SPECIAL ISSUE

Ivan Catton

Transport phenomena in heterogeneous media basedon volume averaging theory

Received: 14 June 2004 / Accepted: 31 January 2005 / Published online: 10 January 2006� Springer-Verlag 2006

Abstract Models are developed to describe transportphenomena in a porous medium that take into accountthe scales and other characteristics of the mediummorphology. Equation sets allowing for turbulence andtwo-temperature or two-concentration diffusion areobtained for non-isotropic porous media with interfaceexchange. The equations differ from known equationsand were developed using an advanced averagingtechnique, hierarchical modeling methodology, andfully turbulent models with Reynolds stresses and fluxesin the space of every pore. The transport equations areshown to have additional integral and differentialterms. The description of the structural morphologydetermines the importance of these terms and the rangeof application of the closure schemes. A natural way totransfer from transport equations in a porous mediawith integral terms to differential equations with coef-ficients that could be experimentally evaluated anddetermined is described.Some numerical results that il-lustrate what occurs when the porosity approachesunity or the porosity approaches zero show that solu-tions smoothly converge to the transport characteristicsof a plane channel. A simple heat sink is modeled as atwo-temperature process allowing solution of the con-jugate problem to be accomplished and is optimizedusing design of experiment (DOE) methods. As a finalexample, acoustic energy absorption in a porous mediais addressed.

List of symbols

b Turbulent fluctuation energycd Mean drag coefficient in the REV (�)cdp Mean form resistance coefficient in REV (�)cf Mean skin friction coefficient in the REV (�)cp Specific heat (J/kg K)dpin Pin fin diameter (m)

dp Porous media hydraulic diameter (m)

dS Interphase differential area (m2)¶Sw Internal surface in the REV (m2)Eeff Heat sink effectivenessff Fanning friction factor~f Value f averaged over D W f

f_

Value f averaged over D W s

f Æ f æ� f, morphologically induced fluctuations of fÆ f æf Value f, averaged over D W f in a REV

H Height of pin fins (m)K Permeability (m2)K Thermal conductivity (W/m K)L Length of base plate (m)L Turbulence mixing length (m)m Porosity (�)Æm æ Averaged porosity (�)P Pitch (m), or Pumping power (W)p Pressure (Pa)q Heat flux (W/m2)Repor Pore Reynolds number (�)Sw Specific surface ¶ Sw/D W (1/m)Swp S?=DX ð1=mÞS? Cross flow projected area of obstacles (m2)T Temperature (K)u, w Velocity in x, z-direction (m/s)V Volume (m3)

Subscripts

c Characteristic valuef fluid phasei Component of turbulent vectorL Laminarm scale value

I. CattonMorin, Martinelli, Gier Memorial Heat Transfer Laboratory,Department of Mechanical and Aerospace Engineering,School of Engineering and Applied Science,University of California, Los Angeles, CA 90095-1597, USAE-mail: catton@scas.ucla.eduFax: +1-310-2064830

Heat Mass Transfer (2006) 42: 537–551DOI 10.1007/s00231-005-0650-9

s Solid phaseT Turbulentw Wall

Greek letters~aT Averaged heat transfer coefficient over ¶ Sw (W/

m2 K)D W Representative elementary volume (REV) (m3)D Wf Pore volume in a REV (m3)D Ws Solid phase volume in a REV (m3)m Kinematic viscosity (m2/s)q Density (kg/m3)mu Dynamic viscosity (kg/ms)

1 Introduction

Determination of flow-variables and scalar transport ina heterogeneous (or porous) media is difficult even whensubject to simplifications allowing the specification ofmedium periodicity or regularity. Linear or linearizedmodels fail to intrinsically account for transport phe-nomena, requiring dynamic coefficient models to correctfor short-comings in the governing models. Allowinginhomogeneities to adopt random or stochastic char-acter further confounds the already daunting task ofproperly identifying pertinent transport mechanisms andpredicting transport phenomena.

Some aspects of the development of transport phe-nomena in porous media are now well understood andhave seen substantial progress in thermal physics andfluid mechanics sciences, particularly in porous mediatransport phenomena. The basis for this progress isvolume averaging theory (VAT) first proposed in thesixties by Anderson and Jackson [1], Whitaker [2] andothers. Many of the important details and examples ofits application are found in books by Kheifets andNeimark [3] and most recently by Whittaker [4].

In most physically realistic cases, highly complex in-tegral-differential equations result. The largest challengeis the insufficient development of closure theory, espe-cially for integral-differential equations. The ability toaccurately evaluate various kinds of medium morphol-ogy irregularities results from the modelling methodol-ogy once a porous medium morphology is assigned.Further, when attempting to describe transport pro-cesses in a heterogeneous media, the correct form of thegoverning equations remains an area where inattentionto procedure by some researchers has led to significantlydifferent equations for the same media (see Refs. [5, 6]).The VAT approach has the following desirable features:

– Effects of interfaces can be included in the modeling.– The effect of morphology of the different phases is

directly incorporated into the field equations.– Interactions are described.– Correct descriptions of effective transport coefficients

result.

– Optimization of materials using hierarchical physicaldescriptions based on the VAT governing equationscan be used to connect top-level properties and mor-phological characteristics to component features.

2 Fundamentals of volume averaging theory (VAT)

The method of volume averaging begins by associatingan averaging volume D W with every point in space. Theaveraging volume is shown in Fig. 1, and is representedin terms of the volumes of the individual phases by

DX ¼ DXf~X ; t� �

þ DXs~X� �

ð1Þ

The volumes of the fluid and solid phases can dependon position and, in addition, the volume of the fluidphase can depend on time. Average values are defined interms of these volumes and these averaged values areassociated with the centroid of DW .

