Effective Thermal Conductivity of Continuous Matrix-spherical Dispersed Phase Composite with Single-point Interfacial Thermal Contact: Free Molecular Gas Conduction in the Gap

Abstract

Analytical solutions are presented for the lower bound of the effective thermal conductivity of a continuous matrix composite for the limiting case of fully debonded spherical inclusions in single-point contact with the matrix, with heat conduction across the interfacial gap occurring by gaseous heat transfer in the free-molecular regime. Imposed temperature gradients parallel and perpendicular to the diameter through the point-of-contact are considered. The composite thermal conductivity in addition to the generally accepted variables identified in the literature at any given pressure is found to be a function of the accommodation coefficients of the surfaces of the interfacial gap, but independent of the width of the gap. When the accommodation coefficient of one of the surfaces approaches zero, the composite thermal conductivity approaches Maxwell's value for a matrix with spherical pores, regardless of the value of the contact conductance. Instantaneous closure of the gap results in a discontinuous increase in the composite thermal conductivity to the value predicted by Maxwell for perfect interfacial thermal contact or the one predicted by Hasselman and Johnson for finite values of the interfacial thermal conductance. Numerical results are presented for a sodaborosilicate glass with spherical nickel inclusions. The composite thermal conductivity for imposed temperature gradients parallel or perpendicular to the diameter through the point of contact is found to differ significantly depending on the values for the interfacial thermal contact conductance and accommodation coefficients.