Hard particle simulations

In a hard sphere approach, particles are assumed to interact through instantaneous binary collisions. This means particle interaction times are much smaller than the free flight time and therefore, hard particle simulations are event (collision) driven. For a comprehensive introduction to this type of simulation, the reader is referred to Allen and Tildesley (1990). Hoomans (2000) used this approach to simulate gas-solid flows in dense as well as fast-fluidized beds. There are three key parameters in such hard sphere models, namely coefficient of restitution, coefficient of dynamic friction and coefficient of tangential restitution. Coefficient of restitution is discussed later in this chapter. Detailed discussion of these three model parameters can be found in Hoomans (2000). 

The investigations show that the microfocus high speed radioscopy system is suitable for monitoring the hard particle transport during laser beam dispersing. It is possible to observe and analyse the processes inside the molten bath with the presented test equipment. As a consequence a basis for correlation with the results of a simulation is available. 

It has not proved possible to develop general analytical hard-core models for liquid crystals, just as for nonnal liquids. Instead, computer simulations have played an important role in extending our understanding of the phase behaviour of hard particles. Frenkel and Mulder found that a system of hard ellipsoids can fonn a nematic phase for ratios L/D >2.5 (rods) or L/D <0.4 (discs) [73] however, such a system cannot fonn a smectic phase, as can be shown by a scaling... 

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. 

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). 

Two of the most common classes of particle-dynamic simulations are termed hard-particle and soft-particle methods. Hard-particle methods calculate particle trajectories in response to instantaneous, binary collisions between particles and allow particles to travel ballistically between collisions. This class of... 

Hoomans B.P.B et al (1996) Discrete particle simulation of bubble and slug formation in a two-dimensional gas-fluidised bed A hard-sphere approach. Chemical Engineering Science, 51, pp. 99-118... 

J. G. Malherbe and S. Amokrane (1999) Asymmetric mixture of hard particles with Yukawa attraction between unhke ones a cluster algorithm simulation... 

The second notable feature of these evolution curves is the pronounced shoulder effect seen on short time scales, particularly for the case where the flow is initiated from a site farthest removed from the reaction center. The appearance of shoulders is related to the fact that, for a particle initiating its motion at a specific site somewhere in the lattice, there is a minimum time required for the coreactant to reach the reaction center this time is proportional to the length of the shortest path, and hence the reactive event cannot occur until (at least) that interval of time has expired. This effect is analogous to the one observed in computer simulations of Boltzmann s H function calculated for two-dimensional hard disks [27]. Starting with disks on lattice sites with an isotropic velocity distribution, there is a time lag (a horizontal shoulder) in the evolution of the system owing to the time required for the first collision between two hard particles to occur. 

Donev, A., Torquato, S., StiUinger, R Neighbor list coUision-driven molecular dynamics simulation for nonspherical hard particles. 1. Algorithmic details. J. Comput. Phys. 202,737-764 (2005). doi 10.1016/j.jcp.2004.08.014... 

The abrasion resistance of plastics can also be determined by the ASTM G 65 dry sand rubber wheel test (7), vriiich is shown schematically in Figure 2. The three-body abrasion produced by this type of test is simitar to that produced by rubbing plastic on a rigid surface with hard particles in the faying surface, but it is probably faster and easier to perform. Test times can be as short as one minute. This test can also be conduct immersed in a slurry (8) if this better simulates the system of interest. 

The simulation process based on the above RSA scheme is very efficient for hard particles adsorbing flat (side-on) at the interface [126-130]. However, in the case of interacting particles of nonspherical shape (e.g., spheroids) adsorbing xmder random orientations, the numerical calculations become rather tedious due to mathematical problems in evaluating the minimum surface-to-surface distance and the interaction energy [15,131]. 

FIG. 35 The reduced maximum coverage of interacting particles / oo (where is the jamming coverage of hard particles) versus the a/Le parameter. The points denote the numerical RSA simulations performed for curve 1, = 1 (spheres) curve 2,A = 0.5 (prolate spheroids, side-on adsorption) curve 3, A = 0.2 (prolate spheroids, side-on adsorption). (From Ref. 12.)... 

The dissipative particle dynamics (DPD) method is a recent variation of the molecular dynamics technique. Here, in addition to Newtonian forces between hard particles, soft forces between particles are also introduced. These pairwise damping and noise forces model slower molecular motions. The dissipative forces also reduce the drift in kinetic energy that occurs in molecular dynamics simulations. These two reasons mean that DPD can be used to model longer time-scale processes, such as hydrodynamic flows or phase separation processes. 

Two of the most common classes of particle-dynamic simulations are termed hard-particle and soft-particle methods. Hard-particle methods calculate particle trajectories in response to instantaneous, binary collisions between particles, and allow particles to follow ballistic trajectories between collisions. This class of simulation permits only instantaneous contacts and is consequently often used in rapid flow situations such as are found in chutes, fluidized beds, and energetically agitated systems. Soft-particle methods, on the other hand, allow each particle to deform elastoplastically and compute responses using standard models from elasticity and tribology theory. This approach permits enduring particle contacts and is therefore the method of choice for mmbler apphcations. The simulations described in this chapter use soft-particle methods and have been validated and found to agree in detail with experiments. 

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