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ADVANCES IN

HEAT TRANSFER

Volume 34

a

This Page Intentionally Left Blank

Advances in

HEATTRANSFER

Serial Editors

James P. Hartnett Thomas F. Irvine, Jr.Energy Resources Center Department of Mechanical Engineering

University of Illinois at Chicago State University of New York at Stony Brook

Chicago, Illinois Stony Brook, New York

Serial Associate Editors

Young I. Cho George A. GreeneDepartment of Mechanical Engineering Department of Advanced Technology

Drexel University Brookhaven National Laboratory

Philadelphia, Pennsylvania Upton, New York

Volume 34

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CONTENTS

Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Transport Phenomena in Heterogeneous Media Based

on Volume Averaging Theory

V. S. Travkin and I. Catton

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1

II. Fundamentals of Hierarchical Volume Averaging

Techniques . . . . . . . . . . . . . . . . . . . . . . . . 4

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

III. Nonlinear and Turbulent Transport in Porous Media . . . . 14A. Laminar Flow with Constant Coefficients . . . . . . . . . . . . . 17

B. Nonlinear Fluid Medium Equations in Laminar Flow . . . . . . . . 19

C. Porous Medium Turbulent VAT Equations . . . . . . . . . . . . 21

D. Development of Turbulent Transport Models in Highly Porous Media . 26

E. Closure Theories and Approaches for Transport in Porous Media . . . 32

IV. Microscale Heat Transport Description Problems and

VAT Approach . . . . . . . . . . . . . . . . . . . . . . . 37A. Traditional Descriptions of Microscale Heat Transport . . . . . . . . 38

B. VAT-Based Two-Temperature Conservation Equations . . . . . . . . 43

C. Subcrystalline Single Crystal Domain Wave Heat Transport Equations . 45

D. Nonlocal Electrodynamics and Heat Transport in Superstructures . . . 46

E. Photonic Crystals Band-Gap Problem: Conventional DMM-DNM and

VAT Treatment . . . . . . . . . . . . . . . . . . . . . . . 52

V. Radiative Heat Transport in Porous and Heterogeneous

Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

VI. Flow Resistance Experiments and VAT-Based Data

Reduction in Porous Media . . . . . . . . . . . . . . . . . 66

VII. Experimental Measurements and Analysis of Internal Heat

Transfer Coefficients in Porous Media . . . . . . . . . . . . 85

VIII. Thermal Conductivity Measurement in a Two-Phase

Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

IX. VAT-Based Compact Heat Exchanger Design and

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

v

B. New Kinds of Heat Exchanger Mathematical Models . . . . . . . . 116

C. VAT-Based Compact Heat Exchanger Modeling . . . . . . . . . . 117

D. Optimal Control Problems in Heat Exchanger Design . . . . . . . . 123

E. A VAT-Based Optimization Technique for Heat Exchangers . . . . . . 124

X. New Optimization Technique for Material Design Based

on VAT . . . . . . . . . . . . . . . . . . . . . . . . . . . 127XI. Concluding Remarks . . . . . . . . . . . . . . . . . . . . 129

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . 131

References . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Two-Phase Flow in Microchannels

S. M. Ghiaasiaan and S. I. Abdel-Khalik

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 145

II. Characteristics of Microchannel Flow . . . . . . . . . . . . 146

III. Two-Phase Flow Regimes and Void Fraction in

Microchannels. . . . . . . . . . . . . . . . . . . . . . . . 147A. Definition of Major Two-Phase Flow Regimes . . . . . . . . . . . 148

B. Two-Phase Flow Regimes in Microchannels . . . . . . . . . . . . 150

C. Review of Previous Experimental Studies and Their Trends . . . . . . 153

D. Flow Regime Transition Models and Correlations . . . . . . . . . 161

E. Flow Patterns in a Micro-Rod Bundle . . . . . . . . . . . . . . 166

F. Void Fraction . . . . . . . . . . . . . . . . . . . . . . . . 169

G. Two-Phase Flow in Narrow Rectangular and Annular Channels . . . . 170

H. Two-Phase Flow Caused by the Release of Dissolved Noncondensables . 178

IV. Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . 180

A. General Remarks . . . . . . . . . . . . . . . . . . . . . . 180

B. Frictional Pressure Drop in Two-Phase Flow . . . . . . . . . . . 180

C. Review of Previous Experimental Studies . . . . . . . . . . . . . 184

D. Frictional Pressure Drop in Narrow Rectangular and Annular Channels . 189

V. Forced Flow Subcooled Boiling . . . . . . . . . . . . . . . 191A. General Remarks . . . . . . . . . . . . . . . . . . . . . . 191

B. Void Fraction Regimes in Heated Channels . . . . . . . . . . . . 192

C. Onset of Nucleate Boiling . . . . . . . . . . . . . . . . . . . 195

D. Onset of Significant Void and Onset of Flow Instability . . . . . . . 198

E. Observations on Bubble Nucleation and Boiling . . . . . . . . . . 205

VI. Critical Heat Flux in Microchannels . . . . . . . . . . . . . . . . . 209A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 209

B. Experimental Data and Their Trends . . . . . . . . . . . . . . . 210

C. Effects of Pressure, Mass Flux, and Noncondensables . . . . . . . . 215

D. Empirical Correlations . . . . . . . . . . . . . . . . . . . . 216

E. Theoretical Models . . . . . . . . . . . . . . . . . . . . . . 221

VII. Critical Flow in Cracks and Slits . . . . . . . . . . . . . . . 224A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 224

B. Experimental Critical Flow Data . . . . . . . . . . . . . . . . 225

vi contents

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

Microchannels . . . . . . . . . . . . . . . . . . . . . . . 230

D. Integral Models . . . . . . . . . . . . . . . . . . . . . . . 232

E. Models Based on Numerical Solution of Differential Conservation

Equations . . . . . . . . . . . . . . . . . . . . . . . . . 236

VIII. Concluding Remarks . . . . . . . . . . . . . . . . . . . . 240

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . 242

References . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Turbulent Flow and Convection: The Prediction of Turbulent Flow

and Convection in a Round Tube

Stuart W. Churchill

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 256A. Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . 257

B. Turbulent Convection . . . . . . . . . . . . . . . . . . . . 259

II. The Quantitative Representation of Turbulent Flow . . . . . 260A. Historical Highlights . . . . . . . . . . . . . . . . . . . . . 260

B. New Improved Formulations and Correlating Equations . . . . . . . . . 294

III. The Quantitative Representation of Fully Developed

Turbulent Convection . . . . . . . . . . . . . . . . . . . . 304A. Essentially Exact Formulations . . . . . . . . . . . . . . . . . 305

B. Essentially Exact Numerical Solutions . . . . . . . . . . . . . . 323

C. Correlation for Nu . . . . . . . . . . . . . . . . . . . . . . 335

IV. Summary and Conclusions . . . . . . . . . . . . . . . . . . 348A. Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . 348

B. Turbulent Convection . . . . . . . . . . . . . . . . . . . . 353

References . . . . . . . . . . . . . . . . . . . . . . . . . . 356

Progress in the Numerical Analysis of Compact Heat

Exchanger Surfaces

R. K. Shah, M. R. Heikal, B. Thonon, and P. Tochon

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 363

II. Physics of Flow and Heat Transfer of CHE Surfaces . . . . . 366A. Interrupted Flow Passages . . . . . . . . . . . . . . . . . . . 366

B. Uninterrupted Complex Flow Passages . . . . . . . . . . . . . . 371

C. Unsteady Laminar versus Low Reynolds Number

Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . 374

III. Numerical Analysis . . . . . . . . . . . . . . . . . . . . . 375A. Mesh Generation . . . . . . . . . . . . . . . . . . . . . . 376

B. Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 376

C. Solution Algorithm and Numerical Scheme . . . . . . . . . . . . 378

viicontents

IV. Turbulence Models . . . . . . . . . . . . . . . . . . . . . 380A. Reynolds Averaged Navier—Stokes (RANS) Equations . . . . . . . . 381

B. Large Eddy Simulation (LES) . . . . . . . . . . . . . . . . . 392

C. Direct Numerical Simulation . . . . . . . . . . . . . . . . . . 395

D. Concluding Remarks on Turbulence Modeling . . . . . . . . . . . 397

V. Numerical Results of the CHE Surfaces . . . . . . . . . . . 397A. Offset Strip Fins . . . . . . . . . . . . . . . . . . . . . . . 398

B. Louver Fins . . . . . . . . . . . . . . . . . . . . . . . . 406

C. Wavy Channels . . . . . . . . . . . . . . . . . . . . . . . 416

D. Chevron Trough Plates . . . . . . . . . . . . . . . . . . . . 425

VI. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 432

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . 434

References . . . . . . . . . . . . . . . . . . . . . . . . . . 435

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . 445

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

viii contents

CONTRIBUTORS

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

S. I. Abdel-Khalik (145), G. W. Woodruff School of Mechanical Engineer-

ing, 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 Engineer-

ing, Georgia Institute of Technology Atlanta, Georgia 30332.

M. R. Heikal (363), University of Brighton, Brighton, United Kingdom.

R. K. Shah (363), Delphi Harrison Thermal Systems, Lockport, New York

14094.

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

P. Tochon (363), CEA-Grenoble, DTP/GRETh, Grenoble, France.

V. S. Travkin (1), Department of Mechanical and Aerospace Engineering,

University of California, Los Angeles, Los Angeles, California 90095.

ix

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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 journals

and university-level textbooks. The series presents review articles on special

topics of current interest. Each contribution starts from widely understood

principles and brings the reader up to the forefront of the topic being

addressed. The favorable response by the international scientific and engin-

eering community to the thirty-four volumes published to date is an

indication of the success of our authors in fulfilling this purpose.

In recent years, the editors have undertaken to publish topical volumes

dedicated to specific fields of endeavor. Several examples of such topical

volumes are Volume 22 (Bioengineering Heat Transfer), Volume 28 (Trans-

port 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.

xi

aa


ADVANCES IN

HEAT TRANSFER

Volume 34

aa


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 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

ISBN: 0-12-020034-1 Copyright � 2001 by Academic Press. All rights of reproduction in any form reserved.

0065-2717/01 $35.00

Further advances in the use of VAT are found in the work of Slattery [6],

Kaviany [7], Gray et al. [8], and Whitaker [9, 10]). Many of the important

details and examples of application are found in books by Kheifets and

Neimark [11], Dullien [12], and Adler [13].

Publications on turbulent transport in porous media based on VAT

began to appear in 1986. Primak et al. [14], Shcherban et al. [15], and later

studies by Travkin and Catton [16, 18, 20, 21], etc., Travkin et al. [17, 19,

22], Gratton et al. [26, 27] and Catton and Travkin [28] present a

generalized development of VAT for heterogeneous media applicable to

nonlinear physical phenomena in thermal physics and fluid mechanics.

In most physically realistic cases, highly complex integral—differential

equations result. When additional terms in the two- and three-phase

statements are encountered, the level of difficulty in attempting to obtain

closure and, hence, effective coefficients, increases greatly. The largest

challenge is surmounting problems associated with the consistent lack of

understanding of new, advanced equations and insufficient development of

closure theory, especially for integral—differential equations. The ability to

accurately evaluate various kinds of medium morphology irregularities

results from the modeling methodology once a porous medium morphology

is assigned. Further, when attempting to describe transport processes in a

heterogeneous media, the correct form of the governing equations remains

an area of continuously varying methods among researchers (see some

discussion in Travkin and Catton [16, 21]).An important feature of VAT is being able to consider specific medium

types and morphologies, lower-scale fluctuations of variables, cross-effects of

different variable fluctuations, interface variable fluctuations effects, etc. It is

not possible to include all of these characteristics in current models using

conventional theoretical approaches. The VAT approach has the following

desirable features:

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

modeling.

2. The effect of morphology of the different phases is incorporated. The

morphology decription is directly incorporated into the field equa-

tions.

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 ‘‘the-

ories’’ presently used for that purpose are only approximate descrip-

tion, and often simply wrong.

5. Correct description of experiments in heterogeneous media—again, at

present the homogeneous presentation of medium properties is used

2 v. s. travkin and i. catton

for this purpose, and explanation of experiments is done via bulk

features. 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 be

used to connect properties and morphological characteristics to com-

ponent 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 two-

phase 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 environ-

mental 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 remedi-

ation actions. Using second-order turbulent models, equation sets were

obtained for turbulent filtration and two-temperature or two-concentration

diffusion in nonisotropic porous media and interphase exchange and micro-

roughness. 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 mor-

phology 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

3volume averaging theory

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 possibili-

ties 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 ��m

��) f�

�, (1)

phase averages of f in each component of the medium,

� f ��m

��

1

��

f (t, x�) d�� m�� f �

�(2)

� f ��m

��

1

��

f (t, x�) d�� m�� f �

�(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 first

and second phase (as liquid and solid) within the REV and over the entire

REV fulfill the conditions


f� g�� f

�� g

�and a

�� a� const (6)

for steady-state phases and

�� f

�t ��

�� f

��t

, f� g�� f� g� (7)

except for the differentiation condition,

f �� f � �

1

��

�

f � d�s�

f� � f � f�, f ��, (8)

where �S�

is the inner surface in the REV, and d�s is the solid-phase,

inward-directed differential area in the REV (d�s� n�dS). The fourth condi-

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 � 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

� f �� f �

��

1

�� f d�s

�, (12)

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�) � ��

��

f (x� , �)p��, x� d�

lim

��

f �� f � �. (13)


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 phase

f can be written as

T�(r�) � h · r

�� T�

�(r�) �T

�, (16)

where h and T�

are constants and T��(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�

)��m�T

�,

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

�T�(r�)�

��T

�(�r

��)�

�� h ·�r

�� m�T

�, (17)

meaning that

�T��(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 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

assumed, however, that the REV volume �� contains the exact number of

spacial 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 of r�

(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 r�

in terms of the position vector,

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


Fig. 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 y�

as 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 x and 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 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


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 x for the two right-hand-side terms in (23) to obtain

T�(x, r

�)�� h� h · xy�

(r�, x)

�. (24)

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

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

�),

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 function

should be periodic. The VAT-based answer should be seen by determination

of the averaged values, which are not averaged, only the REV being used at

the lower scale.

The work by Quintard and Whittaker and the improving of understand-

ing 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, 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 locations x�within 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 REV ��

within ��. It represents the

intrinsic 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 ��.


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

Next, consider a point y�

located an infinitely small distance outside of

the 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 an

infinitely small constant.

Following arguments of Carbonell and Whitaker [31], this point y�

is

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

boundary ��.

Further, suppose that one is approaching the boundary ��

from both

sides by points y��

and y�

. According to Carbonell and Whitaker [31] and

Quintard and Whitaker [29], the values ��

and ��

can be different

when ��

is 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.


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

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. Theoretical Verification of Central VAT Theoremand Its Consequences

When the coefficient of thermal conductivity k�

is a constant value, the

fluid stedy-state conduction regime is described by

k� �(�m�T�

�) � ·�

k�

��

T�d�s��

k�

��

T�· d�s � 0. (27)

The full 1D Cartesian coordinates version of this equation, without any

source, for a fixed solid matrix in is

��x ��m�

�T��

�x ��x �

1

��

T��d�s��

1

��

�T�

�x�

· d�s� 0, (28)

��x ��m�

�T��

�x �� MD��MD

� 0, (29)

where the second and third terms on the right-hand side are the so-called

morphodiffusive terms, MD�

and MD, respectively (see also, for example,

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

equation is of the same form,

��x ��s�

�T

�x ��x �

1

��

T�d�s

��1

��

�T

�x�

· d�s�� 0, (30)

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

�s��T�

�x�

��x �

1

��

T�d�s

��1

��

�T�

�x�

· d�s�� 0. (31)


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 mor-

phologies. 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 equa-

tions and those resulting from a more rigorous development based on VAT

are certain additional terms. The best way to evaluate the need for these

additional more complex terms is to obtain an exact mathematical solution

and compare the results with calculations using the VAT equations. This

will clearly display the need for using the more complex VAT mathematical

statements.

Consider a two-phase heterogeneous medium consisting of an isotropic

continuous (solid or fluid) matrix and an isotropic discontinuous phase

(spherical particles or pores). The volume fraction of the matrix, or f-phase,

is �m��m��

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

m� 1 �m

��

/��, where ��

��

is the volume of the REV.

The constant properties (phase conductivities, k�

and k), stationary (time-

independent) heat conduction differential equations for T�

and T, the local

phase temperatures, are

� · q�� k

� �T

�� 0, � · q

� k

�T

� 0,

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

T��T

, ds

�· q

��

�

� ds�· q

��

�

.

Here q�� k

� T

�and 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 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


knows how to obtain or calculate it. The exact effective thermal coefficient

was obtained using direct numerical modeling (DNM) based on the math-

ematical 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

· (k��

�T �) � 0. (32)

The boundary conditions for this equation are formulated in the same

manner as for a homogeneous medium.

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

three-term integrodifferential equations (28) and (30). To evaluate and

compare solutions to these equations with the DNM results, one needs to

know the local solution characteristics, the averaged characteristics over the

both phases in each cell, and, in this case, the additional morphodiffusive

terms.

An infinite homogeneous isotropic medium containing a three-dimen-

sional (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,

because of the absence of heat sources, leads to the equality of averaged

internal heat flux �q��Q�.

When all the particles have the same radii, the result is the triple periodic

structure used widely, beginning from Rayleigh’s work [39], to evaluate the

effective conductivity of particle-reinforced composites.

The composite medium model consists of the three regions shown in Fig.

3. The half-space lying above the A—A plane has a volume content of the

disperse phase m�m

, and for the half-space below the B—B plane

m�m

�. To define the problem, let m

�m

�. The third part is the

composite layer between the plane boundaries A—A and B—B containing Ndouble 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, mean-

ing that each term was calculated independently using the results of DNM

calculations.

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

��z �k��

�T�z�� 0, (z

�� z� z

�), m

�m

(z),

where k��

(m) is the effective conductivity coefficient.


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

The normalized solution of the both models (VAT and DNM) for the case

of linearly changing porosity m�m��

� z(m��

� m��

), where m

(z

�) �m��

,

m(z

�) �m��

, z

�� 0, z

�� 1, between A—A and B—B and with effective

conductivity coefficients of k��

� 0, 0.2, 1, 10, and 10,000, are presented in

Fig. 4. There is practically no difference (less than 10�) between the

solutions, and what there is is probably because of numerical error accumu-

lation (Travkin and Kushch [34]).Lines 1—5 represent solutions of the one-term equation, respectively,

whereas the points (circles, triangles, etc.) represent the solutions of the VAT

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

ferential terms MD�

and MD

for the composite with varying volume

content of disperse phase. Here the number of screens is nine, corresponding

to a relatively small particle phase concentration gradient.

The coincidence of the results of the exact calculation of the two-equation,

three-term conductive-diffusion transport VAT model (28) and (30) with the

exact DNM solution and with the one-temperature effective coefficient

model for heterogeneous media with nonconstant spatial morphology

clearly demonstrates the need for using all the terms in the VAT equations.

The need for the morphodiffusive terms in the energy equation is further

demonstrated by noting that their magnitudes are all of the same order.

Confirmation of the fact that there is no difference in solutions between

the correct one-term, one-temperature effective diffusivity equation and the


Fig. 4. Comparison of VAT three-term equation particle temperature (symbols) with the

exact analytical based on the effective conductance coefficient obtained by exact DNM (solid

lines).

three-terms, two-temperature VAT equations does not mean that it is better

to take for modeling and analysis the effective diffusivity one-term, one-

temperature equation (see Subsection VI, E and arguments in Sections VII

and VIII). Among other issues one needs to analyze goals of modeling and

to understand that the good solution of the effective diffusivity one-term

one-temperature equation as it was found and described in the preceding

statements means nothing less than the ground of the exact solution of the

VAT problem. Also, it is important that for the exact (or accurate) solution

of conventional diffusivity equation, the effective coefficient needs to be

found, and this means in turn that finding the solution of the two-field

problem is imperative and consequently appears to be the major problem.

Meanwhile, this is the problem that was posed just at the beginning as the

original one.

III. Nonlinear and Turbulent Transport in Porous Media

To a great extent, the analysis of porous media linear transport phenom-

ena are given in the numerous studies by Whitaker and coauthors; see, for


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 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 as

well as nonlinear processes will result in different equations and modeling

features.

The general averaged form of the transport equations will be developed

for permeable interface boundaries between the phases. Two forms of the

right-hand-side Laplacian term will be considered. First, one can have two

forms of the diffusive flux in gradient form that can be written

�� V �� V �

��

��

�

V d�s (34)

or

�� V ��m� V� �

��

�

V� d�s. (35)

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,

�� ( V )�� · ( �m�V� ) �� ·�

1

��

V d�s��

�

V · d�s,

(36)

where the production term V · d�s is a tensorial variable, and the version

with fluctuations in the second integral term

�� ( V )�� · (�m� V� ) � � ·�

1

��

V� d�s��

�

V · d�s,

(37)


Using these two forms of the momentum viscous diffusion term, one can

write two versions of the averaged Stokes equations. The first version is

�V ��

1

��

U�· d�s� 0, U

��V (38)

and

0 �� p��

1

��

pd�s�� · ( �m�V� )

�� ·�1

��

V d�s��

�

V · d�s��m��g� , (39)

and the second version is found by using the following relation for the

pressure gradient:

� �p��

1

��

pd�s� ��m� p� �1

��

p� d�s. (40)

Using the averaging rules developed by Primak et al. [14], Shcherban etal. [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

�V �� V

� �m��

1

��

U��· d�s � 0, (41)

obtained using

V � V �� V�

1

��

V d�s ��V � �m��

1

��

U��· d�s, (42)

and the momentum equation

0 ��m� p� �1

��

p� d�s�� · (�m� V� )

� � ·�1

��

V� d�s��

�

V · d�s� �m��g� . (43)

The third version of these equations is almost never used but can be found

in [21].


A. Laminar Flow with Constant Coefficients

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

�U�

�x�

� 0 (44)

�U�

�t� U

�

�U�

�x�

��1

��

�p�x

�

� ��x

��U

��x

�� S

�(45)

��

�t�U

�

��

�x�

�D�

��x

��

��x

��S�

�

. (46)

Here � represents any scalar field (for example, concentration C) that might

be transported into either of the porous medium phases, and the last terms

on the right-hand side of (45) and (46) are source terms. In the solid phase,

the diffusion equation is

��

�t�D

��x

��

�x

��S�

�

. (47)

The averaged convective operator term in divergence form becomes, after

phase averaging,

��x

�

(U�U�)�

�� (U�U�)�

�� U

�U��

1

��

U�U�· d�s

� [�m�U��U��m�u�

�u��]�

1

��

U�U�· d�s. (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

�

(U�U�)�

��m�U�� U�

��U�

� �U

�� u�

�u��

1

��

U�U�· d�s

��m�U��

��x

�

U��U�

�

1

��

U�· d�s

� �u��u��

1

��

U�U�· d�s. (49)

The divergent and nondivergent forms of the averaged convective term in


the diffusion equation are

� (CU�)�� CU

��

1

��

CU�· d�s

� [�m�C� U�� m�c� u�

��]�

1

��

CU�· d�s

��m�U��

��x

�

C� �C�1

��

U�· d�s� �c� u�

��

1

��

CU�· d�s.

(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

� · (D C )�� D · (�m�C� ) � D ·�

1

��

Cd�s��D

��

C · d�s,

(51)or

� · (D C )��D · (�m� C� ) �D ·�

1

��

c� d�s��D

��

C · d�s,

(52)or

�D · (D C )��D�m� �C� �D ·�

1

��

c� d�s��D

��

c� · d�s.

(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

��m�

�V��t

� �m�V� · V� �V�1

��

V · d�s� �v� v� ��

1

��

V V · d�s�� (�m�p� ) �

1

��

pd�s�� · (�m�V� )

�� ·�1

��

V d�s��

�

V · d�s��m��g� . (54)


with the fluctuation diffusion terms it becomes

��m�

�V��t

� �m�V� · V� �V�1

��

V · d�s� �v� v� ��

1

��

V V · d�s��m� p� �

1

��

p� d�s�� · (�m� V� )

� � ·�1

��

v� d�s��

�

V · d�s��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

��

(�m�V� · V� � �v� v� ��) � � (�m�p� ) �

1

��

pd�s�� · (�m�V� )

��

�

V · d�s� �m��g� , (56)

or

��

(�m�V� · V� � �v� v� ��) ��m� p� ) �

1

��

p� d�s �� · (�m�V� )

��

�

V · d�s ��m��g� . (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,

��

�V�t

�V · V �� p� · [�( V � ( V )*)] � ��g� (58)

��(V, C�, T ),

rather than the constant viscosity Navier—Stokes equations. The following

form of the momentum equation will be used in further developments:

��

�V�t

�V · V �� p� · (2�S ) ��g� (59)

��(V, C�, T ).


The negative stress tensor ��

in this equation is

N��

�� 2�( V ) � 2�S, (60)

and the symmetric tensor S is the deformation tensor

S� ( V ) �1

2( V � ( V )*), (61)

with ( V )* being the transposed diad V.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 convective

terms were established earlier. The averaged nonlinear diffusion term yields

� · (D C )�� · (�m�D� C� ) � ·�D�

1

��

c� d�s�� · (�D� c� �

�) �

1

��

D C · d�s. (64)

The other version of the diffusive terms with the full value of concentration

on the interface surface is

� · (D C )�� · (D� (�m�C� )) � ·�D�

1

��

Cd�s�� · (�D� c� �

�) �

1

��

D C · d�s. (65)

General forms of the nonlinear transport equations can be derived for

impermeable and permeable interface surfaces. The averaged momentum

diffusion term is

��x

�

(2�S )�

�� · (2�S )�� · (�2�S �

�) �

1

��

2�S · d�s

� · 2(�m��S� ��m�� S� �) �

2

��

�S · d�s. (66)


The general nonlinear averaged momentum equation for a porous medium

is

�� m�

�V��t

��m�V� · V� �V�1

��

V · d�s� �v� v� ��

1

��

V V · d�s�� (�m�p� ) �

1

��

pd�s � · 2(�m�� S� � �m�� S� �)

�2

��

�S · d�s��m��g� . (67)

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

meable interfaces follows from Eq. (67),

��

(�m�V� · V� � �v� v� ��)

�� (�m�p� ) �1

��

pd�s� · 2(�m�� S� ��m�� S� �)

��

�

�S · d�s��m��g� . (68)

The averaged nonlinear mass transport equation in porous medium

follows

�m��C�

��t

��m�U�� C�

��

C��

��

U�· d�s� �c�

�u��

1

��

C�U�· d�s

� · (D� (�m�C� )) � ·�D�1

��

Cd�s�� · (�D� c� �

�) �

1

��

D C · d�s��m�S��

. (69)

A few simpler transport equations that can be readily used while main-

taining fundamental relationships in heterogeneous medium transport are

given by Travkin and Catton [21].

C. Porous Medium Turbulent VAT Equations

Turbulent transport processes in highly structured or porous media are

of great importance because of the large variety of heat- and mass-exchange

equipment used in modern technology. These include heterogeneous media

for heat exchangers and grain layers, packed columns, and reactors. In all

cases there occurs a jet or stalled flow of fluids in channels or around the


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 math-

ematical models for turbulent transport in the complex geometrical environ-

ments found in nuclear reactors subchannels where rod-bundle geometries

are considered to be formed by subchannels. Processes in each subchannel

are calculated separately (see Teyssedou et al. [51]). The equations used in

this work has often been obtained from two-phase transport modeling

equations [52] with heterogeneity of spacial phase distributions neglected in

the bulk. Three-dimensional two-fluid flow equations were obtained by Ishii

[52] using a statistical averaging method. In his development, he essentially

neglected nonlinear phenomena and took the flux forms of the diffusive

terms to avoid averaging of the second power differential operators. Ishii

and Mishima [53] averaged a two-fluid momentum equation of the form

��v�

�t� · (�

��v�v�) ��

� p

�� · �

�(��

�)

� ��g � v

�� M

��

�· ��, (70)

where ��

is the local void fraction, ��is the mean interfacial shear stress, ��

�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

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��

Dt��

� p

�� · �

�[�

� u

��

�(u��

u��)]

� ��g �M

��M

��

�· �

�� ( p

�� p

�) �

�. (71)

Here the index i denotes interfacial phenomena and M��

is the volumetric

wall force on phase k. Additional terms in Eq. (70) and (71) are usually

based on separate micro modeling efforts and experimental data.


One of the more detailed derivations of the two-phase flow governing

equations by Lahey and Drew [56] is based on a volume averaging

methodology. Among the problems was that the authors developed their

own volume averaging technique without consideration of theoretical ad-

vancements developed by Whitaker and colleagues [10, 42] and Gray et al.[8] for laminar and half-linear transport equations. The most important

weaknesses are the lack of nonlinear terms (apart from the convection

terms) that naturally arise and the nonexistence of interphase fluctuations.

Zhang and Prosperetti [57] derived averaged equations for the motion of

equal-sized rigid spheres suspended in a potential flow using an equation for

the probability distribution. They used the small particle dilute limit

approximation to ‘‘close’’ the momentum equations. After approximate

resolution of the continuous phase fluctuation tensor M�, the vector

A�(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 unconvec-

tional media. The development by Zhang and Prosperetti [57] is a good

example of the correct application of ensemble averaging. The equations

they derive compare exactly with those derived from rigorous volume

averaging theory (VAT) [24].

Transport phenomena in tube bundles of nuclear reactors and heat

exchangers can be modeled by treating them as porous media [58]. The

two-dimensional momentum equations for a constant porosity distribution

usually have the form [59]

�U�x

��V�y

� 0 (72)

�U�

�x�

�UV

�y��

1

��P�x

� ��

�U� A�V� ��U (73)

�UV

�x�

�V �

�y��

1

��P�y

� ��

�V �A��V� ��V, n� 0, (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�,

and an effective eddy viscosity, ��

, that is not equal to the turbulent eddy

viscosity. These kinds of equations were not designed to deal with non-


linearities induced by the physics of the problem and the medium variable

porosity or to take into account local inhomogeneities.

Some of the more interesting applications of turbulent transport in

heterogeneous media are to agrometeorology, urban planning, and air

pollution. The first significant papers on momentum and pollutant diffusion

in urban environment treated as a two-phase medium were those by Popov

[60, 61]. In these investigations, an urban porosity function was defined

based on statistical averaging of a characteristic function �(x, y, z) for the

surface roughness that is equal to zero inside of buildings and other

structures and equal to unity in an outdoor space. The turbulent diffusion

equation for an urban roughness porous medium after ensemble averaging

is

�m(x�)�C

��

�t�

��x

�

(m(x�)�V

��C

��)

� ��x

�

�(v!�)�(c!

�)��

��x

�

�v��c��

��x

��D�

��C��

�x��, n� 1, 2, 3, 4 . . . ,

(75)

where � � means porous volume ensemble averaging, and m(x�) is porosity.

Closure of the two ‘‘morphological’’ terms, the first and the second terms on

the right-hand side, was obtained using a Boussinesque analogy,

��x

�

�(v!�)�(c!

�)��

��x

�

�v��c��K

��

��C ��

�x�

. (76)

A descriptive analysis of the deviation variables (v!�)�, (c!

�)� and the effective

diffusion coefficient K��

was not given. In many studies of meteorology and

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 �t

� f�V �

1

��P �x

��z

(�u�w�) � (1 �m)c�S(z)�U �U (77)

�V �t

��f�U �

1

��P �y

��z

(�v�w�) � (1 �m)c�S(z)�V �V ,

where (1 �m) is the fraction of the earth surface occupied by forest, m

is

the area porosity due to a tree volume, and f�is a Coriolis parameter.

The averaging technique used by Raupach and Shaw [63] to obtain a

turbulent transport equation for a two-phase medium of agro- and forest


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

function is defined by

� f ��

�1

��

��

f d��, (78)

with ��

being the area within the volume ��

occupied by air. Raupach

et al. [64] and Coppin et al. [65] assumed that the dispersive covariances

were unimportant,

�u! "�u! "��

, (79)

where u! "�

is a fluctuation value within the canopy and u! "�� u�

�. The

contribution of these covariances was found by Raupach et al. [64] to be

small in the region just above the canopy from experiments with a regular

rough morphology. This finding has been explained by Scherban et al. [15],

Primak et al. [14], and Travkin and Catton [16, 20] for regular porous

(roughness) morphology. Covariances are, however, the result of irregular

or random two-phase media. When the surface averaging used by Raupach

and Shaw [64] is used instead of volume averaging, especially in the case of

nonisotropic media, the neglect of one of the dimensions in the averaging

process results in an incorrect value. This result should be called a 2D

averaging procedure, particularly when 3D averaging procedures are re-

placed by 2D for nonisotropic urban rough layer (URL) when developing

averaged transport equations.

Raupach et al. [64—66] 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

�

�u��u�� U �

�

��

��

�

��x

�

U �· d�s

��x

�

u! "�u! "��

1

��

��

�

P d�s, i, j� 1� 3, (80)

where �S�

is 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 by

developing a bulk overall drag coefficient. The second, third, and fifth terms

on the right-hand side of Eq. (80) are represented by a common drag

resistance term. For a stationary fully developed boundary layer, they write


��z �u�w��u! "w! "��

�U �

�z ��1

2C��

S��

U � �, (81)

where C��

is an element drag coefficient and S��

is an element area

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

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

�is 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 averaging

the terms on the right-hand side were overcome by using assumed closure

models for the stress components. As a result, the averaged turbulent

momentum equation, for example, has conventional additional resistance

terms such as the averaged momentum equation developed by Vafai and

Tien [70] for laminar regime transport in porous medium. A major

assumption is the linearity of the fluctuation terms obtained, for example,

by neglect of additional terms in the momentum equation.

A meaningful experimental study by Howle et al. [71] confirmed the

importance of the role of randomness in the enhancement of transport

processes. The results show the very distinct patterns of flow and heat

transfer for two cases of regular and nonuniform 2D structured nonorthog-

onal 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 Modelsin 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 hyperturbu-

lent 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


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 stream-

lines

5. Large-scale vortex effects

6. The anisotropic nature of turbulent transfer and resulting anisotropy

of turbulent viscosity

It is well known that in spacial boundary flows, an important role is

played by the gradients of normal Reynolds stresses and that this is the case

for flows in porous medium channels as well. As a rule, flow symmetry is

not observed in these channels. Therefore, in channel turbulence models, the

shear components of the Reynolds stress tensors have a decisive effect on

the flow characteristics. At present, however, turbulence models that are less

than second-order can not be successfully employed for simulating such

flows (Rodi [72], Lumley [73], and Shvab and Bezprozvannykh [74]).Derivation of the equations of turbulent flow and diffusion in a highly

porous medium during the filtration mode is based on the theory of

averaging of the turbulent transfer equation in the liquid phase and the

transfer equations in the solid phase of a heterogeneous medium (Primak etal. [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 �

�t�U

�

�U �

�x�

��1

��

�p!�x

�

��x

��

�U �

�x�

� u��u��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 on

the right-hand side of (82) and (83) are source terms.


Next we introduce free stream turbulence into the hierarchy Let us

represent the turbulent values as

U�U � u��U ��U�

�� u�

�� u�

�(85)

U �U � � u!� ,

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 proper-

ties of the inhomogeneities of the porous medium configurations, sections,

and boundary surfaces does not mean that the distribution of values of U �

and u��

are altogether independent of the distance to the wall, pressure

distribution, etc. Thus, the values U �

or u��

stand for the values generally

accepted in the turbulence theory, that is, when a plane surface is referred

to, these values are those of a classical turbulent boundary layer. When a

round-section channel is involved, and even if the cross-section of this

channel is not round, but without disturbing nonhomogeneities in the

section, then the characteristics of this regular sections (and flow) may be

considered to be those that could be marked with index k. Hence, if a

channel in a porous medium can be approximately by superposition of

smooth regular (of regular shape) channels, it is possible to give such a flow

its characteristics and designated them with the index k, which stands for

the basic (canonical) values of the turbulent quantities.

Triple decomposition techniques have been used in papers by Brereton

and Kodal [76] and Bisset et al. [77], among others. The latter utilized

triple decomposition, conditional averaging, and double averaging to ana-

lyze the structure of large-scale organized motion over the rough plate.

It should be noted that there are problems where U �and u�

�can be found

from known theoretical or experimental expressions (correlations) where the

definitions of U �

and u��

are equivalent to the solution of an independent

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�

�. (86)

The term u�� u!� appears if the flow is through a nonuniform array of

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

fluctuation vector u��

over a smooth, plain surface.


The following hypothesis about the additive components is developed to

correct 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 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 adop-

tion of a hypothesis about the additive components of functions represen-

ting 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,

��x

�

�U ��

1

��

U �· d�s � 0, (88)

for turbulent filtration (with molecular viscosity terms neglected for

simplicity),

�m��U �

��t

��x

�

(�m�U ��U ��)��

1

��

��x

�

(�m�p!� ) ��x

�

��u��u��

��x

�

��u!��u!��

�1

��

�

p!d�s�1

��

U �U �· d�s

�1

��

u��u��· d�s ��m�S�

��

, i, j � 1—3, (89)

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

�m��

��t

��x

�

(�m�U �� )

��x

�

��u��#��

��x

�

��u! ��#!��

1

��

U �� · d�s

�1

��

u��#��· d�s� �m�S� �

�

, i � 1—3. (90)


Many details and possible variants of the preceding equations with

tensorial terms are found in Primak et al. [14], Scherban et al. [15], and

Travkin and Catton [16, 21]. Using an approximation to K-theory in an

elementary channel (pore), the equation for turbulent diffusion of nth species

takes the following more complex form after being averaged:

�m��C �

��t

��V �� C �

�� v!�

�c!��

� · (K�� (�m�C �

�)) � ·�K� �

1

��

C �d�s�

� · (�k�� c!�

��) �

1

��

K� C

�· d�s

�C ��

��

U �· d�s �

1

��

C �U �· d�s��m�S�

�,

n � 1, 2, 3, 4 . . . . (91)

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

sides of Eq. (89) through (91) do not equal zero, since there could be

penetration through the phase transition boundary changing the boundary

conditions in the microelement to allow for heat and mass exchange through

the interface surface as the values of velocity, concentrations, and tempera-

ture at �S�

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 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 particu-

lar porous medium. The third term on the right-hand side of Eq. (91) can

be associated with the notion of morphodiffusive dispersion of a substance

or heat in a randomly nonhomogeneous medium. The term with S��may also

reflect, specifically, the impact of microroughness from the previous level of

the simulation hierarchy. The importance of accounting for this roughness

has been demonstrated by many studies. The remaining step is to account

for the microroughness characteristics of the previous level.

One-dimensional mathematical statements will be used in what follows

for simplicity. Admission of specific types of medium irregularity or random-

ness requires that complicated additional expressions be included in the

generalized governing equations. Treatment of these additional terms be-

comes a crucial step once the governing averaged equations are written. An

attempt to implement some basic departures from a porous medium with


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 descrip-

tion of the medium morphology is depicted as

��x ��m�(K�

�� )

�U �

�x��x �K� � �u!�

�x��

��x

(��u!� u!� ��)

��m� U ��U �

�x�

�1

��

(K�� )

�U �x

�

· d�s

�1

��

�

p!d�s�1

��

��x

(�m�p!� )

��m�U ��U �

�x�

� u�*��

S�(x) �

1

��

�

p! d�s �1

��

��x

(�m�p!� ), (92)

where K�

is the turbulent eddy viscosity, and u�*��

is the square friction

velocity at the upper boundary of surface roughness layer h�averaged 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 proper

form for the turbulent heat transfer equation in the fluid phase using

one-equation K-theory closure with primarily 1D convective heat transfer is

c��m�U �

�T ��

�x�

��x ��m�(K�

�� k

�)�T �

��x ��

��x �K� � �T �

��x

��

� c��

��x

(�m��T ��u!�

�) �

��x �

(K�� k

�)

��

T ��d�s�

�1

��

(K�� k

�)�T

��x

�

· d�s, (93)

whereas in the neighboring solid phase, the corresponding equation is

��x �(1 ��m�)K

�

�T

�x ��x �K� � �T�

�x

�

��x �

K�

��

T�d�s

��1

��

K�

�T

�x�

· d�s�. (94)

The generalized longitudinal 1D mass transport equation in the fluid

phase, including description of potential morphofluctuation influence, for a


medium morphology with only 1D fluctuations is written

��x ��m�(K�

�D

�)�C �

��x ��

��x �K� �C �

��x

��

��x

(�m��c! ��u!�

�) �

��x �

(K� � D

�)

��

c!��d�s�

�1

��

(K �D

�)�C

��x

�

· d�s��m�S� ��m�U �

�C ��

�x, (95)

whereas the corresponding nonlinear equation for the solid phase is

��x �(1 ��m�)D

�C

�x ��x �D� �c� �x

�

��x �

D

��

C�d�s

��1

��

D

�C

�x�

· d�s�. (96)

E. Closure Theories and Approaches for Transportin 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 equa-

tions for transport. There are only two well-known types of porous media

morphologies for which researchers have had major successes. But even for

these morphologies, namely straight parallel pores and equal-size spherical

inclusions, not enough evidence is available to state that the closure

problems for them are ‘‘closed.’’

One of the few existing studies of closure for VAT type equations is by

Hsu and Cheng [79, 80]. They used a one-temperature averaged equation

[Equation (40a) in Hsu and Cheng [80]) without the morphodiffusive term

· [(k�� k

)T� (� �m�)] � · [(k

� k

�)T� ( �m�)].

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 (seeKheifets and Neimark [11])

c� $ � C �d��

, and j��D� C �, so �c� j� ��$D�� C��

�

d��l

,


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

of pores in a REV. It is not obvious that the length scale, d��

, taken for the

approximation of c� follows from use of l as a scale for the second derivative.

Furthermore, assuming that the variable to be averaged over the REV

changes very slowly over the REV does not mean that it changes very slowly

in the neighborhood of the primary REV.

Various closure attempts for heterogeneous medium transport equations

resulted in various final equations. One needs to know what these equations

are all about. Treatment of the one-dimensional heat conduction equation

with a stochastic function for the thermal diffusivity in a paper by Fox and

Barakat [82] yielded a spatially fourth-order partial differential equation to

be solved. Gelhar et al. [83], after having eliminated the second-order terms

in the species conservation equation for a stochastic media, were able to

develop an interesting procedure for deriving a mean concentration trans-

port equation. The equation form includes an infinite series of derivatives

on the right-hand side of the equation. Analysis of this equation allows the

derivation of the final form of the mass transport equation,

�C� *�t

�U�C� *�x

� (A � a!)U

��C� *�x�

�B�C� *�t�x�

�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 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]:

��m�C

�t� �m�V

�· C� D · (�m� C ).

They transformed the equation with the help of ensemble averaging into a

stochastic transport equation,

��m�C� *�t

��m�u� *� C� *�D · (�m� C� *) ��m��

��

��C� *�x

��x

�

,

where the tensor of the ensemble dispersion coefficient is a correlation

function denoted by

��

�1

2

u��u��*

u�*u�*x� · u�*,

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


reflecting the influence of the stochastic or inhomogeneous nature of the

spatial velocity and concentration fluctuations in the ensemble averaged

stochastic equation developed by Tang et al. [84], has the dispersivity

coefficient fully dependent on the velocity fluctuations. As can be seen by

this equation, the effect of concentration fluctuations was eliminated.

Torquato and coauthors (see, for example, Torquato et al. [85], Miller

and Torquato [86], Kim and Torquato [87]) have been developing means

to characterize the various mathematical dependencies of a composite

medium microstructure in a statistically homogeneous medium. Some of the

quantities considered by Torquato are useful in obtaining resolution to

certain closure problems for VAT developed mathematical models of

globular morphologies. In particular, the different near-neighbor distance

distribution density functions deserve special mention (Lu and Torquato

[88], Torquato et al. [85]).Carbonell and Whitaker [89] combined the methods of volume averaging

and the morphology approach to specify the dispersion tensor for the

problem of convective diffusion for cases where there is no reaction or

adsorption on the solid phase surface,

� D�C�n

� 0, x� �S�,

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,

V� � (V� � V� ) f��D � f�, �n� · f�� n� , x� �S�,

One should note that the spatial deviation functions defined for a periodic

medium are periodic themselves.

Nozad et al. [40] suggested that the same closure scheme be used to

represent the fluctuation terms T��

and T�for a one-temperature model by

using

T�� f� �T ��, T�

� g� �T �� %


for a transient heat conduction problem with constant coefficients in a

two-phase system (stationary). Partial differential equations for f�, g� , �, and

% are found. They obtained excellent predictions of the effective thermal

conductivity for conductivity ratios k� k/k�& 100.

Carbonell [90] attempted to obtain an averaged convective-diffusion

equation for a straight tube morphological model and obtained an equation

with three different concentration variables. This demonstrates that the

averaging procedures, taken too literally, can result in incorrect expressions

or conclusions.

A common form of the averaged governing equations for closure of

multiphase laminar transport in porous media was obtained by Crapiste etal. [41]. They developed a closure approach that led to a complex integro-

differential equation for the spatial deviations of a substance in the void or

fluid phase volume of the macro REV. This means that solving the

boundary value problem for spatial concentration fluctuations, for example,

requires that one obtain a solution to second-order partial differential or

coupled integro differential equations in a real complex geometric volume

within the porous medium.

For a heterogeneous porous medium, this means that the coupled

integrodifferential equation sets for the averaged spatial deviation variables

must be solved for at least two scales. For averaged variables the scales are

the external scale or L domain, and for the spatial deviations it is the volume

of the fluid phase considered at the local (pore) scale. This presents a great

challenge and has not yet been resolved by a real mathematical statement.

To close the reaction-diffusion problem Crapiste et al. [41] made a series

of assumptions: (1) the diffusion coefficient D and the first-order reaction

rate coefficient k�are constant; (2) diffusion is linear in the solid part of the

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 solid

surface averaged concentration are equal, (5) the restriction

k�d�

D 1

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 concen-

trations, which means that the adsorption mechanisms are taken to be

volumetric phenomena.

In our previous efforts we have obtained some results for both morphol-

ogies and demonstrated the strength of morphological closure procedures.


A model of turbulent flow and two-temperature heat transfer in a highly

porous medium was evaluated numerically for a layer of regular packed

particles (Travkin and Catton [16, 20]; Gratton et al. [26, 27]) with heat

exchange from the side surfaces. Nonlinear two-temperature heat and

momentum turbulent transport equations were developed on the basis of

VAT, requiring the evaluation of transport coefficient models. This ap-

proach 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 impor-

tantly, 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

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 correla-

tions. Existing models for variable morphology functions such as porosity

and specific surface were used by Travkin and Catton [20] and Gratton etal. [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 coeffi-

cient 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 morphol-

ogy models (Travkin and Catton [16, 17, 20]; Gratton et al. [26, 27]). It was

shown that the approach taken to close the integral resistance terms in the

momentum equation for a regular structure allows the second-order terms

for the laminar and turbulent regimes to naturally occur. These terms were

taken to be analogous to the Darcy or Forchheimer terms for different flow

velocities. Numerical evaluations of the models show distinct differences in


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 reduc-

es the value of the physical description that results. VAT shows great

promise as a tool for development of models for this type of phenomena

because it becomes possible to include the inherent nonlinearity and

heterogeneity found at the subcrystalline level and reflect the impact at the

upper levels or scales.

A heuristic approach suggested by Tzou [96] lumps all the atomic and

subcrystalline scale phenomena ‘‘into the delayed response in time in the

macroscopic formulation.’’ This approach was proposed by author to close

the existing gap in knowledge and to help engineers develop applications.

Unfortunately, the coupling between the characteristics of the subscale

phenomena and delayed response time is lost. There is an ongoing search

for the transport equations describing many-body systems that exhibit

highly nonequilibrium behavior, including non-Markovian diffusion. The

more exact the description of a physical phenomenon provided by a

mathematical model, the more possibilities there are for innovative improve-

ments in the function of a particular material or device. Our contribution to

the effort is an extensive analysis of existing approaches to the development

of theories for the subcrystalline and atomic scale levels. We have also made

progress in the development of VAT-based tools applicable at the atomic

and nanoscale level for description of transport of heat, mass, and charge in

SiC and superconductive ceramics.

At the subcrystalline scale, we will consider energy transport using a VAT

description for effects of crystal defects and impurities on phonon—phonon

scattering, which has a substantial impact on thermal conductivity. At the

crystal scale, the importance of thermal resistance (different models) due to

various mechanisms— lattice unharmonic resistance and crystal boundary

defects—will be treated. Including these phenomena shows that they have

a major impact on the transport characteristics in critical applications such

as optical ceramics and superconductive ceramics.


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 tem-

perature T"will be presented.

The heat balance equation for the electron temperature is

c�(T )

�T�

�t��U �Q, (97)

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

c�(T ) ��

'2�

�k�n�

k�T�

�

,

and

�� (3n

�/8')� (2'�)�/(2m*).

U is the heat exchange term,

U �2'�

3

m*c�n�(T

��T

")

�(T")

(T��T

")

T"

, T" T

�; (T

�� T

") T

"(98)

U �'�

6

m*c�n�

�(T")

(T��T

")

T"

, T"�T

�; (T

�� T

") T

", (99)

where m* is the effective electronic mass, cis the sound velocity, n

�is the

number of electrons per unit volume, �(T") is the time to traverse a mean

free path of electrons under the condition that the lattice temperature

coincides with the electron temperature and is equal respectively to T".

When the lattice temperature is assumed to be much less than the

temperature of the electrons (an assumption later found to be weak), then

U ��2'�

15

m*c�n�

�(T�)

, (T�T

�; T

" T

�);

'�

6

m*c�n�

�(T�)

, (T�� T

�; T

"T

�).

. (100)


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

��U"

�r�� c�

�U

"��

U

�� ((r �Vt). (101)

This also allowed them to develop the heat exchange term (here U"is the

displacement vector). In this equation, ��M/d is the density of lattice, Mis the mass of the lattice atom, V is the lattice volume, and U is the

interaction constant of the electron with the lattice that appears in the

expression for the time to travel the mean free path.

It was nearly 20 years before needs in different physical fields (namely,

intense short-timespan energy heating in laser applications) brought atten-

tion to this phenomenon and to use it for further technological advances.

Anisimov et al. [100, 101] introduced a simplified two-fluid temperature

model for heat transport in solids,

C�(T

�)�T

��t

� )�T��

��(T

��T

") � f (r, t) (102)

C"

�T"

�t� �

��(T

��T

"), �

��

'�

6

m*c�n�

�(T�)

. (103)

Further development of the idea of a two-field two-temperature model for

energy transport in metals by Qiu and Tien [102—104] used this model.

They modified the energy exchange rate coefficent (heat transfer) model in

a way that uses the coefficient of conductivity K�

in the following formula

instead of time between collisions �(T�):

U �G �'�(n

�ck�)�

K�

. (104)

Tzou et al. [93] used the two-fluid model with two equations for the

electron—phonon transport in metals based on previous works by Anisimov

et al. [101], Fujimoto et al. [105], Elsayed-Ali [106], and others. The

equation for diffusion in an electron gas is a parabolic heat conduction

equation with an exchange term

C�

�T�

�t� · (K

� T

�) �G

��(T�� T

"), (105)

with phonon transport (phonon—electron interactions) for the metal lattice

(just simplified equation) being described by

C"

�T"

�t� G

��(T

��T

"), (106)


where K�

is the thermal conductivity of the electron gas. Using the

Wiedermann—Franz law for the electron—phonon interaction, Qiu and Tien

[102, 103] show that the coupling factor G��

can be approximated by

G��

�'�(n

�ck�)�

K�

, (107)

where c, the speed of sound in solid, is

c�

k�

2'�(6'�n

#)�� T

�, (108)

T�

is the Debye temperature, � is Planck’s constant, and n�

and n#

are the

electronic and atomic volumetric number densities. Assuming constant

thermal properties, the two equations can be combined, yielding a one-

temperature equation

��T"

�x��

��

C��

�T"

�x��t�

1

��

�T"

�t�

1

C��

��T"

�t�, (109)

where the thermal diffusivity of electron gas ��, equivalent thermal diffusiv-

ity ��, and thermal wave speed C

�are defined by

��

K�

C�

, ��

K�

C��C

"

, C��

K�G

C�C"

. (110)

Tzou [96] later proposed a unified two-fluid model to derive the general

hyperbolic equation with two relaxation times ��

and �$,

�T � ��

��t

( �T ) �1

��T�t

��$��T�t�

, (111)

which he argues is the same equation derived from two-step models in

metals. A more complex two-temperatures model was obtained by Gladkov

[107] using parabolic equations

�T�

�t� V

�T�

�x� )

�

��T�

�x��

��(T

��T

�) (112)

and

�T�

�t� )

�

��T�

�x��

��(T

��T

�), (113)

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


for a mobile (liquid) medium results:

�1��T

��t

�1

��

��T�

�t��

V

��

��T�

�t�x�

��

V

��

�T�

�x

� )�

V

��

�T�

�x�

)�)�

��

��T�

�x�� )

�

��T�

�x�. (114)

There are other works (see, for example, Joseph and Preziosi [108]) treating

the two-fluid heat transport and obtaining the same kind of hyperbolic

equation.

1. Equation of Phonon Radiative Transfer

Majumdar [94] suggested an equation for phonon radiative transfer

(EPRT). In three dimensions the equation is

�L��t

� (V��

· I�) �I�(T (x)) � I�

�(�, T ), (115)

where I� is the directional-spectral phonon intensity, V��

is the phonon

propagation speed, and I�(T (x)) is the equilibrium intensity corresponding

to a blackbody intensity at temperatures below the Debye temperature. To

make matters more complex, it should be noted that as stressed by Peterson

[95], ‘‘However, fundamental differences exist between phonon and photon

behavior in the regime where scattering and collisional processes are

important . . . . Even in perfect crystals, the so-called unklapp processes that

are responsible for finite thermal conductivity do not obey momentum

conservation.’’

2. Hyperbolic Heat Conduction Equations

The work by Vernotte, Cattaneo, Morse, and Feshbach that led to the

hyperbolic heat conduction equation was primarily heuristic in nature

(without a first principle physical basis). The final form is often presented as

a telegraph equation (see Joseph and Preziosi [108]),

��T�t�

�1

��T�t

�k

(�*) �T, (116)

or

��T�t�

��T�t

�1

* · (K T ), (117)

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


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�t

� · (nV�) � 0; (118)

conservation of electron momentum,

�V�

�t� (V

�· )V

��

e

m*E�

k�

m*n (nT

�) ��

V�

��; (119)

where the last term ‘‘is the collision and scattering term analogous to the

Darcy term in porous media flow’’; conservation of electron energy,

�W�

�t� · (W

�V�) � �e(nV

�· E) �k

� · (nV

�T�)

� · (k� T

�) � �

(W�� (3/2)k

�T%)

��%

�; (120)

conservation of lattice optical phonon energy,

C%

�T%

�t�

(W�� (3/2)k

�T%)

��%

��C%

(T%� T

#)

�%�#

�; (121)

and conservation of acoustical energy,

C#

�T#

�t� · (k

# T

#) ��C%

(T%�T

#)

�%�#

�. (122)

The last four equations have terms, the last term on the right-hand side, that

qualitatively reflect the collision and scattering rates in each process. Here

��

is the electron momentum relaxation time, ��%

is the electron optical

relaxation time, �%�#

is the optical acoustical relaxation time, and k�

is

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�� ),

· (� �) ��e(N��N

�� n� p) ��eN� (123)

N� � (N��N

�� n� p),


where � is the dielectric constant of Si, N�is the n-doping concentration, N

�is the p-doping concentration, and p is the hole number density.

B. VAT-Based Two-Temperature Conservation Equations

Conservation equations derived using VAT enable one to capture all of

the physics associated with transport of heat at the micro scale with more

rigor than any other method. VAT allows one to avoid the ad hoc

assumptions that are often required to close an equation set. The resulting

equations will have sufficient generality for one to begin to optimize material

design from the nanoscale upward. The theoretical development is briefly

outlined in what follows.

The nonlinear paraboic VAT-based heat conduction equation in one of

the phases of the superstructure (where superstructure is to be determined

as the micro- or nanoscale material’s organized morphology along with its

local characteristics) is

�s��(�c

�)�

�T��

�t� · [�s

��K

�� T

��] � · [�s

��K�

� T�

��]

� ·�K

��

�� T��d�s

��1

�� K

�

�T�

�x�

· d�s�� s

��S

��.

(124)

For constant thermal conductivity, the averaged equation for heat transfer

in the first phase can be written

�s��(�c

�)�

�T��

�t� k

� �(�s

��T�

�) � k

� ·�

1

�� T�d�s

��

k�

�� T

�· d�s

��s

��S

��. (125)

These VAT equations (124) and (125), written for the two phases, will be

seen to yield the same pair of parabolic equations derived by researchers

such as Gladkov [107], but with quite different arguments. Closure to Eq.

(125) is needed for the second and third terms on the right-hand side. The

steps to closure are

1

�� k�

�T�x

�

· d�s��

1

�� k�

�T�n

�

ds · n�

�1

�� q�· d�s

��

��S��

(T �� T

�), (126)


with the heat transfer coefficient, ��

(�S��

), defined in phase 2. This closure

procedure is appropriate for description of fluid—solid medium heat ex-

change and might be considered as the analog to solid—solid heat exchange

found in many works. A more precise integration of the heat flux over the

interface surface, �S��

, yields exact closure for that term in governing

equations for both neighboring phases.

Industry needs to lead one to attempt to estimate, or simulate by

numerical calculation or other methods, the effective transport properties of

heterogenous material. Among the many diverse methods used to do this,

VAT presents itself as an effective tool for evaluating and bringing together

different methods and is useful in providing a basis for comparative

validation of techniques. To demonstrate the value of a VAT-based process,

the effective thermal conductivity will be determined within the VAT

framework. The averagd energy equation in phase 1 of a medium is

k� �(�s

��T�

�) � k

� ·�

1

�� T�d�s

�� · [�q��].

The right-hand-side (‘‘diffusive’’-like) flux is different from that convention-

ally found,

q�� [�k

�� T�

�] ��k

� (�s

��T�

�) �

k�

�� T�d�s

�, (127)

where

k��

� �k� (�s

��T�

�) �

k�

�� T�d�s

�� ( T��)��. (128)

After these transformations, the heat transfer equation for phase 1 becomes

�s��(�c

�)�

�T��

�t� · [k

�� T�

�]� ��

��S��

(T �� T

�) ��s

��S

��.

(129)

This is the same type of heat transport equation routinely used in two-fluid

models. The equation for heat transport in the second phase (if any) would

be the same, and one can easily obtain the hyperbolic type two-fluid

temperature model.

A similar VAT-based equation can be obtained for the heat transfer in

phase 1 when the heat conductivity coefficient is a function of the tempera-

ture or other scalar field (nonlinear) (Eq. (124)), but the effective conductiv-

ity will have an additional term reflecting the mean surface temperature over


the interface surface inside of the REV,

K��

� �K�� (�s

��T�

�) � �s

��K�

� T�

��

K��

�� T�d�s

�� ( T��)��.

(130)Equation (124) simplifies to

�s��(�c

�)�

�T��

�t� · [K

�� T�

�] � ��

��S��

(T �� T

�) ��s

��S

��.

(131)

The third term on the right-hand side of (124) plays a different role when

the interface between two phases is only a mathematical surface without

thickness neglecting the transport within the surface means there is no need

to consider this medium separately. When this is the case, this term can be

equal for the both phases, simplifying the closure problem. The problem

becomes significantly more complicated when transport within the interface

must be accounted for.

C. Subcrystalline Single Crystal Domain Wave HeatTransport 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 descrip-

tion 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

&�t�

��T

&�t

�1

* · (K T

&) �

1

*S��

. (132)

Comparing this equation with the equation developed by Tzou [96] with

two relaxation times, ��

and �$,

�$

��T�t�

��T�t

� � �T � ��t

( �T ) � �S��

, (133)

and the parabolic equation obtained by Gladkov [107] for the model with

two temperatures for constant coefficients,

�1��T

��t

�1

��

��T�

�t��

V

��

��T�

�t�x�

��

V

��

�T�

�x

� )�

V

��

�T�

�x�

)�)�

��

��T�

�x�� )

�

��T�

�x�, (134)


one can see that all belong to the family of VAT two-temperature conduc-

tion problems with nonconstant effective coefficients for the charged carriers,

�T��

�t� a

� · [K

� T�

�] � b

�(T

"� T

�) �S

��

, (135)

and for phonon temperature transport,

�T�"

�t� a

" · [K

" T�

"]� b

"(T�"� T�

�) �S

�

. (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 inSuperstructures

Many microscale heterogeneous heat transport equations and some of the

solutions provided elsewhere (see, for example, [110, 111, 112, 113, 109])required substantial analysis, and many need improvement. Goodson [113],

for example, directly addresses the need to model nonhomogeneous medium

(diamond CVD layer) thermal transport by accounting for the presence of

grains. The Peierls—Boltzmann equation for phonon transport was used

along with information on grain structure. In the present work, the goal is

to give some insight to situations (and those are substantial in number)where the medium cannot be considered as homogeneous even at the

microscale level. For these circumstances, the governing field equations

should be based on conservation equations for a heterogeneous medium, for

example, the VAT governing equations.

The VAT governing equations for heterostructures will be found starting

from a set of governing equations for a solid-state electron plasma fluid.

Phase averaging of the electron conservation equation (118) yields

�n�t �

� � · (nV�)�

�� 0 (137)

where � ��

means averaging over the major phase of the material. The VAT

final form for this equation is

��n��

�t� · �nV

��

1

��

nV�· d�s� 0, (138)


or

��n��

�t� · [�s

��n� V�

�� m�n� V�

��] �

1

�� nV

�· d�s

�� 0, (139)

where �S�

is the ‘‘interface’’ (real or imaginary) of phases and scatterers.

It will be assumed that only immobile scatterers produce phase separ-

ation. 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��t

�

��(V�· )V

��

e

m*�E�

��

k�

m* 1

n (nT

�)�

� ��V�

��.� �(140)

Using the transformation

1

n (nT

�)�

� T��T

�

1

n n

�

� � T�� T

� (ln n)�

�

� � T�� T

� (Z

�)��, Z

�� ln n, (141)

it can be written as

�V��t

�

��(V�· )V

��

e

m*�E�

��

k�

m*� T

�� T

� (Z

�)�

��

V�

��,� �

(142)

where the brackets +�+� define the problem uncertainty in the treatment of

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�

�t �

� � · (nV�V�)�

��

e

m*�nE�

��

k�

m*� (nT

�)��

� ��nV

��,� �(143)


Using

�(V�· )V

�� · (nV

�V�)�

�� V

�( · (nV

�))�

�

� ·�nV�V��

�1

��

(nV�V�) · d�s

�

� �s��V�

� · (nV

�)��V�

�( · (nV

�)) (�)�

�, (144)

Eq. (142) can be written in the VAT form as

��V��

�t� · �nV

�V��

1

��

(nV�V�) · d�s

�

� �s��V�

� · (nV

�)�

��V��( · (nV

�)) (�)�

�

��e

m*�E�

��

k�

m* � �T��

�1

��

T�d�s

��

k�

m* ��s��T�

�� Z��

1

��

�

Z�d�s

�� T��( (Z

�)) (�)�

��,(145)

where the last term on the right-hand side of (142), the scattering and

collision reflection term, has been replaced by a number of terms, each

reflecting interface-specific phenomena, including scattering and collision.

Some manipulation of the convection terms of the conservative form of the

momentum equation has been done to combine the forms of the equations

of mass and momentum.

The second conservative form of the momentum equation is derived in

the form

�n��

�V��

�t� (�nV

��

· )V��

��t

�n� V�� ·�nV�

�V��

�V��

1

��

nV�· d�s��

1

��

(nV�V�) · d�s

�

��e

m*�nE�

��

k�

m* � [�s��n�T�

��s

��n� T�

��] �

1

��

nT�d�s

��,(146)

where a number of the integral terms are scattering and collision terms.

There are other possible forms of the left-hand side of the momentum

equation VAT equations that will not be pursued at this time.


The homogeneous volume averaged electron gas energy equation for a

heterogeneous polycrystal becomes

�W��t

�

�� · (W�V�)�

��e�nV

�· E�

�� k

�� · (nV

�T�)�

�

�� · (k� T

�)��

� ��(W

�� (3/2)k

�T%)

��%

��, (147)

or

��W��

�t� ·�W

�V��

1

��

(W�V�) · d�s

�

��e�nV�·E�

�� k

� · �(nV

�T�)�

��

k�

��

(nV�T�) · d�s

�

� · [k�� (�s

��T

��)] � · [�s

��k�

� T�

��]

� ·�k

��

��

T�d�s

��1

��

k�

�T�

�x�

· d�s�. (148)

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

C%

�T%

�t �

��W

��t �

�

��W

!'�t �

�

, (149)

or

C%

�T%

�t �

�k�

��

(nV�T�) · d�s

��

1

��

(W�V�) · d�s

�

� ·�k

��

��

T�d�s

��1

��

k�

�T�

�x�

· d�s�

� ·�k

��

��

T#d�s

��1

��

k#

�T#

�x�

· d�s�

. (150)

The equation of acoustical phonon energy is

C#

�T#

�t �

� � · (k# T

#)��

� ��C%

(T%� T

#)

�%�#

�� (151)


or

C#

�T#

�t �

� · [k#� (�s

��T

#�)] � · [�s

��k�

# T�

#�]

� ·�k

��

��

T#d�s

��1

��

k#

�T#

�x�

· d�s�.

(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��t

�

� �(V�· )V

��

� �e

m*�E�V

� B�

��

k�

m* 1

n (nT

�)

�

� ��V�

��. (153)

The VAT form of this equation is

��V��

�t� · �nV

�V��

1

��

(nV�V�) · d�s

�

� �s��V�

� · (nV

�)��V�

�( · (nV

�)) (�)�

�

��e

m*(�E�

�� V

� B�

�) �

k�

m* � �T��

�1

��

T�d�s

��

k�

m* ��s��T�

� � z��1

��

�

Z�d�s

�� T��( (Z

�)) (�)�

��.(154)

The Maxwell equations for electromagnetic fields used to develop the

VAT Maxwell equations for electromagnetic fields are

· (��E�) ��

�, · (�

�H�) � 0 (155)

E��

�B�

�t(156)

H�� j

��

��t

(D�

), (157)


with constitutive relationships

B��

�H�, D

��

�E�, j

��

�E�. (158)

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

· [�s��

�E��] � · [�s

��

�E��] �

1

�� (��E�) · d�s

��

��

(159)

(�s��E�

�) �

1

��

d�s� E

��

��t

��H��. (160)

The phase averaged magnetic field equations are

· (�s��

�H��) � · [�s

��

�H��]�

1

��

(��H�) · d�s

�� 0, (161)

and

(�s��H�

�) �

1

��

d�s� H

�

��t

��E��

� [�s��

�E�� s

��

�E��]. (162)

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

wave equations

�E�

��

�E�

�t� �

��

��E�

�t��

��, (163)

which becomes

�(�s��E�

�) � ·�

1

��

E�d�s

��1

��

E�· d�s

�

��

��E��

�t��

��

��E��

�t��

1

��

(�s��

�) �

1

��

�

��d�s

�,

(164)

are the basis for modeling of electric and magnetic fields at the microscale

level in heterostructures.


The primary advantage of the VAT-based heterogenous media elec-

trodynamics equations is the inclusion of terms reflecting phenomena on the

interface surface �S�

that can be used to precisely incorporate multiple

morphological effects occuring at interfaces.

E. Photonic Crystals Band-Gap Problem:

Conventional DMM-DNM and VAT Treatment

One of the possible applications of VAT electrodynamics is the formula-

tion of models describing electromagnetic waves in a dielectric medium of

materials considered to be photonic crystals [118, 116, 117, 119, 126, 120].

The problem of photonic band-gap in composite materials has received

great attention since 1987 [118, 116] because of its exciting promises. The

most interesting applications appear in the purposeful design of materials

exhibiting selective, at least in some wave bands, propagation of electromag-

netic energy [120].

Figotin and Kuchment [122] were the first who theoretically demon-

strated the existence of band-gaps in certain morphologies. Unfortunately,

this problem as presently formulated is based on the homogeneous Maxwell

equations. The most common way to treat such problems has been to seek

a solution by doing numerical experiments over more or less the exact

morphology of interest, a method called detailed micromodeling (DMM),which is often done using direct numerical modeling (DNM) (for example,

see [124]). As a result, questions arise about differences between DMM-

DNM and heterogeneous media modeling (HMM), which is the modeling

of an averaged medium to determine its properties. How the averaging for

HMM is accomplished is often not clear or not done at all.

So, why cannot DMM be self-sufficient in the description of heterogen-

eous medium transport phenomena? The answers can be primarily under-

stood by analyzing, among others [23], the following issues:

1. A basic principal mismatch occurs at the boundaries, causing bound-

ary condition problems. This means that for DMM and for the bulk

(averaged characteristics) material fields, the boundary conditions are prin-

cipally different.

2. The DMM solution must be matched to a corresponding HMM to

make it meaningful at the upper scales. This can only be done for regular

morphologies. Discrete continuum gap closure or mismatching will occur

with DMM-DNM, precluding generalization to the next or higher levels in

the hierarchy.

3. The spatial scaling of heterogeneous problems with the chosen REV

(for DMM) is needed to address large or small deviations in elements


considered that are governed by different underlying physics. When spatial

heterogeneities of the characteristics or morphology are evolving along the

coordinates, DMM cannot be used.

4. Numerical experiments provided by DMM-DNM need to be trans-

lated to a form that implies that the overall spatially averaged bulk

characteristics model random morphologies. It is not clear what kind of

equations are to be used as the governing equations, nor what variables

should be compared. In the case of the local porosity theory [128, 129], for

example, the results of using real porous medium digitized images for

morphological analysis to calculate the effective dielectric constant assumes

that the HMM equations are applicable.

5. Interpretation of the results of DMM-DNM is always a problem. If

results are presented for a heterogeneous continuum, then the previous

point applies. If the results are being used as a solution for some discreteproblem, then the question is how to relate that solution to the continuum

problem of interest or even to a slightly different problem. If the results

obtained are fit into a statistical model, then the phenomena are being

subjected to a statistical averaging procedure that is in most cases only

correct for independent events.

6. The most sought-after characteristics in heterogeneous media trans-

port studies are the effective transport coefficients that can only be correctly

evaluated from

� �j�� * �� (�

��

�)

1

�� d�,

using the DMM-DNM exact solutions for a small fraction of the problems

of interest. The issue is that problems of interest having inhomogeneous,

nonlinear coefficients and, in many transient problems, two-field DMM-

DNM 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


properties. They cannot be treated as mathematical surfaces without any

properties. Imperfections in the internal spacial structures must be treated

as domain morphologies are not perfect at any spacial level.

The insufficiency of a homogeneous wave propagation description of a

heterogeneous medium was addressed in [125] from a pure mathematical

point of view by searching for another type of governing operator that could

better explain the behavior of the frequency spectrum eigenmodes via

‘‘heuristic arguments.’’ The general band-gap formulation should be treated

using the HMM statements developed from the analysis of the VAT

equations. A straightforward description of one of the band-gap problems

is given next.

Representing electromagnetic field components with time-harmonic com-

ponents,

E(x, y, z)ei�t, H(x, y, z)ei�t, i��(�1) , (165)

The equations describing a dielectric medium becomes

· (�E) � 0, ·H � 0, �� 1 (166)

E� �i�H, H� i��!E, (167)

where �! is the complex dielectric ‘‘constant’’ defined by �! � �� i(�/�), and

�� 0(x�), �� (x�), �! � �! (x� , �).Taking the curl of the both sides of the vector equations,

�1

�! H�� (i�E ), ( E) � (�i�H ), (168)

yields

�1

�! H�� i�(�i�H ) ��H (169)

( E ) ��i�(i�! �E) � �! ��E. (170)

This is the set of equations usually used when problems of photonic

band-gap materials are under investigation; see the study by Figotin and

Kuchment [123], p. 1564. These equations can be transformed to

��E(x� ) ��! (x�)E

(x�) (171)

for E-polarized fields and

� ·�1

�! (x� ) H

(x� )��H

(x�) (172)

for H-polarized fields.


The further treatment by Figotin and Kuchment [123] reduces the

mathematics to two equations,

� ·�1

�! (x� ) f

�(x�)�� f

�(x�) (173)

�1

�! (x� )� f

�(x�) � � f

�(x�), (174)

where f�

is the H

or E

polarization determined components of electric or

magnetic fields.

These equations state the eigenvalue problem characterizing the spectrum

of electromagnetic wave propagation in a dielectric two-phase medium,

which is supposed to describe the photonic materials band-gap problem of

EM propagation (see equations on p. 1568 in Figotin and Kuchment [123])There are no spatial morphological terms or functions involved in the

description, just the permittivity, which is supposed to be a space-dependent

function with changes at the interface boundary.

When these equations are phase averaged to represent the macroscale

characteristics of wave propagation in a two-phase dielectric medium, the

equations become

· (*�� (�m

�� f �

��)) � ·�*� �

1

�� f��

d�s��

� · (�*� � f��) �

1

�� *� f

��· d�s

��m

�� f

�� (175)

*(x�) �1

�!

�(�m�� f �

��) � ·�

1

�� f��

d�s��

1

�� f

��· d�s

�

��[�m��

�f��

��m��

�f��

�]. (176)

The three additional terms appear along with the porosity (or volume

fraction) function �m�� as a factor on the right-hand side of each of the

equations. When the dielectric permittivity function is homogeneous in each

of the two phases, then the VAT photonic band-gap equations can be

reduced to one equation in each phase and written in a simpler form,

�(�m�� f�

��) � ·�

1

�� f��

d�s��

1

�� f

��· d�s

�� !

��m

�� f

��

(177)


and

�(�m�� f �

��) � ·�

1

�� f��

d�s��

1

�� f

��· d�s

��!

��m

�� f �

��.

(178)

The equations are almost the same as equations for heat or charge

conductance in a heterogeneous medium. The similarity of the equations

means that the analysis of the simplest band-gap problem should also be

very similar.

Using DMM-DNM, Pereverzev and Ufimtsev [121] found that exact

micromodel solutions among others features can have ‘‘medium . . . internal

generation’’ that might be well characterized by the impact of the additional

terms in the VAT Maxwell equations in both phases and in the combined

electric field and effective coefficient equations; see Sections V and VIII. The

exact closure and direct numerical modeling derived by Travkin and

Kushch [33, 34] demonstrated how important and influential the additional

VAT morphoterms can be (Section I). These terms do not explicitly appear

in either the microscale basic mathematical statements or in microscale field

solutions. The terms appear and become very important when averaged

bulk characteristics are being modeled and calculated.

V. Radiative Heat Transport in Porous and Heterogeneous Media

Radiation transport problems in porous (and heterogeneous) media,

including work by Tien [130], Siegel and Howell [131], Hendricks and

Howell [132], Kumar et al. [133], Singh and Kaviany [134], Tien and

Drolen [135], and Lee et al. [139], are primarily based on governing

equations resulting from the assumption of a homogeneous medium. This

implicitly implies that specific problem features due to heterogeneities can

be decribed using different methods for evaluation of the interim transport

coefficients, as, for example, done by Al-Nimr and Arpaci [136], Kumar and

Tien [137], Lee [138], Lee et al. [139], and Dombrovsky [140]. Although

this kind of approach is legitimate, it presents no fundamental understand-

ing 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


et al. [142], van de Hulst [143], Bohren and Huffman [144], Lorrain and

Corson [145], Lindell et al. [146], and Lakhtakia et al. [147]) approach the

study of transport in disperse media with the emphasis on known scattering

techniques and their improvements.

