in

u 43gy,, Ch

Keywords:Effective thermal conductivity

pery wcon

and theoretical data for packed beds, porous foams and wire screens. Secondly, an general expression for

y of aermal

hydrodynamic mixing of the uid particles passing through pores.This thermal dispersion causes additional heat transfer, whicharises further complications in dealing with transport processesin uid-saturated porous media.

Accurate estimation of effective stagnant thermal conductivityof porous media has been continuously investigated since the pio-

They found that their analysis agrees fairly well with the experi-mental data. Chang [12], on the other hand, developed a compara-tively simple theoretical model based on combined series andparallel conduction within uid saturated wire screens. He con-cluded that his analytical model provides better accuracy thanthe existing correlations. Excellent reviews on this topic may befound in Kaviany [13] and Hsu [14].

In the rst part of this paper, we shall consider macroscopicheat conduction in porous media in a somewhat more generalway by appealing to a volume averaging procedure [1517]. Weshall introduce a unit cell model, which consists of rectangular sol-

* Corresponding author. Address: Department of Mechanical Engineering,Shizuoka University, 3-5-1 Johoku, Hamamatsu 432-8561, Japan. Tel./fax: +81 53478 1049.

International Journal of Heat and Mass Transfer 53 (2010) 32223230

Contents lists availab

International Journal of H

.eE-mail address: tmanaka@ipc.shizuoka.ac.jp (A. Nakayama).mal dispersion conductivity. For the case of pure heat conduction,the thermal dispersion conductivity is absent. In such heat conduc-tion case, it is crucial to estimate the effects of tortuosity on thestagnant thermal conductivity, since the thermal conductivity ofthe solid phase is generally much higher than that of the uidphase, and thus, whether the solid is connected or not inuencesthe heat conduction mode signicantly. Thermal dispersion, onthe other hand, is the spreading of heat caused by variations inuid velocity about the mean velocity. In addition to the molecularthermal diffusion, there is signicant mechanical dispersion in heatand uid ow in a uid-saturated porous medium, as a result of

of an appropriate composite model for the estimation of effectivestagnant thermal conductivity. Such attempts include Rayleigh[2], Deissler and Boegli [3], Kunii and Smith [4], Zehner and Schlun-der [5], Nozad et al. [6], Sahraouni and Kaviany [7] and Hsu et al.[8,9]. Hsu et al. [9] introduced a one-dimensional conduction mod-el based on in-line touching cubes, and carried out an elegant anal-ysis to show good agreement with the experimental data for thecase of packed beds. As for consolidated porous media, Calmidand Mahajan [10,11] conducted a comprehensive study on theeffective thermal conductivity of high porosity brous metalfoams, introducing a one-dimensional heat conduction model.Porous mediaTortuosityDispersion

1. Introduction

An effective thermal conductivitmedium is composed of stagnant th0017-9310/$ - see front matter 2010 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2010.03.004the thermal dispersion conductivity has been sought with help of the two energy equations for solid anduid phases, derived on the basis of a volume averaging theory. It has been shown that the interstitialheat transfer between the solid and uid phases is closely associated with the thermal dispersion. Theresulting expressions for the longitudinal and transverse thermal dispersion conductivities agree wellwith available experimental data and empirical correlations.

2010 Elsevier Ltd. All rights reserved.

uid-saturated porousconductivity and ther-

neering work of Maxwell [1]. Standard treatment of heat conduc-tion in porous media is based largely on the lumped mixturemodel under the assumption of local thermal equilibrium condi-tion. In this treatment, the problem reduces to the constructionAvailable online 26 March 2010angular solids with connecting arms in an in-line arrangement. The validity of the expression for the stag-nant thermal conductivity has been conrmed comparing the present results with available experimentalA synthesis of tortuosity and dispersionof porous media

C. Yang a,b, A. Nakayama a,c,*aDepartment of Mechanical Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsb School of Energy and Power Engineering, Huazhong University of Science and TechnolocDepartment of Civil Engineering, Wuhan Polytechnic University, Wuhan, Hubei 430023

a r t i c l e i n f o

Article history:Received 7 October 2009Received in revised form 19 January 2010Accepted 19 February 2010

a b s t r a c t

Effects of tortuosity and disare investigated analyticalleffective stagnant thermal

journal homepage: wwwll rights reserved.effective thermal conductivity

2-8561, JapanWuhan, Hubei 430074, Chinaina

sion on the effective thermal conductivity of uid-saturated porous mediaith help of a volume averaging theory. Firstly, a general expression for theductivity has been derived using a unit cell model, which consists of rect-

le at ScienceDirect

eat and Mass Transfer

l sevier .com/locate / i jhmt

[15]). Under this condition, the volume average of a certain variable

of Hids with connecting arms in an in-line arrangement. This generalmodel allows us to represent most homogeneous porous media,which may or may not have directional dependence. Packed beds,porous foams and wire screens may easily be described by settinggeometrical parameters in the model accordingly. The terms asso-ciated with the tortuosity in porous media are evaluated, by faith-fully carrying out surface integrations with respect to the localtemperature over the interface between the solid and uid phases.