In the development of volume averaged transportequations, a superficial average and an intrinsic averageare defined. The superficial average of f is given by

ffh i ¼1

DX

Z

DXf

ff ~X ; t� �

dx ð2Þ

and the intrinsic average of f is defined by

~ff ¼1

DXf

Z

DXf

ff ~X ; t� �

dx ð3Þ

The intrinsic and superficial averages are related by

ffh i ¼ mh i~ff ð4Þwhere Æm æ is the porosity in the averaging volume D Wdefined by

mh i ¼ DXf

DXð5Þ

The random porosity function in terms of the averagevalue of mð~xÞh i in the REV and its fluctuations mð~xÞ invarious directions is

mð~xÞ ¼ mð~xÞh i þ mð~xÞ ð6Þ

The different types of averaging over the REV func-tion f are defined by the following averaging operators:The average of the function f over the REV is given by

fh i ¼ 1

DX

Z

DX

f ~X ; t� �

dx¼ mh i~f þ 1� mh ið Þf_

ð7Þ

The phase average of the function f in the fluid

fh if¼DXf

DX1

DXf

Z

DXf

f ~X ; t� �

dx ¼ mh i~f ð8Þ

538

The phase average of the function f in the solid phase

fh is ¼DXs

DX1

DXs

Z

DXs

f ~X ; t� �

dx¼ 1� mh ið Þf_

ð9Þ

The interphase average of each component is

~f ¼ ff gf¼1

DXf

Z

DXf

f ~X ; t� �

dx ð10Þ

f_

¼ ff gs ¼1

DXs

Z

DXs

f ~X ; t� �

dx ð11Þ

When the interface is fixed in space, averaged func-tions for the first and second phase within the REV andover the entire REV fulfill all four of the Reynoldsconditions as well as the following four consequences

~f� �

f¼ ~f ; ~f

� �f¼ f � ~f� �

f¼ 0;

~f ~g� �

f¼ ~f ~g; ~f g

� �f¼ ~f ~g ¼ 0:

The differential condition,

rff gf¼1

DXf

Z

DXf

rfdx ¼ r~f þ 1

DXf

Z

@Sw

f ds! ð12Þ

using the D averaging theorem [4]. The fourth conditionimplies an unchanging porous medium morphology. Allthe above are well known from earlier work [7–9].

At the same time, Æf æf and Æf æs fulfill neither the thirdof the Reynolds conditions nor all the consequences ofthe other Reynolds conditions.

Most of the following analysis is based on averagingtechniques developed by Whitaker [4] who focused onsolving linear diffusion problems and by Travkin andCatton [6] who focused on solving non-linear turbulentdiffusion problems. The divergent and non-divergentforms of the averaged convective term in the diffusionequation are

r TUið Þh if

¼ mh i ~Ui@

@xi~T þr T ui

� �fþ 1

DXZ

@Sw

TUi � ds!� ~T

1

DX

Z

@Sw

Ui � ds! ð13Þ

where ~T ¼ T � T :Applying the VAT rules and definitions to the heat

flux (or mass flux) gradient term with spatially depen-dent heat conductivity kT yields

r � kTrTð Þh if¼ r � mh i~kTr~T

� �þr � kTrT

D E

f

� �

þr � ~kT1

DX

Z

@Sw

Tds!

2

64

3

75þ1

DX

Z

@Sw

kTrT � ds! ð14Þ

Consistent application of these averaging rules anddefinitions to the conservation equations will allow theVAT governing equations to be systematically and rig-orously derived.

3 VAT Based equations describing thermal physicsand fluid mechanics in heterogeneous media

The VAT based thermal physics and fluid mechanicsgoverning equations in heterogeneous porous media de-veloped from the Navier-Stokes equation and the thermalenergy equation are the starting point and the basis forstudying flow and heat transfer in porous media. Theturbulent transport considered here is flow in a highlyporous layer combined with a two energy equation model.

For flow in porous media, the instantaneous velocityfor the turbulent regime is represented by

u ¼ �u x; zð Þ þ u0 x; z; tð Þ

¼ �uk x; zð Þ þ ~ur x; zð Þ þ u00

r x; zð Þ þ u00

k x; z; tð Þ ð15Þ

where

�u x; zð Þ ¼ 1

tT

Z

tT

udt ¼ ~�u x; zð Þ þ u00

r x; zð Þ

¼�uk x; zð Þ þ ~ur x; zð Þ þ u00

r x; zð Þ ð16Þ

k signifies turbulence induced components that are in-dependent of inhomogeneities in spatial dimensions and

Fig. 1 Representative elementary volume (REV)

539

properties resulting from the multitude of porous med-ium channels (pores), and r denotes the fluctuationcontributions due to the porous medium inhomogeneity.The fluctuation velocity u¢r(x,z) due to porous mediuminhomogeneity is u0

0

r x; zð Þ ¼ �uðx; zÞ The temperature isrepresented by

T ¼�T þ T 0 ¼ ~�T x; zð Þ þ T 0r x; zð Þ þ T0

x; z; tð Þ

¼ �Tk x; zð Þ þ ~Tr x; zð Þ þ T 00

r x; zð Þn o

þ T 00

k x; z; tð Þ ð17Þ

3.1 Development of the continuity equation

For incompressible flow, the continuity equation is

@ui

@xi¼ 0 ð18Þ

Using Eq. 12, the averaged continuity equation in thefluid phase for flow in a porous media is written

@ui@xi

f

¼ @

@xiuih if

� �þ 1

DX

Z

@Sw

~�ui þ �ui

� �� d~s ¼ 0 ð19Þ

3.2 Development of the momentum equation

Starting from the momentum transport equation forincompressible flow

uj@ui

@xj¼ � 1

qf

@�p@xiþ @

@xjm@ui

@xj� u0iu

0j

� �þ Fi; ð20Þ

and using Eqs. 12, 13 and 14, the averaged momentumequation is written

mh i~�uj@~�ui@xj� ~�ui

1

DX

Z

@Sw

~�uj þ �uj

� �� d$sþ @

@xj�ui�uj� �

f

þ 1

DX

Z

@Sw

ujui � d$s

¼ � 1

qf

@

@ximh i~�pð Þ � 1

qf

1

DX

Z

@Sw

�pd$s

þ @

@xjmh im @

~�ui

@xj

� �þ 1

DX

Z

@SwL

m@ui

@xj� d$s

þ @

@xj�u0iu0

0j

D E

f� 1

DX

Z

@Sw

u00iu0 0jd$sþ mh i~Fi ð21Þ

where ~FiFi incorporates the impact of microroughnessand augmentation of the previous level of the simulationhierarchy.