The area of neutron transport and radiative transport in heterogeneous

medium being developed by Pomraning [148—151] and Malvagi and

Pomraning [152] treats linear transport in a two-phase (two materials)medium with stochastic coefficients. This approach is the same as that which

has been used to treat thermal and electrical conductivity in heterogeneous

media, and to this point it has not been brought to a high enough level to

include variable properties, their nonlinearities, and cross-field (electrical

and thermal or magnetic) phenomena.

Research by Lee et al. [139] on attenuation of electromagnetic and

radiation fields in fibrous media has shown a high extinction rate for

infrared radiation. The problem is treated as a scattering problem for a

single two-layer cylinder by Farone and Querfeld [153], Samaddar [154],

and Bohren and Huffman [144]. The process of radiative heat transport in

porous media is very similar to propagation of electromagnetic waves in

porous media and will also be evaluated. These two very close fields seem

not to have been considered as a coherent area. Complicated problems of

propagation of electromagnetic waves through the fiber gratings have been

primarily the subject of electrodynamics. The most notable work in this area

is that of Pereverzev and Ufimtsev [121], Figotin and Kuchment [122, 125],

Figotin and Godin [124], Botten et al. [155], and McPhedran et al. [156,

157]. No effort seems to have been made to translate results obtained for

polarized electromagnetic radiation to the area of heat radiative transfer.

Detailed micromodeling (DMM) of electromagnetic wave scattering has

been based on single particles or specific arrangements of particulate media.

Direct numerical modeling (DNM) of the problem seems enables one to do

a full analysis of the fields involved. As already discussed, the analysis of the

results of a DNM is limited in the performance of a scaling analysis, which

is the goal in most situations. Performing DNM without a proper scaling

theory is like performing experiments, often very challenging and expensive;

without proper data analysis, it yields a certain amount of detailed field

results, but not the needed bulk or mean media physical characteristics.

Most recent work on radiative transport is based on linearized radiative

transfer equations for porous media. We first review this work to set the

stage for the development that follows. This radiative transport related work

extends our results in the theoretical advancement of fluid mechanics, heat

transport, and electrodynamics in heterogeneous media (Travkin et al. [19];

Catton and Travkin [28, 158]; Travkin and Catton [20, 159—163]; Travkin

et al. [114, 115]) and provides a means for formulation of radiative


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

c

�I��t

� · (�I�) � [,#(r) � ,

(r)]I�

�,#(r)I �� (T ) �

1

4',(r) �

��p(�� ·�)I�(r, ��)d�� (179)

I�� I� (r, �, t),

with ,#(r) the absorption and ,

(r) scattering coefficients, and for steady

state, using the identity

· (�I�) �� · I� ,in the form

� · I� � [,#(r) �,

(r)]I ��,

#(r)I ��(T ) �

1

4',(r) �

��p(�� ·�)I�(r, ��) d��.

(180)

In terms of a spectral source function S�(s), the equation can be written in

a particularly simple form,

1

-�� · I�� I�� S�(s), (181)

where the extinction coefficient (total cross section—Pomraning [150, 151]

is

-��,#(r) �,

(r).

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)

where the quantities -(r), ,(r), and S(r, �) are taken to be two-states

discrete random variables. By assuming this, one needs to treat the porous


(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 -, ,,

and S are assumed to be random variables, Eq. (182) is treated as an

ensemble-averaged equation (see Malvagi and Pomraning [152] and Pom-

raning and Su [164])

� · (p��

�)�p

�-

��

��p

�S

��

,�

4'p�#��

p��

��

�p��

��

, i�1, 2, j�i

(183)

#��

�(r, ��)d��,

where �� is the conditional ensemble averaged function � at some point r

that is in phase i, and ��

and ��

are the interface ensemble-averaged fluxes.

The solution to this equation is also supposed to be ensemble-averaged. The

overall averaging over the both phases is given by

��(r, �)� � p��

�� p

��

�, (184)

where p�

and p�

are the probabilities of point r being in medium i� 1 or 2,

and �� is the conditional ensemble averaged value of �, when r is in

medium i.Ensemble averaging in this representation is obtained by averaging of

medium features, including coefficients, along a straight line the � direc-

tion—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].


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 accom-

plished 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 math-

ematical 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 equa-

tions for both heterogeneous and homogeneous media.

Major developments in the use of VAT, showing the potential for

application to eletrophysical and acoustics phenomena in heterogeneous

media, are found in Travkin and Catton [21], Travkin et al. [159, 114, 115],

and with experimental applications to ferromagnetism in Ryvkina et al.[160, 162] and Ponomarenko et al. [161].

It has been demonstrated during the past 20 years of VAT-based

modeling in the thermal physics and fluid mechanics area (see Slattery [6];

Whitaker [10]; Kaviany [7]; Gray et al. [8]) that the potential of the

approach is enormous. Substantial success has also been achieved in

analyzing the more narrow phenomena of electromagnetic wave propaga-

tion 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 instru-

ments 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.


The same kind of operators and averaging theorems used in preceding

sections are applied to the following development, involving the rot oper-

ator, in which averaging will result because of the following averaging

theorems:

� f�� f�

��

1

�� d�s

� f (185)

f�� f

��

1

�� d�s

� f� . (186)

Rigorous application to linear and nonlinear electrodynamics and electro-

static problems is described in Travkin et al. [114, 115].

The phase averaging the equation for linear local thermal equilibrium

radiative transfer,

� · I�� -�(r)I�� ,#(r)I �� (T ) �

1

4',(r) �

��p(�� ·�)I� (r, ��) d��, (187)

in phase 1 yields the VAT radiative equation (VARE)

� ·� (I��

1

�� I��d

�s��-� ��I

��

� �,#�

(r)I��(T )��

,�

4'�#

��-� ��I� ��

, i� 1 (188)

#��

p(�� ·�)I�(r, ��) d��,

when it is assumed that ,�

is a constant, as done by Malvagi and

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

� �-� ��I� ��(189)

in (188) is the result of fluctuations correlation inside of medium 1 in the

REV, but it is described by

p��

��

�p��

��

(190)

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


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��d

�s��. (191)

When the coefficients in the radiative transfer equation are dependent

functions, more linearized terms are observed in the corresponding VARE,

� · �I��-� ��I

��

� �,�#�

I� ��(T )��,�

#�I� ��

�1

4'(�,�

�#��,�

�#��)

�� ·�1

�� I��d

�s��-� ��I

��

, i � 1, (192)

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

�s��(�c

�)�

�T��

�t� k

� �(�s

��T�

�) � k

� ·�

1

�� T�d�s

��

k�

�� T

�· d�s

�� ·�q��

��

1

�� q� · d�s

�. (193)

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

· �q�� ·�

4

3- (I

�)

�

� ·�4

3- �

n��T �

' ��

, (194)

where - is the total extinction coefficient. An energy equation similar to Eq.

(193) needs to be written for the fluid-filled volume, phase 1 of the porous

medium. The radiation flux term would be much more complex because of

the spectral characteristics of radiation in a fluid.

Closure is needed for the second, third, and fifth terms in Eq. (193) on the

r.h.s. For convective heat exchange, the last term can be written

k�

�� T�x

�

· d�s��

��S��

(T �� T

�) (195)


by noting that

1

�� k�

�T�x

�

· d�s��

1

�� k�

�T�n

�

ds · n�

�1

�� q�· d�s

��

��S��

(T �� T

�). (196)

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

�� q� · d�s

��

1

�� (T �

�� T �

��)

�1

��

�1

��

� 1�� · d�s�

��((T�

��)� � (T�

��)�)S

��

�1

��

�1

��

� 1� � , (197)

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��

� n��T ��

to close the

integral energy exchange term as follows:

1

�� q� · d�s

��

1

�� (n��T �

��)d�s

��

��(T�

��)�S

��. (198)

Here, ��

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,

��

� ��, (199)


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 HarmonicField Equations

Representing the electromagnetic field components with time-harmonic

components results in

· (��E) ��, · (�

�H) � 0 (200)

E� �i��H, H� i��!

�E. (201)

Here, as outlined earlier, �!�

is the complex dielectric function �!��

��i(�

�/�), and �

��

�(x�), �

��

�(x� ), �

��

�(x� ,�), �!

�� !

�(x� ,�). In many

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

wave formulation of the Maxwell equations with constant phase coefficients

for the magnetic field is

�H� ��

�H�t

��

��H�t�

� 0, (202)

whereas the electric field wave equation is almost the same,

�E��

�E�t

� ��

��E�t�

� ��. (203)

Another form of the equation for E appears in Cartesian coordinates

when electromagnetic fields are time-harmonic functions:

�E� k�E� 0, (204)

Here, the inhomogeneous function k� ��

is the wave number

squared. This equation is often applicable to linear acoustics phenomena.

This category of equations can be transformed to a form legitimate for

application to heterogeneous media problems.

The time-harmonic forms of equations for rot of electromagnetic fields are

(�m��E�

�) �

1

�� d�s

� E

�� i�[�m

��

��H�

��m

��

��H�

��]

(205)

(�m��H�

�) �

1

�� d�s

� H

�� i�[�m

��!�

��E�

��m

��!�

��E��].

(206)


The magnetic field wave form equation with constant coefficients, when

averaged over phase 1, transforms to

�(�m��H�

�) � · �

1

�� H

�d�s

��1

�� H

�· d�s

�

��

��H��

�t��

��

��H��

�t�, (207)

and the electric field wave equation (203) becomes

�(�m��E�

�)� ·�

1

�� E�d�s

��1

�� E� · d�s

�

� ��

��E��

�t��

��

��E��

�t��

1

��

(�m��

�) �

1

��

��d�s

�.

(208)

An analogous form of the averaged equation is obtained for the time-

harmonic electrical field:

�(�m��E�

�) � · �

1

�� E

�d�s

��1

�� E

�· d�s

��m

��k�E�

�� 0.

(209)

It is the naturally appearing feature of the heterogeneous medium elec-

trodynamics 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

averaged electrical field E��, one can seek the Stokes parameters I, Q, U,

and V,

I� �E�"�

E� *"�� E�

��E� *��

(210)

Q� �E�"�

E� *"�� E�

��E� *��

(211)

U�Re[�2E�"�

E� *��] (212)

V � Im[�2E�"�

E� *��], (213)

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


temperature fields; still, the same questions of multiple versions, applicability

of current methods, and variance in interpretation are the present agenda.

VAT-based models were developed recently while addressing the prob-

lems of modeling of electrodynamic properties of a liquid-impregnated

porous ferrite medium (Ponomarenko et al. [161]), coupled electrostatic-

diffusion processes in composites (Travkin et al. [159]), and to analyze heat

conductivity experimental data in high-T�

superconductors (Travkin and

Catton [166]). Powders of ferrites with NFMR frequency in the microwave

range were used as the porous magnetic medium in Ponomarenko et al.[161]. The search for tunable levels of reflection and absorption of elec-

tromagnetic waves was conducted using a few morphologies that were

arbitrarily chosen. Thus, the need for closer consideration of experiment and

models presenting the data using VAT heterogeneous description tools for

both became obvious.

VI. Flow Resistance Experiments and VAT-Based

Data Reduction in Porous Media

It is well known that existing measurements of transport coefficients in

porous (and heterogeneous) media must be used with care. As long as a

complete description of an experiment is provided and the data analysis is

carried out using correct mathematical formulations (models), the relation-

ship 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 re-

duced 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 electri-

cal conductivity and permittivity derived from homogeneous models, prob-

lems 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


and compared. VAT-based analysis is shown to reveal the influence of

morphological characteristics of the medium; to suggest scaling parameters

that allow a wide variety of different porous medium morphologies to be

normalized, often eliminating the need for further experimental efforts; and

to clarify the relationships between differing experimental configurations.

The origin, and insufficiency, of electrical conductivity and momentum

transport ‘‘cross-correlation’’ approaches based on analogies using math-

ematical models without examining the physical foundation of the phenom-

ena will be described and explained.

1. Experimental Assessment of Flow Resistance in Porous Medium

A one-term flow resistance model for porous medium experimental data

analysis often used is

�dp!�

dx� f �

S�

�m��u!� �2

, (214)

where f is some coefficient of hydraulic resistance. On the other hand, most

two-term models used for flow resistance experimental data reduction have

first-order and second-order velocity terms, the Darcy—Forchheimer flow

resistance models. These models were obtained primarily for direct compari-

son with established empirical and semiempirical Darcy and Darcy—For-

chheimer 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! �

dx�

�k�

�m� u!� ��A�m��u!� �. (215)

Similarly, the model given by Vafai and Kim [168] for the middle part of

a porous layer is

�dp!�

dx�

�k�

�m�u!� ��m�

F

k� ��

u!� �, (216)

and the Poulikakos and Renken [169] equation for the turbulent regime is

�dp!�

dx�

�k�

u!� ��Au!� �. (217)

Analysis of a simple idealized morphology where solutions are known will

show that the Darcy and Darcy—Forchheimer or Ergun type model corre-

lations are not matched consistently for any regime. Further, they are also

without theoretical foundation. Thus, problems arise when studies to


improve the description of transport use combined models for flow resis-

tance 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]):

f��

8��

(��U � �)

, ��

�d��p

4L, u�

*�

��

�f�U � �8

(218)

�p

L� �

dp!�

dx�

4

d�

��u�*�

f�

d�

��U � �2

. (219)

The morphology function S�/�m� for a straight equal-diameter tube mor-

phology is

S��

�S�

��

2'R

p�y, �m��

'R�

p�y,

S�

�m��

4

d�

(220)

and an exact expression for the Darcy friction factor is

�p

L� f

�

��

d�

U � �2

, f�

�2d

��U � �

�p

L. (221)

The Fanning friction factor for this specific morphology is (using (220))

�p

L�

f�4 �

4

d��U � �2

� �f�4 ��

S�

�m��U � �2

(222)

f��

d�

2��U � �

�p

L, (223)

and a relationship to the Darcy friction coefficient is (Travkin and Catton

[16, 20])

f��

f�4

. (224)

The friction coefficient c�

for smooth tubes often calculated using the

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

Lh� c�

�

��U � �2

, c��

2��

��U � �

�2u�

*U � �

�2h

��U � �

�p

L, u�

*�

h

��

�p

L.

(225)


The morphology ratio S�/�m� for a porous medium morphology model of

straight equal slits is found as follows:

S��

(2L �y)

( pL �y)�

2

p, �m��

(HL �y)

( pL�y)�

H

p(226)

S�

�m��

2

H�

1

h, d

�� 4h, (227)

yielding the Fanning friction factor,

�dp!�

dx�

�p

L� f

� �1

h��U � �2 �� f

� �S�

�m��U � �2 � (228)

f��

H

��U � �

�p

L(229)

As one can easily see, these flow resistance models are written with the

second power of bulk velocity variable. The convergency of the VAT-based

flow resistance transport models to these classical constructions was dem-

onstrated on several occasions by Travkin and Catton [16, 20, 21, 23] and

Travkin et al. [25].

Exact flow resistance results obtained on the basis of VAT governing

equations by Travkin and Catton [16, 26, 23] for the random pore diameter

distribution for almost the same morphology as was used by Achdou and

Avellaneda [171] demonstrated the wide departure from the Darcy-law-

based treatments. That was shown even for the morphology where a single

pore exists with diameter different from the all others. Meanwhile, by

consistently using the VAT-based procedures (Travkin and Catton [23]),one can easily develop the needed variable, nonlinear permeability coeffi-

cient for Darcy dependency,

k��

� �c� �S�

�m��U �

2��

, (230)

where c�� f

�is derived for this particular morphology using exact analyti-

cal (in the laminar regime) or well-established correlations for the Fanning

friction factor in tubes.

2. Momentum Resistance in 1D Membrane and Porous Layer Transport

The steady-state VAT-based governing equations for laminar transport in


porous media (Travkin and Catton [21]) are

�m�U��U��x

��x

�u� u� ��

1

��

��x

(�m�p�)

��1

��

�

pd�s� ��x �

��m�U��x ��

��

�

�U�x

�

· d�s (231)

and

c��m�U�

�T��

�x� k

�

��x �

��m�T��

�x �� c��

��x

(�m��T��u�

�)

��x �

k�

��

T�d�s��

1

��

k�

�T�

�x�

· d�s (232)

��x �

��s�T

�x ��x �

1

��

Td�s

��1

��

�T

�x�

· d�s�� 0. (233)

The momentum equation for turbulent flow of an incompressible fluid in

porous media based on K-theory can be written in the form (Gratton et al.[26], Travkin and Catton [20])

�m� ��U �

�t�U �

�U �

�x ��1

��

(K�� )

�U �x

�

· d�s��x ��m�(K�

�� )

�U �

�x��

��x ��m� �K� �

�u!�

�x��

��x

(�m��u!� u!� �)

��1

��

�

p!d�s��1

��

��x

(�m�p!� ). (234)

By comparing these equations with conventional mathematical models and

experimental correlations, one can easily see the differences.

The one-dimensional momentum equation for a homogeneous, regular

porous medium is

��x

p��

1

�m��

p�d�s��

�m��

V · d�s. (235)

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

��

�!

�U�x

�

· d�s ��

��

�

��!

· d�s��1

2c�!

(x� )S�!

(x� )[��U� �(x�)],

(236)


with

��!

� ��U�x

�

, u��

��c�!

U� �(x�),

and closure of the form drag resistance integral term using a form drag

coefficient, c��

, is

1

��

pd�s�1

2c��

S��

(x�)[��U� �(x�)]. (237)

For these equations, the specific surface has two parts. The first part, S�!

, is

S�!

(x�) �1

�� !

ds, �1

m�, (238)

where �S�!

is the laminar subregion of the interface surface element �S�,

and

S��

(x� ) �1

��

ds �S�

��, �

1

m�, (239)

where �S��

is the cross flow projected area of the surface of the solid phase

inside the REV. Substitution into the one-dimensional momentum equation

yields

��x

p�� (c

�!(x)S

�!(x) � c

��S��

(x))��U� �(x)

2�m��

p��m�

( �m�). (240)

When the porosity is constant, the flow is laminar and S�!

�S�, the

equation becomes

�dp�dx

��c�� c��

S��

S��

S�

�m��U� �2

� c�(U � , M�) �

S�

�m��U� �2

, (241)

where c�

is the friction factor and c��

the form drag, S��

is the cross flow

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

c�(U � , M�) ��c�� c

��

S��

S��. (242)

Correlations for drag resistance can be evaluated for a homogeneous

porous medium from experimental relationships for pressure drop. For


example, the equation often used for packed beds is

�dp!�

dx� f

� �S�

�m��U � �2

. (243)

The complete VAT version of this equation is

��x

(�m�p� ) �1

��

pd�s ��u��

S�(x)

��m��U�

�U��x

��x �

��m�U��x ��

��x

[��m��u� u�

�]. (244)

If the porosity function is constant (a frequent assumption), the left-hand

side of Eq. (244) reduces to

�dp�dx

� f� �

S�

�m��U� �2

. (245)

Setting Eq. (245) equal to zero recovers equation (243). As a result, data

correlation using Eq. (243) incorporates the right-hand side of Eq. (244)implicitly into the correlation. Friction factor data presented in this way

detracts from objectivity. The correlation can be written to reflect all the

right-hand terms from Eq. (244),

�d(�m�p� )

dx��c�� c

��

S��

S�

�F��F

��F

� (S�(x))

��U� �2

, (246)

where F�, . . . , F

are deduced from the following relationship:

(F��F

��F

) �S� (x)

��U� �2 ��m��

�U�

�U��x

� ��

��x

[�m�u� u� �]

��x �

��m�U��x �. (247)

In the middle part of a porous medium sample, one can assume that the

porosity and flow regime are constant and steady state and then neglect all

terms on the right-hand side of (244). In reality, a large number of

experiments are being carried out under conditions where input—output

zones are present and can add significantly to the value of the friction

coefficient because of the input—output pressure losses. If one wants to

separate the effects of input—output pressure loss from the viscous friction

and drag resistance components inside the porous medium, then taking into

account the terms shown in Eq. (247) is essential. There are correlations that

reflect a dependence on sample thickness as a result of this oversight. An


even more complex situation arises when the flow and temperature inside

the medium are transient, such as one might find in a regenerator, and very

inhomogeneous in space because of sharp gradients. The inhomogeneity in

space and time precludes neglecting the four right-hand terms in Eq. (244).The inhomogeneous terms on the right-hand side of (247) may be

analyzed by scaling. Some of these terms are easily interpreted. For example,

the first term on the right-hand side is the convective term

�S�(x)��U � �2 � F

��m��

�U �

�U �

�x, (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 � �2 � F

��

��x

(��m�u!� u!�

�) (249)

and the momentum diffusion term

�S� (x)��U � �2 � F

� ��

��x �

��m�U �

�x �. (250)

The complete momentum equation written in a proper form for experi-

mental data reduction is

�d(�m�p� )

dx��c�� c

��

S��

S�

�F��F

��F

� (S�(x))

��U� �2

� (c��R

()(S

�(x))

��U� �2

, (251)

where

c�� c

�� c

��

S��

S�

(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�%�

� c��R

(. (254)


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


��1

��

��x

(�m� p�) � �

��x �

��m�U��x �� c

�(x)

S�(x)

�m��U� �2

(255)

or


��1

��

��x

(�m� p�) � �

��x �

��m�U��x �

��k�

�m�U� � ��m�

F

k� ��

U� �, (256)

as is done by many, then the fluctuation term �[�m��u� u� �]/�x is

neglected and the equation

2�m�U��U��x

��1

��

��x

(�m� p�) � 2�

��x �

��m�U��x �

�1

��

pd�s ��

�

�U�x

�

· d�s ��x

[�m��u� u� �] (257)

is being used as the problem’s model instead of


��1

��

��x

(�m� p�) � �

��x �

��m�U��x �

�1

��

�

pd�s ��

�

�U�x

�

· d�s ��x

[�m��u� u� �] (258)

because the model used the coefficient c�(x) determined from

c�(x)

S�(x)

�m(x)��U� �2

� (c�(x) �R

((x))

S�(x)

�m(x)��U� �2

� �c��c��

S��

S�

�F��F

��F

�S�(x)

�m(x)��U� �2

�1

��

�

pd�s��

�

�U�x

�

· d�s��x

[�m��u� u� �]

� ��x �

��m�U��x �� m�U�

�U��x

, (259)


instead of using the coefficient c�(x) determined from

c�(x)

S�(x)

�m(x)��U� �2

��c�� c��

S��

S�

�F��

S�(x)

�m(x)��U� �2

�1

��

�

pd�s ��

�

�U�x

�

· d�s��x

[�m�u� u� �].

(260)

The terms needed for experimental data reduction model should include

all five active terms,

�d(�m�p� )

dx� �c�� c

��

S��

S�

�F��F

�� F

� (S�(x))

��U� �2

� (c�� R

()(S

�(x))

��U� �2

, (261)

with

f�%�

� c��R

(. (262)

The general 1D VAT laminar regime constant viscosity momentum equa-

tion has six terms,


��x

�u� u� ��

1

��

��x

(�m�p�)

��1

��

�

pd�s � ��x �

��m�U��x ��

��

�

�U�x

�

· d�s. (263)

For simplicity, Eq. (263) is written in the following shorthand notation:

UC��UMC

��UP

��UMP

��UD

��UMF

�. (264)

The two right-hand integral terms reflect the morphology-induced flow

resistance of the medium. Three flow resistance models are needed to

properly tie everything together.

a. Flow Resistance Model 1 The first flow resistance model is for the

internal frictional and form drag resistance:

�c��

(U � , M�, x) �S�(x)U� �(x)

2 �� (�c��

S��

(x) � c�!

(x)S�!

(x))U� �(x)

2

��1

��

�

pd�s��

�

�U�x

�

· d�s.

(265)


b. Flow Resistance Model 2 The second flow resistance model reflects the

addition of the fluid fluctuation term UMC�:

�c��

(U � , M� , x) �S�(x)U� �(x)

2 ��c�� c

��

S��

S�

�F��

S�U� �2 �

� (�c��

S��

(x) � c�!

(x)S�!

(x))U� �(x)

2��S�! (x)

U� �2 � F

�

��1

��

pd�s��

�

�U�x

�

· d�s��x

�u� u� ��.

(266)

c. Flow Resistance Model 3 The third flow resistance model reflects all of

the terms responsible for momentum resistance in a porous medium:

c�

(U � , M� , x) �S�(x)U� �(x)

2 ��S� (x)

U� �2 � R

(� c

��(U � , M�) �

S�U� �2 �

��m�U��U��x

� ��x �

��m�U��x �

�1

��

�

pd�s��

�

�U�x

�

· ds��x

�u� u� ��, (267)

where

�s� (x)U� �2 � R

(� (F

��F

��F

) �S�(x)

U� �2 �

��m�U��U��x

��x

[�m�u� u� �]� �

��x �

��m�U��x �. (268)

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�� UMF

�)��

S�(x)U� �(x)

2 � (269)


c�

(U � , M� , x) � (UMP�� UMF

��UMC

�)��

S�(x)U� �(x)

2 � (270)

c��

(U � , M� , x)

� (UC��UD

��UMP

��UMF

�� UMC

�)��

S�(x)U� �(x)

2 �. (271)

Each of the different forms will yield a correlation of a given set of data.

The problem is that the effects of the different characteristics that are

manifested in the terms in the equations are lost from consideration. If

predictive tools are to be developed, consideration must be given to the

impact of the details that the terms reflect.

3. Scaling in Pressure L oss Experiments and Data Analysis

Direct use of any Ergun type friction factor in a Fanning or Darcy friction

factor correlation is incorrect. Ergun [167] suggested two types of friction

factors, one of which is the so-called kinetic energy friction factor f��

, which

differs from the Fanning friction factor by a factor of three for the same

medium:

f��

d�

2��u!� � �

�P

L ��f��3

. (272)

For the same reason, direct implementation of the correlations given by

Kays and London [172] should be treated with care. For example, the

correlations for friction factor (Fanning) given by Kays and London for

flow through an infinite randomly stacked, woven-screen matrix uses surface

porosity p, and specific surface �[1/m] to define a hydraulic radius r�,

r��

p

��

�m�

S�

.

Here the specific surface S�

is defined as the interface surface divided by the

volume of the REV. Unfortunately, the surface porosity �m�and volume

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��

.

This assumption led them to the formula

r��

��m�d

�6(1 ��m�)

. (273)


It would be double this value if a consistent definition were used for all

systems,

d��

4�m�S�

�4�m�

a�(1 ��m�)

�2�m�

3(1 ��m�)d�� 4r

��, (274)

where a�

is the ‘‘particle specific surface’’ (the total particle surface area

divided by the volume of the particle), and

S�� a

�(1 ��m�). (275)

The expression given by (274) is justified when an equal or mean particle

diameter is

d��

6

a�

,

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�,

and d�

and can generally be considered to be spherical particles with

S��

6(1 ��m�)

d�

, d��

2

3

�m�(1 � �m�)

d�. (276)

This expression has the same dependency on equivalent pore diameter as

found for a one-diameter capillary morphology, leading naturally to

S��

6(1 � �m�)

d�

�6(1 ��m�)

�3

2

(1 � �m�)

�m�d��

�4�m�

d�

. (277)

This observation leads to defining a simple ‘‘universal’’ porous medium

scale,

d�� d

�%��

4�m�S�

, (278)

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 poros-

ity �m�, specific surface S�, and pore (particle) diameter d

�can lead to large

errors.


The particle diameter d�

is often used as a length scale when reducing

experimental data. Chhabra [174], for example, writes the friction factor

f��

�d�

�m��u!� �

�p

L, (279)

This friction factor can be related to the friction factor f�, given by Eq.

(6.4-1) of Bird et al. [173], to the Fanning friction fator f�, and to the Ergun

kinetic energy friction factor f��

as follows:

f��

� 2 f�� f

�� 1��m��m� �� f

�3 �

1� �m��m� �. (280)

These models all use different length scales, leading to large uncertainties

and confusion when a correlation must be selected for a particular applica-

tion. Little attention is paid to these differences, often requiring new

experimental data for a new medium configuration.

Only a few of the many issues important to modeling of pressure loss in

porous media are addressed here. As it is known, the two-term quadratic

Reynolds—Forchheimer pressure loss equation is

�P

L� ��U � �m�� -�

�U � ��m��; ��

1

k�

. (281)

By comparison with the simplified VAT (SVAT) momentum equation for

constant morphological characteristics and flow field properties and only

the resistance coefficient c�,

�P

L� c

� �S�

�m��U � �2

, (282)

a set of transfer relationships can be found to transform Ergun-type

correlations and the SVAT expression. The transfer formula (Travkin and

Catton [21]) is

c�� f

��

��

��U �

� -�m��2�m��

S��, (283)

where

�� 1501� �m�)�

d��m�

, -� 1.75(1 ��m�)

d��m�

, (284)

or

c�� f

��

A

Re�%�

�B, A �8��m�

S��

, B� 2-�m�

S�

, (285)


where

Re�%�

�4U � �m��S

�

.

The Ergun energy friction factor relation can be written in terms of the

VAT-based formulae (Travkin and Catton [21]) as

�p

L� f

�� S�

�m��U � �2

. (286)

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

�p

L��150

(1 ��m�)�

d��m� � ��m�U � ��1.75

(1 ��m�)

d��m� � �

��m��U � �, (287)

and if it can be further transformed to the (SVAT) Fanning friction factor,

then

f��

�A*�

Re�

�B*�, A*

��

50(1 � �m�)

�m� �, B*�

�3.5

6� 0.583, (288)

where the particle Reynolds number is

Re�� (U � d

�)/�, (289)

and

f��

�A*��

Re�%�

�B*��

, with A*��

�100

3� 33.33, and B*

��B*

�� 0.583,

(290)where

Re�%�

�U � d

��

�2

3

�m�(1 ��m�)

U � d�

�, and Re

�%� �3(1 ��m�)

2�m� ��Re��

U � d�

�.

(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 correla-

tion is used with arbitrary relationships between porosity �m�, specific

surface S�, and pore (particle) diameter d

�when it was originally developed

for granular media. How unsatisfactory it can be is shown in Fig. 5.


Fig. 5. Fanning friction factor f�

(bulk flow resistance in SVAT for different medium

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

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 momen-

tum equation in the form

�� 6.61 · 10�(d �)��m��, (292)

-� 5.16 · 10�(d �)��m��, (293)


where hydraulic diameter d �(mm) is

d ��

d�[m]

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

f�(Re

�) �

40

Re�

(1 � 2.5 · 10��m��Re�), �m�� 0.83) 0.92, (295)

where

Re��

U � d��m��

.

To transform this correlation, the Reynolds number must be transformed

and the result divided by 4 to yield the Fanning friction factor,

f�(Re

�%�) �

1

4 �40

Re�%�

�m�(1 � 2.5 · 10��m��Re

�%��m�)�, (296)

with

Re�%�

� Re�/�m� (297)

The correlation derived by Beavers and Sparrow [177] seems to be of

little value in the original form,

F�

(R�) �

1

R�

� 0.074, (298)

because the Reynolds number,

R��

U � �m��k�

�, (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

F�

�1

R�

� -�k�, (300)

where


�k��

1

��, R

��

U � �m�

��(301)

Re�%�

�R�

4

S��k

�

�R��

4

S��, R

��Re

�%� �S�

4��, (302)

yields

F�

�1

�U � �m��k

��

�-�k�

or

F�

(R�) �

1

��U � ��m��

�P

�x�, (303)

and when compared to

f�(Re

�%�) �

2�m�

��U � �S

��P

�x�, (304)

one obtains

f�(R

�) � �

1

��m��1

��U � � �

�P

�x� ·��m��(2�m�)

S�

�� F

�(R

�) �

2��m�

S�

�. (305)

This means that the Fanning friction factor, f�, can be assessed from the

friction factor suggested by Ward [178] and Beavers and Sparrow [177],

f�

, from

f�(R

�) �F

�(R

�) �

2��m�

S�

�. (306)

To accomplish the transformation of F�

to f�, the permeability k

�or the

viscous coefficient of resistance � porosity �m� and specific surface S�

must

be known. Estimates of f�

were obtained from measured values of F�

for

FOAMETAL (Beavers and Sparrow sample Type C) using

k�

� 19.01 · 10�� [cm�]� 19.01 · 10�� [m�]

��1

k�

� 0.0526 · 10��1

m�� (307)


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

F�

(R�) �

1

R�

� 0.074 and R�

�Re�%� �

S�

4��F�

(Re�%�

) �1

Re�%��4��S�� 0.074, (308)

then

f�(Re

�%�) ��

1

Re�%��4��S�� 0.074��

2��m�

S�

�. (309)

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

L� �

��m�U � , �

��

1

k�

, U � �d��

32� ��P

�x�. (310)

(b) A characteristic length d�

evaluated by equating the preceding ex-

pressions:

d��

32

��m��

� ��

32k�

�m��

. (311)

(This is only justified for straight parallel capillary morphology where

d�� d

�.)

(c) Critical numbers Re��

and Re��

to distinguish the viscous, transi-

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

L� f

�(Re

�%�) �

S�

�m��U � �2

, (312)


the Fanning friction factor correlations can be easily compared with one

another as they have a common consistent basis. A number of correlations

were transformed and are in Fig. 5. The reason for the spread in the results

is thought to be inadequate accounting for details of the medium.

Analysis of Macdonald et al. [180] reformulated with the help of the

foregoing developed procedures gives the corrected Ergun-like type of

correlation

f�(

�40

Re�%�

� 0.6. (313)

Meanwhile, Souto and Moyne [181], using the DMM-DNM solutions,

came to the number of resistance curves that are separate for each morphol-

ogy. One of them for rectangular rods in VAT terms appears as

f��(

�1

3f��

�54.3

Re�%�

, Re�%�

* 0. (314)

VII. Experimental Measurements and Analysis of Internal

Heat Transfer Coefficients in Porous Media

A VAT-based approach applied to heat transfer in a porous medium

allows one to analyze and measure effective internal heat transfer coefficients

in a porous medium. As noted by Viskanta [182], ‘‘Convective heat and

mass transfer in consolidated porous materials has received practically no

theoretical research attention. This is partially due to the complexity which

arises as a result of physical and chemical heterogeneity that is difficult to

characterize with the limited amount of data that can be obtained through

experiments.’’ Viskanta [182, 183] generalized the data he analyzed for

internal heat transfer coefficient porous ceramic media using a correlation

of the form

Nu�� 2.0� a Re�Pr� , (315)

by assuming that the limiting Nusselt number should be 2.0 when the Redecreases 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 re-

searches 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.


1. Experimental Assessment and Modeling of Heat Exchangein Porous Media

The correct form of the steady-state heat transfer equation in the fluid

phase of a porous media with primarily convective 1D averaged heat

transfer is

c��m�U�

�T��

�x�

k�

��

�T�

�x�

· d�s� k�

��m�T��

�x�

� c��

��x

(�m��u� T��) �

��x �

k�

��

T�d�s�. (316)

Equation (316) can be rewritten as

��S�

(T � T

�) � ��k�

��x �

��m�T��x �� c

��

��x

(�m��u� T� �)

��x �

k�

��

T d�s�� c��m�U�

�T��

�x, (317)

where

k�

��

�T�

�x�

· d�s� ��S�(T

� T

�).

The right-hand side of Eq. (316) can also be written in the form

��S�

(T � T

�) �

��x �K��&

�T��

�x �� S�(T

� T

�) �

��x

[�q��

],

(318)

where the right-hand side (‘‘diffusive’’-like) flux contains more terms than

are conventionally considered:

q��

��K��&

�T��

�x ��m�k

�

�T��

�x� c

��m��u� T�

��

k�

��

T�d�s�. (319)

The corresponding equation for the solid phase is

��x �

��s�T

�x ��x �

1

��

Td�s

��1

��

�T

�x�

· d�s�� 0. (320)

The three terms are written in the following shorthand form:

TD

��T

MD

�� T

ME

�� 0. (321)


Equation (320) can also be written

0��1

k� ��

�S�(T

�� T

) �

��x �k��

�T

�x ��

�S�(T

�� T

) �

��x

[�q��

]. (322)

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

equal),

��S�(T

�� T

) �

k

��

�T

�x�

· d�s�.

Equation (322) has a term that is usually overlooked (the second term on

the right):

q��

��K��

�T

�x ��s�T

�x

�1

��

Td�s

��. (323)

Three heat transfer coefficient models are needed to properly tie every-

thing together. The first model incorporates only the heat transfer coefficient

between the phases.

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

��

��k�

��

�T�

�x�

· d�s��[S�(T

� T

�)], (324)

then the heat transfer equation becomes

c��m�U�

�T��

�x� k

�

��x �

��m�T��

�x ��

S�(T

� T

�), (325)

and when the porosity is constant, the equation becomes

c��U�

�T��

�x� k

�

��x �

�T��

�x ��

S�(T

� T

�)/�m�. (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 experi-

ment. There will be no generality in the results. By redefining ��

, further

medium characteristics can be incorporated into the correlation. The second

model incorporates velocity and temperature fluctuations.


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,

��

��k�

��

�T�

�x�

· d�s � c��

��x

(�m��u� T��))/[S

�(T

� T

��)],

(327)

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

first,

c��m�U�

�T��

�x� k

�

��x �

��m�T��

�x ��

S�(T

� T

��). (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

��

�

�k�

��

�T�

�x�

· d�s� c��

��x

(�m��u� T��) �

��x �

k�

��

T�d�s��

S�(T

� T

�

(329)

The energy equation is again very similar:

c��m�U�

�T��

�x� k

�

��x �

��m�T��

�x ��

S�(T

� T

��). (330)

Each of the models reflects the data obtained for a given medium. Only the

coefficient ��

, however, allows for a complete representation of the par-

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 mathe-

matical model,

c��m�U�

�T��

�x� c

��

��x

(�m�u� T� �)

� k�

��x �

��m�T��x

��x �

k�

��

T�d�s��

�S�(T

� T

�), (331)


can easily be seen by comparison with the definition of ��

. The additional

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

� T

��)], (332)

�k�

��

�T�

�x�

· d�s��[S�(T

� T

�)],

��

� (T�ME

��T

�MC

�)/[S

�(T

� T

��)], (333)

�k�

��

�T�

�x�

· d�s� c��

��x

(�m��u� T��)��[S

�(T

� T

��)],

��

� (T�ME

��T

�MC

��T

�MD

�)/[S

�(T

� T

�)], (334)

�k�

��

�T�

�x�

· d�s� c��

��x

(�m��u� T��) �

��x �

k�

��

T�d�s��

S�(T

� T

�)

.

Substitution of either of the preceding effective coefficients into the

equation

c��m�U�

�T��

�x� k

�

��x �

��m�T��

�x �� c��

��x

(�m��T��u��)

��x �

k�

��

T�d�s��

1

��

k�

�T�

�x�

· d�s, (335)

T�C

�� T

�D

�� T

�MC

�� T

�MC

�� T

�ME

�,

would result having different models for experimental data reduction and

even for experimental setup.

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��

� c��

T��

)

S��

�(T

�T�

�)

, (336)


which accounts for the heat exchange when T��

and T��

are the tempera-

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

M��U � S

��[kg/s]. This definition of heat transfer coefficient corresponds

to the continuum mathematical model of heat exchange in the porous

medium formulated as

�m�(�c�)�U�� T�

��

��+�S�(T

�T�

�), (337)

instead of the correct equation,

�m�(�c�)�U�� T�

�� (�c

�)� ·��T�

�u�� k

� (�m�T�

�)

� k� · �

1

��

T�d�s��

k�

��

T�· d�s. (338)

The last term can be modeled using the heat transfer coefficient given by

k�

��

�T�x

�

· d�s��

�S�(T

�� T

�), (339)

which results from the closure relationship

1

��

k�

�T�x

�

· d�s��

1

��

k�

�T�n

�

ds · n�

�1

��

q�· d�s

��

�S�(T

�� T

�). (340)

Kar and Dybbs measured the temperatures Tand T

�and treated them as

if they were the mean (averaged) temperatures. As a result, they measured

yet another heat transfer coefficient, ��

, that is defined by

��

S�(T

�T�

�) � ��

��+�S�(T

� T�

�)

� (�c�)� · ��T�

�u�� k

� (�m�T�

�)

� k� · �

1

��

T�d�s��

k�

��

T�· d�s. (341)

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 ��

,

��

S�

(T � T�

�) � ��

��+�S�(T

� T�

�)

� (�c�)� ��T�

�u��

k�

��

T�· d�s. (342)


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 ��

.

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 compari-

son of internal porous media heat transfer characteristics. The same scaling

VAT approach used for flow resistance in porous media is used for heat

transfer.


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

Nu��

h�d�

��

� 0.004Re��

Pr� , (343)

where both Nu and Re are based on the mean pore diameter. If a single

hydraulic diameter d�is

d�� d

�%��

4�m�S�

, (344)

then

Re��Re

�%��

4U � �m��S

�

(345)

Nu�%�

(Re�%�

) �h�d�%�

��

�Nu�(Re

�, �m�, S

�) � 0.004Re��

�%�Pr� . (346)

This correlation is shown in Fig. 6. The correlation developed by Rajkumar

[185] for hollow ceramic spheres is

Nu��

h�d�

��

� 1.1 �Re�Pr

d�

L ��

, (347)

with d�� 2.5—3.5 [10�m], 18&Re

�& 980, �m�� 0.38—0.39, Pr� 0.71,

and

Re��

u!� d�

�.

The particle Reynolds number Re�

can be rewritten using

Re��Re

�%��3(1 ��m�)

2�m� �. (348)

Nu�

needs to be transformed to Nu�%�

by relating the particle diameter d�

to

the hydraulic diameter. The result is

Nu�%�

�h�d�

��

� Nu��

2�m�3(1 ��m�)

Nu�(Re

�) �

2�m�3(1 ��m�)

Nu�(x),

(349)

where

x ��3(1 ��m�)

2�m� � Re�%�

.


Then

Nu�%�

�2�m�

3(1 � �m�)Nu

�(x, Pr, d

�, L )

�2�m�

3(1 � �m�)Nu

� ��3(1 ��m�)

2�m� � Re�%�

, Pr, d�, L �. (350)

Achenbach [186] developed the correlation

Nu��(1.18 Re��

�)� ��0.23 �

Re�

�m��

��

��

, (351)

for Pr� 0.71, �m�� 0.387, and 1& (Re�/�m�) & 7.7 10�. The Reynolds

number used by Achenbach is based on hydraulics and

Re�� Re

�%��m�,

and his definition of Nu�is

Nu�%�

(Re�%�

) �Nu�(Re

�%��m�). (352)

A correlation developed for cellular consolidated ceramics by Younis and

Viskanta [187, 188] is

Nu��

�h��d��

��

� �0.0098� 0.11 �d�

L �� Re��

Pr� , (353)

where �m�� 0.83—0.87. The correlation yields an increasing Nu��

when the

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

Nu�%�

�Nu

��(Re

�%��m�)

4�m�. (354)

Viskanta [183] presents a correlation from a study of low porosity media,

0.167&�m�& 0.354, by Galitseysky and Moshaev [189]:

Nu��

�A�m�� (1 ��m�) � Re�Pr. (355)

The coefficient, A given by Viscanta [183] is

A ��37.2 �d�

L �� 0.59� (�m�(1 � �m�))��, (356)

for 0.15& d�/L & 0.23, 10&Re

�& 530, Pr � 0.71. The volumetric Nusselt

number is transformed to the surficial Nusselt number with Eq. (354).


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

�:

Nu��

h�d�

��

� [0.14(4c�Re

�)� � Pr� ]. (357)

Transforming their expression to the general form of the media Nusselt

number yields

Nu�%�

�2�m�

3(1 ��m�)Nu

�(Re

�%��m�). (358)

The heat transfer coefficient given in the Heat Exchanger Design Hand-book [191] is based on a single sphere heat transfer coefficient for the porous

medium,

h��

��

d�

( f�Nu), Nu

� 2� (Nu�

"�Nu�

�)� �, (359)

where

Nu"� 0.664Re� �

�Pr�

Nu��

(0.037Re��

Pr)

(1 � 2.443Re��

(Pr� � 1)),

for 1&Re�& 10�, 0.6&Pr& 10�, and the form coefficient for 0.26

&�m�& 1.0 is

f�� 1 � 1.5(1 ��m�).

Transformation of the Nusselt number yields

Nu�%�

�2�m�

3(1 ��m�)Nu

�(Re

�). (360)

Nu�%�

values at low Reynolds number are unrealistic, leading to the

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


with porosity in the range 0.87&�m�& 0.97,

Nu��

�h��d ��

��

� 0.606Pe��

�m��, (361)

where

Pe��Re

�Pr�

U � �m�d �

a�

, (362)

d �

is in millimeters (see (294)), and Nu��

is the volumetric internal heat

transfer coefficient assessed using

h��q��

h�q�

� ��S

��

�� S�

�. (363)

Also,

Nu�%�

(Re�%�

) �Nu

��(�m� Re

�%�)

4�m�. (364)

The correlation given by Kays and London [172] is

StPr�� 1.4Re��%�

, (365)

which is transformed by

NuPr��

Re�%�

Pr� 1.4Re��

�%�, �Nu

�%�� 1.4Re��

�%�Pr� . (366)

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 ap-

proaches. 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.


VIII. Thermal Conductivity Measurement in a Two-Phase Medium

A majority of thermal conduction experiments are based on a constant

heat flux through the experimental specimen and measurement of interface

temperatures. Data reduction (see, for example, Uher [192]) is accomplished

using

K�QL

A�T, (367)

where Q is the electrical power from heater dissipated through the specimen,

L is the distance used to measure the temperature difference, and A is the

uniform cross-sectional area of the sample.

1. Traditional L ocal and Piecewise Distributed Coefficient HeatConductivity Problem Formulations

In DMM-DNM as, for example, for a dielectric medium, the equation

usually used is

· (k(r) T (r)) � 0, r �, (368)

where the conductivity coefficient function k is

k(r) � k�)��(r) � k

�)��(r), (369)

and )�� 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

��(370)

k�(n · T

�(r)) � k

�(n · T

�(r)), r �S

��. (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 equa-

tions (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.


Conventional theories of treatment of this problem do not specify the

meaning of the field T, assuming that it is the local variable, or �T �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

constant.

In the DMM-DNM approaches the mathematical statement usually deals

with the local fields, and as soon as the boundary conditions are taken in

some way, the problem became formulated correctly and can be solved

exactly, as in work by Cheng and Torquato [193].

Difficulties arise when the result of this solution needs to be interpreted—

and this is in the majority of problem statements in heterogeneous media,

in terms of nonlocal fields, but averaged in some way. The averaging

procedure usually is stated as being done either by stochastic or by spatial,

volumetric integration. Almost all of these averaging developments are done

incorrectly because of a disregard of averaging theorems for differential

operators in a heterogeneous medium. More analysis of this matter is given

in work by Travkin et al. [115].

Further, a more complicated situation arises when the intention is to

formulate and find effective transport coefficients in a heterogeneous me-

dium. Let us consider the conductivity problem in a two-phase medium.

According to most accepted mathematical statements this problem is given

as (368)—(371).

3. Conventional Formulation of the Effective Conductivity Problem in aTwo-Phase Medium

One of the methods of closure of mathematical models of diffusion

processes in a heterogeneous medium is the quasihomogeneous method

(Travkin and Catton [21]). In this case, the transfer process is modeled as

an ideal continuum with homogeneous effective transport characteristics

instead of the real heterogeneous characteristics of a porous medium. This

method of closure of the diffusive terms in the heat and mass diffusion

equations results in certain limitations: (a) the two-phase medium compo-

nents 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��

��k*

�� T �, (372)


when for constant (effective) coefficients it is

�k��

� T �� k

�� T �

��k*

�� T �, (373)

and also

� T �� T �� T �

, (374)

so it might be written as

(k��

)�� j�� (k

��)�� j��

� �� T �. (375)

Here k��

, k��

are the transfer coefficient tensors in each of the phases, and

k*��

is the effective conductivity coefficient. Thus, at least in this case, the

problem of closure has been reduced to finding k*��

.

Applying the closure relation, for example,

k��

� T �� k

�� T �

, (376)

yields the effective stagnant coefficient

k*��

�2k�

��k��

(k��

� k��

), (377)

which represents the lower bound of the effective stagnant conductivity for

a two-phase material from the known boundaries of Hashin—Shtrikman

(see, for example, [196], Kudinov and Moizhes [197]) for equal volume

fraction of phases. Other closure equations for calculating the stagnant

effective conductivity are found in work by Hadley [198] and by Kudinov

and Moizhes [197]. The quasi homogeneous approach has several defects:

(a) The basis for the quasi-homogeneous equations is in question, (b) the

local fluctuation values, as well as inhomogeneity and dispersivity of the

medium, are neglected, and (c) the interdependence of the correlated

coefficients and arbitrary adjustment to fit data significantly reduce the

generality of the results.

4. VAT-Based Considerations for Heterogeneous Media Heat ConductivityExperimental 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 reduc-

tion procedures and pertinent modeling equations.


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

�s��(�c

�)�

�T��

�t� k

� �(�s

��T�

�) � k

� ·�

1

�� T�d�s

��k�

�� T

�· d�s

�,

which needs the closure of the second and the third r.h.s. terms. The latter is

k�

�� T�x

�

· d�s��

��S��

(T �� T

�), (378)

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� �(�s

��T�

�) � k

� ·�

1

�� T�d�s

�� · [�q��],

where the ‘‘diffusive’’-like flux q��

contains some more terms than are

conventionally considered,

q��k

�� (�s

��T�

�) ��k

� (�s

��T�

�) �

k�

�� T�d�s

�, (379)

where the heat flux in phase 1 is determined through the averaged tempera-

ture T��.

So, the effective (not homogeneous) conductivity coefficient in phase 1 is

k��

� k� � (�s

��T�

�) �

1

�� T�d�s

�� ( (�s��T�

�))��

� k� �1�

1

�� T�d�s

��( (�s��T�

�))�. (380)

There is a difference between this introduced coefficient k��

and that

traditionally determined through the flux in phase 1, which is

q�� [�k

�� T

��]��k

� � (�s��T�

�) �

1

�� T�d�s

��. (381)


Arising in this situation is the effective conductivity coefficient determina-

tion

k��

� k� � (�s

��T�

�) �

1

�� T�d�s

�� 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),

q� � [�k��

� T �] ��k��

[� T �� T �

�]

��k��

(�s��T�

�� s

��T�

�) ��k

�� T �. (383)

After those transformations the heat transfer equation in phase 1 becomes

�s��(�c

�)�

�T��

�t� · [k

�� (�s

��T�

�)] � ��

��S��

(T �� T

�). (384)

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

�(�s��T�

�) � ·�

1

�� T�d�s

��1

�� T

�· d�s

�� 0, (385)

one obtains

k��

�� (�s��T�

�) �

1

�� T�d�s

�� ( (�s��T�

�))�� (386)

for the equation

· [k��

(�s��T�

�)] ��

��k�� S

��(T

�� T

�) � 0, (387)

where k��

does not even depend explicitly on the phase heat conductivity

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

· [k ��

(�s��T�

�)] � 0. (388)

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


transport processes, which is not present in conventional composite medium

modeling.

Also, in the more complete and challenging physics of interface transport

modeling as in the third phase, this third interphase exchange term, along

with the second term, is an issue tightly connected to the closure problem

and to the models of interface surface transport.

b. Nonlinear Heat Conductivity of a Pure Phase Material Meanwhile, for

materials such as high-temperature superconductors (HTSC), a constant

heat conductivity coefficient is not a justifiable choice, as the usual analysis

of approaches has shown above. That means complications in treating the

equation with a nonlinear heat conductivity coefficient in phase 1,

�s��(�c

�)�

�T��

�t� · [K

�� (�s

��T

��)] � · [�s

��K�

� T�

��]

� ·�K

��

�� T�d�s

��1

�� K

�

�T�

�x�

· d�s�� s

��S

��,

(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�

��

�K

��

�� T�d�s

�� ( (�s��T�

�))��, (390)

which when inserted in the heat transport equation gives

�s��(�c

�)�

�T��

�t� · [K

�� (�s

��T�

�)] � ��

��S��

(T �

� T �) � �s

��S

��. (391)

Meanwhile, when an experimentalist evaluates his or her experimental

data using the equation

(�c�)�

�T��

�t� · [k

� T�

�] (393)

with the calculation shown earlier of the thermal conductivity coefficient

using experimental data, he or she makes two mistakes:


1. He is confusing the material’s clear homogeneous conductivity coeffi-

cient 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.

2. Doing data reduction as for the modeling equation

(�c�)�

�T��

�t� · [k

�� T�

�] (392)

meaning that

k��

� k�, (394)

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,

��

S��

(T �� 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 —

k��

�/ k�.

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 experi-

mental data modeling are of primary interest.

The standard definition of the effective (macroscopic) conductivity tensor

is determined from

� j��k*�� T �, (396)

in which it is assumed that

�j�� j�� j�

��k

�� T �

�� k

�� T �

��k*

�� T �� k*

�� T �

� �k*��[� T �

�� T �

�]��k*

�� T �

�� k*

�� T �

�, (397)

so, for the usually assumed interface �S��

physics, the effective coefficient is

determined to be

k*�� T �� [k

�� T �

�� k

�� T �

�]

� k� (�m

��T�

�) � k

� (�m

��T,

�) � (k

�� k

�)

1

�� T�d�s

�(398)


or

k*��k�

(�m��T�

�) � k

� (�m

��T�

�) � (k

�� k

�)

1

�� T�d�s

�� T ��,

(399)

or

k*��k� (�m

��T�

�) � k

� (�m

��T�

�) � (k

�� k

�)

1

�� T�d�s

��[� T �

�� T �

�]

,

(400)

which involves knowledge of three different functions, T��, T�

�, T

��S��, in the

volume �. This formula for the steady-state effective conductivity can be

shown to be equal to the known expression

k*�� T �� k

� �T �� (k

�� k

�)

1

�� T d�

� k� �T �� (k

�� 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, (402)

for a field applied parallel to the interface of layers, and

k*�

��.��

�m��

k��

(403)

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 magni-

tude. The same assumptions are effectual when conventional volume aver-

aging 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


both temperature equations (if only two phases are present) for the simplest

case of constant coefficients,

�s��(�c

�)�

�T��

�t� k

� �(�s

��T�

�) � k

� ·�

1

�� T�d�s

��k�

�� T

�· d�s

�

�s��(�c

�)�

�T��

�t� k

� �(�s

��T�

�)�k

� ·�

1

�� T�d�s

��k�

�� T

�· d�s

�,

into one equation by adding one to another, we obtain

�s��(�c

�)�

�T��

�t��s

��(�c

�)�

�T��

�t� · (k

� (�s

��T�

�) � k

� (�s

��T�

�))

� ·�k�

�� T�d�s

��

k�

�� T�d�s

��

k�

�� T

�· d�s

��

k�

�� T

�· d�s

�,

(404)

keeping in mind that the two-phase averaged temperature is

�T �� s��T�

�� s

��T�

�. (405)

One can write down the mixture temperature equation when summation

of the equations gives (when taking into account the boundary condition of

temperature fluxes equality at the interface surface, (k� T

�) � (k

� T

�))

�s��(�c

�)�

�T��

�t��s

��(�c

�)�

�T��

�t

� · (k� (�s

��T�

�) � k

� (�s

��T�

�)) � (k

�� k

�) ·�

1

�� T�d�s

��,(406)

or, written in terms of thermal diffusivities a�

and a�,

��T ��t

� · [a� (�s

��T�

�) � a

� (�s

��T�

�)] � (a

�� a

�) ·�

1

�� T�d�s

��

� �1 �a�

a�

k�

k��

1

�� T

�· d�s

��, a��

k�

(�c�)�

, i� 1, 2, (407)

which has the three different temperatures �T��, T�

�, and T

�(�S

��) (here

�T ��s��T�

��s

��T�

�).


And, assuming only a local thermal equilibrium,

�T ��s��T�

��s

��T�

��T * �T�

�� T�

�, (408)

the mixed temperature equation becomes two-temperature T *, T�(�S

��)

dependable with simplified left hand part of the equation

(�s��(�c

�)��s

��(�c

�)�)�T *

�t� · [(k

� (�s

��T *) � k

� (�s

��T *))]

�(k�� k

�) ·�

1

�� T�d�s

��. (409)

With the two different temperatures, the effective coefficient of conductiv-

ity is equal to

k*��

��[(k� (�s

��T *) � k

� (�s

��T *))]

� (k�� k

�) �

1

�� T�d�s

�� (� T *�)��. (410)

This formula coincides with the effective coefficient of conductivity for the

steady-state effective conductivity in the medium and can be shown to be

equal to the known expression

k*��

�T �� k� �T �� (k

�� k

�)

1

�� T d�. (411)

From this formula an important conclusion can be drawn: that the most

sought-after characteristics in heterogeneous media transport, which are the

effective transport coefficients, can be correctly determined using the con-

ventional definition for the effective conductivity— for example, for the

steady-state problem

�� j �� k*��

�T �� k� �T �� (k

�� k

�)

1

�� T d�, (412)

but only in a fraction of problems, while employing the DMM-DNM exact

solution. The issue is that in a majority of problems, such as for in-

homogeneous, nonlinear coefficients and in many transient problems, hav-

ing the two-field DMM-DNM exact solution is not enough to find effective

coefficients. As shown earlier, only the requirement of thermal equilibrium

warrants the equality of steady-state and transient effective conductivities in

a two-phase medium.


The second form of the same equation with the surface integral of the

fluctuation temperature in phase 1 is

(�s��(�c

�)�� s

��(�c

�)�)�T *

�t� · [(k

��s

�� k

��s

��) (T *)]

� (k�� k

�) ·�

1

�� T��d�s

��, (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):

[(k� (�s

��T *) � k

� (�s

��T *))] � (k

�� k

�) �

1

�� T�d�s

�� [(k

��s

�� k

��s

��) (T *)] � (k

�� k

�) �

1

�� T��d�s

��. (414)

As can be seen, the transient effective diffusivity coefficent a°��

in the VAT

nonequilibrium two-temperature equation (407) can be derived through the

equality

a°��

� T �� a� (�s

��T�

�) � a

� (�s

��T�

�) � (a

�� a

�) �

1

�� T�d�s

�� a� �1�

a�

a�

k�

k��

1

�� T

�· d�s

�� (415)

or

a°��

�T �� a� (�s

��T�

�) � a

� (�s

��T�

�) � (a

�� a

�) �

1

�� T�d�s

�� A,

(416)

where �� is the inverse operator � · ( ��( f )) � f such that if

·A � a� �1�

a�

a�

k�

k��

1

�� T

�· d�s

��, (417)

then

A � �� a� �1�a�

a�

k�

k��

1

�� T

�· d�s

��. (418)


From the preceding expression, the transient effective nonequilibrium

coefficient in a two-phase medium can be determined as

k°��

� a°��

(�s��(�c

�)�� s

��(�c

�)�), (419)

which looks rather inconvenient for analytical or experimental assessment

or numerical calculation. The solution of this problem, which includes as an

imperative part the finding of the effective bulk composite material heat

conductivity (diffusivity), coefficient, is equal to the solution of the exact

two-phase problem. We see that the two-temperature DMM-DNM is not

enough for the convenient construction of the effective coefficient of conduc-

tivity. 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

�)�)��T ��t

� ·��s��k

� � �T ��1

��

T��d�s

��s

��k

� � �T ��1

��

T��d�s

��s

��(�c

�)�

�T

��t

��s��(�c

�)�

�T

��t

� · (�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


three different temperatures resumes:

(�s��(�c

�)��s

��(�c

�)�)��T ��t

� ·��s��k

� � �T ��1

��

T��d�s

��s

��k

� � �T ��1

��

T��d�s

��.(422)

This means that the neglect of the global deviation T

�, T

�terms still does

not remove the requirement of a two-temperature solution.

a. Effective Conductivity Coefficients in a Porous Medium When Phase One

Is a Fluid In phase 1 the VAT equation is written for the laminar regime.

In the work by Kuwahara and Nakayama [199] is given the DMM-DNM

solution of the 2D problem of uniformly located quadratic rods with equal

spacing in both directions. Studies were undertaken of both the For-

chheimer and post-Forchheimer flow regimes.

This work is a good example of how DMM-DNM goals cannot be

accomplished, even if the solution on the microlevel is obtained completely,

if the proper VAT scaling procedures basics are not applied.

The one structural unit—periodic cell in the medium—was taken for

DMM-DNM.

Equations were taken with constant coefficients, and in phase 1 the VAT

equation was written for the laminar regime as

�m�(�c�)�

�T��

�t��m�(�c

�)�U�� T�

�� (�c

�)� · ��T�

�u�� k

� (�m�T�

�)

� k� ·�

1

��

T�d�s��

k�

��

T�· d�s. (423)

Adding this equation to the VAT solid-phase (second phase) two-

temperature equation gives

�m�(�c�)�

�T��

�t��s

��(�c

�)�

�T��

�t��m�(�c

�)�U�� T�

�

� (�c�)� ·��T�

�u�� · (k

� (�m�T�

�) � k

� (�s

��T�

�))

� ·�k�

�� T�d�s

��

k�

�� T�d�s

��

k�

�� T

�· d�s

��

k�

�� T

�· d�s

�, (424)


which reduces because of interface flux equality to

�m�(�c�)�

�T��

�t��s

��(�c

�)�

�T��

�t��m�(�c

�)�U�� T�

�

� · (k� (�m�T�

�) � k

� (�s

��T�

�)) � (�c

�)� ·��T�

�u��

� (k�� k

�) ·�

1

�� T�d�s

��, (425)

which has two averaged temperatures T��

and T��, interface surface integrated

temperature T�(�S

��), and two fields of fluctuations T�

�(x) and u�

�(x),

assuming that the velocity field is also computed and known.

We now write the effective conductivity coefficients for (425) and for the

one-temperature equation when temperature equilibrium is assumed.

In the first case, for the weighted temperature,

�T �� (�m�(�c�)�T�� s

��(�c

�)�T��)/w

�(426)

w��m�(�c

�)�� s

��(�c

�)�� const, (427)

the equation can be written as

w�

��T ��t

��m�(�c�)�U�� T�

�

� · (k� (�m�T�

�) � k

� (�s

��T�

�)) � (�c

�)� ·��T�

�u��

� (k�� k

�) ·�

1

�� T�d�s

��, (428)

where three temperatures are unknown, �T ��, T��, and T�

�, plus the interface

surface temperature integral T�(�S

��) and fluctuation fields T�

�(x) and u�

�(x).

The effective coefficient of conductivity can be looked for is

k°��

� T �� (k� (�m�T�

�) � k

� (�s

��T�

�)) � (�c

�)��T�

�u��

� (k�� k

�) �

1

�� T�d�s

��. (429)

In order to avoid the complicated problems with effective conductivity

coefficient definition in a multitemperature environment, Kuwahara and

Kakayama [199], while performing DMM-DNM for the problem of

laminar regime transport in a porous medium, decided to justify the local

thermal equilibrium condition

�T �� m�T��s

��T�

��T *� T�

�� T�

�,

which greatly changes the one effective temperature equation. This equation


becomes simpler with only one unknown temperature T * and variable field

T��

and is written as

(�m�(�c�)�� s

��(�c

�)�)�T *

�t��m�(�c

�)�U�� T *

� · (k� (�m�T *) � k

� (�s

��T *)) � (�c

�)� ��T�

�u��

� (k�� k

�) ·�

1

�� T�d�s

��, (430)

as the variable temperature and velocity fluctuation fields T��

and u��should

be known, although this is a problem. As long as the definition of the

effective conductivity coefficient is

k*��

� T *�� k� (�m�T *) � k

� (�s

��T *) � (�c

�)��T�

�u��

� (k�� k

�) �

1

�� T�d�s

��, (431)

then the effective conductivity can be calculated subject to known T *, T��,

T�, and u�

�. At the same time, the important issue is that in DMM-DNM the

assumption of thermal equilibrium has no sense at all—as long as the

problem have been already calculated as the two-temperature problem.

To further perform the correct estimation or calculation of effective

characteristics, one needs to know what are those characteristics in terms of

definition and mathematical description or model?

This is the one more place where the DMM-DNM as it is performed now

is in trouble if it does not comply with the same hierarchical theory

derivations and conclusions as the VAT (see also the studies by Travkin etal. [115] and Travkin and Catton [114, 21]).

As shown earlier, only the requirement of thermal equilibrium warrants

the equality of steady-state and transient effective conductivities in a

two-phase medium.

Consequently, if taken correctly, the two-temperature model will intro-

duce more trouble in treatment and even interpretation of the needed bulk,

averaged temperature (as long as this problem is already known to exist and

is treated in nonlinear and temperature-dependent situations) and the

corresponding effective conductivity coefficient (or coefficients).

1. Thus, comparing the two effective conductivity coefficients (429) and

(431), one can assess the difference in the second term form and

consequently, the value of computed coefficients.

Comparing the expressions for one equilibrium temperature and one

effective weighted temperature, as well as for their effective conductiv-

ity coefficients, one can also observe the great imbalance and inequal-

ity in their definitions and computations.


2. Summarizing application of DMM-DNM approach by Kuwahara and

Nakayama [199], it can be said that it is questionable procedure to

make an assumption of equilibrium temperatures when the problem

was stated and computed as via DNM for two temperatures.

3. In the calculation of the effective coefficients of conductivity—stag-

nant thermal conductivity k�; tortuosity molecular diffusion k

�%�; and

thermal dispersion k��

—Kuwahara and Nakayama [199] used a

questionable procedure for calculation of the two last coefficients.

They used one-cell (REV) computation for surface and fluctuation tem-

peratures for periodical morphology of the medium, and at the same time

they used the infinite REV definition for the effective temperature gradient

for their calculation (assigned in the problem); see the expressions for

calculation of these coefficients, (21)—(24) on p. 413. That action means the

mixture of two different scale variables in one expression for effective

characteristics—which is incorrect by definition. If this is used consciously,

the fact should be stated on that matter explicitly, because it alters the

results.

IX. VAT-Based Compact Heat Exchanger Design and Optimization

At the present time, compact heat exchanger (CHE) design is based

primarily on utilization of known standard heat exchanger calculation

procedures (see, for example, Kays and London [172]). Typical analysis of

a heat exchanger design depends on the simple heat balance equations that

are widely used in the process equipment industry. Analytically based

models are composed for a properly constructed set of formulas for a given

spatial design of heat transfer elements that allow, most of the existing heat

transfer mechanisms to be accounted for.

Analogies between heat transfer and friction have been shown by Church-

ill [200] and by Churchill and Chan [201] to be inadequate for describing

many of the HE configurations of interest. This has been suspected for some

time and will seriously affect the use of the ‘‘j-factor’’ in HE modeling and

design.

Modeling of a specific heat exchanger geometry by Tsay and Weinbaum

[202] provides a useful preliminary step and a potential benchmark test

case. Though the study only considered hydrodynamic effects and restricted

itself to consideration of regular media and the creeping flow regime, the

effects of morphology-characteristic variation upon momentum transport

phenomena were explored. The authors show that the overall bed drag

coefficient in the creep flow regime increases dramatically as the inner-

cylinder spacing approaches the order of the channel half-height.


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 three-

temperature transfer in nonisotropic heterogeneous CHE media with ac-

counting for interphase exchange and microroughness.

The development of new optimization problems based on the VAT-

formulated 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 Practicein 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� � dA�T,

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/

�d0

�

�/��/#�

�

�d/

�d0

�

�/��/

�,

where /�, /

�, and /#�

�are dimensionless first and second fluid temperatures


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

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

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

0��

�A(Mc

�)�

z�

L�

, i� 1, 2,

where � is the overall heat transfer coefficient and M is the mass flow rate.

Second-order ordinary differential equations are developed for HE as well

(see, for example, Paffenbarger [206]).


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

the fluids.

Webb in a book [207], and in his invited talk at the 10th International

Heat Transfer Conference [208], distinguishes four basic approaches to

predicting the heat transfer j-factor and the Fanning friction factor f for

heat exchanger design. They are (1) power-law correlations; (2) asymptotic

correlations; (3) analytically based models; and (4) numerical solutions.

Analytically based models are properly constructed set of formulas for a

given spatial construction of heat transfer elements that allows most of the

existing heat transfer mechanisms to be accounted for. Many examples are

given in publications by Webb [207, 208], Bergles [209], and other

researchers.

The major differences between the measured characteristics of air-cooled

heat exchangers with aluminum or copper finned tubes with large height,

small thickness, and narrow-pitch fins, and high-temperature waste heat

recovery exchangers with steel finned tubes with rather low height and

thickness and wide-pitched fins, are given in a paper by Fukagawa et al.[210] Despite the fact that morphology of the heat exchange medium is

essentially the same, the correlations predicting heat transfer and pressure

drop values do not work for both HE types altogether. For this particular

heat exchange morphology, a wide-ranging experiment program is needed

for different ratios of the morphology parameters. There is, at present, no

general approach for describing the dependencies of heat transfer effective-

ness 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


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 men-

tioned 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 equa-

tions (PDE), to which VAT-based HE models convert, can have detailed


solutions of the linear quadratic regulator problem, including conditions for

the convergence of modal approximation schemes. However, for more

general optimal control problems involving PDEs, the main approach has

been to use some method for constructing a particular finite-dimensional

approximating optimal control problem and then to solve this problem by

some method or other (Teo and Wu [213]).It seems that no attention has been given to the optimal control systems

governed by the partial integrodifferential equations like volume averaging

theory equations for HE design.

B. New Kinds of Heat Exchanger Mathematical Models

Our earlier work has shown that flow resistance and heat transfer in HEs

and CHEs can be treated as highly porous structures and that their behavior

can be properly predicted by averaging the transport equations over a

representative elementary volume (REV) in the region neighboring the surface.

The averaging of processes in regular and randomly organized heterogeneous

media and in HE can be performed in different ways. Travkin and Catton [21,

28] discussed alternate forms for the mass, momentum, and heat transport

equations recently presented by various researchers. The alternate forms of the

transport equations are often quite different. The differences among the

transport equation forms advocated by the numerous authors demonstrate the

fact that research on the basic form of the governing equations of transport

processes in heterogeneous media is still an evolving field of study. Derivation

of the equations of flow and heat transport for a highly porous medium during

the filtration mode is based on the theory of averaging by certain REV of the

transfer equation in the liquid phase and transfer equations in the solid phase

of the heterogeneous medium (see, for example, Whitaker [42, 10] for laminar

regime developments, and Shcherban et al. [15], Primak et al. [14], and

Travkin and Catton [16, 21, 23] for turbulent filtration).These models account for the medium morphology characteristics. Using

second-order turbulent models, equation sets are obtained for turbulent

filtration and two-temperature diffusion in nonisotropic porous media with

interphase exchange and micro-roughness. The equations differ from those

found in the literature. They were developed using an advanced averaging

technique, a hierarchical modeling methodology, and fully turbulent models

with Reynolds stresses and fluxes in every pore space.

Independent treatment of turbulent energy transport in the fluid phase

and energy transport in the solid phase, connected through the specific

surface (the solid—fluid interface in the REV), allows for more accurate

modeling of the heat transfer mechanisms between rough surfaces or porous

insert of HE and the fluid phases.


C. VAT-Based Compact Heat Exchanger Modeling

For a pin fin (PFHE), with cross-flow morphology, the governing

equations can be written in the following form:

Momentum equation for the first fluid:

��x ��m

��(K�

��

�)�U �

��x ��

��x �K�

��

�u!��

�x ��

��x

(��u!��u!��)

� �m�� U �

�

U ��

�x�

1

��

(K��

� ��)�U

��x

�

· d�s

�1

��

��

p!�d�s �

1

��

��x

(�m��p!�

�). (432)

momentum equation for the second fluid:

��z ��m

��(K�

��

�)�W �

��z ��

��z �K�

��

�w!��

�z ��

��z

(��w!��w!�

��)

� �m�� W �

�

�W ��

�z�

1

��

(K��

� ��)�W

��x

�

· d�s

�1

��

��

p!�d�s�

1

��

��z

(�m��p!�

�). (433)

Energy equation for the first fluid:

c��

��m

��U �

�

�T ��

�x�

��x ��m

��(K�

�� k

�)�T �

��x �

��z ��m

��(K�

�� k

�)�T �

��z �

��x �K�

��

�T ��

�x ��

��z �K�

��

�T ��

�z ��

� c��

��

��x

(�m��T �

�u!��)

��x �

(K��

� k�)

��

T ��d�s�

��z �

(K��

� k�)

��

T ��d�s�

�1

��

(K��

� k�)�T

��x

�

· d�s. (434)


Energy equation for the solid phase:

��x ��s�K

�

�T

�x ��x �K�

�

�T�

�x ��

��z ��s�K

�

�T

�x ��

��z �K�

�

�T�

�x ��

��x �

K�

��

T�d�s

��

��z �

K�

��

T�d�s

��1

��

K�

�T

�x�

· d�s�� 0. (435)

Energy equation for the second fluid:

c��

��m

��W �

�

�T ��

�z�

��x ��m

��(K�

�� k

�)�T �

��x �

��z ��m

��(K�

�� k

�)�T �

��z �

��x �K�

��

�T ��

�x ��

��z �K�

��

�T ��

�z ��

� c��

��

��z

(�m��T �

�w!�

��)

��x �

(K��

� k�)

��

T ��d�s�

��z �

(K��

� k�)

��

T ��d�s�

�1

��

(K��

� k�)�T

��x

�

· d�s. (436)

The volumes for averaging in equations are ��, ��

, ��

, ��.

A majority of the additional terms in these equations can be treated using

closure procedures developed in previous work (see, for example, Travkin

and Catton [16, 19]), for selected fin geometries and solid matrices of a HE.

Our generic interest, however, is in the theoretical applications of the VAT

governing equations and possible advantages gained by introduction of

irregular or random morphology into heat exchange volumes and surfaces.

Cocurrent parallel flow matrix type CHE morphology can be described

using the next VAT-based set of governing equation.


Momentum equation for the first fluid:

�m��U �

�

�U ��

�x�

1

��

(K��

� ��)�U

��x

�

· d�s �1

��

��

p!�d�s

��1

��

��x

(�m��p!�

�) �

��x

(�m��(K�

��

�)�U �

��z �

��x �K�

��

�u!��

�x ��

��x

(� � u!��u!��). (438)

Momentum equation for the second fluid:

�m��U �

�

�U ��

�x�

1

��

(K��

� ��)�U

��x

�

· d�s �1

��

��

p!�d�s

��1

��

��x

(�m��p!�

�) �

��x

(�m��(K�

��

�)�U �

��x �

��x �K�

��

�u!��

�x ��

��x

(� � u!��u!��). (437)

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

��z ��m�(K�

�� v

�)�U �

��z ��

��z �K�

��

�u!��

�z ��

��z

(��u- ��w!��)

�1

�� S��

K��

�U �

�x�

· dS��1

�� S�!�

v�

�U �

�x�

· d�S

�1

��

��

p!�dS� �

1

��

��p! ��

�x. (439)

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


porous filling or insert that has regular morphology,

��z ��m(z)�(K�

�� v

�)�U �

�(z)

�z ��U�(�

(U �, �S

�, K

��) �U

�(!(U

�, �S

�, v

�)

�U�(�%��

( p!�, �S

�) �

1

��

�(�m(z)�p!��)

�x, (440)

where the three morphology-based terms are defined by

U�(�

(U �,�S

�, K

��) �

1

�� S��

K��

�U �

�x�

· dS� (441)

U�(!