In the second part of this paper, we shall investigate the thermaldispersion conductivity, which becomes predominant for convec-tive ows in porous media. According to Wakao and Kaguei [18],Yagi et al. [19] were the rst to measure the effective longitudinalthermal conductivities of packed bed, taking full account of the ef-fect of thermal dispersion, and eventually found the longitudinalcomponent of the dispersion coefcient much greater than itstransverse component. Since the famous analytical treatment in atube by Taylor [20], a number of theoretical and experimental ef-forts (e.g. Aris [21], Koch and Brady [22], Han et al. [23] and Vort-meyer [24]) were made to establish useful correlations for

Nomenclature

A surface area (m2)Aint interface between the uid and solid (m

2)af specic surface area (1/m)c specic heat (J/kg K)cp specic heat at constant pressure (J/kg K)C size of the touching arm (m)D size of the solid (m)F, f, g prole functions ()H size of the cell (m)hf interfacial heat transfer coefcient (W/m

2 K)k thermal conductivity (W/m K)L external scale (m)nj unit vector pointing outward from the uid side to solid

side ()Pr Prandtl number ()q heat ux (W/m2)Ra;b;c;f ;s thermal resistances

C. Yang, A. Nakayama / International Journalestimating the effective thermal conductivities due to thermal dis-persion (see Kaviany [13]). Furthermore, a series of numericalexperiments were conducted by Kuwahara et al. [25] and Kuwaha-ra and Nakayama [26], assuming a macroscopically uniform owthrough a lattice of rods, so as to elucidate the effects of micro-scopic velocity and temperature elds on the thermal dispersion.It is also worthwhile to mention that Nakayama et al. [27] deriveda thermal dispersion heat ux transport equation from the volumeaveraged version of NavierStokes and energy equations andshowed that it naturally reduces to an algebraic expression forthe effective thermal conductivity based on a gradient-type diffu-sion hypothesis.

In this paper, we shall revisit the volume average version of thetwo energy equations for solid and uid phases, derived on the ba-sis of the volume averaging theory [28]. We shall follow the deni-tion of thermal dispersion heat ux to evaluate the longitudinaland transverse components of the thermal dispersion conductivity,exploiting the derived two energy equations. It will be shown thatthe interstitial heat transfer is closely linked with the thermal dis-persion phenomena. A general expression will be derived using theinter-relationship found between the interstitial heat transfer andthermal dispersion. The validity of general expressions for bothstagnant thermal conductivity and thermal dispersion conductivitywill be conrmed comparing the present results with availableexperimental and theoretical data.2. Macroscopic energy equations

We shall consider the energy equation for the uid phase andthat for the solid matrix phase as follows:

For the uid phase:

qf cpf@T@t qf cpf

@

@xjujT @

@xjkf

@T@xj

1

For the matrix phase:

qscs@T@t @

@xjks

@T@xj

2

where the subscripts f and s stand for the uid phase and solidphase, respectively. In order to obtain the macroscopic energy equa-tion for uid-saturated porous media, we take a control volume Vwithin in the medium, whose length scale V1=3 is much smaller thanthe macroscopic characteristic length, but, at the same time, muchgreater than the structural characteristic length (see e.g. Nakayama

T temperature (K)V representative elementary volume (m3)xi Cartesian coordinates (m)x, y, z Cartesian coordinates (m)e porosity ()g dimensionless vertical coordinate ()q density (kg/m3)

Special symbols~/ deviation from intrinsic averageh/i Darcian averageh/if ;s intrinsic average

Subscripts and superscriptsf uids solid

eat and Mass Transfer 53 (2010) 32223230 3223/ in the uid phase is dened as

h/if 1Vf

ZVf

/dV 3

where Vf is the volume space which the uid phase occupies. Theporosity e Vf =V is the volume fraction of the uid space. Follow-ing Nakayama [15], Cheng [16], Quintard and Whitaker [17] andmany others, we decompose a variable / into its intrinsic averageand the spatial deviation from it:

/ h/if ~/ 4We shall exploit the following spatial average relationships:

h/1/2if h/1if h/2if h~/1~/2if 5@/@xi

f 1e@eh/if@xi

1Vf

ZAint

/ni dA 6

where Aint is the local interfacial area between the uid and solidmatrix, while ni is the unit vector pointing outward from the uidside to solid side. Similar relationships hold for the solid phase,whose intrinsic average is dened as

h/is 1Vs

ZVs

/dV 7

parallel model, namely, ekf 1 eksdij. Thus, it is the tortuosityterm (i.e. the second term) that adjusts the level of the effectivestagnant thermal conductivity from its upper bound to a correctone.

