The effects of mesoscale structures on the macroscopic momentum equations for two-phase flows

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Mesoscale structures (bubbles, clusters and streamers) in two-phase flows, especially in gas–solid fluidized beds significantly affect macroscopic hydrodynamic behavior. For industrial-scale fluidized beds, it is typically impractical to simulate these structures directly due to the excessive resolution required. To model effects of mesoscale structures, the ensemble phase averaging method is extended to derive macroscopic averaged equations and their closures. It is found that added-mass and drag reduction effects due to mesoscale structures play essential roles in the macroscopic equations of motion. Unlike the classical added-mass force, which is proportional to the continuous fluid density, the mesoscale added-mass force is proportional to the mixture density. Thus for gas–solid systems wherein the classical added-mass force is almost always negligible, the mesoscale added-mass force is, in contrast, found to be quite important. Mesoscale drag reduction results from the fact that, in a particle rich region, there is significantly less relative velocity between particle and fluid phases than indicated by the macroscopic relative velocity. Possible effects of the new force terms in the macroscopic equations are examined from a one-dimensional simulation of a fluidized bed. Significant effects from the new terms on vertical pressure gradient and particle volume fraction distributions are observed.