Local Thermal Nonequilibrium
The Local Thermal Nonequilibrium 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 Nonequilibrium Heat Transfer
Nonequilibrium 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 θs and εp=1 − θs respectively, and with an additional source term quantifying exchanged heat between both phases (2.12 and 2.13 in Ref. 13):
(4-43)
(4-44)
In these expressions:
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θs is the solid volume fraction (dimensionless)
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εp is the porosity (dimensionless)
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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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u is the fluid velocity vector (SI unit: m/s). 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.
The Porous Medium feature with Porous medium type set to Local thermal equilibrium 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. 35):
Correlation for the Interstitial Convective Heat Transfer Coefficient
The Porous Medium feature with Porous medium type set to Local thermal equilibrium provides a built-in correlation for qsf for a packed bed of spherical pellets (2.14, 2.15, and 2.16 in Ref. 13):
The specific surface area, Sb (SI unit: 1/m), for a bed packed with spherical particles of average diameter dpe 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. 15):
The Prandtl number, Pr, and particle Reynolds number, Rep, are defined by:
Volumetric and Surface Thermal Conditions
Because each energy equation is multiplied by its volume fraction, θs and εp for solid and fluid phases respectively, a heat source or heat flux defined in one of the phases is also accounted with that ratio. As shown in Equation 4-43 and Equation 4-44, the volumetric heat sources θsQs and εpQf are applied to the energy equations when Heat Source is added as a subnodes of the Porous Matrix or Fluid nodes to specify Qs or Qf.