3. Unit cell model and effective stagnant thermal conductivity

The present unit cell model of rectangular solid is illustrated inFig. 1. We assume that an innite number of these rectangular sol-ids are arranged in a regular fashion. The dimensions of the unitcell and rectangular solid are indicated as Hx; Hy; Hz andDx; Dy; Dz, respectively, while the sizes of touching arms ofsquare crosssection are denoted by Cx; Cy; Cz, respectively.

We shall apply Eq. (13) to this unit model so as to nd out theeffective stagnant thermal conductivity. The axial component ofthe surface integral term in the second term in the rightmostexpression in Eq. (13) may be estimated as follows:

~i 1V

ZAint

Tni dA

* +~i 1

HxHyHz

Z Hz0

Z Hy0

Z Hx0

1Vx;y;z

ZAint

Tni dA

!dxdydz

f Heat and Mass Transfer 53 (2010) 32223230Upon integrating Eqs. (1) and (2) over the local control volume withhelp of the foregoing relationships, we obtain the volume averagedset of the energy equations as follows [28]:

For the uid phase:

qf cpf e@hTif@t qf cpf e

@hujif hTif@xj

@@xj

ekf@hTif@xj

kfV

ZAint

Tnj dA qf cpf eh~u~Tif !

1V

ZAint

kf@T@xj

nj dA 8

For the solid matrix phase:

qscs1 e@hTis@t @

@xj1 eks @hTi

s

@xj ks

V

ZAint

Tnj dA

!

1V

ZAint

kf@T@xj

nj dA 9

where hTif is the intrinsic average of the uid temperature, whilehTis is the intrinsic average of the solid matrix temperature. Obvi-ously, the parenthetical terms on the right hand side of Eq. (8) de-note the diffusive heat transfer, while the last term describes theinterfacial heat transfer between the solid and uid phases. Wecombine the foregoing two energy equations to obtain a single mac-roscopic equation as follows:

@eqf cpf hTif 1eqscshTis@t

eqf cpf@hujif hTif

@xj

@@xj

ekf@hTif@xj1eks @hTi

s

@xjkf ks

V

ZAint

TnjdAeqf cpf h~ueT if !

10where we exploited the continuity of temperature and heat ux atthe interface. Further assuming that the local thermal equilibriumcondition holds between both phases, namely, hTif hTis hTi,the equation reduces to

eqf cpf 1 eqscs@hTi@t qf cpf

@hujihTi@xj

@@xj

ekf 1 eks @hTi@xj kf ks

V

ZAint

Tnj dA eqf cpf h~ujeT if !

11where

h/i 1V

ZV/dV 12

is the Darcian average of the variable / such that huji ehujif is theDarcian velocity vector. Thus, we nd the macroscopic heat uxvector qj and its corresponding total effective thermal conductivitytensor ktotalij as follows:

qiktotalij@hTi@xj

ekf 1eks@hTi@xikf ks 1V

ZAint

Tni dA

* +eqf cpf h~uieT if

13The last term in the rightmost expression describes the thermal dis-persion term, while the second associated with the surface integral

3224 C. Yang, A. Nakayama / International Journal odescribes the effects of the tortuosity on the macroscopic heat ux.Note that the rst term in the expression corresponds to the upperbound of the effective stagnant thermal conductivity based on theZDyDzC2x

TjxDx2 TjxDx2

dAC:V :1

HxDxH2xHyHz

ZDyDzC2x

TjxDx2 TjxDx2 Hx

dAC:V :2

DxH2xHyHz

ZCyHyDy

TjxCy2

TjxCy2

dAC:V :1

HxCyH2xHyHz

ZCyHyDy

TjxCy2Tj

xCy2 Hx

dAC:V :2

CyH2xHyHz

ZCzHzDz

TjxCz2 TjxCz2

dAC:V :1

HxCzH2xHyHz

ZCzHzDz

TjxCz2 TjxCz2 Hx

dAC:V :2

CzH2xHyHz

14

Upon noting that ni is the unit vector pointing outward from theuid side to solid side, we considered the two distinctive controlvolumes, namely, C.V.1 and C.V.2, as shown in Fig. 2, with their cor-responding weights, to evaluate the volume average of the surfaceintegral term associated with the interfacial temperatures.

