Granular flows, energy collisions

The theoretical studies of rapid granular flows are generally based on the assumption that the energy dissipation in a binary particle collision is determined by a constant coefficient of restitution e, the ratio of the relative approach to recoil velocities normal to the point of impact on the particle. However, measurements show that the coefficient of restitution is a strong function of the relative impact velocity [10]. Physically, the energy dissipation relates to the plastic deformation of the particle s surface. Thus, a realistic microscopic model should include the deformation history of the particle s surface. However, such a model might become computationally demanding and thus not feasible. 

In order to solve for the foregoing conservation equations to establish the granular flow field they must be closed by plausible constitutive relations for the stress terms, P, and Pf, the kinetic energy flux, q, and rate of dissipation by inelastic collision, 7, along with suitable boundary conditions. Applying these equations to describe the flow of material in the transverse plane of the rotary kiln will require a true quantification of the actual flow properties, for example velocity, in the various modes of rotary kiln observed and described earlier in Chapter 2. 

In the analysis of flow of granular material, two types of flow can be distinguished. The first is slow frictional flow where the particles remain in continuous contact with each other the internal forces result from Coulomb friction between contacting particles. The second type of flow is much more rapid the particles are not in constant contact with their neighbors. The energy associated with the velocity fluctuations is comparable to that of the mean motion. In this type of flow, the internal forces arise because of the transfer of momentum during collisions between particles. The constitutive relations for this rapid flow are rate-dependent. This type of flow, therefore, is referred to as viscous flow (sometimes just rapid flow). Steady,... 

In Fig. 3, the energy lost due to inelastic collisions and viscous gas are compared. For a mixture flow, compared with inelastic dissipation, the viscous dissipation is relatively small, especially in the region near the boundary wall. Hence it may be acceptable to neglect the effect of the drag force in establishing the boundary conditions. When the granular temperature decreases, the viscous dissipation becomes more important until it is, indeed, greater than the inelastic collision dissipation. 

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