ADVANCES IN HEAT TRANSFER Volume 34
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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
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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 . . . . . . . .
. . . .
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. . . . 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. VATBased TwoTemperature Conservation Equations . . . . . . . . C. Subcrystalline Single Crystal Domain Wave Heat Transport Equations . D. Nonlocal Electrodynamics and Heat Transport in Superstructures . . . E. Photonic Crystals BandGap Problem: Conventional DMMDNM and VAT Treatment . . . . . . . . . . . . . . . . . . . . . .
V. Radiative Heat Transport in Porous and Heterogeneous Media . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Flow Resistance Experiments and VATBased Data Reduction in Porous Media . . . . . . . . . . . . . . . VII. Experimental Measurements and Analysis of Internal Heat Transfer Coefﬁcients in Porous Media . . . . . . . . . . VIII. Thermal Conductivity Measurement in a TwoPhase Medium . . . . . . . . . . . . . . . . . . . . . . . . . IX. VATBased 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
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contents
vi B. C. D. E.
New Kinds of Heat Exchanger Mathematical Models . . VATBased Compact Heat Exchanger Modeling . . . . Optimal Control Problems in Heat Exchanger Design . . A VATBased Optimization Technique for Heat Exchangers
X. New Optimization Technique for on VAT . . . . . . . . . . . . XI. Concluding Remarks . . . . . Nomenclature . . . . . . . . . References . . . . . . . . . . .
. . . . . . . . . . . . Material Design Based . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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116 117 123 124
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127 129 131 133
TwoPhase Flow in Microchannels S. M. Ghiaasiaan and S. I. AbdelKhalik I. Introduction . . . . . . . . . . . . . . . . . . II. Characteristics of Microchannel Flow . . . . . III. TwoPhase Flow Regimes and Void Fraction in Microchannels. . . . . . . . . . . . . . . . . A. Deﬁnition of Major TwoPhase Flow Regimes . . . . B. TwoPhase Flow Regimes in Microchannels . . . . .
. . . . . . . 145 . . . . . . . 146
. . . . . . . . . . . . Review of Previous Experimental Studies and Their Trends . . . Flow Regime Transition Models and Correlations . . . . . . Flow Patterns in a MicroRod Bundle . . . . . . . . . . . Void Fraction . . . . . . . . . . . . . . . . . . . . . TwoPhase Flow in Narrow Rectangular and Annular Channels .
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. . . C. . D. . E. . F. . G. . H. TwoPhase Flow Caused by the Release of Dissolved Noncondensables . IV. Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . A. General Remarks . . . . . . . . . . . . . . . . . . . . . . B. Frictional Pressure Drop in TwoPhase 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 TwoPhase Critical Flow in Microchannels . . . . . . . . . . . . . . . . . . . . D. Integral Models . . . . . . . . . . . . . . . . . . . . E. Models Based on Numerical Solution of Differential Conservation Equations . . . . . . . . . . . . . . . . . . . . . .
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. VIII. Concluding Remarks . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .
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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 . . .
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. 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 . . . . . . . . . . . . . . . . . . . . .
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294
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. 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 .