The temperature differences between the opposing interfacesmay be estimated considering the heat currents, namely,Fig. 1. Unit cell model.

the resistances Ray to Rfz in Eqs. (20b) and (20c). Moreover, theporosity of the unit cell of rectangular solid may be evaluated from

of HQax ; Qbx ; Qcx ; Qsx and Qfx , as indicated in Fig. 1, and their corre-sponding thermal circuits as shown in Fig. 3, such that

TjxDx2 TjxDx2

QaxDxDyDz C2x ks

1Rax

qxHyHzDx1Rax 1Rsx

1Rbx 1Rcx

1Rfx

DyDz C2x ks

15

where

Rax Hx Dx

kf Dx

ks

1

DyDz C2x16a

Rsx HxksC

2x

16b

Rbx Hx Cy

kf Cy

ks

1

CyHy Dy 16c

Rcx Hx Cz

kf Cz

ks

1

CzHz Dz 16d

Rfx Hxkf

1HyHz DyDz CyHy Dy CzHz Dz 16e

The other temperature differences may be evaluated likewise.Substituting these temperature differences into Eq. (14), we

nd

~i 1Z

Tn dA

* +~i 1

Z Hz Z Hy Z Hx 1 ZTn dA

!dxdydz

Fig. 2. Two distinctive control volumes for the surface integral.

C. Yang, A. Nakayama / International JournalV Ainti HxHyHx 0 0 0 Vx;y;z Aint

i

1Rax

1Rax 1Rsx

1Rbx 1Rcx

1Rfx

qx1ks 1kf

HxDxDxH2x

1Rbx

1Rax 1Rsx

1Rbx 1Rcx

1Rfx

qx1ks 1kf

HxCyCyH2x

1Rcx

1Rax 1Rsx

1Rbx 1Rcx

1Rfx

qx1ks 1kf

HxCzCzH2x

17

The foregoing expressions are substituted into the denition of themacroscopic heat ux, namely, Eq. (13) without the dispersionterm, to nd

qxkstagxx@hTi@x ekf 1eks

@hTi@x

kf ks 1ks 1kf

qx1Rax 1Rsx

1Rbx 1Rcx

1Rfx

1

Rax1Dx

Hx

DxHx 1Rbx

1CyHx

CyHx 1Rcx

1 CzHx

CzHx

18

Upon solving for the macroscopic heat ux qx, we haveqx kstagxx@hTi@x ekf 1 eks

1 kfks2

kskf

1Rax

1DxHx DxHx 1Rbx 1CyHx

CyHx 1Rcx 1

CzHx CzHx

1Rax 1Rsx

1Rbx 1Rcx

1Rfx

@hTi@x

19which gives the effective stagnant thermal conductivity as follows:

kstagxx ekf 1 eks

1 kfks2

kskf

1Rax

1DxHx DxHx 1Rbx 1CyHx

CyHx 1Rcx 1

CzHx CzHx

1Rax 1Rsx

1Rbx 1Rcx

1Rfx

20a

Likewise,

kstagyy ekf 1 eks

1 kfks2

kskf

1Ray

1DyHy

DyHy 1Rby

1CzHy

CzHy 1Rcy 1

CxHy

CxHy

1Ray 1Rsy

1Rby 1Rcy

1Rfy

20b

kstagzz ekf 1 eks

1 kfks2

kskf

1Raz

1DzHz DzHz 1Rbz 1CxHz CxHz 1Rcz 1

CyHz

CyHz

1Raz 1Rsz

1Rbz 1Rcz

1Rfz

20c

The set of Eq. (16) should be used with cyclic permutation to obtain

Fig. 3. Thermal circuit.eat and Mass Transfer 53 (2010) 32223230 3225e HxHyHz DxDyDz C2x Hx Dx C2yHy Dy C2z Hz Dz

HxHyHz21

4. Validation of the expression for stagnant thermalconductivity

In order to examine the generality acquired in our mathemati-cal model for estimating the effective stagnant thermal conductiv-ity, we shall apply our model to evaluate the effective stagnantthermal conductivities of various porous media, such as packedbeds, metal foams and wire screens.