(U �, �S

�, v

�) �

1

�� S�!�

v�

�U �

�x�

· dS� (442)

U�(�%��

(p!�, �S

�) ��

1

��

��

p!�dS�. (443)

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:

K��

(z) ��U �

��z �

��

d

dz ��K��

� v��

db�(z)

dz ��f��

(c�)S

��(z)

�m�U � �

�g

T#��K� ��

�T ��

�z �� 2v �db� �

�(z)

dz ��C

�

b��(z)

K��

. (444)

Here, f�(c�) is approximately the friction factor for constant and nearly

constant morphology functions, and the mean eddy viscosity is given by

K��

(z) �C� ��

l(z)b� ��

(z), (445)

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��

��m

��U �

�(z)

�T ��(x, z)

�x�

��z ��m

��(K�

�� k

��)�T �

�(x, z)

�z ��T

�(�$��(T

�, �S

�, K

��) � T

�(!$��(T �, �S

�, k

�)

�1

�� S�!�

k�

�T �

�x�

· dS�, (446)


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

T�(�$��

(T �, �S

�, K

��) �

1

�� S��

K��

�T �

�x�

· dS� (447)

T�(!$��

(T �, �S

�, k

�) �

1

�� S�!�

k�

�T �

�x�

· dS�. (448)

In the solid phase of CHE, the energy equation is

��z �(1 ��m�)K�

�(z)

�T(x, z)

�z ��T($��

(T, �S

�, K

�) � 0, (449)

with the one ‘‘control’’ term

T($��

(T, �S

�, K

�) �

1

��

K�

�T

�x�

· dS�,

where

dS��dS�.

If we apply the closure procedures described earlier, the equation of

motion becomes

��z ��m(z)�K�

��(U � , b, l )

�U ��(z)

�z ��

1

2[c�!

(z, U ��)S

�!(z) � c�

�(z, U �

�)S

��(z) � c

��(z, U �

�)S

�.(z)]U � �

�

�1

��

d�p!��

dx� c

�S�

U � ��

2�

1

��

d�p!��

dx, (450)

where

K��

�K��

� v�,

and the lumped flow resistance coefficient c�

is the complex morphology

dependent function. The energy equation in the jth fluid phase is

c��

��m�U �

�(z)

�T��(x, z)

�x�

��z

(�m(z)�K��

(z)�T �

�(x, z)

�z

� ��(z)S

�(z)(T

(x, z) �T�

�(x, z)), (451)


with (x, z) ��, and the energy equation in the solid phase

��z

(�1 �m(z)�K��

(z)�T

(x, z)

�z

� ��(z)S

�(z)(T

(x, z) �T�

�(x, z))(x, z) ��, (452)

with

P��$ 1 : K�

��$K�

��c��

��

� k��

, (453)

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 mor-

phological models of the porous structure. The pressure gradient term in Eq.

(450) is modeled as a constant value in the layer, or simulated by the local

value of the right-hand side of the experimental correlations. The boundary

conditions for these equations are

z� 0 :U �� 0,

�b�

�z� 0

K�� v, Q

��K�

��

�T��

�z

Q��K

��

�T�

�z(454)

z��

�h�:�U �

��z

� 0,�b

��z

� 0

�T��

�z� 0,

�T�

�z� 0, (455)

where h�

is the half channel width. The control terms in the preceding

equations depend on temperature and velocity distributions as well as on

morphological characteristics of the media.

Comparing the three latest equation (450)—(452) with the equations

derived by Paffenbarger [206] for practically the same structural design of

HE, one will find numerous discrepancies. For example, the energy balance

equations in Paffenbarger’s [206] work have energy conservation terms that

do not match each other.

The VAT-based general transport equations for a single phase fluid in an

HE medium have more integral and differential terms than the homogenized

or classical continuum mechanics equations. Various descriptions of the


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 equa-

tions resulting from the VAT-based analysis are also PPDEs, but they are

nonlinear and have additional integral and integrodifferential terms. The

models presented and the resulting differential equations contain additional

integral and integrodifferential terms not studied in the literature.

The performance of a heat exchanger depends on the design criteria for

optimizing the liquid flow velocity, dimensions of the heat exchanger, the

heat transfer area between hot side and cold side, etc. Thermal optimization

of an HE requires selection of many features— for example, both the

optimum fin spacing and optimum fin thickness, each determined to

maximize total heat dissipation for a given added mass or profile area. These

criteria set the optimal conditions for HE operation. Theoretically, the

optimal dimensions of an HE require a large number of tiny tubelets with

diameters tending to zero with increasing number of tubes. This leads to a

very fine dispersion problem with porous medium— like behavior. Extremely

compact micro heat exchangers with plate—fin cross flow have already been

built. However, the optimization problems involving such designs are more

complex than traditional designs and require new simulation techniques.


E. A VAT-Based Optimization Technique for Heat Exchangers

A variety of optimal control problems that can be formulated in the area


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 equa-

tions have been studied intensively. For example, Ahmed and Teo [214] give

a survey on main results in this field. Questions concerning necessary

conditions for optimality and existence of optimal controls for these

problems have been investigated in work by Ahmed and Teo (215—217] and

Fleming [218]. Moreover, a few results by Teo et al. (1980) on the

computational methods of finding optimal controls are also available in the

literature (Teo and Wu [213]). However, turbulent transport equations in

highly porous media were proposed by Travkin et al. [19] for optimization

problems and developed in more detail in Section IV with additional

‘‘morphlogical’’ as well as integral and integrodifferential terms. Recent

literature studies show optimal control problems involving PPDE either in

general form or in divergence form and propose computational methods

such as variational technique and gradient method (see, for example, Ahmed

and Teo [214]). These studies seems to be helpful for solving various

optimization problems involving integro—differential transport equations

considered by Travkin et al. [19]. However, complete research has to be

done for this class of equations, including analysis of necessary conditions

and existence of optimal control, as well as developing computational

methods for solving various optimal control problems.

Optimal control for some classes of integrodifferential equations has also

been considered in recent years. Da Prato and Ichikawa [219] studied the

quadratic control problems for integrodifferential equations of parabolic

type. A state-space representation of the system is obtained by choosing an

appropriate product space. By using the standard method based on the

Riccati equation, a unique optimal control over a finite horizon and under

a stabilizability condition is obtained and the quadratic problem over an

infinite horizon is solved. Butkovski [220] was the first to discuss the

optimal control problems for distributed parameter systems. The maximum

principle as a set of necessary conditions for optimal control of distributed

parameter systems has been studied by many authors.

Since it is well known that the maximum principle may be false for

distributed parameter systems (see Balakrishnan [221]), there are many

papers that give some conditions ensuring that the maximum principle


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 maxi-

mum principle for relaxed infinite-dimensional optimal control problems.

He considered relaxed optimal control problems described by semilinear

systems ODE and used relaxed controls whose values are finitely additive

probability measures. Under suitable conditions, relaxed trajectories co-

incide with those obtained from differential inclusions. The existence the-

orems 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�)

�)

(�m� f(x�) f

�(x�)

�)

�#�(x�) ��

�

f�(x�) · d�s�

��

[#�(x� ) ( f

�(x� , f

�(x� )))] · d�s, (456)

with controls f�, f

�, f

, f

�. Such statements of the control problem are hardly

seen in the contemporary literature on optimal control distributed-par-

ameter 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 mass-

diffusion 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


yields

k��

�A �

�x��m�(K�

�� )

�A �

�x�K�

�

�a! �

�x�

� ��a!� a!� ��. (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

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

K��

�T �

�x��m�(K�

�� k

�)�T �

�x�K�

�

�T �

�x�

� c��m�T � u!�

�

�(K�

�� k

�)

��

T � d�s�, (458)

where the first and last terms resemble the effective thermal conductivity

coefficient for each phase, using constant coefficients, found in the work by

Nozad et al. [40]. By allowing the control terms to be added to the bulk

transport coefficients, another variation of a mathematical statement for

optimal control can be found.

As far as optimal control problems with PDE dynamics are concerned,

one can find a detailed solution of the linear quadratic regulator problem,

including conditions for the convergence of modal approximation schemes.

However, for more general optimal control problems involving PDE, the

main approach has been to use some method for constructing a particular

finite-dimensional approximating optimal control problem and then to

solve this problem. The relationship between the solutions and stationary

points of the approximating optimal control problem and those of the

original optimal control problem is not established in these papers.

For the models and differential equations describing HEs to be useful, the

additional integral and integrodifferential terms need to be addressed in a

systematic way. VAT has the unique ability to enable the combination of

direct general physical and mathematical problem statement analysis with

the convenience of the segmented analysis usually employed in HE design.

A segmented approach is a method where overall physical processes or

groups of phenomena are divided into selected subprocesses or phenomena

that are interconnected to others by an adopted chain or set of depend-


encies. A few of the obvious steps that need to be taken are the following:

1. Model what increases the heat transfer rate

2. Model what decreases of flow resistance (pressure drop)3. Combine the transport (thermal/mass transfer) analysis and structural

analysis (spatial) and design

4. Find the minimum volume (the combination of parameters yielding a

minimum weight HE)5. Include nonlinear conditions and nonlinear physical characteristics

into analysis and design procedures

The power and convenience of this method is clear, but its credibility is

greatly undermined by variability and freedom of choice in selection of

subportions of the whole system or process. The greatest weakness is that

the whole process of phenomena described by a voluntarily assigned set of

rules for the description of each segment is sometimes done without serious

consideration of the implications of such segmentation. Strict physical

analysis and consideration of the consequences of segmentation is not

possible without a strict formulation of the problem that the VAT-based

modeling supplies. Structural optimization of a plate HE, for example, using

the VAT approach might consist of the following steps: (1) optimization of

the number of plates, plate spacing and fin spacing; (2) optimization of the

fin shape; (3) simultaneous optimization of multiple mathematical state-

ments. This approach also allows consideration and description of hydraul-

ically and thermally developing processes by representing them through the

distributed partial differential systems.

X. New Optimization Technique for Material Design Based on VAT



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


has three control terms,

�s��C

��t

� · (D�� s

��C�

�) � ·�D� �

1

�� S��

C�d�s

�� · (�D�

� c�

�) �

1

�� S��

D� C

�· d�s

�

� · (D�� s

��C�

�) �F

�(C

�, D�

�, M�) �F

�(c�

�, D�

�, M�)

� F

(C�, D

�, M�), (459)

where the morphology characteristics set M� contains many parameters, ��,

such as phase fraction �s�� and specific surface area �S

��,

M�� (�s��, �S

�� , �, �

, . . . ).

The equation for an electrostatic electrical field in a particulate medium

(polycrystalline medium) is

· [�s��

�E��]� · ��

�E��

1

�� (�

�E�) · d�s

��

�,

which becomes

· [�s��

�E��]� F

/�(��

�, E�

�, M�) �F

/�(�

�, E

�, M�) ��

�. (460)

Additional equations are

(�s��E�

�) �

1

�� d�s

� E

�� 0

(�s��E�

�) �F

/(E

�, M�) � 0. (461)

A temperature control equation for the solid phase with the two morphol-

ogy control terms can be written

�T��

�t� a

�

��T��

�z��T

�(��(T

�, �S

��, t, z) � T

�($��(T�, �S

�, t, z), (462)

where

T�(��

� a�

��z �

1

��

T�d�s

��, T�($��

�a�

��

�T�

�x�

· d�s�, (463)

and in the void phase

�T��

�t� a

�

��T��

�z��T

�(��(T

�, t, z) �T

�($��(T

�, t, z)


T�(��

(T�, �S

��, t, z) � a

�

��z �

1

��

T�d�s

�� (464)

T�($��

(T�, �S

��, t, z) �

a�

��

T�· d�s

�. (465)

These terms are not equal and their calculation or estimation presents a

challenge. However, these are the real driving forces that will differentiate

the behavior of one composite from another. Their application will lead to

a direct connection between design goals and morphological solutions.

XI. Concluding Remarks

Determination of the effective parameters in model equations are usually

based on a medium morphology model and there are dozens of associated

quasi-homogeneous and quasi-stochastic methods that claim to accomplish

this. In most cases, quasi-homogeneous and quasi-stochastic methods have

no well treated solutions and, most important, they are not sufficient for

description of the physical process features in heterogeneous media, especial-

ly when treating a multiscale processes.

The hierarchical approach applied to radiative transfer in a porous

medium and to the electrodynamics governing equations (Maxwell’s equa-

tions) in a heterogeneous medium yielded new volume averaged radiative

transfer equations—VAREs. These equations have additional terms reflect-

ing 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, non-

linear, laminar, and turbulent hierarchical transport in nonisotropic hetero-

geneous 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 methodol-

ogy, and fully turbulent models with Reynolds stresses and fluxes. It is worth


noting that nonlocal mathematical modeling is very different from hom-

ogenization modeling. The new integrodifferential transport statements in

heterogeneous media and application of these nonclassical types of equa-

tions is the current issue. The theory allows one to take into consideration

characteristics of multicomponent multiphase composites with perfect as

well as imperfect morphologies and interphases. The transport equations

obtained using VAT involved additional terms that quantify the influence of

the medium morphology. Various descriptions of the porous medium

structural morphology determine the importance of these terms and the

range of application of closure schemes.

Many mathematical models currently in use have not received a critical

review because there was nothing to review them against. The more

common models were compared with the more rigorous VAT-based models

and found deficient in many respects. This does not mean they do not serve

a useful purpose. Rather, they are incomplete and suffer from lack of

generality.

VAT-based modeling is very powerful, allowing random morphology

fluctuations to be incorporated into the VAT-based transport equations by

means of randomly varying morphoconvective and morphodiffusive terms.

Closure of some of the resulting morphofluctuation in the governing

transport equations has been outlined, resulting in some well-developed

closure expressions for the VAT-based transport equations in porous media.

Some of them exploit the properties of available solutions to transport

problems for individual morphological elements, and others are based on

the natural morphological data of porous media.

Statistical and numerical techniques were applied to classical irregular

morphologies to treat the morphodiffusive and morphoconvective terms

along with integral terms. The challenging problem in irregular and random

morphologies is to produce an analytical or numerical evaluation of the

deviations in scalar or vector fields. In previous work, the authors have

presented a few exact closures for predetermined regular and random

porous medium morphologies. The questions related to effective coefficient

dependencies, boundary conditions, and porous medium experiment analy-

sis 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


performance of a heat exchanger depends on the design criteria for optimiz-

ing the liquid flow velocity, dimensions of the heat exchanger, the hea‘t

transfer area between the hot side and cold side, etc. However, the optimiz-

ation problems involving such designs are more complex than for tradi-

tional designs and require new optimal control simulation techniques.



the physics of the process. Many of them have a fairly complicated form,

and the contemporary literature on optimal control considers much simpler

formulations of the optimization problems with constraints in the form of

differential equations. Linear optimal control systems governed by parabolic

partial differential equations (PDEs) are relatively well studied in the

literature. The modeling CHE equations resulting from VAT-based analysis

are also PDEs, but they are nonlinear and have additional integral and

integrodifferential terms.

It is well known that some matrix composites (often porous) represent the

promise for design of a series of materials with highly desirable characteris-

tics such as high temperature accommodation and enhanced toughness.

Their performance is very dependent on the volume fraction of the consti-

tuent 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 contain-

ing the design objectives.

The importance of the physical processes taking place in a heterogeneous

multiscale—multiphase—composite medium creates the need for the develop-

ment of new tools to characterize such media. It leads to the development

of new approaches to describing these processes. One of them (VAT) has

great advantages and is the subject of this review.

Acknowledgments

This work was partly sponsored by the Department of Energy, Office of Basic Energy

Sciences, through the grant DE-FG03-89ER14033 A002.

Nomenclature

a thermal diffusivity [m�/s] c��

mean skin friction coefficient

c�

mean drag resistance coefficient over the turbulent area of

in the REV [-] �S�[-]


c��

mean form resistance coefficient

in the REV [-]

c��

drag resistance coefficient upon

single sphere [-]

c�!

mean skin friction coefficient

over the laminar region inside

of the REV [-]

c�

specific heat [J/(kg · K)]C

�constant coefficient in

Kolmogorov turbulent

exchange coefficient correlation

[-]

d��

character pore size in the cross

section [m]

d�

diameter of ith pore [m]

d�

particle diameter [m]

ds interphase differential area in

porous medium [m�]

D�

molecular diffusion coefficient

[m�/s]; also tube or pore

diameter [m]

D�

flat channel hydraulic diameter

[m]

D

diffusion coefficient in solid

[m�/s]

�S�

internal surface in the REV

[m�]

f� � f �

averaged over ��

value f —intrinsic averaged variable

� f ��

value f, averaged over ��

din

an REV —phase averaged

variable

f� morphofluctuation value of fin a �

�g gravitational constant [1/m�]

H width of the channel [m]

h averaged heat transfer

coefficient over �S�

[W/(m�/K)]; half-width of the

channel [m]

h�

pore scale microroughness

layer thickness [m]

�S�

internal surface in the REV

[m�]

k�

fluid thermal conductivity [W/

(mK)]k

solid phase thermal

conductivity [W/(mK)]K permeability [m�]

K�

turbulent kinetic energy

exchange coefficient [m�/s]

K�

turbulent diffusion coefficient

[m�/s]

K�

turbulent eddy viscosity [m�/s]

K�

effective thermal conductivity

of solid phase [W/(mK)]K�

turbulent eddy thermal

conductivity [W/(mK)]l turbulence mixing length [m]

L scale [m]

�m� averaged porosity [-]

m

surface porosity [-]

n number of pores [-]

n�

number of pores with diameter

of type i [-]

Nu�%� �

h�d�

��

, interface surface

Nusselt number [-]

p pressure [Pa]; or pitch in

regular porous 2D and 3D

medium [m]; or phase function

[-]

Pe�

�Re�Pr, Darcy velocity pore

scale Peclet number [-]

Pe�

�Re�Pr, particle radius Peclet

number [-]

Pr ��a�

, Prandtl number [-]

Q

outward heat flux [W/m�]

Re��

Reynolds number of pore

hydraulic diameter [-]

Re�

��m�u!� d

��

, Darcy velocity

Reynolds number of pore

hydraulic diameter [-]

Re�

�u!� d

��

, particle Reynolds

number [-]

Re�%� �

u!� d�%��

, Reynolds number of

general scale pore hydraulic

diameter [-]

S��

total cross-sectional area

available to flow [m�]

S�

specific surface of a porous

medium �S�/�� [1/m]

S��

�S�/�� [1/m]


S�

�S��

cross flow projected area

of obstacles [m�]

T temperature [K]

T#

characteristic temperature for

given temperature range [K]

T

solid phase temperature [K]

T�

wall temperature [K]

T

reference temperature [K]

U, u velocity in x direction [m/s]

u��

square friction velocity at the

upper boundary of HR

averaged over surface �S�

[m�/s�]

V velocity [m/s]

V�

�u!� �m� Darcy velocity [m/s]

W velocity in z direction [m/s]

Subscripts

e effective

f fluid phase

i component of turbulent vector

variable; or species or pore type

k component of turbulent

variable that designates

turbulent ‘‘microeffects’’ on a

pore level

L laminar

m scale value or medium

r roughness

s solid phase

T turbulent

w 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

��

averaged heat transfer

coefficient over �S�

[W/(m�K)]�� representative elementary

volume (REV) [m]

��

pore volume in a REV [m]

��

solid phase volume in a REV

[m]

��, �

�electric permittivity [Fr/m]

� dynamic viscosity [kg/(ms)] or

[Pas]

��

magnetic permeability [H/m]

� kinematic viscosity [m�/s]; also

�, frequency [Hz]

� density [kg/m]; also �, electric

charge density [C/m]

��

medium specific electric

conductivity [A/V/m]

� electric scalar potential [V]

� particle intensity per unit

energy (frequency)� ensemble-averaged value of ��

interface ensemble-averaged

value of �, with phase j being to

the left

� angular frequency [rad/s]

) magnetic susceptibility [-]

,�#� ,#

absorption coefficient [1/m]

,�� ,

scattering coefficient [1/m]

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Two-Phase Flow in Microchannels

S. M. GHIAASIAAN and S. I. ABDEL-KHALIK

G. W. Woodruff School of Mechanical Engineering

Georgia Institute of Technology

Atlanta, Georgia 30332

I. Introduction

The application of miniature thermal and mechanical systems is rapidly

increasing in various branches of industry. Recent technological advances

have led to extremely fine spatial and temporal thermal load resolutions,

requiring the analysis of microscale heat transfer phenomena where the

system characteristic dimension can be smaller than the mean free path of

the heat carrying particles [1].

In this article the recently published research dealing with gas— liquid

two-phase flow in microchannels is reviewed. Only microchannels with

hydraulic diameters of the order of 0.1 to 1 mm and with long length-to-

hydraulic diameter ratios are considered. In these systems the channel

characteristic dimension is of the same order of magnitude, or smaller than,

the neutral interfacial wavelengths predicted by the Taylor stability analysis.

The review is also restricted to situations where the fluid inertia is significant

in comparison with surface tension. Such microchannels and flow conditions

are encountered in miniature heat exchangers, research nuclear reactors,

biotechnology systems, the cooling of high-power electronic systems, the

cooling of the plasma-facing components in fusion reactors, and the heat

rejection systems of spacecraft, to name a few. The flow through cracks and

slits when such cracks develop in vessels and piping systems containing

high-pressure liquids is another application of two-phase flow in microchan-

nels of interest here. Two-phase flow in capillaries with complex geometry

(porous media) has been reviewed in the recent past [2—4] and will not be

addressed.




0065-2717/01 $35.00

II. Characteristics of Microchannel Flow

For steady-state and developed two-phase flow in a smooth pipe, and

assuming incompressible phases, the variables that can affect the hy-

drodynamics of gas—liquid two-phase flow are �!, �

0, �

!, �

0, �, D, g, 1! , �,

U!�

, and U0�

. Since the minimum number of reference dimensions in

hydrodynamics is three (time, mass, and length), according to the Buckin-

ham theorem eight independent dimensionless parameters can be defined

that in general affect the hydrodynamics of gas— liquid two-phase flow. The

following three dimensionless parameters are particularly important for the

characterization of microchannels:

Eo��gD�

�(1)

We!�

�U�!�

D�!

�(2)

We0�

�U�0�

D�0

�. (3)

The remainder of the dimensionless parameters can be represented as ��/�!,

�0/�

!, 1! , �, and the phasic superficial Reynolds numbers:

Re!�

� U!�

D/�!

(4)

Re0�

� U0�

D/�0. (5)

Note that the phasic Froude numbers Fr0�

�U�0�

/gD and Fr!�

� U�!�

/gD ,

and the capillary number Ca��!U!�

/�, can all be derived by combining

two or more of these dimensionless parameters. When Eo, We!�

, and We0�

are all less than 1, gravitation and inertia are both insignificant in compari-

son with surface tension. Re!�& 1, furthermore, implies small inertia com-

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

tension mostly dominates buoyancy, inertia can be significant. Similar

conditions apply to two-phase flow in microgravity, resulting in important

and useful similarities between the two categories of systems with respect to

hydrodynamics of two-phase flow. In both types of systems the predomin-

ance of the surface tension force on buoyancy leads to the insensitivity of

two-phase hydrodynamics to channel orientation, and in nonseparated

two-phase flow patterns it leads to the suppression of velocity difference

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

between the two phases in the absence of significant acceleration.

Suo and Griffith [9] derived the following criterion for negligible buoy-

ancy effect in two-phase pipe flow:

�!gD�

�& 0.88/3. (6)

Based on an analysis of stratified to nonstratified flow regime transition,

Brauner and Moalem-Maron [10] derived the following criterion for the

predominance of surface tension on buoyancy:

Eo& (2')�. (7)

Experiments with water and air flowing in pipes indicate that the transition

to the surface tension—dominated regime (where flow patterns are not

affected by channel orientation) occurs in the 1&D& 2mm 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(�), (8)

where

��

g��. (9)

The Laplace length scale, �, represents the order of magnitude of the

wavelength of the interfacial waves in Taylor instability, and the latter

instability type is known to govern important hydrodynamic processes such

as bubble and droplet breakup. Equation (8) evidently implies that some

Taylor instability-driven processes may be entirely irrelevant to microchan-

nels. The criterion of Suo and Griffith, Eq. (6), can approximately be recast

as

D� 0.3�. (10)

III. Two-Phase Flow Regimes and Void Fraction in Microchannels

The gas and liquid phases in a two-phase flow system can exist in various

distinct morphological configurations. Flow regimes represent the major

morphological configurations of the phases and are among the most

important characteristics of two-phase flow systems, since they strongly

influence all the hydrodynamic and transport processes, such as pressure

147two-phase flow in microchannels

drop, heat and mass transfer, and flow stability. A methodology for

predicting the two-phase flow regimes is thus required for the design of

two-phase flow systems and for the specification of appropriate closure

relations for two-phase conservation equations.

Flow regimes and conditions leading to their establishment have been

extensively investigated in the past several decades. Methods for predicting

the flow regimes are often based on the flow regime map concept, according

to which the empirically determined ranges of occurrence of all major flow

patterns are specified on a two-dimensional map, with the two coordinates

representing some appropriate hydrodynamic parameters [13, 14]. How-

ever, since the gas— liquid hydrodynamics are affected by a large number of

independent dimensionless parameters, the two-dimensional flow regime

maps are often in disagreement with respect to the parameters they use as

coordinates and their ranges of applicability are limited to the ranges of

their databases. More recently, semianalytical methods, where the flow

regime transition processes are mechanistically or semianalytically modeled,

have been proposed [15—17] and have undergone extension and improve-

ment [18—20]. The existing flow regime maps, as well as the aforementioned

semianalytical models, however, generally do poorly when compared with

experimental two-phase flow regime data representing microchannels.

A. Definition of Major Two-Phase Flow Regimes

Experiments indicate that flow regimes, which are morphologically simi-

lar to the major two-phase flow regimes common in large channels, occur

in microchannels as well. Therefore, a brief review of the major flow regimes

observed in large channels is provided in this section. Detailed explanations

of these flow regimes and their characteristics can be found in various

textbooks and monographs [13, 14] and in more recent review articles [18].

Figure 1 schematically depicts the major flow regimes in common

gas—Newtonian liquid two-phase flow systems in large vertical channels.

Bubbly flow occurs at low gas and liquid flow rates and is characterized by

bubbles distorted-spherical in shape and moving upward in zigzag fashion.

The slug flow regime is characterized by bullet-shaped Taylor bubbles that

have diameters close to the diameter of the channel and are separated from

the channel wall by a thin liquid film, with lengths that can widely vary and

may reach more than 15 times the channel diameter. The Taylor bubbles

are separated by liquid slugs that often contain small entrained bubbles. The

churn flow is established following the disruption of the Taylor bubbles due

to high gas flow rates and is characterized by chaotic oscillations and

churning. In the annular flow pattern a thin liquid film, which can be

smooth or wavy depending on the gas velocity, flows on the wall, while the


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

gas flows through the channel core. The gas often contains dispersed

droplets. In the dispersed-bubbly flow regime, which is established at very

high liquid flow rates, small spherical bubbles with little or no interaction

with each other are mixed with the liquid. Unlike the common bubbly flow

regime, in which the bubble size is controlled by Taylor instability and

aerodynamic forces, the size of the bubbles in the dispersed bubbly flow

regime is dictated by turbulence in the liquid.

The commonly observed two-phase flow patterns associated with the flow

of a gas and a Newtonian liquid in a horizontal large channel are depicted

in Fig. 2. Bubbly flow occurs at high liquid and low gas flow rates and is

followed by plug (elongated bubble), slug, and annular/dispersed flow

patterns as the gas flow rate is increased. The stratified-smooth flow regime

occurs at low liquid and low gas flow rates and is followed by stratified wavy

and annular/dispersed flow regimes as the gas flow rate is increased. The

dispersed-bubbly regime occurs at very high liquid flow rates, and its

characteristics are similar to those of the dispersed-bubbly flow regime in

vertical channels. The plug and slug flow patterns are often referred to

collectively as the intermittent flow pattern, since the distinction between

them is not always clear or important.

The flow patterns in Figs. 1 and 2 only display the major flow regimes

that are easily discernible visually and with simple photographic techniques

and are commonly addressed in flow regime maps and transition models.


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

Numerous subtle variations within some of the flow patterns can be

recognized using more sophisticated techniques, however [21].

B. Two-Phase Flow Regimes in Microchannels

Early studies dealing with two-phase flow in microchannels were mostly

concerned with surface tension-driven flows [5—8]. Two-phase flow regimes

in microchannels under conditions where inertia is significant have been

experimentally investigated by Suo and Griffith [9], Oya [22], Barnea et al.[23], Damianides and Westwater [11], Barajas and Panton [24], Fukano

and Kariyasaki [25], Mishima and Hibiki [26], and Triplett et al. [12].

Two-phase flow patterns in narrow, rectangular channels, some simulating

slits and cracks, have also been reported in [27—33]. Narrow et al. [34] and

Ekberg et al. [35] studied the two phase flow regimes in a micro-rod bundle


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

and in narrow annuli, respectively. Two-phase flow and transport phenom-

ena 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 1.45mm diam-

eter, and in microchannels with semitriangular (triangular with one corner

smoothed) cross sections with 1.09 and 1.49mm hydraulic diameters (seeFig. 3). They identified the flow regimes using high-speed videocameras

recording flow details near the centers of the test sections. Figure 4 displays

typical photographs of the flow patterns identified in their 1.09-mm-

diameter circular test section. The overall flow pattern morphologies ob-

served with the other test sections used by Triplett et al. [12] were similar

to the pictures in Fig. 4.


Fig. 4. Representative photographs of flow patterns in the 1.1-mm-diameter test section of

Triplett et al. [12]. (With permission from [12].)


Bubbly flow (Fig. 4a) was characterized by distinct and distorted (non-

spherical) bubbles, typically considerably smaller in diameter than the

channel. The slug flow (Fig. 4b) is characterized by elongated cylindrical

bubbles. This flow pattern has been referred to as slug by some investigators

[9, 26] and plug by others [11, 24]. Unlike plug flow in larger channels

where the elongated gas bubbles typically occupy only part of the channel

cross section (Fig. 2), the bubbles in slug flow in microchannels appear to

occupy most of the channel cross section [12, 26].

Figures 4c and 4d display the churn flow pattern in the experiments of

Triplett et al. [12], who assumed two processes to characterize churn flow.

In one process, the elongated bubbles associated with the slug flow pattern

become unstable as the gas flow rate is increased and their trailing ends are

disrupted into dispersed bubbles (Fig. 4c). This flow pattern has been

referred to as pseudo-slug [9], churn [26], and frothy-slug by Zhao and

Rezkallah [38] in their microgravity experiments. The second process that

characterizes churn flow is the occurrence of churning waves that period-

ically disrupt an otherwise apparently wavy-annular flow pattern (Fig. 4d).This flow pattern is referred to as frothy slug-annular by Zhao and

Rezkallah [38]. At relatively low liquid superficial velocities, increasing the

mixture volumetric flux leads to the merging of long bubbles that charac-

terize slug flow, and to the development of the slug—annular flow regime

represented by Fig. 4e. In this flow pattern long segments of the channel

support an essentially wavy-annular flow and are interrupted by large-

amplitude 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 represen-

ted by Fig. 4f is established.

C. Review of Previous Experimental Studies and Their Trends

The important studies of microchannel two-phase flow that have ad-

dressed parameter ranges of interest here are reviewed, and their experimen-

tal results are compared, in this section. Table I is a summary of the

experimental investigations reviewed here.

1. General Trends

The study by Suo and Griffith [9] is among the earliest experimental

investigations. They could observe slug—bubbly, slug, and annular flow

patterns. They observed no stratification, attributed its absence to the

predominane of surface tension over buoyancy, and proposed the criterion

in Eq. (6). Oya [22, 39] was concerned with flow patterns and pressure drop


TABLE I

Summary of Experimental Data for Microchannel and Narrow Passage Two-Phase Flow Regimes

U0�

U!�

Author Orientation Channel characteristics Fluids (m/s) (m/s)

Water—N�,

heptane—N�,

Suo and Griffith [9] Horizontal Circular, D � 1.0 and 1.4 mm

heptane—He

Not given Not given

Barnea et al. [23] Horizontal and Glass, circular, D � 4—12.3 mm Water—air 0.04—60 0.002—10

vertical

Triplett et al. [12] Horizontal Pyrex, circular, D � 1.1 and 1.45mm; Water—air 0.02—80 0.02—8.0

semitriangular (Fig. 3), D�� 1.1 and

1.49mm

Damianides and Westwater [11] Horizontal Pyrex, circular, D � 1—5 mm; stack of fins Water—air 0.03—100 0.08—10

Fukano and Kariyasaki [25] Horizontal Circular, D � 1, 2.4, 4.9, 9 and 26mm Water—air 0.1—30 0.02—2

and vertical

Mishima and Hibiki [26] Vertical Pyrex and aluminum, D� 1.05—4.08mm Water—air 0.1—50 0.02—2

Barajas and Panton [24] Horizontal Pyrex, polyethylene, polyurethane, Water—air 0.1—100 0.003—2

fluoropolymer resin

154

Narrow et al. [34] Horizontal Glass, seven-rod bundle, D�� 1.46mm Water—air 0.02—40 003—5.0

Lowry and Kawaji [27] Vertical Rectangular, W� 8 cm, L � 8 cm, S � 0.5, Water—air 0.1—18 0.1—8

1, 2mm

Wambsganss et al. [28] Horizontal Rectangular, W� 19.05mm, L � 1.14m, Water—air 0.05—30 0.2—2

S � 3.18mm

Ali and Kawaji [29] Horizontal/ Rectangular, W� 80mm, L � 240mm, Water—air 0.15—16 0.2—7.0

Vertical S � 1.465mm

Ali et al. [30] Horizontal/ Rectangular, W� 80mm, L � 240mm, Water—air 0.15—16 0.15—6.0

Vertical S � 0.778 and 1.465mm

Mishima et al. [31] Vertical Rectangular, W� 40mm, L � 1.5m, S � 1.07, Water—air 0.02—10 0.1—10

2.45, 5.0 mm

Wilmarth and Ishii [32] Horizontal/ Rectangular, L � 630mm; W� 15mm and Water—air 0.02—8 0.07—4.0

Vertical S � 1mm; W� 20mm and S � 2mm

Fourar and Bories [33] Horizontal Rectangular glass slit, W� 0.5 m, L � 2m, Water—air 0.0—10 0.005—1

S � 1mm; brick slit, W� 14 cm, L � 28 cm,

S � 0.18, 0.4, 0.54 mm

Ekberg et al. [35] Horizontal Glass annuli; D�� 6.6 mm, D

%� 8.6mm; and Water—air 0.02—57 0.1—6.1

D�� 33.2mm and D

%� 35.2mm; L � 35 cm

for both annuli

155

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

resulting from the confluence of air— fossil liquid fuel in short vertical tubes

with 2, 3, and 6mm diameters, and L /D ratios of 20 to 25, and could identify

nine distinct flow patterns. Because of the predominance of entrance effects,

however, Oya’s data may not be representative of fully developed flow

patterns.

In the study by Barnea et al. [23], air—water flow regimes were compared

with the flow regime transition models of Taitel and Dukler [15] for

horizontal flow and Taitel et al. [12] for vertical flow, with some minor

modifications. The latter models predicted their data well. Since the tube

diameters were relatively large, however, their data clearly show the effects

of gravity and test section orientation.

The two-phase flow regime data of Triplett et al. [12] are shown in Fig.

5. Regime transition lines representing a micro-rod bundle (Narrow et al.[34]) are also depicted and are discussed in Subsection E of this part. The

flow patterns representing the four test sections of Triplett et al. are similar,

and none of the test sections supported stratified flow. The depicted flow

patterns indicate the predominance of intermittent (slug, churn, and slug—annular) flow patterns that together occupy most of the maps.

The two-phase flow regimes representing the flow of air—water mixture in

glass tubes with D � 1mm to 2.4mm reported by several authors are


Fig. 6. Comparison among air—water flow regime maps obtained in glass tubes with

D� 1mm. (Symbols represent the test section (a) in Fig. 3 [12]).

depicted in Fig. 6. For the air—water—Pyrex system 1! $ 34° [40], implying

a partially wetting liquid. In Fig. 6, the flow pattern names in capital and

bold letters represent those reported by Damianides and Westwater [11]

and Fukano and Kariyasaki [25], respectively; the lowercase letters are

from Mishima and Hibiki [26]; and the symbols represent the data of

Triplett et al. [12].

Damianides and Westwater [11] were concerned with two-phase flow

patterns in compact heat exchangers. The flow patterns in their 1 and 2mm

diameter tubes, which are of interest here, included dispersed bubbly,

bubbly, plug, slug, pseudoslug, dispersed-droplet, and annular. As noted, the

flow pattern identified as churn by Triplett et al. (Figs. 4c and 4d) appears

to coincide with the flow pattern identified as dispersed by Damianides and

Westwater. Furthermore, the slug and slug—annular regimes in Triplett’s

experiments (Figs. 4e and 4f ) coincide with the plug and slug flow regimes

in Damianides and Westwater, respectively. These differences are evidently

associated with subjective identification and naming of flow patterns, and

the two experimental sets are otherwise in good overall agreement.


Fukano and Kariyasaki [25] reported no significant effect of channel

orientation on the flow patterns for channel diameters 4.9mm and smaller.

Fukano and Kariyasaki identified only three flow patterns: bubbly, inter-

mittent, 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 2.4mm-diameter test sections are

depicted in Fig. 6. Their data are evidently in disagreement with the data of

Triplett et al. [12], and Damianides and Westwater [11], except for the

intermittent-to-bubbly flow transition line, where all three data sets are in

good agreement.

In the investigation by Mishima and Hibiki [26], except for void fraction

measurements, which were carried out in aluminum test sections, all

experiments were performed in Pyrex test sections, and flow regimes were

identified using a high-speed camera. The identified flow regimes were

bubbly, slug, churn, annular, and annular-mist. Mishima and Hibiki com-

pared their data representing 2.05 and 4.08mm 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

2.05mm 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 1.05mm diameter test section were similar to the patterns

for their 2.05mm diameter test section.

2. Effect of Surface Wettability

The experimetal studies just discussed all utilized materials that represen-

ted partially wetting (�& 90°) conditions. In view of the significance of

surface tension, however, the surface wettability can evidently affect the

two-phase flow hydrodynamics in microchannels. Barajas and Panton [24]

conducted experiments with air and water, using four different channel

materials. These included Pyrex (� � 34°), polyethylene (� � 61°), and

polyurethane (�� 74°) as partially wetting; and the FEP fluoropolymer

resin (�� 106°) as a partially nonwetting combination. Figure 7 displays a

summary of their flow regime maps, where the data of Triplett et al. [12]

representing their 1.09mm 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-


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’ 1mm and 2mm

diameter test sections (which were also made of Pyrex), with the excep-

tion of the wavy stratified flow pattern, which did not occur in the 1mm

test section of Damianides and Westwater. With the other partially wetting

test sections, polyethylene and polyurethane, the flow regimes and their

ranges of occurrence were similar to those obtained with Pyrex, with the

difference that with polyethylene and polyurethane the wavy flow pattern

was now replaced with a flow regime characterized by a single rivulet. A

small multirivulet region also occurred on the flow regime map representing

the polyurethane test section. The flow regimes observed with the partially

nonwetting channel FEP fluoropolymer were significantly different,

however, and compared with the partially wetting tubes, the ranges of

occurrence of the rivulet and multirivulet flow patterns were significantly

wider.

3. Flow Regimes in Microgravity

As mentioned earlier in Part II, dimensional analysis indicates that

two-phase flow in common large channels in microgravity has important

similarities with two-phase flow in terrestrial microchannels, since in both


Fig. 8. Comparison of microchannel flow regime data of Triplett et al. for their 1-mm

diameter circular test section (Fig. 3a) with the microgravity data of Zhao and Rezkallah [43]

and Bousman et al. [45].

systems the surface tension predominates buoyancy, while inertia can be

significant.

Experiments conducted aboard aircraft flying parabolic trajectories that

can maintain microgravity (�0.02g) for periods up to 22 seconds have been

reported by several research groups. Based on the published data, empirical

correlations for flow regime transitions applicable over wide ranges of fluid

properties have been proposed by Rezkallah [43] and Jayawardena et al.[44]. Zhao and Rezkallah [38] performed experiments in 9.52mm and

12.7mm diameter tubes, where the regimes associated with water—air

two-phase flow were identified using video cameras. Four major flow

patterns were identified: bubbly, slug, frothy slug—annular, and annular. The

bubbly, slug, and annular flow regimes were morphologically similar to

those described before (see Figs. 4a, 4b, and 4f ). The frothy slug—annular

regime as described by Zhao and Rezkallah, however, is similar to the flow

pattern depicted in Fig. 4d and has been called the pseudo-slug by Suo and

Griffith [9], and churn by others [12, 26]. Figure 8 compares the flow

regimes of Triplett et al. [12] representing their test section (a) (see Fig. 3),with the experimental flow regime transition lines of Zhao and Rezkallah

[38].

Bousman et al. [45] utilized test sections with 12.7 mm and 25.4mm

diameters, and studied the two-phase flow regime, void fraction, and liquid

film thickness in the annular regime, using air—water, air—water� glycerin


(�!� 6 cP, �� 63 dyn/cm), and air—water�Zonyl FPS (a surfactant; re-

sulting in �!� 1 cP, �� 21 dyn/cm) mixtures. Bousman et al. identified four

major flow regimes similar to those reported by Zhao and Rezkallah [38].

Their experimental flow regime transition lines are also depicted in Fig. 8

and are noted to be in relatively poor agreement with the data of Triplett

et al. [12] with respect to the bubbly—slug flow regime transition, and in

good agreement with respect to the transition to annular flow.

D. Flow Regime Transition Models and Correlations

As noted earlier, theoretical arguments and experimental evidence indi-

cate that flow patterns in microchannels should be insensitive to gravi-

tational field, and therefore to channel orientation. Criteria for calculating

the maximum channel size for which gravity is inconsequential have been

proposed in [9, 10] (see Eqs. (6—8)). Equation (6) [9], which provides a

criterion for the dominance of surface tension over buoyancy to the extent

that for channel diameters smaller than the provided limit the two-phase

flow patterns are not affected by the channel orientation, agrees with the

experimental data of [11, 12].

The flow regime map of Mandhane et al. [41], a widely used empirical

flow regime map for common horizontal channels, was compared with

microchannel data by some authors with poor agreement [12, 25].

Barnea et al. [23] invetigated the two-phase flow regimes in small

channels (4 mm�D� 12.3mm) and extended the methodology of Taitel

and Dukler [15] for horizontal flow. In the original model of Taitel and

Dukler [15], transition from stratified to intermittent flow regimes is

assumed to take place when small but finite amplitude disturbances that

occur on the liquid surface grow. The position of the liquid surface (i.e., the

liquid depth in the channel) is predicted from the solution of one-dimen-

sional (1D) gas and liquid momentum conservation equations assuming

steady-state and fully developed stratified flow [15]. Barnea et al. [23]

argued that in very small channels the predominance of surface tension on

gravitational force, and not the interfacial wave instability, is responsible for

regime transition from stratified to intermittent. Based on a simple model,

Barnea et al. proposed that the latter flow regime transition occurs in very

small channels when the liquid depth in the channel, found from the solution

of 1D momentum equations for steady-state and fully developed stratified

flow, satisfies the following equation:

D � h!��

��0(1 �'/4)�

� �. (11)

Barnea et al. [23] also argued that when D is smaller than the right-hand


side of the preceding equation, the aforementioned flow regime transition

occurs when

h!2�1�

'4� D. (12)

The modified Taitel and Dukler [15] models for horizontal flow and the

flow regime transition models of Taitel et al. well predicted the aforemen-

tioned data of Barnea et al. [23]. When applied to smaller channels,

however, the aforementioned semianalytical flow regime transition models

appear to do poorly [11, 12], with the exception of the model of Taitel and

Dukler [15] for the establishment of dispersed bubbly flow.

For the transition to dispersed bubbly flow, Taitel and Dukler [15]

derived the following relation based on a mechanistic model according to

which in the latter regime spherical bubbles have diameters within the size

range of inertial eddies predicted by Kolmogorov’s theory of locally iso-

tropic turbulence, and their diameters are controlled by interation with the

latter eddies:

U!�

�U0�

� 4.0 �D��(�/�

!)��

��!

�g(�

!��

0)

�!

��

�. (13)

This relation is valid as long as �& 0.52, the latter approximately

representing the upper limit of void fraction for the existence of spherical

bubbles. In large horizontal channels, in addition to Eq. (13), another

criterion should be met according to which turbulence dominates over

buoyancy so that bubbles do not gather near the channel top [15, 20]. The

latter criterion is evidently redundant in microchannels where buoyancy

effect is insignificant. Although the basic assumptions for the development

of Eq. (13) are usually not met in microchannels [12], it appears to predict

well the transition line representing the development of bubbly flow [12, 25].

Based on the drift flow model, and arguing that the void fraction is a

suitable parameter for correlating flow regime transitions, Mishima and

Ishii [42] derived expressions for flow regime transition in vertical, upward

flow. The major flow regimes considered were bubbly, slug, churn, and

annular. The transition from bubbly to slug flow was assumed to occur

when a void fraction of 0.3 is reached, and led to

U!�

��3.33

�� 1� U

0��

0.76

C%��g��!2 �

� �. (14)

The slug-to-churn transition was assumed to occur when the liquid slugs

become unstable because of the wake effect caused by the Taylor bubbles.


For this transition, Mishima and Ishii derived

�� 1 � 0.813(C

%� 1)(U

!��U

0�) � 0.35�gD��/�

!

U!�

�U0�

� 0.75 gD��!�gD�

!��

��!

��

(15)

where � is predicted using the following expression provided by the drift flux

model [46, 47]:

��U0�

C%(U

!��U

0�) �V

&�

. (16)

Mishima and Ishii [42] assumed that the transition from churn to annular

flow occurred either because of flow reversal in the liquid films separating

large bubbles from the wall, or because of the disruption of liquid slugs. The

first mechanism is applicable when

D�

�g��

N��![(1 � 0.11C

%)/C

%]�

, (17)

where

N�!� �

! ��! �

g��

. (18)

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

when

U0�

� (� � 0.11) gD��0

. (19)

(Note than � used in the preceding equation must be larger than the

right-hand side of Eq. (15)). The second mechanism causes the regime

transition when

U0�

� ��g��0��

N��! . (20)

The drift flux parameters C

in the preceding expressions should be found

from [48]

C� 1.2—0.2��

0/�

!for round tubes (21)

C� 1.35—0.35��

0/�

!for rectangular ducts. (22)

Mishima and Hibiki [26] compared their experimental data representing


two-phase flow in tubes with 1.05mm to 4.08mm diameters with the

aforementioned flow regime transition models of Mishima and Ishii [42],

apparently with good agreement, despite the fact that the empirical drift flux

model parameters (namely C

and V&�

) of the latter authors may be

inappropriate for microchannels. The representation of V&�

in the derivation

of the foregoing transition models based on expressions valid for larger

channels is evidently in disagreement with experimental data associated

with microchannels, where velocity slip and the buoyancy effect are both

negligible.

Zhao and Rezkallah [38] and Rezkallah [43], in correlating their micro-

gravity flow regime data, argued that when the buoyancy force is negligible

compared with surface tension while inertia is significant, the phasic Weber

numbers are the most appropriate dimensionless parameters for the corre-

lation of flow regime transitions. Based on their experimental data, Zhao

and Rezkallah [38] correlated the bubbly-to-slug transition assuming equal

phasic velocities, and that this transition occurs at a void fraction of

�� 0.18, and derived U!�

� 4.56U0�

accordingly.

Zhao and Rezkallah empirically correlated the flow regime transitions

from slug to frothy slug—annular (equivalent to the slug—annular flow

pattern in Triplett et al. [12]; see Fig. 4e) according to We0�

� 1 and from

frothy slug—annular to annular flow according to We0�

� 20.

The preceding correlations well predicted the experimental data of Zhao

and Rezkallah [38]. When data from several other sources were also

considered, the aforementioned We0�

� constant correlations were found

inadequate [43]. However, correlations in the form We0� constant could

well predict the entire data [43], where

We0� �

0U�0D/�, (23)

with U0

representing the gas phase velocity. The latter We0� constant

correlations are inconvenient for application, however, since the void

fraction is needed for the calculation of U0. Rezkallah [43], however,

showed that the data from several microgravity sources, which covered a

relatively wide range of fluid properties, could be well correlated in a

two-dimensional map using We0�

and We!�

as coordinates, provided that

the entire flow regime map is divided into three regions: the surface tension

region including bubbly and slug flow regimes; the intermediate (transi-

tional) region including the frothy slug—annular regime; and the inertial

region representing annular flow.

The experimental data of Triplett et al. [12] representing all four of their

test sections, and the experimental flow regime transition lines of Damian-

ides and Westwater [11] representing their 1mm diameter test section, are

depicted in a flow regime map with We!�

and We0�

as coordinates in Fig. 9.


Fig. 9. Comparisn of microchannel data representing D$ 1mm with the microgravity flow

regime transition lines of Rezkallah [43]. Capitals: data from (11); Lowercase: data from [43].

As noted, the depicted data agree with the empirical transition lines of

Rezkallah [43] relatively well with respect to the flow regime transitions

among surface tension (bubbly or slug), transitional, and inertia (annular)regions. The transitional region in Fig. 9 includes the churn (Figs. 4c and

4d) and slug—annular (Fig. 4e), also referred to as pseudoslug [9] and frothy

slug—annular [38] flow patterns. The data of Fukano and Kariyasaki [25]

are not shown, since the latter authors defined only three flow patterns

(bubbly, intermittent, and annular).Bousman, McQuillen, and Witte [45], in correlating their microgravity

flow regime data, argued for the use of the void fraction as the criterion for

flow regime transition. The transition from bubbly to slug flow was assumed

to occur when �� 0.4, and transition to annular flow was assumed to take

place when �� 0.70—0.75, with the exact value depending on the liquid type.

They thus divided the entire flow regime map into three regions: bubbly,

transitional (slug and slug/annular), and annular. They calculated the void

fractions used for the derivation of the aforementioned criteria using the

drift flux model [46, 47], Eq. (16), assuming V&�

� 0 because of the absence

of velocity drift in nonseparated flow patterns in microgravity. Based on

their measurement of void fractions in their smaller (12.7mm diameter) test

section, they derived C� 1.21 and assumed that the same value was


applicable to their larger (25.4mm diameter) test section. The �� 0.4

criterion for transition from bubbly to slug flow patterns is in disagreement

with the aforementioned �� 0.18 criterion suggested by Zhao and Rezkal-

lah [38]. It is also in disagreement with the 0.25 to 0.3 value in large

terrestrial channel flow data [17, 42].

Jayawarden et al. [44] considered the aforementioned data of Zhao and

Rezkallah [38] and Bousman et al. [45], as well as data from several other

sources covering fluids with a wide range of properties. Similar to Bousman

et al. [45], they divided the entire flow regime map into three zones: bubbly,

slug/slug—annular, and annular. They argued that the phasic superficial

Weber and Reynolds numbers were the primary dimensionless parameters

that should be used for correlating regime transitions in microgravity. They

demonstrated that the transition from bubbly to slug/slug—annular for the

entire data considered could be correlated as

Re0�

Re!�

� K�Su�� (24)

where

Re0�

�U0�

D/�0

(25)

Re!�

�U!�

D/�!

(26)

Su�Re�

!�We

!�

��D�

!��!

. (27)

Equation (24) was recommended for the range 10�&Su& 10�. For transi-

tion to the annular flow regime, Jayawarden et al. [44] derived

Re0�

Re!�

� K�Su�� (28)

for Su& 10�, and

Re0�

�KSu� (29)

for Su� 10�, where K�� 4,641.6, and K

� 2 10��. Unfortunately, the

available microchannel two-phase flow data represent very narrow ranges

of the Su parameter, and meaningful comparison between the available data

and the aforementioned flow regime transition correlations of Jayawarden

et al. [44] is not feasible at this time.

E. Flow Patterns in a Micro-Rod Bundle

Micro-tube bundles have potential applications in miniature heat ex-

changers. Experimental data dealing with two-phase flow in micro-rod and

-tube bundles are scarce, however.


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

permission from [34].)

Narrow et al. [34] experimentally investigated the air—water two-phase

flow patterns and pressure drop in a horizontal, 23 cm long glass micro-rod

bundle that included seven rods configured as in Fig 10. Their test section was

entirely transparent, had an average hydraulic diameter of 1.29mm, and

included several short components specifically designed to reduce the

entrance and exit effects. They used a high-speed digital video camera near the

test section center to directly view subchannels 11 and 12 in Fig. 10, and

subchannels 5 and 6, which were visible through the transparent rod in front

of them. The flow regime map of Narrow et al. is depicted in Fig. 5. Froth flow

was characterized by the absence of a discernible interfacial geometry. In the

stratified— intermittent flow pattern the upper subchannels in the test section

were in plug or slug flow patterns, while some of the bottom subchannels

carried single-phase liquid. In the annular— intermittent flow pattern (a flow

pattern not reported in the past) the inner subchannels supported an

intermittent (slug or plug) or froth flow pattern, while the flow regime in the

peripheral subchannels was predominantly annular. In the annular—wavy

flow pattern, liquid films flowed on all rods and on the test section wall.

The flow regime maps representing single channels and the micro-rod

bundle depicted in Fig. 5 are in fair agreement with respect to annular and

churn or froth flow patterns. The range of occurrence of the slug—annular


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

flow pattern in single channels coincides with the annular— intermittent flow

pattern in the bundle. Furthermore, the plug/slug and the stratified-inter-

mittent flow patterns in the micro-rod bundle are replaced everywhere with

the slug flow pattern in single channels. The occurrence of the stratified—intermittent regime in the micro-rod bundle, which implies sensitivity to

buoyancy and rod bundle orientation, however, is a crucial difference in

comparison with single microchannels.

Horizontal rod bundles are used in CANDU nuclear reactors. The con-

ditions leading to bundle-wide or partial stratification in the latter rod bundles

may lead to dryout and critical heat flux and have been experimentally studied

in the past [49—51]. The micro-rod bundle flow regime map of Narrow et al.did not agree with the available flow regime maps for large horizontal rod

bundles. Bundle-wide stratification, which can readily occur in large horizon-

tal rod bundles [49—51], was not observed by Narrow et al. [34].

Narrow et al. [34] developed an empirical flow regime map, using void

fraction and mass flux as the coordinates (Fig. 11), where the void fraction

was predicted everywhere using the homogeneous flow assumption whereby

the forthcoming Eq. (32) with S� 1 was applied. For �& 0.25, transition

occurred at a mass flux of G � 1000kg/m�s. At higher void fractions, the

transition could be represented by the following equation:

ln(G) ��4.7�� 8.1. (30)


F. Void Fraction

Void fractions in microchannels have been measured by Kariyasaki et al.[52], Mishima and Hibiki [26], Bao et al. [53], and Triplett et al. [54].

Fukano and Kariyasaki [25] and Mishima and Hibiki [26] also attempted

to measure and correlate the velocity of large bubbles.

Equation (16), which is a result of the drift flux model [46, 47], has long

been utilized for the correlation of void fraction in two-phase channel flow.

In the latter equation the two-phase distribution coefficient, C, is the

cross-sectional average of the product of total volumetric flux and void

fraction, divided by the product of the averages of the two, while the gas

drift velocity, V&�

, is a weighted mean drift velocity of the gas phase with

respect to the mixture. The parameters C

and V&�

represent the global and

local interfacial slip effects, respectively, and are often determined empiri-

cally. Widely used correlations for C

and V&�

, developed based on experi-

mental data for common large channels, have also been used for correlating

microchannel data (Mishima et al. [31]). Correlations applied in this way

include Eq. (22). An empirical correlation based on the drift flux model

formulation has been developed by Chexal and co-workers [55], which is

based on an extensive database, includes a large number of empirically

adjusted parameters, and can address channels with various configurations.

The data base for this correlation does not include microchannels of interest

here, however.

Mishima and Hibiki [26] correlated their void fraction data for upward

flow in vertical channels, as well as the data of Kariyasaki et al. [52], using

the drift flux model [46], Eq (16), with V&�

� 0 for bubbly and slug flow

regimes. The distribution coefficient C, however, was found to be a function

of channel diameter and was correlated according to [52]

C� 1.2� 0.510e��. (31)

where D�

is in millimeters.

When the slip ratio, defined as S� U0/U

!, is known, the void fraction

can be calculated using the fundamental void-quality relation in one-

dimensional two-phase flow:

��x

x �S(�!/�

0)(1 �x)

. (32)

In homogeneous two-phase flow, S � 1. A correlation for the slip ratio,

proposed by the CISE group (Premoli et al. [56]) and recommended by

Hewitt [57], can be represented as

S � 1� B� �

y

1� yB�

yB��

� �(33)


where

y�U!�

/U0�

(34)

B�� 1.578Re��

!(�

!/�0)�� (35)

B�� 0.0273We

!Re��

!(�

!/�

0)�� (36)

Re!

�GD/�!

(37)

We!

�G�D/(��!). (38)

Bao et al. [53] measured pressure drop and void fraction in tubes with

0.74mm to 3.07mm diameters, using air and water mixed with various

concentrations of glycerin. The void fractions were measured using two

solenoid valves located near the two ends of the test sections, which could

be closed simultaneously. Bao et al [53] compared their void fraction data

with predictions of several correlations, all taken from literature dealing

with commonly used large channels, and based on the results they recom-

mended the aforementioned empirical correlations for the slip ratio Sproposed by the CISE group (Premoli et al. [56]).

Butterworth [58] has shown that the void fraction correlations of

Lockhart and Martinelli [59] and several other investigators can be

represented in the generic form

1 � ��

� A �1 �x

x ��

(�0/�

!)$(�

!/�0)� (39)

where A � 0.28, p� 0.64, q � 0.36, r� 0.07 for Lockhart and Martinelli

[59]. Bao et al. [53] found good agreement between their measured void

fractions and the predictions of the Lockhart and Martinelli [59] correla-

tion. Triplett et al. [54] compared their void fraction data, estimated from

photographs taken from their circular test sections, with predictions of the

preceding correlations of Lockhart and Martinelli as presented by Butter-

worth [58], the aforementioned correlation due to the CISE group [56, 57],

and the correlation of Chexal et al. [55]. Figure 12 depicts a typical

comparison between the data of Triplett et al. and the preceding correla-

tions. These comparisons indicated that with the exception of the annular

flow regime where all the tested correlations overpredicted the data, the

homogeneous model provided the best agreement with experiment.

G. Two-Phase Flow in Narrow Rectangular and Annular Channels

Two-phase flow in rectangular channels with (�O(1) mm occurs in the

coolant channels of research reactors, and during critical flow through

cracks that may occur in vessels containing pressurized fluids. Investigations


have been reported by [27—33]. The recent experimental investigations are

sumarized in Table I.

Experiments in vertical narrow channels, using air and water, with

consistent overall results with respect to the two-phase flow regime maps,

have been reported by Kawaji and co-workers [27, 29, 30], Mishima et al.[31], and Wilmarth and Ishii [32]. Minor differences with respect to the

description and identification of the flow patterns exist among these inves-

tigators, however.

Lowry and Kawaji [27] used strobe flash photography and could identify

bubbly, slug, churn, and annular flow regimes. In the bubbly flow the

bubbles were small and near-spherical. The slug flow was characterized by

large irregular and flattened bubbles, while the curn flow pattern contained

large irregularly shaped, as well as small, bubbles. Their flow transition lines

for the establishment of dispersed bubbly and annular flow patterns for the

test sections with (� 1 and 2mm disagreed with the models of Taitel et al.[17]. Ali and Kawaji [29] and Ali et al. [30] performed an extensive

experimental study using room-temperature and near-atmospheric air and

water in rectangular narrow channels with six different configurations:

vertical, cocurrent up and down flow; 45° inclined, cocurrent up and

downflow; horizontal flow between horizontal plates; and horizontal flow

between vertical plates. Their observed flow regimes and flow regime maps,

except for the last configuration, were similar and are displayed in Fig. 13.

The rivulet flow pattern occurred at very low liquid superficial velocities and

was relatively sensitive to the orientation of their test section. The flow

regimes for horizontal flow between vertical plates included bubbly, inter-

mittent, and stratified—wavy, and the flow regime maps for both gap sizes

((� 0.778mm and 1.465mm) were similar to the flow regime maps ob-

served in large pipes.

Mishima et al. [31] identified four major flow regimes in their experi-

ments: bubbly flow, characterized by crushed or pancake-shaped bubbles;

slug flow, represented by crushed slug (elongated) bubbles; churn flow, in

which the noses of the elongated bubbles were unstable and noticeably

disturbed; and annular flow. Figure 14 depicts the flow regimes in their

1.07mm gap test section. The flow regime map for their 2.4mm-gap test

section was similar except for the presence of a small churn region in the

latter. With a gap of (� 5.0mm, however, the flow regime transition

boundaries were displaced in comparison with Fig. 14. The predictions of

the following correlation for the slug—annular flow regime transition, due to

Jones and Zuber [60], are also shown in Fig. 14:

U!�

�1� �

��

U0�

�V&�

. (40)


Fig. 12. Comparison of some void fraction correlations with the measured data of Triplett

et al. [54] for the test section (a) depicted in Fig. 3: (a) homogeneous flow model; (b) Chexal

et al. [55]; (c) Lockhart—Martinelli—Butterworth, Eq. (39) [58]; (d) CISE [56]. (With


where �� 0.8 represents the void fraction for the slug—annular transition.

The correlation of Jones and Zuger [60] was based on air—water experi-

ments in a vertical retangular channel with (� 5mm.

The drift flux model, Eq. (16), with C

found from the aforementioned

correlation of Ishii [48], Eq. (22), and V&�

obtained from the forthcoming

Eq. (41) [60], could well predict all the void fraction data of Mishima et al.[31] for (� 1.07 and 2.45mm, except for the annular flow regime, for which

the data and correlation deviated significantly:

V&�

� (0.23 � 0.13(/W )(��gW /(). (41)

Wilmarth and Ishii [32] studied the two-phase flow regimes in vertical up


Fig. 12. (continued).

flow and horizontal flow between vertical plates. Their flow regimes for

vertical up flow are compared with the aforementioned data of Mishima etal. in Fig. 14, where bubbly and ‘‘cap-bubbly’’ flow patterns have been

combined in the depicted bubbly flow regime zone. In comparing their

vertical flow data with models, Wilmarth and Ishii noted relatively good

agreement for the flow regime transition from bubbly to slug, with the flow

regime transition models of Mishima and Ishii [42], Eq. (14), and Taitel etal. [17].


Fig. 14. Flow patterns in the experiments of Mishima et al. [31] and Wilmarth and Ishii

[32] in test sections with 1mm gap. (With permission from [32].)

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

For horizontal channels (i.e., flow between two horizontal parallel plates),several experimental studies have been published. Differences with respect

to the flow regime description and identification among various authors can

be noted, however. The experimental flow regimes of Ali et al. [30] were

shown in Fig. 13. For the horizontal flow configuration, Wilmarth and Ishii

[32] could identify stratified, plug, slug, dispersed bubbly, and wavy—


Fig. 15. Flow regimes in air—water experiments in horizontal rectangular channels. (With


annular flow patterns. In their experiments in similarly configured narrow

channels, as mentioned before, Ali et al. [30] identified bubbly, intermittent,

and stratified—wavy flow regimes only. The two flow regime maps are

compared in Fig. 15. The wavy—annular flow pattern occurred at the low

U!�

and high U0�

range, which appears to be outside the range of the

experiments of Ali et al. [30]. The two sets of data are qualitatively in

agreement with respect to the bubbly—plug/slug transition.

In the study by Wambsganss et al. [28], air—water tests were conducted

in transparent, horizontal rectangular test sections with two configurations,

one with the 3.18mm side oriented vertically (i.e., flow between two

horizontal plates), and the other with the 19.05mm side oriented vertically

(flow between two vertical plates). Their flow regime transition lines,

furthermore, disagreed with several flow regime maps that are based on

large channel data, including the flow regime map of Mandhane et al. [41].

Using an image processing technique, Wilmarth and Ishii [61] measured

the void fraction and interfacial concentrations in their air—water experi-

ments using vertical rectangular channels with (� 1 and 2mm, and

calculated the drift flux parameters C

and V&�

. The V&�

values have large

uncertainties. For the smaller gap, they found C� 0.81—1 for bubbly flow,

indicating that bubbles moved with a smaller velocity than liquid; and

C� 1 for the slug and churn flow patterns. For the larger channel they

obtained C� 0.4—1 for bubbly flow and C

� 1 and 1.2 for slug and

churn-turbulent flow regimes, respectively.


Fig. 16. Flow regimes of Fourar and Bories [33] for air—water flow between horizontal flat

plates with (� 1 mm. (With permission from [33].)

Fourar and Bories [33] conducted experiments in glass and brick slits,

addressing low-liquid superficial velocities. Figure 16 depicts their experi-

mental flow regime map. Bubbly flow was characterized by small, isolated

bubbles, whereas in the fingering bubbly flow large, flat, and unstable

bubbles were visible. The ‘‘complex’’ flow pattern was chaotic, without an

apparent structure (i.e., similar to froth flow). In the annular regime the

liquid was reported to have flowed in the form of unstable films on the walls

and may refer to the rivulet flow pattern identified by Ali et al. [30]. The

films were replaced by entrained droplets at very low liquid flow rates.

Fourar and Bories [33] also measured the average void fraction in their

test section by careful measurement of its water contents. In all flow regimes

excluding annular, their test section void fraction closely agreed with the

correlation

�� 1��X

1�X��, (42)

with the Martinelli factor, X, to be estimated from

X��!U!�

�0U0��

. (43)*

Recently, Ekberg et al. [35] conducted experiments using two horizontal


Fig. 17. Flow regimes in narrow horizontal annuli. Regime names in capital and small

letters are for the small and large test sections of Ekberg et al. [34], respectively. Regime names

in bold letters are from Osamusali and Chang [63]. (With permission from [35].)

glass annuli with 1.02mm spacing and studied the two-phase flow regimes,

void fraction, and pressure drop. The two-phase flow patterns in vertical

and horizontal large annular channels had earlier been studied by Kelessidis

and Dukler [62] and Osamusali and Chang [63], respectively. Osamusali

and Chang carried out experiments in three annuli, all with D� 4.08 cm,

and with D�/D

� 0.375, 0.5, and 0.625 ((� 4.75, 6.35 and 11.75mm,

respectively), and noted that the flow patterns and their transition lines were

relatively insensitive to D�/D

. The experimental flow regime transition lines

of Ekberg et al. [35] are displayed in Fig. 17, where they are compared with

the experimental results of Osamusali and Chung [63]. These transition

lines disagree with the flow regime map of Mandhane et al. [41]. Stratified

flow occurred in the experiments of Ekberg et al. [35].

Ekberg et al. [35] compared their measured void fractions with the

predictions of the homogeneous mixture model, the correlation of Lockhart

and Martinelli [59] as presented by Butterworth [58], Eq. (39), the

correlations of Premoli et al. [56, 57], Eqs. (33)—(38), and the drift flux

model, Eq. (16), with C� 1.25 and V

&�� 0, following the results of Ali et

al. [30] for narrow channels. The Lockhart—Martinelli—Butterworth cor-

relation best agreed with their data.


H. Two-Phase Flow Caused by the Releaseof Dissolved NONCONDENSABLES

Liquid forced convection in microchannels is an effective cooling mech-

anism for systems with very high volumetric heating such as accelerator

targets, high-power resistive magnets, and high-power microchips and

electronic devices. The widely used correlations for turbulent friction factor

and convection heat transfer in large channels have been shown to be

inadequate for microchannels by some investigators, indicating that the

turbulence characteristics of microchannels may be different from those of

large channels [64—66].

An interesting issue related to the forced flow of liquids in microchannels

is the potential effect of noncondensables dissolved in the liquid. Dissolved

noncondensables typically have a negligible effect in experiments with large

channels where they undergo little desorption. In microchannels, however,

because of the typically large axial pressure drops, significant desorption of

noncondensables is possible. Such desorption will lead to the development

of a two-phase mixture, will increase the convection heat transfer coefficient,

and may be at least partially responsible for the experimental data of some

investigators, which suggest that the widely used correlations representing

heat and mass transfer in large channels underpredict the heat and mass

transfer in microchannels rather significantly [65, 66].

Adams et al. [67, 68] recently studied the effect of dissolved noncondens-

ables on the hydrodynamic and heat transfer processes in a microchannel.

A theoretical model [67] indicated that the release of the dissolved noncon-

densables can have a relatively significant effect on the channel hy-

drodynamics. In [68], experiments were performed in a channel 0.76mm in

diameter with a 16 cm heated length, subject to water forced convection. The

range of experimental parameters were as follows: wall heat flux, 0.5 to

2.5MW/m�; liquid mean velocity at inlet, 2.07 to 8.53m/s; channel exit

pressure, 5.9 bar. The water remained subcooled in all the experiments.

Typical results, depicting the Nusselt numbers at their test section exit

obtained with pure (degassed) water, and with water initially saturated with

air at a pressure equal to the test section exit pressure, are displayed in Fig.

18. The Nusselt numbers, defined as Nu� hD/k!, were calculated assuming

single-phase liquid flow properties. The apparent dependence of the en-

hancement in Nu due to the noncondensables (air) on Re and heat flux in

fact represents the dependence of the results on the pressure drop and the

liquid temperature rise in the test section. Higher pressure drop and higher

liquid temperature rise in the test section both lead to increased desorption

of dissolved noncondensables from the liquid, and therefore to the enhance-


Fig. 18. The enhancement in the local convective heat transfer due to the presence of

dissolved air [68]. (With permission from [68]).

ment of the local heat transfer coefficient. The presence of dissolved air in

water could increase the measured Nu by as much as 17%.

An upper limit of voidage development resulting from the release of

dissolved air from water can be obtained by assuming (a) homogeneous-

equilibrium two-phase flow; (b) the release of dissolved air is accompanied

by evaporation such that the gas—vapor mixture is everywhere saturated

with vapor; and (c) the liquid and the gas—vapor mixture are everywhere at

equilibrium with respect to the concentration of the noncondensable, and

the latter equilibrium can be represented by Henry’s law. Using these

assumptions, Adams et al. [68] showed that void fractions up to several

percent were possible, implying the occurrence of the bubbly flow regime in

their test section. For forced convection heat transfer in developed bubbly

flow, the Nu augmentation factor resulting from the presence of the gas

phase, which increases the flow velocity, can be shown to be of the order of

(1� �)��, with n representing the power of Re in the appropriate single-

phase forced convection heat transfer correlation [69]. The augmentation in

Nu in the data of Adams et al. [68] was significantly higher, however,

indicating that the observed heat transfer enhancement should be primarily

due to the flow field disturbance caused by the formation and release of

microbubbles on the channel walls.


IV. Pressure Drop

A. General Remarks

Pressure drop is among the most essential design parameters for piping

systems. However, frictional pressure drop in microchannels, for both single

and two-phase flow, is not well understood. Turbulence characteristics in

microchannels are known to be different from those in large channels. Thus,

although the predictions of theoretial solutions for laminar flow closely

agree with measured friction factors in microchannels [70, 71], most of the

recent experimental investigations indicate that the well-proven correlations

for turbulent flow in large channels fail to correctly predict frictional

pressure drop in circular and rectangular microchannels [71—74]. Some

investigators, on the other hand, have reported good agreement between

their data and turbulent friction factor correlations for smooth pipes [26],

and for pipes with carefully measured roughness characteristics [70].

Measurement uncertainties and uncertainties associated with roughness

evidently contribute to the disagreement among published experimental

data, while the lack of adequate understanding of turbulence in microchan-

nels is believed to be the main reason for disagreement between existing data

and the commonly used correlations for large channels.

B. Frictional Pressure Drop in Two-Phase Flow

The existing methods and correlations applied in microchannel pressure

drop are mostly similar to those applied to large channels. A brief review of

the principles of two-phase frictional pressure drop modeling, and the

predictive methods that have been applied to microchannels, is provided in

this section.

The simplest method for calculating the two-phase frictional pressure

drop is to assume homogeneous flow and apply an appropriate single-phase

turbulent friction factor correlation using the homogeneous two-phase

mixture properties everywhere. Thus,

��P�z�

��.

��f�.

G�

2��D

, (44)

where

��

x

�0

�1 �x

�!��

. (45)


Now, applying the Blasius correlation for turbulent friction factor, for

example, one can write

f�.

� 0.316Re��.

(46)

where

Re�.

�GD/��.

. (47)

One of the most popular correlations for homogeneous mixture viscosity is

that of McAdams [75]:

��.

��x

�0

�1�x

�!��

. (48)

In the preceding equations x represents the flow quality. In dealing with

a single-component two-phase flow (i.e., the flow of a liquid and its own

vapor), and further assuming thermal equilibrium between the two phases

(the homogeneous equilibrium mixture model, HEM), x�x�$

, where

x�$

� (h � h�

)/h�&

. (49)

The homogeneous flow assumption has limited applicability, however,

and the foregoing method for calculating two-phase frictional pressure drop

is inaccurate in most applications.

The two-phase multiplier method is the most common technique for

correlating two-phase frictional pressure drop in channels [59], according

to which

��P�z�

��.

��!' �

�P�z�

��!'

��0' �

�P�z�

��0'

(50)

or

��P�z�

��.

��! �

�P�z�

��!

��0 �

�P�z�

��0

, (51)

where

��P�z�

��!'

and ��P�z�

��!

represent single-phase frictional pressure gradients in the channel when pure

liquid at mass fluxes G and G(1 � x), respectively, flows in the channel. The

terms

��P�z�

��0

and ��P�z�

��0


are defined similarly when pure gas at mass fluxes G and Gx, respectively,

flows in the channel. Equations (50) and (51) evidently provide four ways

for representing two-phase pressure drop, by correlating any of �!

, �0

,

�!, or �

0. Based on the two-phase multiplier concept, many correlations

have been proposed in the past. These correlations are usually applicable

without restriction to all flow regimes.

Martinelli and co-workers were the first to correlate ��!

and ��0, graphi-

cally in terms of the Martinelli factor X, defined as

X��(�P/�z)

��!(�P/�z)

��0

. (52)

Lockhart and Martinelli’s graphical representation of �0

have been utilized

for the development of algebraic correlations by some investigators [76, 77].

A widely used correlation, suggested by Chisholm and Laird [78], is

��!� 1� C/X� 1/X�, (53)

where C may have values between 10 and 20. An alternative expression for

this correlation is [79]

��0

� 1� CX�X�. (54)

In the foregoing equations the constant C depends on whether the gas and

liquid phases, when flowing alone, are laminar (viscous) or turbulent, and

its recommended values are as follows: C� 20 for turbulent—turbulent flow;

C� 12 for viscous liquid and turbulent gas; C � 10 for turbulent liquid and

viscous gas; and C � 5 for viscous—viscous flow [78]. A correlation repre-

senting ��!'

as a function of flow quality, x, �P��0'

/�P��0'

, and n with nrepresenting the power of Re in the single-phase friction factor correlation,

has also been proposed by Chisholm [80, 14]. Other correlations that

account for the effect of phasic properties and the two-phase mixture mass

flux have been described in [14].

A widely used correlation, proposed by Friedel [81], is based on a vast

database covering an extensive parameter range. For horizontal channels,

the Friedel correlation is

��!'

�A�3.21x��(1�x)��!

�0��

��0

�!��

�1��0

�!��

Fr��.

We��.

,

(55)


where

A � (1 � x)� �x��!f0'

(�0

f!'

)�� (56)

Fr�.

�G�

gD��.

(57)

We�.

�G�D

��.

. (58)

According to Friedel, the single-phase friction factors should be calculated

from

f ��'

� 0.25[0.86859 lnRe�/(1.964 lnRe

�� 3.8215)]�� (59)

when Re�� 1055, where

Re��DG/�

�. (60)

When Re�� 1055, the appropriate laminar Fanning friction factor relation

is used.

The following correlation, derived by Beattie and Whalley [82], is

convenient to use because of its simplicity. In this method, homogeneous

mixture flow is assumed and Eqs. (44)—(47) are applied. The mixture

viscosity, however, is defined as

��.

