_Advances_in_Heat_Transfer_-_Advances_in_Heat_Transfer_Volume_34__2000.pdf

8/11/2019 _Advances_in_Heat_Transfer_-_Advances_in_Heat_Transfer_Volume_34__2000.pdf

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ADVANCES INHEAT TRANSFER

Volume 34


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Advances inHEATTRANSFER

Serial Editors

James P. Hartnett Thomas F. Irvine, Jr.Energy Resources Center Department of Mechanical EngineeringUniversity of Illinois at Chicago State University of New York at Stony BrookChicago, Illinois Stony Brook, New York

Serial Associate EditorsYoung I. Cho George A. GreeneDepartment of Mechanical Engineering Department of Advanced TechnologyDrexel University Brookhaven National LaboratoryPhiladelphia, Pennsylvania Upton, New York

Volume 34

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Copyright 2001 by Academic Press

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CONTENTS

Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . ixPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Transport Phenomena in Heterogeneous Media Basedon Volume Averaging Theory

V. S. T I. C

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1II. Fundamentals of Hierarchical Volume Averaging

Techniques . . . . . . . . . . . . . . . . . . . . . . . . 4A. Theoretical Verification of Central VAT Theorem and Its Consequences . 10

III. Nonlinear and Turbulent Transport in Porous Media . . . . 14

A. Laminar Flow with Constant Coefficients . . . . . . . . . . . . . 17B. Nonlinear Fluid Medium Equations in Laminar Flow . . . . . . . . 19C. Porous Medium Turbulent VAT Equations . . . . . . . . . . . . 21D. Development of Turbulent Transport Models in Highly Porous Media . 26E. Closure Theories and Approaches for Transport in Porous Media . . . 32

IV. Microscale Heat Transport Description Problems andVAT Approach . . . . . . . . . . . . . . . . . . . . . . . 37A. Traditional Descriptions of Microscale Heat Transport. . . . . . . . 38B. VAT-Based Two-Temperature Conservation Equations. . . . . . . . 43C. Subcrystalline Single Crystal Domain Wave Heat Transport Equations. 45

D. Nonlocal Electrodynamics and Heat Transport in Superstructures. . . 46E. Photonic Crystals Band-Gap Problem: Conventional DMM-DNM andVAT Treatment. . . . . . . . . . . . . . . . . . . . . . . 52

V. Radiative Heat Transport in Porous and HeterogeneousMedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

VI. Flow Resistance Experiments and VAT-Based DataReduction in Porous Media . . . . . . . . . . . . . . . . . 66

VII. Experimental Measurements and Analysis of Internal HeatTransfer Coefficients in Porous Media . . . . . . . . . . . . 85

VIII. Thermal Conductivity Measurement in a Two-Phase

Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 96IX. VAT-Based Compact Heat Exchanger Design and

Optimization . . . . . . . . . . . . . . . . . . . . . . . . 111A. A Short Review of Current Practice in Heat Exchanger Modeling. . . 112

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B. New Kinds of Heat Exchanger Mathematical Models. . . . . . . . 116C. VAT-Based Compact Heat Exchanger Modeling. . . . . . . . . . 117D. Optimal Control Problems in Heat Exchanger Design. . . . . . . . 123E. A VAT-Based Optimization Technique for Heat Exchangers. . . . . . 124

X. New Optimization Technique for Material Design Basedon VAT . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

XI. Concluding Remarks . . . . . . . . . . . . . . . . . . . . 129Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . 131References . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Two-Phase Flow in MicrochannelsS. M. G S. I. A-K

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 145II. Characteristics of Microchannel Flow . . . . . . . . . . . . 146

III. Two-Phase Flow Regimes and Void Fraction inMicrochannels. . . . . . . . . . . . . . . . . . . . . . . . 147A. Definition of Major Two-Phase Flow Regimes. . . . . . . . . . . 148B. Two-Phase Flow Regimes in Microchannels. . . . . . . . . . . . 150C. Review of Previous Experimental Studies and Their Trends. . . . . . 153D. Flow Regime Transition Models and Correlations. . . . . . . . . 161E. Flow Patterns in a Micro-Rod Bundle. . . . . . . . . . . . . . 166F. Void Fraction. . . . . . . . . . . . . . . . . . . . . . . . 169G. Two-Phase Flow in Narrow Rectangular and Annular Channels. . . . 170H. Two-Phase Flow Caused by the Release of Dissolved Noncondensables. 178

IV. Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . 180A. General Remarks . . . . . . . . . . . . . . . . . . . . . . 180B. Frictional Pressure Drop in Two-Phase Flow. . . . . . . . . . . 180C. Review of Previous Experimental Studies. . . . . . . . . . . . . 184D. Frictional Pressure Drop in Narrow Rectangular and Annular Channels. 189

V. Forced Flow Subcooled Boiling . . . . . . . . . . . . . . . 191A. General Remarks . . . . . . . . . . . . . . . . . . . . . . 191B. Void Fraction Regimes in Heated Channels. . . . . . . . . . . . 192C. Onset of Nucleate Boiling. . . . . . . . . . . . . . . . . . . 195D. Onset of Significant Void and Onset of Flow Instability. . . . . . . 198E. Observations on Bubble Nucleation and Boiling. . . . . . . . . . 205

VI. Critical Heat Flux in Microchannels. . . . . . . . . . . . . . . . . 209A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 209B. Experimental Data and Their Trends. . . . . . . . . . . . . . . 210C. Effects of Pressure, Mass Flux, and Noncondensables. . . . . . . . 215

D. Empirical Correlations. . . . . . . . . . . . . . . . . . . . 216E. Theoretical Models. . . . . . . . . . . . . . . . . . . . . . 221VII. Critical Flow in Cracks and Slits . . . . . . . . . . . . . . . 224

A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 224B. Experimental Critical Flow Data. . . . . . . . . . . . . . . . 225

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C. General Remarks on Models for Two-Phase Critical Flow inMicrochannels . . . . . . . . . . . . . . . . . . . . . . . 230

D. Integral Models. . . . . . . . . . . . . . . . . . . . . . . 232E. Models Based on Numerical Solution of Differential Conservation

Equations . . . . . . . . . . . . . . . . . . . . . . . . . 236VIII. Concluding Remarks . . . . . . . . . . . . . . . . . . . . 240

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . 242References . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Turbulent Flow and Convection: The Prediction of Turbulent Flow

and Convection in a Round Tube

SW. C

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 256A. Turbulent Flow. . . . . . . . . . . . . . . . . . . . . . . 257B. Turbulent Convection . . . . . . . . . . . . . . . . . . . . 259

II. The Quantitative Representation of Turbulent Flow . . . . . 260A. Historical Highlights. . . . . . . . . . . . . . . . . . . . . 260B. New Improved Formulations and Correlating Equations . . . . . . . . . 294

III. The Quantitative Representation of Fully Developed

Turbulent Convection . . . . . . . . . . . . . . . . . . . . 304A. Essentially Exact Formulations. . . . . . . . . . . . . . . . . 305B. Essentially Exact Numerical Solutions. . . . . . . . . . . . . . 323C. Correlation for Nu . . . . . . . . . . . . . . . . . . . . . . 335

IV. Summary and Conclusions . . . . . . . . . . . . . . . . . . 348A. Turbulent Flow. . . . . . . . . . . . . . . . . . . . . . . 348B. Turbulent Convection . . . . . . . . . . . . . . . . . . . . 353References . . . . . . . . . . . . . . . . . . . . . . . . . . 356

Progress in the Numerical Analysis of Compact HeatExchanger Surfaces

R. K. S, M. R. H, B. T, P. T

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 363II. Physics of Flow and Heat Transfer of CHE Surfaces . . . . . 366

