A Simplified Method for Upscaling Composite Materials with High Contrast of the Conductivity

Abstract

A large class of industrial composite materials, such as metal foams, fibrous glass materials, mineral wools, and the like, are widely used in insulation and advanced heat exchangers. These materials are characterized by a substantial difference between the thermal properties of the highly conductive materials (glass or metal) and the insulator (air) as well as low volume fractions and complex network-like structures of the highly conductive components. In this paper we address the important issue for the engineering practice of developing fast, reliable, and accurate methods for computing the macroscopic (upscaled) thermal conductivities of such materials. We assume that the materials have constant macroscopic thermal conductivity tensors, which can be obtained by upscaling techniques based on the postprocessing of a number of linearly independent solutions of the steady-state heat equation on representative elementary volumes (REVs). We propose, theoretically justify, and computationally study a numerical method for computing the effective conductivities of materials for which the ratio $\delta$ of low and high conductivities satisfies $\delta\ll1$. We show that in this case one needs to solve the heat equation in the region occupied by the highly conductive media only. Further, we prove that under certain conditions on the microscale geometry the proposed method gives an approximation that is $\mathcal{O}(\delta)$-close to the upscaled conductivity. Finally, we illustrate the accuracy and the limitations of the method on a number of numerical examples.