For many general flow situations, the Reynolds stresscan be approximated by

�u00iu00j ¼ mT

@ui

@xjþ @uj

@xi

� �ð22Þ

For one-dimensional parallel flow, under steady-stateconditions and no flow penetration through ¶Sw, Eq. 21has the form

@

@zmh i ~mT þ mð Þ @

~�u@z

� �þ @

@z~mT@~�u@z

f

� �þ @

@z��u �w� �

f

� �

þ 1

DX

Z

@SwT

mT@�u@xi� d$sþ 1

DX

Z

@SwL

m@�u@xi� d$s

� 1

qfDX

Z

@Sw

�pd$s ¼ 1

qf

@ ph if@x

ð23Þ

An equation for one-dimensional steady fully devel-oped turbulent flow in a porous layer with regularpacked bed structure characteristics and an impermeableinterphase surface simplifies to

@

@zm zð Þh i ~mT þ mð Þ @

~�u zð Þ@z

� �þ 1

DX

Z

@SwT

mT@�u@xi� d~s

þ 1

DX

Z

@SwL

m@�u@xi� d~s� 1

qfDX

Z

@Sw

�pd~s

¼ 1

qf

@ m zð Þh i~�pð Þ@x

ð24Þ

3.3 Development of the turbulent kinetic energyequation

Following Rodi [10], the turbulent kinetic energy equa-tion is written

~mT zð Þ @~�u@z

� �2

þ d

dz~mTrbþ m

� �db zð Þdz

� �þ X 0iu

0i

� gT rb

~mT@�T

_

@z

2

4

3

5þ 2mdb1=2 zð Þ

dz

� �2

¼ CDC lb2 zð Þ~mT

ð25Þ

where CD, Cl and r b are empirical coefficients. CDCl �0.08 and r b=1 appear to be reasonable values for theempirical constants. Assuming that most of the meanmotion kinetic energy lost due to interaction of the flowwith the porous medium solid obstacles translates intoincreasing the turbulent fluctuation energy, see Ref. [11],leads to the conclusion that

X 0u0 ¼ cdSw~�u3 ð26Þ

It follows that the equation for the mean turbulentfluctuation energy b(z) can be written in the form

540

~mTðzÞ@~�u@z

� �2

þ d

dz~mTrbþ m

� �db zð Þdz

� �þ cdSw zð Þ

mh i~�u3

� gTrb

~mT@�T

_

@z

2

4

3

5þ 2mdb1=2ðzÞ

dz

� �2

¼ CDC lb2ðzÞ~mT

ð27Þ

where the mean eddy viscosity ~mT can be determinedusing Prandtl’s mixing length theory. By choosing

ffiffiffibp

asthe velocity scale, ~mT can be expressed as

~mT ¼ Cl

ffiffiffibp

LðzÞ ð28Þ

where L is the mixing length scale function defined bythe assumed porous medium structure and

bðzÞ ¼ 1

2u02 þ v02 þ w02� �

ð29Þ

is the turbulent kinetic energy. Eqs. 27 and 28 were usedin the modeling of a regular porous medium filledchannel transport by Travkin and Catton [5, 12] andGratton et al. [13, 14].

3.4 Development of the energy equations

To derive the volume averaged energy equation, theenergy transport equation with turbulent transport,

qfcpfuj@�T@xj¼ @

@xjkf@�T@xj� T 0u0j

� �þ ST; ð30Þ

is the starting point. Using Eqs. 12, 13, and 14, theaveraged energy equation in fluid phase becomes

qfcpf mh i~�uj@~�T@xj� ~�T

1

DX

Z

@Sw

~�uj þ �uj� �

� d~s

8><

>:

9>=

>;

þ qfcpf@

@xj�T �ujD E

fþ 1

DX

Z

@Sw

uj�T � d~s

8><

>:

9>=

>;

¼ @

@xjmh ikf

@~�T@xj

!

þ 1

DX

Z

@SwL

kf@�T@xj� d~s

þ @

@xj

kfDX

Z

@Sw

�Td~s

0

B@

1

CAþ@

@xj�T 0u0j

D E

f

�Z

@Sw

T 0u0d~sþ mh i~ST ð31Þ

The term ST represents a heat source.By analogy with the concept of eddy viscosity, the

turbulent fluctuation terms are proportional to the meantemperature gradients. For two-dimensional heattransfer in a channel and no flow penetration through¶Sw, the above equation simplifies to

cpfqf mh i~�u zð Þ @~�T x; zð Þ@x

¼ mh i @@xi

~kT þ kf� � @~�T x; zð Þ

@xi

!

