the Heaviside function with x as a dummy variable, is the molecular viscosity, ! is the mean dissipation rate, and C is a constant taken equal to 73.5. For a high Reynolds number ﬂow, the RNG-based subgrid scale model reduces to the Smagorinsky—Lilly model. But in the low Reynolds number ﬂow region, the effective viscosity is equal to the molecular viscosity, allowing the RNG model to better predict ﬂow in near-wall regions. Dynamic Procedure (Germano et al. [67], Ferziger [68]). The basic idea of this procedure is to use information available from the smallest scales to determine the coefﬁcient in the SGS model. So, a test ﬁlter wider than the basic LES ﬁlter ( : 2) is introduced and the SGS model is applied to scales between and . The dynamic procedure and the Smagorinsky model are the most widely used subgrid-scale models. Structure Function Model (Me´ tais and Lesieur [69]). The eddy viscosity is computed according to a structure function (42) : 0.063(F () F (r) : [u (x ; r) ; u (x)]. (43) G G This structure function model has been validated for simple geometries (backward-facing step, for example; Fallon [70]), but still requires work for complex ﬂow geometry, especially with thermal effects. C. Direct Numerical Simulation The direct numerical simulation is the simplest way to simulate turbulence because no critical assumptions and no closure equations are needed (Moin and Mahesh [71]). The classical Navier—Stokes, continuity, and heat balance equations are solved directly using a high-order convective scheme. However, this model requires a very ﬁne mesh because the ﬁnest mesh must

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be ﬁner than the smallest scale of turbulence (i.e., Kolmogorov scale : lRe\, where the Reynolds number is based on the mixing length l ). J But the ratio of the smallest scale and the largest scale is a function of the Reynolds number Re\, so the number of grid points in any direction is J Re. In practice, a grid scale on the order of ﬁve times the Kolmogorov J scale is usually sufﬁcient (except in the near-wall turbulence). Xi et al. [72], Mercier and Tochon [20] and Kouidry [73] have used DNS and unsteady models to predict ﬂow ﬁelds for the OSF geometry and corrugated channels, but only for a 2D approach. So the DNS studies are limited by the computational means. For instance, let us review the landmarks of DNS research: 1. The ﬁrst DNS study, by Orszag and Patterson [74] on an isotropic decaying turbulence, used a 32 mesh (Re based on Taylor macroscale equal to 35) 2. Kim et al. [75] studied a turbulent channel ﬂow with a 192;129;160 mesh (Re : 3300) 3. Spalart [76] studied a turbulent boundary layer at Re : 1410 with a 432;80;320 mesh So presently, the main applications of DNS are (1) to provide reliable data for the validation of turbulence models, especially useful where experimental accuracy is low; (2) to provide data for evaluation of subgrid models for LES (i.e., dynamic models); and (3) to conduct some fundamental studies of turbulence. Even modern supercomputers have limited capability for analyzing complex ﬂows in the CHE surfaces. Classical industrial applications also require too much computer capability at present; hence, only some conﬁned geometries have been simulated with the DNS model, and also at relatively low Reynolds numbers: McNab et al. [137] and Tochon and Mercier [139] for corrugated wavy channels, Blomerius et al. [156] for chevron trough plates, and Mercier and Tochon [20] for offset strip-ﬁn geometries. 1. Assessment of Simulation Models Although the LES and DNS techniques have demonstrated several advantages over the RANS approach, these simulation techniques require excessive CPU time and memory for computations of 3D complex ﬂow problems in the CHEs and other industrial problems, particularly at large Reynolds numbers. Therefore, these methods are limited to a large-scale turbulence with relatively low Reynolds numbers such as transition ﬂows in channels/ducts, ﬂow over a bluff body, or CHE surfaces at low Reynolds

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numbers. Because computer capabilities increase 10 times or more every 5 years, the simulation models represent a promising numerical tool for the future. D. Concluding Remarks on Turbulence Modeling For the past 15 years, RANS models (k—) have been used for modeling unsteady laminar, transitional, and turbulent ﬂows in compact heat exchangers. But the poor description of anisotropic ﬂows requires the use of more advanced models (LES and DNS). Most of the work on the development of new turbulence models has been based on simple geometries, such as ﬂat plates and isolated backward-facing steps, rather than the complex geometries encountered in CHE systems. For CHE surfaces, anisotropy, shear ﬂows, and rotational and other body force effects exist that cannot be satisfactorily analyzed by most of the existing turbulence models. Although the foregoing review shows promising results from these models, it is clearly the case that the CHE technology calls for more accurate and dependable models. The focus of such efforts should center around models that can handle the following: ( The multiple and interacting shear layers, separating and reattachment surfaces ( The correct near-wall behavior (ﬂow and heat transfer) in complex geometry situations The LES procedure has received attention as a potential solution to these kinds of difﬁculties, and some progress has been made. However, robust near-wall treatments as well as easy-to-implement averaging procedures for the inhomogeneous turbulence features of CHE systems are current challenges. Until these difﬁculties with LES are resolved, coupled with the inappropriateness of DNS for realistic CHE systems, it would seem that RANS (with problem-speciﬁc parametric optimization) will remain the CFD tool for design and optimization of compact heat exchanger surfaces. RANS models have already provided in-depth knowledge of local phenomena on a relative basis leading to improvements in the design of CHE surfaces.

V. Numerical Results of the CHE Surfaces In this section, the numerical analysis of some important compact heat exchanger surfaces is reviewed. Included are offset strip-ﬁn surfaces, louverﬁn surfaces, wavy-ﬁn surfaces and channels, and chevron plates.

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A. Offset Strip Fins Extensive numerical analysis of offset strip ﬁn geometry (shown in Figs. 1a and 8) has been conducted since 1977. The details of numerical models, operating conditions and geometries of offset strip ﬁns analyzed are provided in Table I. Numerical solutions for the offset strip-ﬁn (OSF) geometry was ﬁrst started with zero ﬁn thickness (( : 0) and an inﬁnite ﬁn height by Sparrow et al. [77]. They varied the nondimensional strip length l/s from 0.2 to 5, where l is the strip length and s is the transverse spacing of offset strip ﬁns. Because of the zero ﬁn thickness, the impingement region at the leading edge of a strip and the recirculating region behind the trailing edge were absent. Hence, the partial differential equations were parabolic and a marching procedure was used. Patankar and Prakash [78] extended the analysis to ﬁnite ﬁn thickness ( in terms of dimensionless (/s : 0.1, 0.2, and 0.3 for a ‘‘fully developed’’ periodic ﬂow for Reynolds number Re in the range 100 & Re & 2000. They found a small recirculating zone behind the trailing edge at low Re or low (/s. At high Re or high (/s, this zone extended from the trailing edge of a strip to the leading edge of the succeeding strip. This constricted the ﬂow to the minimum ﬂow area, thinned boundary layers, and resulted in high j and f factors. An increase in f was more pronounced with increasing (/s, as expected because of the higher form drag. The prediction for the f factor was in reasonable agreement with experimental data, but the predicted j factor was about 100% high, and the slopes of j and f vs Re data were steeper than those for experimental data. Kelkar and Patankar [79] extended their previous work [78] on offset strip ﬁns to investigate the effects of the ﬁn length and aspect ratio. The numerical simulations were performed assuming a zero ﬁn thickness for a 3D control volume. The grid used was 30;20;30. To characterize the geometry, they introduced a parameter -* deﬁned by -* : lRe/s. The main conclusions from their work are as follows: The number of cells necessary to obtain a fully developed ﬂow increases with -*; the 3D aspect of the ﬂow is negligible for low aspect ratios (&0.2); the ﬂow becomes more complex and three-dimensional for higher aspect ratios. They compared their numerical results with the empirical correlation of Wieting [80]. Their numerical model underestimated the friction factors by about 15% and overestimated the Nusselt numbers by 15% compared to those predicted by the Wieting correlation for the entire range of Reynolds numbers and geometrical parameters. The agreement appeared to be better for higher values of the ﬁn length parameter (-* 0.01). Suzuki et al. [81] obtained a numerical solution for combined free and forced convection in laminar ﬂow through staggered arrays of zero thickness

TABLE I Summary of Numerical Models, Operating Conditions, and Geometries of Offset Strip Fins Authors

Model

Grid

Boundary condition

399

Patankar andLaminar Prakash [78] Kelkar and Laminar Prakash [79] Suzuki et al. Laminar [81]

2D: 60;30

2D: 339;27

Constant wall temperature

Suzuki et al. Laminar [82]

2D: 339;31

Constant heat ﬂux

Xi et al. [83]

2D: 500;80

Constant heat ﬂux

Suzuki et al. Unsteady [86] laminar

2D: 380;130

Constant wall temperature

Xi et al. [72]

2D: 380;130

Constant wall temperature

Laminar

Unsteady laminar

Michallon Laminar [87] k— Mizuno et al. Laminar [88] Mercier and DNS Tochon [20] Xi and Shah Unsteady [90] Taminar

Constant heat ﬂux

3D: 30;20;30

Inlet condition Fully developed

Reynolds number, Re

Pr

Geometry

0.7

(/p : 0; 0.1; 0.2; 0.3 l/p : 1

Pressure drop Nusselt number

0.7

b/s : 0.1 to 1

Wieting [80] correlations

125—500

0.7

(:0 l/p : 1

200—1500

0.7

(/p : 0; 0.2; 0.4 l/p : 1

Local Nusselt number

250—1000

0.7

(/l : 0; 0.04; 0.08 l/p : 0.8 to 4

Local Nusselt number

800—5000

0.7

(/l : 0.0314; 0.126

860 and 3430

0.7

(/l : 00314; 0.126

Flow visualization, velocity etc.

300—5000

0.7

20—300

Pressure drop heat transfer Heat transfer

5000

0.7 and 7 7

(/l : 0.14 l/p : 2.2 (/p : 0.064 l/p : 2.61 (/p : 0.14 l/p : 2.2

100—6000

0.7

100—2000

Developing

2D: 40;38 Constant wall 3D: 30;39;10 temperature 3D: 36000 grid Constant heat point ﬂux 2D: 245;141 Constant wall temperature 3D: Constant wall 460;45;45 temperature

Uniform velocity and temperature Uniform velocity and temperature Uniform velocity and temperature Uniform velocity and temperature Uniform velocity and temperature Fully developed Developing Developing and fully developed Uniform velocity and temperature

(/p : 0.064; 0.1 l/p : 1.5; 1.95; 2.61

Validation

Local and overall (literature) Experimental j and f factors from literature

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offset strip ﬁns. Suzuki et al. [82] extended the analysis to ﬁnite thickness ﬁns and also to free-stream turbulence. To predict the turbulent ﬂow at a low Reynolds number, a turbulence model derived from the standard k— model implemented. A relatively good agreement was found between numerical and experimental values of Nu within the range of Reynolds number tested (Re & 800). In addition, experiments were conducted by varying the inlet turbulent intensity, and the effect was found to be very small. Furthermore, the effect of the ﬁn thickness was found to be rather small. The ﬁn length and spacing effects were extensively studied from the heat transfer point of view. Xi et al. [83] provided details on the numerical treatment of the cells adjoining the ﬁn surface and extended the previous study to a larger number of offset strips (up to nine). The effect of ﬁn thickness was studied in detail and appeared to depend on the ﬁn spacing and length ratio. The detailed explanation on heat transfer enhancement in a ﬁn array was also presented. An unsteady ﬂow ﬁeld in the OSF geometries results in heat transfer enhancement. Based on ﬂow visualization results, Mochizuki et al. [84] and Xi et al. [85] reported that the ﬂow in unsteady region has highly periodic velocity ﬂuctuations, and resulting ﬂow instability is strongly dependent on the Reynolds number and geometric parameters, such as ﬁn pitch and ﬁn thickness. In the transition ﬂow region, ﬂow patterns are different for different rows in OSF arrays. As outlined by Xi et al. [83] and Jacobi and Shah [1], ﬂow instabilities can appear in the wake of the ﬁn even at low Reynolds numbers. Suzuki et al. [86] and Xi et al. [72] studied unsteady ﬂow ﬁeld in an inline array of three ﬁns. The ﬂow was assumed uniform at the inlet. The computational grid was 380;130 and was selectively nonuniform: A ﬁner grid was applied near the ﬁn wall surface. To solve the governing equations, a central ﬁnite difference scheme was used for the diffusion terms. In the ﬁner mesh region, a second-order upwind scheme was used to solve the convective terms and a third-order scheme was used in the region far from the wall. The predicted ﬂow ﬁeld was compared with experimental data obtained by Xi et al. [72], and a good agreement was found for the mean streamwise velocity and the rms values of the ﬂuctuating velocities. A close analysis of the ﬂow pattern and heat transfer near the ﬁn wall surface has led to the conclusion that heat transfer enhancement is created by self-sustained ﬂow oscillations. Local instabilities near the wall, created by upstream vortices that impinge on the ﬁn, produce dissimilarity between the momentum transfer and heat transfer, thus reducing the local skin friction. These phenomena have already been observed for an array of cylinders. Unsteady ﬂow simulation is very promising as it gives local instantaneous information. However, to engender conﬁdence in the numerical results, a careful validation should be provided on instantaneous data, not just on the time averaged values.

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Michallon [87] used a standard CFD code (TRIO-VF, developed by the French Atomic Commission) to model an offset strip ﬁn channel. The fully developed ﬂow was obtained by reinjection of the outlet ﬂow conditions as the input in the iterative solution. The numerical simulations were performed for 300 & Re & 4000. For Re & 2000, a laminar model was applied, whereas for higher Reynolds numbers, the standard k— model was used. Two- and three-dimensional analyses were performed, and no signiﬁcant 3D effects were noticed. The reason for this may be related to their computational model, which is only a part of OSF, and boundary conditions in which they used a reinjection procedure. As mentioned earlier, ﬂow patterns are different for different rows in OSF arrays, especially when the ﬂow is unsteady. Therefore, the ﬁn height of the OSF channel, ﬁn thickness, and inlet ﬂow condition need to be taken into consideration for a more precise 3D numerical analysis. The comparison with experimental values obtained by Michallon [87] showed that the friction factors are overestimated by about 25%, and the Nusselt numbers are overpredicted signiﬁcantly, from 30% for low Reynolds numbers (Re & 1000) up to a factor of 2 for high Reynolds numbers (Re 3000). In terms of local interpretation, a recirculation zone was observed at the trailing edge of the ﬁn, but the size of this zone was limited. This result is in agreement with the work of Patankar and Prakash [78]. Mizuno et al. [88] numerically investigated three-dimensional offset strip ﬁns in the laminar low Reynolds number regime (Re & 300). The thermal boundary layer developed on the ﬁn surfaces and the primary surface (parting sheets) cannot be considered to be of the same thickness. On the parting sheet, the thermal boundary layer is thicker and therefore affects heat transfer. To prove this phenomenon, a three-dimensional analysis taking into account conduction in the ﬁns was undertaken [88]. A conventional control volume method was applied, assuming a zero ﬁn thickness. For the ﬂow boundary conditions, a uniform velocity was assumed at the inlet, with a zero velocity gradient at the outlet. The thermal boundary condition was constant wall temperature on the separation plates. In parallel with the numerical work, experimental measurements were performed. The numerical results underestimate the pressure drop by 20% at a Reynolds number of 20 but are in agreement at higher Reynolds numbers (Re : 300). The average Nusselt number is well predicted at Re : 100; below this value, it is overpredicted; and above this value, it is underpredicted. The maximum deviation is around 10%. A parametric study reports on the effects of the thermal conductivity of the ﬂuid and of the ﬁn material, and outlines the fact that for a high value of the ﬂuid thermal conductivity (water), the heat transfer performance is affected by the ﬁn material; i.e., aluminum ﬁns are better than stainless steel ﬁns. The other results of this study concern the thermal boundary layer on the parting sheets; its

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thickness is comparable to the one of a plain rectangular channel. Mercier and Tochon [20] have performed a 2D time-dependent analysis of turbulent ﬂow in an offset-strip ﬁn heat exchanger. The turbulent ﬂow behavior is solved by using both an accurate convective scheme (third order) and a ﬁne grid. The mesh size must be smaller than the smallest turbulent scale, and the computational domain must be greater than the largest scale involved. Due to limited computer capabilities, not all the turbulent scales could be solved, and the method is referred to as pseudo-direct numerical simulations. Two cases considered by Mercier and Tochon [20] are a single ﬁn with a uniform ﬂow upstream, and an array of offset strip ﬁns under developing and fully developed ﬂows. On a single ﬁn, the time-dependent evolution gives fundamental information on the ﬂow structure (see Fig. 4). The ﬂow, hitting the front edge, separates and creates a recirculation zone. At the reattachment point, which oscillates, one part of the ﬂow is convected downward and the other part of the ﬂow upward in the recirculation zone. At the trailing edge, vortices are convected, and a von Ka´rma´n street is formed. These phenomena are in agreement with visual observation performed on similar geometries. When the size of the recirculation zone is compared with data from the literature, good agreement is found. Concerning the array of ﬁns, two basic inlet ﬂow conditions have been studied: developing ﬂow and fully developed (reinjection). For developing ﬂows, the vortices created by the ﬁrst row of ﬁns impinge on the downward ﬁns and suppress all organized structure (highly turbulent ﬂow). The comparison of the calculated time average friction factors and Nusselt numbers with measured values show poor agreement for the developing ﬂow. Applying a reinjection procedure (fully developed ﬂow) gave relatively good agreement with the literature data. Despite a 2D approach, the results appear qualitatively correct, and local and overall information on ﬂow structure and thermal and hydraulic performances can be obtained. For a single ﬁn, the numerical velocity ﬁeld was compared with the data of Ota and Itasaka [89], and the agreement is qualitatively correct with a mean deviation of ;20%. For the array of ﬁns, the predicted values of the friction and Colburn factors are compared to several correlations and results from the open literature. The predicted friction factors are underestimated by 5 to 30%, and the Colburn factors are underestimated by 30 to 50%. Xi and Shah [90] conducted a 3D numerical analysis of OSF geometries (Fig. 8) that differed from that of Michallon [87] and Mizuno et al. [88] in the following aspects: (1) time-dependent, i.e., unsteady laminar ﬂow; (2) ﬁnite ﬁn thickness; and (3) computational domain having upstream ﬂow region (2l), OSF region (9l), and downstream wake region (10l) so that the effects of form drag, velocity defect, and temperature excess in the wake are properly taken into account. In order to suppress an overshoot of the

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solutions and reduce the magnitude of numerical viscosity, a mixed usage of two upwind schemes for convection terms was applied for the momentum equations (third order) and energy equation (second order). The ﬁnite difference equations of fully implicit forms were solved step by step along the time axis with the evaluation of pressure by the SIMPLE algorithm and the alternating direction implicit (ADI) method for each relaxation. In order to optimize the computational time for minimizing the maximum and total residual errors of the computational domain, the following scheme was adopted: (a) The computation was ﬁrst carried for 15,000 time steps. In each time step, one iteration was done. (b) Then, the computation was carried for another 2000 time steps in which ﬁve iterations were done. (c) Finally, the computation was carried out for additional 2000 time steps (with ﬁve iterations per time step) to get the mean values for the results presented here. The computed results of the ﬂow and thermal ﬁelds are little affected by what pattern of spatial distributions used as the initial condition. Refer to Xi and Shah [90] for further numerical details. Three comparisons were made by Xi and Shah [90]: numerical results of Mizuno et al. [88], experimental results of an idealized OSF (Mochizuki et al. [84]), and a real OSF (London and Shah [91]). The 3D numerical results obtained by Xi and Shah [90] are in better agreement with the experimental data of Mizuno et al. than are the the latter group’s own 3D numerical data. The primary reasons are that Xi and Shah considered a ﬁnite (actual) ﬁn thickness of the OSF geometry, considered unsteady laminar ﬂow in the analysis, and employed 2l upstream and 10l downstream regions as part of the OSF domain, whereas Mizuno et al. considered zero ﬁn thickness, steady laminar ﬂow, and employed no upstream and downstream regions outside the OSF region. A comparison between the experimental results of Mochizuki et al. and the 3D numerical results of Xi and Shah for an idealized OSF shows an excellent agreement in the j factors for Re 6000 and in f factors for Re 4000, as shown in Fig. 9. The f factors computed for Re : 6000 have possible numerical convergence error due to high Re where the aforementioned iteration scheme may not be adequate; and a turbulence model may be required as explained in [90]. A comparison of experimental results for the real OSF surface of London and Shah [91] and the 3D numerical results of Xi and Shah for the idealized ﬁn geometry (without burrs at edges, surface roughness, etc.) are shown in Fig 10. For f factors, there is an excellent agreement between experimental and numerical results for Re & 1000. However, the numerical results have lower values than the experimental f factors for 1000&Re&5000 (maximum difference of 35% at Re : 2500). The burrs and bent leading and trailing edges of an OSF and surface roughness due to brazing would result in a larger effective ﬁn thickness. This

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Fig. 9. A comparison of 3D numerical results of Xi and Shah [90] with experimental results of idealized OSF of Mochizuki et al. [84].

Fig. 10. A comparison of 3D numerical results of Xi and Shah [90] with experimental results of a real OSF of London and Shah [91].