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. 363 . 366 . 366 . 371
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. 374 . 375 . 376 . 376 . 378
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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. AbdelKhalik (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), CEAGrenoble, DTP/GRETh, Grenoble, France. P. Tochon (363), CEAGrenoble, 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 universitylevel 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 thirtyfour 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 socalled 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: 0120200341
ADVANCES IN HEAT TRANSFER, VOL. 34 Copyright 2001 by Academic Press. All rights of reproduction in any form reserved. 00652717/01 $35.00
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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 threephase 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, lowerscale ﬂuctuations of variables, crosseffects 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 secondorder turbulent models, equation sets were obtained for turbulent ﬁltration and twotemperature or twoconcentration 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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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 steadystate 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 solidphase, U s : ndS). The fourth condiinwarddirected 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 fphase. 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 socalled ‘‘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
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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 righthandside 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 VATbased 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 twophase 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 stedystate 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 righthand side are the socalled morphodiffusive terms, MD and MD , respectively (see also, for example, Travkin and Catton [21]), The solidphase 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 twophase 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 fphase, is m : m : /, and the volume fraction of ﬁller, or sphase, 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 sphase. 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 twoequation, threeterm 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 threedimensional (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 particlereinforced composites. The composite medium model consists of the three regions shown in Fig. 3. The halfspace lying above the A—A plane has a volume content of the disperse phase m : m , and for the halfspace 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 onedimensional 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 twophase 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 oneterm 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 twoequation, threeterm conductivediffusion transport VAT model (28) and (30) with the exact DNM solution and with the onetemperature 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 oneterm, onetemperature effective diffusivity equation and the
14
v. s. travkin and i. catton
Fig. 4. Comparison of VAT threeterm equation particle temperature (symbols) with the exact analytical based on the effective conductance coefﬁcient obtained by exact DNM (solid lines).
threeterms, twotemperature VAT equations does not mean that it is better to take for modeling and analysis the effective diffusivity oneterm, 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 oneterm onetemperature 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 twoﬁeld 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 righthandside 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 righthand 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 righthand 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 steadystate 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 steadystate 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 massexchange 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 rodbundle 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 twophase transport modeling equations [52] with heterogeneity of spacial phase distributions neglected in the bulk. Threedimensional twoﬂuid 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 twoﬂuid 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 twophase 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 halflinear 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 equalsized 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 twodimensional 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 twophase 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 righthand 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 twophase 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 twophase 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 righthand 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 volumeaveraged turbulent transport equations. As in numerous other studies of multiphase transport, the major difﬁculties of averaging the terms on the righthand 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. Largescale 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 secondorder 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 righthand 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 roundsection channel is involved, and even if the crosssection 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 largescale 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 Ktheory 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 righthand 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 righthand 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 righthand 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. Onedimensional 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 twotemperature treatments. Travkin et al. [19, 26] showed that the proper form for the turbulent heat transfer equation in the ﬂuid phase using oneequation Ktheory 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 wellknown types of porous media morphologies for which researchers have had major successes. But even for these morphologies, namely straight parallel pores and equalsize 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 onetemperature 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 onedimensional heat conduction equation with a stochastic function for the thermal diffusivity in a paper by Fox and Barakat [82] yielded a spatially fourthorder partial differential equation to be solved. Gelhar et al. [83], after having eliminated the secondorder 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 righthand 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 righthand 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 nearneighbor 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 onetemperature 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 twophase 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 convectivediffusion 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 secondorder 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 reactiondiffusion problem Crapiste et al. [41] made a series of assumptions: (1) the diffusion coefﬁcient D and the ﬁrstorder 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 twotemperature 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 twotemperature 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 secondorder 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 manybody systems that exhibit highly nonequilibrium behavior, including nonMarkovian 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 VATbased 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 shorttimespan 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 twoﬂuid 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 twoﬁeld twotemperature 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 twoﬂuid model with two equations for the electron—phonon transport in metals based on previous works by Anisimov et al. [101], Fujimoto et al. [105], ElsayedAli [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 twoﬂuid 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 twostep models in metals. A more complex twotemperatures 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 twoﬂuid 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 directionalspectral 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 socalled 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 righthand 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 ndoping concentration, N L N is the pdoping concentration, and p is the hole number density. B. VATBased TwoTemperature 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 VATbased 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 righthand 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 VATbased 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 righthandside (‘‘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 twoﬂuid models. The equation for heat transport in the second phase (if any) would be the same, and one can easily obtain the hyperbolic type twoﬂuid temperature model. A similar VATbased 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 righthand 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 twotemperature 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 solidstate 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 righthand side of (142), the scattering and collision reﬂection term, has been replaced by a number of terms, each reﬂecting interfacespeciﬁc 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 lefthand 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 VATbased 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 BandGap Problem: Conventional DMMDNM 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 bandgap 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 bandgaps 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 selfsufﬁcient 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 DMMDNM, 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 DMMDNM 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 DMMDNM 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 soughtafter 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 DMMDNM 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, twoﬁeld 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 bandgap formulation should be treated using the HMM statements developed from the analysis of the VAT equations. A straightforward description of one of the bandgap problems is given next. Representing electromagnetic ﬁeld components with timeharmonic 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 bandgap 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 Epolarized ﬁelds and 9 · for Hpolarized 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 twophase medium, which is supposed to describe the photonic materials bandgap 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 spacedependent function with changes at the interface boundary. When these equations are phase averaged to represent the macroscale characteristics of wave propagation in a twophase 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 righthand side of each of the equations. When the dielectric permittivity function is homogeneous in each of the two phases, then the VAT photonic bandgap 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 bandgap problem should also be very similar. Using DMMDNM, 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 AlNimr 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 twophase (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 crossﬁeld (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 twolayer 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
58
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) ?
