Internal energy transport in adiabatic systems: Thermal taylor dispersion phenomena

Abstract

This paper outlines a general theory of thermal convective-conductive Taylor dispersion phenomena applicable to all adiabatic systems, both continuous and spatially periodic. General expressions are obtained for the macroscale thermal propagation velocity vector U ∗ and effective thermal dispersivity dyadic α ∗ in terms of the prescribed thermophysical microscale data. By way of example, we address the problem of heat conduction in a composite medium. As a specific illustration, we show that although a composite medium may be composed of two separately homogeneous materials possessing identical (isotropic) thermal diffusivities α, the effective thermal dispersivity α ∗ of the composite medium may nevertheless differ by many orders of magnitude from the constant diffusivity α characterizing the individual phases; moreover, in contrast to the scalar, isotropic nature of the microscale diffusivity α, the effective macroscale dispersivity α ∗ will generally be anisotropic, depending upon which (if either) of the two homogeneous phases is continuous and which discontinuous! Detailed calculations are also presented for the effective thermal dispersivity dyadic for Darcy flow through the interstices of a two-dimensional packed bed composed of insulated circular cylinders in various spatial configurations. Results are given for the effective thermal dispersivities in terms of the Peclet number and bed porosities. While most of the numerical data were obtained for the Stokes flow case, Re = 0 (Re = Reynolds number), a few isolated calculations were also performed at Reynolds numbers up to about 100.