Chapter 4Macroscopic Conduction Modelsby Volume Averaging for Two-Phase Systems

Benoı̂t Goyeau

The aim here is to describe macroscopic models of conductive heat transfer withinsystems comprising two solid phases, using the method of volume averaging. Thepresentation of this technique largely stems from work by Carbonell, Quintard, andWhitaker [1–3]. The macroscopic conservation equations are set up under the as-sumption of local thermal equilibrium, leading to a model governed by a singleequation. The effective thermal conductivity of the equivalent medium is obtainedby solving the associated closure problems. The case where thermal equilibriumdoes not pertain, leading to a model with two energy conservation equations, is dis-cussed briefly.

4.1 Introduction

We consider heat conduction in a composite system combining two phases. In gen-eral, this configuration may correspond to a porous medium saturated by an im-mobilised phase, but also a composite medium comprising two solid phases. It ismainly the latter case that will be discussed in the following. For simplicity, the twosolid phases will be assumed to be non-deformable, with constant thermophysicalproperties. Under such conditions, the local energy conservation equations withineach of the phases σ and β (see Fig. 4.1) are

(ρcp)σ∂Tσ∂ t

= ∇· (kσ∇Tσ ) in phase σ , (4.1)

and

(ρcp)β∂Tβ∂ t

= ∇·(kβ∇Tβ)

in phase β . (4.2)

The boundary conditions at the fluid–solid interface Aβσ , expressing continuity oftemperature and heat flux, can be written in the form

S. Volz (ed.), Thermal Nanosystems and Nanomaterials, Topics in Applied Physics, 118c

95DOI 10.1007/978-3-642-04258-4 4, © Springer-Verlag Berlin Heidelberg 2009

96 Benoı̂t Goyeau

Fig. 4.1 Volume averaging region V for a system comprising two phases σ and β

Tβ = Tσ , on Aβσ , (4.3)

andnβσ ·kβ∇Tβ = nβσ ·kσ∇Tσ , on Aβσ , (4.4)

respectively, where nβσ is the unit vector normal to the solid–liquid interface, ori-ented toward the phase σ . In general, the complexity of the system microstructuremakes it impossible to determine the local temperature field. The alternative is toderive a macroscopic representation (an equivalent continuum model) which rep-resents insofar as possible the geometry and physics on the scale of the pore. Tothis end, several homogenisation techniques can be put to work. In the present dis-cussion, the local equations will be upscaled by means of the volume averagingtechnique [3].

4.2 Local Volume Averages

We consider an averaging volume V (see Fig. 4.1) whose characteristic size r0 mustsatisfy the scale separation constraints

4 Macroscopic Conduction Models by Volume Averaging for Two-Phase Systems 97

lβ , lσ � r0� L , (4.5)

where lβ and lσ are the characteristic lengths of the local geometry in each phaseand L is the length scale of macroscopic temperature variations (the system scale).If Ψβ is a physical quantity associated with phase β (for example, temperature,concentration, velocity, etc.), the superficial volume average ofΨβ over the volumeV , defined at the center x of V , is given by

〈Ψβ 〉|x =1V

∫VβΨβ (x + yβ )dV , (4.6)

where Vβ is the volume occupied by the phase β in V . In many cases, the intrinsicvolume average ofΨβ turns out to be more representative:

〈Ψβ 〉β |x =1

Vβ

∫VβΨβ (x+ yβ )dV . (4.7)

To simplify the notation, we shall just write 〈Ψβ 〉= 〈Ψβ 〉|x. The intrinsic and super-ficial averages are related by

〈Ψβ 〉= εβ 〈Ψβ 〉β , (4.8)

where εβ is the volume fraction of phase β (corresponding to the porosity in the caseof a saturated porous medium). The averaged conservation equations are obtainedby applying the following spatial and temporal differentiation theorems [4]:

〈∇Ψβ 〉= ∇〈Ψβ 〉+1V

∫Aβσ

nβσΨβdA (4.9)

and ⟨∂Ψβ∂ t

⟩=∂ 〈Ψβ 〉∂ t

− 1V

∫Aβσ

nβσ ·wβσΨβdA , (4.10)

where wβσ is the velocity of the interface Aβσ .