[email protected] 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 ?
[email protected] 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 twostates Q discrete random variables. By assuming this, one needs to treat the porous
volume averaging theory
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 ensembleaveraged 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 ensembleaveraged ﬂuxes. H G The solution to this equation is also supposed to be ensembleaveraged. 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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v. s. travkin and i. catton
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 VATbased 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.
volume averaging theory
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 ?
[email protected] 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 ?
[email protected] 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
v. s. travkin and i. catton
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 ; (, # ; , # ) ?
[email protected] ?
[email protected] 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 ﬂuidﬁlled 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)
volume averaging theory
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)
v. s. travkin and i. catton
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 timeharmonic 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 timeharmonic 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 timeharmonic 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
66
v. s. travkin and i. catton
temperature ﬁelds; still, the same questions of multiple versions, applicability of current methods, and variance in interpretation are the present agenda. VATbased models were developed recently while addressing the problems of modeling of electrodynamic properties of a liquidimpregnated porous ferrite medium (Ponomarenko et al. [161]), coupled electrostaticdiffusion processes in composites (Travkin et al. [159]), and to analyze heat conductivity experimental data in highT 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 VATBased 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. VATbased 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 ‘‘crosscorrelation’’ 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 oneterm 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 twoterm models used for ﬂow resistance experimental data reduction have ﬁrstorder and secondorder 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 equaldiameter 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 VATbased 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 Darcylawbased 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 VATbased 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 wellestablished correlations for the Fanning friction factor in tubes. 2. Momentum Resistance in 1D Membrane and Porous Layer Transport The steadystate VATbased governing equations for laminar transport in
v. s. travkin and i. catton
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 Ktheory 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 onedimensional 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 onedimensional 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 lefthand 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 righthand 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 righthand 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 righthand 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 righthand terms in Eq. (244). The inhomogeneous terms on the righthand side of (247) may be analyzed by scaling. Some of these terms are easily interpreted. For example, the ﬁrst term on the righthand 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 righthand integral terms reﬂect the morphologyinduced 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 socalled 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, wovenscreen 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 .
[email protected] This assumption led them to the formula md N . r :
[email protected] 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 :
[email protected] 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 onediameter 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 :
[email protected] m u! L D This friction factor can be related to the friction factor f , given by Eq. @ (6.41) 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)
[email protected] @ 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 twoterm 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 Erguntype 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 VATbased 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 Ergunlike type of correlation 40 ; 0.6. f : D+ Re NMP
(313)
Meanwhile, Souto and Moyne [181], using the DMMDNM 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 VATbased 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
[email protected], 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 steadystate 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 righthand 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 righthand 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).
v. s. travkin and i. catton
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.