A. Interrupted Flow Passages. . . . . . . . . . . . . . . . . . . 366B. Uninterrupted Complex Flow Passages. . . . . . . . . . . . . . 371C. Unsteady Laminar versus Low Reynolds Number

Turbulent Flow. . . . . . . . . . . . . . . . . . . . . . . 374III. Numerical Analysis . . . . . . . . . . . . . . . . . . . . . 375A. Mesh Generation . . . . . . . . . . . . . . . . . . . . . . 376B. Boundary Conditions. . . . . . . . . . . . . . . . . . . . . 376C. Solution Algorithm and Numerical Scheme. . . . . . . . . . . . 378

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IV. Turbulence Models . . . . . . . . . . . . . . . . . . . . . 380A. Reynolds Averaged NavierStokes (RANS)Equations. . . . . . . . 381

B. Large Eddy Simulation (LES). . . . . . . . . . . . . . . . . 392C. Direct Numerical Simulation. . . . . . . . . . . . . . . . . . 395D. Concluding Remarks on Turbulence Modeling. . . . . . . . . . . 397

V. Numerical Results of the CHE Surfaces . . . . . . . . . . . 397A. Offset Strip Fins. . . . . . . . . . . . . . . . . . . . . . . 398B. Louver Fins . . . . . . . . . . . . . . . . . . . . . . . . 406C. Wavy Channels. . . . . . . . . . . . . . . . . . . . . . . 416D. Chevron Trough Plates. . . . . . . . . . . . . . . . . . . . 425

VI. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 432Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . 434

References . . . . . . . . . . . . . . . . . . . . . . . . . . 435Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . 445Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

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CONTRIBUTORS

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

S. I. A-K(145), G. W. Woodruff School of Mechanical Engineer-ing, Georgia Institute of Technology, Atlanta, Georgia 30332.

I. C (1), Department of Mechanical and Aerospace Engineering,

University of California, Los Angeles, Los Angeles, California 90095.SW. C (255), Department of Chemical Engineering, The

University of Pennsylvania, Philadelphia, Pennsylvania 19104.S. M. G (145), G. W. Woodruff School of Mechanical Engineer-

ing, Georgia Institute of Technology Atlanta, Georgia 30332.M. R. H (363), University of Brighton, Brighton, United Kingdom.R. K. S (363), Delphi Harrison Thermal Systems, Lockport, New York

14094.B. T (363), CEA-Grenoble, DTP/GRETh, Grenoble, France.

P. T (363), CEA-Grenoble, DTP/GRETh, Grenoble, France.V. S. T (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 HeatTransfer, has filled the information gap between regularly published journalsand university-level textbooks. The series presents review articles on specialtopics of current interest. Each contribution starts from widely understoodprinciples and brings the reader up to the forefront of the topic being

addressed. The favorable response by the international scientific and engin-eering community to the thirty-four volumes published to date is anindication of the success of our authors in fulfilling this purpose.

In recent years, the editors have undertaken to publish topical volumesdedicated to specific fields of endeavor. Several examples of such topicalvolumes are Volume 22 (Bioengineering Heat Transfer), Volume 28(Trans-port Phenomena in Materials Processing), and Volume 29 (Heat Transferin Nuclear Reactor Safety). As a result of the enthusiastic response of thereaders, 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 associatedwith Advances in Heat Transfer.Lastly, the editors acknowledge the effortsof the professional staff at Academic Press who have been responsible forthe attractive presentation of the published volumes over the years.

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ADVANCES INHEAT TRANSFER

Volume 34


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Transport Phenomena inHeterogeneous Media Based on

Volume Averaging Theory

V. S. TRAVKIN and I. CATTON

Department of Mechanical and Aerospace EngineeringUniversity of California, Los Angeles

Los Angeles, California 90095

I. Introduction

Determination of flow variables and scalar transport for problemsinvolving heterogeneous (and porous)media is difficult, even when theproblem is subject to simplifications allowing the specification of mediumperiodicity or regularity. Linear or linearized models fail to intrinsicallyaccount for transport phenomena, requiring dynamic coefficient models tocorrect for shortcomings in the governing models. Allowing inhomogeneitiesto adopt random or stochastic character further confounds the already

daunting task of properly identifying pertinent transport mechanisms andpredicting transport phenomena.This problem is presently treated by procedures that are mostly heuristic

in nature because sufficiently detailed descriptions are not included in thedescription of the problem and consequently are not available. The abilityto describe the details, and features, of a proposed material with precisionwill help reduce the need for a heuristic approach.

Some aspects of the development of the needed theory are now wellunderstood and have seen substantial progress in the thermal physics and

in fluid mechanics sciences, particularly in porous media transport phenom-ena. The basis for this progress is the so-called volume averaging theory(VAT), which was first proposed in the 1960s by Anderson and Jackson [1],Slattery [2], Marle [3], Whitaker [4], and Zolotarev and Radushkevich [5].

ADVANCES IN HEAT TRANSFER, VOLUME 34

1ADVANCES IN HEAT TRANSFER, VOL. 34

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Further advances in the use of VAT are found in the work of Slattery [6],Kaviany [7], Grayet al.[8], and Whitaker [9, 10]). Many of the important

details and examples of application are found in books by Kheifets andNeimark [11], Dullien [12], and Adler [13].

Publications on turbulent transport in porous media based on VATbegan to appear in 1986. Primak et al.[14], Shcherbanet al.[15], and laterstudies 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 ageneralized development of VAT for heterogeneous media applicable tononlinear physical phenomena in thermal physics and fluid mechanics.

In most physically realistic cases, highly complex integraldifferential

equations result. When additional terms in the two- and three-phasestatements are encountered, the level of difficulty in attempting to obtainclosure and, hence, effective coefficients, increases greatly. The largestchallenge is surmounting problems associated with the consistent lack ofunderstanding of new, advanced equations and insufficient development ofclosure theory, especially for integraldifferential equations. The ability toaccurately evaluate various kinds of medium morphology irregularitiesresults from the modeling methodology once a porous medium morphologyis assigned. Further, when attempting to describe transport processes in a

heterogeneous media, the correct form of the governing equations remainsan area of continuously varying methods among researchers (see somediscussion in Travkin and Catton [16, 21]).

An important feature of VAT is being able to consider specific mediumtypes and morphologies, lower-scale fluctuations of variables, cross-effects ofdifferent variable fluctuations, interface variable fluctuations effects, etc. It isnot possible to include all of these characteristics in current models usingconventional theoretical approaches. The VAT approach has the followingdesirable features:

1. Effects of interfaces and grain boundaries can be included in themodeling.

2. The effect of morphology of the different phases is incorporated. Themorphology decription is directly incorporated into the field equa-tions.

3. Separate and combined fields and their interactions are describedexactly. No assumptions about effective coefficients are required.

4. Effective coefficients correct mathematical descriptionthose the-

ories presently used for that purpose are only approximate descrip-tion, and often simply wrong.

