Closure of a macroscopic turbulence and non-equilibrium turbulent heat and mass transfer model for a porous media comprised of randomly packed spheres

Turbulent heat and mass transfer in packed beds of spheres is widely encountered in industrial and food storage applications and, as such, modeling of such cases is of interest in design and development. Herein, we propose a closure of the volume- and time-averaged (macroscopic) turbulence, and non-equilibrium turbulent heat and mass transfer equations for randomly packed spheres, based on computational results of flow and heat transfer for a unique geometric model. In this respect, the closure results are derived from pore-level (microscopic) information obtained from numerical simulations of turbulent heat and fluid flow. Turbulence is incorporated at both levels using the k-ɛ model, and the dispersive effects of turbulence are also considered. For the momentum equation, the closure is sought for the Darcy and Forchheimer terms. For the non-equilibrium heat and mass transport equations, we obtain closures for the dispersion, turbulent flux, turbulent dispersion, and interfacial heat and mass transfer terms. The closure results are found to be dependent upon the porosity and Reynolds number. However, the mean sphere diameter and its local variation inside the representative elemental volumes only weakly affect the results. The closure results are presented as power law-based correlations, such that they can be easily implemented in a volume-time-averaged framework.