Analytical models of thermal conductivity in two-phase dispersive media. 1. Theoretical study

It is traditionally believed that various theories and formulas for averaging (homogenization) the properties of inhomogeneous dispersive media, which do not take into account the distance interaction of dispersed particles, are applicable only at low volume concentrations of particles $0 < f_2 < 0.1$. The molecular heat transfer in two-phase dispersive media, both with and without allowance for the interaction of identical spherical particles, is considered in a mathematically rigorous formulation using the method of physical analogy and the concept of the Lorentz local field. It is shown that with an increase in the volume concentration of dispersed particles, the main influence on the effective thermal conductivity coefficient of the medium is exerted by a geometric constraint factor of the carrier phase, which is taken into account by the classical Maxwell's (Clausius-Mossotti) formula. The analytical dependences of the error in the Maxwell's formula, due to the neglected interaction of particles, on the concentration $f_2$ of the particles and the relative thermal conductivity of phases $\lambda_2/\lambda_1$ are obtained. Two corollaries from the Maxwell's formula are derived. The first corollary determines the exact boundaries enclosing the effective thermal conductivity coefficients of homogeneous and isotropic suspensions. They coincide with the known Hashin-Shtrikman bounds. The second corollary gives an exact solution that is invariant with respect to the phase inversion transformation. This solution is used to calculate the effective thermal conductivity coefficient in three-dimensional disordered structurally symmetric two-phase media.