Local Thermal Equilibrium
The local thermal equilibrium hypothesis of Equation 4-24 implies a common temperature, T, for both solid and fluid phase. The Heat Transfer in Porous Media Interface solves for the following version of the heat equation (Ref. 16), reformulated using T:
(4-25)
(4-26)
The different quantities appearing here are:
ρ is the fluid density.
Cp is the fluid heat capacity at constant pressure.
Cp)eff is the effective volumetric heat capacity at constant pressure, defined by
keff is the effective thermal conductivity (a scalar or a tensor if the thermal conductivity is anisotropic).
q is the conductive heat flux.
u is the velocity field, either an analytic expression or computed from a Fluid Flow interface. It should be interpreted as the Darcy velocity, that is, the volume flow rate per unit cross sectional area. The average linear velocity (the velocity within the pores) can be calculated as uf = u/(1−θp), where (1−θp) is the fluid’s volume fraction, or equivalently the porosity.
Q is the heat source (or sink). Add one or several heat sources as separate physics features.
For a steady-state problem the temperature does not change with time, and the terms with time derivatives of Equation 4-25 disappear.
The effective thermal conductivity of the solid-fluid system, keff, is related to the conductivity of the solid, kp, and to the conductivity of the fluid, k, and depends in a complex way on the geometry of the medium. In Ref. 13, three models are proposed for an isotropic medium:
This volume average model provides an upper bound for the effective thermal conductivity.
This reciprocal average model provides a lower bound for the effective thermal conductivity.
This model provides a good estimate as long as k and kp are not too different from each other.
When k and kp are equal the three models give the same effective thermal conductivity.