� ��0� �

!(1 � �

�)(1� 2.5�

�) (61)

where ��, the homogeneous void fraction, is found from Eq. (32) with S � 1.

Beattie and Whalley recommend that the single-phase friction factor be

calculated, for all values of Re�.

, from the Colebrook [83] correlation,

1

� f ��.

� 3.48� 4 log� �2

�D

�9.35

Re�.

� f ��.� (62)

where f �� f /4 represents the Fanning friction factor.

The preceding correlations, as mentioned earlier, do not explicitly account

for flow patterns and have been developed to cover various flow patterns

covered by their databases. The fractional pressure drop, like many other

important hydrodynamic phenomena, depends on flow pattern, however.

Flow regime—dependent models have been derived for the stratified [84]

and annular [85, 86] flow regimes in the past, because of the relatively

simple morphology of the latter flow patterns. These models, in addition to

correlating the wall friction, also account for the gas— liquid interfacial

friction.


C. Review of Previous Experimental Studies

The occurrence of flashing two-phase flow in refrigerant restrictors was

the impetus for early studies of single- and two-phase pressure drop in long

microchannels [87—90]. Two-phase pressure drop in these studies was

generally modeled using the homogeneous flow model. Table II is a

summary of the more recent experimental investigations that are reviewed

here.

Koizumi and Yokohama [91] modeled their experimental data using Eqs.

(44)—(48), where the liquid and vapor phases were assumed to be at

equilibrium everywhere, and the two-phase viscosity was calculated from

��.

��!

based on the argument that the flashing two-phase flow in their

simulated refrigerant restrictor was predominantly bubbly. Although their

calculations were in good agreement with their total measured pressure

drops, they noted that the preceding model was in fact significantly in error

since it did not account for the evaporation delay in their experiments.

Further investigation into the flashing and two-phase flow processes as

associated with refrigerant restrictors was conducted more recently by Lin

et al. [92]. They measured pressure drop with single-phase liquid flow, and

noted that their data could be well predicted using the Churchill [93]

correlation,

f � 8 ��8

Re��

�1

(A �B) ��

, (63)

where

A ��2.457 ln�1

�7

Re��

� 0.27�D��

��(64)

B��37530

Re ��

(65)

Based on an argument similar to that of Koizumi and Yokohama [91], Lin

et al. [92] applied the homogeneous-equilibrium mixture model (Eqs. (44),(45), and (47)) for two-phase pressure drop calculations. For calculating the

two-phase friction factor, f�.

, they used the aforementioned Churchill

correlation (Eq. (63)) by replacing Re with Re�.

, and over a quality range

of 0& x& 0.25 they empirically correlated the two-phase mixture viscosity

according to

��.

��0�!

�0�x�(�

!� �

0), (66)

with n � 1.4 providing the best agreement between model and data.


TABLE II

Summary of Experimental Data for Microchannel and Narrow Passage Two-Phase Flow Pressure Drop

Author Channel characteristics Fluid(s) Flow Range

Koizumi and Yokohama [91] Adiabatic circular channels, D� 1, 1.5mm R-12 Re!� 4.3 10�—6.4 10�

Lin et al. [92] Adiabatic copper tubes, R-12 G� 1.44 10—5.09 10kg/m�s

D� 0.66mm (�� 2�m), 1.17 mm (�� 3.5�m) P��

� 6.3—13.2 bar; �T��

� 0—17K

Ungar and Crowley [94] Adiabatic circular tubes, D� 1.46—3.15 mm Ammonia Re!� 700; 450�Re

0�1.1 10�, 0.09&x&0.98

Bao et al. [53] Glass and copper tubes, D � 0.74—1.9mm Water—air, 15�Re0�2 10; 0.05�Re

!�4 10�

water—aqueous,

glycerin solutions

Bowers and Mudawar Heated copper channels (subcooled boiling) R-113 U!�

�7.7 m/s

[95] D� 0.51, 2.54 mm

Fukano and Kariyasaki [25] Circular channels, D� 1—26mm Water—air 0.02�U!�

�2m/s; 0.1�U0�

�30m/s

Mishima and Hibiki [26] Pyrex and aluminum circular channels, Water—air 0.02�U!�

�2m/s; 0.1�U0�

�50m/s

D� 1.05—4.08mm

Triplett et al. [12] Pyrex circular channels, D � 1.1, 1.45mm; Water—air 0.02�U!�

� 8m/s; 0.02�U0�

�80 m/s

semitriangular channels, D�� 1.1, 1.49mm

Narrow et al. [34] Glass seven-rod bundle, D�� 1.46mm Water—air 0.03�U

!��5m/s; 0.02�U

0��40m/s

Lowry and Kawaji [27] Rectangular, W� 8 cm, L � 8 cm, S � 0.5, 1, 2mm Water—air 0.1�U!�

�8 m/s; 0.1�U0�

�18 m/s

Ali and Kawaji [29] Rectangular, W �80mm, L �240mm, S � 1.465mm Water—air 0.15�U!�

�16m/s; 0.2�U0�

�7.0 m/s

Ali et al. [30] Rectangular, W� 80mm, L � 240mm, S � 0.778 Water—air 0.15�U!�

�16m/s; 0.15�U0�

�6.0m/s

and 1.465mm

Mishima et al. [31] Rectangular, W� 40mm, L � 1.5m, Water—air 0.1�U!�

�10m/s; 0.02�U0�

�10m/s;

S � 1.07, 2.45, 5.0 mm 0.5�x�100

Fourar and Bories [33] Rectangular glass slit, W� 0.5 m, L � 1m, Water—air 0.005�U!�

�1m/s; 0.0�U0�

�10m/s

S � 1mm; brick slit, W� 14 cm, L � 28 cm 0.1�x�40

S � 0.18, 0.4, 0.54 mm

Yan and Lin [96, 97] Circular, D � 2mm; 28 parallel pipes with R-134a Re!� 200—12,000; mean quality� 0.1—0.95

condensation and evaporation

Ekberg et al. [35] Glass annuli; D�� 6.6mm, D

%� 8.63mm; and Water—air 0.1�U

!��6.1m/s; 0.02�U

0��57 m/s

D�� 33.15mm, D

%� 35.2mm; L � 35 cm

185

Ungar and Cornwell [94] conducted experiments at high quality

(0.09&x& 0.98); their data were therefore predominantly in the annular-

dispersed two-phase flow regime. They calculated their experimental two-

phase friction factors using channel-average properties, neglecting the effect

of axial variations in vapor properties and quality, and compared them with

predictions of several correlations. The homogeneous-equilibrium mixture

model (Eqs. (44)—(47)), when applied with McAdam’s correlation for

mixture viscosity, Eq. (48), provided the best agreement with data. An

empirical correlation for vertical annular flow, due to Asali et al. [85], also

predicted their data well.

Bao et al. [53] performed an extensive experimental study using air and

aqueous glycerin solutions with various concentrations and calculated the

experimental friction factors using channel-average properties. They com-

pared their data with the correlations of Lockhart and Martinelli [59],

Chisholm [80], Friedel [81], and Beattie and Whalley [82], apparently

without consideration for the effect of channel roughness. With the excep-

tion of the correlation of Lockhart and Martinelli [59], which did relatively

well for Re!& 1000, all the tested correlations failed to predict the data well.

By implementing the forthcoming simple modification into the correlation

of Beattie and Whalley [82], Eq. (61), the latter correlation predicted all

their data well. The correlation of Beattie and Whalley is based on the

application of the homogeneous flow model, Eqs. (44), (45), and (47), and

the Colebrook—White correlation (Eq. (62)) for the friction factor over the

entire two-phase Reynolds number range. Bao et al. [53] modified the

correlation of Beattie and Whalley [82] simply by using f ��.

� 16/Re�.

in

the Re�.

& 1000 range.

Bowers and Mudawar [95] studied high heat flux boiling in ‘‘mini’’

(D� 2.54mm) and ‘‘micro’’ (D� 0.5mm) channels. They applied the homo-

geneous-equilibrum model, Eqs. (44), (45), and (49), assuming f�.

� 0.02

The homogeneous-equlibrium model could well predict the total experimen-

tal pressure drops. Because of the significance of the acceleration pressure

drop in most of their tests, the accuracy of the homogeneous-equilibrium

model for the calculation of the frictional pressure drop in their experiments

cannot be directly assessed. The good agreement between model-predicted

and measured total pressure drops, nevertheless, may indicate that the

homogeneous-equilibrium model in its entirety is adequate for similar

applications.

The experiments of Fukano and Kariyasaki [25] were described in

Subsection B of Section III (see Table I). They compared their two-phase

pressure drop data with the correlation of Chisholm [78, 79] (Eq. (53) or

(54)), indicating a large discrepancy. The discrepancy was particularly

significant in the intermittent (plug and slug) flow patterns. Fukano and


Kariyasaki [25] indicated that the pressure loss associated with the expan-

sion of the liquid as it flows from the film region surrounding elongated

bubbles into the liquid slugs is a significant component of the total frictional

pressure loss.

Mishima and Hibiki [26] reported that with single-phase liquids, the

Blasius correlation well predicted their turbulent friction factor data, where-

as in the laminar range their data agreed with the Hagen—Poussuille

relation within 2%. For calculating the two-phase frictional pressure drop,

they neglected the acceleration along their test sections, and chose to modify

the Chisholm correlation [78, 79], Eqs. (53) or (54), by empirically correlat-

ing the constant C according to

C� 21(1 � e��) (67)

where the channel diameter, D, is in millimeters.

Triplet et al. [12] measured the frictional pressure drop for air—water flow

in circular and semitriangular microchannels with D�1 1.1 to 1.49mm (see

Fig. 3). They compared their measured frictional pressure drops with the

predictions of the homogeneous mixture method, Eqs. (44)—(48), and the

correlation of Friedel [81], Eqs. (55)—(60). They noted that because of the

significant axial variation of pressure in microchannels, the gas density

cannot be assumed constant. They applied a one-dimensional model, based

on the numerical solution of one-dimensional mass and momentum conser-

vation equations for the calculation of pressure drops, using the aforemen-

tioned correlations for two-phase wall friction. Overall, the homogeneous

mixture model better predicted the data. Both correlations did poorly when

applied to the annular flow regime data, however.

Yan and Lin recently measured the two-phase pressure drop and heat

transfer associated with evaporation [96] and condensation [97] of Refrig-

erant 134a in a horizontal, 28-tube bundle consisting of tubes with 2mm

inner diameter. In calculating the frictional pressure drops for the tubes they

needed to estimate the pressure losses at inlet and exit to their tube bundle,

and the deceleration pressure change associated with the condensing two-

phase flow. Following Yang and Webb [98], who investigated the two-

phase pressure drop associated with the adiabatic two-phase flow in

extruded aluminum tubes, Yan and Lin based the aforementioned estimates

on the test section average quality and an average void fraction. The average

void fraction was calculated using the following slip ratio correlation

originally derived by Zivi [99] based on the assumption of minimum

entropy generation in steady-state, annular two-phase flow:

S� (�!/�

0)� . (68)

Yan and Lin calculated the entrance and exit pressure losses using common-


ly applied contraction and expansion models They correlated the experi-

mental frictional pressure drops obtained in this way using a method

originally suggested by Akers et al. [100], and recently applied by Yang and

Webb [98], according to which

�P�� 4 f

�.

L

D

G��$�

2�!

, (69)

where (Akers et al. [100]):

G�$�

�G[(1 � x�) �x

�(�

!/�0)��], (70)

where x�

is the average channel quality. For condensation, Yan and Lin

[97] obtained

f�.

� 498.3Re��$�

, (71)

where

Re�$�

�G�$�

D/�!. (72)

Using their experimentally measured frictional pressure drops, Yan and

Lin calculated the experimental friction factors in their evaporating two-

phase flow tests based on the homogeneous mixture assumption, with the

homogeneous density ��defined based on channel average quality and void

friction:

�P�� 4 f

�.

L

D

G�

2��

(73)

They, however, correlated the friction factors in terms of the aforementioned

equivalent Reynolds number, defined in Eq. (72), according to

f�.

� 0.11Re��$

. (74)

Recently Narrow et al. [34] investigated the hydrodynamic processes

associated with air—water two-phase flow in a seven-rod micro-rod bundle.

Their experiments were described in Section III, E. They measured the

pressure drop in their experiments and compared their data with the

predictions of the homogeneous mixture model (Eqs. (44)—(48)) and the

correlation of Friedel [81], Eqs. (55)—(58) for two-phase frictional pressure

drop. Neither correlation could satisfactorily predict the data over the entire

flow regime map. The correlation of Friedel could predict most of the data

typically within a factor of 2, except for the data representing very low

superficial velocities. The homogeneous mixture model, on the other hand,

consistently underpredicted the frictional pressure drop for all flow patterns,

ecept for the annular/intermittent and plug/slug flow patterns.


D. Frictional Pressure Drop in Narrow Rectangularand Annular Channels

For laminar, single-phase flow, the measured friction factors in rectangu-

lar channels agree well with theory [101], For turbulent flow, the friction

factor in rectangular channels is known to depend on the Reynolds number

based on hydraulic diameter, as well as on the channel aspect ratio. Jones

[102] derived a simple method that allows for the application of smooth

pipe laminar and turbulent friction factor correlations to rectangular chan-

nels. Accordingly, a laminar equivalent diameter, D!��$�

, is defined as

D!��$�

��*D�, (75)

where the shape factor �* is a function of the aspect ratio, W /(, and is

formulated using the theoretical solution for friction factor in rectangular

channels, such that for laminar flow

f � 64/Re��

, (76)

where the modified Reynolds number Re* is defined according to

Re��

�GD

!��$��

. (77)

The function �* can be found from the analytical solution of Cornish [103].

The following simple correlation, however, agrees with the aforementioned

analytical solution within 2% [102]:

�*$2

3�

11

24

(W �2�

(W �. (78)

Using Eqs. (75)—(78), the turbulent smooth pipe flow correlations can be

applied to rectangular channels.

Kawaji et al. [27, 29, 30], Mishima et al. [31], and Fourar and Bories [33]

have reported two-phase pressure drop data dealing with narrow rectangu-

lar channels (see Table II).Lowry and Kawaji [27] indicated that the wall roughness was only 1.5�m

in their test section and that the Blasius correlation for fully turbulent

single-phase liquid flow did extremely well for their data. This result is

evidently in disagreement with the aforementioned well-known effects of the

aspect ratio on the turbulent friction factor in narrow channels. Lowry and

Kawaji compared their two-phase pressure drop data with the predictions

of the correlation of Lockhart and Martinelli [59] and indicated that the

correlation did not account for the clear dependence of the data on mass

flux. Their experimental two-phase multiplier, furthermore, was a weak

function of U!�

and (, and a strong function of U0�

. The experiments of Ali


et al. [30], described earlier in Section III, G, were performed in narrow

rectangular channels with (� 0.778 mm and 1.465mm, with the following

six orientations: vertical upward and downward flow; 45°-inclined upward

and downward flow; horizontal flow between horizontal plates; and hori-

zontal flow between rectangular plates. With the exception of the last flow

configuration, the effect of orientation on pressure drop was quite small, and

the correlation of Chisholm and Laird [78], Eq. (53), agreed well with their

data using C values between 10 and 20, depending on mass flux. In these

comparisons, for the calculation of the single-phase frictional pressure

gradients, Ali et al. [30] used the following expressions for the D’Arcy

friction factors, which they derived by curve fitting their own experimental

data. For laminar flow (Re & 2300), f � 95/Re for (� 0.778mm, and

f �94/Re for (�1.465mm. For turbulent flow (Re�3500),

f �0.339Re�� for (� 0.778mm, and f � 0.338Re�� for (� 1.465mm.

For the transition 2300&Re& 3500 range, they applied a linear interpola-

tion on a logarithmic scale. For horizontal flow between vertical plates, the

effect of mass flux on the two-phase multiplier was strong, and fixed values

of C could not correlate the data. For the stratified flow regime in the latter

configuration, Ali et al. [30] derived the following expression, based on a

simple separated flow model that can be applied when the two phases are

both either laminar or turbulent:

��!� [1�X� ��]�� (79)

Here, m is the power of Re in the appropriate single-phase friction factor.

This expression well predicted the turbulent—turbulent data of Ali et al.The experiments of Mishima et al. [31] were described in Section III, G.

For single-phase flow pressure drop, their results were in agreement with the

recommendations of Jones [102]. They also noted good agreement between

their data and a correlation proposed by Sadatomi et al. [104]. For

two-phase frictional pressure drop, Mishima et al. chose the correlation of

Chisholm and Laird, Eq. (53), for modification and correlated their data

according to

C � 21 tanh (0.199D�) $ 21[1 � 1056 exp(�0.331D

�)], (80)

where D�

is in millimeters.

Fourar and Bories [33] noted that, except for the annular flow regime,

the frictional pressure drops could be well correlated using

f �96

Re�

, (81)


where the mixture Reynolds number, Re�, is obtained from

Re��

2S��(U

!��U

0�)

��

(82)

��

0� (1� �)�

!(83)

��

!

U!�

U!�

�U0�

. (84)

Annular flow appeared to occur at Re�� 4000 apparently corresponding to

the establishment of ‘‘turbulent’’ mixture flow.

John et al. [105] performed an extensive series of experiments dealing

with critical flow in cracks and slits. The work of John et al. [105] is

discussed in Section VII. In view of the important effect of friction on critical

mass flux, John et al. measured and correlated the single-phase friction

factors in their test sections. They could correlate their data according to

f ��3.39 log�

D�

2�� 0.866�

��(85)

where � represents the surface roughness.

Ekberg et al. [35] measured pressure drops associated with air—water

two-phase flow in two horizontal annuli with (� 2 and 3mm (see Table II).When compared with predictions of a one-dimensional model based on the

homogeneous flow assumption, Friedel’s correlation [81] for two-phase

frictional pressure drop, Eqs. (55)—(60), predicted the experimental data

better than the homogeneous flow wall friction model.

V. Forced Flow Subcooled Boiling

A. General Remarks

Cooling by the flow of a highly subcooled liquid, and subcooled boiling,

are the heat transfer regimes of choice in numerous applications, because of

the extremely high heat fluxes they can sustain at relatively low heated

surface temperatures. The extensive past studies have been reviewed in a

number of textbooks and monographs, including [14, 106—108]. The work

by Tuckerman and Peasa [109], and that pursued by many other investiga-

tors, has shown that the inclusion of networks of microchannels cooled by

subcooled liquids in circuit boards, in particular, can provide effective

cooling for extremely high thermal loads. A good review of the past research

dealing with single-phase flow heat transfer in microchannels can be found

in [110]. An extensive summary is provided by Duncan and Peterson [111].


These experimental studies have also shown that the hydrodynamic and

heat transfer characteristics of microchannels are different from those of the

commonly applied large channels. These differences, the causes of which are

not fully understood, include the Reynolds number range for laminar-to-

turbulent flow pattern transition, and the wall friction factor and forced

convection heat transfer in the transition and fully turbulent regimes.

Despite its evident significance, forced-flow boiling in microchannels with

De� 0.1 to 1mm has attracted little investigation in the past, although heat

transfer in small channels representative of modern compact heat ex-

changers (with D�

values typically in the few-millimeters range) has been

investigated by several investigators recently [112—115]. The limited avail-

able data for microchannels, nevertheless, indicate major differences between

small and commonly used large channels, with respect to the basic bubble

ebullition phenomenology. The experimental data of Wambsganss et al.[114] and Tran et al. [115], for example, indicate that, unlike in large

channels, in small channels used in compact heat exchangers the nucleation

process, and not the forced convection process, is the dominant boiling heat

transfer mechanism at high flow qualities.

In the forthcoming sections, the subcooled boiling phenomena, for which

recent investigations have led to reasonably consistent results, are discussed.

Table III is a summary of the recently published experimental studies

dealing with subcooled boiling phenomena in microchannels.

B. Void Fraction Regimes in Heated Channels

The voidage development in heated channels with subcooled inlet condi-

tions has been studied extensively in the past, and is described in textbooks

[107, 116]. Figure 19 is a schematic of the axial void fraction distribution

along a uniformly-heated channels with subcooled liquid inlet conditions.

This schematic is consistent with experimental observations with water at

high pressures in commonly used large channels [117]. The flow field

upstream of point A is single-phase liquid, and bubbles attached to the wall

can be seen at and beyond point A, referred to as the onset of nucleate

boiling (ONB) point. Bubbles remain predominantly attached to the wall

upstream of point B. Beyond the latter point bubbles can be seen detached

from the wall. Beyond point C, referred to as the onset of significant void

(OSV), or the point of net vapor generation (NVG), the detached bubbles

can survive condensation and a rapid increase in the gradient of the void

fraction curve is observed.

Recent low-pressure experiments with water by Bibeau and Salcudean

[118—120], also carried out in test sections representative of common large

channels, have shown that the phenomenology implied in the schematic of


TABLE III

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

Author Channel characteristics Fluid Parameter range Phenomena studied

Inasaka et al. [130] D� 1, 3mm; L � 1, 3, 5, 10 cm; vertical Water G� 7000, 13,000, 20,000kg/m�s; ONB, OSV, OFI

T��

� 20,60°C; P��

$ 1 bar

Vandervort et al. D� 0.3—2.6mm; L �2.5—66 mm; vertical Water G� 8400—42,700 kg/m�s; ONB

[131] P��

� 1—22 bar

Kennedy et al. D� 1.17, 1.45mm, L � 22 cm; Water G� 800—4500kg/m�s; ONB, OFI

[132] L2

� 16 cm; horizontal P��

� 3.44—10.34 bar

Roach et al. D� 1.17, 1.45mm, circular; Water G� 220—790kg/m�s; OFI

[135] D2

� 1.13mm, semitriangular; P��

� 2.4—9.33 bar

L � 22 cm; L2

� 16 cm; horizontal

Blasick et al. Annuli with r�� 6.4mm and Water G� 85—1,428 kg/m�s; OFI

[139] (� 0.724—1.0mm; L2

� 17.4—19.7 cm; P��

� 3.44—10.34 bar

horizontal

Peng and Wang Rectangular, width� 0.2—0.8mm, Water, U!��

� 0.2—2.1m/s, �T��

� 65—90°C Liquid single-phase and

[153] depth� 0.7mm; L � 4.5 cm; methanol for water; U!��

� 0.2—1.5m/s, subcooled boiling heat

horizontal �T��

�45—50°C for methanol; transfer

P��

$ 1 bar

Peng and Wang Rectangular, width� 0.6mm; Water U��

� 1.5—4.0 m/s; T!��

� 30—60°C; Liquid single-phase and

[110] depth� 0.7mm; L � 6.0 cm P��

$ 1 bar subcooled boiling heat

transfer

Hosaka et al. D� 0.5, 1, 3mm; L /D� 50, vertical R-113 G� 9300—32,000 kg/m�s; Subcooled boiling heat

[155] �T��

�50—80°C; P� 11—24 bar transfer and CHF

193

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

Fig. 19 is not accurate, at least for low-pressure subcooled boiling of water,

where the region of attached voidage (the region beween points A and B in

Fig. 19, where bubbles are presumed to grow and collapse while attached to

the wall) is essentially nonexistent, and bubbles that are detached always

slide on the wall before being injected into the liquid core. The phenomenol-

ogy depicted in Fig. 19, nevertheless, has been the basis of successful

correlations [121, 122] and analytical models [123—125] in the past.

Figure 20 schematically depicts the pressure drop-flow rate characteristics

(the demand curve) of a heated channel subject to a constant heat load

(constant heat flux for a uniformly heated channel). The demand curve can

be used for the analysis of static instabilities [126, 127]. When the channel

is part of a forced or natural circulatory loop, the segment of the heated

channel demand curve with negative slope (between points OFI and S in

Fig. 20) can be unstable, and the onset of flow instability (OFI) point is

defined as the relative minimum point on the demand curve. The occurrence

of OFI is due to the increase in the channel pressure drop which results from


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

subcooled boiling voidage, and in experiments with steady heat flux (or

steady mass flow rate), OFI is known to occur at a flow rate slightly lower

(or a heat flux slightly higher) than the flow rate (or the heat flux) that leads

to OSV. The OSV point can thus be considered as a conservative estimate

of OFI.

C. Onset of Nucleate Boiling

The onset of nucleate boiling (ONB), or boiling incipience, has been

modeled by several authors. A good compilation of the existing correlations

can be found in Marsh and Mudawar [128]. Most of the models are based

on the assumption that at boiling incipience stationary and stable bubbles,

attached to wall crevices, exist, and that the steady-state liquid temperature

profile is tangent to the temperature profile predicted by the Clapeyron

bubble superheat profile. According to the model of Bergles and Rohsenow

[129], one of the most widely used models of this kind, bubbles attached to

the wall at the ONB point are hemispherical, and the heat flux and wall


temperature at ONB are related to each other according to

q"�

� k!

T�� T

!(r�)

r�

(86)

q"� h!'

(T��T

!), (87)

where T!(r�) is the local liquid temperature at a distance r

�away from the

heated surface. The aforementioned tangency criterion, furthermore, re-

quires that

T!(r�) � T

�(88)

�T!

�y ��

��T

��r

�

, (89)

where T�, the bubble temperature, accounts for the superheat required for

mechanical equilibrium between the bubble and its surroundings based on

Clapeyron’s relation:

T��T

#�� T

�T#�

R

Mh��&

ln �1�2�r�P� (90)

An empirical curve fit to the predictions of the above equations for water in

the 1 bar�P� 136 bar, was derived by Bergles and Rohsenow [129] as

q"��'3�

� 5.30P��[1.8(T4

�T#�

)'3�

]�, n� 2.41/P�� (91)

where q"��'3�

is in W/m�, P is in kPa, and temperatures are in K.

The preceding correlation of Bergles and Rohsenow has been compared

with microchannel data with water as the working fluid by Inasaka et al.[130], Vandervort et al. [131], and Kennedy et al. [132].

Inasaka et al. [130] performed experiments in heated tubes with D� 1

and 3 mm, with G � 7000 to 20,000 kg/m�s. The correlation of Bergles and

Rohsenow, Eq. (91), predicted the data of Inasaka et al. relatively well.

Significant scatter can be noted in their comparison results, however, with

maximum dicrepancies of about � 50% in the prediction of q"��'3�

.Vandervort et al. [131] carried out an experimental investigation of

subcooled boiling phenomena in microchannels with diameters in the

D� 0.3 to 2.6mm range, subject to high heat fluxes, with water as the

working fluid. They reported that the correlation of Bergles and Rohsenow

predicted their ONB data well. They also applied the model of Bergles and

Rohsenow (Eqs. (87)—(91)), for the estimation of the released bubble

diameters, and showed that the released bubbles were typically only a few

micrometers in diameter. Using the estimated bubble sizes, they calculated

the order of magnitude of various forces that act on the bubbles, and


Fig. 21. Comparison between the ONB data of Kennedy [132] and the correlation of

Bergles and Rohsenow [129]. (D� 1.17 mm). (With permission from [132].)

showed that the thermocapillary (Marangoni) force and the lift force

resulting from the ambient liquid velocity radient are significant forces that

must be considered in modeling of bubble ebullition phenomena in micro-

bubbles. A more detailed discussion of these forces is presented in the

forthcoming Subsection E of this section.

Kennedy et al. [132] studied the ONB and OFI phenomena in heated

microchannels with D� 1.17 and 1.45mm, using water as the working fluid.

Figure 21 displays the comparison between the ONB data for their 1.17mm

test section and the correlation of Bergles and Rohsenow. The correlation

of Bergles and Rohsenow systematically overpredicted the experimental

data, typically by a factor of 2. The correlation, however, agreed reasonably

well (with a slight systematic overprediction) with the data of Kennedy etal. representing their 1.45mm diameter test section.

Inasaka et al. [130] and Kennedy et al. [132] utilized the pressure drop

characteristic curves of their test sections (similar to Fig. 20) for specifying

the ONB conditions. Inasaka et al. identified the heat flux that led to the

occurrence of ONB at the exit of their test sections, for given (constant) inlet

temperature and mass flux, as the minimum point on the �P/�P!'

versus

q"4

curve, with �P representing the measured pressure drop in the experi-

ment with heated wall, and �P!'

representing the channel pressure drop

with adiabatic single-phase liquid flow only.


Kennedy et al. [132] also identified the ONB point on each pressure drop

versus mass flow rate (demand) curve (Fig. 20) by comparing the experimen-

tal demand curves with pressure drop—flow rate characteristics representing

single-phase liquid flow. The ONB occurs at the point where the slopes of

the two curves deviate. They could also recognize an easily audible whistle-

like sound from their test section, before the onset of flow instability (OFI)occurred, which was evidently due to the appearance of vapor bubbles at

the test section exit and could be attributed to the occurrence of ONB.

Kennedy et al. [132] compared the conditions leading to ONB predicted by

the aforementioned method, with the conditions where the whistlelike sound

was heard, and noted good agreement between the two methods.

Several other correlations for ONB have been proposed in the past. A

good review of these correlations can be found in Marsh and Mudawar

[128]. Most of the correlations, however, are based on data with water only.

Yin and Abdelmessih [133] and Hino and Ueda [134] have proposed

correlations that are based on data with Freon 11 and Freon 113, respec-

tively. With the exception of the correlation of Bergles and Rohsenow [129],

however, these correlations have not been systematically compared with

microchannel experimental data.

D. Onset of Significant Void and Onset of Flow Instability

The onset of significant void (OSV) point is usually identified in experi-

ments with commonly applied large channels by measuring the void fraction

profile along the channel, and defining the OSV as the point downstream of

which the slope of the void fraction profile is significantly high (see Fig. 19).Since OSV occurs only slightly before the onset of flow instability (OFI),however, the conditions leading to OFI can be used for estimating the OSV

conditions. The latter approach has been used by Inasaka et al. [130],

Kennedy et al. [132], and Roach et al. [135] in their experiments with

microchannels.

The OSV phenomenon has been studied extensively in the past, and

several empirical correlations and mechanistic models have been proposed

for its prediction [121—125]. Saha and Zuber [121] have proposed the

following widely used correlation, which has been successful in predicting a

wide range of experimental data dealing with commonly applied channels:

St� 455/Pe for Pe& 70,000 (92)

St� 0.0065 for Pe� 70,000, (93)


Fig. 22. Comparison between the OFI data of Kennedy et al. [132] on the correlation of

Saha and Zuber [121] for OSV. (With permission from [132].)

where

St�q"�

�!U!C.!

(T#�

�T!)

(94)

Pe�GDC

.!k!

(95)

Equation (92) represents OSV in the thermally controlled regime, where,

based on the experimental observations [117], bubbles generated on the wall

roll next to the wall, and are ejected into the bulk flow at the point where Eq.

(92) is satisfied. Equation (93), on the other hand, represents OSV in the

hydrodynamically controlled regime, where bubbles attached to the wall act

as surface roughness, and when the roughness height reaches a characteristic

height the bubbles are detached because of the hydrodynamic effects.

Inasaka et al. [130] compared their OFI data (defined similar to Fig. 20)with the foregoing correlation of Saha and Zuber. For their 3mm diameter

test section the correlation well predicted the data. For their 1mm diameter

test section the correlation of Saha and Zuber agreed with the data

reasonably well for G � 7000kg/m�s (corresponding to the thermally con-

trolled Pe$ 4.8 10�).A similar comparison, between the OFI data and the correlation of Saha

and Zuber for OSV [121], was carried out by Kennedy et al. [132]. Figure

22 depicts the results of Kennedy et al. As noted, consistent with the results


of Inasaka et al. [130], within the experimental parameter range, the

correlation of Saha and Zuber agrees with the data reasonably well in the

20,000�Pe range. The apparent large uncertainty bands in the experiments

represent only �2% uncertainty in heat flux, and are a result of the integral

nature of the experiments of Kennedy et al. [132].

Successful analytical models for OSV, based on the bubble detachment

mechanism, have been proposed by Levy [123], Staub [124], and Rogers etal. [125, 136]. These models, which are very similar in their basic approach,

assume that OSV occurs when the largest bubbles that can be thermally

sustained in the steady-state thermal boundary layer that forms on the

heated surface are detached from the heated surface by the hydrodynamic

and buoyancy forces parallel to the heated surface. The models of Levy

[123] and Staub [124] assume a high liquid mass flux and neglect the effect

of buoyancy force on bubble departure, and are known to do well when

applied to high-pressure data for water. The model of Rogers et al. [125]

considers relatively low mass flux and pressure conditions, takes account of

the buoyancy effect, explicitly accounts for the effect of advancing and

receding contact angles (surface wettability), and is based on a model for

bubble detachment due to Al-Hayes and Winterton [137]. Rogers and Li

[136] recently modified the aforementioned model of Rogers et al. [125],

thereby extending its parameter range of applicability.

Recent experimental studies by Bibeau and Salcudean [118—120] have

cast doubt on the bubble detachment phenomenon as the process respons-

ible for OSV. Careful visual observations in the latter studies have shown

that bubble ejection normal to the wall, and not bubble detachment and

motion parallel to the wall, is responsible for OSV. The aforementioned

detachment models [123—125, 136] do not address bubble ejection at all.

Notwithstanding, these analytical models have been relatively successful in

predicting experimental data. The model of Levy, briefly described next, has

been applied to microchannel data by Inasaka et al. [130].

According to Levy’s model [123], in a fully turbulent subcooled flow field

in a heated channel bubble departure occurs when the drag force on the

bubble (itself obtained from wall frictional stress) becomes equal to the

resistive surface tension force, leading to

y5�

� y�

U*

�!

�C��D

��!

�!

, (96)

where y�

is the distance from the heated wall to the tip of the bubble, and

U*��/�

!. (97)


The coefficient C � 0.015 is an empirically adjusted parameter, and ��

is

obtained from

��

�f!'4

G�

2�!

. (98)

The friction factor, f!'

, is found from

f!'

� 0.022 �1 ��2x10��

D�

�10�

Re!'��

�, (99)

where �/D�� 10�� is assumed to represent the surface roughness caused by

the presence of bubbles at the departure.

Levy [123] assumed that bubbles are at saturation temperature with

respect to the local ambient pressure, and can be sustained if the local liquid

temperature, T!(y

�), is at least at saturation. The liquid temperature

distribution furthermore, was assumed to follow the fully developed, steady-

state turbulent boundary layer temperature profile [138], according to

which

T�

�T!(y5) �Q f (y5, Pr

!), (100)

where y5� yU*/�!

is the dimensionless distance from the wall, and

Q�q"�

�!C.!

U*(101)

f (y5, Pr!) ��

Pr!y5, O� y5� 5

5�Pr!� ln �1�Pr

!�y5

5� 1��, 5& y5� 30.

5Pr!� ln[1� 5Pr

!] � 0.5 ln(y5/30) 30& y5

(102)

In fully developed and steady-state,

T�

�T !� q"

�/h!'

. (103)

Utilizing Eqs. (100) and (102) and requiring that T!( y*

�) �T

#�at OSV, one

gets [123]

(T#�

�T !)'�6

�q"4�'�6h!'

�Q f ( y5�

, Pr!). (104)

Accordingly, based on the model of Levy [123], Eqs. (97) and (104) provide

the relationship between q"��'�6

and the bulk liquid subcooling at OSV.

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


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 � 3mm) reasonably well, and systematically underpredicts q"��'�6

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#�

,

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).


Fig. 24. Cross-sectional geometries of the test sections of Roach et al. [135]. (With


The model of Levy [123] can be corrected for the effect of surface tension

on bubble temperature by requiring that

T��T

!(y5

�) � T

#��T� (105)

where, assuming that the bubble diameter is approximately equal to Y�, and

using Clapeyron’s relation,

�T��2�T

#�Y�2

hfg�

1

��

1

�!�. (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 microchan-

nels cooled with water was recently carried out at the Georgia Institute of

Technology, for the purpose of generating the data bases needed for the

design of the proposed Accelerator Production of Tritium (APT) system

[132, 135, 139]. A summary of the parameter ranges of these experiments is

included in Table III. The OFI data of Kennedy et al. are compared with

the correlation of Saha and Zuber [121] for OSV in Fig. 22.

The experimental data of Roach et al. [135] deal with OFI at very low

flow rates, in channels with the cross-sectional geometries displayed in Fig.

24. Test sections (a) and (b) were uniformly heated circular channels, and


Fig. 25. OFI equilibrium qualities in the experiments of Roach et al. [135]. (With per-

mission from [135].)

test sections (d) and (e) were meant to represent the flow channels in a

micro-rod bundle with triangular array. Test section (d) was uniformly

heated over its entire surface, while the test section (e) was heated over the

surfaces of the surrounding rods. Roach et al. also examined the effect of

dissolved noncondensables on OFI by performing similar experiments with

fully degassed water and with water saturated with air with respect to the

test section inlet temperature and exit pressure. The bulk of the data

indicated that OFI occurred when the coolant at channel exit had a positive

equilibrium quality, indicating that, unlike in large channels and microchan-

nels subject to high heat fluxes and high coolant flow rates, subcooled

voidage was insignificant in these experiments. Figure 25 displays typical

data. These results show that the commonly used models for OFI, which

emphasize subcooled voidage, or use the onset of significant void (OSV) as

an indicator for the eminence of OFI, may be inapplicable for microchannels

under low flow conditions. In comparison with tests with degassed water,

the total channel pressure drops in tests with air-saturated water were

consistently and rather significantly larger, indicating strong desorption of

the noncondensables, which contributed to channel voidage and therefore


increased the total channel pressure drop. The impact of the noncondens-

ables on the conditions leading to OFI was small, however. With all

parameters including heat flux unchanged, the mass fluxes leading to OFI

in air-saturated water experiments were different than in degassed water

experiments, typically by a few percent.

Blasick et al. [139] investigated the OFI phenomenon in uniformly heated

horizontal annuli, using six different test sections, all with an inner radius of

6.4mm, and gap widths in the 0.72—1.00mm range. Among the parametric

effects they examined was the impact of the inner-to-outer surface heat flux

ratio (varied in the 0—� range), which was found to be negligible. Kennedy

et al. [132], Roach et al. [135], and Blasick et al. [139] developed simple

and purely empirical correlations for their OFI data by comparing the flow

and boundary conditions that lead to OFI with those leading to saturation

at the exit of their test sections.

E. Observations on Bubble Nucleation and Boiling

Heterogeneous bubble nucleation and ebullition phenomena in common-

ly 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 nucle-

ation 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,


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—2.5mm diameter. At the relatively low mass flux of G � 5000 kg/m�s,

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� 25,000kg/m�s. 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 � 1.07mm,

L /D � 25, P � 1.2MPa, G� 25,000kg/m�s, and �T��

� 100°C.

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,

and buoyancy, respectively. The forces F��

and F"�

are due to the generated

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�,

which is small for large bubbles and is therefore usually not considered in

bubble ebullition models for large channels, can be estimated from [148]

F��'

D��

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

velocity gradient and can be obtained from [149, 150]

F"�C

'D�

6�!(U

!�U

6)�U

!�r

. (108)


Fig. 26. Magnitudes of various forces acting on microbubble formed at ONB conditions in

a microchannel with D� 1.07mm [131]. (With permission from [131].)

Based on air—water experimental data in a 57.1mm diameter test section,

Wang et al. [150] correlated the coefficient C in the preceding expression as

C � 0.01�0.490

'cot��

log %� 9.3168

0.1963 �, (109)

where

%� e��D�

�U0�U

!� �

dU!

dr ��D�

D

1

Re��

�U0

U��, (110)

where D�

and D are the bubble and channel diameters, respectively, and

Re��D

��U

0�U

!�/�

!(111)

U�

� 1.18(�g /�!)��. (112)

The force F4

is a near-wall force that opposes the contact between the

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].


Fig. 27. Magnitudes of various forces acting on a microbubble detached from the wall in a

microchannel with D� 1.07 mm [131]. (With permission from [131].)

Based on a two-dimensional analysis, Antal et al. [152] derived

F4

�'D

�6

2�!(U

!�U�)�

D�

�C4��C

4� �D�

2y��, (113)

where:

C4�

��0.104� 0.06(U0�U

!) (114)

C4�

� 0.147, (115)

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


rectangular cross sections, 0.2—0.8mm wide, and 0.7mm deep, with near-

atmospheric 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 well-

known 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 dimen-

sions and length sales peculiar for each fluid affect the boiling phenomena

in microchannels.

Evidently, experiments aimed at careful elucidation of the bubble ebul-

lition 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 60MW/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


correlations in the past [157]. The available experimental data are extensive

and cover a wide parameter range. Most of the data, nevertheless, deal with

water only, and until recently data have been scarce for certain parameter

ranges, such as low-flow, low-pressure CHF in small channels.

Some CHF experimental investigations in the past have included channels

with D2

� 0 (1mm) [95, 131, 155, 160—175], and microchannel CHF data

have been included in the databases of some of the widely used CHF

correlations [176—178], apparently without consideration of the important

differences between micro- and large channels. Systematic investigation of

CHF and other boiling/two-phase flow processes in microchannels, how-

ever, 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—3mm 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 D2

� 0 (1mm)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

10.2mm diameters [167, 168]. The CHF experimental investigations in


TABLE IV

Summary of Experimental Data Dealing with CHF in Small Channels

Critical heat

Pressure Mass flux flux

Source Channel characteristics Fluid (MPa) (Mg/m�s) Inlet conditions (MW/m�)

Ornatskiy [160]# D� 0.5mm, L � 14 cm, vertical Water 1.0—3.2 20—90 T��

� 1.5—154°C 41.9—224.5

Ornatskiy and Kichigan D� 2 mm, L � 56mm, vertical Water 1.0—2.5 5.0—30.0 T��

� 2.7 � 204.5°C 6.4—64.6

[161]#

Ornatsky and Vinyarskiy D� 0.4—2.0 mm, L � 11.2—56mm, vertical Water 1.1—3.2 10.0—90.0 T��

� 6.7—155.6°C 27.9—227.9

[162]#

Loomsmore and Skinner D� 0.6—2.4 mm, L � 6.3—150mm, vertical Water 0.1—0.7 3.0—25.0 T��

� 3.2—130.9°C 6.7—44.8

[163]#

Daleas and Bergles D� 1.2—2.4 mm, L /D� 14.9—26, vertical Water 0.2 1.52—3.0 0.31—3.1

[164]�

Subbotin et al. D� 1.63mm, L � 180mm, vertical Helium 0.1—0.2 0.08—0.32 x��2�0.25

[165]

Katto and Yokoya D� 1 mm, L /D� 25—200, vertical Liquid He 0.199 11—10� h�� h

��3.5 to

[166] �7.0 kJ/kg

Boyd [167] D� 3 mm, L /D� 96.9, horizontal Water 0.77 at exit 4.6—40.6 T��

� 20°C 6.25—41.58

Nariai et al. [169, 170] D� 1, 2, 3 mm; L � 1.0—100mm, Water 0.1 6.7—20.9 T��

� 15.4—64°C 4.6—70

vertical

Inasaka and Nariai [171] D� 3 mm, L �100mm, vertical Water 0.3—1.1 4.3—30 T��

� 25—78°C 7.3—44.5

Hosaka et al. [155] D� 0.5, 1, 3mm; L /D� 50, vertical R-113 1.1—2.4 9.3—32.0 �T��

� 50—80°C

Celata et al. [172] D� 2.5mm, L � 100mm, vertical Water 0.6—2.6 10.1—40.0 T��

� 29.8—70.5°C 12.1—60.6

Vandervort et al. D� 0.3—2.6 mm, L � 2.5—66mm, vertical Water 0.1—2.3 8.4—42.7 T��

� 6.4—84.9°C 18.7—123.8

[131, 173]

Bowers and Mudawar D� 0.51, 2.54mm, L � 10mm, horizontal R-113 0.138 at inlet 0.031—0.15 for �T��

� 10—32°C

[95] D � 2.54 mm;

0.12—0.48 for

D� 0.5 mm

Roach et al [174] D� 1.17, 1.45mm, circular; D2

� 1.13mm, Water 0.344—1.043 0.25—1.0 T� 49—72.5°C 0.86—3.7

semitriangular; L � 160mm; horizontal at exit

#From Celata, Cumo, and Mariani [175].

�From Boyd [156].

211

[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 Gin Boyd’s experiments and was correlated accordingly [167]. Boyd [168]

examined the effect of L /D on CHF in a 1.0 cm-diameter channel.

The experimental data of Nariai et al. [169, 170] include subcooled as

well as saturated (two-phase) CHF data. The dependence of CHF on

channel diameter was found to vary with the local quality. When CHF

occurred in subcooled bulk liquid, CHF monotonically increased as Ddecreased. 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 aforemen-

tioned trend of increasing CHF with decreasing D in subcooled liquids by

arguing that smaller bubbles imply a thinner bubble layer and a smaller

void fraction, and lead to a higher CHF. A recent systematic assessment of

the effect of channel diameter on CHF in subcooled flow by Celata et al.[180], based on experimental data from several sources, has confirmed the

aforementioned mechanism. Hosaka et al. [155], in their experiments with

R-113, observed a similar trend and attributed the increase in CHF

associated with decreasing D to the decreasing bubble terminal size.

The trends of the available data, however, indicate that a threshold

diameter exists beyond which the effect of channel diameter on subcooled

flow CHF is negligible. Figure 28 depicts the results of Vandervort et al.[173]. Below a threshold diameter (about 2mm 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 diam-

eter, 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


Fig. 28. Parametric dependencies in the CHF data of Vandervort et al. [131]. (With


opposite trend was reported by Nariai et al. [169, 170] for CHF occurring

when x�$� 0.

The experiments of Celata et al. [172] covered the intermediate and low

pressure range of 0.6—2.6MPa, and were all carried out in the relatively

large 2.5mm 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 subcool-

ing, channel diameter, pressure, and length-to-diameter ratio, dissolved

noncondensable gas, and heated wall material on CHF of subcooled water

flow. Their data, along with data from several other sources, were used for

parametric trend identification by Celata [181]. Vandervort et al. [173]

noted the frequent occurrence of premature burnout in their tests, which

they defined as any thermal failure not directly attributable to CHF or other

obvious failure mechanisms. Premature failure occurred following boiling


Fig. 29. Equilibrium quality at CHF in the experiments of Bowers and Mudawar [95].

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 2.4mm 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 super-

heated 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� 2.54mm) and a ‘‘micro’’

(D� 0.51mm) channel. CHF occurred when the channel exit equilibrum

quality was quite high, typically at x�$

� 0.5 for the larger channel and

x�$

� 1 for the smaller channel. At very low mass fluxes, furthermore,

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� G�L2/(��

�), and L

2is the heated length. For the smaller


(micro) test section (D � 0.51mm) 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 micro-

channels. The latter authors studied boiling heat transfer in a channel with a

0.6mm 0.7mm 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

occurrence of dryout; and x�$

� 1 was noted in many of their tests,

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 system-

atic study of various parametric effects associated with CHF in microchan-

nels 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

increasing pressure [156]. The available data relevant to subcooled CHF in

microchannels (which virtually all represent P&P*��

) indicate that CHF is

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].


Fig. 30. The effect of exit subcooling on CHF in the experiments of Vandervort et al. [173].

(With permission from [173].)

Vandervort et al. [173] and Roach et al. [174] attempted to measure the

effect of dissolved air in subcooled water on CHF. The impact of the release

of the dissolved noncondensables on liquid forced-convection in microchan-

nels 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.


The following simple empirical correlation for subcooled flow CHF was

proposed by Tong [182]:

q" 27h�&

�CG��

�D��

. (116)

Tong [182] correlated the coefficient C in terms of the local equilibrium

quality, x�$

, according to

C � 1.76� 7.433x�$

� 12.222x��$

. (117)

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 condi-

tions. 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 etal. [175] correlated the parameter C according to

C � (0.216� 4.74 10��P)3 (118)

3� 0.825� 0.987x�$

, for �0.1& x�$& 0 (119)

3� 1 for x�$&�0.1 (120)

3� 1/(2� 30x�$

) for x�$� 0, (121)

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& 9.4MPa, 0.3&D& 25.4mm, 0.1& L & 0.61m, 2&G& 90Mg/m�s, and 90&�T

��& 230K)

within �50%. The agreement of the correlation with the microchannel data

included in the database of Celata et al., furthermore, appeared to be

satisfactory. Celata et al. [172, 175] also compared their compiled database

with the predictions of several other empirical correlations, generally with

poor agreement in comparison with the aforementioned modified-Tong

correlation.

Hall and Mudawar [183] have recently assessed the validity of previously

published CHF experimental data and have compiled a qualified CHF

database (referred to as the PU-BTPFL CHF Database). This database

includes experiments representing 0.3�D� 45mm, 10�G� 2484 kg/m�s,

and �2.25�x�$� 1.0, in vertical, upflow tubes, with x

�$representing the

local equilibrium quality at the CHF (i.e., the end of the heated segment of

test sections) point. They compared 25 widely referenced correlations

dealing with CHF in vertical, upflow channels with their database and


showed that the empirical correlations of Caira et al. [184] provided the

most accurate predictions. The correlation of Bowring [178] also showed

relatively good agreement with data. Although the PU-BTPFL Database

and the aforementioned correlations generally address vertical channels with

upflow, the data included in the database that represent small channels

(D� 1 mm) may represent horizontal microchannels as well because of the

small influence of channel orientation with respect to gravity on two-phase

flow in such microchannels.

The correlation of Bowring [178] can be expressed as

q" 27

�A �DGh�

�&x�$

/4

C(122)

where q" 27

is in W/m�, and

A �2.317(h�

�&DG/4)F

�1� 0.0143F

�D� �G

(123)

C �0.077F

DG

1� 0.347F� �

G

1356��

(124)

n � 2.0� 0.5P8

(125)

P�� 0.145P. (126)

P in Eq. (126) is in MPa, and

F�� P��

8exp[20.891(1�P

8)] � 0.917/1.917 (127)

F�� 1.309F

�/P��

8exp[2.444(1�P

8)] � 0.309 (128)

F� P��

8exp[16.658(1�P

8)] � 0.667/1.667 (129)

F��F

P��8

. (130)

The preceding correlation could predict the low and high mass velocity data

of the PU-BTPFL database with mean absolute errors of 21.9 and 53.5%,

respectively. The correlation of Bowring systematically underpredicted the

data of Roach et al. [174], on the average by 36%.

The correlation of Caira et al. [184] can be represented as

q" 27

�� [0.25(h

�� h)

��"��]��1!

1��L��(131)


where

� � y%D��G�� (132)

1! � y�D��G�� (133)

�� y�D��G��. (134)

All parameters are in SI units, and

y� 10,829.55

y��0.0547

y�� 0.713

y � 0.978

y�� 0.188

y�� 0.486

y�� 0.462

y�� 0.188

y�� 1.2

y�� 0.36

y�

� 0.911.

The preceding correlation agreed with the low and high mass velocity data

in the PU-BTPFL Database with mean absolute errors of 16.5 and 22.6%,

respectively. Caira’s correlation agreed with the data of Roach et al. [174],

with an average overprediction of the data by only 18%.

Vandervort et al. [173] developed a statistical correlation based on their

subcooled flow water CHF data. Their correlation, however, includes more

than 20 constants.

The experimental data of Bowers and Mudawar [95], as noted in the

previous section, represented low-flow CHF, where CHF at high equilib-

rium qualities occurred. Noting the insensitivity of their data to inlet

subcooling, Bowers and Mudawar developed the following empirical correl-

ation:

q 27

G��&

� 0.16 �G�L

��

(L /D)��. (135)

This correlation is remarkable for the implied recognition of the importance

of surface tension. The correlation, however, has not been validated


against data from other sources and is unphysical in its monotonic depend-

ence on D.

Shah [185] has developed an empirical CHF correlation based on a vast

pool of data representing upflow in vertical channels with parameter ranges

0.315�D� 37.5mm, 1.3� L /D� 940, 4�G� 29041kg/m�s, 0.0014�Pr� 0.96, and inlet qualities covering �2.6 to 1.0. Shah’s database includes

a wide variety of fluids, and his empirical correlation is dimensionless and

utilizes all important thermophysical properties. Shah’s correlation has two

versions: the upstream condition correlation (UCC), and the local condition

correlation (LCC). The UCC vesion can be expressed as

q" 27

/G��&

� 0.124(D/L/)��(10�/Y )(1 � x

�/) (136)

where x�/

and L/

are the effective inlet equilibrium quality and effective tube

length, respectively, and n is an empirical exponent. When x�� 0, L

/is the

axial distance from the channel inlet and x�/

�x�; when x

�� 0 L

/is equal

to the boiling length (i.e., the axial distance from the point where equilib-

rium quality is equal to zero) and x�/

� 0. Distinction is made between

helium and other fluids. For helium n� (D/L/)�, and for other fluids

n� (D/L/)��, for Y � 10� (137)

n�0.12

(1 �x�/)��

, for Y � 10�. (138)

The parameter Y (Shah’s correlating parameter) is defined as

Y �GDC

�!k!

(��!gD/G�)��(�

!/�

0)��. (139)

The LCC correlation of Shah can be expressed as

q" 27

/Gh�&

� F/FBo

%, (140)

where F/, the entrance effect factor, is the smaller of 1 and [1.54� 0.032(L

�/

D)], with L�representing the axial distance from entrance. Parameters Bo

%and F

are functions of the local quality, reduced pressure, P*

�, and the

parameter Y. Shah recommends that the UCC correlation be used when

Y� 10� or L/� 160/P��

�; otherwise, the correlation version predicting a

lower q" 27

should be chosen. Hosaka et al. [155] compared the predictions

of Shah’s correlation with their data. On the average, the correlation

overpredicted the data only slightly. The correlation, however, has not been

adequately compared with other recent microchannel CHF data. Katto

[159] has indicated that the strong dependence of the parameter Y in Shah’s

CHF correlation on g for high mass flux forced flow may be physically

questionable.


E. Theoretical Models

Theoretical modeling of CHF has undergone great advances in the recent

past. Most of the developed models, however, may be inapplicable to

microchannels.

Weisman [186] has summarized the status of theoretical models for CHF

and indicated that the phenomenology of CHF depends on the two-phase

flow regime. In annular (high quality) flow, film dryout leads to CHF. In

the slug/plug flow regime, CHF appears to occur if the time period

associated with the passage of a gas plug is long enough to allow for the

complete evaporation of the liquid film that is left behind on the wall

following passage of a liquid slug. In highly subcooled flow, CHF is

triggered by the thermal and hydrodynamic processes adjacent to the heated

wall. Two modeling approaches have been pursued for CHF in highly

subcooled flow. Weisman and co-workers [187, 188] suggested that the

coalescence of microbubbles that form on the wall and the occurrence of a

critical void fraction in the bubble layer lead to CHF.

For subcooled or low-quality CHF, based on careful flow visualization,

Lee and Mudawar [189] proposed that CHF in the aforementioned regime

occurs when the liquid sublayer that separated vapor blankets or slugs from

the wall is disrupted. They developed a mechanistic model accordingly.

More recent flow visualization studies further support the latter liquid

sublayer dryout model [190], and models based on this mechanism, and

following the essential elements of the model by Lee and Mudawar [189],

have recently been compared with data including small channel data by

Katto [191, 192] and Celata et al. [193]. The outline of the model, as

elaborated by Katto [191], is now presented.

Figure 31 is a schematic of the flow field [191], where vapor slugs, formed

as a result of the coalescence of smaller bubbles, are separated from the

heated surface by a liquid sublayer. The vapor blankets are assumed to

remain thin because of condensation, and their velocity is assumed to be

closely related to the local ambient fluid velocity. CHF is assumed to occur

when the residence time of a vapor slug over the liquid sublayer is

sufficiently long to allow for its complete evaporation and breakdown.

Katto assumed that the local flow quality can be obtained from the quality

profile fit of Ahmad [194], according to which

x �

x�$

�x�$�%�

exp �x�$

x�$�%�

� 1�1�x

�$�%�exp �

x�$

x�$�%�

� 1�for x

�$�3& x

�$, (141)

where the equilibrium quality at the point of onset of significant void, x�$�%�

,


Fig. 31. Schematic of flow field near the CHF conditions with high subcooling and high

mass flux. (With permission from [191].)

is obtained from the empirical correlation of Saha and Zuber [121]. The

sublayer initial film thickness at the front end of the vapor slug is found

based on an empirical correlation due to Haramura and Katto [195];

(��"�

� 1.705 10�' ��!��

�1��!�

��

��h��&

q"��, (142)

where the boiling heat flux, q"�, is obtained by assuming that the wall heat

flux is the summation of convective and boiling terms, thereby

q"�� q"

�� h

7 (T�

�T !), (143)

where h7

is obtained from the well-known correlation of Dittus and

Boelter, assuming purely liquid flow, and T�

�T !

is obtained from

T�

�T !�

(�%� 1)(T

#��T

!) � q"

�/h7

�%

(144)

�%� 230(q"

�/Gh�

�&)��. (145)

CHF is assumed to occur when


q"��

!(��"�

h��&

/t��

, (146)

where t��

, the residence time of the vapor slug over the film underneath it,

can be found from

t��

�L�

U�

�2'�(�

!��)

�!��U

�

. (147)

Here, the length of the vapor slug has been assumed to be equal to the

neutral wave length according to the Helmholtz stability theory. The

velocity of the vapor slug relative to the film is assumed to follow

U��,U

!�� , (148)

where U!�� is the velocity at the distance (

��"�from the wall, as predicted by

the Karman’s universal turbulent boundary layer velocity profile, based on

a wall frictional shear stress found from

�� ( f /4)G�/2�

�. (149)

The homogeneous density, ��, is found using the flow quality obtained from

Eq. (45), and the D’Arcy friction factor f is found from the Prandtl—Karman correlation:

1/� f � 2.0 log�

(Re� f ) � 0.8 (150)

Re�GD/[��!(1 � �)(1 � 2.5�)]. (151)

Katto empirically correlated the parameter , in Eq. (148) in terms of

Re(��/�!) and � [191, 192].

The model of Katto [191, 192] just described has been shown to predict

experimental data well for channels with D2 1mm, for a variety of fluids

and a pressure range of 0.1 to 2.0MPa. More recently, Celata et al. [193]

pointed out that Katto’s model is unable to calculate the CHF when the

void fraction is larger than 70%. Celata et al. further modified the aforemen-

tioned phenomenological model of Lee and Mudawar [189] and Katto

[191, 192]. They assumed that the thickness of the vapor blanket (slug) is

equal to the bubble departure diameter, and the vapor blanket is always

surrounded with saturated liquid. Celata et al. [193] obtained the vapor

blanket velocity, U�, from the balance between drag and buoyancy forces,

assuming a vertical, upflow configuration. Other essential model elements

were similar to [189, 191]. Celata et al. [193] compared their model with

experimental data covering the following range of parameters, with good

agreement between model and data: 0.2&D& 25.4mm, 25��T�� 255K, 10&G& 9 10�kg/m�s, and 0.1&P& 8.4MPa. The applica-

tion of the aforementioned force balance on vapor blankets in the model of


Celata, however, implies dependence on channel orientation, which may not

apply to microchannels.

CHF under high-quality conditions is primarily due to the film dryout

phenomenon in the annular flow regime. The film dryout models are based

on the solution of two-phase mass, momentum, and energy conservation

equations, whereby the flow rate of the liquid film on the heated wall and

the film thickness are calculated. CHF is assumed to occur when the liquid

film is depleted [159]. Annular-dispersed flow is the flow regime that occurs

during film dryout CHF in commonly used large channels, where entrained

liquid droplets are mixed with the vapor phase and the two-phase flow is

accompanied by continuous entrainment of new droplets from the liquid

film and deposition of droplets on the film. The variation of the film

thickness is evidently affected by evaporation as well as droplet entrainment

and deposition. Dryout CHF models utilize empirical correlations for

droplet entrainment and deposition [196, 197]. A model by Sugawara et al.[198] also accounts for the inhibition of droplet deposition due to the

counter flow of vapor resulting from film evaporation.

Film dryout models have not been systematically applied to microchan-

nels, and the current models [196, 198] employ constitutive relations

associated with interfacial transfer processes and droplet entrainment and

deposition that may not be applicable to microchannels. The recent low-

flow and high-quality CHF data [95, 174] indeed suggest that the dryout

phenomenology in microchannels may be significantly different than in

larger channels.

VII. Critical Flow in Cracks and Slits

A. Introduction

Critical or choked flow represents the maximum discharge rate of a fluid

through an opening connecting a pressurized vessel to a low-pressure

environment. When choking happens, the flow conditions downstream of a

location where critical conditions occur do not affect the flow rate, implying

that the hydrodynamic signals originating downstream are unable to pass

through the critical location.

Critical flow of a compressible fluid can be well predicted by assuming

that the one-dimensional fluid velocity at the critical location is equal to the

local isentropic speed of sound, and knowledge of fluid stagnation proper-

ties is sufficient for calculating the conditions at the critical cross-section.

Critical two-phase flow is considerably more complicated, however, because

of the development of thermal and mechanical nonequilibria between the


phases. The fluid stagnation properties are thus insufficient for uniquely

determining the conditions at the critical cross-section in two-phase flow.

Critical flow can be used for flow control and plays an important role in a

number of hypothetical nuclear reactor accidents. Critical two-phase flow

has been extensively studied in the past. Good reviews can be found in [106,

199, 200].

The critical flow of initially highly subcooled liquids through narrow

cracks and slits is of great interest in relation to the safety of nuclear and

chemical reactors. When cracks occur in high-pressure piping systems, they

often support critical flow and, in accordance with the leak-before-break

concept, their detection and correct characterization are necessary for

prediction and prevention of major leaks. Extensive research effort has been

devoted to the critical flow in cracks and slits in the past 15 years. Most of

the studies have focused on critical flow in slits or simulated cracks with

simple and regular cross-sections. Photomicrographs of typical intergranu-

lar stress corrosion-induced cracks in type 304 stainless steel [201], however,

show that such cracks often have highly tortuous and irregular flow

passages.

Cracks often have hydraulic diameters in the D�& 1 mm range and have

large length-to-diameter ratios; their nucleation, boiling heat transfer, and

two-phase flow characteristics are different from those of large channels.

Consequently, the correlations and models that have been developed in the

past for critical flow in commonly applied large channels do not apply to

cracks without modification.

In the forthcoming discussion, previous experimental studies dealing with

critical flow of initially subcooled liquids in cracks are reviewed in Section

B. The theoretical models are discussed in Sections C—E.

B. Experimental Critical Flow Data

Table V is a summary of the recent experimental data. Collier et al. [202,

203] performed two sets of experiments. In Phase I of their experiments,

they measured critical flow in artificial slits. The geometric and test

parameters associated with their Phase I experiments are summarized in

Table V.

The test sections were produced by splitting flanges and machining crack

faces in the center of each half, then reassembling the two flange halves. The

simulated crack surfaces were roughened by shot blasting them, and the

crack width, (, was adjusted with spacer blocks placed between the two

flange halves. They measured the temperatures and pressures at three

locations along their simulated cracks. In phase II of their experiments,

Collier et al. [202, 203] used a stainless steel pipe with 32.4 cm outside


TABLE V

Summary of Experimental Data for Critical Flow in Cracks and Microchannels

Maximum

inlet Inlet

pressure subcooling Length Roughness

Author Fluid (MPa) (K) Cross-section (mm) (�m)

Collier et al. [202, 203] Water 11.5 33—118 Rectangular, W� 57.2mm; S � 0.2—1.12mm 63.5 0.3—10.2;

simulated crack

Collier et al. [202, 203] Water 11.5 0—72 Rectangular, W � 0.74—27.9mm; 20 Real crack

S� 0.0183—0.247 mm

Amos and Schrock [204] Water 16.2 0—65 Rectangular, W� 20.4mm; S � 0.127—0.381 mm 60—75 Smooth

Kefer et al.# [205] Water 16.0 0—60 Rectangular, W� 19—108mm; 10—33 20—40;

S� 0.097—0.325mm simulated and real

cracks

John et al. [105] Water 14 2—60 Rectangular, W � 80mm; S � 0.2—0.6mm 46 2—150; real cracks

with 240

Nabarayashi et al. Water 7 &30 Real cracks in pipe wall, Dp� 114.3, 216.3mm, 8.6, 12.7 mm 5.4—12.1

[206, 207] W� 60—160mm, S�0.5 mm;

rectangular cracks, W � 30, 60mm, 10, 20, 36mm 3.6—12

S � 0.07—1 mm

Ghiaasiaan et al. [208] Water 7.24 34—258 Circular, D� 0.78mm 0.78mm Not measured

#Cited in John et al. [105].

226

diameter that contained a girth butt weld at midlength with a full circum-

ferential stress-corrosion crack near the welds. The initial circumferential

cracks were deep about 90% of the wall thickness, and through cracks of

varying lengths were obtained by machining away the pipe surface in the

vicinity of the crack.

Amos and Schrock [204] used two stainless steel blocks, with their

surfaces ground flat, to provide flow channels with smooth walls. Various

slit widths were created by adjusting the distance between the stainless steel

blocks. They measured axial pressure variations using several pressure taps.

The experimental data of Kefer et al. [205] are referred to by John et al.[105]. John et al. [105] used two blocks of stainless steel to produce slits

with adjustable distances between the steel blocks in steps of 0.1mm. Using

several pressure taps they measured pressure distribution along their

simulated cracks. To examine the effect of slit surface roughness, they varied

the surface roughness of the steel blocks by shot blasting them with sand

and steel grit. For tests with a real crack, they used blocks of real reactor

pipe steel (20 Mn Mo In 55), with a crack that was produced by cyclic

bending. They measured the width of the cracks at inlet and exit of their test

section after mounting them. In some cases the slit width at exit was slightly

larger than at its inlet.

Nabarayashi et al. [206] and Matsumoto et al. [207] were interested in

fatigue cracks in carbon steel and stainles steel pipes and performed two sets

of experiments. In the ‘‘fundamental’’ tests [206], critical flow of saturated

water and steam in artificial slits with W � 30 and 60mm, L � 10, 20, and36mm, S� 0.07—1 mm was studied. They also conducted experiments in

through-wall cracks [207], initiated by electric discharge machining and

popagated by bending. They performed a careful measurement of surface

roughnesses in the artificial and fatigue-induced cracks. The ranges of

average roughnesses are given in Table V. The maximum surface rough-

nesses varied in the 20—55�m range for their artificial cracks and in the

16—60�m range in the fatigue-induced cracks. The surface roughnesses,

furthermore, did not vary noticeably after tests. They empirically correlated

their data for the effect of bending undulation on the crack pressure loss.

The experimental investigation of Ghiaasiaan et al [208] was concerned

with critical flow of highly subcooled water through a very short capillary.

Some important trends in critical flow data are now described. The

experimental data indicate the occurrence of metastable liquid flow near the

entrance of cracks, due to delayed nucleation (pressure undershoot). Figure

32 is an example pressure profile [204] that shows the effect of delayed

flashing. The solid line represents the liquid single-phase pressure drop, the

dashed line represents the measured pressure profile, and the horizontal line

is the saturation pressure corresponding to the inlet temperature. Flashing


Fig. 32. Pressure profile in a critical flow experiment [204]. (With permission from [204].)

occurs when the slope of the dashed line abruptly deviates from the slope

of the solid line. This type of delay in flashing is also common in large

channels and is typically 1—2K for water [209]. Amos and Schrock [204]

noted that the degree of subcooling before the inception of flashing in-

creased with increasing the stagnation subcooling. An empirical correlation


Fig. 33. Some parametric trends in the experiments of Amos and Schrock [204]. (With


for pressure undershoot has been developed by Alamgir and Lienhard [210]

and has been applied by some investigators in mechanistic modeling of

critical flow in large channels. Amos and Schrock [204] noted that the

majority of depressurization rates in their experiments were outside the

range of the latter correlation.

With respect to the important trends, the available critical flow experi-

mental data indicate that G��

(a) increases with increasing inlet subcooling;

(b) increases with increasing stagnation pressure; (c) decreases with increas-

ing L /D; and (d) is very sensitive to frictional pressure drop and consequent-

ly decreases with increasing the channel surface roughness. Figures 33 and

34 depict typical parametric trends in the experiments of Amos and Schrock

[204] and John et al. [105], respectively. The depicted model predictions are

discussed later.

In Fig. 35 typical data of Ghiaasiaan et al. [208] are depicted and are

compared with some relevant data points from Collier and Norris [202].

The figure demonstrates that critical flow in short capillaries may be


Fig. 34. Some parametric trends in the experiments of John et al. [105]. (With permission

from [105].)

significantly different from that in cracks and slits with large L /D. The

measured G��

in the experiments of [208] are evidently much larger than

those measured by Collier and Norris [202] under similar conditions and

demonstrate the significance of channel length. The data of [208], further-

more, are relatively insensitive to the inlet subcooling. The latter data,

however, represent extremely high inlet subcooling, and their apparent

insensitivity to small variations in inlet temperature may not be representa-

tive of crack critical flow in their typical applications.

C. General Remarks on Models for Two-Phase Critical Flowin Microchannels

Many semianalytical and mechanistic models have been developed for

two-phase critical flow in common large passages. These models can be


Fig. 35. Comparison of the data of Ghiaasiaan et al. [208] with the data of Collier and

Norris [202], the predictions of the correlations of the RETRAN-03 code [219], and the

correlation of Leung and Grolmes [220]. (With permission from [208].)

divided into two broad groups: integral models and models based on

numerical solution of differential conservation equations. The two groups of

models follow similar principles: They all search for conditions that lead to

the maximum possible mass flux without violating the first and second laws

of thermodynamics. In the integral models, the conservation equations are

analytically integrated along the flow passage in order to derive a closed-

form solution for the mass flux. The resulting closed-form solutions, of

course, may need iterative numerical solution. Major simplifying assump-

tions are evidently needed to make closed-form solutions possible. Reviews

of the most widely used integral models can be found in [106, 200].


Mechanistic critical flow models based on the numerical solution of the

one-dimensional differential conservation equation can rigorously account

for the thermal and mechanical nonequilibria and the system integral effects

on local flow properties. In these models, the 1D conservation equations are

numerically solved and, by iteratively varying the passage discharge rate, the

flow rate leading to critical conditions (an infinitely large pressure gradient,

a vanishing determinant of the coefficient matrix for the system of one-

dimensional quasi-linear conservation equations, etc.) is specified. Early

contributions to this method include [211, 212], and a discussion of

mathematical bases can be found in [213]. This technique is particularly

appropriate for cracks and slits with large L /D where the 1D flow assump-

tion is adequate. Models based on direct numerical solution of differential

conservation equations for cracks and slits have been developed by Amos

and Schrock [204], Schrock and co-workers [201, 214, 215], and Feburie etal. [216]. More recent contributions include [217, 218], where the effect of

noncondensables was modeled.

D. Integral Models

Integral models dealing with two-phase critical flow in large passages are

many and are reviewed among others in [106, 200]. These models are

mostly based on assumptions and/or simple models that allow for the

calculation of the velocity slip and the magnitude of thermodynamic

nonequilibrium at the critical cross-section. Most of the widely used

methods for the prediction of two-phase critical flow neglect the frictional

pressure losses, which are often small in commonly applied large passages.

These models overpredict the critical flow rates in small passages and cracks

where such pressure losses are significant.

The isentropic homogeneous-equilibrium model is the simplest among the

models that neglect irreversible losses and leads to

G��

� [2��(h�%� h�

��)]� � (152)

where

x��

��s%� s

�s�&��

(153)

h��

� (h�� xh�

�&)��

(154)

v��

��

� (v��xv

�&)��. (155)

The critical mass flux is obtained by applying (dG/dP)��

� 0, which leads to


2[h�%� h� (s

%, P

��)] �

�v�P�

��

� v�(s%, P

��) � 0. (156)

Note that (�v/�P) also corresponds to the thermodynamic state specified

with (s%, P

��). Equation (156) along with Eqs. (153)—(155) are thus iterative-

ly solved for P��, as well as other thermodynamic properties at the critical

cross-section, whereupon G��

is obtained from Eq. (152).Since the aforementioned iterative solution is cumbersome, simpler corre-

lations for water have been developed [219, 220]. The correlation in [219]

is a simple polynomial-type curve fit to the predictions of Eqs. (152)—(156)for water, and the correlation can be found in [200]. Utilizing an approxi-

mate equation of state for water, Leung and Grolmes [220] derived the

following correlation, based on the isentropic homogeneous-equilibrium

flow assumption for water:

G��

��2w

w � 1��1��1 �1

��2w� 1

2w ��

�� x �P%��%�

� ��

. (157)

Here,

��P

#�(T

%)/P

%��(158)

��C�%

T%P#�

(T%)

��%

��&%

h��&%��, (159)

where T%

is the subcooled liquid stagnation temperature and P%��

is the

stagnation pressure at channel inlet when the effect of channel entrance

irreversible pressure loss is accounted for:

P%��

�P%�K

�� A��

A�� G�

��2�

%

. (160)

Predictions of the aforementioned approximate correlations are com-

pared with typical experimental data of Ghiaasiaan et al. [208] in Fig. 35.

The flow passage in the latter data is very short (L /D$ 1) and their

frictional losses are negligible. Model predictions are of course relatively

sensitive to the entrance pressure loss coefficient, K��

. For sharp-edged

sudden contractions with very large contraction ratio, K��

� 0.5 [221], and

with smooth or conical entrances K��

will be lower. The isentropic homo-

geneous-equilibrium model well predicted the latter data, with standard

deviations of about 10% with K��

� 0.5 and about 13% for K��

� 0.3.

The isentropic homogeneous-equilibrium model is not appropriate for

application to cracks and slits with larger L /D, however, and overpredicts

the data because of the significance of frictional pressure losses in cracks and

slits.


The critical flow of initially saturated liquid of a liquid—vapor mixture in

a channel, with frictional pressure loss, has been modeled by Moody [222].

This model assumes thermal equilibrium between the vapor and liquid

phases and accounts for velocity slip by assuming a slip ratio everywhere

equal to the expression derived by Zivi [99], Eq. (68), for minimum entropy

production in a steady-state liquid—vapor flow. In an earlier, and widely

used, model for critical, isentropic flow, Moody derived the same expression

for the slip ratio at the channel throat. The model of Moody [222] is based

on the integration of the one-dimensional two-phase momentum conserva-

tion equation along a constant-area channel,

�dP/dz� f�!'

L /D��!'

G�

2�!

�d

dz[xU

0� (1 � x)U

!]G, (161)

where f �!'

is an average D’Arcy channel friction factor for liquid flow. The

one-dimensional energy conservation also gives

h�%� h�

!� h�

�&�

1

2G� �

(1 �x)

��!(1 � �)�

�x

��0��. (162)

Note that � and x are related by the slip ratio, according to Eq. (32). Since

the fluid mixture is saturated everywhere, the right-hand side of Eq. (161)can be expanded by replacing d/dz with d/dP(dP/dz), obtaining d/dP of all

properties using appropriate thermodynamics data, and factoring out (dP/dz) in all the terms that include it. Equation (162) is also utilized in the

manipulation of Equation (161), noting that dh%/dz� 0. The final result can

be represented as

�.�

.�

�(P, h%, G )dP � f�

!'

L

D, (163)

where P�

is the pressure at inlet to the channel (i.e., the point beyond which

irreversible pressure losses occur), P�

is the pressure at the location of the

throat, and � is a function that involves the pressure, slip ratio, and

thermodynamic properties and their partial derivatives. The critical mass

flux is obtained by iteratively changing P�

and solving Eq. (163) for G, until

dG/dP�� 0, at which point P

��P

�and G

��G, Moody [222] suggested

the following expression for the two-phase multiplier:

��!'

��1 �x

1� ��. (164)

Nabarayashi et al. [206, 207] compared their measured flow rates with

predictions of the aforementioned model of Moody [222], and noted that

Moody’s model well predicted their data when pressure losses due to the


entrance effect and wall friction, and flow area reduction due to the wall

roughness, were correctly accounted for. The aforementioned model of

Moody [222] well predicted the measured flow rates in their fatigue-induced

cracks as well, once the entrance and frictional pressure losses, flow area

blockage by wall roughness, and pressure losses due to bending undulation

were all included in the calculations.