5. Correct description of experiments in heterogeneous mediaagain, atpresent the homogeneous presentation of medium properties is used

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for this purpose, and explanation of experiments is done via bulkfeatures. Those bulk features describe the field as by classical homo-

geneous medium differential equations.6. Deliberate design and optimization of materials using hierarchical

physical descriptions based on the VAT governing equations can beused to connect properties and morphological characteristics to com-ponent features. What is usually done is to carry out an experimentalsearch by adding a third or a fourth component to the piezoelectricmaterial, for example. This can be done in a more direct, moreobservable way, and with a more correct understanding of the effectsof 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 andshow that the best theoretical tool is the nonlocal description of hierarchical,multiscaled processes resulting from application of VAT. Application ofVAT to radiative transport in a porous medium is based on our advancesin electrodynamics and microscale energy transport phenomena in two-phase heterogeneous media. Some other governing conservation equationsfor 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 thatcan be used in different engineering fields. Applications range from heat andmass exchangers and reactors in mechanical engineering design to environ-mental engineering usage (Travkin et al.[19]). A recent application is inurban air pollution, where optimal control of a pollutant level in acontaminated area is determined, along with the design of an optimalcontrol point network for the control of constituent dispersion and remedi-

ation actions. Using second-order turbulent models, equation sets wereobtained for turbulent filtration and two-temperature or two-concentrationdiffusion in nonisotropic porous media and interphase exchange and micro-roughness. Previous work has shown that the flow resistance and heattransfer over highly rough surfaces or in a rough channel or pipe can beproperly predicted using the technique of averaging the transport equationsover the near surface representative elementary volume (REV). Prescribingthe statistical structure of the capillary or globular porous medium mor-phology gives the basis for transforming the integraldifferential transport

equations into differential equations with probability density functionsgoverning their coefficients and source terms. Several different closuremodels for these terms for some uniform, nonuniform, nonisotropic, andspecifically random nonisotropic highly porous layers were developed. Quite

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different situations arise when describing processes occuring in irregular orrandom morphology. The latest results, obtained with the help of exact

closure modeling for canonical morphologies, open a new field of possibili-ties for a purposeful search for optimal design of spacial heterogeneoustransport structures. A way to find and govern momentum transportthrough a capillary nonintersecting medium by altering its morphometricalcharacteristics 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 randomlyinhomogeneous, the random porosity function can be decomposed intoadditive components: the average value of m(x) in the REV and itsfluctuations in various directions,

m(x) m

(x) m

(x), m

.

The averaged equations of turbulent filtration for a highly porousmedium are similar to those in an anisotropic porous medium. Five types

of averaging over an REV function fare defined by the following averagingoperators arranged in their order of seniority (Primak et al.[14]): averageof fover the whole REV,

f f

f

mf

(1 m

)f

, (1)

phase averages of f in each component of the medium,

f

m

1

f(t, x)d mf

(2)

f

m

1

f(t, x)d mf

(3)

and intraphase averages,

f

f

1

f(t, x)d (4)

f

f

1

f(t, x)d. (5)

When the interface is fixed in space, the averaged functions for the firstand second phase (as liquid and solid)within the REV and over the entireREV fulfill the conditions

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f g

f

g

and a a const (6)

for steady-state phases and

ft

f

t

, fg

fg (7)

except for the differentiation condition,

f

f 1

fds

f f f, f , (8)

where S is the inner surface in the REV, and ds is the solid-phase,inward-directed differential area in the REV (ds ndS). The fourth condi-tion implies an unchanging porous medium morphology.

The three types of averaging fulfill all four of the preceding conditions aswell as the following four consequences:

f

f, f

f f

0 (9)

fg

fg, fg

fg 0 (10)

Meanwhile,fandffulfill neither the third of the conditions,a

a, a

m

a, (11)

nor all the consequences of the other averaging conditions. Futher, thedifferential condition becomes

f

f

1

f ds

, (12)

in accordance with one of the major averaging theoremsthe theorem of

averaging theoperator (Slattery [6]; Gray et al.[8]; Whitaker [10]).If the statistical characteristics of the REV morphology and the averagingconditions with their consequences lead to the following special ergodichypothesis: the spacial averages, (f

, f, and f

), then this theorem

converges with increases in the averaging volume to the appropriateprobability (statistical)average of the function fof a random value withprobability density distribution p. This hypothesis is stated mathematicallyas follows:

f(x)

f(x,)p, x d

lim

f f. (13)

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Quintard and Whitaker [29] expressed some concern about the connec-tion between different scale volume averaged variables, for example,

T

1

T

d (14)

T

1

T

d. (15)

In a truly periodic system it is known that the steady temperature in phasefcan be written as

T

(r

) h r

T

(r

) T

, (16)

wherehandT

are constants andT(r

)is a periodic function of zero mean

over the f-phase. Applying the phase averaging operator to this

function, one finds

T(x)

h

r

(x)

T

(r

)

mT

,

while Quintard and Whitaker [29] obtain (their Eq. (13))

T(r

)

T

(r

)

h r

mT

, (17)

meaning thatT

(r

)

0. (18)

The parameter T(r

)

cannot always be equal to zero, because it

depends on the peculiarities of the chosen REV. In some instances, when theREV is not the volume that contains the known number of exact functionperiods, the averaged function T

(r

)

value should not be zero. If it is

assumed, however, that the REV volumecontains the exact number ofspacial periods, then

T(r

)

0.

Averaging the fluid temperature,T, over

yields the intrinsic average

T(x)

h r

T

h x h y

T

, (19)

because the averaging ofr

(see Fig. 1)results in

r

x y

, (20)

while Quintard and Whitaker [29] obtain (their Eq. (15))

T

T(r

)

h r

T

(21)

They note (see p. 375), now represent rin terms of the position vector,

x, that locates the centroid of the averaging volume, and the relative

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F. 1. Representative elementary averaging volume with the virtual points of representa-tion inside of the REV (Carbonell and Whitaker [31]; Quintard and Whitaker [29]).

position vector yas indicated in Fig. 3 (see Fig. 1), or

r

x y r

(x, y

)

x y

(r

, x)

, (22)

so that Eq. (21)can be written with dependence on both xand r ,T

(x, r

)

T

(x, y

)

h x h y

(r

, x)

T

, (23)

meaning that after averaging, T(r

)continues to be dependent on the

position of the virtual point r, which may have changed location within

the.

To do this, they introduce a so-called virtual REV allowing theaveraged value inside of the REV to be variable (see the remark on p. 354of Quintard and Whitaker [30]: In all our previous studies of multiphase

transport phenomena, we have always assumed that averaged quantitiescould be treated as constants within the averaging volumeand that theaverage of the spatial deviations was zero.We now wish to avoid theseassumptions. . . .), 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 Whit-taker derive the gradient of the average of the function (23), they use its

dependence on xfor the two right-hand-side terms in (23)to obtainT

(x, r

)

h h

xy

(r

, x)

. (24)

Several comments about the Quintard and Whittaker treatment that needto be considered are the following:

1. How the communication of the variables from different spacesrat the

lower scale space and x at the upper scale space is established is notmeaningful. 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 thelower level REV, not only to a point within the lower level REV. When oneconsiders an averaged variable at any point other than the representativepoint xfor a particular REV, then

T(x)

(h x T

(x)

T

),

and for the upper scale, the exact result is

T(x)

(h x T

) h. (25)

3. If a function and its gradient are periodic, then the averaged functionshould be periodic. The VAT-based answer should be seen by determinationof the averaged values, which are not averaged, only the REV being used atthe lower scale.

The work by Quintard and Whittaker and the improving of understand-ing of some basic principles of averaging has led us to state the followinglemma and then point out differences from the work of Whittaker and hiscolleagues.

Lemma. If a function , representing any continuous physical field, isaveraged over the subdomain

, which is the subregion occupied by phase

f(fluid phase) of the REV

in the heterogeneous two-phase medium(Fig.1), and the averaged function (x)

is assumed to have different values at

different locationsxwithin the

, then the averaged function(x)

can

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,p. 375, in the paper by Quintard and Whitaker [29])is located an infinitely

distance from the boundary of the REVwithin. It represents theintrinsic phase averaged value

of variable averaged in the REV

. According to Carbonell and Whitaker [31] and Quintard and

Whitaker [29], its value can be different from

or

.

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

Next, consider a point y

located an infinitely small distance outside ofthe initial REV

within a boundary

. The point y

represents the

averaged value

which belongs to some REV

, as shown in Fig.