4.1. Packed beds

Hsu et al. [9] introduced a unit cell model of touching cubes in aregular arrangement and carried out a lumped mixture modelanalysis and predicted the effective stagnant thermal conductivityof the packed beds. They showed that their model of cubes in a reg-ular arrangement is a rational model to describe heat transfer in

the packed beds. Albeit the difference in its appearance, our gen-eral expression based on the volume averaging theory, when ap-plied to their geometry, turns out to be mathematically identicalto their expression based on the lumped mixture model. Thus, asfor the rst example, we shall predict the effective stagnant ther-mal conductivity of the packed beds. The geometrical parametersfor the packed beds may be set following Hsu et al. [9]:

Hx Hy Hz H; Dx Dy Dz D;Cx Cy Cz C 0:13D 22a

e 1 DH

3 3 0:132 D

H

21 D

H

0:36 22b

We have approximated the collection of the spherical particles inthe packed bed by the collection of the cubes with touching armsof square cross section. Note that its porosity e 0:36 is set tothe same as that of their packed beds. Eq. (22a) together with

our general model to high porosity metal foams, by setting the geo-metrical parameters as

Hx Hy Hz H; Dx Dy Dz Cx Cy Cz D 24a

e 1 3 DH

2 2 D

H

324b

Note that the metal foam structure was approximated by a unit cu-bic cell structure according to Eq. (24a), and that the porosity wasset equal to that of the corresponding metal form. Eq. (24b) forthe porosity of the unit cubic cell may readily be derived as we sub-tract the square rod volume per unit 3H DD2 D3 from theunit volume H3 to obtain the space occupied by the uid per unitcell, then divided it by H3.

In Fig. 5, our prediction based on Eq. (20a) is shown along withCalmidMahajan expression based their hexagonal model [10] forks=kf 357 and 8226, over the porosity range from 0.90 to 0.98. Ascan be seen from the gure, both predictions agree with the exper-

3226 C. Yang, A. Nakayama / International Journal of Heat and Mass Transfer 53 (2010) 32223230(22b) gives

D=H 0:86 and C=H 0:11 23As we substitute foregoing values into Eq. (20a), the ratio of theeffective stagnant thermal conductivity kstagxx=kf may readily beevaluated for each value of the solid to uid thermal conductivityratio, ks=kf . Such a curve showing kstagxx=kf is generated from Eq.(20a) and plotted in Fig. 4 along with the experimental data of Noz-ad et al. [6] for the case of the packed beds. The prediction followsthe experimental data closely over a wide range of the thermal con-ductivity ratio. Note that the size ratio of the touching arm to therectangular solid, namely, C/D, accounts for the contact resistancebetween the particles. Naturally, the resulting stagnant thermalconductivity is fairly sensitive to the ratio C/D. Empirical informa-tion on the contact resistance is needed to set the ratio.

4.2. Metal foams

Our general set of Eqs. (20) can also be used for evaluating theeffective stagnant thermal conductivities of high porosity metalfoams. In reality, the structure of metal foams is quite complex.Therefore, it may not be practical to describe all details of thestructure accurately. Calmid and Mahajan [10] approximated sucha complex structure, introducing a hexagonal structure of metalfoam matrix. In this way, they were able to obtain the expressionfor the effective stagnant thermal conductivity exploiting a one-dimensional heat conduction concept. Their prediction agrees verywell with available experimental data. Alternatively, we may applyFig. 4. Effective stagnant thermal conductivity of packed beds.imental data very well.

4.3. Wire screens

As for the third example, we shall consider wire screens and ap-ply our general model to estimate their effective stagnant thermalconductivities. A multiple layer of square-mesh standard wirescreens is depicted in Fig. 6.

The multiple layers of wire screens of this kind are often usedfor heat pipe applications. Thus, it is essential to estimate the effec-tive thermal conductivity of wire screens, as we use them forvarious heat transfer applications. Chang [12] proposed a compar-atively simple theoretical model based on combined series andparallel conduction and showed that his equation gives fairly goodagreement with available experimental data. Our general Eq. (20a)may also be applied for this case, in order to estimate the effectivethermal conductivity of the multiple layers of wire screens. A care-ful observation of the cross section of the wire screen, as shown inFig. 6 prompts us to set the geometrical parameters as follows:

Hx Hy Hz H; Dx=2 Dy Dz D;Cx 0:032D; Cy D; Cz D 25a

e 1 2 DH

3 0:0322 D

H

21 2D

H

2 D

H

21 D

H

0:70

25bsuch that D=H 0:39. We estimated the touching area asCx 0:032D for a multiple layers of wire screens.Fig. 5. Effective stagnant thermal conductivity of metal foams.

of HC. Yang, A. Nakayama / International JournalNote that the wire screen structure was approximated by a unitcell structure of a rectangular prism 2D D D with four thickarms D D H D=2 and two thin arms 0:032D 0:032DH 2D=2 as given by Eq. (25a), and that the porosity was setequal to that of the corresponding wire screens. Eq. (25b) for theporosity of the unit cell may be derived as we subtract the solidvolume per unit 2D3 0:0322D2H 2D 2D2H D from theunit volume H3 to obtain the space occupied by the uid per unitcell, then divided it by H3.