Based on the experimental data of Collier et al. [202, 203], a homogeneous-

nonequilibrium model was developed at Battelle-Columbus Laboratory,

which is briefly described by Abdollahian et al. [223]. The model was based

on the widely used homogeneous, nonequilibrium model of Henry [224]. In

the model, which was coded into the computer program LEAK, the flow

was assumed to remain single-phase liquid between the entrance and a point

located at a distance of z � 12D downstream from the entrance; the

acceleration and frictional pressure drops between the channel inlet and the

critical cross-section were obtained using channel averaged properties and

assuming homogeneous flow everywhere; the expansion of the vapor phase

was assumed to be isentropic; and quality was assumed to vary linearly

between z� 12D and the throat.

Abdollahian et al. [223] and Chexal et al. [225] modified and improved

the LEAK model and developed the LEAK 01 model by using an isenthal-

pic flow assumption for pressure drop calculations, assuming a linear flow

area variation along the channel, and improving the two-phase frictional

pressure drop calculation. Noting that for subcooled liquid inlet conditions

the models based on the homogeneous-equilibrium flow in cracks appear to

generally predict saturated liquid conditions at the critical cross-section.

Abdollahian et al. [223] also proposed the following simple expression for

the critical mass flux in long racks and slits:

G��

��2[P

%�P

#�(T

%)]

��(1 � f L /D

2) �K

��%��

. (165)

Here, ��

is the average homogeneous two-phase mixture specific volume:

�� !

��x! �!

�&. (166)

The average properties, including �!�

and �!�&

, are calculated at

P � [P%�P

#�(T%)]/2. (167)

The average quality is defined as

x! � [h�%� h�

�(P )]/h�

�&. (168)

Abdollahian et al. used the unrealistically high K��

� 2.7, however, and

obtained f from Karman’s correlation for rough-walled channels. The two


models of Abdollahian et al. (i.e., LEAK-01 and the aforementioned simpli-

fied model) well predicted the data of Collier et al. [201, 203], but were

shown by John et al. [105] to do relatively poorly when applied to the latter

authors’ data, as well as the data of Amos and Schrock [204].

E. Models Based on Numerical Solution of DifferentialConservation Equations

Most of the models dealing with two-phase critical flow in large channels

include constitutive relations that may not apply to cracks and slits. The

models of [209, 217, 226, 227] all apply the two-fluid technique and include

interfacial transfer models of doubtful applicability to microchannels and

cracks where interfacial slip is likely to be suppressed by surface tension.

Equal phasic velocities are assumed in [228, 229]. The latter references,

however, use models for bubble nucleation and relaxation-type delayed

evaporation that may not apply to microchannels.

Several models that specifically address cracks and slits have been

developed and published. These models are now briefly reviewed. Amos and

Schrock [204] and Lee and Schrock [214] modeled critical two-phase flow

in cracks assuming a homogeneous (equal phasic velocities) nonequilibrium

flow, by solving numerically the one-dimensional mixture mass, momentum,

and energy conservation equations. They used empirically adjusted closure

relations, however, to obtain agreement with experimental data. Flashing

was assumed to occur when a pressure undershoot (ie., local pressure lower

than the saturated pressure corresponding to the local pressure) is obtained,

given by

�P��S�P

��!, (169)

where �P��!

is the pressure undershoot associated with bubble nucleation,

as correlated empirically by Alamgir and Lienhard [210]:

�P��!

�C �0.252��(T /T *

��)��(1 � 144��)��

(kT *��)��(1 ��

0/�

!) � (170)

In this equation, kis Boltzmann’s constant, and T *

��is the thermodynamic

critical temperature. Except for 4, the depressurization rate, which must be

in Matm/s, the remaining dimensions are consistent. The correction par-

ameter, C, was empirically correlated by Amos and Schrock in terms of local

velocity [204], and as a function of Reynolds and Jacobs numbers by Lee

and Schrock [214], to obtain agreement between model predictions and


data. The evaporation of the metastable liquid phase (the relaxation of the

liquid phase superheat) was assumed to follow an exponential form, with a

time constant that was also empirically adjusted to fit their data. Critical

flow was assumed to occur when the mixture velocity at the critical

cross-section was equal to the local velocity of sound, and the latter was

represented by an expression by Kroeger [230] representing zero interfacial

slip. Lee and Schrock [214] compared the predictions of their model with

data from several sources. Most of the data represented critical flow in large

channels, however.

The real intergranular cracks have complicated and tortuous configur-

ations. Schrock and Revankar [215] argued that for critical flow in real

intergranular corrosion cracks, the homogeneous-equilibrium model is ap-

propriate, since significant flow separation and thermal nonequilibrium are

unlikely. They developed the fast-running homogeneous-equilibrium code

SOURCE, in which the mixture mass, momentum, and energy conservation

equations were numerically solved using the finite-difference technique, and

the discharge rate was iteratively varied until the mixture velocity at the

critical cross-section was equal to the velocity of sound in homogeneous-

equilibrium two-phase flow. They assumed an equivalent friction factor that,

in addition to wall friction, accounts for the irreversible pressure losses

associated with flow disturbances and tortuosity. Shrock et al. [215]

compared the predictions of SOURCE with 61 out of the 82 BCL data that

were found to be acceptable. An optimum friction factor was found that best

fitted the data associated with each test section, and thereby they developed

a methodology for the prediction of the equivalent friction factor for real

cracks. Since SOURCE predictions showed a systematic dependence on

inlet subcooling, furthermore, to obtain agreement between model and data,

Schrock and Revankar also developed an inlet subcooling correction factor

that must be multiplied by the mass flux predicted by the code in order to

calculate the correct mass flux.

Feburie et al. [216] developed a nonequilibrium model assuming the

existence of three phases: saturated vapor, saturated liquid, and metastable

superheated liquid. The pressure undershoot leading to the initiation of

evaporation was modeled according to [231]

P

P#�

(T!(

)� k

�, (171)

where T!(

is the temperature of the superheated liquid and k�� 0.95 to

0.97. Following the initiation of evaporation, the three phases are assumed

to move at the same velocity (homogeneous flow). Their one-dimensional

steady-state conservation equations for the mixture mass, momentum, and


energy are

d

dz(Au/�

�) � 0 (172)

dP

dz��

�U

dU

dz��

p��

A(173)

GAds

�dz

�GAC.! �ln �

T#�

T!(��

T!(

T#�

� 1��dy

dz

� �p2�!(

q"��!(

T!(

�p��!�

q"��!�

� p��0

q"��0

T#�

�� U �

p��!(

��!(

T!(

�p��!�

��!�

� p��0

��0

T#�

�, (174)

where subscripts L M, L S, and G represent the metastable superheated

liquid, saturated liquid, and saturated vapor, respectively, and (1 � y)represents the metastable liquid mass fraction in the mixture. The last two

terms on the right-hand side of Eq. (174) are the entropy sources due to wall

heat transfer and friction. The parameter p2�!(

represents the wall-super-

heated liquid perimeter through which the heat flux q"��!(

is transferred, and

p��!(

and ��!(

represent the wall-superheated liquid perimeter and its

associated shear stress, respectively. For simplicity, T!(

� constant is as-

sumed once bubble nucleation starts. Other parameters are defined similar-

ly. The mixture entropy is defined as

s�� (1 � y)s

!(� xys

&� (1 �x)ys

!�, (175)

where xy and (1 �x)y are the mass fractions of saturated vapor and

saturated liquid in the mixture, respectively. Other mixture properties,

including ��, are defined similarly. Two additional equations are provided

by the equation of state,

��

�(P, s

�, y), (176)

and a closure relation in the form [232]

dy

dz� k(1 � y) �

P#�

(T!(

) �P

P��

�P#�

(T!(

)��

, (177)

where k is a constant assumed to depend on geometry according to

k� k�

p�

A. (178)


Using the Moby-Dick experimental data, k�� 0.02m. Equations (172)—

(174), (176) (upon differentiation), and (177) constitute five coupled equa-

tions, which can be represented in the form of a quasi-linear set of coupled

ordinary differential equations:

AdY

dz�B (179)

Y� (P, U, ��, s

�, y)�. (180)

Note that if ��

and s�

are known, x and T!(

can be calculated. These

equations can be numerically integrated, and by iteration the critical mass

flow rate, which leads to the vanishing of the determinant of the coefficient

matrix A, can be specified. For wall friction and heat transfer, Feburie et al.used various correlations.

Feburie et al. compared preditions of their model for G��

with 70 data

points from John et al. [105], with agreement within �12%, and with 14

data points from Amos and Schrock [204], with agreement within about

�5%. Sensitivity analysis indicated that the model predictions were rela-

tively sensitive to the magnitude of the constant k�

in Eq. (171), the wall

friction, and entrance pressure loss, and were insensitive to the magnitude

of the constant k�.

Geng and Ghiaasiaan [218] recently developed a homogeneous-equilib-

rium model for the critical flow of an initially subcooled liquid through

cracks and slits, where the effect of a dissolved noncondensable in the inlet

subcooled liquid was accounted for. They assumed that: (a) the solubility of

the noncondensable in the liquid is low; (b) the liquid and vapor gas phases

are everywhere at thermodynamic equilibrium and at equilibrium with

respect to the concentration of the noncondensable; and (c) the vapor—noncondensable gas mixture is everywhere saturated with respect to vapor.

For a channel with uniform cross-sectional area that is inclined with respect

to the horizontal plane by the angle 1! , the mixture mass, momentum, and

energy conservations are

d

dz(��U ) � 0 (181)

��U

dU

dz��

dP

dz� �

�g sin 1! �

p��

A(182)

d

dz �U ��!(1 � �)h��

&h�&�

1

2��U��

�Ug sin 1! � p

�q"�/A,

(183)


where, assuming that the noncondensable is an ideal gas,

��

!(1 � �) ��&�

P �P#�

(T )

(R/M�)T � �, (184)

where T is the local mixture temperature.

Utilizing Henry’s law [233] for the equilibrium between the gas and liquid

phases with respect to the concentration of the noncondensable, Geng and

Ghiaasiaan derived

d

dz �U ��(1 � �)M!��

0�

M!

(P/C2�

) � [1� (Mg/M�)]M

!�� 0 (185)

d

dz �M!��

M�

M&�

P�P#�

(T )

C2�

�� 0. (186)

Geng and Ghiaasiaan expanded Eqs. (181)—(183), (185), and (186) using

thermodynamic relation, and cast the aforementioned equations in the form

represented by Eq. (179), with

Y � (U, P, �, h��, M

!)�. (187)

Geng and Ghiaasiaan applied the correlation of John et al. [105], Eq.

(85), for single-phase wall friction and applied the homogeneous mixture

model, along with McAdams’ correlation (Eqs. (44)—(48) for two-phase

pressure drop. They numerically integrated the foregoing differential con-

servation equations and, by iteratively varying the channel mass flow rate,

specified G��

as the mass flux leading to det A� 0 at the critical cross-

section. Geng and Ghiaasiaan compared the predictions of their model with

the experimental data of Amos and Schrock [204] and John et al. [105]

with satisfactory agreement. To examine the effect of noncondensables, they

chose the data of Amos and Schrock as the basis for parametric calculations.

Figure 36 represents typical results, where model predictions with pure

water and water saturated with nitrogen at inlet are both presented. The

results indicate that desorption of nitrogen (or air) initially dissolved in

subcooled water can reduce the critical mass flux by several percent.

VIII. Concluding Remarks

The recent developments related to gas—liquid two-phase flow, forced-

flow subcooled boiling, and the critical (choked) flow of initially subcooled

liquids, in channels with hydraulic diameters of the order of 0.1 to 1mm,

were reviewed in this article. The hydrodynamic phenomena reviewed


Fig. 36. Comparison between the predictions of the model of Geng and Ghiaasiaan [218]

and the experimental data of Amos and Schrock [204] and the effect of noncondensable gas

on model predictions. (With permission from [218].)

included the two-phase flow regimes, void fraction, and frictional pressure

drop in narrow rectangular and annular passages, a micro-rod bundle, and

microchannels under conditions where surface tension and inertial forces are

both significant. The boiling phenomena reviewed included the onset of

nucleate boiling (ONB), onset of significant void (OSV), onset of flow

instability (OFI), and critical heat flux. The critical (choked) flow of initially

subcooled liquids in capillaries, cracks, and slits was also addressed.

The observed major two-phase flow regimes in microchannels are mor-

phologically similar to the flow regimes in large channels. However, they can

be insensitive to channel orientation and are influenced by surface wettabil-

ity. The commonly applied predictive methods for the flow regime transi-

tions in large channels overall fail to predict the microchannel data well.

Some empirical correlations that have been developed based on the micro-

gravity experiments, however, appear to agree with microchannel data

satisfactorily. The bulk of the existing microchannel data have been ob-

tained with air and water, and more experimental data examining the effects

of fluid properties, in particular the surface tension, surface wettability, and

liquid viscosity, are needed. The existing predictive methods for void

fraction and two-phase frictional pressure drop are also generally inad-

equate for microchannels, in particular for the annular flow regime.


With respect to boiling, the fundamental bubble ebullition phenomena in

microchannels are likely to be different from the qualitatively well-under-

stood bubble phenomena associated with boiling in large channels, because

of the occurrence of very large velocity and temperature gradients in the

former. Bubble ebullition in microchannels needs to be investigated.

The critical heat flux data associated with moderate and high mass fluxes

in microchannels are predicted satisfactorily by some of the recent and

widely used empirical correlations, because of the presence of relevant data

in the data bases of these correlations. The available data associated with

low-flow critical heat flux in microchannels, however, are unlike similar data

in large channels and suggest the occurrence of metastable superheated

liquid prior to dryout. Investigations aimed at the elucidation of the

hydrodyamic and evaporation phenomena associated with annular flow

regime in microchannels are thus needed.

The critical (choked) flow of an initially subcooled liquid in a capillary,

crack, or slit is strongly influenced by pressure drop and can be predicted

using models based on the homogeneous flow assumption.

Nomenclature

A coefficient matrix

A flow area; dimensionless

coefficient

B column vector

C constant

Ca capillary number

C

two-phase distribution

coefficient

Cp specific heat

D diameter

D�

hydraulic diameter

D�, D

%inner, outer diameters

Eo Eotvos number

F force term

F�, F

�, forces acting on a bubble (Figs.

26 and 27)F"�

, F�, F

,

F��, F

�f D’Arcy friction factor

f � Fanning friction factor

Fr Froude number

G mass flux

g gravitational constant

h convection heat transfer

coefficient

h!

liquid depth

h� specific enthalpy

h��&

specific heat of vaporization

k thermal conductivity

k

Boltzmann’s constant

K��

entrance loss coefficient

L length

M molar mass

M mass fraction

N� viscosity number

P pressure

�P�

pressure undershoot

p2, p

4heated and wetted perimeters

Pe Peclet number

Pr Prandtl number

P*��

thermodynamic critical

pressure

P*�

reduced pressure

Q dimensionless heat flux

q"�

wall heat flux

R universal gas constant

r radius

Re Reynolds number

S slip ratio; gap distance

s specific enthalpy


s�&

specific enthalpy of

vaporization

St Stanton number

T temperature

t time

T *��

thermodynamic critical

temperature

U velocity

U* friction velocity

U�

rise velocity

v specific volume

V&�

gas drift velocity

W width

We Weber number

X Martinelli parameter

x quality

Y Shah’s correlation parameter

y distance from wall; the

combined mass fraction of

vapor and saturated liquid

z axial coordinate

Greek Symbols

� Void fraction

- Volumetric quality

( Gap distance; film thickness

� Function in Moody’s critical

flow model

� Roughness

� Pressure ratio

1! function defined in Eq. (133);angle with respect to the

horizontal plane

, dimensionless coefficient

� Laplace length scale

� dynamic viscosity

� density

� surface tension

� shear stress

� kinematic viscosity

% parameter defined in Eq. (110)� two-phase multiplier

�* shape factor defined in Eq. (78)� function defined in Eq. (132);

contact angle

3 function defined in Eqs.

(119)—(121)�

function defined in Eq. (145)� function defined in Eq. (134)� function defined in Eq. (159)

Subscripts

B bubble, vapor blanket

b boiling

cr critical (choked)E effective

eq equilibrium

eqv equivalent

f frictional; saturated liquid

FC forced convection

G gas

g saturated vapor

GS superficial gas

G0 all-gas

H heated

h homogeneous

i inlet

L liquid

L M metastable liquid

L S superficial liquid; saturated

liquid

L 0 all liquid

m average

mod modified

n noncondensable

p pipe

sat saturation

T P two-phase

v vapor

w wall

0 stagnation

( film edge

Superscripts

mean


Abbreviations

CHF critical heat flux

HHF high heat flux

NVG net vapor generation

OFI onset of flow instability

ONB onset of nucleate boiling

OSV onset of significant void

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Turbulent Flow and Convection:

The Prediction of Turbulent Flow and Convection

in a Round Tube

STUART W. CHURCHILL

Department of Chemical Engineering

The University of Pennsylvania

Philadelphia, Pennsylvania 19104

The quantitative prediction of turbulent flow and convection in chan-

nels extends at least from Boussinesq, who in 1877 proposed the eddy-

viscosity model, to Papavassiliou and Hanratty, who in 1997 computed the

rate of transport of molecular species by the turbulent fluctuations using

Lagrangian direct numerical simulation. The history and current state of the

art of such predictions are examined herein. It is concluded that the eddy-

diffusivity, mixing-length, and ,—� models should now be completely aban-

doned in favor of generalized correlating equations for the turbulent shear

stress and the turbulent heat flux density based on semitheoretical asymp-

totic expressions, even though those for the latter quantity are yet uncertain

and incomplete. Correlating equations for the time-averaged velocity dis-

tribution, the friction factor, the time-averaged temperature distribution,

and the heat transfer coefficient may serve as conveniences, but such

expressions are unessential since these four quantities may be determined

numerically with comparable accuracy from simple, single integrals of the

two more elementary quantities. Such direct numerical evaluations reveal

that all of the classical algebraic analogies between heat and momentum

transfer are in significant error functionally as well as numerically, in large

part because of inaccurate representations of the radial variation of the total

heat flux density. In the interests of simplicity and clarity, this presentation

is primarily limited to fully developed heat transfer in a uniformly heated

round tube, but the methodologies and formulations may readily be adapted




0065-2717/01 $35.00

or extended for other one-dimensional flows and other thermal boundary

conditions.

I. Introduction

Progress in engineering and science occurs by discarding old concepts and

correlations in favor of new or improved ones. Turbulent flow and convec-

tion have been somewhat resistant to this process of renewal; many early

concepts and correlations remain enshrined in our current textbooks and

handbooks and in the software for design calculations even though they

long ago became obsolete in terms of accuracy or were shown to be

untenable in a theoretical sense. We should of course recognize and honor

the pioneers of our field and preserve their contributions in a historical

context, but at the same time be alert and aggressive about identifying and

incorporating experimental and theoretical improvements.

Turbulent forced convection is analogous to turbulent flow in many

respects, but is far more complex and demanding analytically, experiment-

ally, and in practice. Under many circumstances, flow may be studied

independently from heat transfer. On the other hand, the detailed analysis

of forced convection requires a quantitative description of the details of the

flow. Also, turbulent convection invokes the Prandtl number as a parameter

as well as several other complexities that are not encountered in the

description of the flow. For these reasons, flow is examined first and

convection thereafter. That division and order is observed in this Introduc-

tion as well as in the presentation as a whole.

In the interests of simplicity and clarity, the detailed descriptions that

follow are limited in the main to fully developed turbulent flow inside a

straight smooth round tube and to fully developed forced convection from

a uniformly or isothermally heated wall. However, the adaptation or

extension of these methodologies and formulations for other one-dimen-

sional flows is examined briefly and shown to be rather straightforward.

Attention is also limited to single-phase Newtonian fluids with invariant

physical properties, including the specific density. The latter restriction

excludes natural convection. Physical property variations are of course often

significant in magnitude and may greatly influence the flow and convection.

However, since the variations of the viscosity and thermal conductivity are

different for every fluid and that of the density for every liquid, accounting

for these effects would eliminate almost all of the generality that is a primary

feature of the new and old developments described herein. At the present

time the best compensation for this oversimplification is to incorporate

empirical corrections in the final idealized results for invariant physical

256 stuart w. churchill

Fig. 1. Self-portrait of Leonardo da Vinci observing turbulent vortices behind a disturbance

in a river. (from Richter [2], Plate XXV).

properties. As computational tools and techniques advance, it may become

feasible to incorporate such variations in direct numerical simulations of the

time-dependent differential equations of conservation or numerical integra-

tions of their time-averaged counterparts.

For convenience, a single standardized notation is utilized throughout

rather than necessarily those that were originally employed in particular

contributions. All symbols are defined when they appear first or in a new

context.

A. Turbulent Flow

Lamb [1], in his classical treatise Hydrodynamics, first published in 1879

but periodically revised by himself through 1932, a span of over half a

century, begins his rather brief treatment of turbulent motion in even the

latest of those editions with the statement, ‘‘It remains to call attention to

the chief outstanding difficulty of our subject.’’ He then proceeds to explain

the reasons for that difficulty, noting that ‘‘the motion becomes wildly

irregular and the tube appears to be filled with interlacing and constantly

varying streams, crossing and recrossing the pipe.’’

257turbulent flow and convection

Leonardo da Vinci in 1515, at age 63, sketched in his notebook, as shown

in Fig. 1 (Plate XXV from Richter [2]), a self-portrait with a river flowing

past obstructions. As indicated by the accompanying text, he was interested

in the pattern of flow in a scientific as well as an artistic sense. His

description has been translated (Richter [2], p. 200) as ‘‘Observe the motion

of the surface of the water which resembles that of hair, and has two

motions, of which one goes on with the flow of the surface, the other forms

the lines of the eddies; thus the water forms eddying whirlpools one part of

which are due to the impetus of the principal current and the other to the

incidental motion and return flow.’’ Even such a universal genius and

perceptive observer as Leonardo may sometimes err; he sketched symmet-

rical pairs of vortices rather than the alternating ones that actually occur.

The later renowned physicists and applied mathematicians who have

written on the subject of turbulent flow include Subrahmanyan Chandrasek-

har [3], Albert Einstein [4], Werner Heisenberg [5], Pyotr Kapitsa [6], Lev

Landau [7], Hendrik Lorentz [8], Isaac Newton [9], Lord Rayleigh [10],

Arnold Sommerfeld [11], George Uhlenbeck [12], Richard von Mises [13],

C. R. von Weizsacker [14], and Yakob Zel’dovich [15]. It is intimidating

and humbling for anyone who undertakes the study of turbulent flow to

realize that even these great scientists made few significant contributions to

this subject outside the special topic of stability.

The focus herein on shear flows and in particular on fully developed flow

in a round tube avoids the necessity of reviewing the statistical develop-

ments that have found applicability primarily in the idealized domains of

isotropic and homogeneous tubulence. Another quotation is appropriate in

this regard. Schlichting [16], in the Author’s Preface of the first German

edition of Boundary L ayer T heory, wrote (in translation) ‘‘No account of the

statistical theories of turbulence has been included because they have not

attained any practical significance for engineers.’’ In a later edition he writes

begrudgingly and defiantly in apparent response to criticisms of that

statement that ‘‘This (the statistical theory of turbulence) admittedly has

contributed to our understanding of turbulent flows but it has not yet

acquired any importance to engineers.’’ Although statistical representations

of turbulence have recently been utilized in the development of approximate

expressions for the kinetic energy of turbulence, ,, and the rate of dissipation

of the energy of turbulence, �, in connection with the ,—� and related

models, the details of that usage are outside the chain of development herein

and only merit this brief mention.

The general partial-differential equations for the conservation of mass and

momentum in time-dependent form are generally presumed to describe

turbulent flow insofar as the fluid may be treated as a continuum. Because

of their complexity, and in particular their nonlinearity, they resisted


numerical as well as analytical methods of solution until roughly the last

decade. Accordingly, as an alternative approach, Sir Osborne Reynolds [17]

in 1895 reduced these expressions to manageable proportions and a tract-

able form by space-averaging. This great simplification has inspired a

century-long development of semitheoretical models and approximate sol-

utions for the reduced equations. In particular, Ludwig Prandtl and his

associates and contemporaries developed a very useful structure for the

prediction and correlation of turbulent shear flows by postulating mechan-

istic models for the unknown term(s) in the time-averaged (rather than

space-averaged) equations of conservation in differential form. This pro-

cedure proved to be so successful in an applied sense that only sporadic and

limited improvements were made over the ensuing half-century. Finally, in

the 1980s and 1990s, a significant breakout occurred in the form of

essentially exact solutions of the general time-dependent equations of

conservation by direct numerical simulation (DNS). Although the results

obtained by this technique are yet very restricted in scope, both intrinsically

and because of their computational demands, this development has invigor-

ated the fields of turbulent flow and convection and has played a key role

in the development of the new formulations that constitute the principal

contribution of this article.

Before the presentation of these improved formulations, a retrospective

assessment of the historical development and validity of the quantitative

descriptions of turbulent shear flow in the classical and current literature is

provided. This survey is limited to representative contributions with long-

lasting consequences, both positive and negative, rather than being exhaus-

tive, since the primary objective is to provide a framework and perspective

for the new improved expressions. This portion of the presentation is

organized chronologically on the mean, but also topically, in part in the

interests of clarity, continuity, and the avoidance of repetition.

B. Turbulent Convection

Although the path of development of a structure for the correlation and

prediction of convection in turbulent flow might have been expected to

follow the path of development for the flow itself because of the similar

structure of the equation of conservation for energy to those for momentum,

this is not found to be the case. Turbulent convection is much more

complicated because of (1) the coupling of the equations for the conserva-

tions of energy and momentum, (2) the possibility of different boundary

conditions, and (3) the appearance of additional parameters.

Reynolds [18] in 1874, and thus 21 years prior to his great contribution

to the development of a simplified structure for turbulent flow by means of


space averaging, derived a simple algebraic equation, now known as the

Reynolds analogy, by postulating equal mass rates of transport of energy and

momentum from the bulk of the fluid stream all the way to a confining

surface by the turbulent eddies. This expression, which is free of any explicit

empiricism and independent of geometry, is only of first-order accuracy at

best, but it lives on after 125 years by virtue of its implicit or explicit

presence within many predictive correlations for heat and mass transfer.

An alternative approach to the prediction of the rate of heat transfer

involves the adaptation of the mechanistic models of Prandtl and others to

represent the primary unknown term (the turbulent heat flux density) in the

time-averaged differential equation for the conservation of energy. Such

modeling has, however, evolved more slowly and less successfully for

convection than for flow because of the inherently more complex behavior

mentioned earlier. Direct numerical simulation has also been utilized for

convection, but even more severe limitations are encountered than for flow

and, as a consequence, less success has been achieved.

In contrast with flow, the historical development of a structure for

turbulent convection is reviewed retrospectively in the light of the new and

improved formulations that prompted this article. Finally, some representa-

tive numerical results obtained by means of these new formulations are

presented and generalized.

II. The Quantitative Representation of Turbulent Flow

Structures for the prediction and correlation of the characteristics of

turbulent flow that are important in the subsequent prediction and correla-

tion of rates of heat transfer are first examined from a historical point of

view. New and improved formulations are then described and compared

with experimental data and numerical predictions. The derivations and final

expressions are presented herein primarily in the context of a round tube,

even though they may have been formulated originally for flow between

parallel plates or for unconfined flow along a flat plate. The adaptation or

extension of these new expressions for other geometries is finally examined

briefly.

A. Historical Highlights

1. The Exact Structure

The partial differential equations for the conservation of momentum in

the flow of a single-phase Newtonian fluid with invariant viscosity and


density may be expressed in cylindrical coordinates as follows:

Radial component:

� ��u

��t

� u�

�u�

�r�

u�r

�u�

�1!�

u��r

� u�

�u�

�z

��p�r

� � ��r �

1

r

��r

(ru�)��

1

r�

��u�

�1! ��

2

r�

�u��1!

��u

��z��g

�(1)

Angular component:

� ��u��t

� u�

�u��r

�u�r

�u��1!

�u�u�r

� u�

�u��z �

� �1

r

�p�1!

�� r �

1

r

��r

(ru�)��1

r�

��u��1! �

�2

r�

�u�

�1!�

��u��z� ��g� (2)

Axial component:

� ��u

��t

� u�

�u�

�r�

u�r

�u�

�1!� u

�

�u�

�z ��

�p�z

�� 1

r

��r �r

�u�

�r ��1

r�

��u�

�1! ��

��u�

�z� ��g�. (3)

Here, t represents time, r, z and 1! the radial, axial and angular coordinates,

g�, g

�, and g� the corresponding components of the gravitational vector, u

�,

u� , and u�

those of the instantaneous velocity vector, p the instantaneous

thermodynamic pressure, � the specific density, and � the dynamic viscosity.

Equations (1)—(3) are generally known as the Navier—Stokes equations in

recognition of their derivation by Navier [19] in 1822 and their refinement

by Stokes [20] in 1845. After repeated attempts to derive or justify these

expressions on the basis of statistical mechanics, Uhlenbeck [12] noted that

‘‘Quantitatively, some of the predictions from these equations surely deviate

from experiment, but the very remarkable fact remains that qualitatively the

Navier—Stokes equations always describe physical phenomena sensibly . . . .

The mathematical reason for this virtue of the Navier—Stokes equations is

completely mysterious to me.’’

The greatest advance ever in the analysis of turbulent flow was made in

1895 by Reynolds [17], who not only conceived of the practical advantages

of space averaging but also derived in detail the required mathematical

procedures for this averaging. He then applied this process to Eqs. (1)—(3).Time averaging, which is generally presumed to be equivalent to space

averaging, followed by specialization for steady fully developed flow, reduces


these equations to

��P�r

�1

r

d

dr(�ru�

�u��) �

�(u��u��)r

� 0 (4)

1

r

d

dr(ru�

�u�� ) �

u��u��r

� 0 (5)

and

��P�z

�1

r

d

dr ��rdu

�dr

� (�ru��u�� 0, (6)

where here, as contrasted with Eqs. (1)—(3), u�

denotes the time-averaged

velocity, and P the time-averaged dynamic pressure that arises from changes

in velocity only; u��, u�

�and u�� the instantaneous fluctuations in velocity

about the time-averaged values; and the superbars the time-averaged values

of products of these fluctuations. The validity of the equations obtained by

space or time averaging has been questioned, but no specific failures have

been demonstrated for conditions such that treatment of the fluid as a

continuum is a valid approximation. Barenblatt and Goldenfeld [21] have

questioned the concept of full development for turbulent shear flows (the

attainment of a velocity field and pressure gradient independent from z), but

such behavior is certainly attained for all practical purposes (see, for

example, Abbrecht and Churchill [22]), and such a postulate has resulted

in many apparently valid asymptotic predictions.

Equations (4)—(6), even if exact, are, in contrast with Eqs. (1)—(3), an

incomplete description of the fluid motion. The terms u��u��, u��u�� , u�

�u�� , and

u��u��, which are known quite appropriately as the Reynolds stresses, repre-

sent the information lost by time averaging since they are indeterminate

from these equations alone. Most of the modeling of turbulent flow has

involved the postulate of empirical expressions for these time-averaged

products of the fluctuating components of the velocity. On the other hand,

the structural gain from time averaging is quite evident, not only from the

relative simplicity of Eqs. (4)—(6) as compared to Eqs. (1)—(3), but also from

their susceptibility to analytical and formal integration with respect to r to

obtain

P� P�� u�

�u��

#

�

(u��u�� u��u��)

dr

r(7)

u��u�� 0 (8)

and

r

2 ��P�z��

du�

dr��u�

�u��. (9)


Equation (7) expresses the radial variation in pressure wholly in terms of

the fluctuations in u� and u�; Eq. (8) indicates that the Coriolis force is zero

at all radii. Here P�

is the pressure on the wall of the pipe at r� a, where

a is the radius of the pipe. Since u��u��and u��u�� are not functions of z in fully

developed flow, it follows that �P/�z is also independent of z and hence a

constant that may be expressed as dP/dz. From a force balance over a

central cylindrical segment of the fluid it may be shown that

��r

2 ��dP

dz�, (10)

where � is the total shear stress in the z-direction imposed on the outer fluid

at any radius by this inner segment. It follows that at r� a,

��

a

2 ��dP

dz�. (11)

Here ��

is the shear stress imposed on the wall in the z-direction. From the

ratio of Eqs. (10) and (11),

��

�r

a. (12)

Now for convenience and simplicity, letting u�� u, u�

�� v�, u�

�� u�, and

a� r � y allows Eq. (9) to be reexpressed as

�� 1�

y

a�� du

dy��u�v� . (13)

From Eq. (13), which is the starting point for all subsequent modeling herein

for flow in a round tube, it is evident that the contribution of the turbulent

fluctuations to the time-averaged velocity is wholly represented by u�v�.Similarly, it is evident from Eq. (7) that the radial variation of the dynamic

pressure is wholly a consequence of u��u��and u��u�� .

2. Dimensional, Asymptotic, and Speculative Analyses

The most useful technique for the development of functional relationships

for the characteristics of turbulent flow has proven to be a combination of

dimensional, asymptotic, and speculative analyses. Here speculation refers to

a tentative postulate whose consequences are ultimately to be tested with

experimental or exact theoretical results. It is unfortunate that the uncer-

tainty implied by this terminology has often discouraged its formal usage.

The techniques and results of dimensional, asymptotic, and speculative

analysis have evolved independently in many different contexts and there is


no general agreement on the priority of the various contributions. The

attributions herein are conceded to be somewhat arbitrary.

Fourier [23] in 1822 established the fundamental basis for dimensional

analysis by noting that all added and equated terms in a complete relation-

ship between the variables must have the same net dimensions. Rayleigh [24,

25] in 1892 illustrated the expression of functional relationships in terms of

dimensionless groups, and in 1915 proposed a general mechanistic process

for the determination of an appropriate minimal set of dimensionless

groupings to describe the behavior defined by a listing of dependent

variables, independent variables, physical properties, and any other relevant

parameters.

Asymptotic dimensional analysis, as used herein, refers to the reduction of

such a listing, and hence of the number of dimensionless groupings, for

limiting conditions or locations or times. Speculative dimensional analysis, as

defined by Churchill [26], refers to the tentative elimination of individual

variables or parameters and thereby reduction of the number of dimension-

less groupings without necessarily any justification or rationale in advance.

This latter procedure may be characterized by the question, ‘‘What if . . .?’’

Insofar as the chosen set of variables and parameters is sufficient and

self-consistent, the results obtained by ordinary and speculative dimensional

analyses are exact. The results from an asymptotic dimensional analysis may

additionally depend on arbitrary constraints. Since the original choice of a

set of variables, whether from a mathematical model or a heuristic listing, is

always a possible source of error, Churchill [27] has proposed that all

processes of dimensional analysis be considered to be speculative and

thereby tentative until confirmed by experimental data or exact theoretical

results.

Since neither analytical nor numerical solutions of the general time-

dependent equations of conservation for conditions resulting in turbulent

shear flow have been accomplished until very recently, these several pro-

cesses of dimensional analysis have proven to be invaluable in terms of

suggesting forms for the efficient correlation of experimental data. Such

forms may be expected to serve the same role for the currently emerging

exact but discrete numerical results.

The first application of dimensional analysis for turbulent flow was by

Reynolds [28, 17], who in 1883 determined the conditions for the onset of

turbulence in flow through a long pipe and then in 1895 surmised that the

transition was characterized by the dimensionless grouping Du��/�, now

known as the Reynolds number. Here D �2a is the diameter of the pipe and

u�

the space- and time-mean velocity. This result and its direct counterparts

for the friction factor and the time-mean velocity distribution are all that

may be inferred from the variables of Eq. (13) by simple dimensional


analysis. On the other hand, Prandtl and his associates and contempories

utilized asymptotic and speculative dimensional analysis with great insight

and success to choose forms of these types for the correlation of experimen-

tal data starting from either Eqs. (1)—(3) or (13). Some such analyses follow.

It may be speculated on purely physical grounds, inferred from Eqs.

(1)—(3), or inferred from Eq. (13) that the local time-mean velocity in steady,

fully developed turbulent flow in a round tube with a radius a may be

expressed in general as

u �#y, ��, a, �, �, (14)

where the notation #x designates an unknown function of any indepen-

dent variable x. [The inference of Eq. (14) from Eq. (13) implies that u�v� isa function of the same variables as u.] It follows from the application of

ordinary dimensional analysis to Eq. (14) that one possible set of dimen-

sionless groupings is

u ��

�# �y(�

��)� �

�,y

a�. (15)

Prandtl [29] introduced the notation u5�u(�/��)� � and y5�y(��

�)� �/�,

which is still used almost universally today, to reexpress Eq. (15) as

u5�# �y5,y

a��# �y5,y5

a5�. (16)

Equation (13) may be reexpressed in this notation as

1�y5

a5�

du5

dy5� (u�v�)5, (17)

where, as may be inferred, (u�v�)5��u�v�/��.

Prandtl next speculated that near the wall u might be essentially indepen-

dent of a, thereby reducing Eq. (16) to

u5�#y5, (18)

which is now known as the universal law of the wall. The limitation of Eq.

(18) to y5 a5 explains the terminology law of the wall, and the depend-

ence only on y5 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

u5� y5�(y5)�

2a5* y5. (19)


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

du5

d(y/a)��

y

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

u5�

� u5��1�� y

a��# �y

a�. (21)

The term u5�

� u5, which characterizes the behavior near the centerline in

the same general sense that u5 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 approxi-

mation. Accordingly, he reexpressed Eq. (18) in terms of the velocity defect,

that is, as

u5�

� u5�#a5�#y5, (22)

and noted that the only functional expression for the velocity defect

satisfying both Eqs. (21) and (22) is

u5�

� u5�B ln �a

y�, (23)

where B is an arbitrary dimensionless coefficient. The necessary counterpart

for the velocity itself is

u5�A �B lny5, (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 Karman [31] postulated that despite this

restriction, Eq. (23) might provide an adequate approximation for the entire

cross-section insofar as integration to determine u5�

is concerned. The result


of such an integration is

u5�

� u5�

�3B

2, (25)

which may be combined with Eq. (24), as specialized for y5� a5, to obtain

�2

f ��

� u5�

�A�3B

2�B lna5. (26)

Here f� 2��/�u�

�is the Fanning friction factor. Equation (26) may of course

also be derived directly by integrating u5 from Eq. (24) over the cross-

section. The derivation of Eq. (26) by von Karman 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 � (3B/2) and B.

For a pipe with a roughness e, the same type of analysis that led to Eq.

(16) results in

u5�# �y5,y

a,e

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:

u5�# �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:

u5�C �B ln �y

e�. (29)

Here, C is a dimensionless arbitrary constant and B is implied to have the

same value as in Eqs. (23) and (24). Integration of Eq. (29) over the

cross-section results in

�2

f ��

� u5�

�C �3B

2�B ln �

a

e�, (30)


but Eq. (25) remains applicable. It may be inferred that the effect of

roughness is simply to decrease u5 for a given value of y5 by the quantity

B lne5�A �C and to decrease u5�

by the same amount for a given value

of a5. Here e5� e(��)� �/� in conformity to the definition of y5.

Murphree [32] and several others used a variety of methods of asym-

ptotic 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:

� �u�v�� y� -�y�� . . . . (31)

Here, ��,-�, . . . are arbitrary dimensional coefficients. Equation (31) may be

reexpressed in terms of the previously defined dimensionless variables as

(u�v�)5 � �(y5) � -(y5)� . . . , (32)

where � and - are dimensionless coefficients. Substitution of (u�v�)5 from Eq.

(32) in Eq. (17), followed by integration from u5� 0 at y5�0, leads to a

corresponding expression for u5, namely,

u5� y5��4

(y5)� �-5

(y5)� �(y5)�

2a5� · · · . (33)

Equation (33) without the term in (y5)�/2a5, which is negligible for typical

values of a5, has also been derived directly by asymptotic expansion.

The recognition on physical grounds that the fraction of the shear stress

due to turbulence, namely ��u�v�/�, is necessarily finite, positive, and less

than unity at the centerline requires, by virtue of Eq. (17), that

u5�

� u5� E �1 �y5

a5��, (34)

and therefore that

(u�v�)5 *�1�2E

a5��1�y5

a5�, (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.


3. Empirical Models

An alternative and supplementary approach to dimensional, speculative,

and asymptotic analyses is the postulate of mechanistic empirical models for

the turbulent shear stress and thereby for the prediction of the local

time-mean velocity distribution and its space mean.

a. The Eddy Viscosity Boussinesq [33] in 1877, and thus before the

identification by Reynolds in 1895 of the relationship between the turbulent

shear stress and the fluctuating components of the velocity, proposed by

analogy to Newton’s law for the viscous shear stress the following expression

for the total shear stress in a shear flow:

�� (��)

du

dy. (36)

Here ��is the eddy viscosity, an empirical quantity that is a function of local

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:

��u�v��

du

dy. (37)

Equation (17), with the turbulent shear stress represented by Eq. (37), may

be rewritten as

1�y5

a5� �1 �

��

du5

dy5. (38)

b. The Mixing Length Another historically important model for the tur-

bulent 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

��u�v��l� �du

dy�du

dy(39)

where l is a mixing length for eddies corresponding to the mean free path of

molecules as defined by the kinetic theory of gases. Although this analogy,

as noted by Bird et al. [35, p. 160], has little physical justification, the

mixing-length model has generally been accorded more respect by analysts

than the eddy viscosity model, apparently because of its mechanistic

rationale, however questionable that may be.

Von Karman [31] speculated on dimensionless grounds that near the

wall, l might be proportional to the distance from the wall; that is, he


proposed the expression

l� ky, (40)

where k is a dimensionless factor that is now generally called the vonKarman constant. Prandtl [29] substituted l from Eq. (40) in Eq. (39) and

in turn the resulting expression for ��u�v� in Eq. (13) to obtain

�� 1�

y

a�� du

dy� k�y��

du

dy��, (41)

which may be reexpressed in the canonical dimensionless form as follows:

1�y5

a5�

du5

dy5� k�(y5)� �

du5

dy5��. (42)

Prandtl [34], starting from Eq. (42), neglected the variation in the total

shear stress with y5/a5, neglected the viscous shear stress, took the square

root of the resulting expression, and integrated indefinitely to derive

u5� A�1

klny5. (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


in du5/dy5 may be solved and then integrated from u5� 0 at y5� 0 to

obtain

u5�1 � [1� (2ky5)�]� �

2ky5�

1

kln2ky5� [1� ky5)�]� �. (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 y5* 0 to Eq. (43), with an

effective value of A* (1/k)(ln4k � 1) as y5*�. The correct limiting

behavior for y5* 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 attribu-

table 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 y5 for a representative value of k is more difficult to assign.

Conversely, accounting for the linear variation of the total shear stress

with y5 but neglecting the viscous shear stress permits derivation of the

following solution for the so-reduced form of Eq. (42) by a process similar

to that used to obtain Eqs. (43) and (44):

u5�1

k �2 �1�y5

a5��

� 2 �1�y5

a5��

� ln ��1� �1�

y5

a5��

��1��1 �y5

a5��

��1� �1�

y5

a5��

��1��1 �y5

a5��

��. (45)

Here u5� 0 at y5� y5

was invoked as an arbitrary boundary condition.

The choice of y5

� exp�Ak 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 y5/a5 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 du5/dy5, and integrating formally from u5� 0 at y5� 0 results


in the following integral expression:

u5� 2 ��5

�1�y5

a5� dy5

1 ��1 ��1�y5

a5� (2ky5)��

. (46)

Equation (46) coincides with Eq. (44) for y5 a5 and represents an

improvement on Eq. (45) for larger values of y5 at the expense of numerical

integration, but fails for y5* a5 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 y5

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.,

l� l#, (47)

where l#

is the limiting value for y5� a5. He then substituted l#

for l in

Eq. (39) and the resulting expression for ��u�v� in Eq. (13), neglected the

viscous term, and integrated from u� u�at y� a to obtain, in dimensionless

form,

u5�

� u5�2

3 �a5

l5#��1 �

y5

a5� �

. (48)

Equation (48) correctly predicts du/dy� 0 at y5� a5, but has a different

power dependence on 1 � (y5/a5) than does Eq. (34).In order to improve upon Eqs. (40) and (45) and thereby on Eqs. (43) and

(48), von Karman [31] postulated that

l� k* �du/dy

d�u/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 lfrom 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

�5�

��5� �1

k* ��1�y5

a5��

� ln �1 ��1 �y5

a5��

��. (50)


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 y5* a5, Eq. (50) may be approxim-

ated by

u5�

� u5�1

2k* �1�y5

a5�, (51)

which not only has a different functional dependence on 1� (y5/a5) than

Eq. (34) but in addition fails to predict du/dy� 0 at y5� a5.

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

l� ky(1 � exp�*y5) (52)

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 ��u�v� in Eq. (13), neglecting the variation in the total shear stress,

solving the resulting quadratic equation for du5/dy5, and integrating

formally from u5� 0 at y5� 0 results in

u5� 2 ��5

dy5

1� (1 � [2ky5(1 � exp�*y5]�)� �. (53)

For y5* 0, Eq. (53) reduces to Eq. (29), but unfortunately with �� 0 and

-� k�*�/5. For large values of y5 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

u5� 2 ��5

�1�y5

a5� dy5

1 ��1 ��1�y5

a5� (2ky5)�(1 � exp�*y5)�]� �. (54)

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� (y5/a5) have a significant role.

c. Other Models Kolmogorov [39] and Prandtl [40] independently con-

jectured on dimensional grounds (local similarity) that

�� c

�,� �l*�, (55)


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)

Launder and Spalding [42] proposed calculating , and � numerically from

differential transport equations formulated as moments of that for momen-

tum, such as Eq. (9), and then in turn calculating ��

from Eq. (57).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 ,—�—u�v� or Reynolds-stress

model, which adds a transport equation for u�v� 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 eddysimulation (LES) method starts from the time-dependent equations of

conservation but introduces arbitrary terms such as those of the ,—� and

Reynolds-stress models as well as utilizing the ,—� model or additional

empirical terms for the region near the wall. For an illustration of the

applicability of this model, again for an annulus, see Satake and Kawamura

[44].

4. T he Experimental Data of Nikuradse

Nikuradse [45—47] in 1930, 1932, and 1933 obtained extensive and

precise sets of experimental data for the time-mean velocity distribution in

the turbulent core and for the axial pressure gradient for the fully developed

flow of water in smooth round tubes for 600&Re& 3.24 10� and in round

tubes with a uniform artificial roughness e corresponding to 15�(a/a)�507

for 600&Re& 10�. Furthermore, he presented his data in tabular as well

as graphical forms, thereby making it readily available in full numerical

detail to subsequent investigators. For more than 60 years these data have


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 y5 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 y5 (the dimensionless distance of each point of measurement from the

wall) in order to force the measured values of u5 in his smooth pipes to

approach the limiting form of Eq. (19) as y5* 0. This discovery by Miller

may readily be confirmed by comparing the values of u5y5 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 y5 but

precludes the use of the tabulated values for the small values. Robertson etal. [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

u5�

—u5�

to 3B/2� 3.75, and Lynn [50] in 1959 discreetly noted ‘‘the

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 thewall, 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 y5� 50, again

even for the region near the centerline. He also found Eq. (29) with C � 8.5and B� 2.5 to be successful for representation of the measured velocity

distribution for the artificially roughened pipes at the larger values of a5.

However, his experimental values for the friction factor in the smooth pipes

were found to be represented better by

�2

f ��

� 2.00� 2.46 ln a5 (58)

than by Eq. (26) with A� 5.5 and B � 2.5, which yields A � (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


Fig. 2. Experimental velocity distribution in fully developed turbulent flow of water in a

127-mm Plexiglas tube (R� Re). (Reprinted with permission from Lindgren and Chao [51],

Figure 1. Copyright 1969 American Institute of Physics.)

the neglect of the boundary layer near the wall, as represented in the limit

by Eq. (33), and of the wake, as represented in the limit by Eq. (34). Both

of these deviations from Eq. (24) are well illustrated by the much later data

of Lindgren and Chao [51] in Fig. 2. On the other hand, the discrepancy in

the coefficient (2.46 as compared to 2.50) is unacceptable on theoretical

grounds. The overly simplified and incongruent expressions for the velocity

distribution and the friction factor that appear unexplained in most of our

current textbooks and handbooks are a legacy of the failure of Nikuradse

to obtain sufficiently precise and accurate values for the velocity distribution

near the wall and near the centerline and to develop correlating equations

encompassing these regions. Nikuradse is, of course, not responsible for the

failure of subsequent investigators and writers to explain and provide a

rational correction for these anomalies.

A similar discrepancy exists for the correlating equations of Nikuradse

[47] for artificially roughened pipe, for which he correlated his experimental


data for the axial pressure gradient for asymptotically large values of the

Reynolds number with the expression

�2

f ��

� 4.92 � 2.46 ln�a

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 u5�

� u5conform 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 ‘‘adjust-

ments’’ 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

a� 0.14� 0.08 �1�

y

a�� 0.06 �1 �

y

a��, (60)

which he attributed to Prandtl and interpreted as an ‘‘interpolation for-

mula’’ between Eq. (40) with k � 0.4 for y5* 0 and Eq. (47) with l#� 0.14

for y5* a5. 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 u5, ��, l, and f based on these data have

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 investi-

gators 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-


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 ex-

plored herein since none of the numerical values determined from the data

of Nikuradse appear in the final expressions for either flow or heat transfer.

5. Power-L aw Models

Power-law models for the velocity distribution and the friction factor

might not have merited attention herein had not a recent attempt been made


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-


tation could be achieved with a straight line equivalent to

f �0.0791

Re� �. (61)

C. Freeman [54] in 1941, in a Foreword to a compilation of the extensive

set of experimental data obtained in 1892 by his father, J. R. Freeman, but

not published until 49 years later, speculated that had Blasius had access to

these values, which extend up to Re� 9 10�, he might have developed a

more general correlating equation and thereby changed the course of history

in applied fluid mechanics. This assertion not only was justified when it was

written, but has proven prophetic.

Prandtl [55] in 1921, and therefore prior to his development of the

mixing-length model, recognized from dimensional considerations that Eq.

(61) implies

u

u�

��y

a��

, (62)

which by virtue of the numerical coefficient of 0.0791 may also be expressed

as

u5� 8.562(y5)� �. (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

u

u�

��y

a��

, (64)

which corresponds to

u5�-(y5)� �. (65)

Here � is an arbitrary dimensionless exponent and - is an arbitrary

dimensionless coefficient. From integration of Eq. (65) over the cross-section

it follows that

-�(1 � �)(1 � 2�)u5

�2�(a5)� �

. (66)

He determined numerical values of � and - as functions of Re and e/a from

his experimental velocity distributions, but abandoned this mode of corre-

lation as inferior to Eqs. (24), (26), (29), and (30).


Nunner [56] in 1956 somewhat revived the power-law model by discover-

ing that the empirical relationship

�� 2 f � � (67)

provides a good approximation for both smooth and roughened pipes.

However, a separate correlation is required for the friction factor as a

function of the Reynolds number and roughness ratio.

Thirty-seven years later, Barenblatt [54], apparently unaware of the work

of Nunner, rationalized the form of the power law for the velocity distribu-

tion 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 u5�

from

Eq. (58) or (59), whichever one is appropriate, is slightly superior to Eq. (24)for 30& y5& 0.1a5. However, it is seriously in error for larger as well as

smaller values of y5. 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� ��

�� 5��, (68)

which implies that a fixed-power dependence of the velocity on the distance

from the wall over the entire cross-section is required to obtain a power-law

dependence of the friction factor on the Reynolds number. It may therefore

be inferred from the previously cited failures of the power-law model for the

velocity near the wall and near the centerline, and more importantly from

the observed dependence of � on Re, that a pure power-law model for the

friction factor cannot have any real range of validity with respect to Re. The

semilogarithmic dependence of the square root of the reciprocal of the

friction factor on the Reynolds number has already been noted to be subject

to a related but numerically less severe defect.

Churchill [58] has recently compared the predictions of the power-law

models for the velocity distribution and the friction factor with experimental

data and other correlating equations. These comparisons support the

preceding qualitative conclusions.

The failure of the power-law models might have been anticipated on the

basis of dimensional analysis. Rayleigh [25] used a power-series expansion


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. Unfor-

tunately, such misinterpretations plague us to this day. Indeed, relationships

in the form of powers other than the unity ordinarily occur only in

asymptotic expressions, such as the limiting form of Eq. (32), or in special

cases, such as with the friction factor (but not the velocity distribution) for

fully developed laminar flow in a round tube.

6. T he Analogy of MacL eod

Before examining some important recent work it is convenient if not

essential to describe a little-known conjecture that suggests a means of

obtaining congruence of the turbulent shear stress and the velocity distribu-

tion for round tubes with their counterparts for flow between parallel plates

of infinite extent. Rothfus and Monrad [59] showed that complete congru-

ence of the velocity profile in fully developed laminar flow in a round tube

with that between parallel plates may be achieved by specifying ��8

� ��.

and a � b, where the subscripts R and P designate round tubes and parallel

plates, respectively, and b is the half-spacing of the plates. This requirement

is excessive; a sufficient condition for u58y5, a5� u5

.y5, b5 is simply

that a5� b5. MacLeod [60] subsequently speculated that this latter

relationship might also hold for fully developed turbulent flow. His specu-

lation is beautifully confirmed in Fig. 5, in which the experimental values of

Whan and Rothfus [61] for u5�

in flow between parallel plates are compared

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 a5 and b5. It may be inferred from Eqs. (17),(38), and (39) that the analogy of MacLeod applies directly to (u�v�)5, �

�/�,

and l5 insofar as it is valid for u5.

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 u5 and (u�v�)5 in flow between parallel


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 (u�v�)5 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 numeri-

cal simulations.

The results from direct numerical simulations for round tubes are current-

ly less extensive and reliable than those for parallel plates because of the

computational complexities associated with curvature. On the other hand,

the experimental measurements for flow in round tubes are more extensive

and reliable than those for parallel plates because of the difficulty of aligning

and supporting plates of sufficient extent and small enough spacing to

minimize side-wall effects. Entrance effects also appear to be more serious.

The law of the wall of Prandtl [Eq. (18)] may be noted to be a special

case of the analogy of MacLeod for the region near the wall, but, on the

other hand, it is presumed to be applicable for all shear flows, not just those

for round tubes and parallel plates.


7. T he Colebrook Equation for the Friction Factorin 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 asso-

ciated 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

that would result in the same asymptotic value of f�for the friction factor

for very large Reynolds numbers as the value of the uniform artificial

roughness of Nikuradse. Thus, on the basis of Eq. (50),

e�a� exp �

4.92� (2/ f�)� �

2.46 �. (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

f ��

��2.46 ln�e�

7.39a�

1

2.25a5�. (70)


He justified the form of Eq. (70) simply by asserting without any rational-

ization 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,

yx�� yx�� y

�x�, (71)

where yx 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 yx in Eq. (71)than (2/ f )� � or f . Then, from Eq. (58) rearranged as

exp �(2/ f )� �

2.46 �� 2.255a5, (72)

yx� 2.255a5, and from Eq. (59) rearranged as

exp �(2/ f )� �

2.46 �� 7.389a

e�

, (73 )

y�x� 7.389a/e

�. A value of n��1 was chosen on the basis of the

experimental data of Colebrook. The result of this procedure may be

expressed as

�2

f ��

� 2.00� 2.46 ln �a5

1� 0.304(e�/a)a5�, (74)

which is exactly equivalent to Eq. (70). Most of the graphical representa-

tions 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 � a5(8/ f )� � or Re f � �� 8� �a5 with e�/a or e

�as a

parameter, accompanied by a table of values of e�for various materials and

conditions. Most of the values of e�that appear in the standard tabulations

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.

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

f ��

� 2.46 ln�2a

e �as a function of

(e5)� �e�(�

�/�)

��

e

a��(a5)�


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 y5 in the argument of the logarithm by

1� 0.304(e�/a)a5.

8. Experimental and Computed Values of u�v� Near the Wall

Difficulty has been encountered in the past in determining � in Eq. (32)or (33), since measurements of very small values of u�v� 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


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 b5 just above

the minimal value of about 145 for fully developed turbulence. The com-

puted values of u�v� 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 b5. This

combination of computational and experimental results for parallel plates

confirms beyond question the form of Eq. (32) and indicates a value

�7 10�� for �.

9. Experimental Values of u�v� 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 u5, from Eq. (35) by virtue of experimental or

computed values of (u�v�)5, or from the derivative of Eq. (34) by virtue of

experimental values of du5/dy5. Unfortunately, all three of these methods

require experimental values of greater accuracy and precision than are

currently available. Even the values of (u�v�)55 computed by direct numer-

ical simulation are marginal in this respect.

It follows from Eqs. (38) and (34) that for y5* a5

��

�a5�

1

a5 �dy5

du5 �1�y5

a5�� 1�*1

2E�

1

a5�

1

2E. (75)

Equation (75) predicts that near the centerline, ��/�a5 approaches an

asymptotic value independent of y5 and essentially independent of a5. Such

behavior has been confirmed (see, for example, Figure 5 of Churchill and

Chan [71]). Groenhof [72] examined and compared experimental determi-

nations of ��/�a5 by five sets of investigators for round tubes and one

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 empiri-

cal constants, are based on the experimental data of Nikuradse [46, 47]

despite their age and indicated limitations. Recently a new, comprehensive


set of experimental data has been obtained for fully developed turbulent

flow in a round tube that challenges the dominant role of the measurements

of Nikuradse. Zagarola [73] in 1996, using modern instrumentation and

carefully controlled conditions, measured the time-averaged velocity and

axial pressure gradient in air flowing through a 129-mm tube with a highly

polished surface. His flows extended from Re� 3.55 10� to 3.526 10� and

thus to higher values than those of Nikuradse, but not to as low ones.

Zagarola conceded that his own measurements of the time-mean velocity

were excessively high for y/a& 0.0155 for all Re. This requires discarding all

of his values in the viscous sublayer (0&y5&10) and all but a few in the

buffer layer (10&y5&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 some-

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 replace-

ment 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 y5 and Refor which this equation is applicable. The plots of u5

�� u5 versus lna/y,

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:

u5�

� u5�1

0.436ln �

a

y�� 1.51. (76)


Fig. 7. Variation with Re of the coefficient k � 1/B in Eqs. (24) and (43) for

50� y5� 0.1a5. (From Zagarola [73], Figure 4.38.)

Fig. 8. Variation of the coefficient k� 1/B in Eq. (23) for u5�

and Eq. (26) for u5�

for various

sets of increasing values of Re. (From Zagarola [73], Figure 4.30.)


Fig. 9. Variation with Re of the constant A of Eqs. (24) and (43) for 50� y5� 0.1a5 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.)


Fig. 11. Comparison of semilogarithmic and power-law representations of the velocity

distribution for 31 10�Re� 4.4 10�. (From Zagarola [73], Figure 4.44.)

Although Zagarola did not present a correlating equation for the region of

the wake, Eq. (24) with A � 6.13 and B� 1/0.436 may be combined with

Eq. (76) to obtain the following expression for the velocity at the centerline

itself:

u5�

� 7.64�1

0.436lna5. (77)

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

u5� 8.7(y5)��, (78)

where 8.7 and 0.137 are purely empirical, represents the measured velocities

better for 30& y5& 500 but much more poorly for y5� 500 than

u5� 6.13�1

0.436lny5. (79).

Both representations are seen to fail for y5& 50. Similar representations

and misrepresentations were found to be provided by these two expressions

for other ranges of Re.


Zagarola represented his experimental data for the friction factor with the

following expression:

�2

f ��

� 3.30�138.5

(a5)��

1

0.436lna5. (80)

He proposed the term 138.5/(a5)�� 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 y5& 0.1a5. Zagarola actually proposed several

different leading coefficients for Eq. (80). The value of 3.30 was chosen here

for consistency with Eqs. (76) and (79).

11. Overall Correlating Equations for the Velocity Distribution

With only a few exceptions, to be noted here, the correlating equations of

the past, as well as those of Zagarola, are for a single regime, primarily ‘‘the

turbulent core near the wall,’’ although such expressions have often been

implied to be applicable for the entire turbulent core. Equations (44)—(46),as well as Eqs. (53) and (54), purport to encompass the boundary layer, but

the first three do not include the higher-order terms of Eq. (33) and the latter

two imply erroneously that �� 0. Despite the presence of a5 in several of

these expressions, none of them approaches Eq. (34) as y5* a5.

Churchill and Choi [75] combined the limiting form of Eq. (19) for

y5* 0 and Eq. (24) with A � 5.5 and B� 2.5 in the form of Eq. (71) and

chose a value of �2 for n on the basis of the experimental data of Abbrecht

and Churchill [22] to obtain

u5�y5

�1 ��y5

2.5 ln9.025y5��

��

. (81)

Here 5.5� 2.5 lny5 is expressed as 2.5 ln9.025y5 for compactness and

to emphasize the presence of a singularity at y5� 1/9.025 � 0.1108. This

singularity may be avoided without a significant effect on the predictions of

u5 for any value of y5 by simply adding unity to the argument of the

logarithm to obtain

u5�y5

�1��y5

2.5 ln1� 9.025y5��

��

. (82)


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 y5. The

deviations for y5& 2 are probably due to experimental error but those for

the largest y5 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

y5� u5� 0.1108 �e0.4u5 � 1� (0.4u5) �(0.4u5)�

2�

(0.4u5)

3 �. (83)

Equation (83) approaches Eq. (24) with A� 5.5 and B � 2.5 for large values

of y5, just as does Eq. (82), but approaches Eq. (33) as y5* 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 y5, but

numerical values of u5 may then readily be obtained by iteration. As

indicated by the absence of a5, Eq. (83) also fails to account for the wake.


Even earlier, Reichardt [76] proposed the following expression for the

entire cross-section, including the region of the wake:

u5� 7.8 �1 � e�y5/11 ��y5

11�� e�0.33y5

� 2.5 ln�3a5(1 � 0.4y5)(2a5 � y5)

2(3a5 � 4a5y5� 2(y5)�) �. (84 )

Equation (84) conforms to Eq. (34) with E � 7.5 for y5* a5 and to Eq.

(24) with A � 5.5 and B� 2.5 for intermediate values of y5. Reichardt

apparently intended it to conform to the first two terms on the right-hand

side of Eq. (33) for y5* 0, but it fails in that respect since the coefficients

of (y5)� and (y5) that result from the expansion of the logarithmic and

exponential terms in series are not quite zero. Furthermore, the correspond-

ing coefficient of the term in (y5)� has the very excessive value of

1.548 10�� as compared to 7 10��/4� 1.75 10�� from the direct nu-

merical simulations. (Values of 18.738 and 0.5071 in place of 11 and 0.33,

respectively, would eliminate the terms in (y5)� and (y5), but would

decrease the coefficient of (y5)� 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 y5& 0.1a5while it is more accurate functionally as well as numerically for

0.1a5& y5� a5.

B. New Improved Formulations and Correlating Equations

The models just described all appear to have a defect or shortcoming and

the correlating equations to be limited in scope or generality. The objective

of the work described in this section has been to develop formulations and

correlating equations that avoid these defects and limitations.

1. Model-Free Formulations

The mixing-length model, as expressed by Eq. (39), was proposed by

Prandtl to facilitate the prediction of the turbulent shear stress in the

time-averaged equation for the conservation of momentum on the premise

that algebraic or differential correlating equations for the mixing length,

such as Eqs. (40), (47), and (49), would be simpler or more general than a

correlating equation for the shear stress itself. The eddy diffusivity model of

Boussinesq, as represented by Eq. (37), may be interpreted to have an

equivalent objective.


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 conse-

quences are generally favorable and in some respects quite surprising. They

started from Eq. (17), which is expressed in terms of (u�v�)5��u�v�/��,

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 u�v� 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

(u�v�)55��u�v�/�, which may be recognized as the local fraction of the

total shear stress due to turbulence. Equation (13) then becomes

�1�y5

a5� [1 � (u�v�)55]�du5/dy5

. (85)

From physical considerations (u�v�)55 must be positive, less than unity, and

greater than zero at all locations within the fluid. This latter characteristic

gives (u�v�)55 a significant advantage in terms of correlation over (u�v�)5,

which is zero at the centerline.

Eliminating du5/dy5 between Eqs. (38) and (85) reveals that

��

�(u�v�)55

1 � (u�v�)55(86)

The eddy viscosity is thus seen to be related algebraically to (u�v�)55, which

is a physically well-defined and unambiguous quantity, and thereby to be

independent of its heuristic diffusional origin. (Boussinesq was either very

intuitive or just lucky.) It further follows from Eq. (85) and the indicated

behavior of (u�v�)55 that ��/� is also finite and positive at all locations

within the fluid, including the centerline.

It similarly follows from Eqs. (39) and (85) that

(l5)� �(u�v�)55

�1�y5

a5� [1� (u�v�)55]�

(87)

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


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 (u�v�)55 as a variable, it

could have been identified much earlier merely by the substitution of

du5/dy5 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 (u�v�)55, which, as noted, is well behaved and

constrained between zero and unity for all values of y5. Formal integration

results in the following expressions for the time-averaged velocity distribution:

u5� ��5

�1 �

y5

a5� [1 � (u�v�)55]dy5

� y5�(y5)�

2a5��

�5

�1�

y5

a5� (u�v�)55dy5 (88)

These may be expressed more compactly as

u5�a5

2 ��

8�

[1 � (u�v�)55]dR��a5

2(1� R�) ��

�

8�

(u�v�)55dR� (89)

The leftmost forms of Eqs. (88) and (89) are more convenient for numerical

integration because the rightmost ones involve small differences of large

numbers, but the latter have the advantage of demonstrating that the effect

of the turbulence is simply to provide a deduction from the well-known

expressions for purely laminar flow at the same value at a5.

Equation (89) may in turn be integrated formally over the cross-sectional

area to obtain the following expression for the space- or mixed-mean

velocity and thereby the friction factor:

�2

f ��

� u5�

�a5

2 ��

��

�

8�

[1 � (u�v�)55]dR�� dR�. (90 )

Equation (90) may be reduced by integration by parts to obtain

�2

f ��

� u5�

�a5

4 ��

[1� (u�v�)55]dR��a5

4�

a5

4 ��

(u�v�)55dR�,

(91)


which involves only a single integral. The rightmost form of Eq. (91) reveals

that the effect of the turbulence on the mixed-mean velocity is also simply

a deduction from the well-known expression (Poiseuille’s law) for purely

laminar flow. This deductibility of an integral term from the expressions for

the time-mean velocity distribution and the mixed-mean velocity in purely

laminar flow may seem to be obvious in retrospect, but such a structure is

not so evident in the analogs of Eqs. (89) and (91) in terms of the eddy

viscosity and the mixing length, and does not appear ever to have been

mentioned in the literature. Also, although it is evident in retrospect that the

double integral of the analogs of Eq. (90) in terms of the eddy viscosity and

the mixing length may be reduced to a single integral by means of

integration by parts, that simplification was apparently never recognized or

implemented because the more complex forms obscure this possibility. It

may be noted that (u�v�)55 has no advantage over (u�v�)5 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

(u�v�)55. Before turning to such expressions several partial precedents for

Eqs. (89) and (91) should be acknowledged. Kampe de Feriet [81] derived

the equivalent of Eq. (89) and the analog of Eq. (91) for parallel plates in

terms of (u�v�)5 but did not implement these expressions; while Bird et al.[35, p. 175], note that the use of a correlating equation for (u�v�)5 rather

than one for the eddy viscosity or the mixing length might lead to a simpler

integration for the velocity distribution.

2. Correlating Equations for the L ocal Turbulent Shear Stress

Despite the advantages of the dimensionless turbulent shear stress in

predicting the velocity distribution and the mixed-mean velocity, as de-

scribed 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 Feriet 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

(u�v�)5� 0.9835R(1 � R). (92 )

Equation (92) is reasonably accurate numerically and functionally for

y5* a5(R * 0), but fails badly for both small and intermediate values of


y5. 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 (u�v�)5 that may be reexpressed for simplicity in terms of

(u�v�)55 as follows:

(u�v�)55��0.7 �y5

10�

��

� �exp ��2.5

y5��2.5

a5 �1�4y5

a5 ��

��

.

(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��8/7,

(u�v�)55

� 0.7 �y5

10�

(94 )

and

(u�v�)55�

� 1�2.5

y5�

2.5

a5 �1 �4y5

a5 �. (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:

u5� 5.5� 2.5 lny5�15

4 �y5

a5��

10

3 �y5

a5�. (96 )

The terms in (y5/a5)� and (y5/a5) were added to the correlating equation

of Nikuradse [46] to encompass the wake. The coefficients 15/4 and �(10/3)were chosen to force du5/dy5* 0 and u5

�� u5* 7.5(1 � y5/a5)� as

y5* a5. 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

du5/dy5 in Eq. (85) from the derivative of Eq. (96) and then simplifying

algebraically results in Eq. (95). The absolute value sign and the approxim-

ation of 1� (2.5/y5) by exp�2.5/y5 in Eq. (93) are merely mathematical

contrivances to avoid singularities in ranges of y5 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


Fig. 13. Representation of experimental data and directly simulated values of (u�v�)55 at

small values of b5. (From Churchill [80], Figure 1.)

extensive and reliable than those for the turbulent shear stress itself.

However, the value of �8/7 for the arbitrary combining exponent is based

on the experimental data of Wei and Willmarth [84] for u�v� in flow between

parallel plates, which appear to be the most accurate ones over a wide range

of values of both y5 and b5. 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 y5 and b5 in Fig. 13 and

with the experimental data of Wei and Willmarth for moderate and large

values of y5 and b5 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 y5� 30 and y5� 0.1a5,

respectively, for Eq. (24) with A � 5.5 and B� 2.5, coincide at a5� 300, a

semilogarithmic regime presumably does not exist for any lesser value of a5.

Equation (93), which implies the existence of a semilogarithmic regime for


Fig. 14. Representation of experimental data of Wei and Willmarth for (u�v�)55 by Eq. (93).(From Churchill and Chan [71], Figure 3.)

the velocity for all values of a5, is therefore of questionable functionality for

intermediate values of y5 for a5& 300 despite the reasonable representa-

tion in Figures 13 and 14.

The recent experimental data of Zagarola [73] for the time-averaged

velocity distribution as discussed in Section II, A, 10 suggest updating Eq.

(96) as

u5� 6.13�1

0.436lny5� 6.824 �

y

a�� 5.314 �

y

a�. (97)

Equation (97) may be seen to be in agreement with Eq. (79) for y5 a5


and with Eq. (77) for y5� a5. For y5* a5, Eq. (97) leads to Eq. (34) with

E� 10.264. Substituting for du5/dy5 in Eq. (85) from the derivative of Eq.

(97), and then simplifying, results in the following updated version of Eq.

(95):

(u�v�)55�

� 1�1

0.436y5�

1

0.436a5 �1�6.95y5

a5 �. (98 )

Combination of Eq. (98) with Eq. (94), again with a combining exponent of

�8/7 and the same mathematical contrivances, results in the final correlat-

ing equation for (u�v�)55, namely

(u�v�)55��0.7 �y5

10��

��

� �exp��1

0.436y5��1

0.436a5 �1 �6.95y5

a5 ��

��

. (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 �8/7 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 y5 and all values of a5� 300 (and perhaps even for a5� 145), and its

incorporation of all of the known theoretical structure, namely, Eq. (32) for

y5* 0, Eq. (35) for y5* a5, and 1�B/y5 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 Distributionand 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


(91), respectively, using values of (u�v�)55 from Eq. (99), generalized correlat-

ing equations for these two quantities may be constructed for convenience.

Churchill and Chan [71, 77] developed such correlating equations using

Eq. (93) for (u�v�)5. 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 u5and u5

�are

u5� ��(y5)�

1� y5� exp�1.75(y5/10)��

��1

0.436ln �

1� 14.48y5

1� 0.301(e/a)a5�� 6.824 �y5

a5�� 5.314 �y5

a5�

��

��

(100)and

u5�

� 3.30�227

a5��

50

a5��

1

0.436ln�

a5

1 � 0.301(e/a)a5�. (101)

Equation (100) has the form of Eq. (71) with u5

adapted from the limiting

form of Eq. (33) with �� 7 10�� and -� 0, and u5�

adapted from Eq. (97).The modified form of u5

and the added value of unity in the argument of

the logarithm of u5�

are simply mathematical contrivances to avoid singul-

arities in ranges of y5 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) ln14.48� 6.13. The combining exponent of �3 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)a5 was included in the

argument of the logarithmic term to extend the applicability of Eq. (100) to

commercial (naturally rough) piping, at least for y5� e5. 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 (a5)�� and (a5)�� are a necessary consequence of u5* y5 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 a5for which these terms are the most significant. The term in (a5)�� 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)a5 was incorporated in the argument of the logarith-

mic term to extend the applicability of Eq. (101) to encompass commercial

(naturally rough) piping.


Numerical integration of Eq. (90) using (u�v�)55 from Eq. (99) predicts

values of the ‘‘constant’’ in Eq. (97) that vary from 5.39 to 5.76 with a5, 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 u5�

by Eq. (100) with the

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 y5 were even less on the whole. The prediction

of u5�

by Eq. (101) was found to have an average absolute deviation of only

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� (y5/a5) is applicable for all

one-dimensional fully developed turbulent flows. However, the variation of

�/��

with distance from a wall is known a priori only for forced flow in a

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 b5 is simply substituted for a5, while the friction factor may be

represented by

�2

f ��

� u5�

� 4.615�155

b5�

1

0.436ln �

b5

1 � 0.301(e/b)b5�. (102)

The value of 4.615 for the constant term as well as that of �155 for the

coefficient are based on computations by Danov et al. [85] using Eq. (99),since experimental values of u5

�for parallel plates of accuracy and modern-

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 u5 in planar Couette flow, for which ��

at all locations

within the fluid. A corresponding correlating equation for (u�v�)55 might be

postulated, but experimental data to test such an expression critically over

a wide range of conditions do not appear to be available.


Churchill [86] also constructed a generalized empirical expression from

which �/��

but not u5 or (u�v�)55 may be estimated for forced flow through

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

u�v� 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 a5 and

b5� 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 signifi-

cantly with the primary dependent variable, the temperature of the fluid,

forcing the use of small overall temperature differences, whereas the viscosity

does not vary with the velocity for ordinary fluids; (2) the mixed-mean

temperature must be determined by integrating the product of the tempera-

ture and the velocity over the cross-section at a series of axial distances,

whereas the mixed-mean velocity is invariant with axial length and may be

determined externally and directly with a flowmeter; (3) the heat flux

density, which is difficult to measure accurately, and/or the temperature


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�

�T

T�

�T�

orT�� T

T�

�T�

and/or h�j�

T��T

�

while T r, T�, T

�, and j

�or T

�are still varying individually; and finally (6)

the experiments must be repeated for a series of different fluids encompass-

ing a wide range of values of the Prandtl number. Corresponding complex-

ities 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 express-

ions 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 con-

sidered 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


be expressed in cylindrical coordinates as follows:

�c� �

�T�t

� u�

�T�r

�u�r

�T�1!

� u�

�T�z�� k �

1

r

��r �r

�T�r ��

1

r�

��T�1! �

��T�z��

�2� ��u

��r �

��

1

r ��u��1!