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in turn has an important effect on the ﬂow instability and in the transition region than in the laminar (low Re) region [90]. For j factors, the numerical results show much larger values than the existing experimental results and correlations (about 50% at Re : 400), except that the slopes of the correlation by Mochizuki et al. [84] for idealized geometries and the numerical results are closer. At present, the reasons for this large difference in numerical vs experimental j factors are not known. The reduction in j factors for the real OSF surface compared numerical values for the idealized geometry may be in part due to passage-to-passage nonuniformity; this nonuniformity has a signiﬁcant effect on j factors and negligible effect on f factors at low Re (Shah [93]). In addition, numerical analysis shows another important difference between the real and idealized OSF geometries as follows. The transition for f factors (deviation from the straight f vs Re line for laminar ﬂow on a log — log plot) begins in the range Re : 1000 — 1200 for the London and Shah surface and in the range Re : 1600 — 2500 for numerical results as shown in Fig. 10. The transition for j factors begins at Re : 1500 — 2000 for the London and Shah surface and between Re values of 2500 and 3300 for the numerical results. The reason is due to burrs and bent edges and surface roughness for real surfaces. Another interesting observation is that the transition in f factors starts at lower Re than that for j factors. This feature indicates that the unsteady ﬂow ﬁrst enhances ﬂow friction and then enhances heat transfer as a function of Re. It has been observed in visualization experiments reported by Mochizuki et al. [84] and Xi et al. [85] that the unsteady ﬂow ﬁrst occurs downstream of ﬁn arrays (affecting only P and f factors) and then progresses upstream into ﬁn arrays as the Reynolds number is increased. A comparison of 2D and 3D numerical results for the Shah and London surface indicate that the effect of 3D geometry is smaller in the laminar ﬂow region (Re 1600) and the 2D computations are quite accurate at low " Re. 1. Concluding Remarks on Offset Strip Fin Performance/Analysis The studies just summarized for offset strip ﬁn geometry indicate that a considerable amount of numerical and experimental work has been conducted on this simple and high-performance compact heat exchanger surface. As outlined by Xi et al. [72] and Mercier and Tochon [20], unsteady analysis is required to accurately calculate turbulent ﬂow in compact geometries. A 3D model is also required to simulate the correct phenomena [90]. Nevertheless, the main physical phenomena can be predicted by these methods, and the predicted values are in relatively good agreement with experimental data.

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406 B. Louver Fins

Since the early 1980s, numerous attempts have been made to develop 2D numerical models of louver ﬁn surfaces. In all louver ﬁn analyses reported in this paper and in the literature, the geometry analyzed is a ﬂat ﬁn as shown in Fig. 11b; no 3D numerical studies are reported for Fig. 11a. The details of the geometries analyzed, numerical models employed, and the Reynolds number range investigated are summarized in Table II. Initially, the models were based on the idealized zero ﬁn thickness and laminar ﬂow with periodic boundary conditions and constant wall temperature. Kajino and Hiramatsu [94] and Tomoda and Suzuki [95] solved the stream function and vorticity equations for incompressible steady laminar 2D ﬂow over ﬂat louver ﬁns using ﬁnite difference methods. They presented streamlines, velocity proﬁles, and Nusselt number distributions on the louver surface for one louver geometry only at a single Reynolds number value. The authors stated that the computational results showed trends similar to the ﬂow visualization results, but no quantitative validation was made. They concluded that although ﬂow visualization is a useful tool for assessing the performance of louver ﬁns, numerical calculations are necessary for more quantitative evaluation. Their main conclusion was that for high-performance ﬁns with small louver pitch, best results are obtained if the ﬁn pitch is matched to that of the louver in such a way that the ﬂuid ﬂows along the louver. However, no evidence was presented in the paper to support this statement, and it would appear that the numerical solution was only used to prove the experimental ﬁndings.

Fig. 11 Typical louver ﬁn geometries: (a) corrugated ﬁn, (b) ﬂat ﬁn.

TABLE II Summary of Geometries, Numerical Models, and Reynolds Number Ranges Covered for Flat Louver Fins Reference Kajino and Hiramatsu [94] Achaichia and Cowell [96] Hiramatsu et al. [98] Ha et al. [100] Baldwin et al. [103] 407

Suga et al. [106] Suga and Aoki [107] Ikuta et al. [108] Achaichia et al. [109] Itoh et al. [110] Atkinson et al. [111] and Drakulic et al. [112] Drakulic´ [113]

Geometry p : 1.0 mm l : 1.0 mm 1!: 26° ( : 0.1 mm (:0 1! : 15° to 35° p /l : 1.0 to 2.5 1!:0° to 50° p /l:1.0 and 2.0 1000 and 1050 ﬁns/m for 1!:23° 1000 ﬁns/m for 1!:31° 1000 ﬁns/m for 1!:31° 1!:15° and 30° l:30 mm 1!:30°, 26° l:10 mm, (:0.8 mm, p :10 mm p /l:1.0, 1.125 and 1.75 1!:20°, 26° and 30°, p /l:0.5 to 1.125, (:0.8 mm l:10 mm, p :10 mm (:0.115 mm, l:1.3 mm, p :1.1, 1.5 and 1.9 mm, 1!:15°, 20°, 25° and 30° 1!:20° to 40°, p /l:1.7. p /l:1.22, (/l:0.625 p:1.5 to 2.5 mm, l:0.9, 1.1 and 1.4 mm, 1!:12° to 28.5°, (:0.05 to 0.1 2D: same as Refs. [111, 112] 3D: p : 8 and 14 mm mm, l:1.1 mm p :2.17 1!:22°, (:0.05 mm

Numerical model 2D, stream function and vorticity, laminar, grid not described

Rel 500

2D, ﬁnite volume, laminar steady, Cartesian grid, periodic boundary conditions, didn’t solve the energy equation

20—1500

2D, ﬁnite difference, body ﬁtted rectangular grid, stream function and vorticity 2D, body ﬁtted generalized grid, laminar steady ﬂow

100—1000

2D, ﬁnite difference, Cartesian grid, staircase louver edge representation 2D, ﬁnite difference, steady laminar ﬂow, overlaid Cartesian grids same as Ref. [106] above 2D, body ﬁtted grid, ﬁnite difference, steady laminar ﬂow 2D, body ﬁtted grid, k9 turbulence model for Rel 1200, energy equation not solved 2D, steady laminar, grid details not given 2D, steady laminar ﬁnite volume, automatic body ﬁtted grid generation Same as Refs. [111, 112] Conjugate heat transfer in the 3D case

176—1006 200—6000 64—450 64—450 417 10—2400 645 and 595 100—3200 2D analysis: 100—3200 3D analysis: 100, 400, and 1600

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A more comprehensive study was carried out by Achaichia and Cowell [96]. Using ﬁnite difference techniques, they modeled one louver in the periodic fully developed region assuming cyclic boundary conditions, laminar steady ﬂow, and zero ﬁn thickness. Since only one louver was modeled, the authors were able to use a ﬁne Cartesian grid normal to the louver surface. Results were presented for ﬁn-to-louver pitch ratios of 1 to 2.5, louver angles 15° to 35°, and louver pitches based Reynolds number Rel from 20 to 1500. The results conﬁrmed the phenomenon ﬁrst identiﬁed by Davenport [97], namely the existence of two distinct ﬂow regimes with duct ﬂow occurring at low Reynolds numbers and louver (aligned) ﬂow prevailing at high Reynolds numbers. This result, which is clearly demonstrated by the sketch of Fig. 5b, has since been accepted and conﬁrmed by a number of other studies both experimentally and numerically [98—100]. From the numerical velocity distributions, Achaichia and Cowell [101] quantiﬁed this ﬂow alignment property of the louvers by introducing the concept of the mean ﬂow angle. They plotted this parameter against the Reynolds number and showed that the ﬂow is aligned with the louver to within a few degrees at high Reynolds numbers, and the degree of alignment begins to fall off as the Reynolds number is decreased. They also found that the Reynolds number at which this trend starts is a function of the ﬁn-to-louver pitch and derived a simple correlation for the mean ﬂow angle in terms of the Reynolds number, the ﬁn-to-louver pitch ratio, and the louver angle. They also found that their model could predict recirculation behind the louvers at high louver angles even for the fully developed periodic case. The effect of this ﬂow-directing behavior is that the friction factor curves exhibit a steep slope at low Reynolds numbers as the less aligned ﬂuid impinges on the louver surface as it ﬂows down the duct between the ﬁns. At high Reynolds numbers, the curve shows a clear ﬂattening as the ﬂow is aligned with the louvers. This ﬂow behavior is also responsible for the ﬂattening of the Stanton number curve observed by Achaichia and Cowell [101]. A ﬂow efﬁciency term was devised by Webb [102] and Webb and Trauger [99] to describe this ﬂow behavior and predict the onset of this ﬂattening in the Stanton number curve. Finite ﬁn thickness models were analyzed in a number of numerical studies on ﬂow through complete louver arrays for a few isolated conﬁgurations and/or ﬂow rates. In all cases, a symmetrical array having two-bankdeep louvers was modeled. Baldwin et al. [103] used a Cartesian grid and solid cells with zero porosity to deﬁne the ﬁn. The louver surface was therefore represented by a series of staircase-type steps with the grid spacing selected to make the solid cells approximate the geometry. A ﬁnite volume solution of the ﬂow equations was carried out using the commercial PHOENICS code. Results were presented for only two louver conﬁgur-

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ations, each at a single Reynolds number, and compared with the LDA measurements of Button et al. [104] and the ﬂow visualization results of Hiramatsu and Ota [105]. The numerical results displayed the same basic phenomena as the experimental studies, showing ﬂow separation after the ﬁrst louver and almost complete alignment after the third. As the ﬂow entered the second bank of louvers, it took longer to become aligned relative to the ﬁrst bank. Suga et al. [106] presented a ﬁnite difference numerical model for a complete louver ﬁn (two louver banks) over a limited range of Reynolds numbers (64Rel450). The model assumed 2D steady laminar ﬂow. They used an elaborate system of overlaid grids to overcome the ﬁnite ﬁn thickness problem and divided the solution domain into regions that were represented by Cartesian grids. Complex communication between the grids was achieved by bilinear interpolation of the dependent variables at the grid boundaries. The authors identiﬁed the possibility of interpolation errors at the false boundaries between grids and stated that they could be reduced by the careful choice of the grid sizes in the overlapping region. Another possible source of error was the use of triangular cells to join the bent parts of the ﬁrst half louver. The predicted velocity distributions in the regions between the louvers were compared with LDV measurements at Rel : 64. However, the computational results did not display the characteristic duct ﬂow expected at such a low Reynolds number. The calculated mean Nusselt number for each louver was also compared with that measured by means of a nickel ﬁlm sensor (which acted both as a heater and a resistance thermometer). A remarkable degree of agreement was shown even though a uniform ﬁn surface temperature was assumed in the processing of the experimental results. Using the same numerical model, Suga and Aoki [107] investigated the effect of grid reﬁnement as expressed by a grid Reynolds number Re , and concluded that a grid independent solution was obtained at values of Re 7. They then carried out a numerical study of the effect of ﬁn parameters on heat transfer performance. The range of the Reynolds number for this study was also 64Rel450, louver angles 20°1!30°, ﬁn thickness to louver pitch ratio 0.04 (/l 0.08, and high-density ﬁns with ﬁn pitch to louver pitch ratio 0.5 p /l 1.125. They conjectured that, provided that the ﬂow is aligned with the louvers, the performance of the ﬁn is dominated by the thermal wake behind the louvers. They then concluded that for each louver angle, there was an optimum value for the louver-pitch-to-ﬁn-pitch ratio that caused the thermal wake behind the louvers to ﬂow along a line halfway between two louvers further downstream. They presented a formula for the calculation of this optimum ratio as a function of the louver angle. It is interesting that the value of the

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optimum ratio was observed to be independent of the Reynolds number in the range Rel 450. Body-ﬁtted coordinates have been used to facilitate the grid generation for a full ﬁn with ﬁnite thickness. Ikuta et al. [108] developed a 2D model using ﬁnite difference techniques in a curvilinear coordinate system with a relatively coarse grid. They displayed the velocity vectors diagram for a single geometry at one value of Reynolds number showing ﬂow separation from the ﬁrst three louvers with complete alignment subsequently. This process was shown to repeat in the second bank of louvers. The oblique grid and coordinate transformation system, adopted by Hiramatsu et al. [98], offered a more versatile method for the ﬁnite difference modeling of louver ﬁn geometries. In this system, two mesh structures were used with an oblique grid in the zone of the inclined louvers and a rectangular grid over the horizontal inlet and turn round parts of the ﬁn. The laminar 2D steady ﬂow model was considered for the analysis. Some mesh reﬁnement was done by increasing the number of grid points near the louver edges. A comparison between the predicted streamlines and ﬂow visualization in a water channel showed good agreement. The streamline plots showed increasing ﬂow alignment with increasing ﬁn-pitch-tolouver-pitch ratio and Reynolds number. The effect of varying the louver angle was not shown. Calculated values of local heat transfer distribution on both ﬁn surfaces were presented for one geometry at a single value of the Reynolds number. The results showed a decrease in the mean value of the mean Nusselt number as the ﬁn pitch increased and with decreasing Reynolds number. This was attributed to the reduced degree of ﬂow alignment under these conditions. Although the calculated friction factors did not agree well with their experiments, the mean Nusselt number showed excellent agreement. Achaichia et al. [109] described a novel body-ﬁtted grid topology that extended over a number of ﬁns, thus simplifying the introduction of ﬁnite ﬁn thickness and geometrical variations. Two-dimensional steady-state ﬁnite volume calculations were performed, using the commercial ﬁnite volume code PHOENICS on a two-bank louver ﬁn array. A number of different louver angles were considered in the range 20° to 40°, a ﬁxed-ﬁn-to-louverpitch of 17 for Rel between 10 and 2400. The ﬂow was assumed to be laminar at low values of Reynolds numbers. The k— turbulence model was applied above the critical Reynolds number of 1200. The mean ﬂow angle was calculated along the ﬁn at different velocities and the results clearly showed a gradual alignment of the ﬂow along the ﬁn. The effect of the Reynolds number on the degree of alignment was also demonstrated and quantiﬁed. The effect of the louver angle on the distribution of the local skin friction on the upper and lower ﬁn surfaces was shown at Rel : 10 and 600.

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At the lower Reynolds number and for all louver angles, high values of local skin friction were observed at the trailing edge of the louvers as the duct directed ﬂow impinged on the surface. This effect was less noticeable at the higher Reynolds number since the degree of ﬂow alignment was much higher. The skin friction was markedly higher for the 40° louver, but only for the lower Reynolds numbers. The authors explained this observation in terms of the increased ﬂow between the louvers (and less duct ﬂow) as a result of the larger gap between them at this angle and concluded that increasing the louver angle has the desired effect of increasing the degree of ﬂow alignment at the same Reynolds number. Itoh et al. [110] reported a difference of about 6% in the value of the average Nusselt number when the temperature dependence of the physical property was included in the solution procedure. Ha et al. [100] deﬁned the governing equations in generalized coordinates in order to overcome grid generation difﬁculties. They used the commercial ﬁnite volume code FLUENT to solve these equations assuming a laminar 2D steady ﬂow model. However, their grid structure produced a coarse distorted mesh between the louvers, where high velocity gradients are expected, thus signiﬁcantly reducing the numerical accuracy of their model. Nevertheless, the authors conﬁrmed the ﬂow directing properties of the louver ﬁn. They also observed a developing ﬂow regime at the inlet with ﬂow separation at both leading and trailing edges of the louvers. Calculated friction factor and Nusselt number results were reported but not validated experimentally. Atkinson et al. [111] and Drakulic´ et al. [112] reported a novel automatic grid generator that produced 2D grids for any louver ﬁn geometry from parametric input of the ﬁn data. The resulting mesh had a block structure with three different types of blocks corresponding to the inlet region of the ﬁn, the individual louvers, and the turnaround region. These blocks were divided into a number of quadrilateral cells. Nearly all the cells were rectangular, giving maximum possible numerical accuracy. With grid lines parallel and normal to the louver, the boundary layer proﬁles and integral parameters could easily be extracted. Two of the most comprehensive numerical studies of louver ﬁn characteristics are by Drakulic´ [113] and Atkinson et al. [114]. In these studies, 2D and 3D numerical modeling was performed for a number of experimentally tested louver ﬁns using the automatic grid generator described earlier. These louver ﬁns were ﬂat ﬁn type used in a ﬂat tube and ﬂat ﬁn construction (see Fig. 9b). The STAR-CD CFD code was used for the analysis for the Reynolds number Rel range of 50—3200 for 2D and Rel : 100, 400, and 1600 for 3D. In the numerical modeling, the computation domain included upstream and downstream region each as l and the louvered ﬁn had two banks of louvers as shown in Fig. 5a with each bank having 5, 10, 13, or 19

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full louvers. A constant wall-temperature boundary condition was applied on the primary surface. A conjugate model of thermal conduction through the ﬁn and the convection from its surface was used. This is because the temperature distribution in the ﬁn and the thermal boundary layer on the tube wall result in complex temperature proﬁles that cannot be predicted using 2D models and constant ﬁn wall temperature. Time-dependent numerical solutions of the louver ﬁn clearly showed vortex shedding from the trailing edges of the ﬁrst two louvers in the upstream and downstream banks, resulting in a rather long unsteady wake behind these louvers. Hot-wire measurements of local velocity distributions in louver ﬁns by Antoniou et al. [115] had showed the same rms values of velocity ﬂuctuations, but the ﬂow was wrongly interpreted as being turbulent. The numerical results clearly showed that this is unsteady laminar ﬂow. The mean Nusselt number curves computed from the time-dependent solutions showed the experimentally observed ﬂattening at high Reynolds numbers. Numerical predictions of local and mean ﬂow and heat transfer parameters were compared with the experimental measurements of Achaichia [116] and Antoniou [117]. The results clearly showed that although reasonable predictions of the overall friction factor could be obtained, the Stanton number was overpredicted. For two ﬂat multilouver ﬁn geometries, experimental results and 2D and 3D numerical computations are shown in Fig. 12 [113]. Reviewing these ﬁgures, it can be seen that while the friction

Fig. 12. A comparison of friction factor and Stanton number experimental results with 2D and 3D numerical results of multilouver ﬁns, having p : 2.17 mm, l : 1.1 mm, and 1! : 22°: (a) p : 8 mm and (b) p : 14 mm [113].

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factors agree with experimental values within 0—10%, the computed Stanton numbers are about 80% higher compared to experimental values at Rel : 400. The results presented in Fig. 12 are correct and supersede those presented in Fig. 10 of [114]. In the numerical analysis, both the air and wall temperatures vary in 3D computations. To obtain the Stanton number from the heat ﬂux, one needs to base it on a difference between the wall (primary surface) temperature and the bulk air temperature. The commercial software does not compute the bulk temperature, only the detailed temperature distribution in the computational domain. Hence, the mean temperature difference for the heat transfer coefﬁcient determination was based on the log-mean temperature difference, where the wall temperature and the air inlet temperatures are known and the outlet bulk temperature of air was computed from the numerical results. Since the heat transfer coefﬁcient based on the LMTD would be different from that based on the wall temperature minus the bulk air temperature, this may be one reason for a large discrepancy in the computed versus measured values of St. However, it cannot account for the large discrepancy; the reasons are not clear, as was the case for the OSF surface [90]. Most of the previous work on multilouver ﬁn geometry is ether experimental or numerical. Only a few studies have combined experimental work on model-scale and full-scale heat exchangers together with the CFD studies on the model scale ﬁn geometry and the ﬂow visualization setup with a number of ﬁns. Beamer et al. [118] reported a study on full-scale heat exchanger performance testing and ﬂow visualization experiments, and a 2D CFD study on one ﬁn (see Fig. 5a) with a periodic boundary condition, and six- and 12-ﬁn geometries to duplicate ﬂow visualization setup and results. The geometric dimensions used in the 2D study of one ﬁn were the same as those of the actual heat exchanger tested. The CFD domain used for one ﬁn (the central ﬁn in Fig. 5a) was extended upstream and downstream of the ﬁn by 20% of the ﬁn length (L in Fig. 1b) to take into account the ﬂow adjustment at louver inlet and exit margins (sections). A known ﬂow rate was applied at the inlet boundary, and pressure boundary conditions were prescribed at the outlet boundary. Periodic boundary conditions were applied on two sides of the complete computational domain (i.e., 1.4 L ); symmetric plane boundary conditions were applied on the top and bottom control volume surfaces (perpendicular to the plane of the paper in Fig. 5a) because of the ﬁnite volume code. Typically 7000 to 27,000 hexahedral cells were used in a single array for ﬂow calculations. Blended upwind differencing with a blending factor of 0.5 was used for the convective differencing scheme. The results obtained from the 2D CFD study include ﬂow development and alignment with the leading bank of louvers, reversal at the center rib, development and alignment with the trailing banks of louvers, and ﬂow

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exit from the ﬁn in a direction parallel to the longitudinal axis. Evidence of ﬂow separation at the leading edge and vorticity in the wake behind the trailing edge of the louvers was observed in the CFD vorticity plots (not shown). The CFD study on the ﬂow visualization setup for 6 and 12 ﬁns duplicated the observed ﬂow phenomena and indicated that even a 12-ﬁn (note that there are 3 ﬁns shown in Fig. 5a) arrangement does not eliminate the wall effects in ﬂow through louver ﬁns in a ﬂow visualization setup. A comparison of ﬂow efﬁciency of their study [118] with those of Webb and Trauger [99] and Cowell et al. [2, 101] is shown in Fig. 13. Note that Achaichia and Cowell [101] did not employ a developing ﬂow region before the beginning of each louver ﬁn in their CFD study. As a result, they overestimated ﬂow efﬁciency as shown in Fig. 13. From this ﬁgure, it is found that the CFD results [118] and ﬂow visualization measurements show good agreement. Both display an easily discernible knee below which the ﬂow efﬁciency drops off rapidly. The data of Webb and Trauger [99] show a similar trend, except that the value of Re for the knee and ﬂow efﬁciency are considerably higher. This rapid dropoff of the ﬂow efﬁciency at low Re is attributable to the thickening of the boundary layer between the louvers, which causes more ﬂuid to ﬂow toward the duct ﬂow region. Thus, the boundary layer growth along the louvers would reduce the heat transfer coefﬁcient (Nu or j factor). And Beamer et al. [118] found a good

Fig. 13. A comparison of computational and experimental results for ﬂow efﬁciency of multilouver ﬁns [118].

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correlation between the dropoff in the measured j factors in a full-scale heat exchanger and the knee for dropoff in the ﬂow efﬁciency of Fig. 13 for Re & 150. Thus, they showed an excellent agreement among the CFD study, ﬂow visualization, and full-scale testing. The reasons for the performance behavior were explained and an idea for further improvement of the ﬁn geometry was suggested. 1. Concluding Remarks on L ouver Fin Performance/Analysis The preceding survey shows that although the performance of louver ﬁns seems to be reasonably well understood, a general correlation for the accurate quantitative prediction of their performance is not available. Also at present, no modeling is available for 3D corrugated louver ﬁn geometry (Fig. 1b). Numerical modeling can be used to provide valuable information on the complex behavior of the ﬂow and heat transfer of these surfaces. However, careful grid generation strategy must be used to accurately predict the details of both ﬂow and thermal boundary layers on the ﬁn surfaces. Two-dimensional models provide a fast tool for the assessment of the relative performance of different geometries. Complete 3D conjugate models are necessary if performance data are to be predicted accurately. Laminar time-dependent models, although computationally demanding, can provide further detailed understanding of some of the unsteady nature of the ﬂow over the louvers. At this stage, it is essential to conﬁrm with precise experimentation that whether the ﬂow through the louvers is laminar unsteady or low Reynolds number turbulent, both from the performance enhancement and numerical analysis points of view. The best potential future beneﬁt of numerical modeling of these surfaces is in providing fundamental understanding of the performance of two promising variants of the basic conﬁgurations. The ﬁrst is inclined louver ﬁns (which acts like an offset strip ﬁn; see Fig. 14, Tanaka et al. [120], and Suzuki et al. [121]),

Fig. 14. Inclined louvers (a counterpart of Fig. 5a).

