Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 333

6.11.2 Equivalence with Dual-Phase-Lagging Heat Conduction

The two-equation model can be used to establish equivalence between the dual-
phase lagging and two-phase-system heat conduction (Wang and Wei 2007a, 2007b,
Wang et al. 2007b). We first rewrite Eqs. (6.273) and (6.274) in their operator form
γ β

⎡
⎣

⎤

T β β

∂
∂ t −

k β Δ + h − k βσ Δ − ha υ

⎦ =

0 . (6.284)

∂
∂ t −

T σ σ

− k βσ Δ − ha υ γ σ k σ Δ + ha υ

We then obtain a uncoupled form by evaluating the operator determinant such that

γ σ ha υ 2

− k βσ Δ −

∂
∂

∂
∂

i

γ β

t −

k β Δ + ha υ

t −

k σ Δ + ha υ T i = 0 ,

(6.285)

where the index i can take β or σ . Its explicit form reads, after dividing by ha υ ( γ β +

γ σ )

F

i

i 2 i

∂ T i

0 ∂

T i

T ∂
∂

+ k

0 ∂

F ( r , t )

i

+ τ = αΔ T i + ατ Δ T i ( r , t )+ τ ,

∂ t ∂ t 2 t ∂ t

(6.286)

where

γ β k σ + γ γ β γ σ

ha υ ( γ β + γ σ ) , τ T =

a k β

τ 0 =

2 k βσ ) , ha υ ( k β + k σ +

k β + k σ +

2 k βσ
γ β + γ σ

k = k β + k σ + 2 k βσ , α = ,

(6.287)

k 2

βσ −

k β k σ
ha υ

( , )

)+ τ 0 ∂

F r t

i

2

F ( r , t = Δ T i .

∂ t

This is the dual-phase-lagging heat-conduction equation with τ

0 and

τ

T as the phase

are the
effective thermal conductivity, capacity and diffusivity of two-phase system, respec-
tively. F

lags of the heat flux and the temperature gradient, respectively. k , ρ c and α

is the volumetric heat source. The reported conductivity and diffusivity
data of two-phase systems (nanofluids, bi-composite media, porous media etc.) in
the literature were based on the Fourier heat conduction and should be reexamined.
Note that Eqs. (6.273) and (6.274) are the mathematical representation of the first
law of thermodynamics and the Fourier law of heat conduction for heat conduc-
tion processes in two-phase systems at the macroscale. Therefore, we have an exact
equivalence between dual-phase-lagging heat conduction and Fourier heat conduc-
tion in two-phase systems. This is significant because all results in these two fields
become mutually applicable. In particular, all analytical methods and results (such
as the solution structure theorems) in the present monograph can be applied to study
heat conduction in two-phase systems such as nanofluids, bi-composite media and
porous media (Wang and Wei 2007a, 2007b, Wang et al. 2007b).

( r , t )