4.3 Averaged Equations

In the case of a composite made up of two solid phases σ and β , and when theinterface Aβσ between them is not moving, application of the volume averagingtheorems (4.9) and (4.10) to (4.1) and (4.2) leads to the expressions

98 Benoı̂t Goyeau

εσ (ρcp)σ∂ 〈Tσ 〉σ∂ t

= ∇·[

kσ

(εσ∇〈Tσ 〉σ + 〈Tσ 〉σ∇εσ +

1V

∫Aβσ

nσβTσdA

)]

+1V

∫Aβσ

nσβ ·kσ∇TσdA (4.11)

and

εβ (ρcp)β∂ 〈Tβ 〉β∂ t

= ∇·[

kβ

(εβ∇〈Tβ 〉β + 〈Tβ 〉β∇εβ +

1V

∫Aβσ

nβσTβdA

)]

+1V

∫Aβσ

nβσ ·kβ∇TβdA . (4.12)

Note that the integrals in (4.11) and (4.12) depend on the local temperatures Tσ andTβ at the field point r. These can be decomposed in the form [5]

Tσ = 〈Tσ 〉σ |r + T̃σ , (4.13)

andTβ = 〈Tβ 〉β |r + T̃β , (4.14)

where T̃σ and T̃β are the local temperature deviations. Introducing these expressionsinto (4.11) and (4.12) yields

εσ (ρcp)σ∂ 〈Tσ 〉σ∂ t

= ∇·[

kσ

(εσ∇〈Tσ 〉σ + 〈Tσ 〉σ∇εσ +

1V

∫Aβσ

nσβ 〈Tσ 〉σ |rdA

)]

+∇·(

kσ1V

∫Aβσ

nσβ T̃σdA

)+

1V

∫Aβσ

nσβ ·kσ∇TσdA

(4.15)

and

εβ (ρcp)β∂ 〈Tβ 〉β∂ t

= ∇·[

kβ

(εβ∇〈Tβ 〉β + 〈Tβ 〉β∇εβ +

1V

∫Aβσ

nβσ 〈Tβ 〉β |rdA

)]

+∇·(

kβ1V

∫Aβσ

nβσ T̃βdA

)+

1V

∫Aβσ

nβσ ·kβ∇TβdA .

(4.16)

Furthermore, it can be shown quite generally that, when the following constraintsare satisfied [3]

lγ � r0 , γ = β ,σ , r02� LεLT 1 , (4.17)

where Lε and LT 1 are defined by

4 Macroscopic Conduction Models by Volume Averaging for Two-Phase Systems 99

∇εγ = O

(ΔεγLε

), ∇∇〈Tγ 〉γ = O

(∇〈Tγ 〉γ

LT 1

), γ = β ,σ , (4.18)

we have〈Tγ〉γ |r ∼ 〈Tγ〉γ |x = 〈Tγ 〉γ . (4.19)

Under such conditions, it is relatively easy to show that

1V

∫Aβσ

nσβ 〈Tσ 〉σ |xdA =−〈Tσ 〉σ∇εσ (4.20)

and1V

∫Aβσ

nβσ 〈Tβ 〉β |xdA =−〈Tβ 〉β∇εβ . (4.21)

Consequently, (4.15) and (4.16) simplify to

εσ (ρcp)σ∂ 〈Tσ 〉σ∂ t

= ∇·[

kσ

(εσ∇〈Tσ 〉σ +

1V

∫Aβσ

nσβ T̃σdA

)]

+1V

∫Aβσ

nσβ ·kσ∇TσdA (4.22)

and

εβ (ρcp)β∂ 〈Tβ 〉β∂ t

= ∇·[

kβ

(εβ∇〈Tβ 〉β +

1V

∫Aβσ

nβσ T̃βdA

)]

+1V

∫Aβσ

nβσ ·kβ∇TβdA . (4.23)

In most conduction problems, the transfer mechanisms can be described using amodel with a single energy conservation equation by applying the principle of localthermal equilibrium to be discussed in the next section.

