Effective thermal conductivity of media with randomly distributed convective gas-filled spheres

Abstract

One of the most well-known models for effective thermal conductivity (kef) is the Maxwell model, which considers a medium with a dilute concentration of spherical inclusions. In this work the Maxwell model is extended to include natural convection in the spheres. We consider a medium with randomly distributed gas-filled (hollow) spheres subjected to a constant heat flux. A method for calculating kef based on the dilute medium approximation is presented and applied to investigate kef, considering a variety of dimensionless parameters of the problem. Based on a large number of kef calculations, an approximate analytical formula is proposed (βc approximation). Furthermore, in many cases the convective effects are minor and can be neglected so that the Maxwell analytical formula can be applied (βMax approximation). The applicability of βc and βMax is studied considering a wide range of dimensionless parameters. It is found that for almost the entire parametric space one of the analytical approximations apply and therefore numerical calculations can be avoided.