where CD0 is the drag coefficient for a single particle of diameter dP and LP is the distance between the centers of the two spheres. Several correlations have been proposed to represent the influence of other particles on the drag coefficient (see a recent review by Enwald et al, 1996 and Appendix 4.2). More information about drag coefficient correlations is included in Part IV when discussing the modeling of different reactor types.

For granular flows (fluid-solid flows), in which particle-particle interactions play a substantial role, it is necessary to introduce additional terms in the basic governing equations. Granular flows may exhibit several sub-regimes such as: (i) an elastic regime, in which the multiphase system behaves like an elastic solid and stress is strain dependent; (ii) a plastic regime, in which stress is independent of strain rate, typical for slow flow conditions; and (iii) a viscous regime, in which stress is dependent on strain rate (see Fig. 4.6 for a schematic representation of these regimes). Several models based on concepts ranging from elasticity to soil mechanics have been proposed. For reactor engineering, the kinetic theory of granular flows will be most useful to model fluid-solid flows in dilute to dense bed regimes. The kinetic theory of granular flows is based on similarities between the flow of a granular material, a population of particles with or without interstitial gas, and the molecules of gas. This treatment uses classical results from the kinetic theory of gases (chapman and Cowling, 1970) to predict the form of transport equations for a granular material. One of the most complete works in the field of kinetic theory of granular flow is Jenkins and Richman (1985). Detailed discussion of the development of models of granular flow is beyond the scope of this book. Readers are referred to Gidaspaw (1994) and some recent papers (Nieuwland et al., 1996; Kuipers and van Swaaij, 1997; Peirano and Leckner, 1998). Kinetic theory based models introduce several additional terms in the solids stresses and, therefore, modify momentum conservation equations for

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