Abstract
Heat transport through a porous medium depends on the local pore geometry and on the heat conductivities of the solid and the saturating fluid. Through upscaling using formal homogenization, the local pore geometry can be accounted for to derive effective heat conductivities to be used at the Darcy scale. We here consider thin porous media, where not only the local pore geometry plays a role for determining the effective heat conductivity, but also the boundary conditions applied at the top and the bottom of the porous medium. Assuming scale separation and using two-scale asymptotic expansions, we derive cell problems determining the effective heat conductivity, which incorporates also the effect of the boundary conditions. Through solving the cell problems, we show how the local grain shape, and in particular its surface area at the top and bottom boundary, affects the effective heat conductivity through the thin porous medium.
Highlights
Heat conduction in porous media is a relevant process in applications ranging from geothermal engineering to various technical applications
We extend the formal homogenization approach applied to fluid flow in thin porous media in Fabricius et al (2016) by incorporating heat transport and focus in particular on the role of the arising effective heat conductivity
We continue with a discussion on the impact of different boundary conditions and grain shapes on the derived effective heat conductivity as well as the impact of the upscaling procedure itself in Sect. 6, before we end with some concluding remarks
Summary
Heat conduction in porous media is a relevant process in applications ranging from geothermal engineering to various technical applications. Formal homogenization allows to derive upscaled equations and corresponding effective quantities following suitable assumptions Within this framework, the effective properties of the porous medium are obtained by solving so-called cell problems at the pore scale. By considering a thin strip as a simplified representation of a porous medium, effective equations have been found by combining asymptotic expansions and transversal averaging for reactive transport (van Noorden 2009), heat transport (Bringedal et al 2015), biofilm growth (van Noorden et al 2010) and two-phase flow (Lunowa et al 2021; Sharmin et al 2020) In these works the thin strip is in practice a channel and does not contain a porous structure. We extend the formal homogenization approach applied to fluid flow in thin porous media in Fabricius et al (2016) by incorporating heat transport and focus in particular on the role of the arising effective heat conductivity. We continue with a discussion on the impact of different boundary conditions and grain shapes on the derived effective heat conductivity as well as the impact of the upscaling procedure itself in Sect. 6, before we end with some concluding remarks
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