þ 1

DX

Z

@SwT

kT@�T@xi� d~s

þ 1

DX

Z

@SwL

kf@�T@xi� d~s ð32Þ

where ~kT is the turbulent heat conductivity.The solid phase energy equation is

@

@xiks x; zð Þ @Ts x; zð Þ

@xi

� �¼ 0 ð33Þ

where k{s is the effective heat conductivity in the solidphase. For two-dimensional heat transfer in the solidphase, the volume averaged solid phase thermal energyequation is written

@

@xi1� mh ið Þks x; zð Þ @Ts x; zð Þ

@xi

� �

þ 1

DX

Z

@SwT

ks@Ts

@xi� d~s1 ¼ 0 ð34Þ

where

d~s1 ¼ �d~s

4 Closure models for VAT equations for regular roughsurfaces

To solve the VAT equations, closure must be obtained.Some morphology simplifications have been exploitedfor laminar fluid flow (see Refs. 15 and 16). Similarmodels can be used for turbulent flow in porous mediaand for diffusion models when fluctuations are ignored(see Refs. 17 and 18 for the laminar regime).

4.1 Closure model for the momentum equation

Travkin and Catton [5, 12] close the integral terms in thetransport equations by integrating over the interphasesurface, or of some other outlined areas of the surface.From a physical view-point, the integration terms inEq. 21 represent momentum loss due to the friction re-sistance over the inter-surfaces. The skin friction re-sistance terms in Eq. 21 are

1

DX

Z

@Sw

ðmTþ mÞ @�u

@xi� d~s

¼ 1

DXqf

Z

@Sw

swL þ swTð Þ � d~s

¼ � 1

2cfLðzÞSwLðzÞ þ ~cdðzÞSwTðzÞð Þ~�u2ðzÞ ð35Þ

541

The pressure drag resistance integral term in Eq. 23or 24 is closed in a manner similar to that for a one-component pressure resistance coefficient over a singleobstacle, according to the definition,

cdp ¼ 2

R@Sw

�pd~s

qf~u2S?ð Þ ;1

qfDX

Z

@Sw

�pd~s ¼ 1

2cdpS?ðzÞ~��u2 ð36Þ

where S? is the cross flow projected area. With the clo-sure given by Eqs. 35 and 36, the simplified momentumEq. 24 becomes

@

@zmðzÞh i ~mT þ mð Þ z; ~�u; b; lð Þ @

~�uðzÞ@z

� �¼ 1

qf

d �ph ifdxþ

~�u2

2

ðcfL þ ~cdÞ z; ~�uð ÞSwðzÞ þ cdp z; ~�uð ÞS?ðzÞ� �

ð37Þ

The pressure gradient term in Eq. 37 is modeled as aconstant value in the layer, or simulated by the local valueof the right hand side of the experimental correlations.

For flow in a channel of height 2h, the boundaryconditions for Eq. 37 are

z ¼ 0; ~�u ¼ 0; ~mT ¼ m and z ¼ h;@~�u@z¼ 0

4.2 Closure model for the thermal energy equations

The integral terms in Eq. 32 represent the heat transferbetween the solid phase and fluid phase in the REV.Closure of the heat exchange integral terms in Eq. 32 isderived from

� 1

DX

Z

@Sw

~kT þ kf� � @�T

@n1� d~s

¼ 1

DX

Z

@Sw

~qT � d~s ¼ �~aTSw ~Tf � Ts

� �ð38Þ

and

~aT ¼qlocal

Ts � ~Tf

¼ Nulocal

dporkf with dpor ¼

4 mh iSw

ð39Þ

The local Nusselt number, Nulocal, can be determinedfrom empirical correlations.

Using the closure models described above, the energyequation in fluid phase is

cpfqf mðzÞh i~�uðzÞ @~�T f x; zð Þ@x

¼ @

@ximðzÞh i ~kTðzÞ þ kf

� � @~�T f x; zð Þ@xi

!

þ ~aTðzÞSwðzÞ Ts x; zð Þ � ~�T f x; zð Þ� �

ð40Þ

with (x, z) 2D W f. It is obvious that heat exchange fromsolid phase to fluid phase is equal to that from the fluid

phase to solid phase, so the integral term in the solidphase energy equation (34) is closed with

1

DX

Z

@SwT

ksT@Ts

@xi�d~s1¼ ~aTðzÞSwðzÞ ~T x;zð Þ�Ts x;zð Þ

� �ð41Þ

and Eq. 34 can be closed with

@

@xi1�mðzÞh ikSTðzÞ

@Ts x; zð Þ@xi

� �

¼ ~aTðzÞSwðzÞ Ts x; zð Þ � ~�T f x; zð Þ� �

; x; zð Þ 2 DXs ð42Þ

In the Eqs. 40 and 42, the coefficient functions andspecific surface functions must be determined by as-sumed real or invented morphological models of theporous structure. The boundary conditions for Eqs. 40and 42 are the following

z¼0 : m~kT@~Tf

@zjz¼þ0¼~aTS? Ts x;0ð Þ� ~Tf x;0ð Þ

� ��mqw;

z¼0 : ksT@Ts

@zjz¼þ0¼~aTS? ~Tf x;0ð Þ�Ts x;0ð Þ

� �� 1�mð Þqw

z¼h :@b@z¼0;

@~Tf

@z¼0;

@Ts

@z¼0

5 Applications of VAT to transport phenomena

Applications of VAT to specific transport phenomenaproblems have only been to some relativelyuncomplicated problems. The modeling approach andmodels described in this review were applied to problemsof transport in highly porous media with indefinite orspecified morphology structures. Several wide-rangingapplications were selected to demonstrate the versatilityof VAT based modeling.

5.1 Flow over regular rough surfaces and regularporous media

In this section, we present simulation results of VATbased transport models and closure models for flow in achannel with rib roughened walls; channel flow acrossspherical beads, and channel flow across square tubebanks.

For flow across spherical beads or tube banks, thebeads or the tubes are regularly arranged and the pitchof the unit is fixed. When the diameter or the side lengthof the unit becomes smaller and smaller, the porosity ofthe channel will approach unity. For flow across squaretube banks, when the side length of the unit becomesbigger and bigger the porosity will approach zero. Sothat the VAT models and SVAT closure scheme can betested for the limiting cases [19]. For the limiting cases,results are available for comparison.

