Abstract
This paper presents the development of a model enabling the analysis of rarefied gas flow through highly heterogeneous porous media. To capture the characteristics associated with the global- and the local-scale topology of the permeable phase in a typical porous medium, the heterogeneous multi-scale method, which is a flexible framework for constructing two-scale models, was employed. The rapid spatial variations associated with the local-scale topology are accounted for stochastically, by treating the permeability of different local-scale domains as a random variable. The results obtained with the present model show that an increase in the spatial variability in the heterogeneous topology of the porous medium significantly reduces the relevance of rarefaction effects. This clearly shows the necessity of considering a realistic description of the pore topology and questions the applicability of the results obtained for topologies exhibiting regular pore patterns. Although the present model is developed to study low Knudsen number flows, i.e. the slip-flow regime, the same development procedure could be readily adapted for other regimes as well.
Highlights
The classification of rarefaction, in terms of the Knudsen number, Kn, which is defined as the ratio of mean free path to the characteristic dimensions of the channel, presented in Barber and Emerson (2006) tells us (i) that the flow is rarefied when Kn ≳ 10−3, (ii) when 10−3 ≲ Kn ≲ 10−1, the flow is in the slip-flow regime and continuum theory incorporating the slip flow boundary condition is applicable, (iii) for 10−1 ≲ Kn ≲ 101, the flow is in the transition regime in which continuum theory is not applicable, and (iv) for Kn ≳ 101, the flow is said to be in the free molecular regime, which admits the collisionless Boltzmann equation to govern the flow behaviour
For the numerical simulations performed by means of the stochastic two-scale model developed in this work, we have chosen a collection of randomly packed balls, to represent a benchmark for topology of heterogeneous porous media
This framework is based on a two-scale approach in which the highly heterogeneous nature of the porous medium is modelled by treating the permeability of local-scale domains as a random variable
Summary
For the numerical simulations performed by means of the stochastic two-scale model developed in this work, we have chosen a collection of randomly packed balls, to represent a benchmark for topology of heterogeneous porous media. It can be used to illustrate the applicability of the model and it makes it possible to study the influence of the spatial variability of the topology on the effect of rarefaction It should, be emphasised that the modelling framework developed can be applied to study the flow of rarefied gases through other more complex types of heterogeneous porous media, provided that the due assumptions are met. When considering more complex geometries, the random variations exhibited by the porous medium can only be expected to be more influential than in the simpler geometry considered here
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