THEORETICAL & APPLIED MECHANICS LETTERS 3, 021011 (2013)

Thermal stretching in two-phase porous media: Physical basis forMaxwell model

Xiaohu Yang,1 Tianjian Lu,2, a) and Tongbeum Kim3, b)

1)School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China2)State Key Laboratory for Mechanical Structure Strength and Vibration, Xi’an Jiaotong University, Xi’an 710049,China3)School of Mechanical, Industrial and Aeronautical Engineering, University of the Witwatersrand, Johannesburg,Wits 2050, South Africa

(Received 23 August 2012; accepted 29 January 2013; published online 10 March 2013)

Abstract An alternate yet general form of the classical effective thermal conductivity model(Maxwell model) for two-phase porous materials is presented, serving an explicit thermo-physicalbasis. It is demonstrated that the reduced effective thermal conductivity of the porous media due tonon-conducting pore inclusions is caused by the mechanism of thermal stretching, which is a combi-nation of reduced effective heat flow area and elongated heat transfer distance (thermal tortuosity).c⃝ 2013 The Chinese Society of Theoretical and Applied Mechanics. [doi:10.1063/2.1302111]

Keywords thermal stretching, effective thermal conductivity, porous media, Maxwell model

The effective thermal conductivity of porous mediahas been a long lasting issue, for which a variety offundamental models have been proposed.1–10 Amongstthese models, the analytical model of Maxwell1 typicallyserves as reference, providing good estimation of the de-pendence of effective thermal conductivity upon poros-ity, especially for relatively low porosities.11,12 Builtupon the Maxwell model, numerous alternative modelshave subsequently been proposed, leading to the groupof “Maxwellian models”.3

Maxwell1 originally aimed to determine analyticallythe effective electrical conductivity of a random disper-sion of spherical particles within a continuous medium,assuming sufficient spatial separation of the spheres toexclude mutual interactions amongst them. Its anal-ogous form in heat conduction is often termed the“Maxwell effective thermal conductivity model” for het-erogeneous two-phase media, expressed as

kekc

= 1− 3vp(1− λ)

2 + λ+ vp(1− λ), (1)

where ke denotes the effective thermal conductivity ofthe unit cell, λ = kp/kc, kc and kp are the thermalconductivity of the continuous phase and the dispersephase, and vp is the volume fraction of the dispersephase or, equivalently, the porosity if the disperse phaseis spherical pores. Equation (1) dictates that the pres-ence of low conducting pores (λ < 1) reduces the effec-tive conductivity and, in the limit when the pores arethermally insulating (kp = 0), ke/kc = 1−3/(1+2/vp).The reduction is significant, with ke/kc = 2/3 whenkp = 0 and vp = 1/4.

Despite the common usage of Maxwell andMaxwellian models at least as reference together withthe classical parallel and series models as well as the rel-atively simple form of these models, their physical basis

a)Corresponding author. Email: tjlu@mail.xjtu.edu.cn.b)Email: tong.kim@wits.ac.za.

has not been fully understood. Further, it has not beenfully clarified how the inclusion of low-conducting porescauses significant reduction in the effective thermal con-ductivity of the bulk media. Macroscopically, it was ini-tially thought that the reduction stems mainly from thereduced volume fraction of the conducting phase as thepores are low-conducting. Whilst some studies9,13,14

demonstrated that the shape and orientation of thepores also affect the effective conductivity, the under-lying physical mechanism still remains unclear. Micro-scopically (at the pore level), it has been argued15,16

that the effective conductivity is reduced due to localtemperature distortion in the continuous phase by thepresence of low- (or non-) conducting pores. However,no clear connection has been established between themacroscopic and microscopic analyses on the observedreduction of effective conductivity. This study, there-fore, aims to provide the physical basis connecting thesetwo very different levels by presenting an alternativeyet more general expression of the Maxwell model forlow porosity two-phase porous media containing non-conducting pores. Focus is placed upon the interplaybetween the reduction in effective heat flow area andlocal temperature distortion.

Consider a two-phase porous medium consisting ofspherical pores of equal size (radius R) regularly dis-tributed (square arrays) in a continuous phase (solid)with no mutual physical contact amongst them. Forsimplicity, it is assumed that the pores are thermallynon-conducting (kp = 0), and natural convection withinthe pore as well as thermal radiation is negligible. Itis further assumed that local temperature distortionaround an individual pore does not affect the neighbor-ing ones. Figure 1 depicts schematically a representa-tive unit cell of the porous medium under steady-stateheat conduction. The wall temperature at x = 0 andx = L is fixed at T1 and T2 (T1 > T2 assumed), re-spectively, and the remaining wall boundaries are sym-metric, where the x-axis coincides with the nominal di-rection of heat conduction. Under such conditions, the

021011-2 X. H. Yang, T. J. Lu, and T. Kim Theor. Appl. Mech. Lett. 3, 021011 (2013)

Symmetry face

Constant heat flow line

Isothermal line

Pore

Imaginary

thermaltube

TT

y

xR

L

L/

l↼y↽

kc

b0/

kp

dx

dy

db↼x↪y↽x⇁dx

db↼x↪y↽x

Fig. 1. Schematic of unit cell for two-phase porous media: spherical pore embedded in a cuboid.

three-dimensional problem may be simplified to an ax-isymmetric problem enabling the thermal analysis onan axisymmetric surface consisting of a circular porecentered in a square continuous medium. Whilst theconstant heat-flow lines perpendicularly intersect theconstant temperature lines (isothermal lines), both aredistorted by the presence of the non-conducting porewithin the unit cell.

