Effective Diffusivity From Analytical Solution for Banks of Cylinders

Benchmarks are provided for the evaluation of the effective coefficient for species diffusion, or electrical/thermal conduction, in structured porous media. The cases considered corresponding to doubly periodic rows of circular cylinders, for which a power series solution has been previously obtained, from complex variable theory. Both inline and staggered geometries are considered for three common configurations: inline-square, rotated-square, and equilateral geometries. From these mathematical solutions, values for the effective conduction/diffusion coefficient are readily constructed. The results are presented in terms of correlations for the ratio of effective-to-bulk conductivity/diffusivity or microstructural parameter, as a function of porosity. It is shown that near identical results with the present analytical analysis are obtained using calculations based on a finite-volume method and also with a previous mathematical analysis for the case of inline-square geometry. The present analytical solutions are also compared with two well-known correlations for random porous media, based on effective medium and percolation theory. It is shown that agreement with the analytical solution is not in general particularly good and depends on the choice of fitting parameters. The present results may be used as canonical data for comparative studies with numerical procedures to enumerate microstructural parameters for arbitrary-shaped occlusions in random geometries.