Numerical Calculation of the Effective Thermal Conductivity of Three-Dimensional Three-Scale Microstructures

摘要

In the present work the effective thermal conductivity of multiscale composite media with 3-D microstructures is determined based on the method of reiterated homogenization and the finite element method. Conventional homogenization has been successfully applied to thermal problems in two-scale media. On the other hand, reiterated homogenization has been applied mainly to formulate problems in heterogeneous media with more than two scales, rather than to calculate effective properties. Here, specifically, the finite element method is employed in conjunction with the appropriate reiterated homogenization expressions, to calculate the effective conductivities of the 3-D three-scale body-centered and face-centered cubic arrays of spheres. At this stage, the instance with a perfect thermal contact at the interface is considered. The results for the effective thermal conductivity gains of the three-scale arrays relative to the two-scale counterparts are given in terms of the problem volume fractions and phase contrast. Within the parameter ranges considered, it is verified that gains in excess of 20% are achieved with the three-scale body-centered and face-centered arrays. Reiterated homogenization is seen to provide a sound theoretical framework to help explain observed enhanced heat conduction in media with multiscale microstructures.