Predictions of the P1 approximation for radiative heat transfer in heterogeneous granular media

The P1 approximation is a computationally efficient model for thermal radiation.Here,we present a P1 formulation in the context of the combined computational fluid dynamics and discrete element method(CFD-DEM),including closures for dependent scattering and coarse-graining.Using available analytical and semi-analytical solutions,we find agreement for steady-state and transient quantities in sizedisperse systems.Heat flux is identified as the most sensitive quantity to predict,displaying unphysical spatial oscillations.These oscillations are due to a temperature slip at the locations of abrupt change in solid fraction.We propose two techniques that mitigate this effect:smoothing of the radiative properties,and pseudo-scattering.Furthermore,using up to a million times enlarged particles,we demonstrate practically limitless compatibility with coarse-graining.Finally,we compare predictions made with our code to experimental data for a pebble bed under vacuum conditions,and in presence of nitrogen.We find that a carefully calibrated simulation can replicate trends observed in experiments,with relative temperature error of less than 10%.