� u��

��

�u�

�z ��

�� u��z

�1

r

��u�

�1! ��

��u

��r

��u

��z�

��

1

r

�u�

�1!� r

��r �

v�r �

�

�. (103)

The only new variables as compared to Eqs. (1)�(3) are the temperature

T, the thermal conductivity k, and the specific heat capacity c�. V iscous

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

�c�u�T�z

�1

r

��r �kr

�T�r

� �c�rT �v��, (104)

where for consistancy with Eq. (13), the substitutions u� u�

and v��u�

have been made. Equation (104) may be integrated formally to obtain

�c�

r ��

u ��T�z� rdr�� k

�T�y

� � c�T �v� . (105)

The terms �k(dT /dy) and �c�T �v� represent the heat flux densities in the

y-direction (negative-r-direction) due to thermal conduction and the turbu-

lent fluctuations, respectively. The integral term on the left-hand side of Eq.

(105) represents the axial heat flux in the central core of fluid with a radius

r. It may also be interpreted, by virtue of Eq. (105) itself, as the total heatflux density j at r in the negative-r-direction. Equation (105) may also be

expressed in the dimensionless form

j

j�

[1� (T �v�)55] �dT 5

dy5, (106)

where

T 5� k(T��T )(�

��)� �/�j

�

(T �v�)55� �c�T �v�/j


and

j

j�

�1

R �8�

(�T /�x)

(�T�/�x) �

u5

u5�� dR�. (107)

This definition of T 5 was chosen in order to achieve the same form for Eq.

(106) as for Eq. (85) and to result in T 5� 0 at y5� 0 in analogy to u5� 0

at y5� 0. The term (T �v�)55 is also analogous to (u�v�)55 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

j�

��c

�a �

#

u ��T�z� rdr�

�c�a

2 ��

u ��T�z� dR��

�c�au

�2 �

�T�

�z �. (108)

Equation (108) may also be considered, as indicated, to define the velocity-

weighted (mixed) mean of the longitudinal temperature gradient.

The only explicit difference between Eqs. (85) and (106) is that j/j�

is given

by Eq. (107) in the latter whereas �/��

is simply equal to 1� (y5/a5) �Rin 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 5 remains zero at the wall, T�

itself may vary. The

thermal boundary condition thus becomes a parameter. An implicit differ-

ence of even greater significance is the dependence of (T �v�)55 on a

parameter Pr � c��/k, called the Prandtl number, as well as on y5 and a5.

It follows that T 5 depends on Pr, and in general so does j/j�. This

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�)55 are too limited in scope and accuracy to support the construction

of a generalized correlating equation in terms of y5, a5, and Pr comparable

to Eq. (98) for (u�v�)55. This deficiency may be alleviated somewhat by

reexpressing Eq. (106) as

j

j�

[1� (u�v�)55]Pr

�Pr

�dT 5

dy5(109)

where

Pr�

Pr�

1� (T �v�)55

1 � (u�v�)55(110)


Since (u�v�)55 is implied to be known in advance, the net effect of this

substitution is to replace (T �v�)55 by Pr�/Pr as an unknown. The quantity

Pr�/Pr, as defined by Eq. (110), may be recognized in physical terms as the

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�)55 as well from the lesser ones associated with (u�v�)55, 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

j�

��1�Pr

Pr��

(u�v�)55

1� (u�v�)55��dT 5

dy5. (111)

Here

Pr�

Pr�

(u�v�)55

(T �v�)55 �1� (T �v�)55

1 � (u�v�)55�. (112)

The quantity Pr�/Pr, as defined by Eq. (112), may be recognized in physical

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

essentially constant for large Pr, which results in a significant simplification.

Elimination of (T �v�)55 between Eqs. (110) and (112) or of j/j�(dy5/dT 5)

between Eqs. (109) and (111) results in

1

Pr�

�(u�v�)55

Pr�

�1� (u�v�)55

Pr. (113)

This relationship between Pr�

and Pr�will subsequently prove very useful.

Since j/j�

differs only moderately from �/�� R, it is convenient to

introduce the variable *, defined by

1 � *�j

j��

j/j�

R�

j/j�

1 � (y5/a5). (114)

Substituting for j/j�

in Eq. (109) from Eq. (114) results in

(1 � *) �1�y5

a5� [1 � (u�v�)55 ]Pr

�Pr

�dT 5

dy5. (115)

Comparison of Eqs. (115) and (85) indicates more explicitly the complica-

tions associated with convection than does comparison of Eqs. (106) and


(85). Substitution for j/j�

from Eq. (111) in Eq. (114) results in

(1 � *) �1�y5

a5��1�Pr

Pr��

(u�v�)55

1 � (u�v�)55��dT 5

dy5. (116)

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 sugges-

ted 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 over-

looked 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:

T 5�a5

2 ��

8�

(1� *) [1 � (u�v�)55] �Pr

�Pr � dR�. (117)

Integration of this expression for T 5, weighted by u5/u5�

, over the cross-

section of the pipe then gives the following expression for the mixed-mean

temperature

T 5�

�a5

2 ��

��

�

8�

(1� *)[1� (u�v�)55] �Pr

�Pr � dR��

u5

u5�� dR�, (118)

from which it follows that

Nu��j�D

k(T��T

�)�

2a5

T 5�

�4

��

��

�

8�

(1 � *)[1 � (u�v�)55] �Pr

�Pr � dR��

u5

u5�� dR�

. (119)

The quantity Nu, called the Nusselt number, may be interpreted as the

dimensionless rate of convective heat transfer. The corresponding express-

ions for T 5 and Nu in terms of Pr�rather than Pr

�are

T 5�a5

2 ��

8�

(1� *)dR�

1 �Pr

Pr��

(u�v�)55

1 � (u�v�)55�(120)


and

4

Nu�

2T 5�

a5��

�

��

8�

(1� *)dR�

1�Pr

Pr��

(u�v�)55

1� (u�v�)55�� u5

u5�� dR�. (121)

Equations (117)—(121) are the final, general integral formulations herein for

the temperature distribution, mixed-mean temperature, and Nusselt num-

ber. However, some reductions are possible for particular boundary condi-

tions 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 T5 by

means of Eqs. (117) or (120) requires (u�v�)55 as a function of y5 and a5,

and * and Pr�

or Pr�as a function of Pr and the thermal boundary condition

as well as of y5 and a5. The evaluation of T5�

by means of Eqs. (118) or

(120), and Nu by means of Eqs. (119) or (121), may further be inferred to

require a relationship for u5 as a function of y5 and a5 and u5�

as a function

of a5. However, these requirements may be relaxed somewhat. First, u5 and

u5�

are given exactly as integral functions of (u�v�)55 by Eqs. (89) and (91),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 ��/�

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 (u�v�)55 and Pr.

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

distance in developing thermal convection following a step in wall tempera-

ture 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 a5 and b5— a severe test of

independence from geometry as well as from the thermal boundary condi-

tion. Thus, the evaluation of T 5, T 5�

, and Nu only requires (u�v�)55 as a

function of y5 and a5, Pr�or Pr

�as a function of (u�v�)55 and Pr, and * as

a function of y5, a5, Pr and the thermal boundary condition. The depend-

ence 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

experimentally.


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��T

T�� T

�

orT��T

T�� T

�

and/or of h� j�/(T

��T

�)

with axial distance. Although the exact point of this attainment is ill defined,

the concept is a useful one in both an analytical and an applied sense. The

majority of the theoretical semitheoretical solutions and correlations in the

literature for the Nusselt number in turbulent flow are for this regime, which

prevails or is closely approached over most of the length of ordinary

industrial heat exchangers.

A uniform heat flux density from the wall to the fluid may be attained

approximately by passing an electrical current axially through the metal

wall of a heat exchanger, which thereby functions as an electrical resistance.

Small deviations from uniform heating of the fluid may then occur because

of end effects, for example, thermal conduction along the tube wall or

nonuniform heat losses to the surroundings. A uniform heat flux density

from the wall to the fluid may also be closely approached in the inner pipe

of a concentric circular double-pipe heat exchanger operated in equal

countercurrent flow. In this case small deviations may be expected due to

variations in the local overall heat transfer coefficent as a consequence of

entrance effects in flow and the variation of the physical properties of the

two fluids with temperature. If the heat transfer coefficent h� j�/(T

�� T

�)

approaches an asymptotic value with axial distance for a uniform heat flux

density, T�

�T�

must as well. Then, if (T�

�T )/(T�� T

�) approaches an

asymptotic value,

��z �

T�

�T

T��T

��*

1

(T�

�T�) �

�T�

�z�

�T�z�* 0 (122)

and

��z

(T��T

�) *

�T�

�z�

�T�

�z* 0. (123)

Hence, for fully developed convection with uniform heating,

�T�z

��T

��z

��T

��z

. (124)

Then from Eqs. (107) and (114)

1 � *�1

R� �8�

�u5

u5�� dR�. (125)


By virtue of Eq. (125), Eq. (118) may be integrated by parts to obtain

8

Nu�

4a5

T 5�

��

(1 � *)�[1� (u�v�)55] �Pr

�Pr � dR�. (126)

The analog of Eq. (126) in terms of Pr�is readily shown to be

8

Nu�

4T 5�

a5��

�

(1� *)�dR�

1 �Pr

Pr��

(u�v�)55

1 � (u�v�)55�. (127)

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 a5, as indicated by Eq. (125). This latter relationship may

be expressed directly in terms of (u�v�)55 by substituting for u5 and u5�

from

Eqs. (89) and (91), respectively, integrating by parts, and simplifying to

obtain

*��1�R�

R� � �8�

[1�(u�v�)55]dR��

8��1� R�

R� � [1� (u�v�)55]dR�

��

[1� (u�v�)55]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 y5 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 u5 approaches a nearly constant value,

1� **u5�

u5�

��

�

[1 � (u�v�)55]dR�

��

[1� (u�v�)55]R�dR�

. (129)

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 a5 and thereby characterizes the magnitude of the deviation of j/j�

from

�/��

� 1/R.

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 (u�v�)55 and one

for Pr�

or Pr�, presumably as a function only of (u�v�)55 and Pr. [Actually,

Eqs. (117), (126), (127), and (128) are also applicable for fully developed


laminar convection with uniform heating as well. For this case, (u�v�)55� 0,

Eq. (113) gives Pr/Pr�� 1, Eq. (128) gives *� 1�R�, and both Eqs. (126)

and (127) give Nu� 48/11.]

In the limit of Pr* 0, Eq. (113) reduces to

Pr

Pr�

� 1� (u�v�)55. (130)

Substitution of this expression in Eq. (117) gives

T 5Pr� 0�a5

2 ��

8�

(1� *)dR��a5

2(1�R�)(1 � *mR�), (131)

where *�8�

is seen to be the integrated-mean value from R� to 1. Since * is

finite and positive for all R � 1 for both laminar and turbulent flow, the

term *�8�

is finite and positive as well. Similarly, substitution ofPr

Pr�

from

Eq. (130) in Eq. (126) gives

8

NuPr� 0�

4T 5�Pr� 0a5

��

(1� *)�dR�� (1 � *)��8�

, (132)

where (1 � *)��8�

is seen to be the integrated mean of (1� *)� over R� from

0 to 1.0. Equation (131) provides an upper bound for T 5/a5 and Eq. (132)a lower bound for Nu for all Pr as a function of a5. In all of these

expressions, 1� * represents the effect of the deviation of j/j�

from �/��(which has often been neglected) while (1 � *)� includes the effect of the

velocity distribution as well.

Insofar as Pr�

approaches a finite value as y5* 0, the corresponding

asymptotic solution may be derived for Pr*�. For this case, the entire

temperature development takes place within the viscous boundary layer

where y5/a5 may be neglected, ** 0, and (u�v�)55 may be approximated by

the first term on the right-hand side of Eq. (32). Equation (116) thereby

reduces to

dT 5

dy5�

1 � �(y5)

1��Pr

Pr�

� 1� �(y5)

. (133)

The function on the right-hand side of Eq. (133) may be integrated

analytically if Pr�is postulated to be invariant with respect to y5. With the


boundary condition u5� 0 at y5� 0, the result is

T 5��

Pr

Pr��

3�� Pr

Pr�

� 1��

1

2ln �

(1 � z)�

1� z � z��

� 3� � tan�� 2z � 1

3� � ��3� �'

6 ��y5

Pr

Pr�

� 1

, (134)

where

z� �� Pr

Pr�

� 1��

y5.

Near the wall for very large values of Pr, the last term on the right-hand

side of Eq. (134) may be dropped. Finally, letting z*� gives the following

expression for the fully developed temperature, which differs negligibly from

the mixed-mean temperature, and thereby:

T 5�

�T 5�

�2a5

Nu�

2'�Pr

Pr��

3 �� Pr

Pr�

� 1��

. (135)

For �� 7 10��, it follows that

Nu�0.07343 �1�Pr

�Pr�

�

�Pr

Pr��

Re �f

2��

* 0.07343 �Pr

Pr��

Re �f

2��

(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�

postulated to be unity, was derived much earlier by Petukhov [92] and

others. The utility of the term

�1�Pr

�Pr�

�

is in providing a first-order correction for the effect of a finite value of Prand 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


convection in any fully developed shear flow and for any thermal boundary

condition, not just for a uniformly heated round tube. As shown subsequent-

ly, the one speculative element in the derivation of Eqs. (134) and (135),namely the attainment of a finite asymptotic value for Pr

�as y5* 0 and Pr

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

as well. Insofar as Pr�� Pr for all y5, Eq. (117) reduces to

T5Pr�� Pr

��Pr�

a5

2 ��

8�

(1 � *)[1� (u�v�)55]dR�, (137)

which, by virtue of Eq. (89), may be expressed as

T 5Pr��Pr

��Pr� u5(1 � *

��8�) (138)

where *��8�

is the integrated mean of *, weighted with respect to 1�(u�v�)55

over R� from R� to 1.0. The deviation of the T 5y5, a5 from u5y5, a5for Pr

��Pr is seen to be wholly a consequence of the factor 1 � * and thus

wholly due to the deviation of j/j�

from �/��. The similarity of the

distribution of T 5 to that of u5, as represented by Eq. (138), is one reason

for the arbitrary definition of T 5 herein. The postulate of Pr�

�Pr��Pr

for all y5 allows the reduction of both Eqs. (126) and (127) to

8

NuPr��Pr

�� Pr

�4 T5

�Pr

��Pr

��Pr

a5��

�

(1� *)�[1�(u�v�)55]dR�.

(139)

Comparison of Eqs. (139) and (91) reveals the following similarity for T 5�

and u5�:

T 5�Pr

��Pr

��Pr� u5

�(1 � *)�

��8�(140)

Here (1 � *)��8�

is the integrated mean of (1 � *)�, weighted with respect

to 1� (u�v�)55, over R� from 0 to 1.0. It follows that

NuPr�

�Pr��Pr�

2a5

u5�(1 � *)�

��8�

�Re( f /2)

(1 � *)��8�

. (141)

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

2� Pr, (142)


but occurs at Pr��Pr

��Pr instead of Pr� 1 and differs by the factor

(1� *)��8�

. The speculative element upon which the derivation of Eqs.

(137)—(141) is based, namely the invariance of Pr�

with y5 for the particular

value of Pr��Pr

��Pr, has some experimental and semitheoretical support

for Pr � 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

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 formula-

tions 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 charac-

terized by

��z �

T��T

T�� T

��

��T�z

T�� T

�

�

(T�

�T )�T

��z

(T�

�T�)�

* 0, (143)

which implies that

�T /�z�T

�/�z

�T�� T

T�� T

�

�T 5

T5�

. (144)

Substituting this expression in Eq. (107) and then that result in Eq. (114)leads to

1� *�1

R� �8�

T 5

T 5��u5

u5�� dR��

1

R�T 5�

u5��8�

T 5u5dR�. (145)

Seban and Shimazaki [93] were apparently the first to identify the equival-

ent 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� * may be inferred from Eq. (145) to be equal to (T 5�/T 5

�)(u5

�/u5�

) and

therefore greater than for uniform heating by the factor T 5�

/T 5�. It follows

that the error due to neglecting * is greater for a uniform wall temperature.

Equations (117)—(121) are directly applicable for a uniform wall tempera-

ture, but because of the dependence of * on T 5, 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

expression *y5 for *y5, T 5y5 may be calculated from Eq. (117), T 5

�from Eq. (118), and then *

�y5 from Eq. (145). These calculations are

repeated, starting with *1y5, and continued until convergence is achieved.

Now consider the three special cases of Pr* 0, Pr*�, and

Pr��Pr

��Pr for a uniform wall temperature. For Pr*�, Eqs. (133)—

(136), which are independent of *, remain directly applicable. For Pr* 0, *must be determined iteratively from Eq. (145) using T 5 from Eq. (131) and

T 5�

from the following reduced form of Eq. (118):

T 5�Pr� 0�

a5

2 ��

��

�

8�

(1� *)dR�� u5

u5�� dR�. (146)

Similarly, for Pr��Pr

��Pr, * must be determined iteratively from Eq.

(144) using T 5 from Eq. (136) and T 5�

from the following reduced form of

Eq. (118):

T 5�Pr

��Pr

��Pr�

a5

2 ��

��

�

8�

(1� *)[1� (u�v�)55]dR��u5

u5�� dR�.

(147)

c. Generalized Expressions An alternative form of expression for T 5�

and

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:

T 5�

�a5

2 ��

[1� *][1� (u�v�)55] �Pr

�Pr � dR�. (148)

From this it follows that

Nu�2a5

T 5�

�2a5

T 5��T 5�

T 5��

4(T 5�

/T 5�)

��

[1� *][1� (u�v�)55] �Pr

�Pr � dR�

. (149)


For Pr* 0, Eq. (149) reduces to

NuPr� 0�2a5

T 5�Pr� 0

�4(T 5

�/T 5

�)

��

[1 � *]dR�

�4(T 5

�/T 5

�)

(1 � *)�8�

, (150)

where here (1 � *)��8�

is the integrated mean of 1� * over R� from 0 to 1,

whereas for Pr�� Pr

��Pr Eq. (150) reduces to

NuPr��Pr

�� Pr�

4(T 5�/T 5

�)

��

(1 � *)[1 � (u�v�)55]dR�

�

T 5�

T 5��u5�

u5�� Re �

f

2�(1 � *)

��8�

,

(151)

where (1 � *)��8�

is the integrated mean of 1�*, weighted by [1�(u�v�)55],

over R� from 0 to 1.0. The factors T 5�

/T 5�

and u5�/u5�

may be expected to

compensate for each other to some extent, although T 5�

/T 5�

is always larger.

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 5�

/T 5�. However, these formulations

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 distribu-

tion as well. Equations (148)—(151) are applicable for both uniform heating

and uniform wall temperature. This approach does not result in an alterna-

tive expression for NuPr*� since the postulate that T 5�

�T 5�

is

inherent for that limiting case. It may be inferred that the effects of these two

thermal boundary conditions are exerted wholly through (T 5�/T 5

�)/

(1 � *)��8�

for Pr � 0 and

T 5�

/T5�

u5�/u5��(1 � *)

��8�

for Pr��Pr

��Pr.

3. Parallel Plates and Other Geometries

Insofar as the analogy of MacLeod is applicable for (u�v�)55 and u5, all

of the previous expressions in Section III for T 5 and dT 5/dy5 are directly

applicable for fully developed convection from parallel plates heated equally


on both surfaces (either uniformly or isothermally) if a5 is simply replaced

by b5 and R by Z � 1 � (y5/b5) wherever they appear. The expressions for

* and for T 5�

are, however, different because they invoke integrations over

a planar rather than a circular area. As an example, for equal uniform

heating on both plates, the expression analogous to Eq. (126) is

12

Nu��

�3T 5

�b5

��

(1� *)�[1� (u�v�)55] �Pr

�Pr � dZ. (152)

where here, as contrasted with Eq. (128),

*�

1 �Z

Z �9�

[1 � (u�v�)55]dZ��

9��1�Z

Z � [1 � (u�v�)55]dZ

��

[1� (u�v�)55]dZ

.

(153)

Again, as for a round tube, * increases monotonically from 0 at Z � 0 to

u5�

/u5�

at Z� 1.

For Pr� 0, Eq. (152) reduces to

Nu��Pr� 0�

12

��

(1 � *)�dZ

�12

(1� *)��9�

, (154)

where (1 � *)��9�

is the integrated-mean value over Z. On the other hand,

for Pr� Pr��Pr, Eq. (152) reduces, by analogy with Eq. (141), to

Nu��Pr�Pr

��Pr

��

12

��

(1 � *)�[1 � (u�v�)55]dZ

�

Re��

f

2�(1 � *)�

��9�,

(155)

where (1 � *)��9�

is the integrated-mean value, weighted by [1� u�v�55],

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 5��5

[1 � (u�v�)55] �Pr

�Pr � dZ (156)


and that

Nu��

j�b

k(T��

�T�)�

b5

T 5�

�1

��

[1 � (u�v�)55] �Pr

�Pr � dZ

. (157)

For Pr� 0, Eq. (157), by virtue of Eq. (130), reduces to simply

Nu�Pr� 0� 1. (158)

The half-spacing b was chosen as the characteristic dimension in order to

achieve this particular, obvious result. For Pr� Pr��Pr

�, Eq. (157) re-

duces to

Nu��

1

��

[1 � (u�v�)55]dZ

�1

1� (u�v�)55]�9

, (159)

where [1� (u�v�)]55 is the integrated-mean value over the channel. For

Pr*�, Eq. (136) is directly applicable in terms of Nu�and Re

�.

Expressions for equal uniform wall-temperatures may readily be for-

mulated by analogy to those for an isothermal round tube in Section III, A,

2, b. but are not included here in the interests of brevity. Expressions for

parallel-plate channels analogous to Eqs. (149)—(151) are also omitted, even

though they are referred to subsequently, since their form is readily inferred.

Equivalent formulations for fully developed convection are possible for all

one-dimensional flows, but their implementation is dependent upon individ-

ual expressions for (u�v�)55, 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�)55, or its exact equival-

ents in terms of the dimensionless turbulent shear stress, (u�v�)55, and Pr�

or Pr�. However, the eddy conductivity, k

�, and its implementation are

described here in some detail because of the widespread use of this quantity

or its exact equivalent, �� k

�/�c

�, the thermal eddy diffusivity, for correla-

tion and prediction in the past and present literature. The eddy conductivity

itself may be defined by

�k�

dT

dy��c

�(T �v�) (160)


and incorporated in the elementary differential energy balance to obtain, in

dimensionless form,

j

j�

��1 �k�

k�dT 5

dy5. (161)

Elimination of dT 5/dy5 between Eqs. (161) and (106) reveals that

k�

k�

(T �v�)55

1� (T �v�)55. (162)

Equations (160)—(162) are directly analogous to Eqs. (37), (38), and (86),respectively, for momentum transfer. Since, from physical considerations,

(T �v�)55 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

interchangeable with (T �v�)55 in this geometry.

Equation (161) has often been expanded as

j

j�

� �1 �c��

k �k�

c��

dT 5

dy5�� 1 �

Pr

Pr��

dT 5

dy5(163)

or as

j

j�

��c��

k �k� k

�c�(� ��

�)� �

��

� ��dT 5

dy5�

Pr

Pr��

dT5

dy5. (164)

where here Pr�� c

��/k�, Pr

�� c

�(� ��

�)/(k � k

�), �

��

�, and

k�� k � k

�. These definitions of Pr

�and Pr

�are consistent in every respect

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

be more constrained in its behavior than k�/k.

Equation (163) may be integrated formally to obtain

T 5��5

�

j

j��

dy5

�1�Pr

Pr��

, (165)

and then T 5 from Eq. (165), weighted by u5/u5�, may be integrated formally

over the cross-section of the round tube to obtain

2a5

Nu�T 5

��

�

��5

�

j

j��

dy5

�1 �Pr

Pr��

u5

u�� dR�. (166)

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


from Eq. (125) for j/ j�, thereby transforming Eq. (165) to

T5

a5� �

�

8��

8�

�u5

u5�� dR��

dR

R �1�Pr

Pr��

(167)

and Eq. (166) to

2

Nu�

T 5�

a5��

�

��

8��

8�

�u5

u5�� dR��

dR

R �1�Pr

Pr��

u5

u5�� dR�.

(168)

Lyon [94] in 1951 recognized that changing the order of integration allows

reduction of Eq. (168) to

Nu �2

��

��

8�

�u5

u5�� dR��

� dR

R �1�Pr

Pr��

, (169)

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 (u�v�)55 rather

than both u5 and ��/� as variables. Lyon further recognized that setting

Pr� 0 in Eq. (169) gives an expression for the lower limiting value of Nuthat varies with a5 (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),

T 5

a5��

�

8��

8�

T 5

T 5��u5

u5�� dR��

dR

R �1�Pr

Pr��

(170)

and

Nu�2

��

��

8��

8�

T 5

T 5��u5

u5�� dR��

dR

R �1 �Pr

Pr��

u5

u5�

dR�

.

(171)


Equations (165)—(171) may be expressed in terms of Pr�

rather than Pr�

simply by replacing

1��Pr

Pr��

by �Pr

Pr��

.

The preceding expressions in terms of ��, u5, and Pr

�or Pr

�are exact but

more cumbersome than the corresponding ones in terms of (u�v�)55, *, and

Pr�

or Pr�. An expression for �

�/� could be derived from a correlating

equation for u5 by virtue of Eq. (38), but in most applications, for some

unexplained reason, separate, incongruent correlating equations have been

used for u5 and ��. Despite appearances to the contrary, the use of (u�v�)55

rather than ��/� does not decrease the number of integrations to determine

values of T 5 and Nu; the integration or integrations of u5/u5�

are simply

performed separately in the process of evaluating *.The ,—� and u�v�—T �v� models do not appear to have a useful role for

convection in round tubes or parallel-plate channels, but the latter one has

promise for circular annuli despite the considerable empiricism involved in

the implementation of the supplementary equations of transport.

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

postulate of invariance of Pr�and Pr

�with y5. In addition, the closed-form

solution for Pr*� is subject to the asymptotic attainment of a finite value

for Pr�as y5* 0. Some uncertainty arises in the numerical evaluation of the

integral expressions for Nu for all finite values of Pr by virtue of the

empirical expression, such as Eq. (99), that is used for (u�v�)55, 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

[Pr�, k

�, k

�or (T �v�)55] is potentially very significant. The uncertainty

associated with expressions for Pr�

or its equivalent extends to all prior

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

in this section.

1. Expressions for the Turbulent or Total Prandtl Number

As noted heretofore, Pr�is presumed on the basis of theoretical conjec-

tures, as well as experimental evaluations, to be the same unique function of


(u�v�)55 (or ��/�) and Pr for all geometries and all thermal boundary

conditions. This presumption, which has been overlooked or denied impli-

citly by many prior investigators, greatly simplifies the task of correlation

for Pr�as well as the integrations for T 5 and Nu.

The high degree of uncertainty of the various expressions for Pr�arises on

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 disappoint-

ing 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 exten-

sive data for the turbulent core for fluids with Pr2 0.7 by means of a simple

algebraic expression, such as

Pr�� 0.85�

0.015

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 y5 and a5 [or (u�v�)55]. Such constrained behavior, insofar as this

prediction is valid, appears to justify the use of Pr�rather than (T �v�)55 or

k�/k or even Pr

�as a variable for correlation. The nominal restriction of Eq.

(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

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

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,

he proposed to set Pr�� 1 in that region.

A more complex empirical expression is that of Notter and Sleicher [97],

which may be rewritten in terms of (u�v�)55 rather than ��/� as follows:

Pr��

1� 90Pr � �u�v�)55

1� (u�v�)55��

�1�10

35�(u�v�)55

1�(u�v�)55��0.025Pr�u�v�)55

1�(u�v�)55��90Pr ��u�v�)55

1�(u�v�)55��

�.

(173)


Equation (173) predicts an asymptotic value of Pr��

�� 0.778 as y5* 0

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 (u�v�)55 even

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 grouptheory to derive

�1.1793�

1

Pr�

1.1793�1

Pr ��

�2.1793�

1

Pr�

2.1793�1

Pr ��

� 1� (u�v�)55 (174)

Equation (174) is implied by the authors to be applicable for all geometries,

all thermal boundary conditions, and all values of Pr and (u�v�)55. Further-

more, they assert that this expression is free of any ‘‘experimentally adjusted

parameters.’’ However, they undermine its credibility somewhat by suggest-

ing, 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

within the fluid stream may be inferred from Eq. (174) itself to be charac-

terized wholly by (u�v�)55. The dependence of Pr�

on Pr and (u�v�)55 only,

if valid, must extend to Pr�by virtue of Eq. (113). The limited experimental

support for these various presumptions has already been discussed in the

paragraph following Eq. (121). In any event, Eq. (174) is attractive in terms

of simplicity and purported generality, and its predictions appear to be

qualitatively if not quantitatively correct. For example, it predicts

Pr�Pr�� Pr

�over the entire cross-section of flow for Pr � 0.848, but on

the other hand an obviously low value of Pr�� 0.39 at the wall for

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 numeri-cal simulations to predict Pr

�for heat transfer between parallel plates for a

series of values of Pr from 0.05 to 2400, but only for b5� 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 develop-

ment serves as the origin of the Lagrangian DNS methodology of Papavas-

siliou and Hanratty. Their predictions appear to be in fair qualitative and


quantitative agreement with Eqs. (172)—(174) for most conditions, but for

Pr� 100 they indicate an increase without limit in Pr�as y5* 0. This latter

prediction, as represented (in thermal terms) by

Pr�

Pr�

1.71

(y5)��, (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

(u�v�)55 (or y5 and a5) and Pr. In the following section the effects of this

uncertainty in Pr�

are avoided insofar as possible by choosing particular

conditions for which its impact is minimal.

Substitution of Pr�from Eq. (172) or (173) and of (u�v�)55 from Eq. (99)

in Eq. (112) would yield a direct analog of (u�v�)55 for convection, that is,

an algebraic expression for (T �v�)55 as a function of y5, a5, and Pr.Substitution of Pr

�from Eq. (110) and again of (u�v�)55 from Eq. (99) in

Eq. (174) or its alternatives would yield much more complex algebraic

expressions for (T �v�)55, again as a function of y5, a5, and Pr. Such

expressions for (T �v�)55 have not been presented herein because they would

obviously be less convenient to apply than those for Pr�

and Pr�. It is

obvious that such expressions for (T �v�)55 would incorporate the consider-

able uncertainties of the generating expressions for Pr�and Pr

�as well as

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

formulations herein for numerical evaluations of Nu but the later evalu-

ations 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 5

�/T 5

�for a5� 500, 1000, 2000, 5000,

and 10,000, as computed using Eq. (99) for (u�v�)55 and in turn Eq. (128)for *, are listed in Table I along with values for laminar and plug flow. The


TABLE II

Computed Characteristics of Fully Developed Turbulent Flow through a Round Tube

(from Yu et al. [100a])

a5 u5�

Re 10� u5�

/u5�

f 10 Re( f /2)

500 17.0 17.0 1.2696 6.920 58.82

1000 18.815 37.63 1.2375 5.650 106.32000 20.518 82.07 1.2148 4.751 194.96

5000 22.69 226.9 1.1926 3.885 440.72

10,000 24.295 485.9 1.1793 3.388 823.21

20,000 25.90 1036 1.1680 2.981 1544.450,000 28.01 2801 1.1552 2.549 3570.15

TABLE I

Computed Thermal Characteristics of Fully

Developed Turbulent Convection in a

Uniformly Heated Round Tube for Pr� 0

Nu

a5 T 5�

/T 5�

(YOC) (KL) (NS)

500 1.862 6.480 6.490 6.82

1000 1.884 6.675 6.695 6.935

2000 1.889 6.808 6.845 7.03

5000 1.911 6.932 6.895 7.175

10,000 1.918 7.004 6.995 7.30

20,000 1.924 7.063

50,000 1.930 7.130

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 2a5u5�

and hence may be used to recover,

if desired, the computed values of u5�

� (2/ f )� �. Values of Nu were obtained

from 2a5/T 5�

and hence can be used to recover the computed values of T 5�

and in turn those of T 5�. The values of Nu attributed to Kays and Leung

[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 u5�

, u5�

/u5�

, Re� 2a5u5�

, f � 2/(u5�

)� and Re( f /2)

� 2a5/u5�

corresponding to the computed values of Nu, etc. in Table I are

listed in Table II.


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)

Nu

a5 T 5�

/T 5�

(YOC) (KL) (NS)

500 1.242 53.63 55.12 52.7

1000 1.222 99.45 102.0 97.1

2000 1.206 185.3 189.5 177.7

5000 1.189 424.1 433.0 401.7

10,000 1.177 796.7 812.4 749.8

20,000 1.167 1502

50,000 1.155 3487

See footnotes in Table I and values of Re�#

and

u5�/u5�

in Table II. Values of KL and NS were

interpolated for both Re�#

and Pr.

Equation (172) implies that Pr �Pr��Pr

�for Pr� 0.8673. Values of Nu

computed for this condition using Eq. (139), and again Eq. (99) for (u�v�)55

and Eq. (128) for *, are listed in Table III. The corresponding values of

T 5�

/T 5�

are also provided. Individual values of T 5�

and T 5�

may be

determined from the tabulated values of Nu and T 5�/T 5

�. The indicated

values of Re( f /2) � 2a5/u5�

may be used to determine values of (1 � *)��8�

.

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 intermedi-

ate to the older ones.

b. Solutions for General Values of Pr For Pr 0.867, Eq. (127) may be

rearranged and approximated by

Nu�8

��

(1 � *)� �1 �Pr

Pr��

u�v�)55

1 � (u�v�)55�� dR�

, (176)

and hence by

Nu�8

(1 � *)��8��1�

Pr

Pr��

u�v�)55

1� (u�v�)55��8�

, (177)

where the weighting factor for the integrated-mean value of the term in


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])

Small Pr

a5 10�� 10� 0.01 0.1 0.7

Nu

500 6.481 6.489 7.073 16.67 48.17

1000 6.675 6.694 7.927 26.70 88.53

2000 6.809 6.848 9.327 44.39 163.75000 6.933 7.035 12.95 90.62 371.5

10,000 7.006 7.210 18.25 159.1 694.020,000 7.068 7.476 27.63 283.3 1302

50,000 7.141 8.154 51.84 618.6 3006

Large Pr

a5 1 10 100 1000 10,000 �

Nu/(0.07343(Pr/Pr�)� Re( f /2)� �)

500 0.7462 0.9227 0.9794 0.9935 0.9950 1.0000

1000 0.6957 0.9066 0.9763 0.9934 0.9953 1.0000

2000 0.6510 0.8903 0.9726 0.9928 0.9953 1.0000

5000 0.5993 0.8688 0.9673 0.9918 0.9952 1.0000

10,000 0.5650 0.8529 0.9631 0.9909 0.9948 1.0000

20,000 0.5341 0.8374 0.9588 0.9899 0.9945 1.0000

50,000 0.4979 0.8176 0.9532 0.9886 0.9936 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

of any error in the values of Pr�used to compute Nu from Eq. (127) for small

values of Pr will be very limited and will continually decrease as Prdecreases below a value of 0.867. Insofar as Eq. (172) is valid, Pr

�only varies

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

significantly only for very large a5 with Pr� 0.001, 0.01, 0.1 and 10� or

greater and therefore are not reproduced herein.


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

value for all conditions, and a number have incorporated the postulate that

u�v� 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

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 (u�v�)55. 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

eliminated.

Notter and Sleicher [97, 111] developed Graetz-type series solutions for

Nu in developing as well as fully developed convection. The correlating

equations that they utilized for u5, ��/�, and Pr

�[Eq. (173)] are almost

certainly more accurate than those used by Kays and Leung, but less

accurate, with respect to u5 and ��/�, than the equivalent values used by Yu

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

(u�v�)55. 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

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

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.


TABLE V

The Thermal Characteristics of Fully

Developed Turbulent Convection in a Round

Tube with Uniform Wall-Temperature for

Pr� 0 (from Yu et al. [100a])

a5 Nu T 5�

/T 5�

500 4.959 2.124

1000 5.055 2.152

2000 5.122 2.171

5000 5.187 2.188

10,000 5.225 2.198

20,000 5.258 2.205

50,000 5.295 2.214

TABLE VI

The Thermal Characteristics of Fully

Developed Turbulent Convection in a Round

Tube with Uniform Wall-Temperature for

Pr�Pr�� 0.8673 Based on Eq. (172)

(from Yu et al. [100a])

a5 Nu T 5��

/T 5��

500 52.07 1.276

1000 97.08 1.251

2000 181.5 1.232

5000 416.9 1.210

10,000 784.8 1.196

20,000 1482 1.184

50,000 3447 1.169

3. Numerical Results for an Isothermal Wall-Temperature

Yu et al. [100a] carried out numerical calculations for fully developed

turbulent convection in a round tube following a discrete step in wall

temperature for the same conditions as those of Tables I, III, and IV. Owing

to the presence of T 5 in Eq. (145) for * and of * in Eq. (120) for T 5, 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 5�

was more efficient computationally than iterative evaluation of the

integrals by quadrature. The results obtained using Eq. (172) for Pr�

are

summarized in Tables V, VI, and VII. The computed values of Nu for


TABLE VII

Predicted Nusselt Numbers for Fully Developed Turbulent Convection in a Round

Tube with Uniform Wall-Temperature with Pr�

based on Eq. (172)(from Yu et al. [100a])

Small Pr

10�� 10� 0.01 0.1 0.7

a5 Nu

500 4.959 4.967 5.488 14.61 46.50

1000 5.055 5.072 6.155 23.89 86.01

2000 5.123 5.157 7.328 40.53 159.75000 5.188 5.275 10.50 84.57 364.1

10,000 5.227 5.402 15.27 150.4 682.020,000 5.262 5.611 23.89 270.6 1282

50,000 5.304 6.173 46.54 596.8 2966

Large Pr

a5 1 10 100 1000 10,000 �

a5 Nu/(0.07343(Pr/Pr�)� Re( f /2)� �)

150 0.8216 0.9440 0.9812 0.9914 0.9930 1.0000

500 0.7270 0.9200 0.9791 0.9935 0.9950 1.0000

1000 0.6811 0.9047 0.9761 0.9934 0.9953 1.0000

2000 0.6394 0.8887 0.9725 0.9928 0.9953 1.0000

5000 0.5904 0.8675 0.9672 0.9918 0.9952 1.0000

10,000 0.5575 0.8518 0.9630 0.9909 0.9948 1.0000

20,000 0.5278 0.8363 0.9588 0.9899 0.9945 1.0000

50,000 0.4928 0.8166 0.9531 0.9886 0.9936 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 a5� 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 (u�v�)55

from Eq. (99) rather than less accurate and incongruent expressions for ��

and u5.


TABLE VIII

Comparison of the Computed Values of Nu for Fully

Developed Turbulent Convection with a Uniform Wall-

Temperature at a5� 5000 with Pr�Based on Eq. (173)

Nu

Pr Yu et al. [100a] Notter and Sleicher* [97]

0 5.187 5.29

10�� 9.350 11.86

10�� 84.96 65.7

0.7 360.2 332

1.0 454.0 443

8 1306� 1220

10� 3572 3436

10 7915 7747

10� 17,119 16,730

*Interpolated semitheoretically with respect to a5.�Interpolated semitheoretically with respect to Pr.

4. Numerical Results for Nu for Parallel-Plate Channels

a. Equal Uniform Heating on Both Plates Danov et al. [85] utilized the

integral formulations of Eqs. (152)—(155) together with (u�v�)55 from Eq.

(99), * from Eq. (153), and Pr�

from Eq. (172) to compute numerical

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��

� 4b5u5�

in Table IX correspond to values

of u5�

determined by integrating (u�v�)55 as given by Eq. (99) with b5substituted for a5. The corresponding values of Re

�� 4u5

�b5, u5

�/u5�

,

f � 2/(u5�

)� and Re��

( f /2) � 4b5/u5�

are also listed in Table X. The values

of Re��

( f /2) were determined from 4b5/u5�

and those of *��9�

from Re��

( f /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 b5 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


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)(from Danov et al. [85])

Nu��

for small values of Pr

Pr

b5 0 0.001 0.01 0.10 0.70 0.867

500 10.43 10.45 11.46 28.93 90.40 101.31000 10.61 10.64 12.76 46.66 166.3 187.95000 10.85 11.03 21.15 162.4 701.6 804.7

10,000 10.93 11.27 30.31 288.0 1314 1515

50,000 11.07 12.78 89.95 1142 5732 6668

Nu��

for large values of Pr

Pr

b5 1.0 10 100 1000 10,000 25,000

500 107.0 280.6 701.8 1535 3335 4531

1000 198.4 547.8 1392 3062 6667 9061

5000 876.2 2672 6927 15,271 33,334 45,288

10,000 1617 5148 13,815 30,466 66,656 90,568

50,000 7176 25,051 68,715 152,087 333,209 452,471

TABLE X

Computed Characteristics of Fully Developed Turbulent Flow between Parallel

Plates per Danov et al. [85]

a5 u5�

u5�

/u5�

Re��

10� f 10 Re��

( f /2)

500 18.558 1.1605 37.116 5.81 107.8

1000 20.312 1.1437 81.249 4.85 196.9

5000 24.132 1.1189 482.64 3.43 828.8

10,000 25.738 1.1113 1029.5 3.02 1554

50,000 29.428 1.0974 5885.5 2.31 6796


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)

Nu��

Nu��

/Pr�

Pr� 0 Pr� 0.867 Pr� 1000

b5 Re��

· 10� K & L D, A & C K & L D, A & C K & L D, A & C

415 30 10.41 10.40 84.90 85.90 99.90 127.5

1204 100 10.66 10.66 212.3 222.0 288.6 368.6

3244 300 10.74 10.81 514.2 543.1 766.5 991.9

9737 1000 10.90 10.93 1392 1478 2305 2967

K & L, Kays and Leung [101]; D, A & C, Danov, Arai and Churchill [85]; b5, 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 u5 and

(91) for u5�

using Eq. (99) for (u�v�)55 are perhaps feasible on demand for each

particular condition of interest, those required for Nu for each value of a5 and

Pr are somewhat more demanding because of the added dependence on * and

Pr�. Correlating equations are therefore convenient for computed values as

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 func-

tionally 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 *.

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


TABLE XII

Predicted Nusselt Numbers for Fully Developed in Turbulent Convection between

Plates at Unequal Uniform Temperatures with Pr�

based on Eq. (172)(from Danov et al. [85])

Nu�

for small values of Pr

Pr

b5 0 0.001 0.01 0.10 0.70 0.867

500 1.0 1.002 1.113 3.337 14.16 16.43

1000 1.0 1.003 1.230 5.609 26.74 31.20

5000 1.0 1.018 2.126 21.86 118.8 139.810,000 1.0 1.036 3.186 40.54 226.9 267.650,000 1.0 1.175 10.83 175.3 1028 1218

Nu�

for large values of Pr

Pr

b5 1.0 10 100 1000 10,000 25,000

500 18.11 66.58 169.7 382.1 833.3 1132

1000 34.49 31.07 338.1 761.3 1665 2264

5000 155.3 630.5 1671 3797 8319 11,307

10,000 180.0 1240 3380 7588 16,624 92,617

50,000 1360 5993 16,656 37,915 82,855 113,154

��/�u�

�is known to be a unique function of D(�

��)� �/�, one of the five

variables in these latter two groupings is redundant in the listing for h. For

example, eliminating ��, u

�, and � individually allows the following three

different dimensionless groupings to be derived:

hD

k�# �

Du��

�,c��

k � or Nu�#Re, Pr (178)

hD

k�# �

D(��)� �

�,c��

k � or Nu�# �Re �f

2��

, Pr� (179)

and

hD

k�# �

D��

�u�

,c��

k � or Nu�# �Re �f

2�, Pr�. (180)

These three expressions are functionally equivalent to one another by virtue

of the relationship between �/�u��

and D(��)� �/�, but on speculative


reduction they lead in some but not all cases to fundamentally different

results. For example, the further speculation of independence of h from Dleads, respectively, to

Nu� Re#Pr (181)

Nu� Re �f

2��

#Pr (182)

and

Nu� Re �f

2� #Pr. (183)

Equation (182) has been shown [see Eq. (136)] to be a valid asymptote for

Pr*�, whereas Eq. (183) provides a first-order expression for

Pr�Pr�� Pr

�(neglecting the dependence on *). On the other hand, Eq.

(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

Pr� 0, but elimination of �, k, �, and ��

individually does not appear to

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 postul-

ated a power dependence of h on each of the dependent variables and

thereby obtained the equivalent of

Nu�ARe�Pr� (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 Refor any condition. Since the friction factor, f, was found in Section II to be

a non-power-function of Re, the proportionality of Nu to Re( f /2) or

Re( f /2)� � does not constitute a power-dependence on Re. Similarly, Nu was

found to be a fixed power of Pr only in the limit of Pr*�.

The use of Eq. (184) for correlation of experimental data has actually

impeded the representation, understanding, and prediction of turbulent

convection, as illustrated in the following paragraphs.

2. Purely Empirical Correlating Equations

Nusselt [102] fitted the constants of Eq. (184) using his own experimental

data for turbulent convection in gases in a round tube and determined an

exponent n� 0.786 for Re and inexplicably the same value for the exponent

of Pr. Based on experimental data for Re� 10� and various gas and liquids


Fig. 15. Determination of power dependence of Nu on Pr for Re� 10�. (From Sherwood

and Petrie [104], Figure 1.)

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� 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

2� 0.023Re��. (186)


Fig. 16. Test of Eq. (185) with experimental data. (From Coulson and Richardson [105], p.

166.)

He thereby inferred that

Nu

RePr� �

f

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 Nuon Re except as a first-order approximation for Pr�O1 and on Pr except

in the asymptotic limit of Pr*�.

3. Numerical Predictions for L ow-Prandtl-Number Fluids

Equations (185) and (187) failed utterly to predict the convective behavior

of liquid metals in nuclear reactors in the 1950s, thereby stimulating the new


theoretical analyses described in the following section. It was soon recog-

nized 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 Graetz-

type series expansion and more accurate values of u5, ��, Pr, and Pr

�than

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 Prbeyond 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 u5 and ��.

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�

0.079Re �f

2��

Pr

[1 �Pr� �]� �(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�"

��exp(2200�Re)/366

Nu�"

��Nu�

0.079Re( f /2)� �Pr�

[1�Pr�� ]� � ��

��

.

(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.)

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

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�


and f /2 on the Reynolds number. Many other analogies for turbulent

convection have been devised by postulating similar mechanisms of trans-

port 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 signif-

icant 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 � (5(Pr � 1)( f /2)� �, (190)

where (5� ((��)� �/�. The major contribution of the Prandtl analogy, Eq.

(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 (5.

Thomas and Fan [114] attempted to improve upon one other deficiency

of the Reynolds analogy by applying the penetration and surface renewalmodel 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

2� Pr� �. (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


boundary layer, and the Thomas and Fan analogy fails because periodic

transient conduction is simply a very poor model for the combination of

turbulent and molecular transport in the boundary layer. Perhaps the

greatest lasting value of these analogies is the understanding provided by

analysis of the reasons for their failure.

5. A Useful Differential Analogy

Reichardt [87] in 1951 derived an analogy based on the differential

momentum and energy balances in time-averaged form. He utilized the eddy

viscosity model for turbulent transport, but his derivation will be outlined

here in terms of (u�v�)55. Taking the ratio of Eqs. (115) and (116),respectively, with Eq. (85) gives

dT 5

du5� (1� *)

Pr�

Pr�

1� *1� (u�v�)55� (Pr/Pr

�)(u�v�)55

. (192)

Integrating the rightmost form from the centerline to the wall results in

T 5�

��u5�

�(1 � *)

1� (u�v�)55��Pr

Pr�� (u�v�)55� du5. (193)

His ingenious expansion of the equivalent integrand in terms of ��may be

rephrased in terms of (u�v�)55 as follows:

T 5�

��u5�

*

1 � (u�v�)55��Pr

Pr�� (u�v�)55

�Pr

�Pr

�

1�Pr

�Pr

1��Pr

Pr��

u�v�55

1�u�v�55�du5.

(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 u5 while (u�v�)55

� 1 for u5* u5�

. He also concluded that the rightmost term was negligible

except very near the wall where du5� dy5. He further postulated Pr�/Pr

to be invariant over the cross-section. Had he utilized the limiting form of

Eq. (93), that is, Eq. (194) for (u�v�)55 in the rightmost term, he would have


obtained

Nu�2a5

T 5�

�2a5

T 5��T 5�

T 5��

�1

(1� *)��5

��Pr

�Pr�

u5�

u5��T 5�

T 5�� Re �

f

2��

1�Pr

�Pr

3 �� 2' �

T5�

T5��

Pr

Pr��

Re �f

2��

, (195)

where (1 � *)��5

is the integrated-mean value over u5. By virtue of Eq. (88),this latter term may also be interpreted as the integrated mean, weighted by

1� (u�v�)55, over R�. Equation (195) is, by virtue of the limits of integration

and several of the approximations, applicable for a uniform wall tempera-

ture as well as for uniform heating.

6. T heoretically Based, Generalized Correlating Equations

On the basis of the asymptotic expressions for Pr� Pr�and Pr*� for

uniform wall temperature, namely, Eqs. (151) and (136), Eq. (195) may be

interpreted as

Nu�1

�Pr

�Pr�

1

Nu�

��1�Pr

�Pr�

1

Nu�

,

(196)

where Nu�

signifies NuPr �Pr� and Nu

�signifies NuPr*�. Accord-

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

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)

with n��1 and limiting solutions of (Pr/Pr�) Nu

�and Nu

�/(1 �Pr

�/Pr).

Equations (195) and (196) are limited to Pr2Pr�, which according to the

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

Nu�Nu

Nu�

�Nu�

�1

1��Pr

�Pr� Pr

��

Nu�

Nu�

(197)


Churchill et al. [117] speculated that

Nu� Nu

Nu��Nu

�1

1��Pr

Pr��Pr�

Nu�

Nu

(198)

might be applicable as a correlating equation for Pr�Pr�. However, Eq.

(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

�.

Accordingly, they introduced an arbitrary coefficient � as a multiplier of

(Pr/Pr��Pr)(Nu

�/Nu

) and evaluated it functionally to provide a continu-

ous derivative. The resulting expression

Nu�Nu

Nu��Nu

�1

1��Pr

Pr�� Pr��

Nu�

Nu��

Nu��

�Nu

Nu�� Nu

�

(199)

where Nu��

�Nu�Pr� Pr

�� 0.07343Re( f /2)� �, has proven as successful

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 a5 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 u5�

from Eq. (102) (with a modified

leading constant of 3.2) reproduce the values of Nu in Tables I and III

almost exactly:

Nu�

8

1�7.7

(u5�

)� �

(200)

Nu��

Re( f /2)

1�185

(u5�

)��

�2a5/u5

�

1�185

(u5�)��

(201)

The choice of u5�

rather than u5�

, u5�

/u5�

, a5 or Re as the independent


variable in Eqs. (200) and (201) is arbitrary since they all bear a one-to-one

correspondence. The leading constant of 3.3 in Eq. (102) was chosen on the

basis of the experimental data of Zagarola [73], while the recommended

value here of 3.2 corresponds more closely to the computed values of u5�

in

Table II and is thereby self-consistent with the computed values of Nu.

b. Isothermally Heated Round Tubes Separate correlating equations might

have been devised for T 5�

/T 5�

and u5�/u5�

as well as for (1 � *)�8�

and

(1� *)��8�

in Eqs. (150) and (151). However, in the interests of simplicity,

the following overall expressions were derived:

Nu�

8

1�1.538

(u5�

)�

(202)

Nu��

Re( f /2)

1�148

(u5�

)� �

�2a5/u5

�

1�148

(u5�)� �

(203)

Equations (202) and (203) reproduce the values of Nu in Tables V and VI,

respectively, almost exactly.

c. Uniform and Equally Heated Parallel Plates The corresponding expres-

sions are

Nu�

12

1�5.71

(u5�

)� �

(204)

and

Nu��

Re( f /2)

1�90

(u5�

)� �

�4b5/u5

�

1�90

(u5�)� �

(205)

Here, Nu and Re are based on a characteristic length of 4b and Eq. (102) is

to be used for u5�

.

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��

Re �f

2�1�

11.707

u5�

�b5/u5

�

1�11.707

u5�

(206)


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

tube in terms of Pr with Pr�estimated from Eq. (172). The agreement is very

good. The slight discrepancy for Pr� 0.01 and a5� 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

more rapid increase beyond Pr�Pr�

followed by a second point of

inflection and a decreased rate of increase approaching a one-third-power

dependence. The changes for Pr�Pr�are smaller than those for Pr&Pr

�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

the predictions of Eqs. (196) and (199) but also the numerically computed

values upon which Eqs. (136) and (200)—(206) are based.

IV. Summary and Conclusions

A. Turbulent Flow

1. A New Model for the Turbulent Shear Stress

The new and improved representatives proposed in Section I for fully

developed turbulent flow in a channel are a direct consequence of the

observation by Churchill and Chan [77] that the local, dimensionless,

time-averaged shear stress, namely (u�v�)5��u�v�/��, constitutes a better

variable for this purpose than traditional mechanistic and heuristic quanti-

ties such as the mixing length and the eddy viscosity. Churchill [80]


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 a5 and b5� 5000. [x, equally and uniformly

heated parallel plates (Nu � 4bu�/v); �, uniformly heated round tube (Nu � 2au

�/v); �,

parallel plates at different uniform temperature (Nu� bu�/v)].

subsequently noted that the local fraction of the shear stress due to

turbulence, namely (u�v�)55��u�v�/�, is an even better choice.

The presentation of new integral formulations and algebraic correlations

for fully turbulent flow based on the time-averaged partial differential

equations of conservation might appear to be atavistic in view of the recent,

presumably exact, solutions of these equations in their unreduced time-dependent form. However, the use of integral and algebraic structures based

on the time-averaged equations may be expected to persist into the


Fig. 20. Representation of numerically predicted values by Yu et al. [100a] of Nu for a

uniform wall temperature by Eqs. (196), (199), and (172).

foreseeable future for two reasons. First, the exact numerical solutions,

which have been attained only by direct numerical simulation, are currently

very limited in scope by their computational requirements and perhaps their

inherent structure. Second, even if these limitations are eventually eliminated

or at least eased by improved computer hardware and software as well as

better inherent representations, or even if the DNS calculations are replaced

or supplemented by some other methodology, the results will be in the form

of discrete instantaneous or time-averaged values of u�, v�, u�v�, and u for a

particular condition and therefore not directly useful for applications such

as the design of hydrodynamic piping. Correlating equations will accord-

ingly 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 (u�v�)55 as a variable was the

realization that, by virtue of integration by parts, u5�

as well as u5 may be

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 Feriet [81] in the context of u�v� and a parallel-


plate channel. This suggestion was first implemented by Pai [82] for both

parallel-plate channels and round tubes, but with very poor representations

for u�v�. The advantage of using u�v� 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 (u�v�)5 was the key to successful

implementation of the integral formulation.

An inherent advantage of correlating equations for (u�v�)5 or (u�v�)55 over

those for u5 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 (u�v�)55, are

inherently more accurate than those of Eqs. (100) and (101).

3. T he Development of Correlating Equations for (u�v�)55, u5, and u5�

The structure of an almost exact correlating equation for (u�v�)55, namely

Eq. (99), was developed by Churchill and Chan [71] from a number of

asymptotic and speculative expressions for the time-averaged velocity as

well as for the time-averaged turbulent shear stress. The empirical coefficient

of the asymptotic solution for y5* 0 was evaluated using the several sets

of results obtained by DNS while those for intermediate values of y5 and

a5 were evaluated from the experimental time-averaged velocity distribu-

tions measured by Nikuradse [46]. These latter constants were subsequently

reevaluated by Churchill [80] using the recent improved measurements of

the time-averaged velocity distribution by Zagarola [73]. The incorporation

of the equivalent of the semilogarithmic expression for the time-averaged

velocity nominally restricts this correlating equation to a5� 300, but it

provides a very good approximation for y5 a5 even down to a5� 145,

the lower limit of fully turbulent flow.

The calculation of u5 and u5�

from the integral formulations using Eq.

(99) for (u�v�)55 is feasible even with a handheld calculator. Hence, separate

correlating equations for u5 and u5�

are not really required. However, such

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 u5 and u5�, respectively, for both smooth and naturally

rough pipe. Since u5�� (2/ f )� �, the correlating equation for u5

�serves as

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 (u�v�)5, u5, u5�

, and the roughness e�

from either

experimentation or numerical simulations. The tabulated values of e�in the


current literature are very old and are almost certainly not representative

for modern piping. The reevaluation of these roughnesses for representative

materials and conditions would appear to have a high priority.

4. T he Analogy of MacL eod

The analogy attributed to MacLeod [60] was crucial to the development

of the just-mentioned correlations for (u�v�)55 and u5 in that it allowed

experimental data and computed values for round tubes and parallel plates

to be used interchangeably. This little-known analogy appears to be

validated within the accuracy of the experimental values for u�v� 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 a5 and

b5& 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

(u�v�)55 as the primary dependent variable was that the mixing length is

unbounded at one location in the fluid in all channels and in addition is

negative over a finite adjacent region in all channels other than round tubes

and parallel plates. Although the eddy viscosity is well behaved in round

tubes and parallel-plate channels, it shares the failure of the mixing length

in all other channels.

How did these anomalies completely escape attention for 75 years? The

explanation has three elements. First, the initial numerical evaluation of the

mixing length by Nikuradse [45, 46] not only was based on experimental

data of insufficient precision but also was conditioned by a preconceived

notion concerning the behavior. Second, the subsequent acceptance of the

mixing length by later investigators is simply inexcusable, since the

aforementioned failures of this concept are readily apparent from a critical

examination of most of their own sets of measurements of the velocity

distribution as well as from the predictions thereof. Third, the anomalies are

much more apparent in terms of (u�v�)55 than in terms of earlier formula-

tions.

When it was first introduced by Launder and Spalding [42], the ,—�model appeared to have great promise for the predictions of turbulent flows,

but it has ultimately proven to have no real utility. For round tubes and

parallel-plate channels the ,—� model, in all of its manifestations, not only

invokes a great deal of empiricism and approximation, but is unneeded. In

all other channels it shares the failure of the eddy-viscosity model, to which


it is directly linked. The ,—�� u�v� model avoids this linkage and thereby

has a possible role, despite its high degree of empiricism, for geometries,

such as circular annuli, in which the variation of the total shear stress is not

known a priori. The large eddy simulation (LES) methodology avoids the

need for time-averaging, at least in the turbulent core, and has a wider range

of applicability than the DNS methodology, but at the price, at least at the

present time, of a considerable degree of empiricism and approximation for

the region near a surface.

Barenblatt [57] and co-workers have recently attempted to resuscitate the

power-law correlation of Nikuradse [46] and Nunner [56] for the time-

averaged velocity, and Zagarola [73] has demonstrated that it is more

accurate than the semilogarithmic model for a narrow range of values of y5.

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 (u�v�)55 and u5.

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 turbu-

lent flow, the development of analogous expressions for fully developed

turbulent convection was undertaken with consideration confidence and

great expectations. Unfortunately, it soon became apparent that turbulent

convection is much more complex than turbulent flow even in the simplest

of contexts, and that the data base, both experimental and computational,

and the known asymptotic structure are much more limited.

Turbulent convection would be expected to be responsive to the same new

numerical methodologies used for flow, such as DNS, but so far the greater

inherent complexity of the behavior has limited the scope and accuracy of

such results.

2. New Differential Models

Time-averaging of the partial differential energy balance, followed by one

integration and expression in terms of dimensionless variables, results in Eq.

(106), in which (T �v�)55� �c�T �v�/j , the fraction of the heat flux density due

to turbulence, is a new variable analogous to (u�v�)55. However, the heat

flux density ratio, j/j�, is a dependent variable, given in general by Eq. (107)

as contrasted with �/��

� 1� y5/a5 for flow. Furthermore, very few data


have been obtained for T �v� or correlated in terms of (T �v�)55. Accordingly,

Eq. (106) was reexpressed as Eq. (109) with the expectation that the

behavior of Pr�/Pr� 1� (T �v�)55/1� (u�v�)55 would be more constrained

than that of (T �v�)55. The terms Pr�/Pr and j/j

�represent the complications

associated with turbulent convection as compared with turbulent flow.

3. T he Heat Flux Density Ratio

For a uniform heat flux from the wall, the heat flux density ratio is a

function only of the time-averaged velocity distribution, and Eq. (109) may

be reduced to Eq. (115) with * given by Eq. (125), which may also be

expressed in terms of (u�v�)55 [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

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

Eq. (113)]. However, an important discovery resulting from the use of

(u�v�)55 and (T �v�)55 as primary variables is that Pr�

and Pr�

bear a

one-to-one correspondence to (u�v�)55 and (T �v�)55 and are therefore

independent of their heuristic diffusional origin.

It has generally been postulated that Pr�

and Pr�are functions only of

(u�v�)55 (or ��/�) and Pr and thus independent of the thermal boundary

condition. For example, this postulate is inherent in the solutions of Notter

and Sleicher [97, 110, 111] for developing thermal convection with both

uniform heating and a uniform wall temperature. It is implied by Eq. (172)of Jischa and Rieke [96], Eq. (173) of Sleicher and Notter [97], and Eq.

(174) of Yahkot et al. [89]. The last imply that their expression is also

independent of geometry. The only direct experimental confirmation of

either postulate appears to be that of Abbrecht and Churchill [22], who

found the eddy conductivity to be independent of length in developing

thermal convection in an isothermal round tube as well as identical to that

of Page et al. [90] for fully developed heat transfer across a parallel-plate

channel at equal values of a5 and b5, 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


theoretical expression for Pr�

or Pr�appears to exist. This is the principal

unresolved problem of turbulent convection, at least in round tubes and

parallel-plate channels, and is worthy of renewed effort, experimentally,

theoretically, and computationally. Thermal calculations by DNS, LES, and

,—�—u�v�—T �v� generate values of (T �v�)55 or the equivalent and therefore

do not require a separate expression for Pr�. However, with the exception

of the predictions of Papavassiliou and Hanratty [98], which are limited to

b5� 150, these methodologies have not yet produced reliable values of

(T �v�)55 or Pr�for a broad range of Pr and (u�v�)55 or y5 and a5.

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

apparently not been achieved before because of the greater complexity of

the formulations in terms of ��/� and u5 as compared to these in terms of

(u�v�)55 only.

Because of the uncertainty in the various expressions for Pr�

and Pr�,

particular attention has been given herein to three cases for which that

uncertainty is eliminated or greatly reduced, namely Pr� 0, Pr�Pr��Pr

�,

and Pr*� while y5* 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

(u�v�)55.

Equation (172) implies a limiting value of Pr�� 0.85 for Pr*� and Eq.

(173) a limiting value of Pr�� 0.78 for y5* 0 for large Pr, but the

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 y5* 0 and Pr*� allows the derivation of an analytical

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 a5 using Eq. (172) for

Pr�. The results of Yu et al., including the three limiting cases described in


the preceding section are summarized in Tables I—VIII. Similar results for

parallel-plate channels, as obtained by Danov et al. [85], are summarized in

Tables IX—XII. Despite the uncertainty associated with the use of Eq. (172)for Pr

�, these values of Nu are presumed to be more accurate than any prior

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.

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�)55 or Pr�. For the same

reason, new correlating equations were not constructed for T 5. Instead

attention was focused directly on Nu.

The integral formulations for Pr� 0 and Pr�Pr��Pr

�and the ana-

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 Pr2 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

Pr2 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 represen-

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

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�.

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a


Progress in the Numerical Analysis of Compact

Heat Exchanger Surfaces

R. K. SHAH

Delphi Harrison Thermal Systems

Lockport, New York 14094

M. R. HEIKAL

University of Brighton

Brighton, United Kingdom

B. THONON AND P. TOCHON

CEA-Grenoble

DTP/GRETh

38054 Grenoble, France

I. Introduction

Compact heat exchangers (CHEs) are characterized by a large heat

transfer surface area per unit volume of the exchanger, resulting in reduced

space, weight, support structure and footprint, energy requirements, and

cost, as well as improved process design, plant layout, and processing

conditions, together with low fluid inventory compared to conventional

designs such as shell-and-tube heat exchangers.

Somewhat arbitrarily, a gas-to-fluid exchanger is referred to as a compactheat exchanger if it incorporates a heat transfer surface with area density

above about 700 m�/m (213 ft�/ft) or the hydraulic diameter D�� 6mm

(1/4 in.) for operating in a gas stream and above about 400 m�/m (122

ft�/ft) for operating in a liquid or phase-change stream. In contrast, a

typical process industry shell-and-tube exchanger has a surface area density

of less than 100 m�/m on one fluid side with plain tubes, and two to three




0065-2717/01 $35.00

Fig. 1. Plate-fin geometries: (a) offset strip fin, and (b) louver fin.

Fig. 2. Tube-fin geometries: (a) wavy fin on round tubes, (b) louver fin on round tubes, and

(c) louver fin on elliptical tubes.

times that with high-fin-density low-finned tubing. A typical plate heat

exchanger has about two times the heat transfer coefficient h or the overall

heat transfer coefficient U compared to that for a shell-and-tube exchanger

for water/water applications. For phase-change applications, even higher

heat transfer coefficients are achieved compared to a shell-and-tube ex-

changer. A laminar flow heat exchanger (also referred to as a meso heatexchanger) has a surface area density on one fluid side greater than about

3000m�/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 15,000m�/m (4570 ft�/ft) or 1 �m�D

�� 100�m. A compact heat exchanger is not necessarily of small bulk and

mass. However, if it did not incorporate a surface of high area density, it

would be much more bulky and massive.

Plate-fin, tube-fin, and rotary regenerators are examples of compact heat

exchangers for gas flow on one or both sides; whereas gasketed, welded,

brazed plate, and printed circuit heat exchangers are examples of compact

heat exchangers for liquid flows. Typical fin geometries used in plate-fin

and tube-fin exchangers are shown in Figs. 1 and 2, and plate geometries

used in plate heat exchangers (PHEs) are shown in Fig. 3. The most

commonly used fin geometries for plate-fin exchangers are offset strip fins

364 r. k. shah et al.

Fig. 3. Plate heat exchanger plate geometries: (a) washboard, (b) zigzag, and (c) chevron.

and louver fins (referred to as multilouver fins in the automobile industry).A considerable amount of experimental results are available in the

literature for flow and heat transfer phenomena in complex flow passages of

compact heat exchanger surfaces. Starting with the description of some of

the complex flows in compact heat exchanger surfaces, it is explained that

flows in compact heat exchanger surfaces are dominated by swirl and

vortices in uninterrupted flow passages, and by boundary layer flows and

wake regions (separation, recirculation, and reattachment) for interrupted

flow passages. Although unsteady laminar flows are relatively easy to

analyze, swirl and low Reynolds number turbulent flows are difficult to solve

numerically because of the lack of appropriate turbulence models. This is

the reason for the very slow progress in the numerical analysis of compact

heat exchanger surfaces.

A comprehensive experimental study of the performance of CHE surfaces

is very expensive because of the high cost of the tools needed to produce a

wide range of geometric variations. Numerical modeling, on the other hand,

has the potential of offering a flexible and cost-effective means for such a

parametric investigation, with the added advantage of reproducing ideal

geometries and boundary conditions, and exploring the performance behav-

ior in specific and critical areas of flow geometry.

Thus, the objective of this work is to provide a comprehensive state-of-

the-art review on numerical studies of single-phase velocity and temperature

fields, and heat transfer and flow friction characteristics of compact heat

exchanger surfaces, as well as to provide specific comparisons to evaluate

the accuracy of numerical work where experimental data are available. The

surfaces include offset strip and louver fins used in plate-fin exchangers,

wavy fins/channels used in tube-fin exchangers and plate heat exchangers,

365numerical analysis of che surfaces

and chevron (stamped) plates used in plate heat exchangers. First, a

description of some of the complex flows in such surfaces is presented. Next,

some highlights are presented for the numerical analysis of compact heat

exchanger surfaces. Since separation, recirculation, and reattachment as well

as large eddies and small-scale turbulence generation are common features

in CHE surfaces, a comprehensive but concise overview of turbulence

models/methods is presented next to illustrate the current capabilities and

limitations of these models. The rest of the paper covers numerical work

reported in the literature on the following CHE surfaces: offset strip fins,

louver fins, wavy fins/channels, and chevron trough plates. For each surface,

the numerical analysis is described in sufficient detail, and comparisons are

presented with experimental measurements where available. Thus, based on

the insight gained from numerical and experimental results, the performance

(fluid flow and heat transfer) behavior of these CHE surfaces is discussed

and summarized. Also, briefly mentioned is the proposed mode of research

that combines numerical analysis, sophisticated experimentation on the

small sample fin geometries, and performance testing of actual heat ex-

changer 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 charac-

terization into laminar unsteady and low Reynolds number turbulent flows.

A. Interrupted Flow Passages

The two most common interrupted fin geometries are the offset strip fin

and louver fin geometries as shown in Fig. 1. Here the fin surface is broken

into a number of small sections. For each section, a new leading edge is

encountered, and thus a new boundary layer development begins, and is

then abruptly disrupted at the end of the fin offset length l. The objective

for such flow passages is not to allow the boundary layers to thicken, thus

resulting in the high heat transfer coefficients associated with thin boundary

layers. However, the interruptions create the wake region, and self-sustained

flow unsteadiness (see Figs. 4 and 5). As a result, the models based on the

boundary layer development are not adequate and do not accurately predict


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).


the heat transfer coefficients (Nusselt number Nu or Colburn factor j ) and

friction factors.

Separation, recirculation, and reattachment are important flow features in

most interrupted heat exchanger geometries [1]. Consider, for example, the

flow at the leading edge of a fin of finite thickness. The flow typically

encounters such a leading edge at the heat exchanger inlet or at the start of

new fins, offset strips, or louvers. For most Reynolds numbers, a geometric

flow separation will occur at the leading edge because the flow cannot turn

the sharp corner of the fin as shown in Fig. 4. Downstream from the leading

edge, the flow reattaches to the fin. The fluid between the separating

streamline (see Fig. 4) and the fin surface is recirculating. This region is

called a separation bubble or recirculation zone. Within the recirculation

zone, a relatively slow-moving fluid flows in a large eddy pattern. The

boundary between the separation bubble and the separated flow (along the

separation streamline) consists of a free-shear layer. Since free shear layers

are highly unstable, velocity fluctuations develop in the free shear layer

downstream from the separation point. These perturbations are advected

downstream to the reattachment region, and there they result in an

increased heat transfer. The fin surface in contact with the recirculation zone

is subject to lower heat transfer because of the lower fluid velocities and the

thermal isolation associated with the recirculation eddy. The separation

bubble increases the form drag, and thus usually represents an increase in

pumping power with no corresponding gain in heat transfer. If the flow does

not reattach to the surface from which it separates, a wake results.

A free shear layer is also manifested in the wake region at the trailing edge

of a fin element. Depending on the Reynolds number and geometry, the wake

from the upstream fins can have a profound impact on the downstream fin

elements. The highly unstable wake can promote strong mixing that destroys

the boundary layers from the upstream fins, causing downstream heat

transfer enhancement. However, at low Reynolds numbers, or for very close

streamwise spacing of fin elements, the shear layers might not be destroyed

or the next fin element might be embedded in the wake of an upstream fin

element. In such cases, the low velocity and near-fin temperature of the wake

will have a detrimental effect on the downstream heat transfer. Wakemanagement in complex heat exchanger passages poses a difficult challenge,

especially at moderate and high Reynolds numbers where numerical simula-

tion 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


low Reynolds numbers (Re1 100) as waviness in the wake of the fin

elements. As the Reynolds number increases, oscillating flow develops in the

wake region. At higher Reynolds numbers, individual strips shed vortices at

regular intervals. These vortices are transverse to the main flow, and as they

are carried through the fin array, they refresh the boundary layer to produce

a time-averaged thinner boundary layer. For deep arrays, vortex shedding

begins at the downstream fins and moves upstream as the flow rate is

increased (see Fig. 6 of [1]). At low Reynolds numbers (less than 400), flow

through the offset strip fin geometry is laminar and nearly steady, and the

boundary layer effects dominate the heat transfer and friction. For inter-

mediate 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 in-

crease 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 identi-

fied 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/bor 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


in a fin geometry. Effectively, at all Reynolds numbers, both the duct flow

and louver flow components exist, but the relative amount depends on the

Reynolds number. Sketches of possible flow patterns in louver fins are

shown in Figs. 5b and 5c, and a flow visualization picture of flow through

louvers is shown in Fig. 5d (see color insert).To effectively exploit high heat transfer associated with short flow lengths

(louvers act as short flat plates), it is important that the fluid follow the

louver (louver flow) rather than passing between two fins (duct flow) to

obtain high Nu (and the resultant high f factors). The degree of the flow

deflection by the louvers is determined by the relative hydraulic resistance

to the flow for the louver flow vs duct flow. This is dependent upon the fin

geometry and the flow Reynolds number. The degree to which the fluid

follows the louvers is sometimes called the flow efficiency, which can be

defined as the mean angle of the flow divided by the louver angle. The

behavior of the flow efficiency and its relation to heat transfer has been

examined by Cowell et al. [2]. For louver angles from 15 to 35° and fin

pitch-to-louver length ratios (p�/l ) from 1 to 2.5, the flow efficiency drops

dramatically for Rel& 100. See typical results presented later in Fig. 14.

Flow efficiency is nearly at its maximum by Rel � 200 and is almost

independent of the Reynolds number for higher flow rates. The flow does

not align with the louver array at the louver inlet and it takes a few louvers

to turn the flow. The heat transfer and pumping power performance is

strongly dependent on this flow-directing properties of the louver array.

Surfaces that cause the flow to follow the louvers, i.e., those with high flow

efficiency, generally perform better than those in which the flow does not

follow the louvers. However, the exact heat transfer performance of these

surfaces is less well understood. Although the qualitative effect of the degree

of flow alignment on heat transfer is accepted, more accurate quantification

of these effects is needed. The degree of flattening of the Stanton number

curve at low Reynolds numbers [2] should be examined further with a view

to determining, more accurately, the critical Reynolds number at which this

flattening starts and the effect of the different geometrical parameters on this

phenomenon.

The experimental work of Chang and Wang [3] demonstrates clearly that

general correlating equations for predicting the heat transfer and pressure

drop performance of these surfaces are far from being achieved. This is

mainly due to the fact that the performance of these surfaces is a function

of a large number of geometric parameters and that a number of variants

of the fin geometry are in use. Additionally, the manufacturing tolerances in

the production of the fins and variations in the test conditions also play a

part in producing different performances for supposedly similar fins at the

same flow conditions.


Fig. 6. (a) Plate-fin exchanger, (b) tube-fin exchanger with flat fins. At section AA: (c) wavy

corrugated passage, and (d) wavy furrowed passage.

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 measure-

ment of the forces acting on the louvers in a 20 : 1 large-scale model of a

typical matrix. The model allows the angle of individual louver rows to be

driven automatically to specific angles. Force data logging and angle control

were performed automatically under computer control implementing opti-

mization strategies for the maximization of heat transfer performance as a

function of louver angles. Results for fixed angle arrays based on known

geometries showed good agreement with previously obtained experimental

data based on thermal experiments on full-size matrix sections. Although

the model used had too few fins to be representative of an infinite array, the

authors demonstrated the viability of such method for the optimization of

variable louver fins.

B. Uninterrupted Complex Flow Passages

In this case, the heat transfer surface (the fin or the prime surface) is not cut,

but convoluted such that the flow passage geometry does not allow the

boundary layer growth. For a plate-fin geometry (Fig. 6a), two flow passages

are possible: wavy corrugated and wavy furrowed cross section (of Fig. 6a) as

shown in Fig. 6c and 6d. For plate heat exchangers, the cross section of plates

having intermating troughs (washboard design) is shown in Fig. 7a and those

of plates having chevron troughs are shown in Figs. 7b and 7c. The physics of

flow of these surfaces is discussed next. The Reynolds number for the plate

heat exchanger is commonly defined with one of two characteristic lengths:

the hydraulic diameter (D�� four times the channel volume divided by the

total heat transfer surface area) or the equivalent diameter (D�� twice the

plate spacing). The ratio D�/D

�characterizes the surface extension ratio

(Adeveloped/Aprojected), and it ranges from 1.1 up to 1.4 for industrial plates.