2, with its center at the point x

(y

) (R

/2), with being aninfinitely small constant.

Following arguments of Carbonell and Whitaker [31], this point yisallowed to be in at least one more REV,

, which has its center x

just

shifted from the point xan infinitely small distance, as does y

from the

boundary.

Further, suppose that one is approaching the boundaryfrom both

sides by pointsy

andy

. According to Carbonell and Whitaker [31] andQuintard and Whitaker [29], the values

and

can be different

whenis reached, which means that the averaged value

experien-

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

have discontinuities at each point of a REV.

9


25/488

The relationships between different scale variables and their points ofrepresentation can be found by noting the following points:

1. There is a fixed relationship between the location of the pointxof 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

the same region (excluding close to boundary regions).2. If there is the value

(x

)

, (x

x

) , then there is another

, and in it

(x) 1

(r)d

1

(r )d, (26)

where

(r

) const.

A. TV CVAT TIC

When the coefficient of thermal conductivity kis a constant value, the

fluid stedy-state conduction regime is described by

k

(mT)

k

T

ds

k

T

ds 0. (27)

The full 1D Cartesian coordinates version of this equation, without anysource, for a fixed solid matrix in is

xm

T

x x

1

T

ds

1

T

x

ds 0, (28)

x

mT

x MD MD 0, (29)

where the second and third terms on the right-hand side are the so-calledmorphodiffusive terms, MD

and MD

, respectively (see also, for example,

Travkin and Catton [21]),The solid-phase equation with constant k

equation is of the same form,

xs

T

x

x

1

T

ds

1

T

x

ds

0, (30)

which can also be written in terms of the fluctuating variable,

sT

x

x

1

T

ds

1

T

x

ds

0. (31)

10 . . .


26/488

Travkin and Catton [16, 18, 20] suggested that the integral heat transferterms 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 changingaveraged surface temperature along the xcoordinate. Equations (28)and(30)can be treated using heat transfer correlations for the heat exchangeintegral term(the last term). Regular dilute arrangements of pores, sphericalparticles, or cylinders have been studied much more than random mor-phologies. Using separate element or cell modeling methods(Sangani andAcrivos [32] and Gratton et al.[26])to find the interface temperature fieldallows 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 transportphenomena in porous media. The primary difference between such equa-tions and those resulting from a more rigorous development based on VATare certain additional terms. The best way to evaluate the need for theseadditional more complex terms is to obtain an exact mathematical solutionand compare the results with calculations using the VAT equations. Thiswill clearly display the need for using the more complex VAT mathematicalstatements.

Consider a two-phase heterogeneous medium consisting of an isotropic

continuous (solid or fluid)matrix and an isotropic discontinuous phase(spherical particles or pores). The volume fraction of the matrix, or f-phase,is m m

/, and the volume fraction of filler, or s-phase, is

m

1 m

/, where

is the volume of the REV.

The constant properties (phase conductivities, kand k

), stationary (time-

independent)heat conduction differential equations for TandT

, the local

phase temperatures, are

q

k

T

0, q

kT

0,

with the fourth (IVth)kind interfacial (f s) thermal boundary conditionsT

T

, ds

q

ds q

.

Here q

k

Tand q

k

T

are the local heat flux vectors,S

is the interfacial surface, and ds

is the unit vector outward to the s-phase.

No internal heat sources are present inside the composite sample, so thetemperature field is determined by the boundary conditions at the externalsurface of the sample. After correct formulation of these conditions, theproblem is completely stated and has a unique solution.

Two ways to realize a solution to this problem were compared (Travkinand Kushch [33, 34]). The first is the conventional way of replacing theactual composite medium by an equivalent homogeneous medium with aneffective thermal conductivity coefficient,k k

(s, k

,k

), assuming one

11


27/488

knows how to obtain or calculate it. The exact effective thermal coefficientwas obtained using direct numerical modeling (DNM)based on the math-

ematical theory of globular morphology multiphase fields developed byKushch(see, for example, [3538]).

Averaging the heat flux,q, and temperature,T, over the REV yieldsq k

T, and for the stationary case there results

(k

T) 0. (32)

The boundary conditions for this equation are formulated in the samemanner as for a homogeneous medium.

The second way is to solve the problem using the VAT two-equation,

three-term integrodifferential equations (28) and (30). To evaluate andcompare solutions to these equations with the DNM results, one needs toknow the local solution characteristics, the averaged characteristics over theboth phases in each cell, and, in this case, the additional morphodiffusiveterms.

An infinite homogeneous isotropic medium containing a three-dimen-sional (3D)array of spherical particles is chosen for analysis. The particlesare arranged so that their centers lie at the nodes of a simple cubic latticewith perioda. The temperature field in this heterogeneous medium is caused

by a constant heat flux Qprescribed at the sample boundaries, which,because of the absence of heat sources, leads to the equality of averagedinternal heat fluxq Q

.

When all the particles have the same radii, the result is the triple periodicstructure used widely, beginning from Rayleighs work [39], to evaluate theeffective conductivity of particle-reinforced composites.

The composite medium model consists of the three regions shown in Fig.3. The half-space lying above the AA plane has a volume content of thedisperse phase m

m

, and for the half-space below the BB plane

m m . To define the problem, let m m . The third part is thecomposite layer between the plane boundaries AA and BB containingNdouble periodic lattices of spheres (screens)with changing diameters.

Solutions to the VAT equations (28) and (30) for a composite withvarying volume content of disperse phase with accurate DNM closure of themicro model VAT integrodifferential terms were obtained implicitly, mean-ing that each term was calculated independently using the results of DNMcalculations.

For the one-dimensional case, Eq. (32)becomes

zk

Tz 0, (z z z), m m(z),

where k

(m) is the effective conductivity coefficient.

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28/488

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

The normalized solution of the both models(VAT and DNM)for the caseof linearly changing porositym

m

z(m

m

), wherem

(z

) m

,

m(z

) m

, z

0, z

1, between AA and BB and with effective

conductivity coefficients ofk

0, 0.2, 1, 10, and 10,000, are presented inFig. 4. There is practically no difference (less than 10) between thesolutions, and what there is is probably because of numerical error accumu-lation (Travkin and Kushch [34]).

Lines 15 represent solutions of the one-term equation, respectively,whereas the points(circles, triangles, etc.)represent the solutions of the VAT

equations with accurate DNM closure of the micro model VAT integrodif-ferential terms MD

and MD

for the composite with varying volume

content of disperse phase. Here the number of screens is nine, correspondingto a relatively small particle phase concentration gradient.

The coincidence of the results of the exact calculation of the two-equation,three-term conductive-diffusion transport VAT model(28)and(30)with theexact DNM solution and with the one-temperature effective coefficientmodel for heterogeneous media with nonconstant spatial morphologyclearly demonstrates the need for using all the terms in the VAT equations.

The need for the morphodiffusive terms in the energy equation is furtherdemonstrated by noting that their magnitudes are all of the same order.

Confirmation of the fact that there is no difference in solutions betweenthe correct one-term, one-temperature effective diffusivity equation and the

13


29/488

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

three-terms, two-temperature VAT equations does not mean that it is betterto take for modeling and analysis the effective diffusivity one-term, one-temperature equation (see Subsection VI, E and arguments in Sections VIIand VIII). Among other issues one needs to analyze goals of modeling andto understand that the good solution of the effective diffusivity one-term

one-temperature equation as it was found and described in the precedingstatements means nothing less than the ground of the exact solution of theVAT problem. Also, it is important that for the exact (or accurate)solutionof conventional diffusivity equation, the effective coefficient needs to befound, and this means in turn that finding the solution of the two-fieldproblem is imperative and consequently appears to be the major problem.Meanwhile, this is the problem that was posed just at the beginning as theoriginal one.