With the thermal boundary conditions prescribedas shown in Fig. 1, Fourier’s law dictates that the totalamount of heat in the unit cell is given by

Qt = keb0∆T

L, (2)

where b0 is the heat transfer area per unit depth, L is theheat transfer length along the x-axis, and ∆T = T1−T2

is the temperature difference imposed across the unitcell, with T (0, y) = T1 and T (L, y) = T2. For non-conducting pore inclusions, the total amount of heattransferred through the unit cell is equal to that con-ducted via the continuous medium, i.e., Qt = Qc.

To analyze the amount of heat transferred throughthe solid phase Qc, consider an infinitely small elementdb(x, y) on an arbitrary and imaginary thermal tube(stretched and distorted by the non-conducting pore)along an arbitrary constant heat-flow line l(y). ForT1 > T2, heat is conducted along the constant heat-flowline from x = 0 to x = L. The amount of local heatdQx(y) transferred nominally along the x-axis but, infact, following the stretched thermal tube is given by

dQx(y) = kc d b(x, y)T1 − T2

l(y), (3)

where db(x, y) is the heat transfer area of the thermaltube per unit depth, and l(y) is the heat transfer lengthalong the thermal tube. Integrating db(x, y) and l(y)of the stretched thermal tube, respectively along thex- and y-axis yields a straight (non-stretched) thermal

tube with the mean width d b(y) and mean length l, as

d b(y) =1

L

∫ L

0

d b(x, y) dx, (4)

l =2

b0

∫ b0/2

0

l(y) d y. (5)

Under steady-state heat conduction, a certain amountof heat transported through a thermal tube is con-stant and invariable with its shape as well as cross-sectional area. The heat transferred through a singlestretched tube can be approximately calculated in thenon-stretched tube with the mean width d b(y) andlength l. Thus, the amount of local heat dQ(y) trans-ferred along the non-stretched thermal tube may be ap-proximately expressed as

dQ(y) = kc d b(y)T1 − T2

l. (6)

Integration of Eq. (6) with respect to y leads to the totalamount of heat conducted in the unit cell domain, as

Qc = 2

∫ b0/2

0

dQ(y) =

2kcT1 − T2

Ll

∫ b0/2

0

∫ L

0

d b(x, y) dx. (7)

For non-conducting pores as considered in the presentstudy, Qt = Qc and hence

kekc

=

(L

l

) ∫ b0/2

0

∫ L

0

d b(x, y) dx

Lb0/2. (8)

This expression states that the effective thermal con-ductivity is determined by the product of:

(i) Change in average heat transfer length alongconstant heat flow lines

L

l. (9a)

021011-3 Thermal stretching in two-phase porous media Theor. Appl. Mech. Lett. 3, 021011 (2013)

(ii) Change in average heat transfer area∫ b0/2

0

∫ L

0

d b(x, y) dx

Lb0/2. (9b)

Equation (9a) is, in fact, identical to the definition of“tortuosity” widely known in fluid mechanics commu-nity, which is defined as the ratio of the distance trav-eled actually by a flow particle to the straight-line dis-tance traveled by the particle without any obstruction.As the tortuosity is typically obtained by averaging overa representative element, the average “thermal tortuos-ity” τ̄ is analogously introduced here as

l̄

L=

2

b0

∫ b0/2

0

l(y) d y

L=

2

b0

∫ b0/2

0

τ(y) d y = τ̄ ,

(10a)

where τ(y) = l(y)/L is the local thermal tortuosity.Equation (9b) indicates that the heat transfer area av-eraged over the unit cell may be expressed as a functionof the void fraction (porosity) vp as∫ b0/2

0

∫ L

0

d b(x, y) dx

b0L/2= 1− vp. (10b)

Substitution of Eq. (10) into Eq. (8) leads to a sim-ple form for the effective thermal conductivity as a func-tion of the porosity and thermal tortuosity, as

kekc

=(1− vp)

τ̄. (11)

Equation (11) is similar in form to the effective dif-fusivity models for mass transfer in porous media,17 al-though the tortuosity is differently defined as the squareof the ratio of actual fluid flow path to the straightpath.18–22 Due to the analogy between heat and masstransfer, a few studies23–26 employed directly the effec-tive diffusivity models to estimate the effective thermalconductivity, without properly validating such usage inheat transfer. The present study, for the first time, es-tablishes the analogy based on physical modeling andclarifies the physical basis of the effective conductivityas a function of porosity and thermal tortuosity.

