Granular flows, energy

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 most gas-particle flow situations occurring in fluidized bed reactors, a standard k — e turbulence model is used to describe the turbulence in the continuous phase whereas a separate transport equation is formulated for the kinetic energy (or granular temperature) of the particulate phase [122, 42, 41, 165, 84, 52]. Further details on granular flows are given in chap 4. 

In this section the application of multiphase flow theory to model the performance of fluidized bed reactors is outlined. A number of models for fluidized bed reactor flows have been established based on solving the average fundamental continuity, momentum and turbulent kinetic energy equations. The conventional granular flow theory for dense beds has been reviewed in chap 4. However, the majority of the papers published on this topic still focus on pure gas-particle flows, intending to develop closures that are able to predict the important flow phenomena observed analyzing experimental data. Very few attempts have been made to predict the performance of chemical reactive processes using this type of model. 

The severe implications of these facts have been partially uncovered in reference [5] as a result of formulating a kinetic theory for granular flow without interaction with the ambient medium. These implications, as well as additional difficulties due to the necessity to calculate the energy supply to the particle fluctuations, make somewhat problematic, at the present state of the art, the formulation of a reliable and sufficiently simple hydrodynamic model even for coarse dispersions. We have succeeded in this respect only at the expense of making certain supplementary assumptions. These assumptions cne ... 

It is therefore common to assume that the state variables that describe the rapid deformation response of granular materials would border on the parameters that describe the behavior of fluids and Coulomb type dissipation of energy (static). In view of the above it is common to find that the theories governing granular flow are formulated around the assumption of a continuum similar in some regard to viscous fluids however, the equilibrium states of the theories are not states of hydrostatic pressure, as would be in the case of fluids, but are rather states that are specified by the Mohr-Coulomb criterion (Cowin, 1974). The advantage of continuum formulations over alternative particulate (stochastic) formulations is that use of continuum is more capable of generating predictive results. Mathematically one... 

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. 

Readers will find this book starts most topics at a very basic level. Some examples of this are the chapters on combustion and on mass and energy balances. Soon, however, the complexities become evident, such as in the chapters on granular flow and segregation. 

In experiments, microscopic, noninvasive observation of the velocity distribution, correlated density fluctuation, or clustering will be helpful to quantify the collective behavior of granular flows under different heating or energy driving conditions. 

With experimental quantification of mesoscale structures, statistical analysis based on nonequilibrium distribution may require novel mathematical skills to unravel the complex dependence of stress-strain relation on the mass exchange between phases. The structure-dependent energy analysis may help elucidate the dependence between energy dissipation and structural parameters, and how to relate such structural-dependent analysis and the nonlinear nonequilibrium thermodynamics remains a challenge, especially for the scale-dependent granular flow or fluidization systems. 

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