When Should Local Thermal Equilibrium and Nonequilibrium Be Considered?
The classical local equilibrium hypothesis in modeling heat transfer in porous media considers pointwise equality of solid and fluid temperatures as said in Equation 4-40. The Local Thermal Equilibrium section below details the derivation of the energy equation considering such assumption that remains accurately sufficient for several applications. Ref. 32 shows for instance that solid and fluid temperatures are equal in steady conduction problems where only prescribed temperature conditions are applied. Most slow motion problems can also assume equality of phase temperatures if volumetric internal heating do not differ in both materials.
In the case of conduction in porous plates, Ref. 33 provides criteria based on the dimensionless Sparrow number, Sp, to indicate if temperature equilibrium is still valid or if a nonequilibrium point of view should be preferred. In Ref. 34, the influence of the Darcy number, Da, and the ratio of phase conductivities is examined for transient heat transfer in packed beds. The Sparrow and Darcy numbers are defined by
where:
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hsf is the interstitial heat transfer coefficient between solid and fluid phases (SI unit: W/(m2·K))
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L is the plate layer thickness (SI unit: m)
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keff is the equivalent thermal conductivity of the porous medium (SI unit: W/(m·K))
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rh is the hydraulic radius (SI unit: m)
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κ is the permeability (SI unit: m2)
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d is the average particle diameter (SI unit: m)
In the situations described in Ref. 33 and Ref. 34, small values of Sp (less than 100 or 500) and large values of Da (from order of magnitude 10-7) indicate discrepancies of temperature in each phase. However, in general, assessing the validity of local thermal equilibrium assumption remains not straightforward in specific situations. The Local Thermal Nonequilibrium approach, described below, makes use of two energy equations, one for each phase of the porous medium, that solve for two temperature fields. It numerically doubles the number of freedom to solve but provides a general framework for heat transfer in porous media where evaluating the validity of the equilibrium hypothesis is not required anymore.