Subgrid Drag Model Derivation from Drift Velocity Equation for Coarse-Grid Euler-Euler Simulations of Fluidized Beds

The industrial-scale simulations of fluidized bed reactors are currently performed using a Two-Fluid modeling approach where both the gas and the particles are treated as inter-penetrating continua. Yet, the size of the mesh used is usually larger than the size of the mesoscale structures that appear in such gas-solids flows, namely clusters of particles. Omitting the influence of these subgrid scale structures on the resolved scales of the flow was shown to give poor predictions of the bed hydrodynamics. The evaluation of the subgrid scale fluid-particle drag term currently relies on the modeling of the so-called drift velocity, due to the correlation between the solid volume fraction and the fluid velocity at the subgrid scale. Functional models based on the resolved particle volume fraction and the size of the coarse grid have been proposed to estimate the drift velocity from resolved quantities. In this study, a novel modeling approach is proposed. First, a transport equation is derived for the drift velocity. Fine-grid simulations of a tri-periodic fluidized bed reactor are performed to obtain mesh-independent results and the data are spatially filtered to perform a priori analyses of the coarse-grid equations. It appears from a budget analysis that the term involving the fluid-particle drag force is dominant in the drift velocity transport equation. An explicit model is derived where the drift velocity is expressed as a function of the resolved slip velocity and an algebraic function of a few moments of the solid volume fraction. Fine-grid simulations also show that the solid volume fraction conditioned by its filtered value and its subgrid variance follows a beta-type distribution. Henceforth, the proposed drift velocity model relies only on the local filtered volume fraction and its variance, for which a distinct transport equation or an algebraic model can be used.