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v. s. travkin and i. catton VIII. Thermal Conductivity Measurement in a TwoPhase 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 crosssectional area of the sample. 1. Traditional L ocal and Piecewise Distributed Coefﬁcient Heat Conductivity Problem Formulations In DMMDNM 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 DMMDNM 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 twophase medium. According to most accepted mathematical statements this problem is given as (368)—(371). 3. Conventional Formulation of the Effective Conductivity Problem in a TwoPhase 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 twophase 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 twophase 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 quasihomogeneous 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. VATBased 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 steadystate 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
volume averaging theory
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 hightemperature 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)
volume averaging theory
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 steadystate 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 twophase 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 twotemperature 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 steadystate 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 soughtafter 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 steadystate 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 DMMDNM exact solution. The issue is that in a majority of problems, such as for inhomogeneous, nonlinear coefﬁcients and in many transient problems, having the twoﬁeld DMMDNM exact solution is not enough to ﬁnd effective coefﬁcients. As shown earlier, only the requirement of thermal equilibrium warrants the equality of steadystate and transient effective conductivities in a twophase medium.
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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 twotemperature 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 twophase 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 twophase problem. We see that the twotemperature DMMDNM 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 twotemperature 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 DMMDNM solution of the 2D problem of uniformly located quadratic rods with equal spacing in both directions. Studies were undertaken of both the Forchheimer and postForchheimer ﬂow regimes. This work is a good example of how DMMDNM 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 DMMDNM. 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 solidphase (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 onetemperature 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 DMMDNM 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 DMMDNM the D G assumption of thermal equilibrium has no sense at all — as long as the problem have been already calculated as the twotemperature 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 DMMDNM 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 steadystate and transient effective conductivities in a twophase medium. Consequently, if taken correctly, the twotemperature 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 temperaturedependent 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.
volume averaging theory
111
2. Summarizing application of DMMDNM 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 onecell (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. VATBased 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 ‘‘jfactor’’ 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 morphologycharacteristic 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 halfheight.
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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. Threephase 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. Secondorder 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 contractedtube 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 jfactor and the Fanning friction factor f for heat exchanger design. They are (1) powerlaw 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 aircooled heat exchangers with aluminum or copper ﬁnned tubes with large height, small thickness, and narrowpitch ﬁns, and hightemperature waste heat recovery exchangers with steel ﬁnned tubes with rather low height and thickness and widepitched 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 wideranging 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 VATbased 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 ﬁnitedimensional 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 secondorder turbulent models, equation sets are obtained for turbulent ﬁltration and twotemperature diffusion in nonisotropic porous media with interphase exchange and microroughness. 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. VATBased Compact Heat Exchanger Modeling For a pin ﬁn (PFHE), with crossﬂow 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 VATbased 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 morphologybased 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 righthand 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 VATbased 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 VATbased 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 VATBased 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 statespace 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 inﬁnitedimensional 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 distributedparameter 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 ﬁnitedimensional 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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127
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 VATbased 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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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 quasihomogeneous and quasistochastic methods that claim to accomplish this. In most cases, quasihomogeneous and quasistochastic 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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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 VATbased 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. VATbased modeling is very powerful, allowing random morphology ﬂuctuations to be incorporated into the VATbased 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 welldeveloped closure expressions for the VATbased 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
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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 VATbased 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 DEFG0389ER14033 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)]; halfwidth 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 crosssectional 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) ensembleaveraged value of interface ensembleaveraged 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
TwoPhase Flow in Microchannels
S. M. GHIAASIAAN and S. I. ABDELKHALIK 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 twophase ﬂow in microchannels is reviewed. Only microchannels with hydraulic diameters of the order of 0.1 to 1 mm and with long lengthtohydraulic 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 highpower electronic systems, the cooling of the plasmafacing 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 highpressure liquids is another application of twophase ﬂow in microchannels of interest here. Twophase ﬂow in capillaries with complex geometry (porous media) has been reviewed in the recent past [2—4] and will not be addressed. 145
ISBN: 0120200341
ADVANCES IN HEAT TRANSFER, VOL. 34 Copyright 2001 by Academic Press. All rights of reproduction in any form reserved. 00652717/01 $35.00
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For steadystate and developed twophase ﬂow in a smooth pipe, and assuming incompressible phases, the variables that can affect the hydrodynamics of gas—liquid twophase 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 twophase 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 twophase ﬂow in microgravity, resulting in important and useful similarities between the two categories of systems with respect to hydrodynamics of twophase ﬂow. In both types of systems the predominance of the surface tension force on buoyancy leads to the insensitivity of twophase hydrodynamics to channel orientation, and in nonseparated twophase ﬂow patterns it leads to the suppression of velocity difference
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between the two phases in the absence of signiﬁcant acceleration. Suo and Grifﬁth [9] derived the following criterion for negligible buoyancy effect in twophase pipe ﬂow: gD * & 0.88/3.
