Permeability correlation with porosity and Knudsen number for rarefied gas flow in Sierpinski carpets

In recent years, application of porous media is highlighted among researchers due to their wide range of usability in micro-scale problems, such as gas reservoirs, micro-filtering, heat exchangers, etc. With this respect, the accurate description of flow behavior using governing equations based on the continuum assumption is not valid since the mean free path is comparable to the characteristics length of the problem. For this purpose, a simple methodology for diffusion reflection boundary condition is developed and validated for two valuable benchmarks, namely micro-channel flow and fractal porous media, where the results were in good agreement with literature. Then, pore-scale simulation of the rarefied gas flow inside the square- and circular-based Sierpinski carpets in the slip and transition flow regimes (based on the developed boundary condition) for different porosities are carried out, using lattice Boltzmann method (LBM). In each case, the effects of different key parameters such as pressure gradient in the x-direction (body force), porosity, and Knudsen number on the velocity contours, velocity profiles, flow structures, and permeability are investigated. It is found that the velocity magnitude is reduced as Knudsen number increases, where the reduction rate of velocity in the slip flow regime is greater than that of the transition one. Also, a correlation between the permeability with porosity and Knudsen number in the Darcy flow regime (Re<<1) is presented. For both geometries, it is observed that the permeability varies logarithmically with Knudsen number and exponentially with the porosity.