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Ideal^)articles2

Non-ideal2 particles2

FIGURE 12.8 Simulated fluidization behavior for ideal and non-ideal particles in a Riser reactor (from Hoomans, 2000).

for non-uniform particle size leads to much better agreement with experimental data as shown in Fig. 12.9. During simulations with uniform particles, some small satellite bubbles appeared above and alongside the main bubble. Such satellite bubbles were not observed in the experiment nor in simulations with non-uniform particles.

When appropriate parameters are used, both, hard-sphere and soft-sphere models lead to similar predictions. Bubble formation results obtained from hard-sphere and soft-sphere models are shown in Fig. 12.10 (from Hoomans, 2000). The main bubble size and position in the bed observed in both simulations agree quite well with the experiment. The shape of the bubble observed in the experiment was more rounded than that observed in simulations. Thus, any of these, hard-sphere or soft-sphere

Experiment

Uniform particles

### Log-normal particle size distribution

FIGURE 12.9 Comparison of simulation results with uniform and with log-normal particle size distribution and experimental observation (from Hoomans, 2000).

Experiment

Uniform particles

### Log-normal particle size distribution

FIGURE 12.9 Comparison of simulation results with uniform and with log-normal particle size distribution and experimental observation (from Hoomans, 2000).

Experiment

Hard-sphere model

Soft-sphere model

FIGURE 12.10 Comparison of simulation results for hard and soft sphere models with experimental results (from Hoomans, 2000).

Experiment

Hard-sphere model

Soft-sphere model

FIGURE 12.10 Comparison of simulation results for hard and soft sphere models with experimental results (from Hoomans, 2000).

models can be applied to gain insight into fluidization behavior. It should be noted that although predicted results of both approaches are almost the same, the computational efficiency of these two models might be different depending on the problem under consideration. Soft-sphere simulations progress at a constant speed (controlled by the time step, which in turn depends on the value of spring stiffness constant) whereas hard-sphere simulations are event driven and their speed depends strongly on system dynamics. More frequent collisions will slow down hard-sphere simulations (controlled by time interval between successive collisions).

DPMs may be used to understand the influence of particle characteristics on bubble formation, cluster formation and so on. These models may provide information regarding bubble size, cluster size, heat and mass transfer from such clusters to reaction engineering models. DPMs may also be used to extrapolate cold-flow data on bubble sizes/cluster sizes to high temperatures and high pressures. Interaction of gas distributor holes and gas-solid dynamics in the fluidized beds can also be examined with these models. Kaneko (2000) used DPMs to examine the behavior of fluidized polymerization reactors. These results are discussed in Section 12.3.

It must be noted here that most industrial fluidized bed reactors operate in a turbulent flow regime. Trajectory simulations of individual particles in a turbulent field may become quite complicated and time consuming. Details of models used to account for the influence of turbulence on particle trajectories are discussed in Chapter 4. These complications and constraints on available computational resources may restrict the number of particles considered in DPM simulations. Eulerian-Eulerian approaches based on the kinetic theory of granular flows may be more suitable to model such cases. Application of this approach to simulations of fluidized beds is discussed below.

12.2.2. Granular Flow Models (GFM)

Granular flow models are based on the interpenetrating continuum assumption. Both, gas and solid phases are modeled as a continuum. In this approach, individual particle trajectories are not simulated but an attempt is made to represent physics of those trajectories and particle-particle interactions using averaged form of governing equations. Because of use of such averaged equations, models based on this approach can be extended to simulate gas-solid flows comprising large number of solid particles. Basic equations of this approach are discussed in Chapter 4. Some recent work on development of CFD based models of fluidized bed reactors is briefly reviewed here. The discussion is divided into riser (fast fluidization regime) and dense bed (bubbling/ turbulent bed regime) simulations.

### Simulations of riser reactors

Riser reactors are used in a variety of applications, with fluid catalytic cracking (FCC) probably the most important one. Evolutionary design changes are constantly being introduced into all components of riser hardware to enhance the performance. Computational fluid dynamics has been used to understand the fluid dynamics of FCC systems and to evaluate alternative hardware configurations (Theologos et al., 1997; Ranade, 1998). Several attempts have been made to model gas-solid flows in vertical pipes (Dasgupta et al., 1998; Kuipers and van Swaaij, 1999 and references cited therein). Most of these attempts were based on the kinetic theory of granular flows (KTGF). Gao et al. (1999) simulated gas-solid flows in risers without using the kinetic theory of granular flows. Their results showed reasonably good agreement with two experimental data sets. However, in general, models based on KTGF require less ad hoc adjustments and have much wider applicability.