4.3.1 Local Thermal Equilibrium and the Single-Equation Model

The characteristic feature of the notion of local thermal equilibrium is expressed bythe approximation

〈Tβ 〉β = 〈Tσ 〉σ . (4.24)

When (4.24) is satisfied, one can add (4.22) and (4.23) to obtain the non-closedmacroscopic expression

100 Benoı̂t Goyeau

〈ρ〉Cp∂ 〈T 〉∂ t

= ∇·(εβ kβ∇〈Tβ 〉β + εσkσ∇〈Tσ 〉σ (4.25)

+kβV

∫Aβσ

nβσ T̃βdA +kσV

∫Aσβ

nσβ T̃σdA

),

where〈ρ〉Cp = εβ (ρcp)β + εσ (ρcp)σ (4.26)

and〈T 〉= εβ 〈Tβ 〉β + εσ 〈Tσ 〉σ . (4.27)

Under these conditions, (4.24) can be written in the form

〈T 〉= 〈Tβ 〉β = 〈Tσ 〉σ . (4.28)

Naturally, the local thermal equilibrium hypothesis depends on a certain number ofconditions being satisfied, as examined in detail by Quintard and Whitaker [2, 3].The aim in the present discussion is not to examine the details of their analysis, butrather to outline the main conclusions. To sum up, the local thermal equilibriumhypothesis is satisfied whenever at least one of the following three conditions holds:

• One of the volume fractions εβ , εσ is zero.• The phases σ and β have rather similar physical properties.• The ratio (lβ/L)2 tends to zero.

4.3.2 Deviation Equations

The aim here is to present a closed form of (4.25) by determining the deviation fieldsT̃β and T̃σ . To do this, the first step is to write down equations for the deviations.

Since the problems are the same for both phases, we shall only consider T̃β in thefollowing. We begin by dividing (4.23) by the volume fraction εβ to give

(ρcp)β∂ 〈Tβ 〉β∂ t

= ∇·(

kβ∇〈Tβ 〉β)

+ ε−1β ∇εβ ·kβ∇〈Tβ 〉β (4.29)

+ε−1β ∇·

(kβV

∫Aβσ

nβσ T̃βdA

)+ε−1β

V

∫Aβσ

nβσ ·kβ∇TβdA .

Equation (4.29) is then subtracted from the local equation (4.2), in which the de-composition (4.14) has been inserted, to obtain

4 Macroscopic Conduction Models by Volume Averaging for Two-Phase Systems 101

(ρcp)β∂ T̃β∂ t

= ∇·(

kβ∇T̃β)− ε−1

β ∇εβ ·kβ∇〈Tβ 〉β (4.30)

−ε−1β ∇·

(kβV

∫Aβσ

nβσ T̃βdA

)−ε−1β

V

∫Aβσ

nβσ ·kβ∇TβdA .

Given (4.21), the last term of (4.30) corresponding to the interface flux can be writ-ten in the form

1V

∫Aβσ

nβσ ·kβ∇TβdA =−∇εβ ·kβ∇〈Tβ 〉β +1V

∫Aβσ

nβσ ·kβ∇T̃βdA , (4.31)

whence (4.30) takes on the simplified form

(ρcp)β∂ T̃β∂ t

=∇·(

kβ∇T̃β)− ε−1

β ∇·(

kβV

∫Aβσ

nβσ T̃βdA

)−ε−1β

V

∫Aβσ

nβσ ·kβ∇T̃βdA .

(4.32)The third term in (4.32) is a non-local contribution to the deviation field. Given thefollowing orders of magnitude

ε−1β ∇·

(kβV

∫Aβσ

nβσ T̃βdA

)= O

(ε−1β kβ T̃β

lβL

)(4.33)

and

∇·(

kβ∇T̃β)

= O

(kβ T̃β

l2β

), (4.34)

and taking into account the scale constraint (4.5), this non-local term can be ne-glected. Under these conditions, (4.32) reduces to

(ρcp)β∂ T̃β∂ t

= ∇·(

kβ∇T̃β)−ε−1β

V

∫Aβσ

nβσ ·kβ∇T̃βdA . (4.35)

Finally, it should be noted that, although the conduction phenomena considered hereare not steady state on the macroscopic scale, the deviation problems can be treatedas quasi-steady provided that the following inequality is satisfied:

αβ t∗

l2β� 1 , (4.36)

where t∗ is the characteristic time scale. Under these conditions, the deviation equa-tions for T̃β and T̃σ become

∇·(

kσ∇T̃σ)

=ε−1σV

∫Aβσ

nσβ ·kσ∇T̃σdA (4.37)

