Granular temperature

Velocity measurement of the dispersed phase in multiphase flow is possible using both PIV and LDV. In PIV, the particles can be masked according to size, and the velocity for each size fraction can be estimated [7]. The turbulent properties, for example, granular temperature, are more difficult to measure because of the low number of particles in the measured volume. With LDV it is also possible to obtain the velocity and size for the dispersed phase, but the turbulent properties for the dispersed phase are still difficult to measure accurately, owing to the low number of particles and also because the position of the particles is not exactly the same aU the time. 

Direct measurement of particle velocity and velocity fluctuations in fluidized beds or riser reactors is necessary for validating multiphase models. Dudukovic [14] and Roy and Dudukovic [28] have used computer-automated radioactive particle tracking (CARPT) to foUow particles in a riser reactor. From their measurements, it was possible to calculate axial and radial solids diffusion as well as the granular temperature from a multiphase KTGF model. Figure 15.10 shows one such measurement... 

In a recent study, Huan et al. [25] performed NM R experiments in vibrofluidized beds of mustard seeds in which the small sample volume allowed pulses short enough that displacements in the ballistic phase were distinguishable from those in the diffusion phase. In this case, the average collision frequency is measured directly, bypassing the uncertainty of the multiplicative factor mentioned above. These workers also measured the height dependence of the granular temperature profile. 

Huan et al. [41] measured the behavior of a small fluidized bed consisting of 45-80 mustard seeds in a small-bore vertical magnet. The small sample size allowed short pulses, and spatial distribution of collision correlation times and granular temperature were measured directly and compared with the hydrodynamic theory of Garzo and Dufty [42], This paper [41] contains an excellent survey of previous experiments on fluidized beds. 

Fig. 19. Simulation results for both the soft-sphere model (squares) and the hard-sphere model (the crosses), compared with the Carnahan-Starling equation (solid-line). At the start of the simulation, the particles are arranged in a FCC configuration. Spring stiffness is K = 70,000, granular temperature is 9 = 1.0, and coefficient of normal restitution is e = 1.0. The system is driven by rescaling.

Fig. 22. The effect of the cohesive force on the excess compressibility. The coefficient of normal restitution is e = 1.0, and granular temperature is T = 1.0. The Hamaker constant is A = 3.0 x 10-12 (circles) and 3.0 x 1CT10 (crosses).

In another class of models, pioneered by Elghobashi and Abou-Arab (1983) and Chen (1985), a particle turbulent viscosity, derived by extending the concept of turbulence from the gas phase to the solid phase, has been used. This is the so-called k—s model, where the k corresponds to the granular temperature and s is a dissipation parameter for which another conservation law is required. By coupling with the gas phase k—s turbulence model, Zhou and Huang (1990) developed a k—s model for turbulent gas-particle flows. The k—s models do not... 

For particles of equal mass, we thus have esps = mn with n the local number density of particles. From the KTGF, the time evolution of the granular temperature is given by... 

Note that in the granular temperature equation Eq. (61), there is one extra term that is absent in the SET, namely the dissipation of fluctuating kinetic energy y. From the KTGF follows that... 

After the new solid volume fractions have been obtained from Eq. (93), new particle pressures are calculated, where after new velocities can be obtained from the coupled momentum equations. Next, new granular temperatures are calculated from the granular energy equations by an iterative procedure described in Section IV.E.l. Finally, the new mass residuals (D ijk and (T>s)v,i are computed and the convergence criteria are checked again. 

If we define Tc as the granular temperature to represent the specific kinetic energy of the velocity fluctuations or the translational fluctuation energy in such a way that... 

Applying Eq. (E5.13) at the wall and using Eq. (5.309) yield the fluctuation-specific kinetic energy or the granular temperature Tc as... 

High velocity, high accuracy motion required to extract granular temperature Motion inside metal-walled vessels of complex shape. Scale-up studies... 

The granular temperature, is obtained by solving its transport equation, which has the form ... 

It can be seen that lower values of particle-particle restitution coefficient predict higher values of centerline solids hold-up. Unfortunately, experimental data concerning solids hold-up was not available for the same operating conditions. The predicted profiles of granular temperature for the two values of restitution coefficient also show significant difference at the region near the symmetry axis. Despite these differences, it can be concluded that the model does not exhibit extreme sensitivity to the value of restitution coefficient. The influence of the value of the speculiarity parameter on... 

---