Thermal macrotransport processes in porous media. A review

Abstract

This paper reviews recent work by the authors involving nonconservative convective-conductive internal energy transport phenomena in porous media. Where appropriate, these heat-transfer results are contrasted and compared with their classical mass-transfer counterparts. Commonalities as well as differences are pointed out, arising from the distinction between molecular diffusivity vs thermal diffusivity. Differences raise from the fact that the latter — in contrast with the former — is a composite material property (derived jointly from separate thermal conductivity and volumetric heat capacity properties). This contrasts with the case of molecular diffusivity, which is a fundamental rather than composite material property. Both adiabatic and nonadiabatic systems are studied, with the latter characterized by a rate of heat loss to the surroundings described by a ‘Newton's law of colling’ heat transfer coefficient, h. Taylor dispersion theory is used to effect the coarse-graining of the thermal problem posed by the microscale equations, thereby producing a macroscale or effective-medium theory of the mean thermal transport process. Various porous media, each possessing a spatially periodic skeletal geometry, are analyzed. General expressions are presented for the macroscale thermal propagation velocity vector Ū ∗ (which is not generally equal to the interstitial Darcy-scale velocity V̄ ∗ of the flowing fluid) and effective thermal dispersivity dyadic α ̄ ∗ in terms of the prescribed microscale data. Additionally, in the nonadiabatic case, an expression is obtained for a third macrotransport coefficient, H ̄ ∗ , representing the effective or overall macroscale heat-transfer coefficient, and distinct from the microscale heat-transfer coefficient h. (The former, macrotransport coefficient represents the same type of macroscale material property as arises in so-called ‘fin’ heat-transfer problems.) Furthermore, it is shown that when solving the transient nonadiabatic microtransport equation for the mean temperature T ̄ , parameterized by the effective-medium phenomenological coefficients H ̄ ∗, H ̄ ∗ and α ̄ ∗ , it becomes necessary to employ a fictitious mean initial temperature distribution in place of the true mean distribution, the latter deriving from the initial microscale distribution. A paradigm is outlined for calculating this fictitious mean initial temperature field from the prescribed initial microscale temperature field. One illustrative example addressed is that of heat conduction in a quiescent composite medium. Specifically, we show that although a composite medium may be composed of two separately homogeneous materials, each possessing identical (and isotropic) thermal diffusivities α, the effective thermal dispersivity α ̄ ∗ of the composite medium may nevertheless differ from the constant diffusivity α characterizing the individual phases by many orders of magnitude; moreover, in contrast to the scalar, isotropic nature of the individual microscale diffusivities α, the effective macroscale dispersivity α ̄ ∗ will generally be anisotropic, possessing a value dependent upon which (if either) of the two homogeneous phases is continuous and which is discontinuous! Detailed results are also summarized for the effective thermal dispersivity dyadic for two-dimensional homogeneous Darcy flow through the interstices of a packed bed composed of circular cylinders in various lattice configurations. In the adiabatic fluid case (corresponding to the cylinders being insulated), results for Ū ∗ and α ̄ ∗ are given for the effective thermal dispersivities in terms of the Peclet number, bed porosities and, where relevant, Reynolds number (Re). While most of the numerical data pertain to the Stokes flow case, Re = 0, a few calculations at Reynolds number up to about 100 are also presented. In the comparable nonadiabatic case (corresponding to noninsulated circular cylinders functioning as heat sources or sinks), numerical results are also presented for the Darcy-scale thermophysical parameters H ̄ ∗ , H̄ ∗ and α ̄ ∗ . In microscale phenomenological data, these calculations show that Ū ∗ and α ̄ ∗ for nonadiabatic systems may differ sensibly from their adiabatic counterparts, as they now also depend functionally upon the heat transfer coefficient h.