102 Benoı̂t Goyeau

and

∇·(

kβ∇T̃β)

=ε−1β

V

∫Aβσ

nβσ ·kβ∇T̃βdA , (4.38)

withT̃β = T̃σ + 〈Tσ 〉σ −〈Tβ 〉β , on Aβσ , (4.39)

and

−nβσ ·kβ∇T̃β =−nβσ ·kσ∇T̃σ + nβσ ·kβ∇〈Tβ 〉β −nβσ ·kσ∇〈Tσ 〉σ , on Aβσ .(4.40)

The system of equations (4.37–4.40) is supplemented by two further boundary con-ditions:

T̃σ = g(r,t) , on Aσe , (4.41)

andT̃β = f (r,t) , on Aβ e , (4.42)

where surfaces Aσe and Aβ e are the boundaries of the volume averaging region V incontact with the environment. At this stage, the functions f and g are unknown.

We observe that the boundary conditions (4.39) and (4.40) contain three sourcesof deviations, viz., 〈Tσ 〉σ −〈Tβ 〉β , ∇〈Tβ 〉β , and ∇〈Tσ 〉σ . Assuming local thermalequilibrium, these three terms can be replaced by the single source term ∇〈T 〉. Un-der these conditions, (4.39) and (4.40) can be written in the form

T̃β = T̃σ , on Aβσ , (4.43)

and

−nβσ ·kβ∇T̃β =−nβσ ·kσ∇T̃σ + nβσ ·(kβ − kσ )∇〈T 〉 , on Aβσ . (4.44)

Generally speaking, the complexity of the local geometry (microstructure) requiresone to choose a periodic volume V that is representative of the composite system.In this case, the boundary conditions at the surfaces Aσe and Aβ e take the form

T̃σ (r + li) = T̃σ (r) , on Aσe , (4.45)

andT̃β (r + li) = T̃β (r) , on Aβ e , (4.46)

where li, i = 1,2,3 are the three vectors defining the unit cell. Finally, it can beshown that the average deviation is zero [3], whence

〈T̃σ 〉σ = 0 , 〈T̃β 〉β = 0 . (4.47)

The second step toward setting up a closed form of (4.25) is to transform the devia-tion problem into a closure problem.

4 Macroscopic Conduction Models by Volume Averaging for Two-Phase Systems 103

4.3.3 Closure Problem

In the framework of local thermal equilibrium, we may assume to a first approxima-tion that T̃σ and T̃β are proportional to the source term ∇〈T 〉, i.e.,

T̃σ = bσ ·∇〈T 〉 (4.48)

andT̃β = bβ ·∇〈T 〉 , (4.49)

where bσ and bβ are the closure variables. Introducing (4.48) and (4.49) into thedeviation equations (4.37) and (4.38), the system (4.43–4.46) leads to the closureproblem

kσ∇2bσ =ε−1σV

∫Aβσ

nσβ ·kσ∇bσdA (4.50)

and

kβ∇2bβ =ε−1β

V

∫Aβσ

nβσ ·kβ∇bβdA , (4.51)

with

bβ = bσ , on Aβσ , (4.52)

−nβσ ·kβ∇bβ =−nβσ ·kσ∇bσ + nβσ (kβ − kσ ) , on Aβσ , (4.53)

bσ (r + li) = bσ (r) , on Aσe , (4.54)

andbβ (r + li) = bβ (r) , on Aβ e . (4.55)

Furthermore, given (4.47), we have

〈bσ 〉σ = 0 , 〈bβ 〉β = 0 . (4.56)

For present purposes, we shall not tackle the general case of solving (4.50–4.56) foran arbitrary unit cell. We shall instead focus on the simpler case of a symmetric unitcell. For this case, using the symmetry conditions and the divergence theorem, weobtain

1V

∫Aβσ

nσβ ·kσ∇bσdA =1V

∫Vσ

kσ∇2bσdA = 0 (4.57)

and1V

∫Aβσ

nβσ ·kβ∇bβdA =1V

∫Vβ

kβ∇2bβdA = 0 . (4.58)

The closure problem (4.50–4.56) then assumes the simplified form [6]

∇2bσ = 0 (4.59)

and

104 Benoı̂t Goyeau

∇2bβ = 0 , (4.60)

with

bβ = bσ , on Aβσ , (4.61)