III. Nonlinear and Turbulent Transport in Porous Media

To a great extent, the analysis of porous media linear transport phenom-ena are given in the numerous studies by Whitaker and coauthors; see, for

14 . . .


30/488

example, [10, 30, 31, 4046], as well as by studies by Gray and coauthors[8, 4750]. Our present work is mostly devoted to the description of other

physical fields, along with development of their physical and mathematicalmodels. Still, the connection to linear and partially linear problem state-ments needs to be outlined.

The linear Stokes equations are

V 0,

0 p V

g, (33)

and although the Stokes equation is adequate for many problems, linear aswell as nonlinear processes will result in different equations and modelingfeatures.

The general averaged form of the transport equations will be developedfor permeable interface boundaries between the phases. Two forms of theright-hand-side Laplacian term will be considered. First, one can have twoforms of the diffusive flux in gradient form that can be written

V

V

V ds (34)

or

V

mV

V ds. (35)

It was pointed out first by Whitaker [42, 43] that these forms allow greaterversatility in addressing particular problems. Using the two averaged forms

of the velocity gradient,(34)and(35), one can obtain two averaged versionsof the diffusion term in Eq. (33), namely,

(V)

(mV) 1

V ds

V ds,

(36)

where the production term V dsis a tensorial variable, and the versionwith fluctuations in the second integral term

(V)

(mV) 1

V ds

V ds,

(37)

15


31/488

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

V

1

U

ds 0, U

V (38)

and

0 p

1

pds (mV)

1

V ds

V ds

m

g, (39)

and the second version is found by using the following relation for thepressure gradient:

p

1

pds mp

1

pds. (40)

Using the averaging rules developed by Primak et al.[14], Shcherban etal.[15] and Travkin and Catton [16, 18] facilitated the development of themomentum equation. By combining equations (37)and (40), one is able towrite the momentum transport equations in the second form with velocityfluctuations

V

Vm

1

U

ds 0, (41)

obtained using

V

V

V1

V ds V

m

1

U

ds, (42)

and the momentum equation

0 mp 1

pds (mV)

1

V d

s

V d

s mg. (43)

The third version of these equations is almost never used but can be foundin [21].

16 . . .


32/488

A. LF CC

The transport equations for a fluid phase with linear diffusive terms are

U

x

0 (44)

U

t U

U

x

1

px

x

U

x S (45)

t U

x

D

x

x

S . (46)Hererepresents any scalar field(for example, concentrationC )that mightbe transported into either of the porous medium phases, and the last termson the right-hand side of(45)and(46)are source terms. In the solid phase,the diffusion equation is

t D

x

x S . (47)

The averaged convective operator term in divergence form becomes, afterphase averaging,

x

(UU

)

(UU

)

U

U

1

U

U

ds

[mUU

mu

u

]

1

U

U

ds. (48)

Decomposition of the first term on the right-hand side of (48) yields

fluctuation types of terms that need to be treated in some way.The nondivergent version of the averaged convective term in the momen-

tum equation is

x

(UU

)

mUU

U

U

u

u

1

U

U

ds

mU

x

U

U

1

U

ds

uu

1

U

U

ds. (49)

The divergent and nondivergent forms of the averaged convective term in

17


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the diffusion equation are

(CU) CU 1

CU d

s

[mCU mcu

]

1

CU

ds

mU

x

CC 1

U

ds cu

1

CU

ds.

(50)

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

(DC )

D (mC) D 1

Cds

D

C ds,

(51)or

(DC )

D (mC) D

1

cds

D

C ds,

(52)or

D (DC )

DmC D 1

cds

D

c ds.

(53)

Other forms of Eq. (52), using the averaging operator for constantdiffusion 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 totreat cases where Stokes flow is invalid. Two versions of the momentumequation will result. The equation without the fluctuation terms is

m

Vt

mV V V 1

V ds vv

1

V V ds

(mp) 1

pd

s (mV)

1

V ds

V ds m

g. (54)

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34/488

with the fluctuation diffusion terms it becomes

mVt

mV V V 1V ds vv 1

V V ds

mp

1

pds (mV)

1

vds

V ds m

g. (55)

The steady-state momentum transport equations for systems with imper-

meable interfaces can readily be derived from Eq. (54)and (55). They are

(mV V vv) (mp)

1

pds (mV)

V ds m

g, (56)

or

(mV V vv )

mp) 1pds (mV)

V ds m

g. (57)

B. NFME LF

To properly account for Newtonian fluid flow phenomena within a

porous medium in a general way, modeling should begin with the NavierStokes equations for variable fluid properties,

Vt

V V p [(V (V)*)] g (58) (V, C

, T),

rather than the constant viscosity NavierStokes equations. The followingform of the momentum equation will be used in further developments:

Vt

V V p (2S ) g (59) (V, C

, T).

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The negative stress tensorin this equation is

N

2(V)

2S, (60)and the symmetric tensor Sis the deformation tensor

S (V)1

2(V (V)*), (61)

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

t U

x

x(

(x,

, V)

x S

(62)

and

t

x

x S . (63)

Here and are scalar fields and nonlinear diffusion coefficients for

these fields. The averaging procedures for transport equation convectiveterms were established earlier. The averaged nonlinear diffusion term yields

(DC )

(mDC) D 1

cds

(Dc)

1

DC ds. (64)

The other version of the diffusive terms with the full value of concentrationon the interface surface is

(DC ) (D(mC))

D

1

Cd

s

(Dc

)

1

DC ds. (65)

General forms of the nonlinear transport equations can be derived forimpermeable and permeable interface surfaces. The averaged momentumdiffusion term is

x

(2S )

(2S ) (2S )

1

2S d

s

2(mS m S)

2

S ds. (66)

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36/488

The general nonlinear averaged momentum equation for a porous mediumis

m

Vt

mV V V 1

V ds vv

1

V V ds

(mp) 1

pds 2(mS mS

)

2

S ds m

g. (67)

The steady-state momentum transport equations for systems with imper-meable interfaces follows from Eq. (67),

(mV V vv)

(mp) 1

pds 2(mS mS

)

S ds m

g. (68)

The averaged nonlinear mass transport equation in porous mediumfollows

mC

t

mUC

C

U

ds c

u

1

C

U

ds

(D(mC)) D 1

Cds

(Dc) 1

DC d

s mS . (69)

A few simpler transport equations that can be readily used while main-taining fundamental relationships in heterogeneous medium transport aregiven by Travkin and Catton [21].

C. PMTVAT E

Turbulent transport processes in highly structured or porous media are

of great importance because of the large variety of heat- and mass-exchangeequipment used in modern technology. These include heterogeneous mediafor heat exchangers and grain layers, packed columns, and reactors. In allcases there occurs a jet or stalled flow of fluids in channels or around the

21


37/488

obstacles. There are, however, few theoretical developments for flow andheat exchange in channels of complex configuration or when flowing around

nonhomogeneous bodies with randomly varied parameters. The advancedforms of laminar transport equations in porous media were developed in apaper by Crapiste et al.[41]. For turbulent transport in heterogeneousmedia, there are few modeling approaches and their theoretical basis andfinal modeling equations differ.

