Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 328

where V REV , V β and V σ are the volumes of the REV, -phase

in REV, respectively. We should note that the superficial temperature is evaluated at
the centroid of the REV, whereas the phase temperature is evaluated throughout the
REV. Neglecting variations of (

β -phase in REV and σ

c ) within the REV and considering the system to
be rigid so that V β and V σ are not functions of time, the volume-averaged form of
Eqs. (6.253) and (6.254) are

ρ

T
β

) β ∂

t = ∇ · (

T β ) , ( ρ c k β ∇ (6.259)

∂

and

) σ ∂

T σ
∂

( ρ c

t = ∇ · (

k σ ∇ T σ ) , (6.260)

where angle brackets indicate superficial quantities such as

1
V REV

T B = T β d V ,

V
β

and

1
V REV

T σ =

.

T σ d V

V
σ

The superficial average, however, is an unsuitable variable because it can yield er-
roneous results. For example, if the temperature of the -phase were constant, the

superficial average would differ from it (Quintard and Whitaker 1993). On the other
hand, intrinsic phase averages do not have this shortcoming. These averages are
defined by

β

T β β

1
V β

= T β d V , (6.261)

V

β

and

1
V σ

T σ σ =

T σ d V . (6.262)

V
σ

Also, intrinsic averages are related to superficial averages by
T β = ∈ β T β β

, (6.263)

and

T σ σ , T σ = ∈ σ (6.264)