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Particle-Particle interaction parameters

Contact theory/material properties

Direct solution of Navier-Stokes equations

FIGURE 12.6 Hierarchy of models for the simulation of gas-solids fluidized bed reactors.

equations, lattice-Boltzmann models and contact theory to obtain all the necessary closure laws and other parameters required for granular flow models. However, with the present state of knowledge, complete a priori simulations are not possible. It is necessary to use these different models judiciously, combined with key experiments, to obtain the desired engineering information about gas-solid flows in industrial equipment. Direct solution of Navier-Stokes equations or lattice Boltzmann methods are too computation intensive to simulate even thousands of solid particles, rather than millions of particles. DPMs are usually used to gain an insight into various vexing issues such as bubble or cluster formations and their characteristics or segregation phenomena. A few hundred thousand particles can be considered in such DPMs. The understanding developed and simulation results are either directly or indirectly used to develop granular flow models. Applications of DPM and GFM to simulate gassolid flows in riser and bubbling/turbulent bed reactors are discussed in the following sections.

12.2.1. Discrete Particle Models (DPM)

In discrete particle models, continuous phase (gas phase) is modeled using Eulerian framework. The trajectories of dispersed phase (solid phase) are then modeled in a Lagrangian framework. Acceleration of individual particles of dispersed phase is calculated from a force balance over that particle. Particle trajectories are then simulated using the flow field of gas phase. Basic governing equations are described in Chapter 4. Based on a large number of particle trajectories, desired characteristics of the gas-solid flow can be evaluated. The basic framework can be extended to include two-way coupling between gas and solid phases as well as four-way coupling to include particle-particle interactions. For most fluidized bed reactor applications, it is necessary to include the influence of particle collisions on the dynamics of gassolid flows. To simulate particle-particle collisions, two approaches may be used: in the first approach, particle interaction times are assumed to be very small compared to the free flight times (hard-sphere approach) and in the second, interaction times are assumed to be large compared to the free flight times (soft-sphere approach). Hoomans (2000) applied a hard-sphere approach to model bubbling fluidized beds and riser flows; Kaneko (2000) applied a soft-sphere approach to simulate bubbling fluidized bed reactors. Some of these results illustrating the influence of key model parameters are discussed below.

Before implementing hard-sphere or soft-sphere models, several issues related to formulations of various terms need to be sorted out. For hard-sphere models, key parameters are coefficient of restitution, coefficient of tangential restitution and coefficient of friction. For soft-sphere models, key parameters are normal spring stiffness, tangential spring stiffness and damping coefficient (Hoomans, 2000). In principle, soft-sphere models reduce to hard-sphere models in the limit of very high spring stiffness. In practice, however, soft-sphere models cannot be applied for very high values of spring stiffness due to computational constraints. Higher values of spring stiffness require lower and lower values of time step and may inordinately increase demands on computational resources. For most simulations based on soft-sphere models, an arbitrary low value of spring stiffness is specified. Fortunately, the actual magnitude of the spring stiffness parameter does not significantly affect the simulated fluidization behavior and low value can be safely used for most simulations. Simulation results

FIGURE 12.7 Simulated results for three values of spring stiffness ((a) 8 N m-1; (b) 800 N m-1; (c) 80 000 N m-1). Soft sphere approach; number of particles = 14 000, u = 3 umf (from Kaneko, 2000).

(Kaneko, 2000) obtained for three values of spring stiffness constant (spread over four orders of magnitude) are shown in Fig. 12.7. It can be seen that if the objective is to understand the macroscopic behavior of the fluidized bed, low values of spring stiffness can be used for faster simulations. It must, however, be remembered that when such artificially low values of spring stiffness constant are used, the predicted values of contact time between solid particles are not realistic. When the objective is to understand local particle to particle heat or mass transfer, it is important to make accurate predictions of particle contact times. For such cases, it is necessary to use realistic values of spring stiffness constant at the expense of increased computational resources.

DPMs can also be used to understand the influence of particle properties on fluidization behavior. It has been demonstrated that ideal particles with restitution coefficient of unity and zero coefficient of friction, lead to entirely different flu-idization behavior than that observed with non-ideal particles. Simulation results of gas-solid flow in a riser reactor reported by Hoomans (2000) for ideal and nonideal particles are shown in Fig. 12.8. The well-known core-annulus flow structure can be observed only in the simulation with non-ideal particles. These comments are also applicable to simulations of bubbling beds. With ideal collision parameters, bubbling was not observed, contrary to the experimental evidence. Simulations with soft-sphere models with ideal particles also indicate that no bubbling is observed for fluidization of ideal particles (Hoomans, 2000). Apart from the particle characteristics, particle size distribution may also affect simulation results. For example, results of bubble formation simulations of Hoomans (2000) indicate that accounting

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