Soft-sphere approach

A 2D soft-sphere approach was first applied to gas-fluidized beds by Tsuji et al. (1993), where the linear spring-dashpot model—similar to the one presented by Cundall and Strack (1979) was employed. Xu and Yu (1997) independently developed a 2D model of a gas-fluidized bed. However in their simulations, a collision detection algorithm that is normally found in hard-sphere simulations was used to determine the first instant of contact precisely. Based on the model developed by Tsuji et al. (1993), Iwadate and Horio (1998) incorporated van der Waals forces to simulate fluidization of cohesive particles. Kafui et al. (2002) developed a DPM based on the theory of contact mechanics, thereby enabling the collision of the particles to be directly specified in terms of material properties such as friction, elasticity, elasto-plasticity, and auto-adhesion. 

The large body of literature on calculations of structures of proteins, polypeptides, polysacharides and polynucleotides is not reviewed. These calculations, in so far as they apply to the force-field approach, are necessarily based on highly simplified fields (hard- or soft sphere approach) and furthermore, due to limits in computer memory capacity and speed, full relaxation of the atomic coordinates of such large molecules (> 75 atoms) is as yet unattainable. 

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

In DPMs, each particle is tracked individually and all collisions are calculated, thus providing a more reliable and detailed representation of the fluidized bed. The model was introduced by Hogue and Newland (1994), Hoomans et al. (1996), and Tsuji et al. (1992), and it employs either a hard-sphere approach for dilute systems or a soft-sphere approach for dense fluidized beds. 

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

Hoomans, B. P. B., Kuipers, J. A. M., and van Swaaij, W. P. M., Discrete particle simulation of a two-dimensional gas-fluidised bed Comparison between a soft sphere and a hard sphere approach. Submitted for publication (1998). 

Several models have been developed to describe these phenomena quantitatively, the main difference being the interaction potential between the particles. There are two major approaches the hard sphere and the soft sphere. The hard sphere assumes that the only interaction between particles is a strong repulsion at the point of contact. The soft sphere is more realistic and assumes a potential with a barrier and a primary minimum like in DLVO theory (Figure 11.8). 

The approximately constant value of the volumes of the VD polyhedra allows us to consider atoms as soft (easy-to-deform) spheres of constant volumes. Approaching of the two atoms due to their chemical interaction is accompanied by mutual deformations of their spheres (Fig. 3f), which, in the end, leads to the transformation of the spheres into VD polyhedra. The shapes of the VD polyhedra are controlled by the arrangement of atoms in the stmcture that can thus be considered as a close packing of soft spheres. 

There are two main approaches for the numerical simulation of the gas-solid flow 1) Eulerian framework for the gas phase and Lagrangian framework for the dispersed phase (E-L) and 2) Eulerian framework for all phases (E-E). In the E-L approach, trajectories of dispersed phase particles are calculated by solving Newton s second law of motion for each dispersed particle, and the motion of the continuous phase (gas phase) is modeled using an Eulerian framework with the coupling of the particle-gas interaction force. This approach is also referred to as the distinct element method or discrete particle method when applied to a granular system. The fluid forces acting upon particles would include the drag force, lift force, virtual mass force, and Basset history force.Moreover, particle-wall and particle-particle collision models (such as hard sphere model, soft sphere model, or Monte Carlo techniques) are commonly employed for this approach. In the E-E approach, the particle cloud is treated as a continuum. Local mean... 

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