Use of a Downscaling Procedure for Macroscopic Heat Source Modeling in Porous Media

Abstract

This work deals with the challenges brought by nonlinear heat sources when upscaling heat transfer in porous media. These difficulties are exemplified through applying the volume averaging method to a simple convective heat transfer problem in a porous medium featuring a nonlinear heterogeneous heat source. The most general solution proposed requires the availability of an estimated value of the heat source in the averaging volume, which can be obtained through a multiscale approach making use of a downscaling methodology. The downscaling methodology yields pore-scale governing equations in a sub-domain of the porous medium allowing to deal with the lack of information about the thermal behavior of the sub-domain’s vicinity. Solving the downscaled equations allows to reconstruct the temperature field and the heat source in the sub-domain with a good accuracy. This approximated reconstruction of the temperature field in sub-domain makes it possible to compute accurate estimates of the macroscopic heat source. In practice, the computational cost of the multiscale approach can be reduced by storing the results of the downscaling procedure in a table which takes as entries the limited number of macro-scale dependencies of the downscaled problem’s solution. In an example, the resulting heat source table is used as an input in a heuristic macro-scale transport model and compared to classic approaches. The use of the heat source reconstructed by downscaling results in a significant improvement of the accuracy of the macro-scale solution when the temperature driving the heat source significantly deviates from the macro-scale temperature.