Sinclair and Jackson (1989) used the kinetic theory of granular flows to simulate gas-solid flows in risers. Their model was found to exhibit extreme sensitivity with respect to the value of restitution coefficient, es. Nieuwland et al. (1996) also observed such an extreme sensitivity. Bolio et al. (1995) reported that such extreme sensitivity could be overcome by including a gas phase turbulence model. Despite these studies, there are no systematic guidelines available to make appropriate selection of models and model parameters (such as laminar versus turbulent, values of restitution coefficients and specularity coefficient and interphase drag coefficients) to simulate gas-solid flow in industrial risers. It is observed from most of the available studies that the range of gas and solid fluxes investigated is not directly relevant to the operating range of industrial riser reactors. The influence of riser diameter, particle size and density, solids flux on various flow characteristics (pressure drop, solid volume fraction profiles and so on) has not been studied systematically. Here we report computational experiments discussed by Ranade (1999) to evolve possible guidelines for modeling gas-solid flows in riser reactors.

A two-fluid model with kinetic theory of granular flows was used to formulate the governing transport equations for gas-solid flows in riser. For details of the model equations, refer to the discussion in Chapter 4. Although in many industrial reactors, gas-solid flow in a riser may not be fully developed (except at the top region of the riser), it is always beneficial to start by developing relevant models to simulate fully developed gas-solid flow in a vertical riser. After adequate validation of such a base model, it can then be extended to simulate developing flow of gas and solid mixture. Ranade (1999) modeled fully developed flow by considering a very short riser with periodic (translationally) boundaries. With this approach, it is not necessary to model the large height of the riser reactor to ensure a fully developed state of flow. The computational grid used in his simulations is shown in Fig. 12.11. In order to resolve steeper gradients near the wall, a finer grid was used in the near wall region. The computational model was mapped on to a commercial CFD code, FLUENT version 4.5 (Fluent Inc., USA) with the help of user-defined subroutines.

For each simulation, superficial gas and solid velocities were specified as input parameters. Computations were started by setting the initial guess equal to these specified velocities. After each time step (of 0.001 s), all the variables except fluid pressure at the inlet, were set from the values calculated at the corresponding outlet computational cells. When setting the gas and solid axial velocity at the inlet, a correction was made to enforce the specified net gas and solid fluxes. At the riser wall, boundary conditions proposed by Sinclair and Jackson (1989) are recommended and were used for solids axial velocity and granular temperature. For the gas phase, the usual no slip boundary conditions (with wall functions) were used. To estimate the interphase drag force, a correlation proposed by Wen and Yu (1966) was used. For the gas-solid flows considered in this work, the contributions of lift and virtual mass forces were negligible. The kinetic theory of granular flows was used to calculate other relevant properties (such as solids viscosity and pressure). A standard k-e model was used to simulate gas phase turbulence. In order to consider solid phase turbulence, the time-averaged granular temperature equation was solved. Additional terms including

Symmetry axis

Periodic boundaries Wall.

Symmetry axis

Periodic boundaries Wall.

dissipation of solid phase turbulence, correlation between fluctuations of granular temperature and solids phase volume fraction were considered in the model. The basic governing equations are discussed in Chapter 4. Using these governing equations and the proposed boundary conditions, transient simulations were carried out until the fully developed steady state results were obtained.

Using a similar model, Bolio et al. (1995) reported good agreement between model predictions and the experimental data of Tsuji et al. (1984). Instead of repeating those simulations, we report here a comparison of simulated gas-solid flow with the experimental data of Yang (1991). The value of solid flux used in these experiments was also rather low (10 kg m-2 s-1). The comparison is shown in Fig. 12.12. It can be seen that the centerline gas velocity predicted without considering the turbulence model, is significantly higher than that reported by Yang (1991). Predicted results after considering the turbulence model show much better agreement with experimental data (Fig. 12.12). The predicted radial profiles of solid hold-up are shown in Fig. 12.13. In order to verify that the predicted results are not unreasonably sensitive to the value of particle-particle restitution coefficient, gas-solid flow simulations were also carried out with a restitution coefficient of 0.95. Comparison of predicted radial profiles of gas and solid velocity for these two values of restitution coefficient indicate that predicted results are not unduly sensitive to the value of restitution coefficient.

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

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