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

One of the most well-known models for effective thermal conductivity (k_ef) 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 k_ef based on the dilute medium approximation is presented and applied to investigate k_ef, considering a variety of dimensionless parameters of the problem. Based on a large number of k_ef calculations, an approximate analytical formula is proposed (β_c approximation). Furthermore, in many cases the convective effects are minor and can be neglected so that the Maxwell analytical formula can be applied (β_Max approximation). The applicability of β_c and β_Max 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.