542

5.2 Flow in a channel with rib roughened walls

The two morphologies shown in Figs. 2 and 3 wereanalyzed. The specific area for the rectangular ribs is:

Swi¼ 2

Pr; zi\hr

Swi¼ Wr þ 2ðhr � zi�1Þ

Prhi; zi�16hr6zi

Swi! 0; zi�1 > hr

ð43Þ

and that for half cylindrical ribs is,

Swi¼ 2hrðarcsinðzi=hrÞ� arcsinðzi�1=hrÞÞ

hiPr; zi\hr

Swi¼ hr p� 2arcsinðzi�1=hrÞ½ �

hiPr; zi�16hr6zi

Swi! 0; zi�1 > hr

ð44Þ

The porosity function for a rectangular rib is

mh ii¼ 1� Wr

Pr; zi\hr and mh ii! 1; ziþ1 > hr ð45Þ

and for a half cylindrical rib

mh ii¼ 1�12h2r ðai�1�ai� sinai�1þ sinaiÞ

Prhi; zi6hr ð46Þ

mh ii! 1; ziþ1 > hr

where a {i=2 arccos (z{i /h{r)

Results for fully developed incompressible turbulentflow are shown in Fig. 4. The parameters chosen forboth the rectangular rib wall and semi-cylindrical ribwall are h{r=20 mm, W{r=40 mm, P{r=200 mm, andh=100 mm. The Reynolds number is 5,00,000 for bothof the calculations. The velocity in the channel of a halfcylindrical rib wall is about 15% higher than that in achannel of rectangular rib walls.

5.3 Flow across a lattice of spherical beads

The parameters chosen for the simulation of flow acrossspherical beads, Fig. 5, in the channel are pitch P =20 mm and channel height 2h = 200 mm The diameterof the beads dp varies from 20 to 0.001 mm. The overalldrag resistance for flow across a lattice of sphericalbeads [20], can be expressed as a function of porosityand porous Reynolds number by

cd ¼25

121� mh ið Þ

� �Cd; sph: ð47Þ

where Cd, sph, is an ever changing function of

Repor ¼4 mh i~�umSw

ð48Þ

For a table giving Cd, sph (see Watanabe [20] or Hu[24]).

Figure 6 shows the velocity distributions across thechannel with different size beads at the same pressuregradient for laminar flow. When the diameter of thebeads is large, the disturbance of the porosity across thechannel is large and the flow resistance plays an im-portant role. As a result, the disturbance of the velocityprofile is large. When the porosity approaches unity thedisturbance of velocity profile disappears and the velo-city distribution approaches the theoretical distribution.In contrast to many models, the VAT model limitsproperly.

5.4 Channel flow over rod bundles

The height and pitch chosen for this case are the same asthose of the previous case. The average porosity for thiscase varies from 1 to 0 as the size of the tube changes

Fig. 2 2D rectangular rib type roughness

Fig. 3 2D semi-cylindrical rib type roughness Fig. 4 Dimensionless fluid velocity profile across the channel

543

from 0 to 20 mm. The friction factor for flow acrosssquare tube banks is developed from Souto and Moyne[21] (Fig. 7). From their work one can deduce that theFanning friction factor is

ff ¼1

3

Dp0

DL0¼ 102:388

3:0 � 32 Repor

¼ 54:3

Repor

As shown in Fig. 8 the momentum resistance of theVAT model is in very close agreement with the results ofSouto and Moyne [21].

The calculated velocity distribution for differentchannel porosities within the square tube banks at thesame pressure gradient demonstrates that the VATmodel does not have limitations as the porosity rangesfrom zero to unity. Calculations for < m> rangingfrom 0.05 to 0.75 and the results are shown in Fig. 9.Others can be found in [19]. The VAT model is applic-able for both laminar and turbulent cases.

5.5 Heat sink

The conservation equations to be solved are the result ofapplying VAT to the Navier-Stokes equations and theenergy equations for both fluid and solid. The char-acteristic parameters used to scale the governing equa-tions are the following:

uc ¼ � lc

qf

dpdx

!12

; lc ¼ dpor ¼4 mh iSw

; mc ¼ lcuc;

Tc ¼ Tb � Tm

The resulting dimensionless equations are the mo-mentum equation:

@

@z�m� z�ð Þh i m�T þ

1

Repor

!@~�u�

@z

!

¼ 2ff ~�u2

m0þ 1

m0; ð49Þ

turbulent kinetic energy equation

~m�T@~�u�

@z�

� �2

þ @

@z�~m�TPrTþ 1

Repor

!@b�

@z�

" #

þ ff ~�u3

m0

þ 2

Repor

@b�1=2

@z�

� �2

¼ Clb�2

~m�T; ð50Þ

the relationship between the eddy viscosity and theturbulent kinetic energy

~m�T ¼ ClL� z�ð Þffiffiffiffiffib�p

; ð51Þ

fluid phase energy equation

~�u�@~�T �

@x�¼ @

@z�m�ðz�Þh i m�T

PrTþ 1

Repor

!@~�T �

@z�

" #

� 4Nu

Pepor~T � � T �s� �

; ð52ÞFig. 6 Velocity profile for laminar flow across a lattice of sphericalbeads

Fig. 5 Channel flow across alatice of spherical beads

544

and the solid phase energy equation

@

@x�1� m0 m� z�ð Þh ið Þ @T �s

@x�

� �

þ @

@z�1� m0 m� z�ð Þh ið Þ @T �s

@zs

� �

¼ 4Nukk

ksT �s � ~T �f� �

ð53Þ

To do meaningful optimization of heat removal de-vices, the additional terms need to be treated so that thelowest level morphology effects are carried to the upperscale, which is of interest to the end user of such a de-vice. To be convinced of the importance of such an ef-fort, one only needs to review the literature or attempt tomake sense out of the wide variation in reported results.

