Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 327

tems. First, they define the microscale problem by the first law of thermodynamics
and the Fourier law of heat conduction (Fig. 6.8)

) β ∂

T β
∂

( ρ c

t = ∇ · (

k β ∇ T β ) , in the β -phase (6.253)

) σ ∂

T σ
∂

( ρ c

t = ∇ · (

k σ ∇ T σ ) , in the σ -phase (6.254)

T

β =

T σ , at the β − σ interface A (6.255)

βσ

n βσ · k β = n βσ · k σ ∇ T σ , at the β − σ interface A βσ (6.256)

Here , c and k are the density, specific heat and thermal conductivity, respectively.

Subscripts

ρ

β and σ refer to the β -and σ -phases, respectively. A represents the

βσ

interface contained in the REV, n βσ is the outward-directed surface
normal from the

area of the β − σ

n σβ (Fig. 6.8). To be
thorough, Quintard and Whitaker (1993) have also specified the initial conditions
and the boundary conditions at the entrances and exits of the REV; however, we
need not do so for our discussion.
Next Quintard and Whitaker (1993) apply the superficial averaging process to
Eqs. (6.253) and (6.254) to obtain,

β -phase toward the σ -phase, and n βσ = −

1
V REV

) β ∂

T β
∂

1
V REV

( ρ c d V = ∇ ( k β ∇ T β ) d V , (6.257)

t V

β

V

β

and

1
V REV

) σ ∂

T σ
∂

1
V REV

σ ( ρ

c d V =

σ ∇ · (

k σ ∇ T σ ) d V , (6.258)

t V V

Fig. 6.8 Rigid two-phase system