The lack of a sound theoretical basis affects the development of math-ematical models for turbulent transport in the complex geometrical environ-ments found in nuclear reactors subchannels where rod-bundle geometriesare considered to be formed by subchannels. Processes in each subchannel

are calculated separately (see Teyssedou et al.[51]). The equations used inthis work has often been obtained from two-phase transport modelingequations [52] with heterogeneity of spacial phase distributions neglected inthe bulk. Three-dimensional two-fluid flow equations were obtained by Ishii[52] using a statistical averaging method. In his development, he essentiallyneglected nonlinear phenomena and took the flux forms of the diffusiveterms to avoid averaging of the second power differential operators. Ishiiand Mishima [53] averaged a two-fluid momentum equation of the form

vt (

v

v)

p

( )

g v

M

, (70)

whereis the local void fraction,

is the mean interfacial shear stress,

is the turbulent stress for the kth phase,is the averaged viscous stress forthe kth phase,

is the mass generation, and M

is the generalized

interfacial drag. Using the area average in the second time averagingprocedure, 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 ofvarious kinds of spatial phase distributions. The equations used by Laheyand Lopez de Bertodano [54] and Lopez de Bertodano et al.[55] are verysimilar, with the momentum equation being

D

u

Dt

p

[

u

(u

u

)]

g M

M

( p

p

)

. (71)

Here the index idenotes interfacial phenomena and M

is the volumetricwall force on phase k. Additional terms in Eq. (70)and (71)are usuallybased on separate micro modeling efforts and experimental data.

22 . . .


38/488

One of the more detailed derivations of the two-phase flow governingequations by Lahey and Drew [56] is based on a volume averaging

methodology. Among the problems was that the authors developed theirown volume averaging technique without consideration of theoretical ad-vancements developed by Whitaker and colleagues [10, 42] and Grayet al.[8] for laminar and half-linear transport equations. The most importantweaknesses are the lack of nonlinear terms (apart from the convectionterms)that naturally arise and the nonexistence of interphase fluctuations.

Zhang and Prosperetti [57] derived averaged equations for the motion ofequal-sized rigid spheres suspended in a potential flow using an equation forthe probability distribution. They used the small particle dilute limit

approximation to close the momentum equations. After approximateresolution of the continuous phase fluctuation tensor M

, the vector

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 thefluctuating particle volume flux tensor M

. . . . This missing information

cannot be supplied internally by the theory without a specification of theinitial conditions imposed on the particle probability distribution. Theyalso considered the case of finite volume fractions for the linear problemwhere the problem equations were formulated for inviscid and unconvec-

tional media. The development by Zhang and Prosperetti [57] is a goodexample of the correct application of ensemble averaging. The equationsthey derive compare exactly with those derived from rigorous volumeaveraging theory (VAT)[24].

Transport phenomena in tube bundles of nuclear reactors and heatexchangers can be modeled by treating them as porous media [58]. Thetwo-dimensional momentum equations for a constant porosity distributionusually have the form [59]

Ux

Vy

0 (72)

Ux

UV

y

1

Px

U AVU (73)

UVx

Vy

1

Py

V AVV, n 0, (74)

where the physical quantities are written as averaged values and the solidphase effects are included in two coefficients of bulk resistance, A

and A

,

and an effective eddy viscosity,

, that is not equal to the turbulent eddyviscosity. These kinds of equations were not designed to deal with non-

23


39/488


40/488

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

f

1

f d, (78)

with

being the area within the volumeoccupied by air. Raupach

et al.[64] and Coppin et al.[65] assumed that the dispersive covarianceswere unimportant,

u

u

, (79)

where u

is a fluctuation value within the canopy and u

u

. Thecontribution of these covariances was found by Raupach et al.[64] to besmall in the region just above the canopy from experiments with a regularrough 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 irregularor random two-phase media. When the surface averaging used by Raupachand Shaw [64] is used instead of volume averaging, especially in the case ofnonisotropic media, the neglect of one of the dimensions in the averagingprocess results in an incorrect value. This result should be called a 2Daveraging procedure, particularly when 3D averaging procedures are re-placed by 2D for nonisotropic urban rough layer (URL)when developingaveraged transport equations.

Raupach et al.[6466] later introduced a true volume averaging pro-cedure within an air volume

that yielded the averaged equation for

momentum conservation

U

t U

x

(U)

1

x

P x

uu

U

x

U

ds

x

u

u

1

Pds, i, j 1 3, (80)

where Sis interfacial area. Development of this equation is based on

intrinsic averaged values of or U

, whereas averages of vector field

variables over the entire REV are more correct (Kheifets and Neimark

[11]). Raupach et al.[64] next simplified all the closure requirements bydeveloping a bulk overall drag coefficient. The second, third, and fifth termson the right-hand side of Eq. (80)are represented by a common dragresistance term. For a stationary fully developed boundary layer, they write

25


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zuwu w

U

z 1

2C

S

U, (81)

where C

is an element drag coefficient and S

is an element areadensity 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 fourregimes for regular spherical packing, and that only when the Reynoldsnumber based on pore diameter, Re

, exceeded 350 could the flow regime

be considered to be turbulent flow. The Fand et al.[67] investigation of arandomnly packed porous medium made up of single size spheres showed

that the fully developed turbulent regime occurs whenRe 120, whereReis particle Reynolds number.Volume averaging procedures were used by Masuoka and Takatsu [69]

to derive their volume-averaged turbulent transport equations. As in numer-ous other studies of multiphase transport, the major difficulties of averagingthe terms on the right-hand side were overcome by using assumed closuremodels for the stress components. As a result, the averaged turbulentmomentum equation, for example, has conventional additional resistanceterms such as the averaged momentum equation developed by Vafai and

Tien [70] for laminar regime transport in porous medium. A majorassumption 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 theimportance of the role of randomness in the enhancement of transportprocesses. The results show the very distinct patterns of flow and heattransfer for two cases of regular and nonuniform 2D structured nonorthog-onal porous media. Their experimental results clearly demonstrate theinfluence of nonuniformity of the porous structure on the enhancement of

heat transfer.

D. D TTMHPM

Fluid flow in a porous layer or medium can be characterized by severalmodes. Let us single out from among them the three modes found in ahighly porous media. The first is flow around isolated ostacle elements, orinside an isolated pore. The second is interaction of traces or a hyperturbu-

lent mode. The third is fluid flow between obstacles or inside a blockedinterconnected swarm of channels (filtration mode). The models developedby Scherbanet al.[15], Primaket al.[14], and Travkin and Catton [1621]are primarily for nonlinear laminar filtration and hyperturbulent modes in

26 . . .


42/488

nonlinear transport.Specific features of flows in the channels of filtered media include the

following:

1. Increased drag due to microroughness on the channel boundarysurfaces

2. Gravity effects3. Free convection effects4. The effects of secondary flows of the second kind and curved stream-

lines5. Large-scale vortex effects

6. The anisotropic nature of turbulent transfer and resulting anisotropyof turbulent viscosity

It is well known that in spacial boundary flows, an important role isplayed by the gradients of normal Reynolds stresses and that this is the casefor flows in porous medium channels as well. As a rule, flow symmetry isnot observed in these channels. Therefore, in channel turbulence models, theshear components of the Reynolds stress tensors have a decisive effect onthe flow characteristics. At present, however, turbulence models that are less

than second-order can not be successfully employed for simulating suchflows(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 ofaveraging of the turbulent transfer equation in the liquid phase and thetransfer equations in the solid phase of a heterogeneous medium (Primaketal.[14] and Scherban et al.[15])over a specified REV.

The initial turbulent transport equation set for the first level of thehierarchy, microelement, or pore, was taken to be of the form (see, for

example, Rodi [72] and Patel et al.[75])U

t

U

U

x

1

px

x

U

x

uu

S (82)

t

U

x

x

D

x

u

S (83)U

x

0. (84)

Here and its fluctuation represent any scalar field that might be

transported into either of the porous medium phases, and the last terms onthe right-hand side of(82)and (83)are source terms.

