When considering heat transfer in a porous medium at the microscopic scale, two heat transfer equations can be established, for the solid and fluid phases. For undeformed immobile solids, Equation 4-16 simplifies into:

and for a fluid domain where pressure work and viscous dissipation are neglected, Equation 4-18 becomes:

When this hypothesis can be assumed, the heat transfer equation for porous media is derived from the mixture rule on energies appearing in solid and fluid heat transfer equations (see Ref. 13). This rule applies by multiplying the equation of the solid domain by the solid volume fraction, θs, multiplying the fluid equation by the porosity, εp, and summing resulting equations.

The theory for this hypothesis is detailed in the Local Thermal Equilibrium section below. Otherwise, the Local Thermal Nonequilibrium section describes the theory for modeling heat transfer in porous media using two temperatures.

In the particular case of a packed bed of pellets, and under some assumptions about thermal conductivities ratio, the local thermal nonequilibrium model can be tuned to capture more precisely some thermal effects within the pellets. This is done by replacing the macroscale heat equation for the solid phase by an ordinary differential equation for the microscale variation of temperature in the pellets along the radial coordinate. See the Packed Bed of Pellets section for more details.