Macroscopic models for heterogeneous reactions in porous media
Federico Municchi, Matteo Icardi · arXiv (Cornell University) · 2019
Derivation of coarse grained models for advection-diffusion processes in the presence of dominant surface reactions using homogenisation theory or volume averaging is often deemed unfeasible \citep{VALDESPARADA20112177,BATTIATO201118} due to the strong coupling between scales that characterise such systems. In this work, we show how this problem can be circumvented by applying and extending the methods presented in \citet{ALLAIRE2007523,Mauri_1991}. Such process relies on the decomposition of the solution with the principal eigenvalue and eigenfunction of the non self-adjoint linear advection-diffusion operator, and on a two-scale asymptotic expansion. A set of cell problems and the corresponding macroscopic equation are obtained. The whole procedure is implemented and the cell problems are solved for arbitrarily complex periodic structures using the open-source finite volume library OpenFOAM\textsuperscript{\textregistered}. We provide details on the implementation and test the methodology for two-dimensional periodic arrays of spheres, in a wide range of Peclet and surface Damk{o}hler numbers. Results are compared to fully resolved numerical simulations, demonstrating the accuracy and generality of the upscaling approach. Noticeably, the effective transport parameters (dispersivity and velocity) can be significantly different from the non-reactive (conserved scalar) case, as the heterogeneity introduced by the reaction strongly affects the micro-scale profiles.