Granular flow defined

Due to the hyperbolicity and nonlinearity of the model equations, associated with possible shocks in granular flows over non-trivial topography, numerical solutions with the traditional high-order accuracy methods are often accompanied with numerical oscillations of the depth profile and velocity field. This usually leads to numerical instabilities unless these are properly counteracted by a sufficient amount of artificial numerical diffusion. Here, a non-oscillatory central (NOC) difference scheme with a total variation diminishing (TVD) limiter for the cell reconstruction is employed, see e.g. [4], [12] we obtain numerical solutions without spurious oscillations. In order to test the model equations, we consider an ideal mountain subregion in which the talweg is defined by the slope function... 

Chapter 10 contains a literature survey of the basic fluidized bed reactor designs, principles of operation and modeling. The classical two- and three phase fluidized bed models for bubbling beds are defined based on heat and species mass balances. The fluid dynamic models are based on kinetic theory of granular flow. A reactive flow simulation of a particular sorption enhanced steam reforming process is assessed. 

The probability on the right-hand side is again defined with respect to the ensemble of all realizations of the granular flow. The dynamical behavior of the NDF will be determined by the right-hand sides of Eqs. (4.1)-(4.3), and we will return to this aspect later. First, however, we will look at methods to estimate the NDF that are based on a single realization of the granular flow. 

The formal definition of the NDF given in Eq. (4.11) is mathematically consistent, but difficult to implement in practice. It is therefore useful to define methods for estimating the NDF from a single realization of the granular flow. Note that mathematically a statistical estimate is a random variable, and thus should not be confused with the NDF, which is deterministic. In order to distinguish the estimated NDF from n, we will denote the estimate by h. Thus, for example, if the estimate is unbiased then (n) = n, where the expected value is taken with respect to the multi-particle joint PDF / defined in Eq. (4.7). 

On the other hand, for systems with low particle-number density and low collision frequencies the estimator h will yield a poor representation of the NDE. Nonetheless, this does not imply that the NDE cannot be defined for such systems. Indeed, it is still precisely defined by Eq. (4.9). Instead it simply states that it will be extremely difficult to estimate the NDF using a single realization of the granular flow. The practical consequence of this statement is that it will be difficult to validate closure models for the terms in the GPBE (using either DNS of the microscale system or experimental measurements) for systems for which the standard deviation of the estimator is large. 

Granular flow behavior extends beyond the critical state as defined by traditional soil mechanics literature. As a result of rapid deformation associated with the flow, inertia as well as shear-rate effects must be considered. 

The nomenclature is consistent with existing granular flow literature (Lun et al., 1984 Johnson and Jackson, 1987 Natarajan and Hunt, 1998) but key parameters in these constitutive equations can be defined using the analogy of fluid flow as follows ... 

Pair distribution function in collisional theory Terms defined in granular flow constitutive equations... 

One can see as granular densities and pressures grow very quickly near the plane of jet interaction. Thus, solids deceleration is carried out in granular shock waves. The rapid decrease in axial components of particle velocities confirms a wavy nature of the granular flow. Radial particle velocity distributions on the jet periphery demonstrate the gas influence on the particle removal from the milling zone. This influence is observed for particles, which are smaller than 10 pm. The intensity of particle chaotic motion (relative particle-particle velocities) drops quickly with decrease in the particle diameters below 15 pm. This drop is caused by particle deceleration in a viscous gas (if collisions are elastic) and additionally by chaotic particle-particle collisions (if collisions are inelastic). This collisional intensity decrease causes a maximum of the relative particle-particle chaotic velocity at some distance from the plane of symmetry that is more explicit for inelastic collisions. Partial particle nonelasticity defines considerable drop in the chaotic velocity. The formation of a maximum of the collisional capacity at some distance from the plane of symmetry means that the maximal probability of particle fragmentation has to be also there. 

---