Simple second-order expression: For the porosity dependence of thermal conductivity

In a series of recent publications it has been shown that there is an almost complete formal analogy in the micromechanical relations describing the effective shear viscosity of suspensions and the effective tensile modulus of porous media [1–5]. In particular, the Coble-Kingery approach [6] to describe the nonlinear porosity dependence of the effective elastic moduli has been recalled [1, 3] and, based on the corrected version of the Coble-Kingery relation for the effective tensile modulus [1], a new relation, the simplest with percolation threshold (critical porosity), has been proposed for the porosity dependence of the effective tensile modulus [5]. We emphasize again, that the aforementioned analogy originates from the possibly analogous microstructures or microstructural models of suspensions and porous media, i.e., it concerns micromechanical relations and is therefore of an entirely different character than the well-known continuum mechanical analogy between linear elasticity and linear viscosity [7]. In this paper we use the Coble-Kingery approach [6] to derive a handy formula for the porosity dependence of the effective thermal conductivity k. Probably the earliest formula applicable to this property in composites of the matrix-inclusion type is the Maxwell approximation of 1873 [7–9], which describes a model material with spherical inclusions. For a two-phase material it can be written in the form