27


43/488

Next we introduce free stream turbulence into the hierarchy Let usrepresent the turbulent values as

U U u U

U

u

u

(85)

U U u ,

where the index kstands for the turbulent components independent ofinhomogeneities of dimensions and properties of the multitude of porousmedium channels (pores), and rstands for contributions due to the porousmedium inhomogeneity. Being independent of the dimensions and proper-

ties of the inhomogeneities of the porous medium configurations, sections,and boundary surfaces does not mean that the distribution of values ofU

and uare altogether independent of the distance to the wall, pressure

distribution, etc. Thus, the values Uor u

stand for the values generally

accepted in the turbulence theory, that is, when a plane surface is referredto, these values are those of a classical turbulent boundary layer. When around-section channel is involved, and even if the cross-section of thischannel is not round, but without disturbing nonhomogeneities in thesection, then the characteristics of this regular sections (and flow)may be

considered to be those that could be marked with index k. Hence, if achannel in a porous medium can be approximately by superposition ofsmooth regular(of regular shape)channels, it is possible to give such a flowits characteristics and designated them with the index k, which stands forthe basic (canonical)values of the turbulent quantities.

Triple decomposition techniques have been used in papers by Breretonand Kodal [76] and Bisset et al.[77], among others. The latter utilizedtriple decomposition, conditional averaging, and double averaging to ana-lyze the structure of large-scale organized motion over the rough plate.

It should be noted that there are problems where Uanducan be foundfrom known theoretical or experimental expressions(correlations)where thedefinitions of U

and u

are equivalent to the solution of an independent

problem (for example, turbulent flow in a curved channel). The same thingcan 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

. (86)

The term u uappears if the flow is through a nonuniform array ofobstacles. If all the obstacles are the same and ordered, then ucan be takenequal to 0. Naturally, the term u

in this particular case does not equal the

fluctuation vector uover a smooth, plain surface.

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The following hypothesis about the additive components is developed tocorrect the foregoing deficiencies:

U U u U

U

u

, u u

U U

U

U

U, u u

(u) 0, u

u

0. (87)

It should be noted that solutions to the equations for the turbulentcharacteristics may be influenced by external parameters of the problem,namely, by the coefficients and boundary conditions, which themselves cancarry information about porous medium morphological features. The adop-

tion of a hypothesis about the additive components of functions represen-ting turbulent filtration facilitates the problem of averaging the equationsfor the Reynolds stresses and covariations of fluctuations (flows)in poresover the REV.

After averaging the basic initial set of turbulent transport equations overthe REV and using the averaging formalism developed in the works byPrimaket al.[14], Shcherbanet al.[15], and Primak and Travkin [78], oneobtains equations for mass conservation,

x

U 1U ds 0, (88)

for turbulent filtration (with molecular viscosity terms neglected forsimplicity),

mU

t

x

(mUU

)

1

x

(mp) x

uu

x

u

u

1

pds 1

U U d

s

1

u

u

ds mS

, i, j 13, (89)

and for scalar diffusion (with molecular diffusivity terms neglected),

m

t

x

(mU

)

x

u

x

u

1

U d

s

1

u

ds mS

, i 13. (90)

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Many details and possible variants of the preceding equations withtensorial terms are found in Primaket al.[14], Scherban et al.[15], and

Travkin and Catton [16, 21]. Using an approximation to K-theory in anelementary channel(pore), the equation for turbulent diffusion ofnth speciestakes the following more complex form after being averaged:

mC

t

V

C

vc

(K(mC

)) K

1

C

ds

(kc

)

1

KC

ds

C

U

ds 1

C

U

ds mS

,

n 1, 2, 3, 4 . . . . (91)

In the more general case, the momentum flux integrals on the right-hand

sides of Eq. (89)through (91) do not equal zero, since there could bepenetration through the phase transition boundary changing the boundaryconditions in the microelement to allow for heat and mass exchange throughthe interface surface as the values of velocity, concentrations, and tempera-ture atS

do not equal zero (see also Crapiste et al.[41]). The first term

on the right-hand side of Eq. (91)is the divergence of the REV averagedproduct of velocity fluctuations and admixture concentration caused byrandom morphological properties of the medium being penetrated and isresponsible for morphoconvectional dispersion of admixture in this particu-

lar porous medium. The third term on the right-hand side of Eq. (91)canbe associated with the notion of morphodiffusive dispersion of a substanceor heat in a randomly nonhomogeneous medium. The term withS

may also

reflect, specifically, the impact of microroughness from the previous level ofthe simulation hierarchy. The importance of accounting for this roughnesshas been demonstrated by many studies. The remaining step is to accountfor the microroughness characteristics of the previous level.

One-dimensional mathematical statements will be used in what followsfor simplicity. Admission of specific types of medium irregularity or random-

ness requires that complicated additional expressions be included in thegeneralized governing equations. Treatment of these additional terms be-comes a crucial step once the governing averaged equations are written. Anattempt to implement some basic departures from a porous medium with

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strictly regular morphology descriptions into a method for evaluation ofsome of the less tractable, additional terms is explained next.

The 1D momentum equation with terms representing a detailed descrip-tion of the medium morphology is depicted as

xm(K )

U

xxK u

x

x

(uu)

m UU

x

1

(K

)

Ux

ds

1

pd

s

1

x(mp)

mUU

x

u*

S(x)

1

pds

1

x

(mp), (92)

where Kis the turbulent eddy viscosity, and u

*is the square friction

velocity at the upper boundary of surface roughness layer haveraged over

interface surface S.

General statements for energy transport in a porous medium require

two-temperature treatments. Travkin et al.[19, 26] showed that the properform for the turbulent heat transfer equation in the fluid phase usingone-equation K-theory closure with primarily 1D convective heat transfer is

c

mUT

x

xm(K k)

T

x xK T

x

c

x

(mT

u)

x

(K

k)

T

ds

1

(K

k)

T

x

ds, (93)

whereas in the neighboring solid phase, the corresponding equation is

x(1 m)K

T

x

xK T

x

x

K

T d

s

1

K

T

x d

s . (94)

The generalized longitudinal 1D mass transport equation in the fluidphase, including description of potential morphofluctuation influence, for a

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medium morphology with only 1D fluctuations is written

xm(K D )C

x

xK C

x

x

(mc

u)

x

(K

D)

c

ds

1

(K

D

)C

x

ds mS

mUC

x

, (95)

whereas the corresponding nonlinear equation for the solid phase is

x(1 m)D

Cx

xD

cx

x

D

Cds

1

D

C

x

ds

. (96)

E. CT A TPM

Closure theories for transport equations in heterogeneous media havebeen the primary measure of advancement and for measuring success inresearch on transport in porous media. It is believed that the only way toachieve substantial gains is to maintain the connection between porousmedium morphology and the rigorous formulation of mathematical equa-tions for transport. There are only two well-known types of porous mediamorphologies for which researchers have had major successes. But even forthese morphologies, namely straight parallel pores and equal-size sphericalinclusions, not enough evidence is available to state that the closure

problems for them are closed.One of the few existing studies of closure for VAT type equations is by

Hsu and Cheng [79, 80]. They used a one-temperature averaged equation[Equation(40a)in Hsu and Cheng [80])without the morphodiffusive term

[(k

k)T(m)] [(k

k

)T(m)].

The reasoning often applied to the morphoconvective term closureproblem in averaged scalar and momentum transport equations is that theterms needing closure may be negligible. The basis for this reasoning is (seeKheifets and Neimark [11])

c C d

, and j DC , socj

DC

dl ,

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where lis the characteristic length associated with averaging volume (see,for example, the work of Lehner [81] and others)andd

the mean diameter

of pores in a REV. It is not obvious that the length scale, d , taken for theapproximation ofcfollows from use oflas a scale for the second derivative.Furthermore, assuming that the variable to be averaged over the REVchanges very slowly over the REV does not mean that it changes very slowlyin the neighborhood of the primary REV.