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which offer a signiﬁcant improvement in the ratio of heat transfer to pressure drop. A second area of potential performance improvement stems from the recognition that the ﬁrst few louvers in an array provide inferior performance. The possibility of varying louver angle within an array suggests itself as an area worthy of further consideration. Experimental studies of these innovations are prohibitively expensive, but they lend themselves well to numerical analysis. C. Wavy Channels 1. Corrugated Wavy Channels Considerable amount of numerical investigation has been conducted on this channel geometry used in a plate-ﬁn exchanger as well as in a plate exchanger with the chevron angle of 90°. The numerical methods, operating parameters and the geometries studied are given in Table III. A reference book by Sunde´n and Faghri [122] provides more details on the numerical methods and analysis of wavy channels discussed later and other selected compact heat exchangers. Asako and Faghri [123] developed a solution methodology for laminar ﬂow and heat transfer in a corrugated duct. A ﬁnite volume scheme was developed to predict fully developed ﬂow, heat transfer coefﬁcients, and friction factors in a corrugated channel. The basic method was an algebraic nonorthogonal coordinate transformation, which mapped the corrugated channel into a rectangular domain. The governing equations of continuity, momentum, and energy were solved assuming constant thermophysical properties and excluding natural convection effects. The details on the transformation of the conservation equations are provided by Faghri et al. [124]. The boundary conditions used were constant wall temperature and periodic ﬂow at the inlet and outlet of the domain. As the ﬂow was assumed to be laminar, no turbulence model was used. Grid size effects were studied and the maximum change in the Reynolds and Nusselt numbers, between a 18;34 mesh and 26;50 ﬁne mesh, were within 3 and 5%. The numerical results were compared with data from the open literature, but as the Re range does not coincide, only qualitative agreement can be found. For the range of the Reynolds number studied (100 & Re & 1000), the numerical results for the friction factor indicate transition from laminar to turbulent ﬂow regime. The slope of the friction factor vs Reynolds number curve increases from approximately 90.5 to almost 0 at Re : 1000. This latter value is representative of fully developed turbulent ﬂow in a rough duct. For heat transfer, in most of the cases studied, the Nusselt number increases with the Reynolds number. These results indicate that for wavy corrugated channels,

TABLE III Summary of Numerical Methods, Operating Conditions, and Geometries of Corrugated Wavy Channels

Author

Method

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Asako and Faghri [123] Xin and Tao [128] Hugonnot [9] Yang et al. [126]

Finite volume

Ergin et al. [130]

Finite volume

Ergin et al. [132]

Model

Grid

Finite volume

Laminar

Finite difference

Laminar

Finite difference

Laminar k— Low-Re model k—

Cartesian Transformed Cartesian Transformed Cartesian

Finite volume

Low-Re model

Kouidry [73]

Finite volume

McNab et al. [137] Tochon and Mercier [139]

Finite volume Finite volume

Transformed Cartesian Polar-Cartesian

Boundary condition

Reynolds number

Fully developed

100—1500

Fully developed

Prandtl number

Pitch/height ratio 1—4

100 —1000

0.7, 4, and 8 0.7

Fully developed

150 —10,000

7

3.33

Constant wall temperature Adiabatic

Fully developed

100—2500

0.7

2.5—6

Fully developed

500—7000

0.7

1.4

Transformed Cartesian

Adiabatic

Fully developed

500 —7000

0.7

1.4

DNS

Curvilinear

Adiabatic

Fully developed developing ﬂow

5000—10,000

7

3.33

3D: 30500 meshes DNS

Cartesian

Constant wall temperature Constant wall temperature

Fully developed

250 —4000

0.7

3—60

Developing ﬂow

5000—10,000

7

3.33

Curvilinear

Constant wall temperature Constant wall temperature Adiabatic

Inlet conditions

3—6

Validation Pressure drop, Nusselt number No validation Pressure drop, visualization Pressure drop, visualization Local velocities, velocity ﬂuctuations Local velocities, velocity ﬂuctuations Local velocities, velocity ﬂuctuations Pressure drop, Nusselt number Pressure drop, Nusselt number

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the transition from purely laminar to unsteady laminar and to turbulent ﬂow occurs at low Reynolds numbers and that the channel geometry has a strong effect on the transition. Asako et al. [125] assessed the heat transfer and pressure drop characteristics of a similar corrugated duct with rounded corners. Computations were carried out in the Reynolds number range 100 Re 1000 for several geometric conﬁgurations. It was determined that the change in heat transfer rates caused by rounding the corners of the sharp cornered corrugated duct depended on the speciﬁc ﬂow conditions, geometry, and performance constraints. To address the use of a laminar model when turbulence is developing, Yang et al. [126] have extended the work of Asako and Faghri [123] by using a low Reynolds number turbulence model. The same numerical method as Asako and Faghri [123] was used, but the source terms in the conservation equations were modiﬁed to take into account turbulent effects in the diffusion coefﬁcients. The turbulence was modeled according to the Lam and Bremhorst [58] model, which is a low Reynolds number form of the k— formulation. The details of the equations are given in Yang et al. [126]. In parallel with this numerical work, experiments were performed in order to validate the model. The test section was a sharp cornered corrugated channel. The comparison was limited to the laminar model of Asako and Faghri [123], as the low Reynolds number turbulent ﬂow model does not allow the modeling of geometries with sharp corners. In term of the ﬂow pattern, the size of the recirculation areas was well predicted. It must be noted, however, that the recirculation region reaches a maximum size at a Reynolds number of 500; the size decreases because of the high diffusion in turbulent ﬂow for higher Reynolds numbers. The same trends were observed by Hugonnot [9] in a wavy channel. For friction factors for a sharp edge channel, there is a good agreement with the laminar model for Reynolds numbers up to 500, beyond which the friction factors are overestimated. Below this transition Reynolds number and for various operating and geometric parameters, the predicted friction factors for the laminar and turbulent models are the same. This suggests that the low Reynolds number turbulent ﬂow model is stable even for laminar ﬂows. Beyond the transition Reynolds number, the friction factors predicted by the turbulent model are higher than those predicted by the laminar model. For the Nusselt numbers, the same trends are observed. Yang et al. [126] claim that the low Reynolds number turbulent ﬂow model can be used to predict friction factors and Nusselt number, but no comparison with experimental data is given. Garg and Maji [127] applied a ﬁnite difference scheme (SIMPLEC algorithm) to laminar ﬂow through a wavy corrugated channel. They presented results for both developing and fully developed ﬂow for Reynolds

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numbers ranging from 100 to 500. The channel dimensions were varied, and for one chosen conﬁguration, the detailed behavior of velocity, pressure, and enthalpy for developing laminar ﬂow were presented. Xin and Tao [128] performed numerical simulation in wavy channel by applying a laminar model and using a ﬁnite difference method. The domain was gridded by a combination of polar and Cartesian coordinates. In the bends, a polar coordinate was applied, and a Cartesian coordinate in the straight duct between two bends. A fully developed ﬂow was obtained by periodic boundary conditions at the inlet and outlet of the domain and a constant wall temperature boundary condition. Several geometric and operating parameters were studied, but no validation with experimental results was reported. The channel studied was quite similar to the one used by Be´rieziat [129], and comparisons can be made through the streamline velocity proﬁles. Xin and Tao [128] claimed that the relative strength and size of the recirculation zone increase with the Reynolds number. This result is in partial disagreement with the experimental data of Be´reiziat [129] and the ﬂow visualizations of Hugonnot [9], which indicate that the recirculation zones increase up to a Reynolds number of 200, then vortices are generated downstream of the corrugation. For higher Reynolds numbers (Re 350), the ﬂow becomes unstable and turbulence is developing. According to Be´reiziat [129], the ﬂow becomes fully developed turbulent for Reynolds numbers above 2000. Applying a laminar model for ﬂows developing turbulence leads to inaccurate predictions, and the conclusions of Xin and Tao are only valid in the low Reynolds number range (Re & 350). Hugonnot [9] performed numerical simulations in a 2D corrugated channel using a laminar model at Re : 150 and a k— turbulence model at Re : 10,000. The choice of the model was dictated by the previous observations on the ﬂow structure. A ﬁnite difference method was applied and the ﬂow was considered adiabatic. The results were compared with ﬂow visualizations and pressure drop measurements in a similar geometry. The size and the position of the recirculation zones were well predicted at both Re : 150 and Re : 10,000. The pressure drop data were compared at Re : 10,000, and a good agreement was found for the overall pressure drop, but differences can be noticed on the pressure proﬁle essentially in the recirculation zone. This result is not surprising, as it is well known that conventional k— models are not accurate in separated ﬂows, and even if the ﬂow pattern is qualitatively correct, the wall shear stress can be inaccurately predicted. The comparison of the Hugonnot [9] and Xin and Tao [128] results for low Reynolds numbers is only qualitative and similar. Ergin et al. [130] applied the method developed by Faghri and coworkers [123, 125, 126] to the study of the effect of interwall spacing on turbulent ﬂow in a sharp-edged corrugated channel. The ﬂow was consider-

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ed turbulent (500 & Re & 7000) and the k— model was adopted for turbulent closure. Experiments were also conducted to validate the numerical model. At Re : 2000, the ﬂow characteristics given by the model were in agreement with the visualization experiments and with previous experimental studies [126, 131]. The comparison of the local velocity proﬁles and turbulent kinetic energy revealed that the proposed method gave accurate prediction of the velocity proﬁles, but the prediction of the turbulent kinetic energy was relatively poor. As a result, the friction factors were underpredicted by approximately 33%. Ergin et al. [132] extended the previous work by applying the Lam— Bremhorst low Reynolds number turbulence model. For a Reynolds number of 2000, the standard k— model and the low-Re turbulence model give similar predictions of the ﬂow ﬁeld. The comparisons of the predicted friction factors showed a good agreement with the low-Re turbulence model in the range 500 & Re & 3000, whereas the standard k— model is more accurate for higher Reynolds numbers (Re 3000). As a consequence, the standard and low-Re turbulence model cannot be applied in the whole range of the Reynolds number, and this implies selecting a priori the model for the range of the Reynolds number. To overcome the problem of simulations of unstable turbulent ﬂow, Kouidry [73] performed direct numerical simulation of turbulent ﬂow in a corrugated channel. The geometry was identical to the one of Hugonnot [9] and Be´reiziat et al. [133]. In parallel to this numerical work, local velocity measurements by laser anemometry were performed. The DNS method requires a very ﬁne mesh in order to reproduce the smallest turbulent structures according to the Kolmogorov scale. Based on experimental data, the smallest turbulent structure was about 95 m, but limitations of the workstation led to a 269 m by 212 m grid (293;95 meshes). As a result, the smallest turbulent structure was not represented, and the method used was called pseudo-DNS. The TRIO code based on a ﬁnite volume method, developed by the CEA (French Atomic Commission), was used. The conservation equations were solved on a control volume with a semiimplicit time scheme and a ﬁne third-order space scheme (Quick-Sharp). The advantage of DNS methods compared to time average methods (k— models) is that they are independent of the ﬂow conﬁguration and that local instantaneous information is available. Developing ﬂow with a ﬂat inlet velocity proﬁle and periodic ﬂows were studied. The numerical results were compared to the experiments. For the mean velocity proﬁles, the agreement was relatively good in the fully developed ﬂow, but differences linked to the inlet boundary conditions were observed for the developing ﬂow. The turbulence intensity was analyzed through the mean square velocity ﬂuctuations. The experimental ﬂuctuations were 30 to 50% around the mean

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velocity, while the numerical simulations gave ﬂuctuations up to 150%. Kouidry [73] claimed that these differences came from the number of time steps required for the second-order average and by the fact that the turbulent dissipation was underestimated by the 2D modeling. Despite these differences, the ﬂow pattern and the vortices are well predicted in terms of the size and growth. Farhanieh and Sunde´n [134] investigated numerically a three-dimensional fully developed laminar ﬂow in a corrugated square duct. A ﬁnite volume method was applied and the Reynolds number range from 30 to 1000. At constant pressure drop, the grid size effect was studied and 70;18;28 grid points were selected. The numerical model was checked on a straight square duct, and the predicted friction factor and Nusselt number were in good agreement with analytical values. A parametric study on the corrugated duct geometry was performed, but no comparison was given with experiments. Asako et al. [135] studied laminar forced convective heat transfer and ﬂuid ﬂow characteristics in a wavy duct with a trapezoidal cross-section. The algebraic coordinate transformation (Faghri et al. [124]) was applied to map the trapezoidal cross-section onto a rectangular one. The numerical predictions of the friction factors were compared for a fully developed ﬂow in a straight trapezoidal duct, and the agreement with the data of Shah and Bhatti [136] was within 0.46%. The numerical simulations showed that the enhancement of heat transfer and pressure drop due to waviness depends strongly on the Reynolds number. To optimize this geometry for industrial applications, the authors suggest studying the effect of the wave length, wave height and corner angle. McNab et al. [137] computed the ﬂow over herringbone corrugated (sharp-cornered wavy) channels, using a commercial STAR-CD CFD code. A 3D approach was adopted and a laminar model was used for Reynolds numbers (based on D ) below 1500, and a high-Re k— model for higher Reynolds numbers. For Reynolds numbers above 600, difﬁculties appeared in obtaining a fully converged solution. The authors suggested that ﬂow unsteadiness may occur for such low Reynolds numbers inducing pressure ﬂuctuations. The maximum ﬂuctuation (14%) was observed for a Reynolds number of 1500, and it was decided not to use time-dependent modeling. The computed j and f values were compared with the measurements of Abou-Madi [138] and are in relatively good agreement in the turbulent regime 17 to 27%, but in laminar regime the differences are 33% for the friction factor and 54% for the Colburn factor. The authors indicate that for such low-Re unsteady ﬂow, steady-state modeling may not be appropriate for low Reynolds number and that the recirculation zone is not well predicted by the k— model in turbulent ﬂow.

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Tochon and Mercier [139] have extended the work of Kouidry [73] using an approach based on a large eddy simulation without a subgrid model of turbulent ﬂow. The advantage of using LES is to produce directly the main part of coherent structures for turbulent regimes, for a better understanding of the ﬂow pattern. A curvilinear coordinate system has been adopted and the reﬁned mesh is 90 m (or ﬁve nondimensional wall units) near the wall in the spanwise directions. The mesh is nearly constant in the streamwise direction and the size is 200 m (or 15 nondimensional wall units). As the Kolmogorov scale is approximately 95 m for the considered geometry (Kouidry [73]), the chosen mesh seems to be sufﬁcient to describe the main part of the dissipative structures: no subgrid is needed. The mechanism of eddy generation and heat transfer have been analyzed in the developing ﬂow (Figs. 15 and 16). The results are qualitatively and quantitatively compared with experimental data obtained on a speciﬁc experimental rig and from open literature, and the differences in the heat transfer coefﬁcient and friction factor do not exceed 10%. The application of numerical modeling of a number of different highperformance heat transfer surfaces has been presented by Atkinson et al. [140]. The 2D and 3D models for louvered ﬁns showed that accurate calculations of overall heat transfer could only be achieved by using 3D models, which incorporate the effects of tube surface area and ﬁn resistance. Accurate predictions of ﬂow and heat transfer over corrugated ﬁns in the laminar ﬂow region were also presented. However, they showed that the high-Re k— model with log-law wall functions gave poor predictions of heat transfer for the turbulent-ﬂow regime. They suggested that low-Re forms of the model would be more accurate in this regime. 2. Wavy Furrowed Channels Sobey [141] conducted a numerical study of ﬂow through furrowed channels in conjunction with a related experimental investigation [142]. Sobey calculated the ﬂow patterns obtained using both steady and pulsatile inﬂow. The effects of varying dimensional parameters and Reynolds number were examined. Furthermore, the ﬂow structures that occur in a channel with arc-shaped walls were compared with the patterns induced by sinusoidal curved wavy walls. An oscillatory ﬂow in various types of wavy passages was examined further in subsequent studies performed by Sobey [143, 144]. Sparrow and Prata [145] examined a family of periodic ducts, using both numerical and experimental methods. The periodic duct is a tube consisting of a succession of alternately converging and diverging conical sections. Numerical simulations were carried out for fully developed laminar ﬂow in

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Fig. 15. Numerical results for time-dependent evolution of vorticity in a 2D wavy channel [139].

the Reynolds number range Re : 100—1000, for various duct conﬁgurations. The resulting data indicated that the periodic furrowed tube is not conducive to heat transfer enhancement for steady laminar ﬂow. Garg and Maji [127] used a ﬁnite difference method to solve the governing equations for steady laminar ﬂow and heat transfer in a furrowed wavy channel. Calculations were performed using various wall amplitudes for Re : 100—500. Both the developing and fully developed ﬂow regions were analyzed. The local Nusselt number was observed to ﬂuctuate sinusoidally in the fully developed region. Moreover, the Nusselt number

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Fig. 16. Numerical results for time-dependent evolution of temperature in a 2D wavy channel [139].

increased with the Reynolds number, unlike a constant value for laminar ﬂow through a straight channel. Blancher et al. [146] analyzed convective heat transfer by a spectral method for an expanding vortex in a wavy furrowed channel. The ﬂow is assumed laminar fully developed with the Reynolds number up to 200. An algebraic transformation of the coordinate was applied to reduce the physical domain to a rectangular computational domain. The governing equations were solved introducing the stream functions, with the details provided on the mathematical models and transformations. The method is limited to Reynolds numbers above 200, as the ﬂow becomes unsteady for higher values. The numerical results (ﬂow pattern, wall shear stress, and vorticity proﬁles), compared to those of Sobey [141], are in good agreement. Guzman and Amon [147] reported the transition to chaos for fully developed ﬂow in furrowed wavy channels. Speciﬁcally, the Ruelle— Takens—Newhouse scenario for the onset of chaos was veriﬁed using direct

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numerical simulations. The results were illustrated for various Reynolds numbers using velocity time signals, Fourier power spectra, and phase space trajectories. Guzman and Amon [148] extended their previous study by using DNS to calculate dynamical system parameters. Dynamical system techniques, such as time-delay reconstructions of pseudophase spaces, Poincare maps, autocorrelation functions, fractal dimensions, and Eulerian Lyapunov exponents, were employed to characterize laminar, transitional, and chaotic ﬂow regimes. Also, 3D simulations were performed to determine the effect of the spanwise direction on the route of transition to chaos. They have shown that the transition to chaos takes place at Re : 500 following a quasi-periodic frequency locking route. Wang and Vanka [149] applied a ﬁnite volume method to the study of convective heat transfer in wavy furrowed channels. To solve the governing equations, an accurate numerical scheme was applied on a 256;128 calculation domain. The simulations were performed for both steady and unsteady regimes. The ﬂow was observed steady until Re $ 180. Afterward, self-sustained oscillations appeared and led to destabilization of the laminar ﬂow. Physical interpretation of the transition was given and the ﬂow structure was analyzed for Reynolds numbers up to 1000. The friction factor calculations, compared to the experimental results of Nishimura et al. [150], slightly underpredicted the measurements but showed a good agreement with experimental trend. No comparison with heat transfer experiments was given in the paper. D. Chevron Trough Plates Chevron plates with a corrugation angle of 90° can be considered as a 2D wavy corrugated geometry. Numerous investigations have been published on this speciﬁc geometry and are discussed in the previous section. In most of the cases, the corrugation angles are between 30° and 60° in commercial chevron plates. As observed from the experimental work, the ﬂow remains mainly in the furrow of the corrugation for the corrugation angle of 30°; whereas for higher corrugation angles (60°), the ﬂow is almost 3D and highly turbulent even for very low Reynolds numbers (Re : 200). Flow simulation in cross corrugated structures has been reported by Fodemsky [151], Ciofalo et al. [152], and Hessami [153]. The numerical and ﬂow conditions are summarized in Table IV. Fodemsky [151] used a standard k— model to predict ﬂow and heat transfer in a corrugated channel. A body ﬁtted Cartesian grid was applied to model the channel geometry, and particular attention was paid to the grid in the corners where the upper and lower plates were in contact. The results were provided for corrugation angles between 15° and 20°, and the

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comparison with experimental results was only qualitative. The simulation provided the heat transfer coefﬁcient ﬁeld and outlined large heterogeneities. The highest values were located at the top of the corrugations where the cross section is minimized. This result is in agreement with the data of Gaiser and Kottke [11]. As for the 2D modeling, k— models are not accurate to predict separated ﬂow where recirculations zones exist, and the results are only qualitative. The study of Ciofalo et al. [152] is more extensive. Both numerical and experimental work was performed and several numerical methods were applied on a large range of operating and geometric conditions (see Table IV). Similar to Fodemsky’s work, a body-ﬁtted grid was used to model the complex geometry. The same grid was used for all the models, except for the standard k— model, where the wall function must be respected. This implies that the center of the control volume near the wall lies far out from the viscous sublayer. Ciofalo et al. [152] used a conventional deﬁnition, and the thickness of the viscous sublayer was based on plain tube correlations. This assumption is incorrect because the thickness of the viscous sublayer is not constant, a result of the large recirculation area. The results were compared to pressure drop measurements of Focke and co-workers [7, 10]. Laminar and k— computations were unsatisfactory, as they respectively underpredicted and overpredicted the friction factors for the complete range of Reynolds numbers investigated (1000 & Re & 5000). The best agreement with the experimental data was obtained with the low-Re turbulence model or LES; the difference was about <50% for the complete range of Reynolds numbers and for various corrugation angles. The agreement was better for Reynolds numbers over 2000 and for corrugation angles between 15° and 30°. For the average Nusselt number, the laminar and standard k— models are not satisfactory and the best agreement was found with the low-Re turbulence model and LES. These two models underpredicted the average Nusselt number by 20%, but the corrugation angle effect was well taken into account. For the local ﬂow structure and heat transfer, the comparison was made with experiments performed by Stasiek et al. [14] on a similar corrugated channel. The Nusselt number vs Re experimental results were reasonably well predicted by the low-Re turbulence model and LES, whereas the laminar and k— models were not satisfactory. The best agreement was obtained with LES. No quantitative comparison of the ﬂow structure was obtained and the accuracy of the different models was not fully validated. Nevertheless, it appears that LES allowed reproducing the ﬂow structure according to Ciofalo et al. [152]. Finally, Ciofalo et al. compared LES and DNS and observed that for their conﬁguration, the DNS models did not improve the results signiﬁcantly. Hessami [153] performed a three-dimensional study of heat transfer and