In this work, we do not solve the complete set ofequations with the turbulent kinetic energy equation.Instead, we rely on the correlations for heat transfer tobridge the gap. The main reason for this is that thepresently available correlations for heat transfer andfriction include turbulence and we would be doublecounting some of the effects. At a future date we willpursue a more complete effort.

5.6 Simulation validation

To illustrate the validity of the present mathematicalmodel and numerical scheme, the calculated local Nus-selt numbers were compared with correlations fromZukauskas [22]. To compare with Zukauskas, the Nus-selt number is defined as

Nu ¼ ~adporkf¼ dpor

kf

1

DXSw ~T � Ts� �

R

@Sw

kfd�Tdxi

d~s ð54Þ

Eq. 54 is simplified with the help of Eq. 52 to

Nupor ¼~adpor

kf

¼ dporkf

cpf qf mh i ~u zð Þ @~T x; zð Þ@x � @

@z m zð Þh i kf@~T x; zð Þ@z

h i

Sw zð Þ Ts x; zð Þ � ~T x; zð Þ� �

ð55Þ

Fig. 7 Channel flow acrosssquare tube banks

Fig. 8 Rod bundle friction factor

545

Using the calculated temperature distribution, Nu{poris found by substituting temperature values into equa-tion (55). The pin fin surface heat transfer results werecompared with those obtained from Zukauskas [22], andthe end wall and pin values with Rizzi and Catton [23].Comparison with the results of Rizzi and Catton areshown in Fig. 10. More complete comparisons can befound in Ref. [24]. As can be seen, the comparisons areexcellent.

5.7 Optimization parameters

The heat transfer per unit volume and heat sinkeffectiveness are chosen for optimization. The heat sinkeffectiveness is a measure of how much heat transferresults from the pumping power applied (Fig. 11). Thepumping power per unit volume is defined

Pp ¼PX¼ _mDp

qf X¼ ff Re

3

porhmyzi

S�all4

hmi4

!l3

128q2f

Wm3

� �ð56Þ

The heat transfer rate per unit volume is given by

Hr ¼Sba�wX¼ Nub

kf S�all4hmi S�b ;

Wm3K

� �ð57Þ

The heat sink effectiveness is the ratio of heat removalper unit volume to pumping power per unit volume,

Eeff ¼Hr

Pp¼ Nub

ffRe3por32

S�bhmyzi

hmi3

S�3all

!kf q2

f

l3

" #

;1

K

� �ð58Þ

The variables in these equations are defined in Table 1.Six independent geometric parameters (see Table 2)

can be chosen for the simple heat sink.The parameters in the VAT equations are related to

the geometric parameters by the following relationships:

mh i¼1�p4

d2

P 2; Sw¼p

d2

P 2

1

d; dpor¼

4

pP 2

d2�1

� �dpin ð59Þ

and

Nui ¼hintdpor

kfð60Þ

where Nu{i is the pin-flow interface Nusselt numberand can be obtained from any number of differentexperimental or numerical sources. Different values areused on the bottom or base surface and the pinsthemselves.

When a problem becomes multi-dimensional, (6D or8D for turbulent flow) it is best to use statistical designof experiment (DOE) methodology. This was done fol-lowing the steps outlined in Fig. 12. A numerical simu-lation was carried out based on statistical selection of theparameter values followed by a statistical analysis of theresults to develop a response surface. This procedurewas implemented using a commercial computer codebased on DOE.

546

When the optimization variable is chosen, the variablesand their ranges are systematically defined. Next, thenumerical experiment design type is selected, e.g. a clas-sical two level,mixed level, or nested level. The design typeused in this work is the classical two level design. Theclassical two level designs are based on standard ortho-gonal arrays that contain two levels for each experimentalvariable. It enables estimation of the effects of some or allterms in a second order model of the general form

Eeff ¼ a0 þ a1X1 þ � � � � þanXn

þ a1; 1X 21 þ a1; 2XxX2 þ � � � � þan; n�1XnXn�1 þ an; nX 2

n

The independent variables Xi are the VAT designvariables. Based on the design type and design variables,experimental design options will be created. Eachoption is a set of input parameters for numerical simu-lation. Description of what was done to obtain the‘‘experimental results’’ from the VAT based laminaror turbulent transport equations for flow in aspecific porous media is described elsewhere (seeRef. 12).

After numerical simulation, the numerical results arerigorously analyzed using statistical analysis tools andgraphics tools. These tools include non-linear response/error analysis, experimental error analysis, regressionanalysis, residuals analysis, two-dimensional graphing,three dimensional response surface graphing, and multiresponse optimization.

6 Results

Three different optimization goals were explored. First,minimum thermal resistance was sought by minimizingthe difference between the maximum plate temperatureand the inlet air temperature, Tw,max � Tin, for a 75 wheat sink. This is followed by maximizing the heat re-moval per unit volume, Hr and then maximizing theoverall effectiveness, E{eff. Following the proceduresoutlined in Fig. 12, the DOE results yield a responsesurface for the parameter ranges given in Table 3. Theresponse surface given is used and maximizing Eeff

yields the desired maximum effectiveness. The valuesgiven in Table 4 are maxima or minima obtained fromthe above equation and its counterparts not shownhere. Figure 13 is an example of a heat sink effective-ness response surface. The three-dimensional figureshows Eeff as a function of two variables when the othervariables are fixed. Although limited by the range of thevariables, the trend of the response surface is clearlyshown in the Fig. 13.

