Effective conductivity of Voronoi's closed- and open-cell foams: analytical laws and numerical results

The conductive heat transfer in high-porosity cellular materials is generally treated by defining a homogeneous effective thermal conductivity. Numerous empirical and semiempirical models as well as numerical investigations have already been conducted to estimate this conductivity. These previous investigations were based on simplifications of the morphology of the cellular structure and/or of the method of solution of the heat transfer problem. Moreover, they were developed specifically for a type of foam, thus limiting their range of applicability. In order to improve the theoretical knowledge on this field, we have developed an innovative approach combining Voronoi methods for the generation of representative cellular materials and the finite element method (FEM) for solving the conductive heat transfer. The structures generated are able to reproduce the discriminating details of the microstructure and cover the whole range of open-cell or closed-cell foams commonly used in scientific or industrial applications. The influence of the structural parameters on the effective conductivity is analyzed. Based on this assessment, new simplified analytical relations are deduced accounting for the composition and structural parameters of the material. The validity of these laws has been verified by comparisons with "tomographic" results obtained from 3D tomographic data of real open-cell and closed-cell foams. The analytical correlations are potentially very useful for numerous applications.