(6)
Based on an analysis of stratiﬁed to nonstratiﬁed ﬂow regime transition, Brauner and MoalemMaron [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 instabilitydriven processes may be entirely irrelevant to microchannels. The criterion of Suo and Grifﬁth, Eq. (6), can approximately be recast as
III. TwoPhase Flow Regimes and Void Fraction in Microchannels The gas and liquid phases in a twophase 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 twophase 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 twophase ﬂow regimes is thus required for the design of twophase ﬂow systems and for the speciﬁcation of appropriate closure relations for twophase 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 twodimensional 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 twodimensional 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 twophase ﬂow regime data representing microchannels. A. Definition of Major TwoPhase Flow Regimes Experiments indicate that ﬂow regimes, which are morphologically similar to the major twophase 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 twophase ﬂow systems in large vertical channels. Bubbly ﬂow occurs at low gas and liquid ﬂow rates and is characterized by bubbles distortedspherical in shape and moving upward in zigzag fashion. The slug ﬂow regime is characterized by bulletshaped 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 dispersedbubbly 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 twophase 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 stratiﬁedsmooth 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 dispersedbubbly regime occurs at very high liquid ﬂow rates, and its characteristics are similar to those of the dispersedbubbly 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. TwoPhase Flow Regimes in Microchannels Early studies dealing with twophase ﬂow in microchannels were mostly concerned with surface tensiondriven ﬂows [5—8]. Twophase 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]. Twophase 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 microrod bundle
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151
and in narrow annuli, respectively. Twophase 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 highspeed videocameras recording ﬂow details near the centers of the test sections. Figure 4 displays typical photographs of the ﬂow patterns identiﬁed in their 1.09mmdiameter 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. Crosssectional geometry of the test sections of Triplett et al. [12].
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Fig. 4. Representative photographs of ﬂow patterns in the 1.1mmdiameter test section of Triplett et al. [12]. (With permission from [12].)
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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 pseudoslug [9], churn [26], and frothyslug 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 wavyannular ﬂow pattern (Fig. 4d). This ﬂow pattern is referred to as frothy slugannular 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 wavyannular 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 twophase 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 TwoPhase 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, sevenrod 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 twophase ﬂow regime data of Triplett et al. [12] are shown in Fig. 5. Regime transition lines representing a microrod 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 twophase 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 microrod bundle (Narrow et al. [34]).
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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 twophase 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, disperseddroplet, 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 mmdiameter 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 intermittenttobubbly 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 highspeed camera. The identiﬁed ﬂow regimes were bubbly, slug, churn, annular, and annularmist. 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 twophase 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 twophase ﬂow in common large channels in microgravity has important similarities with twophase 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 (, 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 steadystate,
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 ) : 514 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. U14 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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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" U14 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).
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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 crosssectional geometries displayed in Fig. 24. Test sections (a) and (b) were uniformly heated circular channels, and
Fig. 24. Crosssectional geometries of the test sections of Roach et al. [135]. (With permission from [135].)
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test sections (d) and (e) were meant to represent the ﬂow channels in a microrod 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 airsaturated 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].)
twophase 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 airsaturated 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 innertoouter 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,
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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.
