Local Thermal Non-Equilibrium
The Local Thermal Non-Equilibrium Interface implements heat transfer in porous media for which the temperatures into the porous matrix and the fluid are not in equilibrium.
Equations for Local Non-Equilibrium Heat Transfer
Non-equilibrium heat transfer in porous media for binary systems of rigid porous matrix and fluid phase are governed by a set of two equations. These are the usual heat equations for solids and fluids, multiplied by the volume fractions θp and (1 − θp) respectively, and with an additional source term quantifying exchanged heat between both phases (2.12 and 2.13 in Ref. 13):
(4-25)
In these expressions:
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θp is the solid volume fraction (SI unit: 1)
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ρs and ρf are the solid and fluid densities (SI unit: kg/m3)
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Cp, s and Cp, f are the solid and fluid heat capacities at constant pressure (SI unit: J/(kg·K))
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qs and qf are the solid and fluid conductive heat fluxes (SI unit: W/m2)
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ks and kf are the solid and fluid thermal conductivities (SI unit: W/(m·K))
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qsf is the interstitial convective heat transfer coefficient (SI unit: W/(m3·K))
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Qs and Qf are the solid and fluid heat sources (SI unit: W/m3)
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uf is the fluid velocity vector (SI unit: m/s)
The fluid velocity is often deduced from a porous velocity up, coming for example from Darcy’s law or Brinkman equations, according to:
so that the heat equations in the fluid domain reduces to:
(4-26)
The Local Thermal Non-Equilibrium multiphysics coupling adds the exchanged opposite heat sources qsf(Tf − Ts) and qsf(Ts − Tf) that one phase receives from or releases to the other when respective temperatures differ. The porous temperature, T, has the following definition (Ref. 32):
Correlation for the Interstitial Convective Heat Transfer Coefficient
The Local Thermal Non-Equilibrium multiphysics feature provides a built-in correlation for qsf in the spherical pellet bed configuration (2.14, 2.15, and 2.16 in Ref. 13):
The specific surface area, asf (SI unit: 1/m), for a bed packed with spherical particles of radius rp is:
The interstitial heat transfer coefficient, hsf (SI unit: W/(m2·K)), satisfies the relation:
where β = 10 for spherical particles, and Nu is the fluid-to-solid Nusselt number derived from following correlation (Ref. 14):
The Prandtl number, Pr, and particle Reynolds number, Rep, are defined by:
Volumetric and Surface Thermal Conditions
Because the Local Thermal Non-Equilibrium multiphysics coupling multiplies each energy equation by its volume fraction, θp and (1 − θp) for solid and fluid phases respectively, a heat source or heat flux defined in a couple heat transfer interface is also accounted with that ratio. As shown in Equation 4-25 and Equation 4-26, the volumetric heat sources θpQs and (1 − θp)Qf are applied to the energy equations while the Heat Source features of each physics interface specify Qs and Qf.