−nβσ ·∇bβ =−nβσ ·κ∇bσ + nβσ (1−κ) , on Aβσ , (4.62)

bγ(r + li) = bγ(r) , on Aγe , γ = β ,σ , (4.63)

and〈bσ 〉σ = 0 , 〈bβ 〉β = 0 , (4.64)

where κ is the ratio of the conductivities in the two phases, viz.,

κ = kσ/kβ . (4.65)

When κ = 0, the problem is precisely as would be obtained for a diffusion problemin a porous medium with zero diffusion in the solid phase (phase σ ) [3].

4.3.4 Closed Form

The non-closed form of the single-equation model is

〈ρ〉Cp∂ 〈T 〉∂ t

= ∇·(εβ kβ∇〈Tβ 〉β + εσkσ∇〈Tσ 〉σ +

kβV

∫Aβσ

nβσ T̃βdA

+kσV

∫Aσβ

nσβ T̃σdA

). (4.66)

Substituting (4.48) and (4.49) into (4.66), with

1V

∫Aβσ

nσβdA = 0 ,1V

∫Aβσ

nβσdA = 0 , (4.67)

leads to the following closed macroscopic form:

〈ρ〉Cp∂ 〈T 〉∂ t

= ∇·(Keff·∇〈T 〉)

, (4.68)

where

Keff =(εβ kβ + εσkσ

)+

kβ − kσV

∫Aβσ

nβσbβdA . (4.69)

It should be noted that that the last term in (4.69) corresponds to a contributionrelated to the tortuosity of the interface Aβσ . Determination of the effective conduc-tivity tensor (4.69) for a given composite structure thus depends on the field of theclosure variable bβ which solves the differential system (4.59–4.64) for the samestructure. Finally, it can be shown that the effective conductivity tensor (4.69) is

4 Macroscopic Conduction Models by Volume Averaging for Two-Phase Systems 105

symmetric, i.e.,Keff = KT

eff , (4.70)

where the superscript T denotes transposition.

4.3.5 Local Thermal Non-Equilibrium

When the local thermal equilibrium hypothesis is not satisfied, Quintard and Whi-taker [2] propose a closed form of (4.22) and (4.23) which leads to the followingtwo-equation model:

εσ (ρcp)σ∂ 〈Tσ 〉σ∂ t

= ∇·(

Kσβ ·∇〈Tβ 〉β +Kσσ ·∇〈Tσ 〉σ)−avh

(〈Tσ 〉σ −〈Tβ 〉β

)(4.71)

and

εβ (ρcp)β∂ 〈Tβ 〉β∂ t

= ∇·(Kββ ·∇〈Tβ 〉β +Kβσ ·∇〈Tσ 〉σ

)−avh

(〈Tβ 〉β −〈Tσ 〉σ

).

(4.72)It can be shown in general that

Kβσ = Kσβ (4.73)

and alsoKβσ � Kββ , Kσσ . (4.74)

Under these conditions, the two-equation system becomes

εσ (ρcp)σ∂ 〈Tσ 〉σ∂ t

= ∇· (Kσσ ·∇〈Tσ 〉σ )−avh(〈Tσ 〉σ −〈Tβ 〉β

)(4.75)

and

εβ (ρcp)β∂ 〈Tβ 〉β∂ t

= ∇·(Kββ ·∇〈Tβ 〉β

)−avh

(〈Tβ 〉β −〈Tσ 〉σ

), (4.76)

where h is the effective interface exchange constant, obtained by solving an associ-ated closure problem.

References

1. R.G. Carbonell, S. Whitaker: Heat and Mass Transfer in Porous Media (Martinus Nijkoff, Dor-drecht, 1984) pp. 121–198

2. M. Quintard, S. Whitaker: Adv. Heat Transfer 23, 369 (1993)3. S. Whitaker: The Method of Volume Averaging, Vol. 13 (Kluwer Academic Publishers, 1999)4. S. Whitaker: Ind. Eng. Chem 12, 12 (1969)5. W.G. Gray: Chem. Eng. Sci. 3, 229 (1975)6. I. Nozad, R.G. Carbonell, S. Whitaker: Chem. Eng. Sci. 40, 843 (1985)