Table 1 Optimization variablesVariable Definition Physical meaning

ff Influence of media resistance to flow

ReporUdpor

m ¼ t Um t 4ðmÞSw

Media Reynolds number

Pe{porUdporaf¼ t

Udporcpf qf

kfMedia Peclet number

< m > DXf

DXfþDXsChannel porosity

Nu{bqwdporðTb�TinÞkf

Heat exchange between phases

Table 2 Geometric parameters

1. Pin diameter d{pin2. Pin pitch P3. Base plate thickness tb4. Base plate width W5. Base plate length L6. Channel (pin) height H

Fig. 10 Comparison of simulated pin fin channel local Nusseltnumber with Rizzi and Catton [23]

Fig. 11 Heat sink model

547

6.1 Acoustic energy absorption

The equations describing acoustic propagation arevolume averaged to produce continuum models forheterogeneous multiphase systems. The resultsfrom the use of VAT are compared with experimentaldata.

The linearized conservation equations for single-phase compressible flow in a rigid porous medium aremass and momentum equation conservation

@p0

@tþ q0c20r � u0 ¼ 0 ; in the fluid - - phase ð62Þ

@u0

@t¼ � 1

q0

rp0 þ m0r2u0; in the fluid - - phase ð63Þ

with boundary conditions

u0 ¼ 0 ; at @Sw and u0 ¼ f x; tð Þ; at @Sfe ð64Þwhere ¶S{w is the inter-facial area of fluid-solid interfaceand ¶S{fe represents all fluid entrances and exits on therepresentative volume averaged volume, Fig. 1.

The VAT version of Eq. 62 is easily found to be

@

@tmf p0f gf

� �þ q0c

20r � mf u0f gf

� �¼ 0 ð65Þ

The VAT version of the momentum equation, Eq. 63, is

mf@

@tu0f gf ¼�

1

q0

mfr p0f gf þ1

DX

Z

@Sw

~p0dS

0

B@

1

CA

þ m mfr2 u0f gf þþ1

DX

Z

@Sw

ðr~u0Þ � dS

0

B@

1

CA

þ m u0f gf � r2mf þrmf � r u0f gf

� �ð66Þ

To simplify Eq. 66, it is assumed that porosity isconstant yielding

mf@

@tu0f gf ¼�

1

q0

mfr p0f gf þ m0mfr2 u0f gf

þ 1

DX

Z

@Sw

� 1

q0

~p0dS þ m0r~u0 � dS� �

ð67Þ

Closure of the integral terms in Eq. 67 are given byEqs. 35 and 36. Equation 67 becomes

mf@

@tu0f gf ¼ �

1

q0

mfr p0f gfþm0mfr2 u0f gf

þ 1

2cd u0f g2f :

ð68Þ

Table 3 Heat sink variables and their ranges

Min value Variable Max value

5000 Repor 100000.04 (in.) dpin 0.40 (in.)1.0 (in.) H 4.0 (in.)1.0 (�) P/dpin 2.0 (�)2 (in.) L 6 (in.)0.08 (in.) tb 0.40 (in.)

Table 4 Optimal values of parameters

Parameter Min (Tw,max � Tin.) Max (Hr) Max (Eeff)

Repor 10,000 10,000 7500dpin 0.1387 0.04 0.04H 1.00 1.00 1.00P/dpor 1.3534 1.0 2.0L 6.00 2.728 2.0tb 0.243 0.087 0.158

Fig. 12 Optimizationprocedure

548

Fig. 13 Effectiveness responsesurface for varying finheight and base thickness(Repor = 7500, dpin = 0.22 in.,P/dpin =1.5)

Fig. 14 Comparison of VATbased predictions of acousticattenuation with experimentaldata

549

The nature of acoustic flow is that pressure-dropchanges dynamically, meaning that pressure drop varieslocally from its minimum to its maximum value as thewave propagates in the medium. Using a model for thedrag coefficient developed by Hersh [26] for fibrous bulkmaterials with very long, small diameter fibers, the dragcoefficient cd is

cd ¼ �8m00 1� mf� �

d2Sw

VnLm; n

d

VpLp; n

d

!1

u0f gfð69Þ

where L{v,n /d and L{p,n /d are length scales associatedwith the fiber orientation. For fibers parallel to the flow

Lm; n

d¼ 1=3:94 1� mf

� �0:4131þ 27 1� mf

� �3h ið70Þ

and for the normally oriented fibers

Lp; n

d¼ 1=16

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� mf

q1þ 14:75 1� mf

� �3h ið71Þ

The specific surface area and porosity for the REV are

Sw ¼pds2

andmf ¼ 1� p4

d2

s2ð72Þ

Plane wave accoustic propagation through porousmedia will be evaluated. The set of governing equations,Eqs. 65 and 68, are reduced to one-dimensional equa-tions,

@

@tp0f gfþq0c20

@

@xu0f gf¼ 0 ð73Þ

@

@tu0f gf¼ �

1

q0

@

@xp0f gfþm0

@2

@x2u0f gf�

cd

2mfu0f g2f ð74Þ

Numerical experiments were carried out for Kevlar29 because data were available from Hersh [26]. Pressuredrop as a function of material thickness is given in theirwork. The mean diameter of the Kevlar fibers isd=0.0013 mm and the corresponding porosity isÆm{fæ=0.965.

Different material thicknesses of the porous mediawere studied with different frequency values and theresults are shown in Fig. 14. The computed resultscompare very well with the measured data. The next stepwill be to optimize the material for broadband absorp-tion.

7 Concluding remarks

The main purpose of this brief review is to demonstratethe value of VAT based mathematical models of channelflow in a porous medium.

Studies of limiting cases where the porosity in achannel approaches unity and a channel full of porousmedia make it clear that there are some mistakes inother studies. The numerical results show that the

model developed in this paper is applicable to thestudy of flow in channels with rough walls or inchannels filled with a regular porous matrix, and de-monstrates that the simplest morphological propertiesof a porous layer such as porosity function and spe-cific surface along with closure models allows thetransport fields in the channel to be determined. Thisparticularly important when the problem is a conjugateproblem because changing flow and heat transfercoefficients is directly coupled to conduction throughthe solid media.