Our prediction based on the general Eq. (20a) is illustrated inFig. 7, which indicates fairly good agreement among our predic-tion, Changs prediction and the experimental data of the wirescreens reported by VanSant and Malet [29]. This suggests thatour general set of Eqs. (20) is valid even for the prediction of effec-tive stagnant thermal conductivity tensors in anisotropic porousmedia.

5. Expressions for thermal dispersion conductivity

Even when the local thermal equilibrium holds at the macro-scopic scale, the interstitial heat transfer takes place between thesolid and uid phases. In other words, the interface at the porescale is never at local thermal equilibrium. In fact, it is this interfa-cial heat transfer at the local non-thermal equilibrium that controlsthe spatial distribution of the temperature at the pore scale andthus the thermal dispersion activities.

Upon introducing the interstitial heat transfer coefcient hf andthe effective porosity tensor, eij, as function of the stagnant thermal

conductivity tensor, kstagij kstagxx i ;~i kstagyy ~j;~j kstagzz ~k;~k, the

two energy equations (8) and (9) may be rewritten, as follows:

Fig. 6. Schematic views of wire screens.For the uid phase:

qf cpf e@hTif@t qf cpf

@hujihTif@xj

@@xjejkkf ekdisjk

@hTif@xk

af hf hTif hTis

26

For the solid matrix phase:

qf cpf 1 e@hTis@t @

@xjdjk ejkks

@hTis@xk

af hf hTis hTif

27where the effective porosity tensor is dened as

ejk edjk ekf 1 eksdjk kstagjk

ks kf 28

such that ejkkf djk ejkks kstagjk , while the dispersion thermalconductivity tensor is dened by

Fig. 7. Effective stagnant thermal conductivity of a multiple layer of wire screens.

eat and Mass Transfer 53 (2010) 32223230 3227qf cpf h~ujeT if kdiskj @hTif@xk 29Moreover, the interstitial heat transfer between the uid and solidphases is modeled according to Newtons cooling law as

1V

ZAint

kf@T@xj

nj dA af hf hTif hTis

30

where af and hf are the specic surface area and interfacial heattransfer coefcient, respectively.

5.1. Longitudinal thermal dispersion conductivity

The energy Eq. (26) at the steady state may be written along themacroscopic ow direction x as follows:

qf cpf hui@hTif@x af hf hTif hTis 31

We have dropped the diffusion term, since the convection term onthe left hand side predominates over the diffusion term, as the ther-mal dispersion becomes appreciable. A magnitude analysis revealsthat the diffusive term in Eq. (26) may be neglected when the Pecletnumber based on an external scale L and dispersion thermal con-ductivity kdis, namely, qf cpf huiL=kdis, is sufciently large. As will beshown later (see Eq. (46)), we have kdis qf cpf huiD for convective

ows. Hence, qf cpf huiL=kdis L=D 1 and Eq. (31) may readily besatised for most convective ow cases. Exploiting the foregoingequation, the longitudinal thermal dispersion term may be evalu-ated as follows:

qf cpf h~ueT if qf cpf huif hTif hTishf 1g 1if qf cpf hui

2

eaf hfhf 1g 1if @hTi

f

@x32

Hence,

kdisxx qf cpf hui2

eaf hfhf 1g 1if 33

where

u huif f g 34a

Kagueis empirical equation [18] for the interstitial heat transfercoefcient for packed beds, namely,

hfDkf 2:0 1:1Pe0:6D =Pr0:27 38

we obtain the longitudinal thermal dispersion in the laminarregime as follows:

kdisxxkf 0:15 Pe

2D

2:0 1:1Pe0:6D =Pr0:27: Laminar regime 39

The foregoing equation gives kdisxx / Pe2D for small Peclet number,which is consistent with Taylor and Aris analysis [20,21]. The equa-tion also implies the existence of the transition regime from laminarto turbulent, in which we have kdisxx / Pe1:4D .

