Local Thermal Equilibrium
The local thermal equilibrium hypothesis of Equation 4-40 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:
(4-41)
(4-42)
The different quantities appearing here are:
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ρf is the fluid density.
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Cp,f is the fluid heat capacity at constant pressure.
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(ρCp)eff is the effective volumetric heat capacity at constant pressure, defined by
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εp is the porosity.
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θs is the solid matrix volume fraction.
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ρs is the solid matrix density.
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Cp,s is the solid matrix heat capacity at constant pressure.
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keff is the effective thermal conductivity (a scalar or a tensor if the thermal conductivity is anisotropic).
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q is the conductive heat flux.
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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/εp, where εp is the fluid’s volume fraction, or equivalently the porosity.
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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 4-41 disappear.
The effective thermal conductivity of the solid-fluid system, keff, is related to the conductivity of the solid, ks, and to the conductivity of the fluid, kf, and depends in a complex way on the geometry of the medium. In Ref. 13, three models are proposed for an isotropic medium:
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If the heat conduction occurs in parallel in the solid and the fluid, then the effective thermal conductivity is the weighted arithmetic mean of the conductivities kf and ks:
This volume average model provides an upper bound for the effective thermal conductivity.
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If the heat conduction takes place in series, with all of the heat flux passing through both solid and fluid, then the effective thermal conductivity is the weighted harmonic mean of the conductivities kf and ks:
This reciprocal average model provides a lower bound for the effective thermal conductivity.
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A last estimate is given by the weighted geometric mean of kf and ks:
This model provides a good estimate as long as kf and ks are not too different from each other.
When kf and ks are equal the three models give the same effective thermal conductivity.