To estimate the effective thermal conductivity usingthe present model as expressed explicitly in Eq. (11),the average thermal tortuosity which is strongly af-fected by the pore shape, size and volume fraction ofpore inclusions needs to be calculated. With the poros-ity fixed at vp = 0.15, Fig. 2(a) displays the constantheat flow lines around the pore obtained from steady-state two-dimensional numerical simulations with Flow-3DTM. Upon image processing of Fig. 2(a), the trans-verse distribution of local thermal tortuosity τ(y) can beobtained, as shown in Fig. 2(b). In the vicinity of thenon-conducting pore, the constant heat-flow lines are

T/K b0/

T T

y

x L

R

l↼y↽

Constant heat flow lines

1.0 1.1 1.2 1.3 1.4 1.5

1.0

0.8

0.6

0.4

0.2

0

τ

y/b

vp/.

vp/.

(a) Distortion of local temperature field and constant heat flow lines obtained from numerical simulations with constant wall temperature at x = 0 and x=L for v

p= 0.15

(b) Transverse distribution of local thermal tortuosity for v

p= 0.15 and v

p= 0.60

Fig. 2. Thermal tortuosity around a non-conducting circu-lar pore.

elongated by about 25 % in comparison with those nearthe unit cell edge, y = b0/2. Even for such low poros-ity levels (vp = 0.15), the constant heat-flow lines aresignificantly stretched and distorted by the presence ofthe pore. Contrast to the low porosity case (vp = 0.15)where the local thermal tortuosity decreases dramati-cally as the value of 2y/b0 is increased from 0 to 0.2,the local thermal tortuosity decreases more mildly asthe porosity is increased to 0.60 shown in Fig. 2(b).

For idealized pore topologies of Fig. 1, Fig. 3presents the normalized effective conductivity ke/kc ob-tained from Eq. (11) as a function of porosity. Theeffective conductivity estimated from the present anal-ysis exhibits a trend in excellent agreement with theMaxwell model as well as experimental data.8 It maybe inferred from the present results that the presence ofnon-conducting pores reduces the effective conductivitydue to the combined effect of two mechanisms: (1) re-duced effective heat flow area (reflected macroscopicallyas porosity), and (2) increased thermal tortuosity (re-flected locally as stretched and distorted heat conduc-tion paths). The thermal tortuosity itself is stronglyaffected by the porosity.

021011-4 X. H. Yang, T. J. Lu, and T. Kim Theor. Appl. Mech. Lett. 3, 021011 (2013)

1.0

0.8

0.6

0.4

0.2

0

vp

kp/ke/kc

Parallel model

Upper limit for

a circular pore

Maxwell model1

Present analysis

Experimental data8

0 0.2 0.4 0.6 0.8 1.0

Fig. 3. Effective thermal conductivity (ke) plotted as afunction of void fraction (vp) for non-conducting pores (kp =0).

2.0

1.6

1.2

0.8

0.4

0

vp

kp/

τ

Present analysis

Valid rangeCalculation base on

numerical simulation

0 0.2 0.4 0.6 0.8 1.0

Fig. 4. Average thermal tortuosity (τ̄) plotted as a functionof void fraction (vp) for non-conducting pores (kp = 0).

Thermal tortuosity, as a measure of an imaginaryheat conduction length, can provide a better under-standing of transport phenomena in porous media. Forsome porous media e.g., close-celled metal foams, aretypically saturated with air, heat transferred throughfluid phase (air) can be reasonably neglected comparedwith that through solid phase (ks ≫ kf). In this case,Eq. (1) is reduced to

kekc

=2(1− vp)

2 + vp. (12)

It follows from Eqs. (11) and (12) that the average ther-mal tortuosity depends on the porosity, expressed ex-plicitly as

τ̄ = 1 +1

2vp. (13)

Figure 4 compares the analytical prediction of Eq. (13)with that numerically calculated, with good agreementachieved in relatively low porosity ranges. The discrep-ancy at higher porosities (e.g., vp > 0.50) is attributed

mainly to the violation of the assumptions associatedwith the Maxwell model. Notice finally that, based onnumerical simulation results, Koponen et al.27 reportedthe linear dependence of “tortuosity” upon “porosity”for fluid flow in two-dimensional porous media contain-ing randomly distributed rectangular pores.

This study provides the physical basis for the clas-sical Maxwell model or Maxwellian models that arecapable of estimating the effective thermal conductiv-ity of two-phase porous media at relatively low poros-ity ranges. An alternate yet general expression forthese models demonstrates that the presence of non- (orvery low-) conducting pores reduces the effective ther-mal conductivity of the porous material mainly via themechanism of thermal stretching: (1) macroscopically,the reduced effective conducting area (1 − vp), and (2)microscopically, the elongated heat flow length (τ̄).

This work was supported by the National 111 Project

of China (B06024), the National Basic Research Program

of China (2011CB610305), the Major International Joint

Research Program of China (11120101002) and the National

Natural Science Foundation of China (51206128).

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