Fig. 7. Cross-section of two neighboring plates: (a) intermating troughs, (b) and (c)chevron troughs.

1. Wavy Corrugated and Furrowed Channels

Corrugated and furrowed channels, as shown in Fig. 6, differ from plain

channels of constant cross-section. Wavy geometries provide little advan-

tage at low Reynolds numbers, and maximum advantage at transitional

Reynolds numbers. However, at higher Reynolds numbers, periodic shed-

ding 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 Re1 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 etal. [6] found that a streamwise—Gortler-like—vortex system forms in the

transitional Reynolds number range. Although the impact on heat transfer

and pressure drop is not completely clear, such a vortex system is known to

increase heat transfer.

Wavy passages clearly offer heat transfer enhancement over plain channel

passages; however, they do not offer a performance advantage (heat transfer

relative to the pressure drop) over interrupted passages.

2. Intermating and Chevron Trough Plates

The high heat transfer coefficients obtained in plate heat exchangers are

a direct result of the corrugated plate patterns. A cross-section of one

corrugation along the flow length of an intermating trough design is shown


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—Gortler vortices are generated. The flow separation and vortices are

responsible for the high performance of these surfaces. The increases in j, f,and j/ f are generally higher than those for flow over a plate having a dimple

surface.

In chevron plate (see Fig. 3c with - defined there) design, the flow

geometry is 3D and quite complex. The typical cross-sectional geometries

for chevron plates at -� 90° are shown in Figs. 7b and 7c. In other

geometries, the furrows in the bottom plate have continuous path at angle

- and the mating top plate has furrows at an angle 180° �-; thus, the fluid

moves in different directions in the flow passages of mating plates. Because

of the criss-crossing (three-dimensional, 3D) nature of corrugations of the

mating plates, the secondary flows induced are swirl flows, which are

generally superior in terms of an increase in heat transfer over friction.

Hence, the relative performance of chevron plates is superior to all other

corrugation patterns and thus it is now most commonly used heat transfer

surface in plate heat exchangers. A much better understanding of the flow

patterns in chevron plates and subsequent enhancement is now available

[7—9]. Flow visualization by Focke and Knibbe [10] and Hugonnot [9] in

a larger-scale channel clearly shows recirculation areas downstream of the

corrugation edges. These areas are large at low Reynolds numbers, but the

transition to turbulent flow (which occurs at Re �$ 200) reduces the size of

these areas. The recirculation area induces degradation of the kinetic energy

of mean flow and reduces heat transfer. Experimental information on local

heat transfer coefficient distribution has been obtained by Gaiser and

Kottke [11]. They indicate that the pitch-to-hydraulic diameter ratio and

the angle of corrugation have some influence. They observed that the heat

transfer coefficient distribution is more homogeneous at high corrugation

angles, but there are still some weak areas.

Local measurements by laser pulse and thermographic analysis in a

two-dimensional (2D) channel [12] show poor heat transfer coefficients

downstream of the corrugation. It can also be seen that the local Nusselt

number tends to be more homogeneous while increasing the Reynolds

number. Bereiziat 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 thermoch-

romic method. Their results (corrugations angles& 30°) show that the heat

transfer coefficients are linked to the flow pattern and to the mixing intensity

in the channel with variations of �50% where measured.


C. Unsteady Laminar versus Low Reynolds Number Turbulent Flow

In the numerical analysis as well as in the heat exchanger design, it is

essential to characterize whether the flow is unsteady laminar or low

Reynolds number turbulent flow. In case of numerical analysis, the unsteady

laminar flow is much easier and accurate to analyze compared to the low

Reynolds number turbulent flow. For the latter case, the large eddy

simulation model and the direct numerical simulation (refer to Section IV

for turbulence models) could be used for analyzing the flow. For heat

exchanger design, the pressure drop associated with the unsteady laminar

flow is lower than that for the low Re turbulent flow. As we understand

today, the following is the characterization of these flows.

Unsteady laminar flow may be seen in the wake of a bluff body, or a

streamlined body inclined to a flow, when the Reynolds number is sufficient-

ly low. The unsteady motion is orderly and contains structures (vortices)that are generated continuously at a characteristic frequency and are of a

size closely related to the width of the body projected normal to the flow.

In contrast, low Reynolds number turbulent flow contains haphazard or

chaotic motions in addition to the underlying steady (or unsteady) flow.

These chaotic motions encompass a range of length and time scales, the

range being related to the Reynolds number, and so, unlike unsteady

laminar flow behind a bluff body, cannot be characterized by a single size

and frequency.

In DNS studies, we have access to time and space values; the statistical

analysis of fluctuations can provide information on the flow structure. At

the moment, these statistical calculations can only give information for a

specific case, and the criteria found for the change in the regime cannot be

extended to other geometries. But such analysis could be generalized to

experiments and DNS calculations to find objective criteria based on time

and space fluctuations.

As far as we know, only a few authors have tried to apply these methods

to numerical modeling, but for experimental work this has already been

done for two-phase flow structures. It is essential that sophisticated experi-

mentation and careful numerical analysis be conducted to characterize what

type of flows occur at low Re in interrupted fin geometries, wavy fins/

channels, and chevron plates for its far-reaching impact on compact heat

exchanger design.

Classical theories of turbulence require very high Reynolds number as a

prerequisite for the occurrence of the phenomenon. However, recent obser-

vations from compact heat exchanger studies (Jacobi and Shah [1] or Focke

and Knibbe [10]) seem to suggest that the Reynolds number need not be

very high to have turbulence. This observation, which we will refer to as low


Reynolds number turbulence, certainly deserves more studies in the context

of the classical notions of turbulence. This is particularly necessary since

some of the standard turbulence modeling procedures are based on the

assumption of an infinitely high Reynolds number.

III. Numerical Analysis

Numerical analysis of compact heat exchanger surfaces started about 20

years ago with significant progress in this time frame. However, the

problems analyzed numerically are simpler models of the real complex flows

within the CHE surfaces. As a result, many of the phenomena observed in

CHE surfaces by flow visualization and partial/full-scale testing have not yet

been duplicated by the numerical analysis. In spite of this, CFD (computa-

tional fluid dynamics) codes allow some modeling of 2D or 3D flows for

better understanding of the basic mechanisms of heat transfer and pressure

drop in CHE surfaces. These methods can be used as guidelines for a

parametric design study and the study of new geometries.

The most common numerical approaches used for the analysis of CHE

surfaces are the finite difference and finite volume methods, and only in

some cases, finite element. The algorithm used for the solution of the partial

differential equations is the pressure-based method because of the low Mach

numbers in CHEs. Also, in most analyses, structured grid is used for the

analysis. Unstructured, adaptive, and composite grids have been rarely used

in analyzing compact heat exchanger surfaces. Refer to Heikal et al. [15] for

the governing equations, solution algorithm, 2D and 3D models for numeri-

cal meshes, boundary conditions, and the determination of performance

parameters (such as Nu, St, Re, h, f ) for multilouver fin geometries.

As mentioned earlier, the flow and heat transfer performance of CHE

surfaces is mainly dictated by the boundary layer behavior over the

interruptions or in complex flow passages, and flow separation, recircula-

tion, reattachment, and vortices in the wake region. Careful consideration

must therefore be given to the grid used. Adequate grid refinement is needed

to capture the boundary layer growth and separation, and this is not always

possible with moderate computing resources. One must also determine

whether steady-state solutions are adequate or a time-dependent model is

needed to capture the correct flow behavior. The decision to perform 2D or

3D modeling must also be made and assessed against grid size requirement

and speed of computation. However, the accurate prediction of local and

overall Nu (or j ) and f factors for CHE surfaces will only be possible by

analyzing time-dependent 3D flows.


A. Mesh Generation

The numerical solution of the governing CFD equations in arbitrarily

shaped regions requires the generation of numerical grids. A grid is a

discrete representation of the continuous field phenomena being modeled. It

is the structure on which the numerical solution is built (Thompson et al.[16]) and should therefore accurately represent the geometrical configur-

ation of the domain and the physics of the problem. The mesh density and

structure have a significant influence on the accuracy and stability of the

solution. The optimum mesh should be fine enough to reduce the discretiz-

ation error and resolve flow and heat transfer details, especially in the areas

of sharp gradients. It is important to keep the grid as orthogonal as possible,

and to avoid cell aspect ratios significantly larger or smaller than unity.

The finite-difference CFD algorithms for complex geometries require grid

generation techniques that transform a curvilinear nonuniform grid into a

uniform rectangular one in the computational space (structured grids). The

boundary conditions can be accurately represented, in this case, as some

coordinate line (or surface in 3D) coincides with a boundary of the physical

region (body-fitted coordinates). Although the body-fitted structured grids

are widely used for both finite-difference and finite-volume algorithms, the

meshing of the complex geometries found in most compact heat exchanger

surfaces can consume considerable time and effort. Finite-volume methods

enable the use of unstructured grids, which allow more meshing flexibility.

Also, Cartesian grids with boundary cells aligned to the surface are among

current trends in grid generation (Anderson [17], Melton [18]). The grid

generation strategy is determined according to the size and location of flow

features such as shear layers, separated regions, boundary layers, and mixing

zones. For wall-bounded flows, the grid size at the wall can affect the

accuracy of the computed shear stress and heat transfer coefficient. One

must address the specific requirements of the wall functions used (seeSection IV). For example, using a classical k—� model, the grid point closer

to the wall must be inside the buffer zone of the boundary layer. Because of

the strong interaction of the mean flow and turbulence, the numerical results

for turbulent flows tend to be more susceptible to grid dependency than

those for laminar flows.

B. Boundary Conditions

The boundary conditions are very important for computational fluid

dynamic techniques as they govern the solutions. Usually, inlet conditions

are uniform bulk velocity (based on the specified flow rate) and temperature

or fixed velocity/temperature distribution, although time-dependent condi-


tions are becoming more common. No-slip velocity conditions are used at

wall as a flow condition, whereas uniform temperature or heat flux at wall

is specified as a thermal one. For the outlet, a zero spatial derivative in a

direction normal to the boundary is specified (Shaw [19]). As the pressure

is obtained by the solution of the Navier-Stokes equation, a uniform

arbitrary pressure is usually fixed at the outlet of the computational domain.

However, this condition is sometimes unsuitable when the reattachment

point of a separated flow is near the outlet or when an eddy structure exists

through it. For these cases, special conditions are used: uniform streamwise

pressure gradient (Mercier and Tochon [20]) or Sommerfeld radiative

conditions (Orlanski [21]) in the outer part of the domain. For the lateral

part, two conditions could be used: periodicity or symmetry (free slip

condition). The former is based on a direct pressure coupling between the

two lateral sides and is well suited for deviated flows (louver fins, for

example). The latter is usually used for spatially developed flows inside a

symmetrical geometry (offset strip fins, for example). To simulate fully

developed flows, a cyclic condition could be used. In this case, the velocity

and temperature profiles at the outlet of the domain are placed at the inlet

at each time step.

With the use of (�u/�x) � 0 and v� 0 for the outflow condition, a longer

wake region downstream of the surface is required to get reasonable results,

especially for unsteady flow. For example, to simulate flow past a cylinder

at Re� 300, the flow becomes unsteady with vortices in the downstream

region. If we set the length of the downstream wake region as smaller than

20D (D is the cylinder diameter), the simulation may diverge or the results

for flow performance (such as Strouhal number, flow friction, or pressure

drop) may differ from what they should be by using the preceding outflow

condition. This is because the actual flow cannot meet the condition of

(�u/�x) � 0 and v� 0 at the boundary. Thus, if we use (�u/�x� 0 and

v� 0) for the downstream boundary, the downstream wake region would

be longer (for example, 30D) for more accurate results. If we use the

boundary layer approximations for the outflow condition, the downstream

wake region in the foregoing example can be reduced to lower than 20D for

accurate results, thus reducing both the number of grid cells and the

computation time. The 3D boundary layer momentum and energy equa-

tions for the outflow condition are

��t

(�u) ��x

(�u�) ��y

(�vu) ��z

(�wu) ��P�x

��y ��

�u�y��

��z ��

�u�z�

��t

(�v) ��x

(�uv) ��y

(�v�) ��z

(�wu) ��y ��

�v�y��

��z ��

�v�z�


��t

(�w) ��x

(�uw) ��y

(�vw) ��z

(�w�) ��y ��

�w�y��

��z ��

�w�z�

��t

(�c�T ) �

��x

(�c�uT ) �

��y

(�c�vT ) �

��z

(�c�wT )

��y ��

�T�y��

��z ��

�T�z�. (1)

These equations are solved in the last cell near the boundary, and hence the

following terms are not present in the preceding equations (they are present

when solving the boundary layer equations in the interior domain):

��u�x�

� 0,��v�x�

� 0,��w�x�

� 0,�P�y

� 0,�P�z

� 0,��T�x�

� 0. (2)

The pressure at the outflow boundary is assumed uniform and used to

compute the pressure correction P� at all interior grid points. Then the

corrected P at the last node before the outflow boundary is used to solve

Eq. (3), the finite difference form of the foregoing boundary layer equations,

for the last cell near the boundary to get a better value of u at the outflow

boundary node. This iteration between correction P� and refined value of ucontinues until the convergence, yielding the correct values of u and pressure

field. Kieda et al. [22] implemented the boundary layer approximation for

the velocity components, and Xi [23] extended the concept by the compu-

tation of the pressure field.

The foregoing boundary conditions and the computational domain em-

ployed by Xi and Shah [90] are shown as an example in Fig. 8 for a 3D

analysis of the offset strip fin geometry.

C. Solution Algorithm and Numerical Scheme

The fidelity of the results from computational fluid dynamics techniques

for turbulent flows is largely determined by the solution algorithm and the

numerical scheme. This is true for Reynolds averaged numerical simulation

(RANS), large eddy simulation (LES), and direct numerical simulation

(DNS). LES methods need to solve accurately motions over a wide range

of scales (although not wide as for DNS) and require spatial and temporal

discretization schemes that are at least second-order accurate. Generally, the

time discretization schemes used are the Adams—Bashford, Runge—Kutta,

or Leap Frog schemes. Mostly explicit schemes are used except viscous

terms, which are treated implicitly when they require smaller time steps. For

flows of practical interest, mostly finite-volume methods are used, since


Fig. 8. Boundary conditions and the computational domain for an offset strip fin geometry

analyzed by Xi and Shah [90].

finite-element applications need a higher cost per node in terms of computer

memory and CPU time are requirements. Moreover, the sub-grid-scale-

stress (SGS) effect is essentially dissipative. So, mostly second-order central

differencing schemes are used for the discretization of convection terms,

since the numerical error is dispersive. Third-order (such as the Quick

scheme described by Leonard [24]) or fifth-order upwind differencing

schemes are used too, but Moin [25] finds them too dissipative in flow-past-

cylinder calculations.

The DNS calculations are usually based on the finite difference or spectral

element scheme for the calculation of flows in CHEs, since their geometry


is very complex. In particular, the pseudospectral method, which is com-

monly used for the DNS of homogeneous flows, cannot be used for CHE

calculations because of the presence of solid walls. And, as for LES models,

explicit schemes or part implicit schemes are used for the DNS methods.

IV. Turbulence Models

In the design of compact heat exchangers, complex geometries are often

used to promote high heat transfer rates. These geometries involve non-

straight channels or ducts as described in Section II earlier. The complex

flow phenomena in such systems have profound influence on the heat

transfer and flow friction. Indeed, even though the mean flow is represented

by a Reynolds number based on the mean velocity and hydraulic diameter,

Lane and Loehrke [26] and Ota et al. [27] defined a Reynolds number

based on half the height of the surface roughness profile to describe the

separated flow. According to these authors, when this specific Reynolds

number is greater than 300, for discrete rib roughness, the flow separates at

the leading edge of the fins in a laminar manner but reattaches in a turbulent

way. This phenomenon creates coherent eddy structures, which increase the

local heat transfer. Thus, because of the complex geometries of compact heat

exchangers, even for flows at low Reynolds numbers based on the hydraulic

diameter (at so-called classical laminar or transitional flows), some turbu-

lent phenomena can appear. Although the ‘‘global turbulence’’ that is

observed in a pipe flow for Re� 2300 is also found in many heat ex-

changers, the quasi-coherent structures of the type of von Karman streets

are more important in the CHE surfaces. The selection of an appropriate

models to calculate low-Re turbulent flow, transition flow, and turbulent

flows is one of the key factors in obtaining reliable prediction of flow friction

and heat transfer in CHEs. For engineering applications, obtaining empiri-

cal data for heat transfer and fluid flow is quite cumbersome and costly; as

a result, the scope of evaluating various geometries and operating par-

ameters is limited. Therefore, the development of reliable computational

techniques is required to evaluate heat transfer rates and pressure drops for

the CHE surfaces.

In this section, the most commonly used turbulence models/methods for

computational fluid dynamics analyses are described. Although not all

turbulence models are commonly used in CHE analysis, it is imperative that

we provide a comprehensive but concise review to challenge the CFD and

CHE researchers to advance the CFD technology for CHE applications.

The methods for calculating turbulence can be divided into the following

three broad categories.


� Reynolds averaged Navier—Stokes (RANS) models of turbulence such as

the k—� model or Reynolds stress closure model (RSM), which consists

of the second moment turbulence modeling� L arge eddy simulation (L ES) techniques� Direct numerical simulation (DNS) techniques

For more information on turbulence models, refer to Rodi [28], Halbaeck etal. [29], and Wilcox [30]. We introduce the equations governing the total

(mean plus fluctuating) flow and the heat transfer, then describe the

turbulence models/methods.

A. Reynolds Averaged Navier—Stokes (RANS) Equations

In this method [31], the instantaneous solution variables in the governing

equations (Navier—Stokes equations, continuity, and energy) are decom-

posed into their mean and fluctuating components. For an incompressible

fluid, the instantaneous velocity obeys the following equation (in Cartesian

tensor form):

�u�

�t�

��x

�

(u�u�) � �

1

��p�x

�

��x

��

�u�

�x�

��u

��x

��. (3)

The velocity components and scalar quantities such as pressure are decom-

posed as

u��U

�� u�

�(4)

p�P � p�, (5)

where U�

and P are the mean components and u��

and p� the fluctuating

components. By substituting u��and p� of Eqs. (4) and (5) into Eq. (3) and

time (or ensemble) averaging, the mean velocity equations can be written as

DU�

Dt��

1

��P�x

�

��x

��

�U�

�x�

��U

��x

�

�2

3(��

�U�

�x��

��x

�

(�u��u��). (6)

The mean continuity equation for an incompressible fluid can be written as

��x

�

(�U�) � 0. (7)

Equations (6) and (7) are called the Reynolds-averaged Navier—Stokes

equations. They have the same form as the laminar Navier—Stokes equa-

tions with the velocities and other variables representing time-averaged (or

ensemble-averaged) values. However, an additional term appears in Eq. (6),which represents the effect of turbulence and is called the Reynolds stress


tensor: (�u��u��). This term needs to be modeled in order to close the system

of equations. Several approaches already exist for this purpose: (1) eddy

viscosity models (EVMs), (2) algebraic stress models (ASMs), and (3)Reynolds stress transport models (RSMs). These are now briefly described.

All these approaches require a special treatment of turbulent flows near the

wall, and some of the models for wall effects are summarized in Subsection

4 of this section.

1. Eddy V iscosity Models (EV M)

This is the most common way to model the Reynolds stresses. It is based

on the Boussinesq hypothesis, which assumes that the Reynolds stresses are

related to the mean velocity gradients by the empirical formula

u��u��

2

3k(

��

� ��U

��x

�

��U

��x

�� (8)

where

(��

0

1

for i� j

for i� j, (9)

and k is the turbulent kinetic energy. Equation (8) is valid for incompressible

fluid only; Eq. (7) is already incorporated in Eq. (8). In this approach, a

turbulent viscosity v�

is introduced and needs to be determined. The

advantage of this method is the relatively low computational cost associated

with the calculation of the turbulent viscosity using one of the following

three methods. The first method does not need an additional equation,

whereas the other two need one and two additional equations, respectively.

a. Zero-Equation Models In these models, no additional differential equa-

tions are needed to obtain the turbulent viscosity v�, which is defined as a

function of the mean flow. The Baldwin—L omax model [32] is one such

model based on the Prandtl mixing-length model. In this model, two

different expressions are given for the turbulent viscosity.

� For the inner zone (0� y� y�),

�� l�� (10)

where the mixing length l and the vorticity �� are given by

l �Ky �1� exp ��y5

A5�� (11)

�� U

��x

�

��U

��x

��

�U�

�x�

��U

��x

��

�U�

�x�

��U

��x

��

(12)

with A5� 26, and the von Karman constant K� 0.42.


� For the outer region (y�� y�(),

��F(�U�

��U�

��U�

�) (13)

where y is the distance normal to the wall, y�the inner zone (viscous

sublayer) thickness, and ( the boundary layer thickness. More complex

models for the outer region are also published.

The main drawback of these models is that the mixing length is only

computed inside the boundary layer of the flow. It is therefore difficult to

generalize the model for the complex geometries found in CHEs. So, this

model is rarely used for CHE applications.

b. One-Equation Models In this method, v�

of Eq. (10) is given by an

additional transport equation. Usually, it is the transport equation for the

turbulent kinetic energy k. In this case, v�� k� �l, and an ad hoc specifica-

tion for l is still needed. Spalart and Allmaras [33] have developed an

example of such models.

This model was designed specifically for aerospace applications involving

wall-bounded flows and has been shown to give good results for boundary

layers subjected to adverse pressure gradients. In its original form, the

Spalart—Allmaras model is a low Reynolds number model, requiring the

viscous-affected region of the boundary layer to be properly solved. When

the mesh resolution is not sufficiently fine, some commercial CFD software

uses specific wall functions. However, one-equation models are often

criticized for their inability to accommodate rapid changes in the length

scale, such as might be necessary when the flow changes abruptly from a

wall-bounded to a free shear flow, a phenomenon that is encountered

frequently in CHEs. For this reason, this model is rarely used in CHE

applications.

c. Two-Equations Models In these models, two separate transport equa-

tions determine independently the turbulent velocity and length scales.

These models are usually implemented as k—�, k—�, or k—l. In the k—�models, the two transport equations are for the turbulent kinetic energy kand its dissipation rate �. The k—� models are based on the Boussinesq

hypothesis and assume that the turbulence is isotropic. As a result, they are

expected to perform poorly in curved geometries and flows with directional

influence. For example, they cannot calculate the flows shown later in Figs.

15 and 16 because of the rotational body forces. The ‘‘standard’’ k—� model

is used for practical engineering flow calculations as proposed by Launder

and Spalding [34]. This is an economic and numerically robust model,

which gives reasonably accurate results for a wide range of turbulent flows

that do not involve too much rotational flow. Nevertheless, it is commonly


used in complex flows of industrial applications and heat transfer simula-

tions that have rotational flows. It is a semiempirical model whose strengths

and weaknesses have become known [35]. Several improvements have been

made to obtain better performance with this model.

T he Standard k—� Model. The standard k—� model proposed by

Launder and Spalding [34] determines the turbulent kinetic energy k and

its dissipation rate � from the following transport equations:

Dk

Dt�

��x

��

��

��

�k�x

��G

��G

�� (14)

D�Dt

��x

��

��

��x

��C

��k

(G��C

�G�) �C

��k

. (15)

In these equations, G�

represents the generation of turbulent kinetic energy

by the mean velocity gradients, G�

is the generation of turbulent kinetic

energy by buoyancy, C�� , C�� , C� are constants, and �

�, �� are the turbulent

Prandtl numbers for k and �, respectively. The turbulent viscosity is given by

��C

�

k�

�(16)

where C�

is a constant often equal to 0.09 in practical applications.

The standard k—� model is commonly used for analyzing flow inside most

CHE geometries; for example, corrugated wavy channels (Hugonnot [9]

and Ergin et al. [130]), louver fins (Achaichia et al. [109]), offset strip fins

(Michallon [87]), or chevron trough plates (Fodemsky [151]), Ciofalo et al.[152], or Hessami [153]).

T he RNG k—� Model. The RNG-based k—� turbulence model is derived

from the instantaneous Navier—Stokes equations using a rigorous statistical

technique called renormalization group theory. The model equations are

Dk

Dt�

��x

��

��

�k�x

��G

��G

�� (17)

D�Dt

��x

��

��

��x

��C

��k

(G��C

�G�) �C

��k

�R. (18)

Here the term R is given by

R �C

��(1 � �/�

)

(1 �-�)

��k

, (19)

where � is given by

��k

� ��U

��x

�

��U

��x

��U

��x

��

. (20)


The quantities -

and �

are constants having values as 0.012 and 4.38,

respectively. For the RNG k—� model, the eddy viscosity expression is

�� 1��

C��

� � k

�� . (21)

The RNG theory and its application to turbulence are described by Yakhot

and Orszag [37]. The scale elimination procedure in the RNG theory results

in a differential equation for turbulent viscosity, which is integrated to

obtain an accurate description of how the effective turbulent transport varies

with the effective Reynolds number and near-wall flows. In the high

Reynolds number limit, the expression of turbulent viscosity is the same as

in the standard k—� model.

The RNG model is similar in form to the standard k—� model, but

includes the following improvements:

� The RNG model has an additional model term R in the � equation that

significantly improves the accuracy for rapidly strained flows� The effect of swirl on turbulence is included in the RNG model,

enhancing accuracy for swirl flows� The RNG theory provides an analytical formula for turbulent Prandtl

numbers, whereas the standard k—� model uses constant values� Whereas the standard k—� model is a high Reynolds number model,

the RNG theory provides an analytically derived differential formula

for effective viscosity that takes into account low Reynolds number

effects

Thus, the RNG model is more accurate and reliable for a wider class of

flows than the standard k—� model. It is well suited for corrugated fin

surfaces where the hydraulic diameters are small and the Reynolds numbers

are low. Because this kind of modeling is relatively recent, few computations

on CHE geometries have been performed. Sunden [157] used this model for

chevron trough plates geometry and obtained more accurate results than

those with the standard k—� model.

However, the accuracy for predicting the turbulent flows using the RNG

model is reported as poorer than that for two other k—� models for vortex

shedding behind the bluff objects, such as square rods and circular tubes in

a heat exchanger. In these cases, the separated flows are not well predicted

as in chevron or corrugated plate geometry. Such vortex shedding has low

frequency modulations. Saha et al. [38] compared three turbulence models

to capture the essence of time-averaged flow quantities in a vortex shedding


dominated flow field through the turbulence models in two dimensions.

They used the Launder and Spalding [34] standard k—� model, the

Kato—Launder k—� model [39], and the RNG k—� model of Yakhot et al.[40]. In terms of the parameters such as the Strouhal number and lift and

drag coefficients, the predictions due to the Kato—Launder and the standard

k—� models were close to each other, and reasonably close to experiments

of Lyn et al. [41]. However, the predictions due to RNG k—� models were

not close to the experimental values. A detailed comparison of velocity

profiles revealed the Kato—Launder model to have the closest agreement

with the experiments. A comparison between the computations and the

experiment were also made for the time averaged kinetic energy variation

along the centerline of the domain of interest. The Kato—Launder model

predicted the peak value of turbulent kinetic energy in good agreement with

the experiments. The peak value of the turbulent kinetic energy due to the

RNG k—� model showed a significant departure from the experimental

value. Thus, the comparison of these turbulence models indicates that the

accuracy of the models may depend upon the geometry investigated; a more

thorough investigation is needed for establishing the utility of specific

models for specific geometries.

T he Realizable k—� Model. The realizable k—� model (Shih et al. [42])is a recent development that satisfies certain mathematical constraints on

the normal stress consistent with the physics of turbulent flows contrary to

the standard and RNG k—� models. In practice, the principle differences

between the realizable model and the other k—� model are that the former

contains a different formulation for the turbulent viscosity and that another

transport equation has been used for the dissipation rate. This equation has

been derived from an exact equation for the transport of the mean-square

vorticity fluctuation. The transport equations for the turbulent kinetic

energy k and for the dissipation rate � are

Dk

Dt�

��x

��

��

��

�k�x

��G

��G

�� (22)

D�Dt

��x

��

��

��x

��C

��k

C�G�

�C��

��

k ��C

�S� (23)

where S is a scalar value of the strain tensor. The turbulent viscosity v�is

calculated from Eq. (16), but C� is no longer a constant. It is given by

C��

1

A�A

�

U*k

�

(24)


where A� 4.04, and A

�and U* [U* defined in Eq. (29)] are functions of

both the mean strain and rotation rates, the angular velocity of the system

rotation, and the turbulence field (k and �) [42].

This model is more accurate for predicting in the spreading rate of planar

or round jets than the standard k—� model. It is likely to provide superior

performance for flows involving rotation, boundary layers under strong

adverse pressure gradients, separation, and recirculation. At present, this

model still requires validation for industrial applications and remains a field

of research.

T he k�—�� Model for L ow Reynolds Number. Hwang and Lin [43]

proposed an improved low Reynolds number k�—�� turbulence model to

describe thermal field. By adopting the Boussinesq approximation, the

turbulent heat flux is approximated as:

� u��T ��

�

�T�

�x�

(25)

where ��is the thermal diffusivity and T

�is the mean temperature.

In the standard k—� model, ��is adopted to be proportional to the ratio

of the turbulent viscosity and the Prandtl number. Most calculations of

CHE have used constant turbulent Prandtl number Pr�; it might be more

appropriate to use a variable one for CHE. See Kays and Crawford [44] for

variable Pr�. In the k�—�� model, the thermal diffusivity is expressed as a

function of the velocity scale and of the thermal and mechanical time scales

to take into account the variations of the turbulent Prandtl number and the

fact that the thermal diffusivity is not necessarily related to the eddy

diffusivity. Moreover, to give a correct asymptotic behavior in the vicinity

of the wall, the dissipation rate is decomposed into two parts in this model

(Jones and Launder [36]):

�� (26)

Here �� is the dependent variable in the dissipation rate equation (23) and

�� 2�(��k/�x�)�. With this decomposition, �� reaches zero at the wall and

�� equals � for y5� 15. This model has been validated on experimental and

DNS data for duct flows.

For CHE applications, the preceding linear eddy viscosity model by

Launder and Sharma [45] simplifies the wall boundary condition of the

dissipation equation. However, one of its main limitations is that it gives far

too large near-wall length scales in impinging or recirculation flows. As a

remedy to this, Yap [46] introduced an extra source term into the dissi-


pation equation. With Yap’s correction, near-wall turbulent length scale can

be reduced in a separated flow, particularly near the flow reattachment

point around where the maximum heat transfer occurs. Some nonlinear

eddy viscosity (k—�) models have been developed to essentially capture the

nonisotropic behavior of the flows as encountered in shear flow, recircula-

tion flow, or swirling flow [47].

For CHE applications, low Reynolds number models such as the k�—��model have been used for internal flows inside chevron trough plates

(Ciofalo et al. [152], Hessami [153], or Sunden [157]) and corrugated wavy

channels (Yang et al. [126] and Ergin et al. [132]). However, it still requires

validation for shear flows, which are encountered in offset strip fin and

louver fin geometries.

Other Common Eddy V iscosity Models. Many other models, based on

the eddy viscosity concept, exist such as the k—� model and the shear stress

transport (SST) model. For more details, refer to Wilcox [30] or Menter

[48].

2. Algebraic Stress Models (ASM) or Nonlinear Eddy V iscosity Models(NL EVMs)

The ASM or nonlinear eddy viscosity model (NLEVM) [49—51] is an

intermediary model between the EVM and the RSM. In this model,

Reynolds stresses are represented as a tensor polynomial expansion in terms

of the mean strain rate and rotation rate tensors. The expansion coefficients

are determined from the simplified differential Reynolds stress transport

equation. The ASM is less sensitive to rotation than the EVM and need less

computational cost than the RSM, but this kind of model is still not

commonly used in industrial applications because they require too many

empirical parameters, which have to be adjusted for each application.

3. Reynolds Stress Models (RSM)

In this method, the Reynolds stress is determined by solving the differen-

tial transport equations for each components of Reynolds stresses (Launder

et al. [52]; Gibson and Launder [53] ; Launder [54]). Taking moments from

the exact momentum equations may derive the exact form of the Reynolds

stress transport equation. The momentum equations are multiplied by a

fluctuating property and then are time-averaged. But the Reynolds stress

transport equation contains several unknown terms that need to be modeled


in order to close the equations:

��t

(u��u��)

(1)

��x

�

(U�u��u��)

(2)

��x

��u��u��u��

P

�

((��

u�� (

��u��)�

(3)

��x

��

��x

�

(u��u��)�

(4)

��u��u��U

��x

�

� u��u��

�U�

�x��

(5)

�-(g�u��1! �� g

�u��1! �)

(6)

�P

��u�

��x

�

��u�

��x

��

(7)

�2��u�

��x

�

�u��

�x�

(8)

� 2��(u��u��

� u��u��

)

(9)

. (27)

Here, - the volumetric expansion coefficient, 1! � is the fluctuating fluid

temperature, and � the rotation vector. In the preceding equation, the

following terms do not require any models: (1) local time derivative; (2)convection; (4) molecular diffusion; (5) stress production; and (9) production

by system rotation. However, in order to close the equation set, the

following terms need to be modeled: (3) turbulent diffusion due to triple

correlations and pressure fluctuations; (6) buoyant production; (7) pressure

strain; and (8) dissipation.

Since the RSM takes into account the effects of streamline curvature,

swirl, rotation, and rapid changes in the strain rate in a more rigorous

manner than other RANS models, it supposedly gives more accurate results

for complex flows. But the RSM predictions are still limited by the closure

assumptions used for various terms of Eq. (27), especially the pressure-strain

and the dissipation-rate terms, designated as (7) and (8).The RSM does not always yield results superior to the simpler models in

all classes of flows. Compared with the k—� models, the RSM requires

additional memory and CPU time because of a large number of transport

equations computed. Furthermore the RSM could need more iterations

than k—� models because of the strong coupling between the Reynolds

stresses and the mean flow.

The use of the RSM is interesting when the flow features studied are the

results of strong anisotropy in the Reynolds stresses (cyclone flows, rotating

flows, etc.). At present, these models are still not used for industrial


applications and remain a field of research. Indeed, they are based on too

many adjustable parameters (unknown quantities) that could be determined

for simple geometries but that are not available for complex ones. Also, it

lacks the generality of the model assumptions, which researchers have tried

to overcome by including higher-order terms in the model equations.

4. Models for Wall Effects

Turbulent flows are significantly affected by the presence of walls as the

mean velocity field is affected through the no-slip condition at the wall.

Numerous experiments have shown that the near-wall region can be largely

subdivided into three layers:

� The viscous sublayer, where the flow has laminar properties and the

viscosity has a dominant role in the momentum and heat transfer� The buffer region, where the effects of molecular viscosity and turbu-

lence are equally important� The fully turbulent layer, where the effects of turbulence are dominant

There are two standard methods to take into account wall effects in

numerical simulations: wall-function modeling and the use of low Reynolds

number turbulence models.

a. Wall-Function Models Wall functions are a collection of semiempirical

formulas and functions that link the solution variables at the near-wall cells

and the corresponding parameters on the wall. They are composed of laws

of the wall for mean velocity and temperature, and formulas for near-wall

turbulent quantities. For industrial flows, Launder and Spalding [35] wall

functions can be used. Therefore, the law of the wall for the mean velocity is

U*�1

Kln (Ey*), (28)

where

U*�UC� �

�k� �

��/�

, y*�yC� �

�k� �

�, K � 0.42, E � 9.81, (29)

and y is the normal distance from the wall. The value of C�

is nearly a

constant, often equal to 0.09 in practical applications. Nevertheless, for the

RNG model, Eq. (24) gives the correct value. This logarithmic law for the

mean velocity is known to be valid for y*� 30—60. For lower y* values, one

can apply the laminar stress—strain relationship that can be written as

U*� y*.


Similar to this law of the wall for the mean velocity, the law of the wall

for temperature can comprise two equations: a linear law for the thermal

conduction sublayer where conduction is important, and a logarithmic law

for the turbulent region where effects of turbulence dominate conduction.

In high Reynolds number flows, the wall function approach substantially

saves computational resources, as the viscosity affected near-wall region

does not need to be solved. The wall-function approach is economical,

robust, and reasonably accurate. It is a practical option for the near-wall

treatments for industrial flow simulations. Some variations of the wall

functions based on the same concept are used in all CFD codes.

b. Turbulence Models for Low Reynolds Number Flows When low

Reynolds number effects are important in the flow domain, the hypothesis

underlying the wall functions cease to be valid. Therefore, models such as

the two-layer model (Iacovides and Launder [55], Rodi [56]) can be used.

Similar to Rodi’s model [56], Chen and Patel’s model [57] resolves the near

wall region by the transport equation for k only while the energy dissipation

and the eddy viscosity are prescribed in an algebraic manner. In these

models, the k—� model is combined with one equation model near the wall

so that the dissipation rate and the turbulence viscosity near the wall are

calculated with the prescribed length scales l�and l� as

�� C

��kl

�, ��

k �

l�, (30)

where the length scales l� and l�contain the damping effects in the near wall

region and are calculated from

l��K

C ��

y[1� exp(�0.236y*)] (31)

l��

K

C ��

y[1� exp(�0.016y*)], (32)

where y*� y�k/v is the dimensionless distance [a definition different from

that in Eq. (29)] and y is the normal distance from the wall. C�� 0.09 and

K� 0.42, respectively. The Chen and Patel model [57] is observed to be

robust from the point of view of numerical stability and also capable of

predicting separated flows and flows over rough surfaces.

Another commonly used low Reynolds number turbulence model is by

Lam and Bremhorst [58].


5. Assessment of RANS Models

� Among the RANS models, the k—� model is the standard model used

for practical engineering flow calculations as it gives reasonably

accurate results for a wide range of turbulent flows without demanding

excessive CPU time and memory.� Other eddy viscosity models do not provide accurate enough results

for turbulent flows, even though the Spalart—Allmaras [33] model or

the k—� model [30, 48] yields good results for boundary layers

subjected to adverse pressure gradients and is well adapted for aero-

space applications.� Moreover, comparing with the k—� models, the RSM requires signifi-

cant additional memory and CPU time. The RSM does not always

give results superior to those of the k—� models because of the large

number of adjustable parameters (unknown quantities). The use of

RSM is interesting when the flow features studied are the results of

strong anisotropy in the Reynolds stresses (cyclone flows, rotating

flows, etc.). ASMs are less sensitive to rotation than EVMs and need

less computational cost than the RSM, but this kind of model is still

not commonly used for industrial applications. Again the reason is that

there are too many parameters that need to be adjusted for each

application.

B. Large Eddy Simulation (LES)

The main idea of this model is to compute the large-scale turbulence and

to model the smaller scales. Here lies a profound similarity between RANS

and LES models. The only difference between them lies in the definition of

a small scale. In RANS models, the effect of all eddies is simulated by the

turbulence model, whereas in LES models, the large scales are simulated and

the scales smaller than the grid size or the filter width are modeled [59]. All

large-scale structures in both RANS and LES models are determined by

solving the governing equations. Numerical models are applied to subscale

structures. As a result, a set of filtered equations with subscale correlations

is obtained. The subscale structures have relatively low energy and their

structure is expected to be rather universal. In the case when the grid size

is less than the Kolmogorov scale ��(�lRe� �

", where the Reynolds number

is based on the mixing length l ), the fluid transport properties are controlled

exclusively by molecular processes, the need for modeling of subscale

processes disappears and LES models turn into DNS models, when suffi-

ciently small time steps are used. The major interest in the LES models is

the cases when RANS models perform poorly but DNS models are


prohibitively computer intensive. For such flows, which are encountered in

CHEs, the k—� models plus wall functions cannot provide accurate predic-

tions of heat transfer to meet the CHE designer’s requirements. In addition,

the more advanced nonlinear second moment models and near-wall sub-

layer models usually result in unsatisfactory numerical stability in many

engineering applications. In some CHE applications where turbulence

mixing can be an important factor influencing the performance, LES can be

used, as it provides not only mean flow mixing characteristics but also

subscale mixing, which can be very valuable. However, LES suffers from its

high computational cost. The choice therefore between RANS (EVM and

ASM) and LES is really dependent on the balance between accuracy and

computational cost (both memory and speed requirements). At present,

RANS is the only engineering tool for design of industrial CHEs. However,

from the point of view that higher and higher accuracy will be required for

CHE modeling and more and more powerful computers will be available to

users in the near future, it might be wise for researchers to investigate the

suitability of these methods for the prediction of the performance of heat

exchanger surfaces. As an example, Ciofalo et al. [152] used the LES

method for a chevron trough plate geometry and obtained the best predic-

tion of friction and heat transfer coefficients compared with other classical

models. However, this kind of approach is still not widespread. LES

methods have also been used with success for the modeling of turbulent

flows in complex geometries (Rodi et al. [60], Moin [61]).LES requires some averaging of the variables (e.g., velocity) to obtain the

solved quantities. For this, various approaches are used and summarized

next.

a. Schumann’s Approach [62] In this approach, instantaneous quantities

(velocities, temperature, etc.) are averaged over a control volume defined by

the numerical grid leading to a piecewise constant distribution of the

velocity. This method is implicitly used in finite volume methods, but the

averaged velocity results from the discretization and changes with it.

b. Filter Approach (Leonard [63]) This approach applies a low-pass filter

(top-hat filter or Gaussian filter) to the solved quantities. It leads to an

averaged velocity that is now a continuous function of the position and is

independent of the physical scale separation and discretization. However, in

practice, the filter does not appear explicitly in many LES codes. In fact, it

is implicit since scales smaller than the grids are automatically disregarded.

So, it is recommended to use a filter width larger than the mesh size in order

to remove the link between the subgrid scale length and the grid size because

the solutions become grid-independent.


The filtered averaged equations for constant fluid density are

�u�

�x�

� 0 (33)

�u�

�t�

��x

�

(u�u�) �

��x

��p!��

��x

�

(2�S�� !

��), (34)

where

S ��

1

2 ��u

��x

�

��u

��x

��. (35)

The subgrid scale stress �!�� u

�u�� u!

�u!�represents the effect of unresolved

(subgrid scale) motion on the resolved one and is modeled using subgrid

scale (SGS) models.

The task is to determine the subgrid-scale stress and hence to simulate the

effect of the unresolved motion on the resolved motion. This effect is mainly

dissipative, i.e., energy transfer is globally from large to small scales, but

locally and instantaneously transfer could be in the other direction (back-

scatter).Eddy V iscosity Models. Most SGS models presently use the eddy-

viscosity (Boussinesq) concept:

��

1

3(��

� �2��S ��. (36)

The task of SGS is now to determine turbulent viscosity ��. From a

dimensional analysis,

�� lq

��, (37)

where l is the length scale of unresolved motion (and not the mixing length

here), and q��

the velocity scale of unresolved motion. Most active unresol-

ved scales are those closest to the cutoff point, so the natural l in LES is

usually the grid size. For determining q��

, there are various approaches,

similar to the RANS modeling, as follows.

Smagorinsky Model (Smagorinsky [64], Lilly [65]). Similar to the

Prandtl mixing model, the Smagorinsky model is related to the gradient of

the space average velocity:

q��

��S ��S ��. (38)

More recently, the subgrid viscosity is directly related to the strain tensor

[59] as

��C

��S

��, (39)


where C�is the Smagorinsky constant (C

�is often taken as 0.1 in practical

applications) and � is the filter width or subgrid scale.

Renormalization Group (Yakhot et al. [66]). The renormalization

group theory can be used to derive a model for the subgrid-scale eddy

viscosity, which results in an effective subgrid viscosity given by

��

(�) � � �1�H �0.12�! ��

�(2')�� C��

� (40)

where

�! $ 2��

(�)�S ��, H(x) � �

x x� 0

0 x� 0. (41)

Here, H is the Heaviside function with x as a dummy variable, � is the

molecular viscosity, �! is the mean dissipation rate, and C is a constant taken

equal to 73.5. For a high Reynolds number flow, the RNG-based subgrid

scale model reduces to the Smagorinsky—Lilly model. But in the low

Reynolds number flow region, the effective viscosity is equal to the molecu-

lar viscosity, allowing the RNG model to better predict flow in near-wall

regions.

Dynamic Procedure (Germano et al. [67], Ferziger [68]). The basic

idea of this procedure is to use information available from the smallest scales

to determine the coefficient in the SGS model. So, a test filter wider than the

basic LES filter (�� 2�) is introduced and the SGS model is applied to

scales between � and �� . The dynamic procedure and the Smagorinsky

model are the most widely used subgrid-scale models.

Structure Function Model (Metais and Lesieur [69]). The eddy viscosity

is computed according to a structure function

�� 0.063��F

�(�) (42)

F�(r) � [u

�(x � r) � u

�(x)]�. (43)

This structure function model has been validated for simple geometries

(backward-facing step, for example; Fallon [70]), but still requires work for

complex flow geometry, especially with thermal effects.

C. Direct Numerical Simulation

The direct numerical simulation is the simplest way to simulate turbu-

lence because no critical assumptions and no closure equations are needed

(Moin and Mahesh [71]). The classical Navier—Stokes, continuity, and heat

balance equations are solved directly using a high-order convective scheme.

However, this model requires a very fine mesh because the finest mesh must


be finer than the smallest scale of turbulence (i.e., Kolmogorov scale

�� lRe� �

", where the Reynolds number is based on the mixing length l ).

But the ratio of the smallest scale and the largest scale is a function of the

Reynolds number Re� �"

, so the number of grid points in any direction is

Re �"

. In practice, a grid scale on the order of five times the Kolmogorov

scale is usually sufficient (except in the near-wall turbulence). Xi et al. [72],

Mercier and Tochon [20] and Kouidry [73] have used DNS and unsteady

models to predict flow fields for the OSF geometry and corrugated channels,

but only for a 2D approach. So the DNS studies are limited by the

computational means. For instance, let us review the landmarks of DNS

research:

1. The first DNS study, by Orszag and Patterson [74] on an isotropic

decaying turbulence, used a 32 mesh (Re based on Taylor macroscale

equal to 35)2. Kim et al. [75] studied a turbulent channel flow with a 192 129 160

mesh (Re� 3300)

3. Spalart [76] studied a turbulent boundary layer at Re� 1410 with a

432 80 320 mesh

So presently, the main applications of DNS are (1) to provide reliable data

for the validation of turbulence models, especially useful where experimental

accuracy is low; (2) to provide data for evaluation of subgrid models for

LES (i.e., dynamic models); and (3) to conduct some fundamental studies of

turbulence. Even modern supercomputers have limited capability for ana-

lyzing complex flows in the CHE surfaces. Classical industrial applications

also require too much computer capability at present; hence, only some

confined geometries have been simulated with the DNS model, and also at

relatively low Reynolds numbers: McNab et al. [137] and Tochon and

Mercier [139] for corrugated wavy channels, Blomerius et al. [156] for

chevron trough plates, and Mercier and Tochon [20] for offset strip-fin

geometries.

1. Assessment of Simulation Models

Although the LES and DNS techniques have demonstrated several

advantages over the RANS approach, these simulation techniques require

excessive CPU time and memory for computations of 3D complex flow

problems in the CHEs and other industrial problems, particularly at large

Reynolds numbers. Therefore, these methods are limited to a large-scale

turbulence with relatively low Reynolds numbers such as transition flows in

channels/ducts, flow over a bluff body, or CHE surfaces at low Reynolds


numbers. Because computer capabilities increase 10 times or more every 5

years, the simulation models represent a promising numerical tool for the

future.

D. Concluding Remarks on Turbulence Modeling

For the past 15 years, RANS models (k—�) have been used for modeling

unsteady laminar, transitional, and turbulent flows in compact heat ex-

changers. But the poor description of anisotropic flows requires the use of

more advanced models (LES and DNS). Most of the work on the develop-

ment of new turbulence models has been based on simple geometries, such

as flat plates and isolated backward-facing steps, rather than the complex

geometries encountered in CHE systems. For CHE surfaces, anisotropy,

shear flows, and rotational and other body force effects exist that cannot be

satisfactorily analyzed by most of the existing turbulence models. Although

the foregoing review shows promising results from these models, it is clearly

the case that the CHE technology calls for more accurate and dependable

models. The focus of such efforts should center around models that can

handle the following:

� The multiple and interacting shear layers, separating and reattachment

surfaces� The correct near-wall behavior (flow and heat transfer) in complex

geometry situations

The LES procedure has received attention as a potential solution to these

kinds of difficulties, and some progress has been made. However, robust

near-wall treatments as well as easy-to-implement averaging procedures for

the inhomogeneous turbulence features of CHE systems are current chal-

lenges. Until these difficulties with LES are resolved, coupled with the

inappropriateness of DNS for realistic CHE systems, it would seem that

RANS (with problem-specific parametric optimization) will remain the

CFD tool for design and optimization of compact heat exchanger surfaces.

RANS models have already provided in-depth knowledge of local phenom-

ena on a relative basis leading to improvements in the design of CHE

surfaces.

V. Numerical Results of the CHE Surfaces

In this section, the numerical analysis of some important compact heat

exchanger surfaces is reviewed. Included are offset strip-fin surfaces, louver-

fin surfaces, wavy-fin surfaces and channels, and chevron plates.


A. Offset Strip Fins

Extensive numerical analysis of offset strip fin geometry (shown in Figs.

1a and 8) has been conducted since 1977. The details of numerical models,

operating conditions and geometries of offset strip fins analyzed are pro-

vided in Table I.

Numerical solutions for the offset strip-fin (OSF) geometry was first

started with zero fin thickness ((� 0) and an infinite fin height by Sparrow

et al. [77]. They varied the nondimensional strip length l/s from 0.2 to 5,

where l is the strip length and s is the transverse spacing of offset strip fins.

Because of the zero fin thickness, the impingement region at the leading edge

of a strip and the recirculating region behind the trailing edge were absent.

Hence, the partial differential equations were parabolic and a marching

procedure was used. Patankar and Prakash [78] extended the analysis to

finite fin thickness ( in terms of dimensionless (/s� 0.1, 0.2, and 0.3 for a

‘‘fully developed’’ periodic flow for Reynolds number Re in the range

100&Re& 2000. They found a small recirculating zone behind the trailing

edge at low Re or low (/s. At high Re or high (/s, this zone extended from

the trailing edge of a strip to the leading edge of the succeeding strip. This

constricted the flow to the minimum flow area, thinned boundary layers,

and resulted in high j and f factors. An increase in f was more pronounced

with increasing (/s, as expected because of the higher form drag. The

prediction for the f factor was in reasonable agreement with experimental

data, but the predicted j factor was about 100% high, and the slopes of jand f vs Re data were steeper than those for experimental data.

Kelkar and Patankar [79] extended their previous work [78] on offset

strip fins to investigate the effects of the fin length and aspect ratio. The

numerical simulations were performed assuming a zero fin thickness for a

3D control volume. The grid used was 30 20 30. To characterize the

geometry, they introduced a parameter -* defined by -* � lRe/s. The main

conclusions from their work are as follows: The number of cells necessary

to obtain a fully developed flow increases with -*; the 3D aspect of the flow

is negligible for low aspect ratios (&0.2); the flow becomes more complex

and three-dimensional for higher aspect ratios. They compared their nu-

merical results with the empirical correlation of Wieting [80]. Their numeri-

cal model underestimated the friction factors by about 15% and

overestimated the Nusselt numbers by 15% compared to those predicted by

the Wieting correlation for the entire range of Reynolds numbers and

geometrical parameters. The agreement appeared to be better for higher

values of the fin length parameter (-*� 0.01).Suzuki et al. [81] obtained a numerical solution for combined free and

forced convection in laminar flow through staggered arrays of zero thickness


TABLE I

Summary of Numerical Models, Operating Conditions, and Geometries of Offset Strip Fins

Boundary Inlet ReynoldsAuthors Model Grid condition condition number, Re Pr Geometry Validation

Patankar andLaminar 2D: 60 30 Constant heat Fully 100—2000 0.7 (/p�� 0; 0.1; 0.2; 0.3 Pressure drop

Prakash flux developed l/p�� 1 Nusselt number

[78]Kelkar and Laminar 3D: 30 20 30 Developing 0.7 b/s� 0.1 to 1 Wieting [80]

Prakash correlations[79]

Suzuki et al. Laminar 2D: 339 27 Constant wall Uniform 125—500 0.7 (� 0[81] temperature velocity and l/p

�� 1

temperatureSuzuki et al. Laminar 2D: 339 31 Constant heat Uniform 200—1500 0.7 (/p

�� 0; 0.2; 0.4 Local Nusselt

[82] flux velocity and l/p�� 1 number

temperatureXi et al. Laminar 2D: 500 80 Constant heat Uniform 250—1000 0.7 (/l� 0; 0.04; 0.08 Local Nusselt

[83] flux velocity and l/p�� 0.8 to 4 number

temperatureSuzuki et al. Unsteady 2D: 380 130 Constant wall Uniform 800—5000 0.7 (/l� 0.0314; 0.126

[86] laminar temperature velocity andtemperature

Xi et al. Unsteady 2D: 380 130 Constant wall Uniform 860 and 3430 0.7 (/l� 00314; 0.126 Flow visualization,[72] laminar temperature velocity and velocity etc.

temperatureMichallon Laminar 2D: 40 38 Constant wall Fully 300—5000 0.7 (/l� 0.14 Pressure drop

[87] k—� 3D: 30 39 10 temperature developed l/p�� 2.2 heat transfer

Mizuno et al. Laminar 3D: 36000 grid Constant heat Developing 20—300 0.7 (/p�� 0.064 Heat transfer

[88] point flux and 7 l/p�� 2.61

Mercier and DNS 2D: 245 141 Constant wall Developing 5000 7 (/p�� 0.14 Local and overall

Tochon temperature and fully l/p�� 2.2 (literature)

[20] developedXi and Shah Unsteady 3D: Constant wall Uniform 100—6000 0.7 (/p

�� 0.064; 0.1 Experimental j and

[90] Taminar 460 45 45 temperature velocity and l/p�� 1.5; 1.95; 2.61 f factors from

temperature literature

399

offset strip fins. Suzuki et al. [82] extended the analysis to finite thickness

fins and also to free-stream turbulence. To predict the turbulent flow at a

low Reynolds number, a turbulence model derived from the standard k—�model implemented. A relatively good agreement was found between

numerical and experimental values of Nu within the range of Reynolds

number tested (Re& 800). In addition, experiments were conducted by

varying the inlet turbulent intensity, and the effect was found to be very

small. Furthermore, the effect of the fin thickness was found to be rather

small. The fin length and spacing effects were extensively studied from the

heat transfer point of view. Xi et al. [83] provided details on the numerical

treatment of the cells adjoining the fin surface and extended the previous

study to a larger number of offset strips (up to nine). The effect of fin

thickness was studied in detail and appeared to depend on the fin spacing

and length ratio. The detailed explanation on heat transfer enhancement in

a fin array was also presented.

An unsteady flow field in the OSF geometries results in heat transfer

enhancement. Based on flow visualization results, Mochizuki et al. [84] and

Xi et al. [85] reported that the flow in unsteady region has highly periodic

velocity fluctuations, and resulting flow instability is strongly dependent on

the Reynolds number and geometric parameters, such as fin pitch and fin

thickness. In the transition flow region, flow patterns are different for

different rows in OSF arrays. As outlined by Xi et al. [83] and Jacobi and

Shah [1], flow instabilities can appear in the wake of the fin even at low

Reynolds numbers. Suzuki et al. [86] and Xi et al. [72] studied unsteady

flow field in an inline array of three fins. The flow was assumed uniform at

the inlet. The computational grid was 380 130 and was selectively nonuni-

form: A finer grid was applied near the fin wall surface. To solve the

governing equations, a central finite difference scheme was used for the

diffusion terms. In the finer mesh region, a second-order upwind scheme was

used to solve the convective terms and a third-order scheme was used in the

region far from the wall. The predicted flow field was compared with

experimental data obtained by Xi et al. [72], and a good agreement was

found for the mean streamwise velocity and the rms values of the fluctuating

velocities. A close analysis of the flow pattern and heat transfer near the fin

wall surface has led to the conclusion that heat transfer enhancement is

created by self-sustained flow oscillations. Local instabilities near the wall,

created by upstream vortices that impinge on the fin, produce dissimilarity

between the momentum transfer and heat transfer, thus reducing the local

skin friction. These phenomena have already been observed for an array of

cylinders. Unsteady flow simulation is very promising as it gives local

instantaneous information. However, to engender confidence in the numeri-

cal results, a careful validation should be provided on instantaneous data,

not just on the time averaged values.


Michallon [87] used a standard CFD code (TRIO-VF, developed by the

French Atomic Commission) to model an offset strip fin channel. The fully

developed flow was obtained by reinjection of the outlet flow conditions as

the input in the iterative solution. The numerical simulations were per-

formed for 300&Re& 4000. For Re& 2000, a laminar model was applied,

whereas for higher Reynolds numbers, the standard k—� model was used.

Two- and three-dimensional analyses were performed, and no significant 3D

effects were noticed. The reason for this may be related to their computa-

tional model, which is only a part of OSF, and boundary conditions in

which they used a reinjection procedure. As mentioned earlier, flow patterns

are different for different rows in OSF arrays, especially when the flow is

unsteady. Therefore, the fin height of the OSF channel, fin thickness, and

inlet flow condition need to be taken into consideration for a more precise

3D numerical analysis. The comparison with experimental values obtained

by Michallon [87] showed that the friction factors are overestimated by

about 25%, and the Nusselt numbers are overpredicted significantly, from

30% for low Reynolds numbers (Re& 1000) up to a factor of 2 for high

Reynolds numbers (Re� 3000). In terms of local interpretation, a recircula-

tion zone was observed at the trailing edge of the fin, but the size of this

zone was limited. This result is in agreement with the work of Patankar and

Prakash [78].

Mizuno et al. [88] numerically investigated three-dimensional offset strip

fins in the laminar low Reynolds number regime (Re& 300). The thermal

boundary layer developed on the fin surfaces and the primary surface

(parting sheets) cannot be considered to be of the same thickness. On the

parting sheet, the thermal boundary layer is thicker and therefore affects

heat transfer. To prove this phenomenon, a three-dimensional analysis

taking into account conduction in the fins was undertaken [88]. A conven-

tional control volume method was applied, assuming a zero fin thickness.

For the flow boundary conditions, a uniform velocity was assumed at the

inlet, with a zero velocity gradient at the outlet. The thermal boundary

condition was constant wall temperature on the separation plates. In

parallel with the numerical work, experimental measurements were per-

formed. The numerical results underestimate the pressure drop by 20% at a

Reynolds number of 20 but are in agreement at higher Reynolds numbers

(Re� 300). The average Nusselt number is well predicted at Re � 100;

below this value, it is overpredicted; and above this value, it is underpredic-

ted. The maximum deviation is around 10%. A parametric study reports on

the effects of the thermal conductivity of the fluid and of the fin material,

and outlines the fact that for a high value of the fluid thermal conductivity

(water), the heat transfer performance is affected by the fin material; i.e.,

aluminum fins are better than stainless steel fins. The other results of this

study concern the thermal boundary layer on the parting sheets; its


thickness is comparable to the one of a plain rectangular channel.

Mercier and Tochon [20] have performed a 2D time-dependent analysis

of turbulent flow in an offset-strip fin heat exchanger. The turbulent flow

behavior is solved by using both an accurate convective scheme (third order)and a fine grid. The mesh size must be smaller than the smallest turbulent

scale, and the computational domain must be greater than the largest scale

involved. Due to limited computer capabilities, not all the turbulent scales

could be solved, and the method is referred to as pseudo-direct numerical

simulations. Two cases considered by Mercier and Tochon [20] are a single

fin with a uniform flow upstream, and an array of offset strip fins under

developing and fully developed flows. On a single fin, the time-dependent

evolution gives fundamental information on the flow structure (see Fig. 4).The flow, hitting the front edge, separates and creates a recirculation zone.

At the reattachment point, which oscillates, one part of the flow is convected

downward and the other part of the flow upward in the recirculation zone.

At the trailing edge, vortices are convected, and a von Karman street is

formed. These phenomena are in agreement with visual observation per-

formed on similar geometries. When the size of the recirculation zone is

compared with data from the literature, good agreement is found. Concern-

ing the array of fins, two basic inlet flow conditions have been studied:

developing flow and fully developed (reinjection). For developing flows, the

vortices created by the first row of fins impinge on the downward fins and

suppress all organized structure (highly turbulent flow). The comparison of

the calculated time average friction factors and Nusselt numbers with

measured values show poor agreement for the developing flow. Applying a

reinjection procedure (fully developed flow) gave relatively good agreement

with the literature data. Despite a 2D approach, the results appear qualitat-

ively correct, and local and overall information on flow structure and

thermal and hydraulic performances can be obtained. For a single fin, the

numerical velocity field was compared with the data of Ota and Itasaka

[89], and the agreement is qualitatively correct with a mean deviation of

�20%. For the array of fins, the predicted values of the friction and

Colburn factors are compared to several correlations and results from the

open literature. The predicted friction factors are underestimated by 5 to

30%, and the Colburn factors are underestimated by 30 to 50%.

Xi and Shah [90] conducted a 3D numerical analysis of OSF geometries

(Fig. 8) that differed from that of Michallon [87] and Mizuno et al. [88] in

the following aspects: (1) time-dependent, i.e., unsteady laminar flow; (2)finite fin thickness; and (3) computational domain having upstream flow

region (2l ), OSF region (9l ), and downstream wake region (10l) so that

the effects of form drag, velocity defect, and temperature excess in the wake

are properly taken into account. In order to suppress an overshoot of the


solutions and reduce the magnitude of numerical viscosity, a mixed usage of

two upwind schemes for convection terms was applied for the momentum

equations (third order) and energy equation (second order). The finite

difference equations of fully implicit forms were solved step by step along

the time axis with the evaluation of pressure by the SIMPLE algorithm and

the alternating direction implicit (ADI) method for each relaxation. In order

to optimize the computational time for minimizing the maximum and total

residual errors of the computational domain, the following scheme was

adopted: (a) The computation was first carried for 15,000 time steps. In each

time step, one iteration was done. (b) Then, the computation was carried for

another 2000 time steps in which five iterations were done. (c) Finally, the

computation was carried out for additional 2000 time steps (with five

iterations per time step) to get the mean values for the results presented here.

The computed results of the flow and thermal fields are little affected by

what pattern of spatial distributions used as the initial condition. Refer to

Xi and Shah [90] for further numerical details.

Three comparisons were made by Xi and Shah [90]: numerical results of

Mizuno et al. [88], experimental results of an idealized OSF (Mochizuki etal. [84]), and a real OSF (London and Shah [91]). The 3D numerical results

obtained by Xi and Shah [90] are in better agreement with the experimental

data of Mizuno et al. than are the the latter group’s own 3D numerical data.

The primary reasons are that Xi and Shah considered a finite (actual) fin

thickness of the OSF geometry, considered unsteady laminar flow in the

analysis, and employed 2l upstream and 10l downstream regions as part of

the OSF domain, whereas Mizuno et al. considered zero fin thickness,

steady laminar flow, and employed no upstream and downstream regions

outside the OSF region.

A comparison between the experimental results of Mochizuki et al. and

the 3D numerical results of Xi and Shah for an idealized OSF shows an

excellent agreement in the j factors for Re� 6000 and in f factors for

Re� 4000, as shown in Fig. 9. The f factors computed for Re � 6000 have

possible numerical convergence error due to high Re where the aforemen-

tioned iteration scheme may not be adequate; and a turbulence model may

be required as explained in [90]. A comparison of experimental results for

the real OSF surface of London and Shah [91] and the 3D numerical results

of Xi and Shah for the idealized fin geometry (without burrs at edges, surface

roughness, etc.) are shown in Fig 10. For f factors, there is an excellent

agreement between experimental and numerical results for Re& 1000.

However, the numerical results have lower values than the experimental ffactors for 1000&Re&5000 (maximum difference of 35% at Re � 2500).The burrs and bent leading and trailing edges of an OSF and surface

roughness due to brazing would result in a larger effective fin thickness. This


Fig. 9. A comparison of 3D numerical results of Xi and Shah [90] with experimental results

of idealized OSF of Mochizuki et al. [84].

Fig. 10. A comparison of 3D numerical results of Xi and Shah [90] with experimental results

of a real OSF of London and Shah [91].


in turn has an important effect on the flow instability and in the transition

region than in the laminar (low Re) region [90]. For j factors, the numerical

results show much larger values than the existing experimental results and

correlations (about 50% at Re � 400), except that the slopes of the correla-

tion by Mochizuki et al. [84] for idealized geometries and the numerical

results are closer. At present, the reasons for this large difference in

numerical vs experimental j factors are not known. The reduction in j factors

for the real OSF surface compared numerical values for the idealized

geometry may be in part due to passage-to-passage nonuniformity; this

nonuniformity has a significant effect on j factors and negligible effect on ffactors at low Re (Shah [93]). In addition, numerical analysis shows another

important difference between the real and idealized OSF geometries as

follows. The transition for f factors (deviation from the straight f vs Re line

for laminar flow on a log — log plot) begins in the range Re� 1000 — 1200

for the London and Shah surface and in the range Re� 1600 — 2500 for

numerical results as shown in Fig. 10. The transition for j factors begins at

Re� 1500 — 2000 for the London and Shah surface and between Re values

of 2500 and 3300 for the numerical results. The reason is due to burrs and

bent edges and surface roughness for real surfaces. Another interesting

observation is that the transition in f factors starts at lower Re than that

for j factors. This feature indicates that the unsteady flow first enhances flow

friction and then enhances heat transfer as a function of Re. It has been

observed in visualization experiments reported by Mochizuki et al. [84] and

Xi et al. [85] that the unsteady flow first occurs downstream of fin arrays

(affecting only �P and f factors) and then progresses upstream into fin

arrays as the Reynolds number is increased.

A comparison of 2D and 3D numerical results for the Shah and London

surface indicate that the effect of 3D geometry is smaller in the laminar flow

region (Re�

� 1600) and the 2D computations are quite accurate at low

Re.

1. Concluding Remarks on Offset Strip Fin Performance/Analysis

The studies just summarized for offset strip fin geometry indicate that a

considerable amount of numerical and experimental work has been conduc-

ted on this simple and high-performance compact heat exchanger surface.

As outlined by Xi et al. [72] and Mercier and Tochon [20], unsteady

analysis is required to accurately calculate turbulent flow in compact

geometries. A 3D model is also required to simulate the correct phenomena

[90]. Nevertheless, the main physical phenomena can be predicted by these

methods, and the predicted values are in relatively good agreement with

experimental data.


Fig. 11 Typical louver fin geometries: (a) corrugated fin, (b) flat fin.

B. Louver Fins

Since the early 1980s, numerous attempts have been made to develop 2D

numerical models of louver fin surfaces. In all louver fin analyses reported

in this paper and in the literature, the geometry analyzed is a flat fin as

shown in Fig. 11b; no 3D numerical studies are reported for Fig. 11a. The

details of the geometries analyzed, numerical models employed, and the

Reynolds number range investigated are summarized in Table II. Initially,

the models were based on the idealized zero fin thickness and laminar flow

with periodic boundary conditions and constant wall temperature. Kajino

and Hiramatsu [94] and Tomoda and Suzuki [95] solved the stream

function and vorticity equations for incompressible steady laminar 2D flow

over flat louver fins using finite difference methods. They presented stream-

lines, velocity profiles, and Nusselt number distributions on the louver

surface for one louver geometry only at a single Reynolds number value.

The authors stated that the computational results showed trends similar to

the flow visualization results, but no quantitative validation was made. They

concluded that although flow visualization is a useful tool for assessing the

performance of louver fins, numerical calculations are necessary for more

quantitative evaluation. Their main conclusion was that for high-perform-

ance fins with small louver pitch, best results are obtained if the fin pitch is

matched to that of the louver in such a way that the fluid flows along the

louver. However, no evidence was presented in the paper to support this

statement, and it would appear that the numerical solution was only used

to prove the experimental findings.


TABLE II

Summary of Geometries, Numerical Models, and Reynolds Number Ranges Covered for Flat Louver Fins

Reference Geometry Numerical model Rel

Kajino and Hiramatsu p�� 1.0 mm l� 1.0 mm 2D, stream function and vorticity, laminar, grid not described 500

[94] 1! � 26°(� 0.1 mm

Achaichia and Cowell (� 0 2D, finite volume, laminar steady, Cartesian grid, periodic 20—1500[96] 1! � 15° to 35° boundary conditions, didn’t solve the energy equation

p�/l� 1.0 to 2.5

Hiramatsu et al. [98] 1! �0° to 50° 2D, finite difference, body fitted rectangular grid, stream 100—1000p�/l�1.0 and 2.0 function and vorticity

Ha et al. [100] 1000 and 1050 fins/m for 1! �23° 2D, body fitted generalized grid, laminar steady flow 176—10061000 fins/m for 1! �31°1000 fins/m for 1! �31°

Baldwin et al. [103] 1! �15° and 30° 2D, finite difference, Cartesian grid, staircase louver edge 200—6000l�30 mm representation

Suga et al. [106] 1! �30°, 26° 2D, finite difference, steady laminar flow, overlaid 64—450l�10mm, (�0.8 mm, p

��10 mm Cartesian grids

p�/l�1.0, 1.125 and 1.75

Suga and Aoki [107] 1! �20°, 26° and 30°, same as Ref. [106] above 64—450p�/l�0.5 to 1.125, (�0.8 mm

l�10 mm, p��10 mm

Ikuta et al. [108] (�0.115 mm, l�1.3 mm, 2D, body fitted grid, finite difference, steady laminar flow 417p��1.1, 1.5 and 1.9 mm,

1! �15°, 20°, 25° and 30°Achaichia et al. [109] 1! �20° to 40°, p

�/l�1.7. 2D, body fitted grid, k�� turbulence model for Rel� 1200, 10—2400

energy equation not solvedItoh et al. [110] p

�/l�1.22, (/l�0.625 2D, steady laminar, grid details not given 645 and 595

Atkinson et al. [111] p��1.5 to 2.5 mm, 2D, steady laminar finite volume, automatic body fitted grid 100—3200

and l�0.9, 1.1 and 1.4 mm, generationDrakulic et al. [112] 1! �12° to 28.5°, (�0.05 to 0.1

Drakulic [113] 2D: same as Refs. [111, 112] Same as Refs. [111, 112] 2D analysis:3D: p

�� 8 and 14 mm Conjugate heat transfer in the 3D case 100—3200

p��2.17 mm, l�1.1 mm 3D analysis:

1! �22°, (�0.05 mm 100, 400, and 1600

407

A more comprehensive study was carried out by Achaichia and Cowell

[96]. Using finite difference techniques, they modeled one louver in the

periodic fully developed region assuming cyclic boundary conditions,

laminar steady flow, and zero fin thickness. Since only one louver was

modeled, the authors were able to use a fine Cartesian grid normal to the

louver surface. Results were presented for fin-to-louver pitch ratios of 1 to

2.5, louver angles 15° to 35°, and louver pitches based Reynolds number Relfrom 20 to 1500. The results confirmed the phenomenon first identified by

Davenport [97], namely the existence of two distinct flow regimes with duct

flow occurring at low Reynolds numbers and louver (aligned) flow prevail-

ing at high Reynolds numbers. This result, which is clearly demonstrated by

the sketch of Fig. 5b, has since been accepted and confirmed by a number

of other studies both experimentally and numerically [98—100]. From the

numerical velocity distributions, Achaichia and Cowell [101] quantified this

flow alignment property of the louvers by introducing the concept of the

mean flow angle. They plotted this parameter against the Reynolds number

and showed that the flow is aligned with the louver to within a few degrees

at high Reynolds numbers, and the degree of alignment begins to fall off as

the Reynolds number is decreased. They also found that the Reynolds

number at which this trend starts is a function of the fin-to-louver pitch and

derived a simple correlation for the mean flow angle in terms of the

Reynolds number, the fin-to-louver pitch ratio, and the louver angle. They

also found that their model could predict recirculation behind the louvers

at high louver angles even for the fully developed periodic case.

The effect of this flow-directing behavior is that the friction factor curves

exhibit a steep slope at low Reynolds numbers as the less aligned fluid

impinges on the louver surface as it flows down the duct between the fins.

At high Reynolds numbers, the curve shows a clear flattening as the flow is

aligned with the louvers. This flow behavior is also responsible for the

flattening of the Stanton number curve observed by Achaichia and Cowell

[101]. A flow efficiency term was devised by Webb [102] and Webb and

Trauger [99] to describe this flow behavior and predict the onset of this

flattening in the Stanton number curve.

Finite fin thickness models were analyzed in a number of numerical

studies on flow through complete louver arrays for a few isolated configur-

ations and/or flow rates. In all cases, a symmetrical array having two-bank-

deep louvers was modeled. Baldwin et al. [103] used a Cartesian grid and

solid cells with zero porosity to define the fin. The louver surface was

therefore represented by a series of staircase-type steps with the grid spacing

selected to make the solid cells approximate the geometry. A finite volume

solution of the flow equations was carried out using the commercial

PHOENICS code. Results were presented for only two louver configur-


ations, each at a single Reynolds number, and compared with the LDA

measurements of Button et al. [104] and the flow visualization results of

Hiramatsu and Ota [105]. The numerical results displayed the same basic

phenomena as the experimental studies, showing flow separation after the

first louver and almost complete alignment after the third. As the flow

entered the second bank of louvers, it took longer to become aligned relative

to the first bank.

Suga et al. [106] presented a finite difference numerical model for a

complete louver fin (two louver banks) over a limited range of Reynolds

numbers (64�Rel�450). The model assumed 2D steady laminar flow. They

used an elaborate system of overlaid grids to overcome the finite fin

thickness problem and divided the solution domain into regions that were

represented by Cartesian grids. Complex communication between the grids

was achieved by bilinear interpolation of the dependent variables at the grid

boundaries. The authors identified the possibility of interpolation errors at

the false boundaries between grids and stated that they could be reduced by

the careful choice of the grid sizes in the overlapping region. Another

possible source of error was the use of triangular cells to join the bent parts

of the first half louver. The predicted velocity distributions in the regions

between the louvers were compared with LDV measurements at Rel � 64.

However, the computational results did not display the characteristic duct

flow expected at such a low Reynolds number. The calculated mean Nusselt

number for each louver was also compared with that measured by means of

a nickel film sensor (which acted both as a heater and a resistance

thermometer). A remarkable degree of agreement was shown even though a

uniform fin surface temperature was assumed in the processing of the

experimental results.

Using the same numerical model, Suga and Aoki [107] investigated the

effect of grid refinement as expressed by a grid Reynolds number Re�, and

concluded that a grid independent solution was obtained at values of

Re��7. They then carried out a numerical study of the effect of fin

parameters on heat transfer performance. The range of the Reynolds

number for this study was also 64�Rel�450, louver angles 20°�1! �30°,

fin thickness to louver pitch ratio 0.04�(/l� 0.08, and high-density fins

with fin pitch to louver pitch ratio 0.5� p�/l�1.125. They conjectured that,

provided that the flow is aligned with the louvers, the performance of the

fin is dominated by the thermal wake behind the louvers. They then

concluded that for each louver angle, there was an optimum value for the

louver-pitch-to-fin-pitch ratio that caused the thermal wake behind the

louvers to flow along a line halfway between two louvers further down-

stream. They presented a formula for the calculation of this optimum ratio

as a function of the louver angle. It is interesting that the value of the


optimum ratio was observed to be independent of the Reynolds number in

the range Rel� 450.