TABLE IV Summary of Numerical Methods, Operating Conditions, and Geometries of Chevron Trough Plates

427

Author

Method

Fodemsky [151] Ciofalo et al. [152]

Finite difference

k—

Finite volume

Laminar, k— Low—Re LES

Hessami [153]

Finite volume

Blomerius et al. [156] Sunde´n [157]

Finite volume Finite volume

Model

Laminar, k— Low—Re DNS laminar Low—Re RNG-k—

Grid

Boundary condition

Body ﬁtted Constant wall Cartesian 12 temperature Body ﬁtted Constant wall Cartesian 24 temperature

Inlet conditions

Reynolds number

Prandtl number

Fully developed

1600—3000

0.7

10—10

Fully developed developing ﬂow

Body ﬁtted Constant wall Fully developed Cartesian 32 temperature developing ﬂow Body ﬁtted Constant wall Developed Cartesian temperature and heat ﬂux 0.1;0.1;0.05 Constant wall Fully developed mm temperature 25;25;25 Constant wall Fully developed Cartesian temperature body ﬁtted

Corrugation angle

Validation

2

15°—20°

No

0.7

2—4

15°—75°

Nu, f Local Nu

200 —6000

7

3.85—5.86

30°, 45°, 60°

No

150—2000

0.7

3.5, 4.5, 5

45°

Nu and f

400—10000

4.33 and 12.8

3

0°, 22.5°, and 45°

Nu and f

Pitch/height

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pressure drop in a cross-corrugated heat exchanger, using the commercial CFD software CFX 4.1 (a version of FLOW3D software). Three models were considered: laminar, standard k— and low-Re k— turbulent models. A unit cell was represented by a 22;16;22 grid. Preliminary investigations showed additional blocks or cells were required at the inlet and outlet of the unit cell to obtain a stable solution. The ﬂow boundary condition was a fully developed velocity proﬁle, but no precision was given on the establishment ﬂow regime. Three corrugation angles (30°, 45° and 60°) and three pitch-toheight ratios (3.85, 4.55, and 5.56) were considered; the Reynolds numbers were covered up to 6000. For a corrugation angle of 60° (hard plates), comparison of the three models showed large differences. For heat transfer, both the laminar and standard k— models gave a very low dependency of the Nusselt number with the Reynolds number (Nu . Re ); in contrast, the low-Re turbulence model was very sensitive (Nu . Re ). These dependencies with the Reynolds number were not physical and may have come from the grid or boundary conditions used in this study. The trends on predicted Nu and f were not in agreement with the experimental work of Focke et al. [154] and Thonon et al. [12], and neither the Reynolds number effect nor the corrugation angle effect could be predicted by the model used by Hessami [153]. Other geometric parameters were studied by Hessami, but the main result of this study was that the boundary conditions and the numerical model have a large inﬂuence on the results, and that great care should be taken in modeling such complex geometries. Sawyers et al. [155] studied numerically, 2D (sinusoidal), and 3D (eggcarton, i.e., sinusoidal in the transverse direction as well) corrugated (wavy) channels. To avoid unsteady ﬂow, the study was limited to low Reynolds numbers (Re & 250). Both algebraic and numerical methods were applied to predict pathlines, pressure drop, and heat transfer. The authors analyzed several geometric conﬁgurations, but no comparison with experimental results was provided. Blomerius et al. [156] presented numerical results on 3D ﬂow ﬁeld and heat transfer in a corrugated channel. The Navier—Stokes were directly solved by the code, using an implicit—explicit scheme for the convective ﬂuxes. A reinjection procedure was adopted to obtain a fully developed ﬂow. Depending on the geometry, 80,000 to 150,000 grid points were used. The sensitivity of the results (heat transfer and pressure drop) with the gridding was up to 10% at Re : 1500. The corrugation angle was ﬁxed at 45°, and the pitch-to-height ratio was 3.5, 4.5, and 5. This latter value was selected for a comparison with the experimental results of Gaiser and Kottke [11]. For Reynolds numbers below 180—270 (depending on the pitch-to-height ratio), the ﬂow was fully laminar; for higher Reynolds numbers, the ﬂow became time-periodic self-sustained oscillatory, requiring time-dependent

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calculations. For high Reynolds numbers, a time averaging procedure was adopted to estimate heat transfer and pressure. At Re : 2000, the deviation between the predicted and experimental results was respectively 5 and 8% for the Nusselt number and friction factor. Sunde´n [157] analyzed the ﬂow structure and thermal and hydraulic performances of plate and frame heat exchangers using CFD. The geometry studied was representative of an industrial plate heat exchanger with two different corrugation angles (22.5° and 45°). A single unit cell was considered (volume between four contact points of chevron plates) to provide information on friction factors and Nusselt numbers. A steady laminar ﬂow model was applied for Reynolds numbers up to 2000 and a low-Reynolds k— or an RNG—k— model with wall functions for turbulent ﬂows. As outlined in Section IV, these turbulent models are more accurate for low Reynolds number unsteady ﬂows or turbulent ﬂows. A 3D body ﬁtted mesh was used, and great care was paid to model the contact points. To avoid singularities, the contact line between the upper and lower plates was replaced by a surface contact. For one of the geometries modeled, the numerical results were compared with experiments. The predicted Nusselt number was underpredicted by at most 25% while the friction factor was underpredicted by 17—40%. The author suggested using a more advanced turbulence model to get a higher accuracy. 1. L ocal Analysis of Flow and Heat Transfer Phenomena in Corrugated Channels Based on the numerical analysis, local information on the ﬂow structure is now available and allows analyzing and qualitatively characterizing heat transfer mechanisms in complex geometries. The experimental studies (see Section II, B) have shown complex ﬂow patterns and transition from steady laminar to unsteady laminar and turbulent ﬂow at low Reynolds numbers. The numerical studies have conﬁrmed these observations, but the mechanism of eddy creation can only be more precisely deﬁned using unsteady numerical models. The 2D studies were performed by Tochon and Mercier [139], and 3D studies by Ciafolo et al. [152] and Blomerius et al. [156]. The information gained from these studies is discussed next. For the 2D geometry, near the entrance and at each corrugation, the ﬂow separates from the wall and generates a large recirculation zone. Because of turbulent instabilities, this zone grows larger and larger and then breaks suddenly in two parts. One part creates a vortex, which is released in the ﬂow (eddy no. 1 in Fig. 15) in the downward direction, as the other part remains in the protected region. Interactions between the rotating eddies in the center part lead to a pairing (eddy nos. 4 and 5 in Fig. 15). The same

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process is repeated. When detached from the wall, eddies are convected in the ﬂow, under the inﬂuence of other eddies. They can keep their coherent structure for several microseconds. Their lifetime has been found to be a little longer numerically than in the experiment. This discrepancy can be explained by the limitation of the two-dimensional model, although turbulence is three-dimensional in nature. This 3D character is especially true in the region of an adverse pressure gradient, where partial disorganization of the coherent structures seems to occur in the experimental visualization. Nevertheless, the main mechanisms of eddy creation and motion are very similar to those observed experimentally. This alternative generation of vortices on the lower and upper walls of the domain of calculation liberates coherent eddies, with either clockwise or anticlockwise rotational motion. As a result, it is also apparent that this geometry is able to ensure good turbulent mixing of the ﬂuid after only a few corrugations. Concerning the thermal aspects (Fig. 16), the alternating vortices lead to a uniform temperature inside the channel by mixing the warm liquid of the recirculating areas with the main ﬂow. Indeed, local investigations show that the heat transfer coefﬁcient is low inside the recirculating areas by means of the conduction process and is high at the reattachment point. So, because of the turbulent behavior of the ﬂow, the hot pocket oscillates, becomes unstable, and is convected to the bulk ﬂow. These new eddies interact with each other and generate large coherent structures that create a fully developed turbulent ﬂow. So the vortex shedding from the warm area near the wall to the core is the main mechanism for heat transfer. For 3D geometries, the ﬂow is more complex to analyze, as the 3D effects are present even for low corrugation angles. Ciafalo et al. [152] have investigated a cross-corrugated channel, using an LES model, with the corrugation angle between 18.5° and 30°. For these typical angles, according to Focke and Knibbe [10], a furrow ﬂow should exist, and the mixing intensity should increase with the Reynolds number. From the velocity ﬂuctuations, a Strouhal number can be deduced (Sr : F;t, where F is the frequency and t the time necessary to cross a unitary cell at the average velocity U). For all values of corrugation angles (18.5° to 30°) and Reynolds numbers (780 to 4250), a constant value of Sr : 3 was found. A detailed examination of the instantaneous and ﬂuctuating ﬂow ﬁelds (at Re : 2450) indicated that the eddies were highly three-dimensional in the central part of the cell, having their largest component in the vertical direction (normal to the channel). These eddies were generated by mixing of two furrow ﬂows (upper and lower plates). The intensity of the ﬂuctuations was found to increase with both Reynolds number and corrugation angle. Blomerius et al. [156] investigated a corrugated channel with a 45°

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corrugation angle. According to previous visualization experiments, the ﬂow structure is strongly affected by the Reynolds number and becomes highly three-dimensional. A Fourier analysis was performed on the u-component of the velocity (main ﬂow direction). For Reynolds numbers of 255 and 425, a dominant frequency with corresponding Strouhal numbers of 0.5 and 1.3 was found. For a higher Reynolds number based on the hydraulic diameter (Re : 1700), no dominant frequency could be detected and the ﬂow was aperiodic, which is the characteristic of turbulent ﬂow. For a low Reynolds number (Re : 180), the midplane shear layer was relatively stable, whereas for a higher Reynolds number (Re : 1800) considerable mixing between the two streams was noticed. The maximum velocities were nearly twice the average ﬂuid velocity. This higher mixing at Re : 1800 led to a much more homogeneous temperature ﬁeld over the whole cross-section. In the region just downstream from the contact point, the ﬂuid temperature was almost the same as the wall temperature, indicating a stagnant zone. This size of this region was strongly dependent on the Reynolds number. The foregoing information is fundamental for selecting the most appropriate geometry for given process conditions or to develop new geometries that could limit the size of these stagnant and recirculation areas. In the food industry, pasteurization and sterilization are often achieved in plate heat exchangers, and it is fundamental to have an equal (homogeneous) residence time for all ﬂuid particles and no dead zones where fouling could initiate. The numerical studies have shown that for Reynolds numbers higher than 1500 and for corrugation angles higher than 45°, the recirculation zones are limited in size and the ﬂuid temperature is much more homogeneous. Furthermore, these channels have a very high local shear stress, which help mitigating fouling. 2. Summary of 2D and 3D Modeling of Wavy Channels and Chevron Trough Plates For 2D geometries (corrugated wavy channels), several grid schemes have been applied, but the major difference lies in the numerical method used. Laminar models are limited to very low Reynolds numbers (Re & 350) as instabilities occurs for higher Reynolds numbers. Furthermore, in fully developed turbulent regimes, conventional turbulent models can only give overall information, as the ﬂow is inaccurately predicted in the recirculation zones. As a result, to cover the entire range of the Reynolds number and to be independent of ﬂow patterns, advanced methods that enable prediction of separated ﬂows must be applied. The low Reynolds number turbulent ﬂow model [126] appears to constitute progress in the simulation of ﬂows

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in complex geometries, but its validity is not established. DNS methods [73] are interesting, but 2D models underestimate the turbulent diffusion and efforts must be applied to 3D modeling (i.e., the realistic chevron plates). Concerning 3D modeling, the studies have covered a wide range of corrugation angles and Reynolds numbers, and the agreement compared to experimental results (average values of the friction factor and Nusselt number) is within 10 to 50%. Local heat transfer coefﬁcients and wall shear stress are still not accurately predicted. Hessami [153] and Sunde´n [157] used various turbulence models, but the accuracy was relatively poor. Nevertheless, the general trends for heat transfer and pressure drop were well predicted. Ciofalo et al. [152] used several numerical models and claimed that the low Reynolds number turbulent ﬂow model or LES gave satisfactory results. The latter method is computationally more expensive but provides instantaneous information. Blomerius et al. [156] used a DNS method and claimed a maximum deviation of 5 and 8%, respectively for Nusselt numbers and friction factors. These results are encouraging, and advanced turbulent models implemented in commercial CFD codes should be tested for complex geometries such as cross-corrugated heat exchangers. As computational costs will decrease, LES or DNS methods must be developed, and careful validation with local measurements (velocity and turbulence intensity) must be performed. Progress in modeling ﬂow and heat transfer will enable study of complex geometries. More sophisticated experiments will be required for validation of modeling and numerical results. A comparison of CFD simulations between researchers, using different research and commercial codes and techniques on standardized geometries, and a comparison with validated laboratory and industrial data must be an objective for CFD to be a design, analysis, and optimization tool for heat exchangers.

VI. Conclusions A comprehensive review is made of numerical analysis of some of the important surface geometries of compact heat exchangers: offset and louver ﬁn geometries used in plate-ﬁn exchangers, wavy corrugated and furrowed ﬁns/channels in tube-ﬁn and plate heat exchangers, and chevron plates in plate heat exchangers. Speciﬁc understanding obtained from the numerical studies of each ﬁn/plate geometry is summarized at the end of each ﬁn/plate geometry section. Also, some of the known physics of ﬂow from experimentation and ﬂow visualization is presented to further assist in the reﬁnement in the numerical analysis. Although numerical analysis has progressed

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considerably in the past 20 years, most of the numerical analyses reported are 2D and/or simpler than the real complex ﬂows. Most of the ﬂows encountered in CHE applications are unsteady, often with ﬂow separation and recirculation zones, and the Reynolds number range is typically 100—2000 for most compact heat exchanger surfaces. The separation and reattachment of the ﬂow and vortices in the wake of the ﬁns affect both heat transfer and pressure drop, and these mechanisms cannot be predicted with time average modeling (conventional k— models). Unsteady laminar model requires a very ﬁne description of the geometry, and computation capacity limits its application to 2D modeling. There is no clear evidence as yet that the ﬂows in this Reynolds number range are unsteady laminar or low Reynolds number turbulent ﬂows, although it is possible that gradually the unsteady laminar ﬂows associated with the interruptions become low Reynolds number turbulent ﬂows. Hence, more sophisticated experimentation is required to determine the ﬂow characteristics. At the same time, RSM and LES models should be further developed to simulate the local ﬂow and heat transfer characteristics of compact heat exchanger surfaces. See Section IV for speciﬁc conclusions/recommendations on turbulence models. Since the art of compact heat exchanger surfaces has reached an asymptotic level, further enhancement and innovation in those surfaces will come from the detailed accurate numerical analysis. Numerical analysis is also needed to study the effects of minor changes in the ﬁn proﬁle (such as those due to the aging of manufacturing tools). Manufacturers of heat exchangers in the automotive industry are active in this area, but the numerical methods proposed in commercial software require some careful validation with experimental data. A similar conclusion was also reported earlier (which came to the authors’ attention) by Baggio and Fornasieri [158]. It is a challenge to numerical analysts to numerically simulate highly complex ﬂows accurately to provide a basic understanding of ﬂow and heat transfer phenomena for their exploitation in designing new and improved heat transfer surfaces.

Acknowledgments We gratefully acknowledge Mr. J. P. Chevalier of CEA-Grenoble, DTP/GRETh, Grenoble, France, for his assistance in preparing Section IV on turbulence models. We appreciate very much a careful and extensive review of our chapter by Dr. F. Ladeinde of Thaerocomp Technical Corp of Stony Brook, NY, USA. We are also thankful to Prof. W. Rodi of the University of Karlsruhe, Germany, Dr. S. Sazhin of the University of Brighton, UK, Prof. G. Biswas of the Indian Institute of Technology Kanpur, India, and Dr. G. Xi of Daikin Industries, Inc., Osaka, Japan, for their critical review of Section IV on turbulence models.

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Nomenclature A A> A b

c N D

D D E F F f G

G

G

h j K k l l m5 Nu P

heat transfer surface area on one ﬂuid side, m constant of Eq. (11), m minimum free ﬂow area on one ﬂuid side of a heat exchanger, m spacing between plates for corrugated ﬁns, mean gap, or plate spacing in a plate heat exchanger, m speciﬁc heat of ﬂuid at constant pressure, J/kg K particular derivative: D : ; u· , 1/s t Dt equivalent diameter, D :2b, m hydraulic diameter, D :4V /A :4A L /A, m constant of Eq. (28) frequency, 1/s structure function deﬁned in Eq. (42), m/s Fanning friction factor, dimensionless ﬂuid mass velocity based on minimum free ﬂow area, G:m5 / A , kg/ms generation of turbulent kinetic energy due to buoyancy, m/s generation of turbulent kinetic energy due to the mean velocity gradients, m/s convective heat transfer coefﬁcient, W/mK Colburn factor, St Pr, dimensionless constant of Eq. (28) kinetic energy of turbulence per unit mass, J/kg (or m/s) mixing length, m offset strip length, louver, length, or louver pitch, m ﬂuid mass ﬂow rate, kg/s Nusselt number, hD /k, dimensionless mean pressure component of ﬂuid static pressure, Pa

Pr p p D p q R Re

Re Rel

Re r S Sr St s T t t t

U u, v, w u, v, w V x Y

ﬂuid Prandtl number, dimensionless ﬂuctuating pressure component, Pa ﬁn pitch, m tube pitch of ﬂat tubes in Fig. 11b, m velocity scale, m/s correction term in Eq. (18), m/s Reynolds number based on hydraulic diameter, GD /, dimensionless Reynolds number based on the equivalent diameter, GD /, dimensionless Reynolds number based on the interrupted length, Gl/, dimensionless Reynolds number based on the mixing length and mean velocity, Ul/, dimensionless correlation distance used for structure function, m strain tensor, m/s Strouhal number, Sr : Ft , dimensionless Stanton number, St : h/Gc , dimensionless ﬁn spacing, s : p 9 (, m D ﬂuid temperature, K time, s characteristic time, s characteristic time, characteristic length of corrugation divided by the bulk velocity, see Figs. 15 and 16, s mean velocity component, m/s ﬂuctuating velocity components, m/s velocity components, m/s void volume on one ﬂuid side, m axial direction, m contribution of the ﬂuctuating dilatation in the dissipation rate, m/s

numerical analysis of che surfaces y y>

normal distance from the wall, m wall coordinate y( /)/v, dimensionless

-

-* ( 1! 1!

wall shear stress, Pa rotation velocity, 1/s vorticity, s\

Subscripts

Greek Letters -

435

thermal diffusivity, m/s cubic dilatability in Section IV, 1/K corrugation angle for chevron plates, measured from the axis parallel to the plate length, -90°, (see Fig. 3), deg lRe/s, dimensionless ﬁlter width, m boundary layer thickness in Section IV, ﬁn thickness in all other sections, m dissipation rate of turbulent kinetic energy, J/kg s (or m/s) ﬂuid temperature, K louver angle for multilouver ﬁns (see Fig. 5a), deg molecular viscosity, m/s turbulent viscosity, m/s ﬂuid density, kg/m turbulent Prandtl number subgrid scale strain, m/s

eff i j k

effective axial direction index lateral direction index vertical direction index related to the dissipation rate related to the kinematic viscosity

Abbreviations ASM CFD DNS EVM LES NLEVM OSF RANS RNG RSM SGS

algebraic stress model computational ﬂuid dynamics direct numerical simulation eddy viscosity model large eddy simulation nonlinear eddy viscosity model offset strip ﬁn Reynolds averaged numerical simulation renormalization group Reynolds stress model subgrid scale

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130. Ergin, S., Ota, M., Yamaguchi, H., and Sakamoto, M. (1996). A numerical study of the effect of interwall spacing on turbulent ﬂow in a corrugated channel, HTD-Vol. 333, 2, 47—54. ASME, New York. 131. Sparrow, E. M., and Comb J. W. (1983). Effect of interwall spacing and ﬂuid inlet conditions on corrugated wall heat exchanger, Int. J. Heat Mass Transfer 26, 993—1005. 132. Ergin, S., Ota, M., Yamaguchi, H., and Sakamoto, M. (1997). Analysis of periodically fully developed turbulent ﬂow in a corrugated duct using various turbulence models and comparison with experiments. JSME Centennial Grand Congress, Int. Conf. on Fluid Eng., 1527—1532, Tokyo, Japan. 133. Be´reiziat, D., Devienne, R., and Lebouche´, M. (1995). Local ﬂow structure for nonNewtonian ﬂuids in a periodically corrugated wall channel. J. Enhanced Heat Transfer 2, 71—77. 134. Farhanieh, B. and Sunde´n, B. (1992). Laminar heat transfer and ﬂuid ﬂow in streamwiseperiodic corrugated square ducts for compact heat exchangers. In Compact Heat Exchanger for Power and Process Industries, HTD-Vol. 201, pp. 37—49. ASME, New York. 135. Asako, Y., Faghri, M., and Sunde´n, B., (1996). Laminar ﬂow and heat transfer characteristics of a wavy duct with a trapezoidal cross section for heat exchanger application. In 2nd European Thermal Conference (G. P. Celata, P. Di Marco, and A. Mariani, Eds.), Vol. 2, pp. 1097—1104. Edizioni ETS, Italy. 136. Shah, R. K., and Bhatti, M. S., (1987). Laminar convective heat transfer in ducts. In Handbook of Single-Phase Heat Transfer (S. Kakaç, R. K. Shah, and W. Aung, Eds.). John Wiley, New York, Chapter 3. 137. McNab C. A., Atkinson K. N., Heikal M. R., and Taylor N. (1998). Numerical modelling of heat transfer and ﬂuid ﬂow over herringbone corrugated ﬁns. Heat Transfer 1998, Proc. 11th Int. Heat Transfer Conf. 6, 119—124. 138. Abou-Madi (1998). A computer model for mobile air conditioning system. Ph. D. Thesis, University of Brighton, Brighton, UK. 139. Tochon, P., and Mercier, P. (1999). Thermal—hydraulic investigations of turbulent ﬂows in compact heat exchangers. In Compact Heat Exchangers and Enhancement Technology for the Process Industries (R. K. Shah, Ed.). Begell House Inc., New York, pp. 97—101. 140. Atkinson, K. N., Drakulic´, R., Heikal, M. R., and McNab, C. A. (1998). Applications of numerical ﬂow modelling to high performance heat transfer surfaces. Heat Transfer 1998, Proc. 11th Int. Heat Transfer Conf. 5, 333—338. 141. Sobey, I. J. (1980). On ﬂow through furrowed channels: Part 1. Calculated ﬂow patterns. J. Fluid Mech. 125, 359—373. 142. Stephanoff, K. D., Sobey, I. J., and Bellhouse, B. J. (1980). On ﬂow through furrowed channels. Part 2. Observed ﬂow patterns. J. Fluid Mech. 125, 359—373. 143. Sobey, I. J. (1982). Oscillatory ﬂows at intermediate Strouhal number in asymmetric channels. J. Fluid Mech. 125, 359—373. 144. Sobey, I. J. (1983). The occurrence of separation in oscillatory ﬂow. J. Fluid Mech. 134, 247—257. 145. Sparrow, E. M., and Prata, A. T. (1983). Numerical solutions for laminar ﬂow and heat transfer in a periodically converging—diverging tube, with experimental conﬁrmation. Num. Heat Transfer 6, 441—461. 146. Blancher, S., Batina, J., Creff, R., and Andre, P., (1990). Analysis of convective heat transfer by a spectral method for an expanding vortex in a wavy-walled channel. Proc. 9th Int. Heat Transfer Conference, Heat Transfer 1990, 2, 393—398. 147. Guzman, A. M., and Amon, C. H. (1994). Transition to chaos in converging—diverging channel ﬂows: Ruelle—Takens—Newhouse scenario. Phys. Fluids 6, 1994—2002.