[email protected] 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)
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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 nearwall 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. abdelkhalik
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 twodimensional 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
twophase 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. abdelkhalik
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 lowﬂow, lowpressure 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/twophase 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 R113 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
R113
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
[email protected]GL : 29.8—70.5°C GL T : 6.4—84.9°C GL T
[email protected]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. abdelkhalik
[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 cmdiameter channel. The experimental data of Nariai et al. [169, 170] include subcooled as well as saturated (twophase) 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 R113, 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
twophase flow in microchannels
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 lengthtodiameter 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. abdelkhalik
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].
twophase 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].
216
s. m. ghiaasiaan and s. i. abdelkhalik
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 forcedconvection 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.
twophase 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)
[email protected] within :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)
266
stuart w. churchill
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 crosssection insofar as integration to determine u> is concerned. The result K
turbulent flow and convection
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 crosssection results in
2 3B a : u> : C 9 ; B ln , K f 2 e
(30)
stuart w. churchill
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.
turbulent flow and convection
269
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 timemean 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 mixinglength 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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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 soreduced 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 Reynoldsstress 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 timedependent equations of conservation but introduces arbitrary terms such as those of the ,— and Reynoldsstress 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 timemean 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 127mm 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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277
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. PowerL aw Models Powerlaw 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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279
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 mixinglength 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 crosssection 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 powerlaw 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. Thirtyseven 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 ﬁxedpower dependence of the velocity on the distance from the wall over the entire crosssection is required to obtain a powerlaw dependence of the friction factor on the Reynolds number. It may therefore be inferred from the previously cited failures of the powerlaw model for the velocity near the wall and near the centerline, and more importantly from the observed dependence of on Re, that a pure powerlaw 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 powerlaw 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 powerlaw models might have been anticipated on the basis of dimensional analysis. Rayleigh [25] used a powerseries 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 littleknown 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 halfspacing 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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283
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 sidewall 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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285
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 timeaveraged velocity and axial pressure gradient in air ﬂowing through a 129mm 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 timemean 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 powerlaw 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 higherorder 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 crosssection, 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 righthand 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. ModelFree Formulations The mixinglength model, as expressed by Eq. (39), was proposed by Prandtl to facilitate the prediction of the turbulent shear stress in the timeaveraged 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 welldeﬁned 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 timeaveraged 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 wellknown expressions for purely laminar ﬂow at the same value at a>. Equation (89) may in turn be integrated formally over the crosssectional area to obtain the following expression for the space or mixedmean 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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297
which involves only a single integral. The rightmost form of Eq. (91) reveals that the effect of the turbulence on the mixedmean velocity is also simply a deduction from the wellknown expression (Poiseuille’s law) for purely laminar ﬂow. This deductibility of an integral term from the expressions for the timemean velocity distribution and the mixedmean 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 mixedmean 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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298
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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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
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300
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 timeaveraged 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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302
(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 onedimensional 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 mixedmean temperature must be determined by integrating the product of the temperature and the velocity over the crosssection at a series of axial distances, whereas the mixedmean 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 ydirection (negativerdirection) due to thermal conduction and the turbulent ﬂuctuations, respectively. The integral term on the lefthand 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 negativerdirection. 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
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(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 mixedmean 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
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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, mixedmean 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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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 doublepipe 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 integratedmean 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 righthand 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 righthand 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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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 righthand side of Eq. (134) may be dropped. Finally, letting z ;  gives the following expression for the fully developed temperature, which differs negligibly from the mixedmean 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 ﬁrstorder 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
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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
turbulent flow and convection
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 integratedmean 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 integratedmean 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)
stuart w. churchill
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 halfspacing 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 integratedmean value over the channel. For Pr ; , Eq. (136) is directly applicable in terms of Nu and Re . @ @ Expressions for equal uniform walltemperatures 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 parallelplate 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 onedimensional 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)
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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 crosssection 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
stuart w. churchill
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)
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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 parallelplate 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 closedform 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
stuart w. churchill
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 crosssection 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
326
stuart w. churchill
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
turbulent flow and convection
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
stuart w. churchill
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 integratedmean value of the term in Nu :
turbulent flow and convection
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.