A similar procedure can be taken to estimate the longitudinal

3228 C. Yang, A. Nakayama / International Journal of Heat and Mass Transfer 53 (2010) 32223230and

T hTis hTif hTisgg 34bWe shall consider a convective ow through cubes in a regulararrangement as shown in Fig. 8, where the dimensionless coordi-nate g is dened as

g 2y=H D 35The cubes are assumed to have no touching arms, since the effectsof such arms on the thermal dispersion may be negligibly small forpacked beds of our interest. For the rst approximation, we may as-sume the following prole functions describing the laminar fullydeveloped velocity and temperature proles:

f g 321 g2 36a

and

gg 5165 6g2 g4 36b

Noting that h/if 12R 11 /dg such that hf if hgif 1, and substitut-

ing the foregoing proles into (33), we readily obtain

kdisxxkf 314qf cpf hui2eaf hf kf

128e1 e

Pe2Dhf Dkf

37where PeD qf cpf huiD=kf is Peclet number based on the Darcianvelocity and particle diameter. Note also af 61 e=D for spher-ical particles of size D. Assuming e 0:4 and using Wakao andFig. 8. Cubes in a regular arrangement and prthermal dispersion for the case of fully turbulent ow, using thewall laws as

u us 1j lnn B

40

and

T hTis qwrTusqf cpf

1jlnn A

41

where us and qw are the friction velocity and wall heat ux, respec-tively, and n usn=mf is the dimensionless distance measuredfrom the wall surface n HD2 y. j is the von-Karman constantwhile both B and A are the empirical constants. It is easy to nd

~u usjln f 1 42

and

eT qwrTusqf cpfj

ln f 1 43

where

f 1 g 44Using these prole functions (42) and (43), we have

qf cpf h~ujeT if rTqwj2 hln f 12if rTqwj2 rTqf cpf huij2af @hTif

@x45

where we used Eq. (31) to eliminate qw hf hTif hTis in favor ofthe temperature gradient. Setting j and rT to 0.41 and 0.9, respec-evailing velocity and temperature elds.

6. Validation of the expressions for longitudinal and transversethermal dispersion conductivities

In Fig. 9, our results for the case of packed beds are presented interms of the ratio of the total longitudinal effective thermal con-ductivity to the uid thermal conductivity, namely,

ktotalxxkf kstagxx ekdisxx

kf54

The present results obtained for the cases of (Pr, ks=kf ) = (0.71,53.28) and (7.02, 2.30) are compared with the experimental data re-ported by Fried and Combarnous [31] and the empirical equationekdisxx=kf 0:5PeD proposed by Wakao and Kaguei [18] for high Pec-let number. The stagnant thermal conductivity kstag was evaluatedusing the general Eq. (20a) along with Eqs. (22a) and (22b), whichwas also added to their experimental data to show the total longi-tudinal effective thermal conductivity. Fairly good agreement canbe seen between the present correlation and the experimental data.

Likewise, the correlations established for the transverse thermaldispersion conductivity are presented along with the experimentaldata reported by Fried and Combarnous [31] for the transversethermal dispersion conductivity. The present expressions appearto be in good accord with the experimental data (see Fig. 10).

of Heat and Mass Transfer 53 (2010) 32223230 3229tively, according to Launder and Spalding [30], we obtain for theturbulent regime:

kdisxxkf rTqf cpf hui

j2af kf rT61 ej2 PeD 1:5PeD : Turbulent regime

46Thus, the thermal dispersion conductivity increases in proportion toPeclet number for the turbulent regime. It should be noted that, forthe case of turbulent ow, the turbulence thermal conductivityshould be added in addition to the thermal dispersion conductivity.However, the recent study of Nakayama and Kuwahara [31] giveskturb=kf 0:0233PeD=1 e for the case of packed beds, which ismuch smaller than the dispersion thermal conductivity, and thus,can be dropped for the rst approximation.