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148. Guzman, A. M., and Amon, C. H. (1996). Dynamical ﬂow characterization of transitional and chaotic regimes in converging-diverging channels. J. Fluid Mech. 321, 25—57. 149. Wang, G., and Vanka, S. P. (1995). Convective heat transfer in periodic wavy passages. Int. J. Heat Mass Transfer 38, 3219—3230. 150. Nishimura, T., Ohori, Y., and Kawamura, Y. (1984). Flow characteristics in a channel with symmetric wavy wall for steady ﬂow. J. Chem. Eng. Jap. 17, 466—471. 151. Fodemsky, T. R. (1990). Computer simulation study of thermal hydraulic performance of corrugated ducts. Proc. 9th Int. Heat Transfer Conf., Heat Transfer 1990, 3, 241—246. 152. Ciofalo, M., Stasiek, J., and Collins, M. W. (1996). Investigation of ﬂow and heat transfer in corrugated passages — II. Numerical results. Int. J. Heat Mass Transfer 39, 165—192. 153. Hessami, M. A. (1997). Numerical study of heat transfer and pressure loss in crosscorrugated plate heat exchangers. In Proc. 3rd ISHMT-ASME Heat Mass Trans. Conf. (G. Biswas, S. Srinivasa Murthy, K. Muralidhar, and V. K. Dhir, Eds.). Tata-McGrawHill, New Delhi, pp. 795—802. 154. Focke, W. W., Zachariades, J., and Olivier, I. (1985). The effect of the corrugation inclination angle on the thermohydraulic performance of plate heat exchanger. Int. J. Heat Mass Transfer 28, 1469—1479. 155. Sawyers, D. R., Sen, M., and Chang, H-S. (1998). Heat transfer enhancement in three dimensional corrugated channel ﬂow. Int. J. Heat Mass Transfer 41, 3559—3573. 156. Blomerius, H., Ho¨sken, C., and Mitra, N. K. (1999). Numerical investigation of ﬂow ﬁeld and heat transfer in cross-corrugated ducts. ASME J. Heat Transfer 121, 314—321. 157. Sunde´n, B. (1999). Flow and heat transfer mechanisms in plate-frame heat exchangers. In Heat Transfer Enhancement of Heat Exchangers (S. Kakaç, Ed.), NATO ASI Series E Applied Sciences 355, pp. 185—206. Kluwer Academic Publishers, Netherlands. 158. Baggio, E., and Fornasieri, E. (1994). Air-side heat transfer and ﬂow friction: theoretical aspects. In Recent Developments in Finned Tube Heat Exchangers, (Ch. Marvillet, General Editor). DTI, Energy Technology, Denmark, pp. 91—159.

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AUTHOR INDEX Numerals in parentheses following the page numbers refer to reference numbers cited in the text.

A Abbrecht, P. H., 262(22), 292(22), 310(22), 354(22) Abdel-Khalik, S. I. (Chapter Author), 145, 147(12), 150(12; 34; 35), 151(12), 152(12), 153(12), 154(12), 155(35), 156(12; 34), 157(12), 158(12), 159(12), 160(12), 161(12), 162(12), 164(12), 167(34), 168(34), 176(35), 177(34; 35), 185(12; 35), 187(12), 188(34), 191(35), 193(132; 135; 139), 196(132), 197(132), 198(132; 135), 199(132), 200(132), 202(132; 135; 139), 203(135; 139), 204(135), 205(132; 135; 139), 210(174), 211(174), 214(174), 215(174), 216(174), 218(174), 219(174), 224(174), 226(208), 227(208), 230(208), 231(208), 233(208) Abdelmessih, A. H., 198(133) Abdollahian, D., 225(199), 235(223; 225) Abdullah, Z., 194(125), 198(125), 200(125), 202(125) Abou-Madi, 421(138) Abraham, M. A., 151(36; 37) Abriola, L. M., 15(49) Achaichia, A., 370(2), 407(96; 109), 408(96; 101), 410(109), 412(116), 414(101) Achdou, Y., 69(171) Achenbach, E., 91(186), 92(186), 95(186) Acrivos, A., 11(32) Adler, P. M., 2(13) Adzerikho, K. S., 57(142), 60(142) Ahmad, S. Y., 221(194) Ahmed, N. U., 124(214; 215; 216; 217), 125(214; 225) Akers, W. W., 187(100) Akhiezer, A. I., 38(99) Akira, T., 415(120) Alamgir, M. D., 229(210), 236(210)

Al-Daini, A. J., 407(103) Alfredsson, P. H., 381(29) Al-Hayes, R. A. M., 200(137) Ali, M. I., 150(29; 30), 155(29; 30), 171(29; 30), 174(30), 176(30), 177(30), 185(29; 30), 189(29; 30), 190(30) Ali, M. M., 372(5) Allmaras, S., 383(33), 392(33) Al-Nimr, M. A., 56(136) Amon, C. H., 424(147), 425(148) Amos, C. N., 225(199), 226(204), 227(204), 228(204), 229(204), 232(204), 236(204), 239(204), 240(204), 241(204) Anderson, J. D., 376(17) Anderson, T. B., 1(1) Andre, P., 424(146) Anisimov, S. I., 39(100; 101) Anita, S., 125(224) Antal, S. P., 208(152) Antonia, R. A., 28(77) Antoniou, A., 412(115; 117) Aoki, H., 407(106; 107), 409(106; 107) Arai, N., 303(85), 333(85), 334(85), 335(85), 336(85), 346(117), 349(85), 356(85) Ardron, K. H., 232(212) Arkhipov, V. V., 210(165), 211(165) Armstrong, R. C., 91(191), 94(191), 95(191) Arpaci, V. S., 56(136) Asako, Y., 388(126), 416(123), 417(123; 126), 418(123; 125; 126), 419(123; 125; 126), 420(126), 421(135), 431(126) Asano, Y., 379(27) Ashikhmin, S. R., 81(176), 82(176) Atkinson, K. N., 396(137), 407(111; 112), 411(111; 112; 114), 413(114), 417(140), 421(137; 140) Avellaneda, M., 69(171) Aziz, K., 148(13), 158(41), 161(41), 175(41), 177(41)

445

author index

446 B

Baggio, E., 433(158) Balakrishnan, A. V., 124(221), 125(221) Baldwin, B. S., 382(32) Baldwin, S. J., 407(103) Baliga, B. R., 398(77) Barajas, A. M., 150(24), 153(24), 154(24), 158(24), 159(24), 166(24) Barakat, R., 33(82) Barenblatt, G. I., 262(21) Barnea, D., 148(19), 150(23), 154(23), 156(23), 158(23), 161(23), 163(23) Batchelor, G. K., 274(41) Batina, J., 424(146) Beamer, H. E., 413(118), 414(118) Beavers, G. S., 81(177), 82(177), 83(177), 84(177) Behringer, R. P., 26(71) Bejan, A., 112(203) Bellhouse, B. J., 422(142) Bellows, K. D., 413(118), 414(118) Beran, M. J., 98(196) Be´reiziat, D., 373(13), 417(133), 419(129) Bergles, A. E., 114(209), 193(131), 194(126), 195(129), 196(129; 131), 197(129), 198(129), 206(131), 207(131), 208(131), 210(131; 164; 173), 211(131; 164; 173), 212(173; 179), 213(131; 173), 215(131; 173), 216(173), 219(173) Bezprozvannykh, V. A., 26(74) Bhatti, M. S., 421(136) Bibeau, E. L., 192(118), 200(118) Bilicki, Z., 236(229) Bird, R. B., 77(173), 78(173), 79(173), 269(35), 280(35), 351(35) Bisset, D. K., 28(77) Biswas, A. K., 385(38) Black, H. S., 192(112) Blain, C. A., 2(8), 5(8), 23(8), 60(8) Blancher, S., 424(146) Blasick, A. M., 193(139), 202(139), 203(139), 205(139) Blasius, H., 279(53) Blinkov, V. N., 236(228) Blomerius, H., 396(156), 427(156), 428(156), 429(156), 430(156), 432(156) Boelter, L. M. K., 338(103) Bohren, C. F., 57(144), 60(144) Bolle, L., 236(229)

Bolstad, M. M., 184(89) Bories, S., 150(33), 155(33), 171(33), 176(33), 185(33), 189(33), 190(33) Bornea, D., 148(17), 156(17), 166(17), 171(17), 173(17) Botten, L. C., 52(126), 57(155; 156; 157) Bouassinesq, J., 269(33) Boure´, J. A., 194(126), 232(213) Bowers, M. B., 185(95), 186(95), 210(95), 211(95), 214(95), 215(95), 219(95), 224(95) Bowring, R. W., 218(178) Boyd, R. D., 209(156; 157), 210(156; 157; 167; 168; 178), 211(156; 167), 212(167; 168), 215(156; 157), 216(157) Brauner, N., 147(10), 161(10) Breaux, D. K., 310(90), 354(90) Bremhorst, K., 391(58), 418(58) Brereton, G. J., 28(76) Bretherton, F. B., 146(8), 150(8) Breuer, M., 393(60) Burkhart, T. H., 275(49) Burns, J. A., 115(211; 212) Butkovski, A. G., 124(220) Butterworth, D., 112(204), 115(204) Button, B. L., 407(104) Buyevich, Y. A., 59(165)

C Caceres, M. O., 37(92) Caira, M., 218(184) Calata, G. P., 213(181), 215(181) Camarero, R., 22(51) Carbonell, R. G., 7(31), 8(31), 9(31), 15(31; 40; 46), 18(46), 34(40; 46; 89), 35(90), 107(40), 126(40) Carey, V. P., 191(107), 192(107), 205(107) Caruso, G., 218(184) Catton, I. (Chapter Author), 1, 2(16; 17; 18; 19; 20; 21; 22; 26; 27; 28), 3(19; 21), 10(21), 11(16; 18; 20; 26), 15(18), 16(16; 18; 21), 21(21), 23(24), 25(16; 20), 26(16; 17; 18; 19; 20; 21), 30(16; 21), 31(19; 26), 36(16; 17; 20; 26; 27), 51(114), 52(23), 57(19; 20; 28; 114; 158; 159; 161; 163), 60(21; 114), 61(114), 62(21), 65(21; 28), 66(166), 68(16; 20), 69(16; 20; 21; 23; 25;

author index 26), 70(16; 17; 20; 21), 71(16; 20), 79(21), 80(21), 81(23), 96(21), 97(21), 102(21), 110(21), 111(114), 116(16; 20; 21; 23; 28), 118(16; 19), 119(16; 20), 123(16; 17; 23), 124(19) Celata, G. P., 210(172; 175), 211(172; 175), 212(180), 213(172; 175), 217(172; 175), 221(193), 223(193) Cerro, R. L., 151(36; 37) Chan, C., 111(201), 287(71), 295(71; 77; 78), 298(71), 300(71), 302(71; 77), 326(100), 348(77), 351(71; 77), 355(100) Chandrasekhar, S., 258(3) Chang, H.-S., 428(155) Chang, Y. J., 371(3) Chao, J., 276(51) Chen, G., 46(110; 111) Chen, H. C., 391(57), 399(57) Chen, Z.-H., 184(92), 185(92) Chen, Z.-Y., 184(92), 185(92) Cheng, H., 96(193), 97(193) Cheng, P., 32(79; 80), 145(4) Chexal, B., 235(223; 225) Chhabra, R. P., 78(174), 79(174) Chiffelle, R. J., 37(93), 39(93) Choi, B., 292(75), 293(75) Choi, H. Y., 309(88) Christ, C. L., 240(233) Churchill, S. W. (Chapter Author), 111(200; 201), 184(93), 255, 262(22), 264(26; 27), 281(58), 285(64; 65), 287(71), 288(58), 292(22; 75), 293(75), 295(71; 77; 78; 79), 296(80), 298(71; 83), 299(80), 300(71), 302(58; 71; 77), 303(58; 83; 85), 304(86), 310(22), 314(91), 326(100; 100a), 327(100a), 329(100a), 330(100a), 331(100a), 332(100a), 333(85; 100a), 334(85), 335(85), 336(85), 340(100a; 112), 342(112), 346(117), 348(77), 349(85; 100a), 350(100a), 351(71; 77; 80), 354(22), 355(100; 100a), 356(85) Ciofalo, M., 384(152), 388(152), 393(152), 423(14), 425(152), 426(14; 152), 428(152), 429(152), 430(152), 432(152) Colburn, A. P., 338(106) Cole, R., 191(108), 205(108) Colebrook, C. F., 274(63), 286(63) Collier, J. G., 148(14), 182(14), 191(14) Collier, R. P., 202(202), 225(202; 203), 226(203), 230(202), 231(202), 235(203;

447

202), 236(203) Collins, M. W., 384(152), 388(152), 393(152), 423(14), 425(152), 426(14; 152), 428(152), 429(152), 430(152), 432(152) Comb, J. W., 420(131) Coppin, P. A., 25(64; 65; 66) Cornish, R. J., 189(101) Cornwell, J. D., 185(94), 186(94) Corson, D. R., 57(145), 60(145) Coulson, J. M., 338(105), 339(105) Cowell, T. A., 370(2), 371(4), 375(15), 407(96; 109), 408(96; 101), 410(109), 411(114), 412(115), 413(114), 414(101) Cox, S. G., 371(4) Cox, S. J., 52(120) Craft, T. J., 388(51) Crapiste, G. H., 15(41), 22(41), 30(41), 35(41), 64(41) Crawford, M. E., 387(44) Creff, R., 424(146) Crosser, O. K., 187(100) Cumo, M., 210(172; 175), 211(172; 175), 212(180), 213(172; 175), 217(172; 175), 221(193), 223(193) Curtain, R. F., 125(222)

D Dagan, R., 236(227) Daleas, R. S., 210(164), 211(164) Damianides, C. A., 147(11), 150(11), 153(11), 154(11), 157(11), 158(11), 159(11), 161(11), 162(11), 164(11) Danckwerts, P. V., 343(116) Danov, S. N., 303(85), 333(85), 334(85), 335(85), 336(85), 349(85), 356(85) Da Prato, G., 124(219) Davenport, C. J., 407(103), 408(97) Dawes, D. H., 57(156) Deans, H. A., 187(100) Deev, V. I., 210(165), 211(165) Devienne, R., 373(13), 417(133) Dhir, V. K., 145(3) Dittus, R. W., 338(103) Dix, G. E., 192(117), 199(117) Dobson, D. C., 52(120) Dombrovsky, L. A., 56(140), 60(140) Dowling, M. F., 193(132; 139), 196(132),

author index

448

197(132), 198(132), 199(132), 200(132), 202(132; 139), 203(139), 205(132; 139) Downar-Zapolski, Z., 236(229) Downie, J. H., 371(4) Drakulic, R., 367(113), 375(15), 407(111; 112; 113), 411(111; 112; 113; 114), 412(113), 413(114), 417(140), 421(140) Drew, D. A., 22(55), 23(56), 206(148; 149) Drolen, B. L., 56(135) Dukler, A. E., 148(15; 16; 17; 18), 156(15; 17), 161(15), 162(15), 166(17), 169(16), 171(17), 173(17) Dullien, F. A. L., 2(12), 81(180), 85(180) Duncan, A. B., 191(111) Dybbs, A., 26(68), 89(184), 91(184), 92(184), 95(184)

E Eckelmann, H., 287(70) Edwards, R. V., 26(68) Einav, S., 386(41) Einstein, A., 258(4), 325(4) Ekberg, N. P., 150(35), 155(35), 176(35), 177(35), 185(35), 191(35) Elias, E., 225(200), 231(200), 232(200), 233(200), 236(227) Elperin, T., 325(97a) El-Sayed, M. S., 81(180), 85(180) Elsayed-Ali, H. E., 39(106), 46(106) Ergin, S., 67(167), 77(167), 81(167), 384(130), 388(132), 417(130; 132), 419(130), 420(130)

Fattorini, H. O., 125(223) Feburie, V., 232(216), 237(216) Fedoseev, V. N., 91(190), 94(190), 95(190) Ferziger, J. H., 393(60), 395(68) Figotin, A., 43(124; 125), 45(124), 52(122; 124), 54(123; 125), 55(123), 57(122; 124; 125) Flaherty, J. E., 208(152) Fleming, W. H., 124(218) Flik, M. I., 46(112) Focke, W. W., 373(7; 8; 10), 374(10), 426(7; 10), 428(154), 430(10) Fodemsky, T. R., 384(151), 425(151), 428(151) Fornasieri, E., 433(158) Fourar, M., 150(33), 155(33), 171(33), 176(33), 185(33), 189(33), 190(33) Fourier, J. B., 264(23) Fox, R. F., 33(82) France, D. M., 150(28), 155(28), 171(28), 175(28), 192(113; 114; 115) Franco, J., 236(229) Freeman, J. R., 280(54), 281(54) Friedel, L., 191(105), 226(105), 227(105), 229(105), 230(105), 236(105), 239(105), 240(105) Fritz, A., 232(211) Fujii, M., 226(206), 227(206) Fujimoto, J. G., 39(105) Fukagawa, M., 114(210) Fukano, T., 150(25), 154(25), 157(25), 159(25), 161(25), 162(25), 165(25), 169(25), 185(25), 186(25), 187(25) Fushinobu, K., 37(91) Futagami, S., 400(85), 405(85)

G F Faghri, M., 388(126), 416(122; 123; 124), 417(123; 126), 418(123; 125; 126), 419(123; 125; 126), 420(126), 421(124; 135), 431(126) Fairbrother, F., 146(5), 150(5) Fallon, B., 393(70) Fan, L. T., 343(114) Fand, R. M., 26(67) Farhanieh, B., 421(134) Farone, W. A., 57(153)

Gaiser, G., 373(11), 426(11), 428(11) Galitseysky, B. M., 91(189), 93(189) Galloway, J. E., 221(190) Garg, C. K., 418(127), 423(127) Garrels, R. M., 240(233) Gatski, T. B., 386(40), 388(50) Gelhar, L. W., 33(83) Geng, H., 232(217; 218), 236(217), 239(218), 241(218) Georgiadis, J. G., 26(71) Germore, 395(67)

author index Ghiaasiaan, S. M. (Chapter Author), 145, 147(12), 150(12; 34; 35), 151(12), 152(12), 153(12), 154(12), 155(35), 156(12; 34), 157(12), 158(12), 159(12), 160(12), 161(12), 162(12), 164(12), 167(34), 168(34), 176(35), 177(34; 35), 183(86), 185(12; 35), 187(12), 188(34), 191(35), 193(132; 135; 139), 196(132), 197(132), 198(132; 135), 199(132), 200(132), 202(132; 135; 139), 203(135; 139), 204(135), 205(132; 135; 139), 210(174), 211(174), 214(174), 215(174), 216(174), 218(174), 219(174), 224(174), 226(208), 227(208), 230(208), 231(208), 232(217; 218), 233(208), 236(217), 239(218), 241(218) Ghosh, D., 413(118), 414(118) Gibson, M. M., 388(53) Ginzburg, V. L., 38(98) Giot, M., 232(211; 216), 237(216) Gladkov, S. O., 40(107), 43(107), 45(107) Gmitter, T. J., 52(117) Godin, Yu. A., 43(124), 45(124), 52(124), 57(124) Goldenfeld, N., 262(21) Goodson, K. E., 46(112; 113) Gortyshov, Yu. F., 81(175; 176), 82(176), 91(175), 94(175) Gose, G. C., 231(219), 233(219) Gotoh, N., 226(207), 227(207) Govan, A. H., 224(196) Govier, F. W., 148(13) Graham, R. W., 191(106), 205(106), 209(106), 225(106), 231(106), 232(106) Granger, S., 232(216), 237(216) Gratton, L., 2(17; 19; 26; 27), 3(19), 11(26), 26(17; 19), 31(19; 26), 36(17; 26; 27), 57(19), 69(26), 70(17), 118(19), 123(17), 124(19) Gray, W. G., 2(8), 5(8), 15(47; 48; 49; 50), 23(8), 60(8) Gregory, G. A., 158(41), 161(41), 175(41), 177(41) Gridnev, S. A., 57(163) Grifﬁth, P., 147(9), 150(9), 153(9), 154(9), 160(9), 161(9), 165(9) Groeneveld, D. C., 209(158), 216(158) Groenhof, H., 287(72) Grolmes, M. A., 231(220), 233(220) Grzesik, J., 56(139), 57(139)