330
stuart w. churchill
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 Graetztype 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.
turbulent flow and convection
331
TABLE V The Thermal Characteristics of Fully Developed Turbulent Convection in a Round Tube with Uniform WallTemperature 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 WallTemperature 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 WallTemperature 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 WallTemperature 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>.
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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 ParallelPlate 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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334
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
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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
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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 ﬁrstorder 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 nonpowerfunction of Re, the proportionality of Nu to Re( f /2) or Re( f /2) does not constitute a powerdependence 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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stuart w. churchill
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.)
turbulent flow and convection
339
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 ﬁrstorder approximation for Pr : O 1 and on Pr except in the asymptotic limit of Pr ; . 3. Numerical Predictions for L owPrandtlNumber 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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340
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.)
342
stuart w. churchill
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
turbulent flow and convection
343
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
stuart w. churchill
344
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 timeaveraged 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 crosssection. Had he utilized the limiting form of Eq. (93), that is, Eq. (194) for (uv)>> in the rightmost term, he would have
turbulent flow and convection
345
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 integratedmean 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)
346
stuart w. churchill
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 :
turbulent flow and convection
347
variable in Eqs. (200) and (201) is arbitrary since they all bear a onetoone 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 selfconsistent 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)
348
stuart w. churchill
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 onethirdpower 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, timeaveraged 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]
turbulent flow and convection
349
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 timeaveraged 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 timeaveraged equations may be expected to persist into the
350
stuart w. churchill
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 timeaveraged 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 parallelplate 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 timeaveraged velocity as well as for the timeaveraged 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 timeaveraged velocity distributions measured by Nikuradse [46]. These latter constants were subsequently reevaluated by Churchill [80] using the recent improved measurements of the timeaveraged velocity distribution by Zagarola [73]. The incorporation of the equivalent of the semilogarithmic expression for the timeaveraged 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 justmentioned 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 littleknown 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 parallelplate 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 parallelplate 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 eddyviscosity 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 timeaveraging, 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 coworkers have recently attempted to resuscitate the powerlaw 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 Timeaveraging 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 timeaveraged 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 onetoone 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 parallelplate 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 parallelplate 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 parallelplate 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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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). Reynoldsnumber 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, PrenticeHall, 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 heattransfer 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. McGrawHill 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 liquidmetal 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 walltemperature 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 CEAGrenoble 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 shellandtube heat exchangers. Somewhat arbitrarily, a gastoﬂuid 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 phasechange stream. In contrast, a typical process industry shellandtube 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: 0120200341
ADVANCES IN HEAT TRANSFER, VOL. 34 Copyright 2001 by Academic Press. All rights of reproduction in any form reserved. 00652717/01 $35.00
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Fig. 1. Plateﬁn geometries: (a) offset strip ﬁn, and (b) louver ﬁn.
times that with highﬁndensity lowﬁnned 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 shellandtube exchanger for water/water applications. For phasechange applications, even higher heat transfer coefﬁcients are achieved compared to a shellandtube 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. Plateﬁn, tubeﬁn, 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 plateﬁn and tubeﬁn 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 plateﬁn exchangers are offset strip ﬁns
Fig. 2. Tubeﬁn 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 costeffective 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 stateoftheart review on numerical studies of singlephase 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 plateﬁn exchangers, wavy ﬁns/channels used in tubeﬁn 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 smallscale 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 selfsustained 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 slowmoving ﬂuid ﬂows in a large eddy pattern. The boundary between the separation bubble and the separated ﬂow (along the separation streamline) consists of a freeshear 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 nearﬁn 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 timeaveraged 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 pitchtolouver 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 ﬂowdirecting 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 largescale 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 fullsize 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 plateﬁn 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) Plateﬁn exchanger, (b) tubeﬁn exchanger with ﬂat ﬁns. At section AA: (c) wavy corrugated passage, and (d) wavy furrowed passage.
