Local Thermal Equilibrium
By making the local thermal equilibrium hypothesis for solid and fluid phases (Equation 4-40), The Heat Transfer in Porous Media Interface solves for the following version of the heat equation (Ref. 17), reformulated using a common temperature, 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, Ref. 14, and Ref. 18, the following models are proposed:
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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 good estimate is given by the weighted geometric mean of kf and ks, as long as kf and ks are not too different from each other:
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When the porous medium is composed of solid spherical inclusions in a fluid phase, the effective thermal conductivity can be expressed as
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On the opposite, when the porous medium is composed of fluid spherical inclusions in a solid phase, the effective thermal conductivity can be expressed as
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In the case of a wrapped screen, the effective thermal conductivity of the solid-fluid system can be defined as
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Finally, when the porous medium is made of sintered metal fibers, the effective thermal conductivity of the solid-fluid system is defined as
Note that when kf and ks are equal, all these models give the same effective thermal conductivity. In addition, the effective conductivity is equal to kf when the porosity is 1, and to ks when the porosity is 0.