## Particle Turbulence in Two-Fluid-Models

It is practical to investigate macro-scale process units by using averaged equations of motion, which are usually referred to as two-fluid model (TFM). However, the TFM approach requires considerably fine grids since the minimum stable sizes of clusters and shear bands are around ten particle diameters. This, in turn, implies that when using TFM the numerical analysis of an industrial scale reactor of several meters height, which contains in the order of O(>10^{12}) particles, would require at least in the order of O(10^{9}) computational cells to resolve all relevant structures. Thus, due to computational limitations a fully resolved simulation of industrial scale reactors is still unfeasible. It is, therefore, common to use coarse grids to reduce the demand on computational resources of the numerical analysis at the macro-scale.

Such “coarse-graining” procedures inevitably neglect the small (unresolved) scales, which leads, for example, to a considerable overestimation of the bed expansion in the case of fluidised beds of fine particles (Figure 1). In particular, the unresolved part of gas-solid drag force appears to be the dominating contribution to the solid momentum equations.