449

Gschwind, P., 372(6) Gutjahr, A. L., 33(83) Guzman, A. M., 424(147), 425(148)

H Ha, M. Y., 407(100), 408(100), 411(100) Hadley, G. R., 98(198) Hagiwara, Y., 396(72), 399(72; 86; 83), 400(72; 83; 85; 86), 405(72; 85) Halbaeck, M., 381(29) Hall, D. D., 217(183) Hanjalic, K., 274(43) Hanratty, T. J., 287(68), 325(98), 326(99), 355(98) Haramura, Y., 222(195) Hardy, P., 238(232) Harimizu, Y., 226(206), 227(206) Hassanizadeh, S. M., 15(50) Hayashi, M., 415(121) Healzer, J., 225(199) Heikal, M. R. (Chapter Author), 363, 370(2), 371(4), 375(15), 396(137), 407(109; 111; 112), 410(109), 411(111; 112; 114), 412(115), 413(114), 417(140), 421(137; 140) Heisenberg, W., 258(5) Hendricks, T. J., 56(132) Heng, L., 326(100), 355(100) Henningson, D. S., 381(29) Henry, R. E., 235(224) Hessami, M. A., 384(153), 388(153), 425(153), 426(153), 427(153), 428(153), 432(153) Hewiit, G. F., 224(196) Hibiki, T., 150(26; 31), 153(26), 154(26), 155(31), 157(26), 158(26), 160(26), 163(26), 169(26; 31), 171(31), 172(31), 174(31), 180(26), 185(26; 31), 187(26), 189(31), 190(31) Higbie, R., 343(115) Hijikata, K., 37(91) Hilfer, R., 53(128) Himeno, R., 407(108), 409(108) Hino, R., 198(134) Hinze, J. O., 277(52) Hirai, E., 398(81; 82), 399(81), 400(82), 407(82)

author index

450

Hiramatsu, M., 406(94), 407(94; 98), 408(98), 409(105), 410(98) Hirata, M., 193(155), 209(155), 210(155), 211(155), 212(155), 215(155) Hiyashi, T., 415(121) Hopkins, N. E., 184(90) Hori, M., 399(88), 401(88), 402(88), 403(88), 407(88) Horimizu, Y., 226(207), 227(207) Hosaka, S., 193(155), 209(155), 210(155), 211(155), 212(155), 215(155) Ho¨sken, C., 396(156), 427(156), 428(156), 429(156), 430(156), 432(156) Howell, J. R., 56(131; 132), 60(131) Howle, L., 26(71) Hsu, C. T., 32(79; 80) Hsu, Y. Y., 191(106), 205(106), 209(106), 225(106), 231(106), 232(106) Hu, H. Y., 209(154) Hu, K., 2(22), 69(25) Huang, L. J., 413(118), 414(118) Huffman, D. R., 57(144), 60(144) Hugonnot, P., 373(9), 384(9), 417(9), 418(9), 419(9), 420(9) Hwang, C. B., 387(43)

I Iacovidea, H., 391(55) Ichikawa, A., 124(219) Idelchik, I. E., 233(221) Ikuta, S., 407(108), 409(108) Ileslamlou, S., 221(188) Ilyushin, B. B., 388(49) Imas, Ya. A., 39(100) Inaoka, K., 399(86), 400(86) Inasaka, F., 193(130), 196(130), 197(130), 198(130), 199(130), 200(130), 201(130), 202(130), 203(130), 210(169; 170; 171), 211(169; 170; 171), 212(169; 170; 180), 213(169; 170), 217(169; 170) Ince, N. Z., 388(51) Ippen, E. P., 39(105) Ishii, M., 22(52; 53), 150(32), 155(32), 158(42), 162(42), 163(42), 164(42), 166(42), 171(32), 172(32), 173(32; 42), 174(32), 205(142) Ishimaru, T., 407(98), 408(98), 410(98)

Ishiyama, T., 226(206), 227(206) Israeli, M., 395(66) Itasaka, M., 402(89) Ito, K., 115(211) Itoh, M., 415(120) Itoh, S., 407(110), 411(110) Iwabuchi, M., 114(210)

J Jackson, R., 1(1) Jacobi, A. M., 368(1), 369(1), 372(1), 374(1), 400(1), 413(118), 414(118) Janssen, E., 225(199) Jendrzejczyk, J. A., 150(28), 155(28), 171(28), 175(28), 192(113; 114) Jensen, M. K., 193(131), 196(131), 206(131), 207(131), 208(131), 210(131; 173), 211(131; 173), 212(173), 213(131; 173), 215(131; 173), 216(173), 219(173) Jensen, P. J., 231(219), 233(219) Jeter, S. M., 150(35), 155(35), 176(35), 177(35), 185(35), 191(35), 193(132; 135; 139), 196(132), 197(132), 198(132; 135), 199(132), 200(132), 202(132; 135; 139), 203(135; 139), 204(135), 205(132; 135; 139), 210(174), 211(174), 214(174), 215(174), 216(174), 218(174), 219(174), 224(174) Jischa, M., 324(96), 345(96), 354(96) Johansson, A. V., 381(29) John, H., 191(105), 226(105), 227(105), 229(105), 230(105), 236(105), 239(105), 240(105) John, S., 52(118; 119) Jones, O. C., 205(145) Jones, O. C., Jr., 189(102), 190(102), 206(150), 207(150), 236(228) Jordan, R. C., 184(89) Joseph, D. D., 41(108)

K Kaganov, M. I., 38(97), 39(97) Kajino, M., 406(94), 407(94) Kampe´ de Fe´riet, J., 297(81), 350(81)

author index Kang, J. K., 407(100), 408(100), 411(100) Kang, S., 115(211; 212) Kanzaka, M., 114(210) Kapeliovich, B. L., 39(101) Kapitsa, P. L., 258(6) Kar, K. K., 89(184), 91(184), 92(184), 95(184) Kariyasaki, A., 150(25), 154(25), 157(25), 159(25), 161(25), 162(25), 165(25), 169(25), 185(25), 186(25), 187(25) Kasagi, N., 193(155), 209(155), 210(155), 211(155), 212(155), 215(155) Kashcheev, V. M., 23(59) Kastner, W., 226(205), 227(205), 229(205) Kato, M., 386(39) Kato, Y., 407(110), 411(110) Katto, Y., 209(159), 210(166), 211(166), 216(159), 220(159), 221(191; 192), 222(191; 195), 223(191; 192), 224(159) Kaufman, S. J., 340(108), 341(108) Kaviany, M., 2(7), 56(134), 60(7) Kawaji, M., 150(27; 29; 30), 155(27; 29; 30), 171(27; 29; 30), 174(30), 176(30), 177(30), 185(27; 29; 30), 189(27; 29; 30), 190(30) Kawamura, H., 274(44) Kawamura, Y., 425(150) Kays, W. M., 77(172), 81(172), 91(172), 95(172), 111(172), 324(95a), 327(101), 328(101), 330(101), 333(101), 335(101), 387(44) Kazantseva, N. E., 57(161), 60(161), 66(161) Kedoh, M., 415(120) Kefer, V., 226(205), 227(205), 229(205) Kelkar, K. M., 398(79), 399(79) Kells, L. C., 286(66) Kennedy, J. E., 193(132), 196(132), 197(132), 198(132), 199(132), 200(132), 202(132), 205(132) Khan, E. U., 23(58) Kharitonov, V. V., 91(190), 94(190), 95(190) Kheifets, L. I., 2(11), 25(11), 32(11) Khodyko, Yu. V., 39(100) Khoroshun, L. P., 97(194; 195) Kichigan, A. M., 210(161), 211(161) Kieda, S., 378(22), 398(81), 399(81) Kim, B. Y. K., 26(67) Kim, I. C., 34(87) Kim, J., 287(67), 396(75) Kim, K. C., 407(100), 408(100), 411(100) Kim, K. H., 407(100), 408(100), 411(100) Kim, K. I., 407(100), 408(100), 411(100)

451

Kim, P. H., 205(143) Kim, S. J., 67(168) Klausner, J. F., 205(146) Kleeorin, N., 325(97a) Knibbe, P. G., 373(7; 10), 374(10), 426(7; 10), 430(10) Koak, S. H., 407(100), 408(100), 411(100) Kobayashi, T., 407(110), 411(110) Kocamustafaogullari, G., 205(142) Kodal, A., 28(76) Koizumi, H., 184(91), 185(91) Kokorev, V. I., 91(190), 94(190), 95(190) Kolar, R. L., 2(8), 5(8), 23(8), 60(8) Kolmogorov, R. R., 273(38) Kottke, E. V., 373(11), 426(11), 428(11) Kottke, V., 372(6) Kouidry, F., 396(73), 417(73), 420(73), 421(73), 422(73), 432(73) Kra¨tzer, W., 226(205), 227(205), 229(205) Kroeger, P. G., 237(230) Kuchment, P., 43(125), 52(122), 54(123; 125), 55(123), 57(122; 125) Kudinov, V. A., 98(197) Kudo, K., 399(88), 401(88), 402(88), 403(88), 407(88) Kumar, S., 56(133; 137) Kunevich, A. P., 81(176), 82(176) Kurbatskii, A. F., 388(49) Kuroda, M., 407(110), 411(110) Kurshin, A. P., 84(179) Kushch, V. I., 11(33; 34), 12(35; 36; 37; 38), 13(34), 102(33; 34) Kuwahara, F., 108(199), 109(199), 111(199) Kwok, C. C. K., 184(92), 185(92)

L Lackme, C., 237(231) Lahey, R. T., Jr., 22(55), 23(56), 206(148; 149; 150), 207(150), 208(152) Lahey, T. R., Jr., 22(54), 192(116) Lai, J., 42(109) Lakhtakia, A., 57(147), 60(147) Lam, A. C. C., 26(67) Lam, C. K. G., 391(58), 418(58) Lamb, H., 257(1) Landau, L. D., 258(7) Lane, J. C., 379(26)

452

author index

Launder, B. E., 274(42; 43), 352(42), 383(34), 384(34; 35), 386(34; 39), 387(45), 388(51; 52; 53; 54), 390(35), 391(55) Lazarek, G. M., 192(112) Lebouche´, M., 373(13), 417(133) Lee, C. H., 221(189), 223(189) Lee, N., 148(16), 169(16) Lee, P. C. Y., 15(47; 48) Lee, R. C., 205(141) Lee, S. C., 56(138; 139), 57(139) Lee, S. J., 22(55), 206(150), 207(150) Lee, S. Y., 225(201), 232(201; 214; 215), 236(201; 214), 237(214; 215) Legg, B. J., 25(64; 65; 66) Lehner, F. K., 33(81) Leijsne, A., 2(8), 5(8), 23(8), 60(8) Lelluche, G. S., 225(200), 231(200), 232(200), 233(200) Leonard, A., 393(63) Leonard, B. P., 379(24) Lesieur, M., 393(69) Leung, J. C., 231(220), 233(220) Leung, R. Y., 327(101), 328(101), 330(101), 333(101), 335(101) Levec, J., 15(46), 18(46), 34(46) Levy, S., 194(123), 198(123), 200(123), 201(123), 202(123), 203(123), 206(123) Li, J.-H., 200(136), 202(136) Li, R.-Y., 184(92), 185(92) Lienhard, J. H., 229(210), 236(210) Lifshitz, I. M., 38(97), 39(97) Lightfoot, E. N., 77(173), 78(173), 79(173), 269(35), 280(35), 351(35) Lilly, D. K., 394(65) Lin, C. A., 387(43) Lin, L., 205(147) Lin, S., 184(92), 185(92) Lin, T. -F., 185(96; 97), 187(96; 97), 188(97) Lindell, I. V., 57(146), 60(146) Lindgren, E. R., 276(51) Liou, W. W., 386(42) Liu, J., 39(105) Loehrke, R. J., 379(26) Lomax, H., 382(32) London, A. L., 77(172), 81(172), 91(172), 95(172), 111(172), 403(91), 404(91) Loomsmore, C. S., 210(163), 211(163) Lopez de Bertodano, M., 22(54; 55) Lorentz, H. A., 258(8) Lorrain, P., 57(145), 60(145)

Lowry, B., 150(27), 155(27), 171(27), 185(27), 189(27) Lu, B., 34(85; 88) Lubarsky, B., 340(108), 341(108) Lulinkski, Y., 150(23), 154(23), 156(23), 158(23), 161(23), 163(23) Lumley, J. L., 26(73) Luo, K., 42(109) Lyn, D. A., 386(41) Lynn, S., 275(50), 277(50) Lyon, R. N., 322(94), 340(94) Lyons, S. L., 287(68)

M Macdonald, I. F., 81(180), 85(180) MacLeod, A. L., 282(61), 283(61) Mahesh, K., 393(71) Maji, P.K., 418(127), 423(127) Majumdar, A., 37(91; 94), 41(94), 42(109), 56(133) Malbagi, F., 57(152), 59(152), 61(152), 63(152) Mali, P., 238(232) Mandane, J. M., 158(41), 161(41), 175(41), 177(41) Marchessault, R. N., 146(6), 150(6) Marcy, G. P., 184(88) Mariani, A., 210(172; 175), 211(172; 175), 212(180), 213(172; 175), 217(172; 175), 221(193), 223(193) Marle, C. M., 1(3) Marsh, W. J., 195(128), 198(128) Martin, H., 112(205) Martin, J. D., 275(49) Martinelli, R. C., 201(138), 340(107) Marvillet, C., 373(12), 428(12) Mason, S. G., 146(6), 150(6) Mastin, C. W., 376(16) Masuoka, T., 26(69) Matsumoto, K., 226(206; 207), 227(206; 207) Matsuo, T., 114(210) Matsuzaki, K., 407(98), 408(98), 410(98) Mayﬁeld, M. E., 225(203), 226(203), 235(203), 236(203) McBeth, R. V., 210(176; 177) McClure, J. A., 231(219), 233(219) McFadden, J. H., 231(219), 233(219)

author index McLaughlin, J. B., 287(68) McLeond, D., 194(125), 198(125), 200(125), 202(125) McNab, C. A., 396(137), 417(140), 421(137; 140) McPhedran, R. C., 52(126), 57(155; 156; 157) Mei, R., 205(146) Melton, J. E., 376(18) Menter, F., 388(48), 392(48) Mercier, P., 377(20), 396(20; 139), 399(20), 402(20), 405(20), 422(139), 423(139), 424(139), 429(139) Me´tais, O., 393(69) Michallon, E., 384(87), 399(87), 401(87), 402(87), 407(87) Mikol, E. P., 184(87) Miller, B., 275(48) Miller, C. A., 34(86) Millikan, C. B., 266(30) Mishima, K., 22(53), 150(26; 31), 153(26), 154(26), 155(31), 157(26), 158(26; 42), 160(26), 162(42), 163(26; 42), 164(42), 166(42), 169(26; 31), 171(31), 172(31), 173(42), 174(31), 180(26), 185(26; 31), 187(26), 189(31), 190(31) Mitra, N. K., 396(156), 427(156), 428(156), 429(156), 430(156), 432(156) Mizuno, M., 399(88), 401(88), 402(88), 403(88), 407(88) Moalem-Maron, D., 147(10), 161(10) Mochizuki, S., 400(84), 403(84), 404(84), 405(84) Moin, P., 287(67), 379(25), 393(61; 71), 396(75) Moizhes, B. Ya., 98(197) Monrad, C. C., 282(59) Moody, F. J., 192(116), 234(222), 235(222) Morega, A. M., 112(203) Morioka, M., 399(88), 401(88), 402(88), 403(88), 407(88) Moser, R., 287(67), 396(75) Moshaev, A. P., 91(189), 93(189) Motai, T., 114(210) Movchan, A. B., 57(155) Mow, K., 81(180), 85(180) Moyne, C., 81(181), 85(181) Mudawar, I., 185(95), 186(95), 195(128), 198(128), 210(95), 211(95), 214(95), 215(95), 217(183), 219(95), 221(189; 190), 223(189), 224(95)

453

Muller, J. R., 226(208), 227(208), 230(208), 231(208), 233(208) Muralidhar, K., 385(38) Muravev, G. B., 81(175), 91(175), 94(175) Murphree, W. V., 268(32)

N Nabarayashi, T., 226(206; 207), 227(206; 207) Nadyrov, I. N., 81(175; 176), 82(176), 91(175), 94(175) Naff, R. L., 33(83) Nakamura, H., 418(125), 419(125) Nakamura, S., 226(207), 227(207) Nakayama, A., 108(199), 109(199), 111(199) Nariai, H., 193(130), 196(130), 197(130), 198(130), 199(130), 200(130), 201(130), 202(130), 203(130), 210(169; 170; 171), 211(169; 170; 171), 212(169; 170; 180), 213(169; 170), 217(169; 170) Narrow, T. L., 150(34), 156(34), 167(34), 168(34), 177(34), 188(34) Navier, C.-L. M. N., 261(19) Naviglio, A., 218(184) Neimark, A. V., 2(11), 25(11), 32(11) Newton, I., 258(9) Nicorovici, N. A., 52(126), 57(155; 156; 157) Nigmatulin, B. I., 236(228) Nikuradse, J., 274(45; 46; 47), 275(45; 46; 47), 276(47), 277(45; 45), 278(45), 279(46), 280(45; 46; 47), 281(46), 284(47), 286(47), 287(46; 47), 297(46), 298(46), 351(46), 352(45; 46), 353(46) Nishihara, A., 415(121) Nishihara, H., 150(31), 155(31), 169(31), 171(31), 172(31), 174(31), 185(31), 189(31), 190(31) Nishimura, T., 425(150) Nogotov, E. F., 57(142), 60(142) Nomoﬁlov, E. V., 23(59) Norris, D. M., 225(202), 226(202), 230(202), 231(202), 235(202; 223; 225) Norris, P. M., 145(1) Notter, R. H., 324(97), 327(97), 328(97), 330(97; 111), 332(97), 333(97), 340(97; 109; 110; 111), 354(97; 110; 111) Nozad, I., 15(40), 34(40), 107(40), 126(40) Nunner, W., 281(56), 353(56)

author index

454 Nusselt, W., 337(102) Nydahl, J. E., 205(141)

O Ohori, Y., 425(150) Okawa, W. J., 379(27) Olek, S., 236(227) Olivier, I., 428(154) Orczag, S. A., 310(89), 325(89), 345(89), 354(89) Orlanski, I., 377(21) Ornatskiy, A. P., 210(160; 161; 162), 211(160; 161; 162) Orszag, P. A., 396(74) Orszag, S. A., 385(37), 386(40), 392(59), 394(59), 395(66) Orszag, S. D., 286(66) Ota, K., 409(105) Ota, M., 384(130), 417(130), 419(130), 420(130) Ota, T., 379(27), 402(89) Oto, M., 388(132), 417(132) Oya, T., 150(22), 153(22; 39) Ozisik, M. N., 37(93), 39(93) Ozoe, H., 326(100a), 327(100a), 329(100a), 330(100a), 331(100a), 332(100a), 333(100a), 340(100a), 349(100a), 350(100a), 355(100a)

P Paffenbarger, J., 113(206), 122(206) Page, F., Jr., 310(90), 354(90) Pai, S. I., 297(82), 351(82) Pantankar, S. V., 398(77; 78; 79), 399(78; 79), 401(78) Panton, R. L., 150(24), 153(24), 154(24), 158(24), 159(24), 166(24) Papavassiliou, D. V., 325(98), 355(98) Park, J.-H., 386(41) Park, T. Y., 407(100), 408(100), 411(100) Patel, V. C., 27(75), 391(57), 399(57) Patterson, G. S., 396(74) Paulsen, M. P., 231(219), 233(219) Peasa, R. F., 191(109) Pei, B. S., 221(187)

Peng, X. F., 191(110), 193(110; 153), 208(110; 153), 209(110; 154), 215(110; 153) Perel’man, T. L., 39(101) Pereverzev, S. I., 56(121), 57(121) Peterson, 231(219), 233(219) Peterson, G. P., 191(111) Peterson, R. B., 37(95), 41(95), 50(95) Petrie, J. M., 338(104) Petukhov, B. S., 314(92) Phan, R. T., 26(67) Pisano, A. P., 205(147) Plumb, O. A., 15(44; 45), 18(44) Poirer, D., 194(125), 198(125), 200(125), 202(125) Pomeranchuk, I., 38(99) Pomraning, G. C., 57(148; 149; 150; 151; 152), 58(150; 151), 59(148; 151; 152; 164), 61(152; 164), 63(152; 164) Ponomarenko, A. T., 51(114; 115), 57(114; 115), 60(114; 115; 159; 160; 161), 61(114; 115), 66(159; 161), 97(115), 111(114; 115) Pope, D. B., 225(203), 226(203), 235(203), 236(203) Popov, A. M., 24(60; 61) Poulikakos, D., 67(169) Pourquie`, M., 393(60) Prakash, C., 398(78), 399(78), 401(78) Prandtl, L., 265(29), 269(24), 270(24; 29), 272(24; 37), 273(40), 280(55), 343(113) Prata, A. T., 416(124), 421(124), 422(145) Preziosi, L., 41(108) Primak, A. V., 2(14; 15), 4(14; 15), 16(14; 15), 25(14; 15), 26(14; 15), 27(14; 15), 29(14; 15; 78), 30(14; 15), 116(14; 15) Pritchard, A. J., 125(222) Prosperetti, A., 23(57)

