Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 329

where ∈ β and ∈ σ are the volume fractions of the β -and σ -phases with ∈ β = ϕ ,

∈ σ = for a rigid two-phase system.

Quintard andWhitaker (1993) substitute Eqs. (6.263) and (6.264) into Eqs. (6.259)
and (6.260) to obtain

1 − ϕ with a constant porosity ϕ

) β ∂ T β β
∂

= ∇ · (

β ) ,

∈ β ( ρ c k

β ∇

T (6.265)

t

and

T σ σ
∂

) σ ∂ ∈ σ ( ρ c = ∇ · ( k σ ∇ T σ ) . (6.266)

t

Next Quintard and Whitaker (1993) apply the spatial averaging theorem (Theo-
rem 5.2 in Wang et al. 2007b) to Eqs. (6.265) and (6.266) and neglect variations
of physical properties within the REV. The result is

k β

∈ β ∇ T β β + T β β ∇ ∈ β + n βσ T β d A ) β ∂ T β β

∂

1
V REV

∈ β ( ρ c = ∇ ·

t A

βσ

accumulation conduction

1
V REV

+ n βσ · k β ∇ T β d A ,

A

βσ

interfacial flux

(6.267)

and

k σ

n βσ T σ d A ) σ ∂ T σ σ

∂

1
V REV

T σ σ + T σ σ ∇ ∈ σ + ∈ σ ( ρ c = ∇ · ∈ σ ∇

t

A

βσ

accumulation conduction

1
V REV

+ n βσ · k σ ∇ T σ d A .

A

βσ

interfacial flux

(6.268)

By introducing the spatial decompositions T β = T β β

T σ σ +
T σ and by applying scaling arguments and Theorem 5.2 in Wang et al. 2007b,
Eqs. (6.267) and (6.268) are simplified into (Quintard and Whitaker 1993)

T β

+ and T σ =