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 size-disperse 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%.