Q Qin, T. Q., 145(1) Qiu, T. Q., 39(102; 103; 104), 40(102; 103; 104) Quereshi, Z. H., 193(132), 196(132), 197(132), 198(132), 199(132), 200(132), 202(132), 205(132) Querfeld, C. W., 57(153) Quintard, M., 6(29), 7(29; 30), 8(29), 9(29), 15(30)

author index R Radushkevich, L. V., 1(5) Rajkumar, M., 91(185), 92(185), 95(185) Ramadhyani, S., 372(5) Ramakers, F. J. M., 205(140) Rangarajan, R., 52(119) Raupach, M. R., 24(63), 25(64; 65; 66), 28(77) Rayleigh, Lord, 258(10), 264(24; 25), 281(25) Rayleigh, R. S., 12(39) Reece, G. J., 388(52) Regele, A., 372(6) Reichardt, H., 294(76), 296(76), 298(76), 309(87), 344(87), 348(87) Reimann, J., 191(105), 226(105), 227(105), 229(105), 230(105), 236(105), 239(105), 240(105) Reiss, H., 56(141) Renken, K., 67(169) Revankar, S. T., 225(201), 232(201; 215), 236(201), 237(215) Reynolds, A. J., 324(95) Reynolds, O., 259(17; 18), 261(17), 264(17; 28), 315(18), 343(18) Rezkallah, K. S., 153(38), 160(38), 161(38), 164(38), 165(38), 166(38) Richardson, J. F., 338(105), 339(105) Richter, H. J., 228(209), 236(209) Richter, J. P., 257(2), 258(2) Rieke, H. B., 324(96), 345(96), 354(96) Roach, G. M., Jr., 193(132), 196(132), 197(132), 198(132), 199(132), 200(132), 202(132), 205(132), 210(174), 211(174), 214(174), 215(174), 216(174), 218(174), 219(174), 224(174) Robertson, J. M., 275(49) Roch, G. M., Jr., 193(135), 198(135), 202(135), 203(135), 204(135), 205(135) Rodi, W., 26(72), 27(75), 381(28), 386(41), 387(28), 388(52), 391(56), 393(60) Rogachevskii, I., 325(97a) Rogers, J. T., 200(136), 202(136) Rogers, T. J., 194(125), 198(125), 200(125), 202(125) Rohsenow, W. M., 23(58), 195(129), 196(129), 197(129), 198(129), 309(88) Romanov, G. S., 39(100) Rothfus, R. R., 282(59; 61; 62), 283(61) Rotstein, E., 15(41), 22(41), 30(41), 35(41), 64(41)

455

Rotta, J. C., 269(35), 280(35), 351(35) Rubenstein, J., 34(85) Rutledge, J., 287(69) Ryvkina, N. G., 51(115), 57(115; 160; 161; 162), 60(115; 160; 161; 162), 61(115), 66(161), 97(115), 111(115)

S Sadatomi, M., 150(29; 30), 155(29; 30), 171(29; 30), 174(30), 176(30), 177(30), 185(29; 30), 189(29; 30), 190(30) Sadatomi, Y., 190(104) Sadowski, D. L., 147(12), 150(12; 34), 151(12), 152(12), 153(12), 154(12), 156(12; 34), 157(12), 158(12), 159(12), 160(12), 161(12), 162(12), 164(12), 167(34), 168(34), 177(34), 185(12), 187(12), 188(34), 226(208), 227(208), 230(208), 231(208), 233(208) Sage, B. H., 310(90), 354(90) Saha, P., 194(121), 198(121), 199(121), 203(121), 222(121) Sakamoto, M., 384(130), 388(132), 417(130; 132), 419(130), 420(130) Salcudean, M., 192(118; 119; 120), 194(125), 198(125), 200(118; 119; 120; 125), 202(125) Samaddar, S. N., 57(154) Sangani, A. S., 11(32) Saruwatari, S., 190(104) Sasaki, Y., 407(108), 409(108) Satake, S., 274(44) Sato, T., 378(22), 398(81; 82), 399(81), 400(82), 407(82) Sato, Y., 190(104) Sawyers, D. R., 428(155) Scheurer, G., 27(75) Schlichting, H., 68(170), 258(16) Schlinger, W. G., 310(90), 354(90) Schrock, V. E., 225(201), 226(204), 227(204), 228(204), 229(204), 232(201; 204; 214; 215), 236(201; 204; 214), 237(214; 215), 239(204), 240(204), 241(204) Schuerger, M. J., 415(121) Schumann, U., 393(62) Schwartz, F. W., 33(84), 34(84) Schwellnus, C. F., 236(226)

456

author index

Scott, P. M., 225(203), 226(203), 235(203), 236(203) Seban, R. A., 316(93) Sen, M., 428(155) Senecal, V. E., 282(62) Seynhaever, J. M., 232(216), 237(216) Shabanskii, V. P., 38(98) Shabbir, A., 386(42) Shah, A. K., 385(38) Shah, M. M., 220(185) Shah, R. K. (Chapter Author), 363, 368(1), 369(1), 372(1), 373(8), 374(1), 378(90), 379(90), 399(90), 400(1), 402(90), 403(90; 91), 404(90; 91), 405(90; 93), 413(90), 421(136) Sharma, B. I., 387(45) Shaw, C. T., 377(19) Shaw, D. A., 326(99) Shaw, R. H., 24(63) Shcherban, A. N., 2(14; 15), 4(14; 15), 16(14; 15), 25(14; 15), 26(14; 15), 27(14; 15), 29(14; 15), 30(14; 15) Sherwood, T. K., 338(104) Shevchenko, V., 57(161; 162), 60(161; 162), 66(161) Shi, Z., 42(109) Shih, T. H., 386(42) Shimazaki, T. T., 316(93) Shimura, T., 193(130), 196(130), 197(130), 198(130), 199(130), 200(130), 201(130), 202(130), 203(130), 210(169), 211(169), 212(169), 213(169), 217(169) Shin, T. S., 205(145) Shinagawa, T., 407(106), 409(106) Shinoda, M., 346(117) Shoukri, M., 236(226) Shultze, H. D., 207(151) Shvab, V. A., 26(74) Siegel, R., 56(131), 60(131) Sihvola, A. H., 57(146), 60(146) Simoncini, M., 221(193), 223(193) Singh, B. P., 56(134) Skinner, B. C., 210(163), 211(163) Slattery, J. C., 1(2), 2(6), 5(6), 60(6) Sleicher, C. A., 287(69), 324(97), 327(97), 328(97), 330(97; 111), 332(97), 333(97), 340(97; 109; 110; 111), 354(97; 110; 111) Smagorinsky, J. S., 394(64) Smith, L., 33(84), 34(84) Snoek, C. W., 209(158), 216(158) Sobey, I. J., 422(141; 142; 143; 144), 424(141)

Sommerfeld, A., 258(11) Sonin, A. A., 23(58) Souto, H. P. A., 81(181), 85(181) So¨zen, M., 145(2) Spalart, P., 383(33), 392(33), 396(76) Spalding, B. D., 91(191), 94(191), 95(191) Spalding, D. B., 274(42), 292(74), 293(74), 352(42), 383(34), 384(34; 35), 386(34), 390(35) Sparrow, E. M., 81(177), 82(177), 83(177), 84(177), 398(77), 416(124), 420(131), 421(124), 422(145) Spedding, P. L., 150(21) Spence, D. R., 150(21) Speziale, C. G., 381(31), 386(40), 388(50) Staroselsky, I., 392(59), 394(59) Stasiek, J., 384(152), 388(152), 393(152), 423(14), 425(152), 426(14; 152), 428(152), 429(152), 430(152), 432(152) Staub, F. W., 194(124), 198(124), 200(124), 202(124) Stephanoff, K. D., 422(142) Stewart, W. E., 77(173), 78(173), 79(173), 269(35), 280(35), 351(35) Stokes, G. G., 261(20) Stralen, S. V., 191(108), 205(108) Strutt, J. W., 258(10), 264(24; 25), 281(25) Stubbs, A. E., 146(5), 150(5) Stuben, F. B., 225(203), 226(203), 235(203), 236(203) Su, B., 59(164), 61(164), 63(164) Subbotin, V. I., 23(59), 91(190), 94(190), 95(190), 210(165), 211(165) Suga, K., 388(47), 407(106; 107), 409(106; 107) Sugawara, S., 224(197; 198) Sulaiman, Y., 407(109), 410(109) Sunde´n, B., 385(157), 388(157), 416(122), 421(134; 135), 428(157), 429(157), 432(157) Suo, M., 147(9), 150(9), 153(9), 154(9), 160(9), 161(9), 165(9) Suzuki, K., 378(22), 396(72), 398(81; 82), 399(72; 81; 83; 86), 400(72; 82; 83; 85; 86), 405(72; 85), 406(95), 407(82), 415(121)

T Taborek, J., 91(191), 94(191), 95(191) Taitel, Y., 148(15; 16; 17; 18; 20), 150(23),

author index 154(23), 156(15; 17; 23), 158(23), 161(15; 23), 162(15; 20), 163(23), 166(17), 169(16), 171(17), 173(17) Takagi, M., 407(108), 409(108) Takatsu, Y., 26(69) Tanaka, K., 407(108), 409(108) Tanaka, T., 415(120) Tanaka, Y., 226(206; 207), 227(206; 207) Tanatarov, L. V., 38(97), 39(97) Tang, D., 33(84), 34(84) Tao, W. Q., 417(128), 419(128) Tapucu, 22(51) Taylor, G. I., 146(7), 150(7) Taylor, N., 396(137), 421(137) Tchmutin, I. A., 57(159; 160; 161; 162; 163), 60(159; 160; 161; 162), 66(159; 161) Teo, K. L., 116(213), 124(213; 214; 215; 216; 217), 125(214) Teyssedou, A., 22(51) Thangam, S., 386(40) Theofanous, T. G., 59(165) Thochon, P. (Chapter Author), 363 Thomas, L. C., 343(114) Thome, J. R., 148(14), 182(14), 191(14) Thompson, B., 210(176) Thompson, J. F., 376(16) Thonon, B. (Chapter Author), 363, 373(12), 428(12) Thulasidas, M. A., 151(36; 37) Tien, C. L., 26(70), 39(102; 103; 104), 40(102; 103; 104), 46(110), 56(130; 133; 135), 145(1) Tochon, P., 377(20), 396(20; 139), 399(20), 402(20), 405(20), 422(139), 423(139), 424(139), 429(139) Todreas, N. E., 23(58) Tomoda, T., 406(95) Tong, L. S., 194(126), 217(182) Torquato, S., 34(85; 86; 87; 88) Tran, T. N., 192(113; 114; 115) Trauger, P., 408(99), 414(99) Travkin, V. S. (Chapter Author), 1, 2(14; 15; 16; 17; 18; 19; 20; 21; 22; 26; 27; 28), 3(19; 21), 4(14; 15), 10(21), 11(16; 18; 20; 26; 33; 34), 13(34), 15(18), 16(14; 15; 16; 18; 21), 21(21), 23(24), 25(14; 15; 16; 20), 26(14; 15; 16; 17; 18; 19; 20; 21), 27(14; 15), 29(14; 15; 78), 30(14; 15; 16; 21), 31(19; 26), 36(16; 17; 20; 26; 27), 51(114; 115), 52(23), 57(19; 20; 28; 114; 115; 158; 159; 160; 161; 162; 163), 60(21;

457

114; 115; 159; 160; 161), 61(114; 115), 62(21), 65(21; 28), 66(159; 161; 166), 68(16; 20), 69(16; 20; 21; 23; 25; 26), 70(16; 17; 20; 21), 71(16; 20), 79(21), 80(21), 81(23), 96(21), 97(21; 115), 102(21; 33; 34), 110(21), 111(114; 115), 116(14; 15; 16; 20; 21; 23; 28), 118(16; 19), 119(16; 20), 123(16; 17; 23), 124(19) Tretyakov, S. A., 57(146), 60(146) Tribus, M., 340(109) Triplett, K. A., 147(12), 150(12), 151(12), 152(12), 153(12), 154(12), 156(12), 157(12), 158(12), 159(12), 160(12), 161(12), 162(12), 164(12), 185(12), 187(12) Troﬁmov, V. P., 57(142), 60(142) Tsay, R., 111(202) Tuckermann, D. B., 191(109) Tura, R., 407(104) Tzou, D. Y., 37(93; 96), 39(93), 40(96)

U Udell, K. S., 205(147) Ueda, T., 198(134) Uehara, K., 210(170), 211(170), 212(170), 213(170), 217(170) Uﬁmtsev, P. Y., 56(121), 57(121) Uher, C., 96(192) Uhlenbeck, G., 258(12), 261(12) Unal, H. C., 194(122), 198(122), 205(144) Ungar, K. E., 185(94), 186(94) Usagi, R., 285(65)

V Vafai, K., 26(70), 67(168), 145(2) van de Hulst, H. C., 57(143), 60(143) Vandervort, C. L., 193(131), 196(131), 206(131), 207(131), 208(131), 210(131; 173), 211(131; 173), 212(173), 213(131; 173), 215(131; 173), 216(173), 219(173) van Dreist, R. R., 273(38) Vanka, S. P., 425(149) Van Stralen, S. J. D., 205(140) Varadan, V. K., 57(147), 60(147) Varadan, V. V., 57(147), 60(147)

author index

458

Vidil, R., 373(12), 428(12) Vinyarskiy, L. S., 210(162), 211(162) Viskanta, R., 85(182; 183), 91(182; 183; 187), 93(183; 186; 187), 95(187; 188) Vittanen, A. J., 57(146), 60(146) von Ka´rma´n, T., 266(31), 269(31), 272(31) von Mises, R., 258(13) von Weizsa¨cker, C. F., 258(14) Voskoboinikov, V. V., 91(190), 94(190), 95(190)

W Wacholder, E., 236(227) Wambsganss, M. W., 150(28), 155(28), 171(28), 175(28), 192(113; 114; 115) Wang, B.-X., 191(110), 193(110; 153), 208(110; 153), 209(110; 154), 215(110; 153) Wang, C. C., 371(3) Wang, C. Y., 145(4) Wang, G., 425(149) Wang, S. K., 206(150), 207(150) Ward, J. C., 83(178) Warsi, Z. U. A., 376(16) Webb, R. L., 114(207; 208), 187(98), 188(98), 408(99; 102), 414(99) Wei, T., 299(84) Weinbaum, S., 111(202) Weisman, J., 221(186; 187; 188) Westacott, J. L., 231(219), 233(219) Westphal, F., 191(105), 226(105), 227(105), 229(105), 230(105), 236(105), 239(105), 240(105) Westwater, J. W., 147(11), 150(11), 153(11), 154(11), 157(11), 158(11), 159(11), 161(11), 162(11), 164(11) Whan, G. A., 282(60), 352(60) Whitaker, S., 1(4), 2(9; 10), 5(10), 6(29), 7(29; 30; 31), 8(29; 31), 9(29; 31), 15(10; 30; 31; 40; 41; 42; 43; 44; 45), 18(42; 44), 22(41), 23(10; 42), 30(41), 34(40; 89), 35(41), 60(10), 64(41), 107(40), 116(10; 42), 126(40) White, P. R. S., 407(103) White, S., 56(139), 57(139) Wieting, A. R., 398(80) Wilcox, D. C., 381(30), 388(30), 392(30) Willmarth, W. W., 299(84)

Wilmarth, T., 150(32), 155(32), 171(32), 172(32), 173(32), 174(32) Winterton, R. H. S., 200(137) Wio, H. S., 37(92) Wright, C. C., 407(104) Wu, Z. S., 116(213), 124(213)

X Xi, G. N., 378(23; 90), 379(90), 396(72), 399(72; 83; 86; 90), 400(72; 83; 85; 86), 402(90), 403(90), 404(90), 405(72; 85; 90), 413(90) Xiang, X., 125(225) Xin, R. C., 417(128), 419(128)

Y Yablonovitch, E., 52(116; 117) Yadigaroglu, G., 194(127) Yagi, Y., 400(84), 403(84), 404(84), 405(84) Yahkot, A., 310(89), 325(89), 345(89), 354(89) Yahkot, V., 310(89), 325(89), 345(89), 354(89) Yakhot, A., 395(66) Yakhot, M., 386(39) Yakhot, V., 385(37), 386(40), 392(59), 394(59), 395(66) Yamada, T., 24(62) Yamaguchi, H., 384(130), 388(132), 417(130; 132), 419(130), 420(130) Yamaguchi, Y., 388(126), 417(126), 418(126), 419(126), 420(126), 431(126) Yan, Y.-Y., 185(96; 97), 187(96; 97), 188(97) Yang, C.-Y., 187(98), 188(98) Yang, L. C., 388(126), 417(126), 418(126), 419(126), 420(126), 431(126) Yang, S. R., 205(143) Yang, W.-J., 400(84), 403(84), 404(84), 405(84) Yao, G., 183(86) Yap, C. R., 387(46) Yin, S. T., 198(133) Yoda, M., 150(35), 155(35), 176(35), 177(35), 185(35), 191(35) Yokohama, K., 184(91), 185(91) Yokoya, S., 210(166), 211(166)

author index Younis, L. B., 91(187), 93(186; 187), 95(187; 188) Yu, B., 326(100a), 327(100a), 329(100a), 330(100a), 331(100a), 332(100a), 333(100a), 340(100a), 349(100a), 350(100a), 355(100a) Yu, F., 81(175; 176), 82(176), 91(175), 94(175) Yur’ev, Yu. S., 23(59)

Z Zachariades, J., 428(154) Zagarola, M. V., 288(73), 289(73), 290(73), 291(73), 300(73), 302(73), 347(73), 351(73), 353(73)

459

Zel’dovich, Ya. B., 258(15) Zeng, L. Z., 205(146) Zhang, D. Z., 23(57) Zhao, L., 153(38), 160(38), 161(38), 164(38), 165(38), 166(38) Zhu, J., 386(42) Zijl, W., 205(140) Zivi, S. M., 187(99), 234(99) Zolotarev, P. P., 1(5) Zuber, N., 194(121), 198(121), 199(121), 203(121), 222(121) Zummo, G., 221(193), 223(193)

a This Page Intentionally Left Blank

SUBJECT INDEX A Asymptotic dimensional analysis, 264 269

B Baldwin-Lomax model, 382—383 Bandgaps, 53—56 Boiling nucleate, 195—198 subcooled forced ﬂow bubble nucleation, 205—209 general issues, 191—192 instability, 198—205 nucleate onset, 195—198 signiﬁcant void, 198—205 void fractions, 192—195 Boundaries CHE surfaces, 376—377 gain, 53—54 Boundary L ayer T heory, 258 Bubble nucleation, 205—209

C Chandrasekhar, Subrahmanyan, 258 Channels furrowed, 372 heat transfer, 429—430 wavy corrugated, 372, 416—422 furrowed, 422—425 via chevron plates, 429—430 CHE. see Compact heat exchange Chevron plates description, 425—429 local analysis, 429—430 wavy channels via, 431—432 CHF. see Critical heat ﬂux Closure theories, 32—37 Colburn analogy, 342—343

Colebrook equation, 284—286 Compact heat exchange characterization, 363 models control problems, 123 current practice, 113—116 development, 111—112 optimization, 124—127 VAT-based equations, 117—122 optimization, 127—128 surfaces chevron plates, 425—430 experiments, 365—366 interrupted ﬂow passages general, 366—367 louver ﬁns, 369—371 offset strip ﬁns, 368—369 louver ﬁns, 406—416 numerical analysis boundary conditions, 376—377 general issues, 375 mesh generation, 376 solution algorithm, 376—377 offset strip ﬁns, 398—405 turbulence models DNS, 392—395 eddy viscosity, 381—388 general issues, 380—381 LES, 392—395 Reynolds number ﬂow, 391—392 Reynolds stress, 388—390 wall effects, 390—392 uninterrupted complex chevron plates, 372—373 furrowed channel, 372 general issues, 371 intermating plates, 372—373 Reynolds number, 374—375 unsteady laminar, 374—375 wavy corrugated channel, 372 wavy channels corrugated, 372, 416—422 furrowed, 422—425 Composite media, 103—108

461

462

subject index

Conductivity composite media, 103—108 hyperbolic heat, 41—42 pure phase media, 101—103 two-phase media conventional formulation, 97—98 local distribution, 96 piecewise distribution, 96 VAT considerations, 99—101 Conservation differential, 236—240 electron, 46—47 energy, 259 mass, 29, 258—259 momentum, 258—259 two-temperature, 43—45 Convection integrals, 311—317 turbulent alternative models, 320—323 correlating equations, 356 differential models, 353—354 differentials, 305—309 geometry formulations, 318—320 heat ﬂux density ratio, 354 initial perspectives, 353 integrals equations, 309—310 expressions, 317—318 isothermal wall, 331—332 Nu correlations differential analogy, 344—345 dimensional analysis, 335—337 empirical, 337—339 integral, 355 low-Prandtl-number ﬂuids, 339—342 mechanistic analogies, 342—344 numerical, 355—356 theoretically-based, 344—348 parallel plates channels, 333, 335 geometries, 318—320 Prandtl number convection, 323—326 elimination, 354—355 ﬂuids, 339—342 structure development, 259—260 uncertainty, 323—324 uniformly heated tube Nu values, 330

particular conditions, 326—328 Pr values, 328—329 Corrugated channels heat transfer, 429—430 wavy, 372, 416—422 Cracks microchannel differential conservation, 236—240 experiments, 225—230 general issues, 224—225 integral models isentropic homogeneous-equilibrium, 232—233 LEAK, 235—236 Moody’s, 234—235 numerical models, 236—240 Critical heat ﬂux microchannels empirical correlations, 216—220 experiments, 210—215 general issues, 209—210 mass, 215—216 noncondensables, 215—216 pressure, 215—216 theoretical models, 221—224 trends, 210—215 Crystals photonic, bandgap, 53—56 subcrystalline single, 45—46

D Detailed micromodeling description, 52 ﬂuid phase one, 108 porous media conductivity, 108, 110 radiative heat transport heterogeneous media, 57—58 porous media, 57—58 thermal conductivity, 97 -VAT, mismatches, 52—53 Differential conservation microchannel, 236—240 turbulent convection, 305—309 Direct numerical modeling description, 52 ﬂuid phase one, 108, 110 porous media conductivity, 108, 110 radiative heat transport, 57—58

subject index thermal conductivity, 97 VAT mismatches, 52—53 veriﬁcation, 12—13 Direct numerical simulation CHE surfaces, 395—397 convection, 259, 260 ﬂow, 259 Dissipation, 306 Distribution, 292—294 DMM. see Detailed micromodeling DNM. see Direct numerical modeling DNS. see Direct numerical simulation Drift ﬂux model, 172—173, 175 Dynamic procedure model, 395