Body-fitted coordinates have been used to facilitate the grid generation

for a full fin with finite thickness. Ikuta et al. [108] developed a 2D model

using finite difference techniques in a curvilinear coordinate system with a

relatively coarse grid. They displayed the velocity vectors diagram for a

single geometry at one value of Reynolds number showing flow separation

from the first three louvers with complete alignment subsequently. This

process was shown to repeat in the second bank of louvers.

The oblique grid and coordinate transformation system, adopted by

Hiramatsu et al. [98], offered a more versatile method for the finite

difference modeling of louver fin geometries. In this system, two mesh

structures were used with an oblique grid in the zone of the inclined louvers

and a rectangular grid over the horizontal inlet and turn round parts of the

fin. The laminar 2D steady flow model was considered for the analysis.

Some mesh refinement was done by increasing the number of grid points

near the louver edges. A comparison between the predicted streamlines and

flow visualization in a water channel showed good agreement. The stream-

line plots showed increasing flow alignment with increasing fin-pitch-to-

louver-pitch ratio and Reynolds number. The effect of varying the louver

angle was not shown. Calculated values of local heat transfer distribution

on both fin surfaces were presented for one geometry at a single value of the

Reynolds number. The results showed a decrease in the mean value of the

mean Nusselt number as the fin pitch increased and with decreasing

Reynolds number. This was attributed to the reduced degree of flow

alignment under these conditions. Although the calculated friction factors

did not agree well with their experiments, the mean Nusselt number showed

excellent agreement.

Achaichia et al. [109] described a novel body-fitted grid topology that

extended over a number of fins, thus simplifying the introduction of finite

fin thickness and geometrical variations. Two-dimensional steady-state finite

volume calculations were performed, using the commercial finite volume

code PHOENICS on a two-bank louver fin array. A number of different

louver angles were considered in the range 20° to 40°, a fixed-fin-to-louver-

pitch of 17 for Rel between 10 and 2400. The flow was assumed to be

laminar at low values of Reynolds numbers. The k—� turbulence model was

applied above the critical Reynolds number of 1200. The mean flow angle

was calculated along the fin at different velocities and the results clearly

showed a gradual alignment of the flow along the fin. The effect of the

Reynolds number on the degree of alignment was also demonstrated and

quantified. The effect of the louver angle on the distribution of the local skin

friction on the upper and lower fin surfaces was shown at Rel� 10 and 600.


At the lower Reynolds number and for all louver angles, high values of local

skin friction were observed at the trailing edge of the louvers as the duct

directed flow impinged on the surface. This effect was less noticeable at the

higher Reynolds number since the degree of flow alignment was much

higher. The skin friction was markedly higher for the 40° louver, but only

for the lower Reynolds numbers. The authors explained this observation in

terms of the increased flow between the louvers (and less duct flow) as a

result of the larger gap between them at this angle and concluded that

increasing the louver angle has the desired effect of increasing the degree of

flow alignment at the same Reynolds number.

Itoh et al. [110] reported a difference of about 6% in the value of the

average Nusselt number when the temperature dependence of the physical

property was included in the solution procedure.

Ha et al. [100] defined the governing equations in generalized coordinates

in order to overcome grid generation difficulties. They used the commercial

finite volume code FLUENT to solve these equations assuming a laminar 2D

steady flow model. However, their grid structure produced a coarse distorted

mesh between the louvers, where high velocity gradients are expected, thus

significantly reducing the numerical accuracy of their model. Nevertheless,

the authors confirmed the flow directing properties of the louver fin. They

also observed a developing flow regime at the inlet with flow separation at

both leading and trailing edges of the louvers. Calculated friction factor and

Nusselt number results were reported but not validated experimentally.

Atkinson et al. [111] and Drakulic et al. [112] reported a novel automatic

grid generator that produced 2D grids for any louver fin geometry from

parametric input of the fin data. The resulting mesh had a block structure

with three different types of blocks corresponding to the inlet region of the

fin, the individual louvers, and the turnaround region. These blocks were

divided into a number of quadrilateral cells. Nearly all the cells were

rectangular, giving maximum possible numerical accuracy. With grid lines

parallel and normal to the louver, the boundary layer profiles and integral

parameters could easily be extracted.

Two of the most comprehensive numerical studies of louver fin character-

istics are by Drakulic [113] and Atkinson et al. [114]. In these studies, 2D

and 3D numerical modeling was performed for a number of experimentally

tested louver fins using the automatic grid generator described earlier. These

louver fins were flat fin type used in a flat tube and flat fin construction (seeFig. 9b). The STAR-CD CFD code was used for the analysis for the

Reynolds number Rel range of 50—3200 for 2D and Rel � 100, 400, and

1600 for 3D. In the numerical modeling, the computation domain included

upstream and downstream region each as l and the louvered fin had two

banks of louvers as shown in Fig. 5a with each bank having 5, 10, 13, or 19


Fig. 12. A comparison of friction factor and Stanton number experimental results with 2D

and 3D numerical results of multilouver fins, having p�� 2.17 mm, l� 1.1 mm, and 1! � 22°:

(a) p�� 8 mm and (b) p

�� 14 mm [113].

full louvers. A constant wall-temperature boundary condition was applied

on the primary surface. A conjugate model of thermal conduction through

the fin and the convection from its surface was used. This is because the

temperature distribution in the fin and the thermal boundary layer on the

tube wall result in complex temperature profiles that cannot be predicted

using 2D models and constant fin wall temperature. Time-dependent nu-

merical solutions of the louver fin clearly showed vortex shedding from the

trailing edges of the first two louvers in the upstream and downstream

banks, resulting in a rather long unsteady wake behind these louvers.

Hot-wire measurements of local velocity distributions in louver fins by

Antoniou et al. [115] had showed the same rms values of velocity fluctu-

ations, but the flow was wrongly interpreted as being turbulent. The

numerical results clearly showed that this is unsteady laminar flow. The

mean Nusselt number curves computed from the time-dependent solutions

showed the experimentally observed flattening at high Reynolds numbers.

Numerical predictions of local and mean flow and heat transfer par-

ameters were compared with the experimental measurements of Achaichia

[116] and Antoniou [117]. The results clearly showed that although

reasonable predictions of the overall friction factor could be obtained, the

Stanton number was overpredicted. For two flat multilouver fin geometries,

experimental results and 2D and 3D numerical computations are shown in

Fig. 12 [113]. Reviewing these figures, it can be seen that while the friction


factors agree with experimental values within 0—10%, the computed Stanton

numbers are about 80% higher compared to experimental values at

Rel � 400. The results presented in Fig. 12 are correct and supersede those

presented in Fig. 10 of [114]. In the numerical analysis, both the air and

wall temperatures vary in 3D computations. To obtain the Stanton number

from the heat flux, one needs to base it on a difference between the wall

(primary surface) temperature and the bulk air temperature. The commer-

cial software does not compute the bulk temperature, only the detailed

temperature distribution in the computational domain. Hence, the mean

temperature difference for the heat transfer coefficient determination was

based on the log-mean temperature difference, where the wall temperature

and the air inlet temperatures are known and the outlet bulk temperature

of air was computed from the numerical results. Since the heat transfer

coefficient based on the LMTD would be different from that based on the

wall temperature minus the bulk air temperature, this may be one reason

for a large discrepancy in the computed versus measured values of St.However, it cannot account for the large discrepancy; the reasons are not

clear, as was the case for the OSF surface [90].

Most of the previous work on multilouver fin geometry is ether experi-

mental or numerical. Only a few studies have combined experimental work

on model-scale and full-scale heat exchangers together with the CFD studies

on the model scale fin geometry and the flow visualization setup with a

number of fins. Beamer et al. [118] reported a study on full-scale heat

exchanger performance testing and flow visualization experiments, and a 2D

CFD study on one fin (see Fig. 5a) with a periodic boundary condition, and

six- and 12-fin geometries to duplicate flow visualization setup and results.

The geometric dimensions used in the 2D study of one fin were the same as

those of the actual heat exchanger tested. The CFD domain used for one fin

(the central fin in Fig. 5a) was extended upstream and downstream of the

fin by 20% of the fin length (L�

in Fig. 1b) to take into account the flow

adjustment at louver inlet and exit margins (sections). A known flow rate

was applied at the inlet boundary, and pressure boundary conditions were

prescribed at the outlet boundary. Periodic boundary conditions were

applied on two sides of the complete computational domain (i.e., 1.4 L�);

symmetric plane boundary conditions were applied on the top and bottom

control volume surfaces (perpendicular to the plane of the paper in Fig. 5a)because of the finite volume code. Typically 7000 to 27,000 hexahedral cells

were used in a single array for flow calculations. Blended upwind differenc-

ing with a blending factor of 0.5 was used for the convective differencing

scheme. The results obtained from the 2D CFD study include flow develop-

ment and alignment with the leading bank of louvers, reversal at the center

rib, development and alignment with the trailing banks of louvers, and flow


Fig. 13. A comparison of computational and experimental results for flow efficiency of

multilouver fins [118].

exit from the fin in a direction parallel to the longitudinal axis. Evidence of

flow separation at the leading edge and vorticity in the wake behind the

trailing edge of the louvers was observed in the CFD vorticity plots (not

shown). The CFD study on the flow visualization setup for 6 and 12 fins

duplicated the observed flow phenomena and indicated that even a 12-fin

(note that there are 3 fins shown in Fig. 5a) arrangement does not eliminate

the wall effects in flow through louver fins in a flow visualization setup. A

comparison of flow efficiency of their study [118] with those of Webb and

Trauger [99] and Cowell et al. [2, 101] is shown in Fig. 13. Note that

Achaichia and Cowell [101] did not employ a developing flow region before

the beginning of each louver fin in their CFD study. As a result, they

overestimated flow efficiency as shown in Fig. 13. From this figure, it is

found that the CFD results [118] and flow visualization measurements

show good agreement. Both display an easily discernible knee below which

the flow efficiency drops off rapidly. The data of Webb and Trauger [99]

show a similar trend, except that the value of Re for the knee and flow

efficiency are considerably higher. This rapid dropoff of the flow efficiency

at low Re is attributable to the thickening of the boundary layer between

the louvers, which causes more fluid to flow toward the duct flow region.

Thus, the boundary layer growth along the louvers would reduce the heat

transfer coefficient (Nu or j factor). And Beamer et al. [118] found a good


Fig. 14. Inclined louvers (a counterpart of Fig. 5a).

correlation between the dropoff in the measured j factors in a full-scale heat

exchanger and the knee for dropoff in the flow efficiency of Fig. 13 for

Re& 150. Thus, they showed an excellent agreement among the CFD study,

flow visualization, and full-scale testing. The reasons for the performance

behavior were explained and an idea for further improvement of the fin

geometry was suggested.

1. Concluding Remarks on L ouver Fin Performance/Analysis

The preceding survey shows that although the performance of louver fins

seems to be reasonably well understood, a general correlation for the

accurate quantitative prediction of their performance is not available. Also

at present, no modeling is available for 3D corrugated louver fin geometry

(Fig. 1b). Numerical modeling can be used to provide valuable information

on the complex behavior of the flow and heat transfer of these surfaces.

However, careful grid generation strategy must be used to accurately predict

the details of both flow and thermal boundary layers on the fin surfaces.

Two-dimensional models provide a fast tool for the assessment of the

relative performance of different geometries. Complete 3D conjugate models

are necessary if performance data are to be predicted accurately. Laminar

time-dependent models, although computationally demanding, can provide

further detailed understanding of some of the unsteady nature of the flow

over the louvers. At this stage, it is essential to confirm with precise

experimentation that whether the flow through the louvers is laminar

unsteady or low Reynolds number turbulent, both from the performance

enhancement and numerical analysis points of view. The best potential

future benefit of numerical modeling of these surfaces is in providing

fundamental understanding of the performance of two promising variants of

the basic configurations. The first is inclined louver fins (which acts like an

offset strip fin; see Fig. 14, Tanaka et al. [120], and Suzuki et al. [121]),


which offer a significant improvement in the ratio of heat transfer to

pressure drop. A second area of potential performance improvement stems

from the recognition that the first few louvers in an array provide inferior

performance. The possibility of varying louver angle within an array

suggests itself as an area worthy of further consideration. Experimental

studies of these innovations are prohibitively expensive, but they lend

themselves well to numerical analysis.

C. Wavy Channels

1. Corrugated Wavy Channels

Considerable amount of numerical investigation has been conducted on

this channel geometry used in a plate-fin exchanger as well as in a plate

exchanger with the chevron angle of 90°. The numerical methods, operating

parameters and the geometries studied are given in Table III. A reference

book by Sunden and Faghri [122] provides more details on the numerical

methods and analysis of wavy channels discussed later and other selected

compact heat exchangers.

Asako and Faghri [123] developed a solution methodology for laminar

flow and heat transfer in a corrugated duct. A finite volume scheme was

developed to predict fully developed flow, heat transfer coefficients, and

friction factors in a corrugated channel. The basic method was an algebraic

nonorthogonal coordinate transformation, which mapped the corrugated

channel into a rectangular domain. The governing equations of continuity,

momentum, and energy were solved assuming constant thermophysical

properties and excluding natural convection effects. The details on the

transformation of the conservation equations are provided by Faghri et al.[124]. The boundary conditions used were constant wall temperature and

periodic flow at the inlet and outlet of the domain. As the flow was assumed

to be laminar, no turbulence model was used. Grid size effects were studied

and the maximum change in the Reynolds and Nusselt numbers, between a

18 34 mesh and 26 50 fine mesh, were within 3 and 5%. The numerical

results were compared with data from the open literature, but as the Re

range does not coincide, only qualitative agreement can be found. For the

range of the Reynolds number studied (100&Re �& 1000), the numerical

results for the friction factor indicate transition from laminar to turbulent

flow regime. The slope of the friction factor vs Reynolds number curve increa-

ses from approximately �0.5 to almost 0 at Re �

� 1000. This latter value

is representative of fully developed turbulent flow in a rough duct. For heat

transfer, in most of the cases studied, the Nusselt number increases with the

Reynolds number. These results indicate that for wavy corrugated channels,


TABLE III

Summary of Numerical Methods, Operating Conditions, and Geometries of Corrugated Wavy Channels

Boundary Reynolds Prandtl Pitch/height

Author Method Model Grid condition Inlet conditions number number ratio Validation

Asako and Faghri Finite volume Laminar Transformed Constant wall Fully developed 100—1500 0.7, 4, 1—4 Pressure drop,

[123] Cartesian temperature and 8 Nusselt number

Xin and Tao Finite difference Laminar Polar-Cartesian Constant wall Fully developed 100—1000 0.7 3—6 No validation

[128] temperature

Hugonnot [9] Finite difference Laminar Cartesian Adiabatic Fully developed 150 —10,000 7 3.33 Pressure drop,

k—� visualization

Yang et al. [126] Finite volume Low-Re Transformed Constant wall Fully developed 100—2500 0.7 2.5—6 Pressure drop,

model Cartesian temperature visualization

Ergin et al. [130] Finite volume k—� Transformed Adiabatic Fully developed 500—7000 0.7 1.4 Local velocities,

Cartesian velocity

fluctuations

Ergin et al. [132] Finite volume Low-Re Transformed Adiabatic Fully developed 500—7000 0.7 1.4 Local velocities,

model Cartesian velocity

fluctuations

Kouidry [73] Finite volume DNS Curvilinear Adiabatic Fully developed 5000—10,000 7 3.33 Local velocities,

developing flow velocity

fluctuations

McNab et al. Finite volume 3D: 30500 Cartesian Constant wall Fully developed 250—4000 0.7 3—60 Pressure drop,

[137] meshes temperature Nusselt number

Tochon and Finite volume DNS Curvilinear Constant wall Developing flow 5000—10,000 7 3.33 Pressure drop,

Mercier [139] temperature Nusselt number

417

the transition from purely laminar to unsteady laminar and to turbulent

flow occurs at low Reynolds numbers and that the channel geometry has a

strong effect on the transition.

Asako et al. [125] assessed the heat transfer and pressure drop character-

istics of a similar corrugated duct with rounded corners. Computations were

carried out in the Reynolds number range 100�Re �� 1000 for several

geometric configurations. It was determined that the change in heat transfer

rates caused by rounding the corners of the sharp cornered corrugated duct

depended on the specific flow conditions, geometry, and performance

constraints.

To address the use of a laminar model when turbulence is developing,

Yang et al. [126] have extended the work of Asako and Faghri [123] by

using a low Reynolds number turbulence model. The same numerical

method as Asako and Faghri [123] was used, but the source terms in the

conservation equations were modified to take into account turbulent effects

in the diffusion coefficients. The turbulence was modeled according to the

Lam and Bremhorst [58] model, which is a low Reynolds number form of

the k—� formulation. The details of the equations are given in Yang et al.[126]. In parallel with this numerical work, experiments were performed in

order to validate the model. The test section was a sharp cornered

corrugated channel. The comparison was limited to the laminar model of

Asako and Faghri [123], as the low Reynolds number turbulent flow model

does not allow the modeling of geometries with sharp corners. In term of

the flow pattern, the size of the recirculation areas was well predicted. It

must be noted, however, that the recirculation region reaches a maximum

size at a Reynolds number of 500; the size decreases because of the high

diffusion in turbulent flow for higher Reynolds numbers. The same trends

were observed by Hugonnot [9] in a wavy channel. For friction factors for

a sharp edge channel, there is a good agreement with the laminar model for

Reynolds numbers up to 500, beyond which the friction factors are overes-

timated. Below this transition Reynolds number and for various operating

and geometric parameters, the predicted friction factors for the laminar and

turbulent models are the same. This suggests that the low Reynolds number

turbulent flow model is stable even for laminar flows. Beyond the transition

Reynolds number, the friction factors predicted by the turbulent model are

higher than those predicted by the laminar model. For the Nusselt numbers,

the same trends are observed. Yang et al. [126] claim that the low Reynolds

number turbulent flow model can be used to predict friction factors and

Nusselt number, but no comparison with experimental data is given.

Garg and Maji [127] applied a finite difference scheme (SIMPLEC

algorithm) to laminar flow through a wavy corrugated channel. They

presented results for both developing and fully developed flow for Reynolds


numbers ranging from 100 to 500. The channel dimensions were varied, and

for one chosen configuration, the detailed behavior of velocity, pressure, and

enthalpy for developing laminar flow were presented.

Xin and Tao [128] performed numerical simulation in wavy channel by

applying a laminar model and using a finite difference method. The domain

was gridded by a combination of polar and Cartesian coordinates. In the

bends, a polar coordinate was applied, and a Cartesian coordinate in the

straight duct between two bends. A fully developed flow was obtained by

periodic boundary conditions at the inlet and outlet of the domain and a

constant wall temperature boundary condition. Several geometric and

operating parameters were studied, but no validation with experimental

results was reported. The channel studied was quite similar to the one used

by Berieziat [129], and comparisons can be made through the streamline

velocity profiles. Xin and Tao [128] claimed that the relative strength and

size of the recirculation zone increase with the Reynolds number. This result

is in partial disagreement with the experimental data of Bereiziat [129] and

the flow visualizations of Hugonnot [9], which indicate that the recircula-

tion zones increase up to a Reynolds number of 200, then vortices are

generated downstream of the corrugation. For higher Reynolds numbers

(Re �� 350), the flow becomes unstable and turbulence is developing.

According to Bereiziat [129], the flow becomes fully developed turbulent for

Reynolds numbers above 2000. Applying a laminar model for flows devel-

oping turbulence leads to inaccurate predictions, and the conclusions of Xin

and Tao are only valid in the low Reynolds number range (Re �& 350).

Hugonnot [9] performed numerical simulations in a 2D corrugated

channel using a laminar model at Re �

� 150 and a k—� turbulence model

at Re �

� 10,000. The choice of the model was dictated by the previous

observations on the flow structure. A finite difference method was applied

and the flow was considered adiabatic. The results were compared with flow

visualizations and pressure drop measurements in a similar geometry. The

size and the position of the recirculation zones were well predicted at both

Re �

� 150 and Re �

� 10,000. The pressure drop data were compared at

Re �

� 10,000, and a good agreement was found for the overall pressure

drop, but differences can be noticed on the pressure profile essentially in the

recirculation zone. This result is not surprising, as it is well known that

conventional k—� models are not accurate in separated flows, and even if the

flow pattern is qualitatively correct, the wall shear stress can be inaccurately

predicted. The comparison of the Hugonnot [9] and Xin and Tao [128]

results for low Reynolds numbers is only qualitative and similar.

Ergin et al. [130] applied the method developed by Faghri and co-

workers [123, 125, 126] to the study of the effect of interwall spacing on

turbulent flow in a sharp-edged corrugated channel. The flow was consider-


ed turbulent (500&Re �& 7000) and the k—� model was adopted for

turbulent closure. Experiments were also conducted to validate the numeri-

cal model. At Re �

� 2000, the flow characteristics given by the model were

in agreement with the visualization experiments and with previous experi-

mental studies [126, 131]. The comparison of the local velocity profiles and

turbulent kinetic energy revealed that the proposed method gave accurate

prediction of the velocity profiles, but the prediction of the turbulent kinetic

energy was relatively poor. As a result, the friction factors were underpredic-

ted by approximately 33%.

Ergin et al. [132] extended the previous work by applying the Lam—Bremhorst low Reynolds number turbulence model. For a Reynolds number

of 2000, the standard k—� model and the low-Re turbulence model give

similar predictions of the flow field. The comparisons of the predicted

friction factors showed a good agreement with the low-Re turbulence model

in the range 500&Re �& 3000, whereas the standard k—� model is more

accurate for higher Reynolds numbers (Re �� 3000). As a consequence, the

standard and low-Re turbulence model cannot be applied in the whole range

of the Reynolds number, and this implies selecting a priori the model for the

range of the Reynolds number.

To overcome the problem of simulations of unstable turbulent flow,

Kouidry [73] performed direct numerical simulation of turbulent flow in a

corrugated channel. The geometry was identical to the one of Hugonnot [9]

and Bereiziat et al. [133]. In parallel to this numerical work, local velocity

measurements by laser anemometry were performed. The DNS method

requires a very fine mesh in order to reproduce the smallest turbulent

structures according to the Kolmogorov scale. Based on experimental data,

the smallest turbulent structure was about 95 �m, but limitations of the

workstation led to a 269 �m by 212 �m grid (293 95 meshes). As a result,

the smallest turbulent structure was not represented, and the method used

was called pseudo-DNS. The TRIO code based on a finite volume method,

developed by the CEA (French Atomic Commission), was used. The

conservation equations were solved on a control volume with a semi-

implicit time scheme and a fine third-order space scheme (Quick-Sharp).The advantage of DNS methods compared to time average methods (k—�models) is that they are independent of the flow configuration and that local

instantaneous information is available. Developing flow with a flat inlet

velocity profile and periodic flows were studied. The numerical results were

compared to the experiments. For the mean velocity profiles, the agreement

was relatively good in the fully developed flow, but differences linked to the

inlet boundary conditions were observed for the developing flow. The

turbulence intensity was analyzed through the mean square velocity fluctu-

ations. The experimental fluctuations were 30 to 50% around the mean


velocity, while the numerical simulations gave fluctuations up to 150%.

Kouidry [73] claimed that these differences came from the number of time

steps required for the second-order average and by the fact that the

turbulent dissipation was underestimated by the 2D modeling. Despite these

differences, the flow pattern and the vortices are well predicted in terms of

the size and growth.

Farhanieh and Sunden [134] investigated numerically a three-dimen-

sional fully developed laminar flow in a corrugated square duct. A finite

volume method was applied and the Reynolds number range from 30 to

1000. At constant pressure drop, the grid size effect was studied and

70 18 28 grid points were selected. The numerical model was checked on

a straight square duct, and the predicted friction factor and Nusselt number

were in good agreement with analytical values. A parametric study on the

corrugated duct geometry was performed, but no comparison was given

with experiments.

Asako et al. [135] studied laminar forced convective heat transfer and

fluid flow characteristics in a wavy duct with a trapezoidal cross-section.

The algebraic coordinate transformation (Faghri et al. [124]) was applied

to map the trapezoidal cross-section onto a rectangular one. The numerical

predictions of the friction factors were compared for a fully developed flow

in a straight trapezoidal duct, and the agreement with the data of Shah and

Bhatti [136] was within 0.46%. The numerical simulations showed that the

enhancement of heat transfer and pressure drop due to waviness depends

strongly on the Reynolds number. To optimize this geometry for industrial

applications, the authors suggest studying the effect of the wave length, wave

height and corner angle.

McNab et al. [137] computed the flow over herringbone corrugated

(sharp-cornered wavy) channels, using a commercial STAR-CD CFD code.

A 3D approach was adopted and a laminar model was used for Reynolds

numbers (based on D�) below 1500, and a high-Re k—� model for higher

Reynolds numbers. For Reynolds numbers above 600, difficulties appeared

in obtaining a fully converged solution. The authors suggested that flow

unsteadiness may occur for such low Reynolds numbers inducing pressure

fluctuations. The maximum fluctuation (14%) was observed for a Reynolds

number of 1500, and it was decided not to use time-dependent modeling.

The computed j and f values were compared with the measurements of

Abou-Madi [138] and are in relatively good agreement in the turbulent

regime 17 to 27%, but in laminar regime the differences are 33% for the

friction factor and 54% for the Colburn factor. The authors indicate that for

such low-Re unsteady flow, steady-state modeling may not be appropriate

for low Reynolds number and that the recirculation zone is not well

predicted by the k—� model in turbulent flow.


Tochon and Mercier [139] have extended the work of Kouidry [73] using

an approach based on a large eddy simulation without a subgrid model of

turbulent flow. The advantage of using LES is to produce directly the main

part of coherent structures for turbulent regimes, for a better understanding

of the flow pattern. A curvilinear coordinate system has been adopted and

the refined mesh is 90 �m (or five nondimensional wall units) near the wall

in the spanwise directions. The mesh is nearly constant in the streamwise

direction and the size is 200 �m (or 15 nondimensional wall units). As the

Kolmogorov scale is approximately 95 �m for the considered geometry

(Kouidry [73]), the chosen mesh seems to be sufficient to describe the main

part of the dissipative structures: no subgrid is needed. The mechanism of

eddy generation and heat transfer have been analyzed in the developing flow

(Figs. 15 and 16). The results are qualitatively and quantitatively compared

with experimental data obtained on a specific experimental rig and from

open literature, and the differences in the heat transfer coefficient and

friction factor do not exceed 10%.

The application of numerical modeling of a number of different high-

performance heat transfer surfaces has been presented by Atkinson et al.[140]. The 2D and 3D models for louvered fins showed that accurate

calculations of overall heat transfer could only be achieved by using 3D

models, which incorporate the effects of tube surface area and fin resistance.

Accurate predictions of flow and heat transfer over corrugated fins in the

laminar flow region were also presented. However, they showed that the

high-Re k—� model with log-law wall functions gave poor predictions of heat

transfer for the turbulent-flow regime. They suggested that low-Re forms of

the model would be more accurate in this regime.

2. Wavy Furrowed Channels

Sobey [141] conducted a numerical study of flow through furrowed

channels in conjunction with a related experimental investigation [142].

Sobey calculated the flow patterns obtained using both steady and pulsatile

inflow. The effects of varying dimensional parameters and Reynolds number

were examined. Furthermore, the flow structures that occur in a channel

with arc-shaped walls were compared with the patterns induced by

sinusoidal curved wavy walls. An oscillatory flow in various types of wavy

passages was examined further in subsequent studies performed by Sobey

[143, 144].

Sparrow and Prata [145] examined a family of periodic ducts, using both

numerical and experimental methods. The periodic duct is a tube consisting

of a succession of alternately converging and diverging conical sections.

Numerical simulations were carried out for fully developed laminar flow in


Fig. 15. Numerical results for time-dependent evolution of vorticity in a 2D wavy channel

[139].

the Reynolds number range Re� 100—1000, for various duct configur-

ations. The resulting data indicated that the periodic furrowed tube is not

conducive to heat transfer enhancement for steady laminar flow.

Garg and Maji [127] used a finite difference method to solve the

governing equations for steady laminar flow and heat transfer in a furrowed

wavy channel. Calculations were performed using various wall amplitudes

for Re� 100—500. Both the developing and fully developed flow regions

were analyzed. The local Nusselt number was observed to fluctuate

sinusoidally in the fully developed region. Moreover, the Nusselt number


Fig. 16. Numerical results for time-dependent evolution of temperature in a 2D wavy

channel [139].

increased with the Reynolds number, unlike a constant value for laminar

flow through a straight channel.

Blancher et al. [146] analyzed convective heat transfer by a spectral

method for an expanding vortex in a wavy furrowed channel. The flow is

assumed laminar fully developed with the Reynolds number up to 200. An

algebraic transformation of the coordinate was applied to reduce the

physical domain to a rectangular computational domain. The governing

equations were solved introducing the stream functions, with the details

provided on the mathematical models and transformations. The method is

limited to Reynolds numbers above 200, as the flow becomes unsteady for

higher values. The numerical results (flow pattern, wall shear stress, and

vorticity profiles), compared to those of Sobey [141], are in good agreement.

Guzman and Amon [147] reported the transition to chaos for fully

developed flow in furrowed wavy channels. Specifically, the Ruelle—Takens—Newhouse scenario for the onset of chaos was verified using direct


numerical simulations. The results were illustrated for various Reynolds

numbers using velocity time signals, Fourier power spectra, and phase space

trajectories.

Guzman and Amon [148] extended their previous study by using DNS

to calculate dynamical system parameters. Dynamical system techniques,

such as time-delay reconstructions of pseudophase spaces, Poincare maps,

autocorrelation functions, fractal dimensions, and Eulerian Lyapunov expo-

nents, were employed to characterize laminar, transitional, and chaotic flow

regimes. Also, 3D simulations were performed to determine the effect of the

spanwise direction on the route of transition to chaos. They have shown that

the transition to chaos takes place at Re� 500 following a quasi-periodic

frequency locking route.

Wang and Vanka [149] applied a finite volume method to the study of

convective heat transfer in wavy furrowed channels. To solve the governing

equations, an accurate numerical scheme was applied on a 256 128

calculation domain. The simulations were performed for both steady and

unsteady regimes. The flow was observed steady until Re$ 180. Afterward,

self-sustained oscillations appeared and led to destabilization of the laminar

flow. Physical interpretation of the transition was given and the flow

structure was analyzed for Reynolds numbers up to 1000. The friction factor

calculations, compared to the experimental results of Nishimura et al. [150],

slightly underpredicted the measurements but showed a good agreement

with experimental trend. No comparison with heat transfer experiments was

given in the paper.

D. Chevron Trough Plates

Chevron plates with a corrugation angle of 90° can be considered as a 2D

wavy corrugated geometry. Numerous investigations have been published

on this specific geometry and are discussed in the previous section. In most

of the cases, the corrugation angles are between 30° and 60° in commercial

chevron plates. As observed from the experimental work, the flow remains

mainly in the furrow of the corrugation for the corrugation angle of 30°;

whereas for higher corrugation angles (60°), the flow is almost 3D and

highly turbulent even for very low Reynolds numbers (Re �

� 200). Flow

simulation in cross corrugated structures has been reported by Fodemsky

[151], Ciofalo et al. [152], and Hessami [153]. The numerical and flow

conditions are summarized in Table IV.

Fodemsky [151] used a standard k—� model to predict flow and heat

transfer in a corrugated channel. A body fitted Cartesian grid was applied

to model the channel geometry, and particular attention was paid to the

grid in the corners where the upper and lower plates were in contact. The

results were provided for corrugation angles between 15° and 20°, and the


comparison with experimental results was only qualitative. The simulation

provided the heat transfer coefficient field and outlined large heterogeneities.

The highest values were located at the top of the corrugations where the

cross section is minimized. This result is in agreement with the data of

Gaiser and Kottke [11]. As for the 2D modeling, k—� models are not

accurate to predict separated flow where recirculations zones exist, and the

results are only qualitative.

The study of Ciofalo et al. [152] is more extensive. Both numerical and

experimental work was performed and several numerical methods were

applied on a large range of operating and geometric conditions (see Table

IV). Similar to Fodemsky’s work, a body-fitted grid was used to model the

complex geometry. The same grid was used for all the models, except for the

standard k—� model, where the wall function must be respected. This implies

that the center of the control volume near the wall lies far out from the

viscous sublayer. Ciofalo et al. [152] used a conventional definition, and the

thickness of the viscous sublayer was based on plain tube correlations. This

assumption is incorrect because the thickness of the viscous sublayer is not

constant, a result of the large recirculation area. The results were compared

to pressure drop measurements of Focke and co-workers [7, 10]. Laminar

and k—� computations were unsatisfactory, as they respectively underpredic-

ted and overpredicted the friction factors for the complete range of Reynolds

numbers investigated (1000&Re& 5000). The best agreement with the

experimental data was obtained with the low-Re turbulence model or LES;

the difference was about �50% for the complete range of Reynolds numbers

and for various corrugation angles. The agreement was better for Reynolds

numbers over 2000 and for corrugation angles between 15° and 30°. For the

average Nusselt number, the laminar and standard k—� models are not

satisfactory and the best agreement was found with the low-Re turbulence

model and LES. These two models underpredicted the average Nusselt

number by 20%, but the corrugation angle effect was well taken into

account. For the local flow structure and heat transfer, the comparison was

made with experiments performed by Stasiek et al. [14] on a similar

corrugated channel. The Nusselt number vs Re experimental results were

reasonably well predicted by the low-Re turbulence model and LES,

whereas the laminar and k—� models were not satisfactory. The best

agreement was obtained with LES. No quantitative comparison of the flow

structure was obtained and the accuracy of the different models was not fully

validated. Nevertheless, it appears that LES allowed reproducing the flow

structure according to Ciofalo et al. [152]. Finally, Ciofalo et al. compared

LES and DNS and observed that for their configuration, the DNS models

did not improve the results significantly.

Hessami [153] performed a three-dimensional study of heat transfer and


TABLE IV

Summary of Numerical Methods, Operating Conditions, and Geometries of Chevron Trough Plates

Boundary Reynolds Prandtl Corrugation

Author Method Model Grid condition Inlet conditions number number Pitch/height angle Validation

Fodemsky Finite difference k—� Body fitted Constant wall Fully developed 1600—3000 0.7 2 15°—20° No

[151] Cartesian 12 temperature

Ciofalo et al. Finite volume Laminar, Body fitted Constant wall Fully developed 10—10� 0.7 2—4 15°—75° Nu, f

[152] k—� Cartesian 24 temperature developing flow Local Nu

Low—Re

LES Body fitted Constant wall Fully developed

Cartesian 32 temperature developing flow

Hessami Finite volume Laminar, Body fitted Constant wall Developed 200 —6000 7 3.85—5.86 30°, 45°, 60° No

[153] k—� Cartesian temperature

Low—Re and heat flux

Blomerius Finite volume DNS 0.1 0.1 0.05 Constant wall Fully developed 150—2000 0.7 3.5, 4.5, 5 45° Nu and f

et al. [156] mm temperature

Sunden Finite volume laminar 25 25 25 Constant wall Fully developed 400—10000 4.33 and 3 0°, 22.5°, and Nu and f

[157] Low—Re Cartesian temperature 12.8 45°

RNG-k—� body fitted

427

pressure drop in a cross-corrugated heat exchanger, using the commercial

CFD software CFX 4.1 (a version of FLOW3D software). Three models

were considered: laminar, standard k—� and low-Re k—� turbulent models.

A unit cell was represented by a 22 16 22 grid. Preliminary investigations

showed additional blocks or cells were required at the inlet and outlet of the

unit cell to obtain a stable solution. The flow boundary condition was a fully

developed velocity profile, but no precision was given on the establishment

flow regime. Three corrugation angles (30°, 45° and 60°) and three pitch-to-

height ratios (3.85, 4.55, and 5.56) were considered; the Reynolds numbers

were covered up to 6000. For a corrugation angle of 60° (hard plates),comparison of the three models showed large differences. For heat transfer,

both the laminar and standard k—� models gave a very low dependency of

the Nusselt number with the Reynolds number (Nu�Re��); in contrast,

the low-Re turbulence model was very sensitive (Nu� Re��). These de-

pendencies with the Reynolds number were not physical and may have come

from the grid or boundary conditions used in this study. The trends on

predicted Nu and f were not in agreement with the experimental work of

Focke et al. [154] and Thonon et al. [12], and neither the Reynolds number

effect nor the corrugation angle effect could be predicted by the model used

by Hessami [153]. Other geometric parameters were studied by Hessami,

but the main result of this study was that the boundary conditions and the

numerical model have a large influence on the results, and that great care

should be taken in modeling such complex geometries.

Sawyers et al. [155] studied numerically, 2D (sinusoidal), and 3D (egg-

carton, i.e., sinusoidal in the transverse direction as well) corrugated (wavy)channels. To avoid unsteady flow, the study was limited to low Reynolds

numbers (Re �& 250). Both algebraic and numerical methods were applied

to predict pathlines, pressure drop, and heat transfer. The authors analyzed

several geometric configurations, but no comparison with experimental

results was provided.

Blomerius et al. [156] presented numerical results on 3D flow field and

heat transfer in a corrugated channel. The Navier—Stokes were directly

solved by the code, using an implicit—explicit scheme for the convective

fluxes. A reinjection procedure was adopted to obtain a fully developed flow.

Depending on the geometry, 80,000 to 150,000 grid points were used. The

sensitivity of the results (heat transfer and pressure drop) with the gridding

was up to 10% at Re� 1500. The corrugation angle was fixed at 45°, and

the pitch-to-height ratio was 3.5, 4.5, and 5. This latter value was selected

for a comparison with the experimental results of Gaiser and Kottke [11].

For Reynolds numbers below 180—270 (depending on the pitch-to-height

ratio), the flow was fully laminar; for higher Reynolds numbers, the flow

became time-periodic self-sustained oscillatory, requiring time-dependent


calculations. For high Reynolds numbers, a time averaging procedure was

adopted to estimate heat transfer and pressure. At Re� 2000, the deviation

between the predicted and experimental results was respectively 5 and 8%

for the Nusselt number and friction factor.

Sunden [157] analyzed the flow structure and thermal and hydraulic

performances of plate and frame heat exchangers using CFD. The geometry

studied was representative of an industrial plate heat exchanger with two

different corrugation angles (22.5° and 45°). A single unit cell was considered

(volume between four contact points of chevron plates) to provide informa-

tion on friction factors and Nusselt numbers. A steady laminar flow model

was applied for Reynolds numbers up to 2000 and a low-Reynolds k—�or an RNG—k—� model with wall functions for turbulent flows. As outlined

in Section IV, these turbulent models are more accurate for low Reynolds

number unsteady flows or turbulent flows. A 3D body fitted mesh was used,

and great care was paid to model the contact points. To avoid singularities,

the contact line between the upper and lower plates was replaced by a

surface contact. For one of the geometries modeled, the numerical results

were compared with experiments. The predicted Nusselt number was

underpredicted by at most 25% while the friction factor was underpredicted

by 17—40%. The author suggested using a more advanced turbulence model

to get a higher accuracy.

1. L ocal Analysis of Flow and Heat Transfer Phenomenain Corrugated Channels

Based on the numerical analysis, local information on the flow structure

is now available and allows analyzing and qualitatively characterizing heat

transfer mechanisms in complex geometries. The experimental studies (seeSection II, B) have shown complex flow patterns and transition from steady

laminar to unsteady laminar and turbulent flow at low Reynolds numbers.

The numerical studies have confirmed these observations, but the mechan-

ism of eddy creation can only be more precisely defined using unsteady

numerical models. The 2D studies were performed by Tochon and Mercier

[139], and 3D studies by Ciafolo et al. [152] and Blomerius et al. [156]. The

information gained from these studies is discussed next.

For the 2D geometry, near the entrance and at each corrugation, the flow

separates from the wall and generates a large recirculation zone. Because of

turbulent instabilities, this zone grows larger and larger and then breaks

suddenly in two parts. One part creates a vortex, which is released in the

flow (eddy no. 1 in Fig. 15) in the downward direction, as the other part

remains in the protected region. Interactions between the rotating eddies in

the center part lead to a pairing (eddy nos. 4 and 5 in Fig. 15). The same


process is repeated. When detached from the wall, eddies are convected in

the flow, under the influence of other eddies. They can keep their coherent

structure for several microseconds. Their lifetime has been found to be a

little longer numerically than in the experiment. This discrepancy can be

explained by the limitation of the two-dimensional model, although turbu-

lence is three-dimensional in nature. This 3D character is especially true in

the region of an adverse pressure gradient, where partial disorganization of

the coherent structures seems to occur in the experimental visualization.

Nevertheless, the main mechanisms of eddy creation and motion are very

similar to those observed experimentally. This alternative generation of

vortices on the lower and upper walls of the domain of calculation liberates

coherent eddies, with either clockwise or anticlockwise rotational motion.

As a result, it is also apparent that this geometry is able to ensure good

turbulent mixing of the fluid after only a few corrugations. Concerning the

thermal aspects (Fig. 16), the alternating vortices lead to a uniform tempera-

ture inside the channel by mixing the warm liquid of the recirculating areas

with the main flow. Indeed, local investigations show that the heat transfer

coefficient is low inside the recirculating areas by means of the conduction

process and is high at the reattachment point. So, because of the turbulent

behavior of the flow, the hot pocket oscillates, becomes unstable, and is

convected to the bulk flow. These new eddies interact with each other and

generate large coherent structures that create a fully developed turbulent

flow. So the vortex shedding from the warm area near the wall to the core

is the main mechanism for heat transfer.

For 3D geometries, the flow is more complex to analyze, as the 3D effects

are present even for low corrugation angles. Ciafalo et al. [152] have

investigated a cross-corrugated channel, using an LES model, with the

corrugation angle between 18.5° and 30°. For these typical angles, according

to Focke and Knibbe [10], a furrow flow should exist, and the mixing

intensity should increase with the Reynolds number. From the velocity

fluctuations, a Strouhal number can be deduced (Sr�F t, where F is the

frequency and t the time necessary to cross a unitary cell at the average

velocity U). For all values of corrugation angles (18.5° to 30°) and Reynolds

numbers (780 to 4250), a constant value of Sr� 3 was found. A detailed

examination of the instantaneous and fluctuating flow fields (at Re� 2450)indicated that the eddies were highly three-dimensional in the central part

of the cell, having their largest component in the vertical direction (normal

to the channel). These eddies were generated by mixing of two furrow flows

(upper and lower plates). The intensity of the fluctuations was found to

increase with both Reynolds number and corrugation angle.

Blomerius et al. [156] investigated a corrugated channel with a 45°


corrugation angle. According to previous visualization experiments, the flow

structure is strongly affected by the Reynolds number and becomes highly

three-dimensional. A Fourier analysis was performed on the u-component

of the velocity (main flow direction). For Reynolds numbers of 255 and 425,

a dominant frequency with corresponding Strouhal numbers of 0.5 and 1.3

was found. For a higher Reynolds number based on the hydraulic diameter

(Re� 1700), no dominant frequency could be detected and the flow was

aperiodic, which is the characteristic of turbulent flow. For a low Reynolds

number (Re� 180), the midplane shear layer was relatively stable, whereas

for a higher Reynolds number (Re� 1800) considerable mixing between

the two streams was noticed. The maximum velocities were nearly twice

the average fluid velocity. This higher mixing at Re� 1800 led to a much

more homogeneous temperature field over the whole cross-section. In the

region just downstream from the contact point, the fluid temperature was

almost the same as the wall temperature, indicating a stagnant zone. This

size of this region was strongly dependent on the Reynolds number. The

foregoing information is fundamental for selecting the most appropriate

geometry for given process conditions or to develop new geometries that

could limit the size of these stagnant and recirculation areas. In the food

industry, pasteurization and sterilization are often achieved in plate heat

exchangers, and it is fundamental to have an equal (homogeneous) residence

time for all fluid particles and no dead zones where fouling could initiate.

The numerical studies have shown that for Reynolds numbers higher than

1500 and for corrugation angles higher than 45°, the recirculation zones are

limited in size and the fluid temperature is much more homogeneous.

Furthermore, these channels have a very high local shear stress, which help

mitigating fouling.

2. Summary of 2D and 3D Modeling of Wavy Channels

and Chevron Trough Plates

For 2D geometries (corrugated wavy channels), several grid schemes have

been applied, but the major difference lies in the numerical method used.

Laminar models are limited to very low Reynolds numbers (Re& 350) as

instabilities occurs for higher Reynolds numbers. Furthermore, in fully

developed turbulent regimes, conventional turbulent models can only give

overall information, as the flow is inaccurately predicted in the recirculation

zones. As a result, to cover the entire range of the Reynolds number and to

be independent of flow patterns, advanced methods that enable prediction

of separated flows must be applied. The low Reynolds number turbulent

flow model [126] appears to constitute progress in the simulation of flows


in complex geometries, but its validity is not established. DNS methods [73]

are interesting, but 2D models underestimate the turbulent diffusion and

efforts must be applied to 3D modeling (i.e., the realistic chevron plates).Concerning 3D modeling, the studies have covered a wide range of

corrugation angles and Reynolds numbers, and the agreement compared to

experimental results (average values of the friction factor and Nusselt

number) is within 10 to 50%. Local heat transfer coefficients and wall shear

stress are still not accurately predicted.

Hessami [153] and Sunden [157] used various turbulence models, but the

accuracy was relatively poor. Nevertheless, the general trends for heat

transfer and pressure drop were well predicted. Ciofalo et al. [152] used

several numerical models and claimed that the low Reynolds number

turbulent flow model or LES gave satisfactory results. The latter method is

computationally more expensive but provides instantaneous information.

Blomerius et al. [156] used a DNS method and claimed a maximum

deviation of 5 and 8%, respectively for Nusselt numbers and friction factors.

These results are encouraging, and advanced turbulent models implemented

in commercial CFD codes should be tested for complex geometries such as

cross-corrugated heat exchangers.

As computational costs will decrease, LES or DNS methods must be

developed, and careful validation with local measurements (velocity and

turbulence intensity) must be performed. Progress in modeling flow and

heat transfer will enable study of complex geometries. More sophisticated

experiments will be required for validation of modeling and numerical

results. A comparison of CFD simulations between researchers, using

different research and commercial codes and techniques on standardized

geometries, and a comparison with validated laboratory and industrial data

must be an objective for CFD to be a design, analysis, and optimization tool

for heat exchangers.

VI. Conclusions

A comprehensive review is made of numerical analysis of some of the

important surface geometries of compact heat exchangers: offset and louver

fin geometries used in plate-fin exchangers, wavy corrugated and furrowed

fins/channels in tube-fin and plate heat exchangers, and chevron plates in

plate heat exchangers. Specific understanding obtained from the numerical

studies of each fin/plate geometry is summarized at the end of each fin/plate

geometry section. Also, some of the known physics of flow from experimen-

tation and flow visualization is presented to further assist in the refinement

in the numerical analysis. Although numerical analysis has progressed


considerably in the past 20 years, most of the numerical analyses reported

are 2D and/or simpler than the real complex flows. Most of the flows

encountered in CHE applications are unsteady, often with flow separation

and recirculation zones, and the Reynolds number range is typically

100—2000 for most compact heat exchanger surfaces. The separation and

reattachment of the flow and vortices in the wake of the fins affect both heat

transfer and pressure drop, and these mechanisms cannot be predicted with

time average modeling (conventional k—� models). Unsteady laminar model

requires a very fine description of the geometry, and computation capacity

limits its application to 2D modeling. There is no clear evidence as yet that

the flows in this Reynolds number range are unsteady laminar or low

Reynolds number turbulent flows, although it is possible that gradually the

unsteady laminar flows associated with the interruptions become low

Reynolds number turbulent flows. Hence, more sophisticated experimenta-

tion is required to determine the flow characteristics. At the same time, RSM

and LES models should be further developed to simulate the local flow and

heat transfer characteristics of compact heat exchanger surfaces. See Section

IV for specific conclusions/recommendations on turbulence models. Since

the art of compact heat exchanger surfaces has reached an asymptotic level,

further enhancement and innovation in those surfaces will come from the

detailed accurate numerical analysis. Numerical analysis is also needed to

study the effects of minor changes in the fin profile (such as those due to the

aging of manufacturing tools). Manufacturers of heat exchangers in the

automotive industry are active in this area, but the numerical methods

proposed in commercial software require some careful validation with

experimental data. A similar conclusion was also reported earlier (which

came to the authors’ attention) by Baggio and Fornasieri [158]. It is a

challenge to numerical analysts to numerically simulate highly complex

flows accurately to provide a basic understanding of flow and heat transfer

phenomena for their exploitation in designing new and improved heat

transfer surfaces.

Acknowledgments

We gratefully acknowledge Mr. J. P. Chevalier of CEA-Grenoble, DTP/GRETh, Grenoble,

France, for his assistance in preparing Section IV on turbulence models. We appreciate very

much a careful and extensive review of our chapter by Dr. F. Ladeinde of Thaerocomp

Technical Corp of Stony Brook, NY, USA. We are also thankful to Prof. W. Rodi of the

University of Karlsruhe, Germany, Dr. S. Sazhin of the University of Brighton, UK, Prof. G.

Biswas of the Indian Institute of Technology Kanpur, India, and Dr. G. Xi of Daikin Industries,

Inc., Osaka, Japan, for their critical review of Section IV on turbulence models.


Nomenclature

A heat transfer surface area on

one fluid side, m�

A5 constant of Eq. (11), m

A

minimum free flow area on one

fluid side of a heat exchanger,

m�

b spacing between plates for

corrugated fins, mean gap, or

plate spacing in a plate heat

exchanger, m

c�

specific heat of fluid at constant

pressure, J/kg K

D particular derivative:

D

Dt��

��t

� u� · ��, 1/s

D�

equivalent diameter, D��2b, m

D�

hydraulic diameter, D��4V /A

�4AL /A, m

E constant of Eq. (28)F frequency, 1/sF�

structure function defined in

Eq. (42), m�/s�

f Fanning friction factor,

dimensionless

G fluid mass velocity based on

minimum free flow area, G�m5 /A

, kg/m�s

G�

generation of turbulent kinetic

energy due to buoyancy, m�/sG

�generation of turbulent kinetic

energy due to the mean velocity

gradients, m�/sh convective heat transfer

coefficient, W/m�K

j Colburn factor, St Pr� ,dimensionless

K constant of Eq. (28)k kinetic energy of turbulence per

unit mass, J/kg (or m�/s�)l mixing length, m

l offset strip length, louver,

length, or louver pitch, m

m5 fluid mass flow rate, kg/s

Nu Nusselt number, hD�/k,

dimensionless

P mean pressure component of

fluid static pressure, Pa

Pr fluid Prandtl number,

dimensionless

p� fluctuating pressure

component, Pa

p�

fin pitch, m

p�

tube pitch of flat tubes in Fig.

11b, m

q��

velocity scale, m/s

R correction term in Eq. (18),m�/s

Re Reynolds number based on

hydraulic diameter, GD�/�,

dimensionless

Re �

Reynolds number based on the

equivalent diameter, GD�/�,

dimensionless

Rel Reynolds number based on the

interrupted length, Gl/�,

dimensionless

Re�

Reynolds number based on the

mixing length and mean

velocity, �Ul/�, dimensionless

r correlation distance used for

structure function, m

S strain tensor, m�/s

Sr Strouhal number, Sr� Ft��

,

dimensionless

St Stanton number, St� h/Gc!,

dimensionless

s fin spacing, s� p�� (, m

T fluid temperature, K

t time, s

t��

characteristic time, s

�t characteristic time,

characteristic length of

corrugation divided by the bulk

velocity, see Figs. 15 and 16, s

U mean velocity component, m/s

u�, v�, w� fluctuating velocity

components, m/s

u, v, w velocity components, m/s

V void volume on one fluid side,

m

x axial direction, m

Y�

contribution of the fluctuating

dilatation in the dissipation

rate, m�/s


y normal distance from the wall,

m

y5 wall coordinate y(�"/�)� �/v,

dimensionless

Greek Letters

��

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)1! fluid temperature, K

1! 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�

�"

wall shear stress, Pa

� rotation velocity, 1/s

� vorticity, s��

Subscripts

eff effective

i axial direction index

j lateral direction index

k vertical direction index

� related to the dissipation rate

� related to the kinematic

viscosity

Abbreviations

ASM algebraic stress model

CFD computational fluid dynamics

DNS direct numerical simulation

EVM eddy viscosity model

LES large eddy simulation

NLEVM nonlinear eddy viscosity model

OSF offset strip fin

RANS Reynolds averaged numerical

simulation

RNG renormalization group

RSM Reynolds stress model

SGS subgrid scale

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a


AUTHOR INDEX

Numerals in parentheses following the page numbers refer to reference numbers cited in the text.

A

Abbrecht, P. H., 262(22), 292(22), 310(22),354(22)

Abdel-Khalik, S. I. (Chapter Author), 145,

147(12), 150(12; 34; 35), 151(12), 152(12),153(12), 154(12), 155(35), 156(12; 34),157(12), 158(12), 159(12), 160(12),161(12), 162(12), 164(12), 167(34),168(34), 176(35), 177(34; 35), 185(12; 35),187(12), 188(34), 191(35), 193(132; 135;

139), 196(132), 197(132), 198(132; 135),199(132), 200(132), 202(132; 135; 139),203(135; 139), 204(135), 205(132; 135;

139), 210(174), 211(174), 214(174),215(174), 216(174), 218(174), 219(174),224(174), 226(208), 227(208), 230(208),231(208), 233(208)

Abdelmessih, A. H., 198(133)Abdollahian, D., 225(199), 235(223; 225)Abdullah, Z., 194(125), 198(125), 200(125),

202(125)Abou-Madi, 421(138)Abraham, M. A., 151(36; 37)Abriola, L. M., 15(49)Achaichia, A., 370(2), 407(96; 109), 408(96;

101), 410(109), 412(116), 414(101)Achdou, Y., 69(171)Achenbach, E., 91(186), 92(186), 95(186)Acrivos, A., 11(32)Adler, P. M., 2(13)Adzerikho, K. S., 57(142), 60(142)Ahmad, S. Y., 221(194)Ahmed, N. U., 124(214; 215; 216; 217),

125(214; 225)Akers, W. W., 187(100)Akhiezer, A. I., 38(99)Akira, T., 415(120)Alamgir, M. D., 229(210), 236(210)

Al-Daini, A. J., 407(103)Alfredsson, P. H., 381(29)Al-Hayes, R. A. M., 200(137)Ali, M. I., 150(29; 30), 155(29; 30), 171(29; 30),

174(30), 176(30), 177(30), 185(29; 30),189(29; 30), 190(30)

Ali, M. M., 372(5)Allmaras, S., 383(33), 392(33)Al-Nimr, M. A., 56(136)Amon, C. H., 424(147), 425(148)Amos, C. N., 225(199), 226(204), 227(204),

228(204), 229(204), 232(204), 236(204),239(204), 240(204), 241(204)

Anderson, J. D., 376(17)Anderson, T. B., 1(1)Andre, P., 424(146)Anisimov, S. I., 39(100; 101)Anita, S., 125(224)Antal, S. P., 208(152)Antonia, R. A., 28(77)Antoniou, A., 412(115; 117)Aoki, H., 407(106; 107), 409(106; 107)Arai, N., 303(85), 333(85), 334(85), 335(85),

336(85), 346(117), 349(85), 356(85)Ardron, K. H., 232(212)Arkhipov, V. V., 210(165), 211(165)Armstrong, R. C., 91(191), 94(191), 95(191)Arpaci, V. S., 56(136)Asako, Y., 388(126), 416(123), 417(123; 126),

418(123; 125; 126), 419(123; 125; 126),420(126), 421(135), 431(126)

Asano, Y., 379(27)Ashikhmin, S. R., 81(176), 82(176)Atkinson, K. N., 396(137), 407(111; 112),

411(111; 112; 114), 413(114), 417(140),421(137; 140)

Avellaneda, M., 69(171)Aziz, K., 148(13), 158(41), 161(41), 175(41),

177(41)

445

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)Bereiziat, 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;

446 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;

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),

447author index

197(132), 198(132), 199(132), 200(132),202(132; 139), 203(139), 205(132; 139)

Downar-Zapolski, Z., 236(229)Downie, J. H., 371(4)Drakulic, R., 367(113), 375(15), 407(111; 112;

113), 411(111; 112; 113; 114), 412(113),413(114), 417(140), 421(140)

Drew, D. A., 22(55), 23(56), 206(148; 149)Drolen, B. L., 56(135)Dukler, A. E., 148(15; 16; 17; 18), 156(15; 17),

161(15), 162(15), 166(17), 169(16),171(17), 173(17)

Dullien, F. A. L., 2(12), 81(180), 85(180)Duncan, A. B., 191(111)Dybbs, A., 26(68), 89(184), 91(184), 92(184),

95(184)

E

Eckelmann, H., 287(70)Edwards, R. V., 26(68)Einav, S., 386(41)Einstein, A., 258(4), 325(4)Ekberg, N. P., 150(35), 155(35), 176(35),

177(35), 185(35), 191(35)Elias, E., 225(200), 231(200), 232(200),

233(200), 236(227)Elperin, T., 325(97a)El-Sayed, M. S., 81(180), 85(180)Elsayed-Ali, H. E., 39(106), 46(106)Ergin, S., 67(167), 77(167), 81(167), 384(130),

388(132), 417(130; 132), 419(130),420(130)

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)

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

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)

448 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)

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)

449author index

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)Hosken, 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 Feriet, J., 297(81), 350(81)

450 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)

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)Kratzer, 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)

451author 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)

452 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)

Metais, O., 393(69)Michallon, E., 384(87), 399(87), 401(87),

402(87), 407(87)Mikol, E. P., 184(87)Miller, B., 275(48)Miller, C. A., 34(86)Millikan, C. B., 266(30)Mishima, K., 22(53), 150(26; 31), 153(26),

154(26), 155(31), 157(26), 158(26; 42),160(26), 162(42), 163(26; 42), 164(42),166(42), 169(26; 31), 171(31), 172(31),173(42), 174(31), 180(26), 185(26; 31),187(26), 189(31), 190(31)

Mitra, N. K., 396(156), 427(156), 428(156),429(156), 430(156), 432(156)

Mizuno, M., 399(88), 401(88), 402(88),403(88), 407(88)

Moalem-Maron, D., 147(10), 161(10)Mochizuki, S., 400(84), 403(84), 404(84),

405(84)Moin, P., 287(67), 379(25), 393(61; 71),

396(75)Moizhes, B. Ya., 98(197)Monrad, C. C., 282(59)Moody, F. J., 192(116), 234(222), 235(222)Morega, A. M., 112(203)Morioka, M., 399(88), 401(88), 402(88),

403(88), 407(88)Moser, R., 287(67), 396(75)Moshaev, A. P., 91(189), 93(189)Motai, T., 114(210)Movchan, A. B., 57(155)Mow, K., 81(180), 85(180)Moyne, C., 81(181), 85(181)Mudawar, I., 185(95), 186(95), 195(128),

198(128), 210(95), 211(95), 214(95),215(95), 217(183), 219(95), 221(189; 190),223(189), 224(95)

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)

453author index

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)O� zisik, 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)

454 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)

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)

455author 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)Sozen, 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)Sunden, 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),

456 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;

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)

457author index

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 Karman, T., 266(31), 269(31), 272(31)von Mises, R., 258(13)von Weizsacker, 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)

458 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)

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)

459author index

a


SUBJECT INDEX

A

Asymptotic dimensional analysis, 264 269

B

Baldwin-Lomax model, 382—383

Bandgaps, 53—56

Boiling

nucleate, 195—198

subcooled

forced flow

bubble nucleation, 205—209

general issues, 191—192

instability, 198—205

nucleate onset, 195—198

significant void, 198—205

void fractions, 192—195

Boundaries

CHE surfaces, 376—377

gain, 53—54

Boundary L ayer T heory, 258

Bubble nucleation, 205—209

C

Chandrasekhar, Subrahmanyan, 258

Channels

furrowed, 372

heat transfer, 429—430

wavy

corrugated, 372, 416—422

furrowed, 422—425

via chevron plates, 429—430

CHE. see Compact heat exchange

Chevron plates

description, 425—429

local analysis, 429—430

wavy channels via, 431—432

CHF. see Critical heat flux

Closure theories, 32—37

Colburn analogy, 342—343

Colebrook equation, 284—286

Compact heat exchange

characterization, 363

models

control problems, 123

current practice, 113—116

development, 111—112

optimization, 124—127

VAT-based

equations, 117—122


surfaces

chevron plates, 425—430

experiments, 365—366

interrupted flow passages

general, 366—367

louver fins, 369—371

offset strip fins, 368—369


numerical analysis

boundary conditions, 376—377

general issues, 375

mesh generation, 376

solution algorithm, 376—377


turbulence models

DNS, 392—395

eddy viscosity, 381—388


LES, 392—395

Reynolds number flow, 391—392

Reynolds stress, 388—390

wall effects, 390—392

uninterrupted complex


furrowed channel, 372

general issues, 371

intermating plates, 372—373

Reynolds number, 374—375

unsteady laminar, 374—375

wavy corrugated channel, 372

wavy channels


furrowed, 422—425

Composite media, 103—108

461

Conductivity

composite media, 103—108

hyperbolic heat, 41—42

pure phase media, 101—103

two-phase media

conventional formulation, 97—98

local distribution, 96

piecewise distribution, 96

VAT considerations, 99—101

Conservation

differential, 236—240

electron, 46—47

energy, 259

mass, 29, 258—259

momentum, 258—259

two-temperature, 43—45

Convection

integrals, 311—317

turbulent

alternative models, 320—323

correlating equations, 356

differential models, 353—354

differentials, 305—309

geometry formulations, 318—320

heat flux density ratio, 354

initial perspectives, 353

integrals


expressions, 317—318

isothermal wall, 331—332

Nu correlations

differential analogy, 344—345

dimensional analysis, 335—337

empirical, 337—339

integral, 355

low-Prandtl-number fluids, 339—342

mechanistic analogies, 342—344

numerical, 355—356

theoretically-based, 344—348

parallel plates

channels, 333, 335

geometries, 318—320

Prandtl number

convection, 323—326

elimination, 354—355

fluids, 339—342

structure development, 259—260

uncertainty, 323—324

uniformly heated tube

Nu values, 330

particular conditions, 326—328

Pr values, 328—329

Corrugated channels


wavy, 372, 416—422

Cracks

microchannel

differential conservation, 236—240



integral models

isentropic homogeneous-equilibrium,

232—233

LEAK, 235—236

Moody’s, 234—235

numerical models, 236—240

Critical heat flux

microchannels

empirical correlations, 216—220



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

462 subject index

thermal conductivity, 97

VAT

mismatches, 52—53

verification, 12—13

Direct numerical simulation


convection, 259, 260

flow, 259

Dissipation, 306

Distribution, 292—294

DMM. see Detailed micromodeling

DNM. see Direct numerical modeling

DNS. see Direct numerical simulation

Drift flux model, 172—173, 175

Dynamic procedure model, 395

E

Eddy viscosity

description, 269

filter approach model, 394

one-equation models, 383

two-equation models

advantages, 383—384

low Reynolds numbers, 387

realizable k—�, 386—387

RNG k—�;, 384—386

standard k—�, 383—384

zero-equation models, 382—383

Einstein, Albert, 258

Electrodynamics, nonlocal

VAT-governing equations

photonic crystals bandgap, 54—55

superstructures

acoustical phonon, 49—50

electromagnetic, 50—51

electron conservation, 46—47

fluid momentum, 47—48

gas energy, 49

longitudinal phonon, 49

Electron conservation, 46—47

Electron gas energy, 49

Ensemble averaging, 59

F

Filtered media, 27

Fins

louver, 369—371, 406—416

offset strip, 368—369, 398—406

Flow

CHE surface

interrupted

general, 366—367



complex passages

uninterrupted



general issues, 371





forced

subcooled boiling




linear models, 1

microchannel

annular, 170—177

characteristics, 146—147

CHF




mass, 215—216


pressure, 215—216


trends, 210—215

conditions, 147—148

correlations, 161—166

in cracks


experimental data, 225—230


integral models, 232—236


definition, 148—149

microgravity, 159—161

narrow rectangular, 170—177

noncondensable release, 178—179

pressure drop

experiment review, 184—191

fractional, 180—183

463subject index

Flow (Continued)general issues, 180

regimes, 150—153

rod bundle patterns, 166—168

in slits






subcooled boiling







surface wettability, 158—159

transitions models, 161—166

trends, 153, 156


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



dimensional, 263—269

dimensional models, 273—274

eddy viscosity, 269

exact structure, 260—263

friction factor



geometry correlations, 303—304

MacLeod analogy, 282—283, 352

mixing length, 269—273

model-free formulations, 294—295

near center line values, 287

near wall values, 286—287

new formulations, 303—304

Nikuradse data, 274—276

numerical simulations, 274

power-law models, 278—282

recapitulation, 304

rough piping, 283—286

shear stress

correlating equations, 351—352

integral formulations, 350—351

limited models, 352—353

MacLeod analogy, 352

new model, 348—349

obsolete models, 352—353

shear stress correlations, 299—301

speculative analyses, 263—269

study, history, 257—259

velocity


distribution, 292—294

Zagarola data, 287—292

Flux, 311—316, 354

Fractions

heated channels, 192—195

microchannels, 169—170

Friction


factors, 301—303

Furrowed channels, 372

G

Gas energy, electron, 49

Gas-to-fluid exchanger, 463—464

Geometry


turbulent flow, 303—304

Grain boundaries, 53—54

H

Harmonic field equations, 64—65

Heat

conductivity

composite media, 103—108

hyperbolic, 41—42

porous media, 108—111

pure phase media, 101—103

two-phase media


effective modeling, 96—97




464 subject index

critical flux

microchannels




mass flux, 215—216


pressure, 215—216


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


transport

CHE models, 124—125

microscale, 37—43

wave transport

CHE models




VAT-based, 117—122

crystal, 45—46

superstructures



electron conversion, 46—47


gas energy, 49


Heat transfer

CHE surface

interrupted

general, 366—367






general issues, 371





coefficient, 335, 337

in corrugated channels, 429—430

porous media, coefficients

assumptions, 85

models, 86—89


Heisenberg, Werner, 258

Heterogeneous media

2-phase, 11



issues, 57—58


Heterogeneous media modeling, 52—53

Highly porous media

turbulent transport

model development

additive components, 29

first level hierarchy, 27

free stream, 28

mass conservation, 29

momentum equations, 30—32

scalar diffusion, 29

separate obstacle, 28

theoretical bases, 26—27

High-temperature superconductors, 101

HMM. see Heterogeneous media modeling

Homogeneous isotropic media, 11

HTSC. see High-temperature

superconductors

Hydrodynamics, 257

Hyperbolic heat conduction, 41—42

I

Integral models

isentropic homogeneous-equilibrium, 232—233

LEAK, 235—236

Moody’s, 234—235

Isentropic homogeneous-equilibrium model,

232—233

K

Kapitsa, Pyotr, 258

465subject index

L

Laminar flow


nonlinear fluid medium

concentration value, 20

homogeneous phase diffusion, 20

mass transport, 21

momentum diffusion, 20—21

Navier-Stokes equations, 19—20

steady-state momentum, 21

porous media

VAT

diffusion equation, 18

divergence form, 17

fluid phase, 17

impermeable interface, 18


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


convection, 353

Law of the wall, 265, 283

LEAK model, 235—236

Linear models, 1

Linear particle transport, 58—59

Linear Stokes equations, 15—16

Lorentz, Hendrik, 258

Louver fins, 406—416

M

MacLeod analogy, 282—283, 352

Mass flux, 215—216

Mesh generation, 376

Microchannel flow

characteristics, 146—147

two-phase media

annular, 170—177

CHF




mass, 215—216


pressure, 215—216


trends, 210—215



in cracks










pressure drop



general issues, 180


in slits






subcooled boiling









trends, 153, 156


Micro-rod bundles, 166—1687

Microscale heat transport

heuristic approach, 37

traditional descriptions

coupling factor, 40

elastic lattice vibration, 38—39

heat balance, 38

hyperbolic heat conduction, 41—42

in metals, 39—40

phonon radiative transfer, 41

466 subject index

in solids, 39

two-fluid model, 40

Mises, Richard von, 258

N

Navier-Stokes equations

description, 261

Reynolds averaged

algebraic stress models, 388


eddy viscosity models, 382—388

stress models, 388—392

Newton, Isaac, 258

Noncondensables, 215—216

Nucleate boiling, 195—198

Nucleation, bubble, 205—209

Nu values

convection

parallel-plate channels, 333, 335

uniformly heated tube, 330

correlations



empirical equations, 337—339



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


PDE. see Partial differential equations

Phonon, 41, 49—50

Photography, strobe flash, 171

Photonic crystals bandgap, 52—56

Plates

chevron


local analysis, 429—430

wavy channels via, 431—432

exchanger, 364

fine heat exchangers, 114—115

isothermal, 347—348

parallel


equal uniform heating, 333

uniformly heated, 347

Porous media

flow resistance


momentum in 1D membrane

equations, 69—75

model 1, 75

model 2, 75

model 4, 75—76



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


liquid-impregnated, 66

nonlinear transport, 15—17


issues, 57—58

linear transfer, 58—59


transport

closure theories, 32—37

linear/nonlinear, 15—17

Power-law models, 278—282

Prandtl, Ludwig, 259

Prandtl analogy, 343

Prandtl number


467subject index

Prandtl number(Continued)fluids, 339—342

tubes, 328—329

turbulent, 354—355

values, 328—329

Pressure

drop, microchannel flow



general issues, 180

microchannel flow, 215—216

Pressure loss experiments, 77—80

PU-BTPFL-CHF Database, 217—219

R

Radiative transport

heterogeneous media


issues, 57—58


phonon, 41

porous media

issues, 57—58

linear transfer, 58


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


transport averaging, 3

Resistance, flow

porous media


momentum in 1D membrane, 69—75



REV. see Representative elementary volume

Reynolds, Sir Osborne, 259

Reynolds analogy, 260, 343

Reynolds numbers

CHE surfacess, 374—375, 391—392


Navier-Stokes equations


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

Scaling, 77—80

Shear stress

local, equations, 299—301

turbulent flow







Significant void, 198—205

Slits

microchannel




integral models

isentropic homogeneous-equilibrium,

232—233

LEAK, 235—236

Moody’s, 234—235


Smagorinsky model, 394—395

Sommerfeld, Arnold, 258

Space averaging, 261—262

Spalart-Allmaras model, 383

Speculation, 263—264

Stress

algebraic models, 388

Reynolds, 262, 388—390

shear


turbulent flow

468 subject index







Strobe flash photography, 171

Structure function model, 395

Subcooled boiling

forced flow






Subcrystalline single crystals, 45—46

Surfaces

CHE


interrupted flow passages

general, 366—367




numerical analysis

general issues, 375

mesh generation, 376



turbulence models

algebraic stress, 388

DNS, 392—395



LES, 392—395

Reynolds number flow, 391—392






general issues, 371





wavy channels


furrowed, 422—425

wettability, 158—159

T

Temperatures


logitudinal phonon, 49

uniform wall, 316—317

Transfer, heat

CHE surface

interrupted

general, 366—367






general issues, 371





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





VAT-based, 117—122

superstructures





gas energy, 49


linear particle, 58—59

microscale heat

heuristic approach, 37

traditional descriptions

469subject index

Transport (Continued)coupling factor, 40

elastic lattice vibration, 38—39

heat balance, 38

hyperbolic heat conduction, 41—42

in metals, 39—40

phonon radiative transfer, 41

in solids, 39

two-fluid model, 40

porous media

closure theories, 32—37

nonlinear, 15—17

raditative heat

issues, 57—58

linear transfer, 58—59


VAT, 3

radiative heat

heterogeneous media


issues, 57—58


Tubes

round, 346—347

uniformly heated

Nu values, 330



Turbulence

CHE surfaces

DNS, 395—397

LES, 392—395

models

algebraic stress, 388





zero-equation, 382—383

Turbulent convection

alternative models, 320—323

correlating equations, 356

differentials, 305—309

geometry formulations, 318—320

heat flux density ratio, 354

initial perspectives, 353

integrals

general equations, 309—310

generalized expressions, 317—318

uniform wall

heat flux, 311—316

temperature, 311—316

models, 353—354

Nu correlations



integral, 355



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


Nu values, 330



Turbulent flow

asymptotic analysis, 263—69




dimensional models, 273—274

eddy viscosity, 269

exact structure, 260—263

friction factor



geometry correlations, 303—304

MacLeod analogy, 282—283

mixing length, 269—273

model-free formulations, 294—295

near centerline values, 287

near wall values, 286—287

new formulations, 303—304

Nikuradse data, 274—276

numerical simulations, 274

470 subject index

power-law models, 278—282

recapitulation, 304

rough piping, 283—286

shear stress








speculative analyses, 263—269

study, history, 257—259

velocity distribution



Zagarola data, 287—292

Turbulent transport

porous media


nonlinear, 14—17


theory, 26—27

Two-phase media

microchannel flow

annular, 170—177

CHF




mass, 215—216


pressure, 215—216


trends, 210—215



in cracks, 232—236









pressure drop



general issues, 180

regimes, 150—153, 156


in slits






subcooled boiling









trends, 153, 156


thermal conductivity





Two-temperature conservation, 43—45

U

Uhlenbeck, George, 258

V

VAT. see Volume averaging theory

Velocity


distribution, 292—294

Vinci, da Leonardo, 258

Viscosity

dissipation, 264—269, 306

eddy

description, 393

filter approach model, 394

one-equation models, 383

two-equation models

advantages, 383

low Reynolds numbers, 387

471subject index

Viscosity (Continued)realizable k—�, 386—387

RNG k—�, 384—386

standard k—�, 383—384

zero-equation models, 382—383

Viscous shear stress law, 269

Void fractions, 169—170, 192—195

Volume averaging theory

development, 1—2

electrodynamics, nonlocal, 46—47

features, 1—2

heat wave transport

CHE models

design problems, 123




subcrystalline crystal, 45—46

superstructures





gas energy, 49


heterogeneous media, 99—101

highly porous medium turbulent

model development

additive components, 29

first level hierarchy, 27

free stream, 28

mass conservation, 29


scalar diffusion, 29

separate obstacle, 28


nonlinear fluid medium

laminar flow

concentration value, 20

homogeneous phase diffusion, 20

mass transport, 21

momentum diffusion, 20—21

Navier-Stokes equations, 19—20

steady-state momentum, 21

photonic crystals bandgap

DMM-DMN mismatches, 52—53

governing equations, 54—55

porous media

data reduction, 66—67

internal heat transfer

assumptions, 85

models, 86—89


laminar flow

diffusion equation, 18

divergence form, 17

fluid phase, 17

impermeable interface, 18


solid phase, 17

linear Stokes equations, 15

momentum in 1D membrane, 75



porous medium transport, 32—37

porous medium turbulent, 21—26


basis, 3

heterogeneous media


issues, 57—58


porous media

issues, 57—58

linear transfer, 58


theorem verification

1D Cartesian coordinate version, 11

1-dimensional cases, 12

DMA, 12—132

integral terms, 11

2-phase heterogeneous medium, 11—12

3-phase homogeneous medium, 12

solid-phase equation, 10

steady-state conduction, 11

two-temperature conservation, 43—45

W

Wall

effects, model

features, 390

function, 390—391

turbulence, 391—392

isothermal, 331—332

law of, 265, 283

near, convection, 286—287

472 subject index

uniform density, 311—316

uniform temperature, 316—317

Wavy channels


furrowed, 422—425

via chevron plates, 431—432

Weizscker, C.R. von, 258

Wettability, surface, 158—159

Z

Zel’dovich, Yakob, 258

Zero-equation models, 382—383

473subject index

a


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