5.2. Transverse thermal dispersion conductivity

The transverse thermal dispersion must be taken into accountcorrectly, when we evaluate the heat transfer from the boundingwall. Let us consider the energy Eq. (26) close enough to the solidsurface for the convection to be negligible, but sufciently awayfrom the solid surface, such that the transverse thermal dispersiondominates over the stagnant thermal diffusion:

ekdisyyd2hTifdy2

af hf hTif hTis 47

A magnitude analysis reveals that the convection terms in Eq. (26)vanish near the wall due to no-slip conditions and at the same timethat transverse diffusion overwhelms the longitudinal one forL=d2 L=D2 1 where d is the thermal boundary layer thick-ness. Hence, Eq. (47) may be satised for most near wall ow cases.Upon assuming hTif hTis at innity, we integrate the foregoingequation and obtain

dhTifdy

af hfekdisyy

shTif hTis 48

Thus, the transverse thermal dispersion term may be evaluated asfollows:

qf cpf h~veT if qf cpf huif hTif hTishFg 1if qf cpf huif

ekdisyyaf hf

shFg 1if dhTi

f

dy49

Hence,

kdisyy qf cpf hui2

eaf hfhFg 1if 2 50

where

~v huif Fg 51such that hFif 0. It is interesting to note that Eq. (50) obtained forthe transverse thermal dispersion conductivity is identical to Eq.(33) obtained for the longitudinal thermal dispersion conductivity,except the difference in the multiplicative constants, namely,

hFg 1if 2 and hf 1g 1if . Fried and Combarnous experi-mental data [32] suggest that hf 1g 1if is about 20 times lar-ger than hFg 1if 2 such that we propose the followingequations:

kdisyykf 0:0075 Pe

2D

2 1:1Pe0:6D =Pr0:27: Laminar regime 52

C. Yang, A. Nakayama / International Journalkdisyykf 0:075PeD : Turbulent regime 53Fig. 9. Longitudinal effective thermal conductivity of packed beds.Fig. 10. Transverse effective thermal conductivity of packed beds.

The foregoing comparison suggests that the present expressionsfor both longitudinal and transverse thermal conductivities are va-lid for the packed beds. The present expressions as function of theinterstitial heat transfer coefcient is quite general, in the sensethat they can be applied for other porous media as their correla-tions for the interstitial heat transfer coefcient are available.

7. Conclusions

A volume averaging theory was exploited to evaluate both stag-nant thermal conductivity and thermal dispersion conductivitywithin porous media. For the stagnant thermal conductivity, a gen-eral unit cell model, consisting of rectangular solids with connect-ing arms in an in-line arrangement, was proposed to describe mosthomogeneous porous media. The resulting expression for the stag-nant thermal conductivity has been validated by comparing thepresent results with available experimental and theoretical data

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[8] C.T. Hsu, P. Cheng, K.W. Wong, Modied ZehnerSchlunder models forstagnant thermal conductivity of porous media, Int. J. Heat Mass Transfer 37(1994) 27512759.

[9] C.T. Hsu, P. Cheng, K.W. Wong, A lumped parameter model for stagnantthermal conductivity of spatially periodic porous media, ASME Trans. J. HeatTransfer 117 (1995) 264269.

[10] V.V. Calmid, R.L. Mahajan, The effective thermal conductivity of high porositybrous metal foams, ASME Trans. J. Heat Transfer 121 (1999) 466471.

[11] V.V. Calmid, R.L. Mahajan, Forced convection in high porosity metal foams,ASME Trans. J. Heat Transfer 122 (2000) 557565.

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[13] M. Kaviany, Principles of Heat Transfer in Porous Media, Springer-Verlag, NewYork, 1991. pp. 115151.

[14] C.T. Hsu, Heat conduction in porous media, in: K. Vafai (Ed.), Handbook ofPorous Media, Marcel Dekker, Inc., New York, 2000, pp. 171200.

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3230 C. Yang, A. Nakayama / International Journal of Heat and Mass Transfer 53 (2010) 32223230for packed beds, porous foams and wire screens. As for the thermaldispersion conductivity, a general expression has been derivedwith help of the two energy equations for solid and uid phases.It has been revealed that the interfacial heat transfer at the localnon-thermal equilibrium controls the spatial distribution of themacroscopic temperature and thus the thermal dispersion activi-ties. The resulting expressions for the longitudinal and transversethermal dispersion conductivities agree well with available exper-imental data and empirical correlations.

Acknowledgements

The authors express their sincere thanks to Prof. W. Liu of Huaz-hongUniversity of Science andTechnology for supporting this study.Thisworkhas beenpartially supported by theNational KeyBasic Re-search Development Program of China (2007CB 206903).

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A synthesis of tortuosity and dispersion in effective thermal conductivity of porous mediaIntroductionMacroscopic energy equationsUnit cell model and effective stagnant thermal conductivityValidation of the expression for stagnant thermal conductivityPacked bedsMetal foamsWire screens

Expressions for thermal dispersion conductivityLongitudinal thermal dispersion conductivityTransverse thermal dispersion conductivity

Validation of the expressions for longitudinal and transverse thermal dispersion conductivitiesConclusionsAcknowledgementsReferences