Various closure attempts for heterogeneous medium transport equationsresulted in various final equations. One needs to know what these equationsare all about. Treatment of the one-dimensional heat conduction equationwith a stochastic function for the thermal diffusivity in a paper by Fox and

Barakat [82] yielded a spatially fourth-order partial differential equation tobe solved. Gelharet al.[83], after having eliminated the second-order termsin the species conservation equation for a stochastic media, were able todevelop an interesting procedure for deriving a mean concentration trans-port equation. The equation form includes an infinite series of derivativeson the right-hand side of the equation. Analysis of this equation allows thederivation of the final form of the mass transport equation,

C*

t U

C*

x (A a

)U

C*

x B

C*

tx BU

C*

x ,

where the most important term is the second term on the right-hand side.In the derivation of this equation, the stochastic character of the existingassigned fields of velocity, concentration, and dispersion coefficients wereassumed.

A simple form of the advective diffusion equation with constant diffusioncoefficients was developed without sorption effects by Tang et al.[84]:

mC

t

m

V

C D

(

m

C ).

They transformed the equation with the help of ensemble averaging into astochastic transport equation,

mC*t

mu*C* D (mC*) m

C*x

x

,

where the tensor of the ensemble dispersion coefficient is a correlationfunction denoted by

12

uu

*

u*u*x u*,

with u* being the ensemble averaged velocity. The additional term,

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reflecting the influence of the stochastic or inhomogeneous nature of thespatial velocity and concentration fluctuations in the ensemble averaged

stochastic equation developed by Tang et al. [84], has the dispersivitycoefficient fully dependent on the velocity fluctuations. As can be seen bythis equation, the effect of concentration fluctuations was eliminated.

Torquato and coauthors (see, for example, Torquato et al.[85], Millerand Torquato [86], Kim and Torquato [87])have been developing meansto characterize the various mathematical dependencies of a compositemedium microstructure in a statistically homogeneous medium. Some of thequantities considered by Torquato are useful in obtaining resolution tocertain closure problems for VAT developed mathematical models of

globular morphologies. In particular, the different near-neighbor distancedistribution density functions deserve special mention (Lu and Torquato[88], Torquato et al.[85]).

Carbonell and Whitaker [89] combined the methods of volume averagingand the morphology approach to specify the dispersion tensor for theproblem of convective diffusion for cases where there is no reaction oradsorption on the solid phase surface,

DC

n 0, xS

,

and considered a constant diffusion coefficient and constant porositym,which greatly simplifies the closure problems. They expressed the spatialdeviation function as

cf( r) C,

where f is a vector function of position in the fluid phase. Averagedequations of convective diffusion are the same as the convective heat transferequation given by Levec and Carbonell [46] with the exclusion of fluxsurface integral term. The closure technique used in their paper is analogousto a turbulence theory scheme, helping them to derive the closure equationfor the spatial deviation function in the form of a partial differentialequation,

V (V V )f Df, n f n, xS,

One should note that the spatial deviation functions defined for a periodicmedium are periodic themselves.

Nozad et al.[40] suggested that the same closure scheme be used torepresent the fluctuation terms Tand T

for a one-temperature model by

using

T

fT , T gT

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for a transient heat conduction problem with constant coefficients in atwo-phase system (stationary). Partial differential equations for f, g,, and

are found. They obtained excellent predictions of the effective thermalconductivity for conductivity ratios k k

/k

100.

Carbonell [90] attempted to obtain an averaged convective-diffusionequation for a straight tube morphological model and obtained an equationwith three different concentration variables. This demonstrates that theaveraging procedures, taken too literally, can result in incorrect expressionsor conclusions.

A common form of the averaged governing equations for closure ofmultiphase laminar transport in porous media was obtained by Crapiste et

al.[41]. They developed a closure approach that led to a complex integro-differential equation for the spatial deviations of a substance in the void orfluid phase volume of the macro REV. This means that solving theboundary value problem for spatial concentration fluctuations, for example,requires that one obtain a solution to second-order partial differential orcoupled integro differential equations in a real complex geometric volumewithin the porous medium.

For a heterogeneous porous medium, this means that the coupledintegrodifferential equation sets for the averaged spatial deviation variables

must be solved for at least two scales. For averaged variables the scales arethe external scale or L domain, and for the spatial deviations it is the volumeof the fluid phase considered at the local (pore)scale. This presents a greatchallenge and has not yet been resolved by a real mathematical statement.

To close the reaction-diffusion problem Crapiste et al.[41] made a seriesof assumptions: (1)the diffusion coefficient Dand the first-order reactionrate coefficient k

are constant; (2)diffusion is linear in the solid part of the

porous medium,(3)the spatial concentration fluctuation is linearly depend-ent on the gradient of the intrinsic averaged concentration and the averaged

concentration itself, (4) the intrinsic averaged concentration and solidsurface averaged concentration are equal, (5)the restriction

kd

D 1

should be satisfied; and(6)spatial fluctuations of the intrinsic concentrationand the surface concentration fluctuations are equal. The fourth and sixthassumptions are equivalent to an equality of surface and intrinsic concen-

trations, which means that the adsorption mechanisms are taken to bevolumetric phenomena.

In our previous efforts we have obtained some results for both morphol-ogies and demonstrated the strength of morphological closure procedures.

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A model of turbulent flow and two-temperature heat transfer in a highlyporous medium was evaluated numerically for a layer of regular packed

particles (Travkin and Catton [16, 20]; Gratton et al.[26, 27])with heatexchange from the side surfaces. Nonlinear two-temperature heat andmomentum turbulent transport equations were developed on the basis ofVAT, requiring the evaluation of transport coefficient models. This ap-proach required that the coefficients in the equations, as well as the form ofthe equations themselves, be consistent to accurately model the processesand morphology of the porous medium. The integral terms in the equationswere dropped or transformed in a rigorous fashion consistent with physicalarguments regarding the porous medium structure, flow and heat transfer

regimes (Travkin and Catton [20]; Travkin et al.[17]). The form of theDarcy term as well as the quadratic term was shown to depend directly onthe assumed version of the convective and diffusion terms. More impor-tantly, both diffusion (Brinkman)and drag resistance terms in the finalforms of the flow equations were proven to be directly connected. Theserelations follow naturally from the closure process. The resulting necessityfor transport coefficient models for forced, single phase fluid convection ledto their development for nonuniformly and randomly structured highlyporous media.

A regular morphology structure was used to determine the characteristicmorphology functions, (porosity m, and specific surface S

)that were

used in the equations in the form of analytically calculable functions. A firstapproximation for the coefficients, for example, drag resistance or heattransfer, was obtained from experimentally determined coefficient correla-tions. Existing models for variable morphology functions such as porosityand specific surface were used by Travkin and Catton [20] and Gratton etal.[27] to obtain comparisons with other work in a relatively high Reynoldsnumber range.

All the coefficient models they used were strictly connected to assumed(or admitted)porous medium morphology models, meaning that the coeffi-cient values are determined in a manner consistent with the selectedgeometry. Comparison of modeling results was sometimes difficult becauseother models utilized mathematical treatments or models that do not allowa complete description of the medium morphology; see Travkin and Catton[16]. Closures were developed for capillary and globular medium morphol-ogy models(Travkin and Catton [16, 17, 20]; Gratton et al.[26, 27]). It wasshown that the approach taken to close the integral resistance terms in the

momentum equation for a regular structure allows the second-order termsfor the laminar and turbulent regimes to naturally occur. These terms weretaken to be analogous to the Darcy or Forchheimer terms for different flowvelocities. Numerical evaluations of the models show distinct differences in

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the overall drag coefficent among the straight capillary and

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