Coarse-graining of Two-fluid Models for Fluidized Gas-particle Suspensions

Gas-particle flows in fluidized beds and circulating fluidized beds are inherently unstable, and they manifest fluctuations in velocities and local suspension density over a wide range of length and time scales. In riser flows, these fluctuations are associated with the random motion of the individual particles and with the chaotic motion of particle clusters. Coarse-grid simulations for industrial scale gas-particle flows will clearly not resolve the structures which exist on sub-grid length scales; however, these small-scale unresolved structures are known to affect the resolved flow characteristics. We pursue a filtered equations approach in which the influence of the small scale structures appears as residual correlations for which constitutive models should be constructed. In our present study, we examine the filter-width dependence of correlations in the filtered two-fluid model through threedimensional (3-D) calculations and demonstrate that the filtered drag coefficient extracted from the analysis of computational data generated through highly resolved 3-D simulations of a kinetic theory based two-fluid model decreased systematically with increasing filter width. We also show that both 2-D and 3-D simulations yield qualitatively similar results. NOMENCLATURE d particle diameter ρg gas density ρs particle density ep coefficient of restitution μg gas viscosity Pg pressure φ particle phase volume fraction vt terminal settling velocity s σ particle phase stress tensor g σ gas phase stress tensor f gas-particle interaction force g gravitational acceleration d rate of strain tensor