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Fig. 7. Crosssection 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 crosssection. 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¨rtlerlike — 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 crosssection 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 crosssectional 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 crisscrossing (threedimensional, 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 largerscale 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 pitchtohydraulic 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 twodimensional (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
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 
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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, HTDVol. 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, HTDVol. 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 SinglePhase 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. AbouMadi (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 wavywalled 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 convergingdiverging 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 ISHMTASME Heat Mass Trans. Conf. (G. Biswas, S. Srinivasa Murthy, K. Muralidhar, and V. K. Dhir, Eds.). TataMcGrawHill, 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, HS. (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 crosscorrugated ducts. ASME J. Heat Transfer 121, 314—321. 157. Sunde´n, B. (1999). Flow and heat transfer mechanisms in plateframe 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). Airside 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) AbdelKhalik, 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) AbouMadi, 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)
AlDaini, A. J., 407(103) Alfredsson, P. H., 381(29) AlHayes, 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) AlNimr, 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) DownarZapolski, 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) ElSayed, M. S., 81(180), 85(180) ElsayedAli, 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) MoalemMaron, 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 BaldwinLomax 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 VATbased 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 twophase 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 twotemperature, 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 lowPrandtlnumber ﬂuids, 339—342 mechanistic analogies, 342—344 numerical, 355—356 theoreticallybased, 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 homogeneousequilibrium, 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 oneequation models, 383 twoequation models advantages, 383—384 low Reynolds numbers, 387 realizable k—, 386—387 RNG k—;, 384—386 standard k—, 383—384 zeroequation models, 382—383 Einstein, Albert, 258 Electrodynamics, nonlocal VATgoverning 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 modelfree formulations, 294—295 near center line values, 287 near wall values, 286—287 new formulations, 303—304 Nikuradse data, 274—276 numerical simulations, 274 powerlaw 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 Gastoﬂuid 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 twophase 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 VATbased, 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 2phase, 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 Hightemperature superconductors, 101 HMM. see Heterogeneous media modeling Homogeneous isotropic media, 11 HTSC. see Hightemperature superconductors Hydrodynamics, 257 Hyperbolic heat conduction, 41—42
I Integral models isentropic homogeneousequilibrium, 232— 233 LEAK, 235—236 Moody’s, 234—235 Isentropic homogeneousequilibrium 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 NavierStokes equations, 19—20 steadystate 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 twophase 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 Microrod 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 twoﬂuid model, 40 Mises, Richard von, 258
N NavierStokes 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 parallelplate channels, 333, 335 uniformly heated tube, 330 correlations differential analogy, 344—345 dimensional analysis, 335—337 empirical equations, 337—339 lowPrandtlnumber 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 liquidimpregnated, 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 Powerlaw 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 PUBTPFLCHF Database, 217—219
Reynolds numbers CHE surfacess, 374—375, 391—392 louver ﬁns, 369—370 NavierStokes 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 homogeneousequilibrium, 232—233 LEAK, 235—236 Moody’s, 234—235 numerical models, 236—240 Smagorinsky model, 394—395 Sommerfeld, Arnold, 258 Space averaging, 261—262 SpalartAllmaras 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 VATbased, 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 twoﬂuid 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 zeroequation, 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 lowPrandtlnumber 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 modelfree 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 powerlaw 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 Twophase 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 Twotemperature 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 oneequation models, 383 twoequation models advantages, 383 low Reynolds numbers, 387
472
subject index
Viscosity (Continued) realizable k—, 386—387 RNG k—, 384—386 standard k—, 383—384 zeroequation 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 NavierStokes equations, 19—20 steadystate momentum, 21 photonic crystals bandgap DMMDMN 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 1dimensional cases, 12 DMA, 12—132 integral terms, 11 2phase heterogeneous medium, 11—12 3phase homogeneous medium, 12 solidphase equation, 10 steadystate conduction, 11 twotemperature 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 Zeroequation models, 382—383
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