The governing equations developed using rigorousVAT methods are used to optimize surface transportprocesses in support of heat transport technology. Acombination of VAT based equations and the theory ofstatistical design were used to effectively begin treating6D or 8D optimization problems. The importance ofthe parameters was unexpected and is a clear demon-stration of both the need for the type of modelingdeveloped here and the need to exercise it. The VATbased model yields solutions to a conjugate heat sinkproblem rapidly enough that the 70 cases needed todevelop the response surface could be done on a lap-top.

The acoustic energy absorption results demonstratethe usefulness of the VAT based approach tomodeling acoustical processes in heterogeneous mate-rials. The model will be changed to more closely re-present the physical problem presented when one isinterested in the performance of a material and itsability to absorb acoustic energy. Experiments areplanned to strengthen the closure relationships andallow more generality in the types of materialsconsidered.

It is worth noting that non-local mathematicalmodeling is very different from homogenization. Thenew integro-differential transport statements in hetero-geneous media and applications of these non-classicaltypes of equations is a current issue. The theory allowsone to take into consideration characteristics of a mul-ticomponent multiphase composite with perfect as wellas imperfect morphologies and interphases. The trans-port equations obtained using VAT involved additionalterms that quantify the influence of the medium mor-phology and interfaces between phases. Various de-scriptions of the porous media structural morphologydetermine the importance of these terms and the rangeof application of closure schemes. Examples of VATapplication ranging from those presented in this work toelectrodynamics to studies of transport phenomena atsubcrystalline and atomic scales can be found in a reviewarticle by Travkin and Catton [27]. The VAT showsgreat promise as a tool for development of models forthis type of phenomena because it becomes possible toinclude the inherent non-linearity and heterogeneityfound at the subcrystalline level and reflect the impact atthe upper levels or scales.

550

References

1. Anderson TB, Jackson R (1967) A fluid mechanical descriptionof fluidized beds. Int Eng Chem Fundam 6:527–538

2. Whitaker S (1967) Diffusion and dispersion in porous media.AICHE J 13:420–427

3. Kheifets LI, Neimark AV (1982) Multiphase processes in por-ous media. Nadra, Moscow

4. Whitaker S (1999) The method of volume averaging. KluwerAcademic Publishers, Boston

5. Travkin VS, Catton I (1992) Models of turbulent thermal dif-fusivity and transfer coefficients for a regular packed bed ofspheres. Fundam Heat Transfer Porous Media ASME HTD193:15–23

6. Travkin VS, Catton I Porous media transport de-scriptions—non-local, linear and nonlinear against effectivethermal/fluid properties. Adv Coll Interface Sci 76–77:389–443

7. Slattery JC (1980) Momentum, energy and mass transfer incontinua. Krieger, Malabar

8. Whitaker S (1977) Simultaneous heat, mass and momentumtransfer in porous media: a theory of drying. Adv HeatTransfer 13:119–203

9. Gray WG, Leijnse A, Kolar RL, Blain CA (1993) Mathema-tical tools for changing spatial scales in the analysis of physicalsystems. CRC Press, Boca Raton

10. Rodi W (1984) Turbulence models and their applications inhydraulics—a state of the art review. Int Assoc Hydraulic Res

11. Monin RS, Yaglom AM (1975) Statistical fluid mechanics.MIT Press, Cambridge, MA

12. Travkin VS, Catton I (1995) A two-temperature model forturbulent flow and heat transfer in a porous layer. J Fluids Eng117:181–188

13. Gratton L, Travkin VS, Catton I (1995) The impact of mor-phology irregularity on bulk flow and two-temperature heattransport in highly porous media. Proc ASME/JSME ThermalEng Joint Conf 3:339–346

14. Gratton L, Travkin VS, Catton I (1996) The influence ofmorphology upon two-temperature statements for convectivetransport in porous media. J Enhanced Heat Transfer 3:129–145

15. Kheifets LI, Neimark AV (1982) Multiphase processes in por-ous media. Nedra, Moscow

16. Dullien FAL (1992) Porous media. Fluid transport and porestructure. Academic Press, NY

17. Shvidler ML (1986) Conditional averaging of unsteady perco-lation fields in random composite porous media. Fluid Dyn21(50):735–740

18. Shvidler ML (1989) Average description of immiscible fluidtransport in porous media with small-scale inhomogeneity.Fluid Dyn 24(6):902–909

19. Travkin VS, Catton I, Hu K (1998) Channel flow in porousmedia in the limit as porosity approaches unity, HTD 361-1.ASME 277–284

20. Watanabe H (1989) Drag coefficient and voidage function onfluid flow through granular packed beds. Int J Eng Fluid Me-chan 2(1):93–108

21. Souto HP, Moyne C (1997) Dispersion in two dimensionalperiodic porous media. Phys Fluids 9(8):2243–2263

22. Zukauskas A (1987) Convective heat transfer in cross flow. In:Kakac S, Shah RK, Aung W (eds) Handbook of single-phaseconvective heat transfer, Wiley, NY

23. Rizzi M, Catton I (2002) Experimental results for endwall andpin fin heat transfer coefficients. In: Proceedings of the 12thINHTC

24. Hu K (2002) Flow and heat transfer over rough surfaces inporous media. PhD Thesis, UCLA

25. Travkin VS, Hu K, Catton I (2001) Multi-variant optimizationin heat sink design. In: Proceedings of NHTC’01 35th nationalheat transfer conference, Anaheim, CA, June 10–12

26. Hersh AS (1980) Acoustical behavior of homogeneous bulkmaterials. In: AIAA 6th aeroacoustics conference, Hartford,CN

27. Travkin VS, Catton I (1999) Transport phenomena in hetero-geneous media based on volume averaging theory. Adv HeatTransfer 34:1–143

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