Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 332

The choice between the one-equation model and the two-equation model has
been well discussed by Quintard and Whitaker (1993) and Whitaker (1999). They
have also developed methods of determining the effective thermal conductivity ten-
sor K ef f in the one-equation model and the four coefficients K ββ , K βσ = K σβ ,

K σσ ,and ha υ in the two-equation model. Their studies suggest that the coupling
coefficients are on the order of the smaller of K ββ and K σσ . Therefore, the cou-
pled conductive terms should not be omitted in any detailed two-equation model of
heat conduction processes. When the principle of local thermal equilibrium is not
valid, the commonly-used two-equation model in the literature is the one without
the coupled conductive terms (Glatzmaier and Ramirez 1988)

) β ∂ T β β
∂

K

β β

ββ · ∇ T

T σ σ − T ∈ β ( ρ c = ∇ · + ha υ , (6.280)

β β

t

and

T σ σ − T β β

T σ σ
∂

= ∇ · K σσ · ∇ T σ σ −

) σ ∂ ∈ σ ( ρ c ha υ . (6.281)

t

On the basis of the above analysis, we now know that the coupled conductive terms
K βσ · ∇ T σ σ and K σβ · ∇ T β β cannot be discarded in the exact representation of

the two-equation model. However, we could argue that Eqs. (6.280) and (6.281) rep-
resent a reasonable approximation of Eqs. (6.271) and (6.272) for a heat conduction
process in which ∇ T

β

β and

T σ σ are sufficiently close to each other. Under ∇

these circumstances K in Eq. (6.280) would be given by K

ββ +

K while K σσ

ββ βσ

in Eq. (6.281) should be interpreted as K σβ + k σσ . This limitation of Eqs. (6.280)

and (6.281) is believed to be the reason behind the paradox of heat conduction
in porous media subject to lack of local thermal equilibrium analyzed by Vadasz
2005a, b, c. For an isotropic system with constant physical properties of the two
phases, Eqs. (6.280) and (6.281) reduce to the traditional formulation of heat con-
duction in two-phase systems (Vadasz 2005a, b, c, Bejan 2004, Bejan et al. 2004,
Nield and Bejan 2006)
γ β ∂ T β β
∂

T σ σ − T β β k e β Δ T β β = + ha υ , (6.282)

t

and

T σ σ − T β β

T σ σ
∂

γ σ ∂

T σ β −

= k e σ Δ ha υ , (6.283)

t

where we introduce the equivalent effective thermal conductivities k e β = k β + k βσ

and k e σ = -phases, respectively, to take the above note

into account. To describe the thermal energy exchange between solid and gas phases
in casting sand, Tzou (1997) has also directly postulated Eqs. (6.282) and (6.283)
(using k β and k σ rather than k eβ and k eσ ) as a two-step model, parallel to the two-
step equations in the microscopic phonon-electron interaction model (Qiu and Tien
1993, Kaganov et al. 1957, Anisimòv et al. 1974).

k σ + k σβ for the β -and σ