Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 330

) β ∂ T β β
∂

k β

T β d A

∈ β ∇ T β β

1
V REV

∈ β ( ρ c = ∇ · + n βσ

t A

βσ

k β ∇ T β β d A

1
V REV

+

n βσ ·

A
βσ

1
V REV

T β d A

+ n βσ · k β ∇ , (6.269)

A
βσ

and

k σ

∈ σ ∇

T σ d A

T σ σ
∂

) σ ∂

1
V REV

T σ σ + ∈ σ ( ρ c = ∇ · n σβ

t

A
βσ

1
V REV

T σ σ d A + n σβ · k σ ∇

A
βσ

1
V REV

T σ d A

+

n σβ ·

.

k σ ∇ (6.270)

A
βσ

After developing the closure for T β and T σ , Quintard and Whitaker (1993) obtain a
two-equation model

) β ∂ T β β
∂

K ββ · ∇ T β β

T σ σ

∈ β ( ρ c = ∇ · + K βσ · ∇

t

T σ σ − T β β

+ ha υ , (6.271)

and

K σσ · ∇

K σβ · ∇ T β β

T σ σ
∂

) σ ∂ T σ σ +

∈ σ ( ρ c = ∇ ·

t

T σ σ − T β β

− ha υ , (6.272)

where h and a υ come from modeling of the interfacial flux and are the film heat
transfer coefficient and the interfacial area per unit volume, respectively. K ββ , K σσ ,
K βσ and K σβ are the effective thermal conductivity tensors, and the coupled ther-
mal conductivity tensors are equal

K βσ = K σβ .