Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 326

6.11 Heat Conduction in Two-Phase Systems

In this section we develop an exact equivalence between dual-phase-lagging heat
conduction and Fourier heat conduction in two-phase systems. This equivalence
builds the intrinsic relationship between the two heat-conduction processes, pro-
vides an additional tool for studying the two heat-conduction processes, and demon-
strates the possibility of the presence of thermal waves and resonance in two-phase-
system heat conduction, a phenomenon that has been observed experimentally. We
also discuss the mechanism responsible for such thermal waves and resonance.

6.11.1 One- and Two-Equation Models

The microscale model for heat conduction in two-phase systems is well-known. It
consists of the field equation and the constitutive equation. The field equation comes
from the conservation of energy (the first law of thermodynamics). The commonly-
used constitutive equation is the Fourier law of heat conduction for the relation be-
tween the temperature gradient

T and the heat flux density vector q (Wang 1994).
For transport in two-phase systems, the macroscale is a phenomenological scale
that is much larger than the microscale of pores and grains and much smaller than
the system length scale. Interest in the macroscale rather than the microscale comes
from the fact that a prediction at the microscale is complicated due to the complex
microscale geometry of two-phase systems such as porous media, and also because
we are usually more interested in large scales of transport for practical applications.
Existence of such a macroscale description equivalent to the microscale behavior
requires a good separation of length scales and has been well discussed by Auriault
(1991).
To develop a macroscale model of transport in two-phase systems, the method
of volume averaging starts with a microscale description (Wang 2000b, Whitaker
1999). Both conservation and constitutive equations are introduced at the mi-
croscale. The resulting microscale field equations are then averaged over a repre-
sentative elementary volume (REV), the smallest differential volume resulting in
statistically meaningful local averaging properties, to obtain the macroscale field
equations. In the process of averaging, the multiscale theorems areusedtoconvert
integrals of gradient, divergence, curl, and partial time derivatives of a function into
some combination of gradient, divergence, curl, and partial time derivatives of in-
tegrals of the function and integrals over the boundary of the REV (Wang 2000b,
Wang et al. 2007, Whitaker 1999). The readers are referred to Wang (2000b), Wang
et al. (2007b) and Whitaker (1999) for the details of the method of volume averaging
and to Wang (2000b) and Wang et al. (2007b) for a summary of the other methods
of obtaining macroscale models.
Quintard and Whitaker (1993) use the method of volume averaging to develop
one- and two-equation macroscale models for heat conduction in two-phase sys-

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