E Eddy viscosity description, 269 ﬁlter approach model, 394 one-equation models, 383 two-equation models advantages, 383—384 low Reynolds numbers, 387 realizable k—, 386—387 RNG k—;, 384—386 standard k—, 383—384 zero-equation models, 382—383 Einstein, Albert, 258 Electrodynamics, nonlocal VAT-governing equations photonic crystals bandgap, 54—55 superstructures acoustical phonon, 49—50 electromagnetic, 50—51 electron conservation, 46—47 ﬂuid momentum, 47—48 gas energy, 49 longitudinal phonon, 49 Electron conservation, 46—47 Electron gas energy, 49 Ensemble averaging, 59

F Filtered media, 27 Fins

louver, 369—371, 406—416 offset strip, 368—369, 398—406 Flow CHE surface interrupted general, 366—367 louver ﬁns, 369—371 offset strip ﬁns, 368—369 complex passages uninterrupted chevron plates, 372—373 furrowed channel, 372 general issues, 371 intermating plates, 372—373 Reynolds number, 374—375 unsteady laminar, 374—375 wavy corrugated channel, 372 forced subcooled boiling general issues, 191—192 nucleate onset, 195—198 void fractions, 192—195 linear models, 1 microchannel annular, 170—177 characteristics, 146—147 CHF empirical correlations, 216—220 experiments, 210—215 general issues, 209—210 mass, 215—216 noncondensables, 215—216 pressure, 215—216 theoretical models, 221—224 trends, 210—215 conditions, 147—148 correlations, 161—166 in cracks differential conservation, 236—240 experimental data, 225—230 general issues, 224—225 integral models, 232—236 numerical models, 236—240 deﬁnition, 148—149 microgravity, 159—161 narrow rectangular, 170—177 noncondensable release, 178—179 pressure drop experiment review, 184—191 fractional, 180—183

463

464

subject index

Flow (Continued) general issues, 180 regimes, 150—153 rod bundle patterns, 166—168 in slits differential conservation, 236—240 experiments, 225—230 general issues, 224—225 integral models, 232—236 numerical models, 236—240 subcooled boiling bubble nucleation, 205—209 general issues, 191—192 instability, 198—205 nucleate onset, 195—198 signiﬁcant void, 198—205 void fractions, 192—195 surface wettability, 158—159 transitions models, 161—166 trends, 153, 156 void fractions, 169—170 resistance porous media experimental assessment, 67—69 momentum in 1D membrane, 69—75 pressure loss, 77—80 simulation procedures, 80—84 turbulent asymptotic, 263—269 CHE surfaces, 374—375 Colebrook equation, 284—286 dimensional, 263—269 dimensional models, 273—274 eddy viscosity, 269 exact structure, 260—263 friction factor correlations, 301—303 description, 283—286 geometry correlations, 303—304 MacLeod analogy, 282—283, 352 mixing length, 269—273 model-free formulations, 294—295 near center line values, 287 near wall values, 286—287 new formulations, 303—304 Nikuradse data, 274—276 numerical simulations, 274 power-law models, 278—282 recapitulation, 304 rough piping, 283—286

shear stress correlating equations, 351—352 integral formulations, 350—351 limited models, 352—353 MacLeod analogy, 352 new model, 348—349 obsolete models, 352—353 shear stress correlations, 299—301 speculative analyses, 263—269 study, history, 257—259 velocity correlations, 301—303 distribution, 292—294 Zagarola data, 287—292 Flux, 311—316, 354 Fractions heated channels, 192—195 microchannels, 169—170 Friction convection, 284—286 factors, 301—303 Furrowed channels, 372

G Gas energy, electron, 49 Gas-to-ﬂuid exchanger, 463—464 Geometry convection, 318—320 turbulent ﬂow, 303—304 Grain boundaries, 53—54

H Harmonic ﬁeld equations, 64—65 Heat conductivity composite media, 103—108 hyperbolic, 41—42 porous media, 108—111 pure phase media, 101—103 two-phase media conventional formulation, 97—98 effective modeling, 96—97 local distribution, 96 piecewise distribution, 96 VAT considerations, 99—101

subject index critical ﬂux microchannels empirical correlations, 216—220 experiments, 210—215 general issues, 209—210 mass ﬂux, 215—216 noncondensables, 215—216 pressure, 215—216 theoretical models, 221—224 trends, 210—215 ﬂux, 311—316, 354 radiative, transport heterogeneous media issues, 57—58 nonlocal volume, 60—64 porous media harmonic ﬁeld equations, 64—65 issues, 57—58 linear transfer, 58 nonlocal volume, 60—64 transport CHE models, 124—125 microscale, 37—43 wave transport CHE models control problems, 123 current practice, 113—116 development, 111—112 VAT-based, 117—122 crystal, 45—46 superstructures acoustical phonon, 49—50 electromagnetic, 50—51 electron conversion, 46—47 ﬂuid momentum, 47—48 gas energy, 49 longitudinal phonon, 49 Heat transfer CHE surface interrupted general, 366—367 louver ﬁns, 369—371 offset strip ﬁns, 368—369 uninterrupted complex chevron plates, 372—373 furrowed channel, 372 general issues, 371 intermating plates, 372—373 Reynolds number, 374—375 unsteady laminar, 374—375

465

wavy corrugated channel, 372 coefﬁcient, 335, 337 in corrugated channels, 429—430 porous media, coefﬁcients assumptions, 85 models, 86—89 simulation procedures, 90—94 Heisenberg, Werner, 258 Heterogeneous media 2-phase, 11 radiative heat transport harmonic ﬁeld equations, 64—65 issues, 57—58 nonlocal volume, 60—64 Heterogeneous media modeling, 52—53 Highly porous media turbulent transport model development additive components, 29 ﬁrst level hierarchy, 27 free stream, 28 mass conservation, 29 momentum equations, 30—32 scalar diffusion, 29 separate obstacle, 28 theoretical bases, 26—27 High-temperature superconductors, 101 HMM. see Heterogeneous media modeling Homogeneous isotropic media, 11 HTSC. see High-temperature superconductors Hydrodynamics, 257 Hyperbolic heat conduction, 41—42

I Integral models isentropic homogeneous-equilibrium, 232— 233 LEAK, 235—236 Moody’s, 234—235 Isentropic homogeneous-equilibrium model, 232—233

K Kapitsa, Pyotr, 258

subject index

466 L Laminar ﬂow CHE surfaces, 374—375 nonlinear ﬂuid medium concentration value, 20 homogeneous phase diffusion, 20 mass transport, 21 momentum diffusion, 20—21 Navier-Stokes equations, 19—20 steady-state momentum, 21 porous media VAT diffusion equation, 18 divergence form, 17 ﬂuid phase, 17 impermeable interface, 18 momentum equations, 18—19 solid phase, 17 Landau, Lev, 258 Large eddy simulations CHE surfaces basic features, 392—393 DNS, 395—397 ﬁlter approach, 393—395 numerical scheme, 378—380 Schumann’s approach, 393 solution algorithm, 378—380 convection, 353 Law of the wall, 265, 283 LEAK model, 235—236 Linear models, 1 Linear particle transport, 58—59 Linear Stokes equations, 15—16 Lorentz, Hendrik, 258 Louver ﬁns, 406—416

M MacLeod analogy, 282—283, 352 Mass ﬂux, 215—216 Mesh generation, 376 Microchannel ﬂow characteristics, 146—147 two-phase media annular, 170—177 CHF empirical correlations, 216—220 experiments, 210—215

general issues, 209—210 mass, 215—216 noncondensables, 215—216 pressure, 215—216 theoretical models, 221—224 trends, 210—215 conditions, 147—148 correlations, 161—166 in cracks differential conservation, 236—240 experimental data, 225—230 general issues, 224—225 integral models, 232—236 numerical models, 236—240 deﬁnition, 148—149 microgravity, 159—161 narrow rectangular, 170—177 noncondensable release, 178—179 pressure drop experiment review, 184—191 fractional, 180—183 general issues, 180 rod bundle patterns, 166—168 in slits differential conservation, 236—240 experiments, 225—230 general issues, 224—225 integral models, 232—236 numerical models, 236—240 subcooled boiling bubble nucleation, 205—209 general issues, 191—192 instability, 198—205 nucleate onset, 195—198 signiﬁcant void, 198—205 void fractions, 192—195 surface wettability, 158—159 transitions models, 161—166 trends, 153, 156 void fractions, 169—170 Micro-rod bundles, 166—1687 Microscale heat transport heuristic approach, 37 traditional descriptions coupling factor, 40 elastic lattice vibration, 38—39 heat balance, 38 hyperbolic heat conduction, 41—42 in metals, 39—40 phonon radiative transfer, 41

subject index in solids, 39 two-ﬂuid model, 40 Mises, Richard von, 258

N Navier-Stokes equations description, 261 Reynolds averaged algebraic stress models, 388 description, 381—382 eddy viscosity models, 382—388 stress models, 388—392 Newton, Isaac, 258 Noncondensables, 215—216 Nucleate boiling, 195—198 Nucleation, bubble, 205—209 Nu values convection parallel-plate channels, 333, 335 uniformly heated tube, 330 correlations differential analogy, 344—345 dimensional analysis, 335—337 empirical equations, 337—339 low-Prandtl-number ﬂuids, 339—344 mechanistic analogies, 342—343 theoretically based components, 345—346 interpretation, 348 isothermal plates, 347—348 parallel plates, 347 round tubes, 346—347 structure, 345—346 test, 348 integral formulations, 355 numerical solutions, 355—356

O Offset strip ﬁns, 398—406

P Partial differential equations CHE models, 115—116

optimization, 126—127 PDE. see Partial differential equations Phonon, 41, 49—50 Photography, strobe ﬂash, 171 Photonic crystals bandgap, 52—56 Plates chevron description, 425—429 local analysis, 429—430 wavy channels via, 431—432 exchanger, 364 ﬁne heat exchangers, 114—115 isothermal, 347—348 parallel convection, 318—320 equal uniform heating, 333 uniformly heated, 347 Porous media ﬂow resistance experimental assessment, 67—69 momentum in 1D membrane equations, 69—75 model 1, 75 model 2, 75 model 4, 75—76 pressure loss, 77—80 simulation procedures, 80—84 heat transfer coefﬁcients assumptions, 85 ﬂuid phase one, 108—111 models conventional, 87 correct form, 86—87 full energy equation, 88—89 nonlinear ﬂuctuations, 88 simulation procedures, 90—94 liquid-impregnated, 66 nonlinear transport, 15—17 radiative heat transport issues, 57—58 linear transfer, 58—59 nonlocal volume, 60—64 transport closure theories, 32—37 linear/nonlinear, 15—17 Power-law models, 278—282 Prandtl, Ludwig, 259 Prandtl analogy, 343 Prandtl number convection, 323—326

467

subject index

468 Prandtl number(Continued) ﬂuids, 339—342 tubes, 328—329 turbulent, 354—355 values, 328—329 Pressure drop, microchannel ﬂow experiment review, 184—191 fractional, 180—183 general issues, 180 microchannel ﬂow, 215—216 Pressure loss experiments, 77—80 PU-BTPFL-CHF Database, 217—219

Reynolds numbers CHE surfacess, 374—375, 391—392 louver ﬁns, 369—370 Navier-Stokes equations description, 381—382 eddy viscosity models, 382—388 stress models, 388—390 wall effect models, 390—392 offset strip ﬁns, 368 Reynolds stress, 27 Rough piping, 284—286 Round tubes, 346—347

S R Radiative transport heterogeneous media harmonic ﬁeld equations, 64—65 issues, 57—58 nonlocal volume, 60—64 phonon, 41 porous media issues, 57—58 linear transfer, 58 nonlocal volume, 60—64 VAT basis, 3 Rayleigh, Lord, 258 Renormalization group model, 395 Representative elementary volume averaging types differentiation conditions, 5—6 ﬁxed space, 4—5 lemma, 8—9 porous medium, 4 scale variables, 10 virtual, 7—8 heat transfer, 44—45 transport averaging, 3 Resistance, ﬂow porous media experimental assessment, 67—69 momentum in 1D membrane, 69—75 pressure loss, 77—80 simulation procedures, 80—84 REV. see Representative elementary volume Reynolds, Sir Osborne, 259 Reynolds analogy, 260, 343

Scaling, 77—80 Shear stress local, equations, 299—301 turbulent ﬂow correlating equations, 351—352 integral formulations, 350—351 limited models, 352—353 MacLeod analogy, 352 new model, 348—349 obsolete models, 352—353 Signiﬁcant void, 198—205 Slits microchannel differential conservation, 236—240 experiments, 225—230 general issues, 224—225 integral models isentropic homogeneous-equilibrium, 232—233 LEAK, 235—236 Moody’s, 234—235 numerical models, 236—240 Smagorinsky model, 394—395 Sommerfeld, Arnold, 258 Space averaging, 261—262 Spalart-Allmaras model, 383 Speculation, 263—264 Stress algebraic models, 388 Reynolds, 262, 388—390 shear equations, 299—301 turbulent ﬂow

subject index correlating equations, 351—352 integral formulations, 350—351 limited models, 352—353 MacLeod analogy, 352 new model, 348—349 obsolete models, 352—353 Strobe ﬂash photography, 171 Structure function model, 395 Subcooled boiling forced ﬂow bubble nucleation, 205—209 general issues, 191—192 instability, 198—205 nucleate onset, 195—198 void fractions, 192—195 Subcrystalline single crystals, 45—46 Surfaces CHE chevron plates, 425—430 interrupted ﬂow passages general, 366—367 louver ﬁns, 369—371 offset strip ﬁns, 368—369 louver ﬁns, 406—416 numerical analysis general issues, 375 mesh generation, 376 solution algorithm, 378—380 offset strip ﬁns, 398—406 turbulence models algebraic stress, 388 DNS, 392—395 eddy viscosity, 381—388 general issues, 380—381 LES, 392—395 Reynolds number ﬂow, 391—392 Reynolds stress, 388—390 wall effects, 390—392 uninterrupted complex chevron plates, 372—373 furrowed channel, 372 general issues, 371 intermating plates, 372—373 Reynolds number, 374—375 unsteady laminar, 374—375 wavy corrugated channel, 372 wavy channels corrugated, 372, 416—422 furrowed, 422—425 wettability, 158—159

469 T

Temperatures isothermal wall, 331—332 logitudinal phonon, 49 uniform wall, 316—317 Transfer, heat CHE surface interrupted general, 366—367 louver ﬁns, 368—370 offset strip ﬁns, 368—369 uninterrupted complex chevron plates, 372—373 furrowed channel, 372 general issues, 371 intermating plates, 372—373 Reynolds number, 374—375 unsteady laminar, 374—375 wavy corrugated channel, 372 coefﬁcient, 335, 337 corrugated channels, 429—430 Transport averaging REV description, 3 differentiation conditions, 5—6 ﬁxed space, 4—5 lemma, 8—9 porous medium, 4 scale variables, 10 virtual, 7—8 heat wave CHE models control problems, 123 current practice, 113—116 development, 111—112 optimization, 124—125 VAT-based, 117—122 superstructures acoustical phonon, 49—50 electromagnetic, 50—51 electron conversion, 46—47 ﬂuid momentum, 47—48 gas energy, 49 longitudinal phonon, 49 linear particle, 58—59 microscale heat heuristic approach, 37 traditional descriptions

470

subject index

Transport (Continued) coupling factor, 40 elastic lattice vibration, 38—39 heat balance, 38 hyperbolic heat conduction, 41—42 in metals, 39—40 phonon radiative transfer, 41 in solids, 39 two-ﬂuid model, 40 porous media closure theories, 32—37 nonlinear, 15—17 raditative heat issues, 57—58 linear transfer, 58—59 nonlocal volume, 57—58 VAT, 3 radiative heat heterogeneous media harmonic ﬁeld equations, 64—65 issues, 57—58 nonlocal volume, 60—64 Tubes round, 346—347 uniformly heated Nu values, 330 particular conditions, 326—328 Pr values, 328—329 Turbulence CHE surfaces DNS, 395—397 LES, 392—395 models algebraic stress, 388 eddy viscosity, 383—388 general issues, 380—381 Reynolds stress, 388—390 wall effects, 390—392 zero-equation, 382—383 Turbulent convection alternative models, 320—323 correlating equations, 356 differentials, 305—309 geometry formulations, 318—320 heat ﬂux density ratio, 354 initial perspectives, 353 integrals general equations, 309—310 generalized expressions, 317—318

uniform wall heat ﬂux, 311—316 temperature, 311—316 models, 353—354 Nu correlations differential analogy, 344—345 dimensional analysis, 335—337 integral, 355 low-Prandtl-number ﬂuids, 339—342 mechanistic analogies, 342—344 theoretically based components, 345—346 interpretation, 348 isothermal plates, 347—348 parallel plates, 347 round tubes, 346—347 structure, 345—346 test, 348 parallel plates different uniform temperatures, 333, 335 equal uniform heating, 333 MacLeod analogy, 318—320 Prandtl number, 323—326, 354—355 uncertainty, 323—324 uniformly heated tube isothermal wall, 331—332 Nu values, 330 particular conditions, 326—328 Pr values, 328—329 Turbulent ﬂow asymptotic analysis, 263—69 CHE surfaces, 374—375 Colebrook equation, 283—286 dimensional analysis, 263—269 dimensional models, 273—274 eddy viscosity, 269 exact structure, 260—263 friction factor correlations, 301—303 description, 283—286 geometry correlations, 303—304 MacLeod analogy, 282—283 mixing length, 269—273 model-free formulations, 294—295 near centerline values, 287 near wall values, 286—287 new formulations, 303—304 Nikuradse data, 274—276 numerical simulations, 274

subject index power-law models, 278—282 recapitulation, 304 rough piping, 283—286 shear stress correlating equations, 351—352 correlations, 299—301 integral formulations, 350—351 limited models, 352—353 MacLeod analogy, 352 new model, 348—349 obsolete models, 352—353 speculative analyses, 263—269 study, history, 257—259 velocity distribution correlations, 301—303 description, 292—294 Zagarola data, 287—292 Turbulent transport porous media momentum equations, 22—26 nonlinear, 14—17 theoretical bases, 21—22 theory, 26—27 Two-phase media microchannel ﬂow annular, 170—177 CHF empirical correlations, 216—220 experiments, 210—215 general issues, 209—210 mass, 215—216 noncondensables, 215—216 pressure, 215—216 theoretical models, 221—224 trends, 210—215 conditions, 147—148 correlations, 161—166 in cracks, 232—236 differential conservation, 236—240 experimental data, 225—230 general issues, 224—225 numerical models, 236—240 deﬁnition, 148—149 microgravity, 159—161 narrow rectangular, 170—177 noncondensable release, 178—179 pressure drop experiment review, 184—191 fractional, 180—183

471

general issues, 180 regimes, 150—153, 156 rod bundle patterns, 166—168 in slits differential conservation, 236—240 experiments, 225—230 general issues, 224—225 integral models, 232—236 numerical models, 236—240 subcooled boiling bubble nucleation, 205—209 general issues, 191—192 instability, 198—205 nucleate onset, 195—198 signiﬁcant void, 198—205 void fractions, 192—195 surface wettability, 158—159 transitions models, 161—166 trends, 153, 156 void fractions, 169—170 thermal conductivity conventional formulation, 97—98 local distribution, 96 piecewise distribution, 96 VAT considerations, 99—101 Two-temperature conservation, 43—45

U Uhlenbeck, George, 258

V VAT. see Volume averaging theory Velocity correlations, 301—303 distribution, 292—294 Vinci, da Leonardo, 258 Viscosity dissipation, 264—269, 306 eddy description, 393 ﬁlter approach model, 394 one-equation models, 383 two-equation models advantages, 383 low Reynolds numbers, 387

472

subject index

Viscosity (Continued) realizable k—, 386—387 RNG k—, 384—386 standard k—, 383—384 zero-equation models, 382—383 Viscous shear stress law, 269 Void fractions, 169—170, 192—195 Volume averaging theory development, 1—2 electrodynamics, nonlocal, 46—47 features, 1—2 heat wave transport CHE models design problems, 123 development, 111—112 equations, 117—122 optimization, 127—128 subcrystalline crystal, 45—46 superstructures acoustical phonon, 49—50 electromagnetic, 50—51 electron conversion, 46—47 ﬂuid momentum, 47—48 gas energy, 49 longitudinal phonon, 49 heterogeneous media, 99—101 highly porous medium turbulent model development additive components, 29 ﬁrst level hierarchy, 27 free stream, 28 mass conservation, 29 momentum equations, 30—32 scalar diffusion, 29 separate obstacle, 28 theoretical bases, 26—27 nonlinear ﬂuid medium laminar ﬂow concentration value, 20 homogeneous phase diffusion, 20 mass transport, 21 momentum diffusion, 20—21 Navier-Stokes equations, 19—20 steady-state momentum, 21 photonic crystals bandgap DMM-DMN mismatches, 52—53 governing equations, 54—55 porous media data reduction, 66—67

internal heat transfer assumptions, 85 models, 86—89 simulation procedures, 90—94 laminar ﬂow diffusion equation, 18 divergence form, 17 ﬂuid phase, 17 impermeable interface, 18 momentum equations, 18—19 solid phase, 17 linear Stokes equations, 15 momentum in 1D membrane, 75 pressure loss, 77—78 simulation procedures, 80—84 porous medium transport, 32—37 porous medium turbulent, 21—26 radiative heat transport basis, 3 heterogeneous media harmonic ﬁeld equations, 64—65 issues, 57—58 nonlocal volume, 60—64 porous media issues, 57—58 linear transfer, 58 nonlocal volume, 60—64 theorem veriﬁcation 1D Cartesian coordinate version, 11 1-dimensional cases, 12 DMA, 12—132 integral terms, 11 2-phase heterogeneous medium, 11—12 3-phase homogeneous medium, 12 solid-phase equation, 10 steady-state conduction, 11 two-temperature conservation, 43—45

W Wall effects, model features, 390 function, 390—391 turbulence, 391—392 isothermal, 331—332 law of, 265, 283 near, convection, 286—287

subject index uniform density, 311—316 uniform temperature, 316—317 Wavy channels corrugated, 372, 416—422 furrowed, 422—425 via chevron plates, 431—432 Weizscker, C.R. von, 258

473

Wettability, surface, 158—159

Z Zel’dovich, Yakob, 258 Zero-equation models, 382—383

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