Collision restituting

Figure A3.1.7. Direct and restituting collisions in the relative coordinate frame. The collision cylinders as well as the appropriate scattering and azimuthal angles are illustrated.

Thus die increase of particles in our region due to restituting collisions with an impact parameter between b and b + Ab and azimuthal angle between e and e + de (see figure A3.1.7 can be obtained by adjusting the expression for the decrease of particles due to a small collision cylinder ... 

Figure A3.1.8. Schematic illustration of tire direct and restituting collisions.

We now show that when H is constant in time, the gas is in equilibrium. The existence of an equilibrium state requires the rates of the restituting and direct collisions to be equal that is, that there is a detailed balance of gain and loss processes taking place in the gas. 

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). 

At high-particle number densities or low coefficients of normal restitution e, the collisions will lead to a dramatical decrease in kinetic energy. This is the so-called inelastic collapse (McNamara and Young, 1992), in which regime the collision frequencies diverge as relative velocities vanish. Clearly in that case, the hard-sphere method becomes useless. 

The contact force between two particles is now determined by only five parameters normal and tangential spring stiffness kn and kt, the coefficient of normal and tangential restitution e and et, and the friction coefficient /if. In principle, kn and k, are related to the Young modulus and Poisson ratio of the solid material however, in practice their value must be chosen much smaller, otherwise the time step of the integration needs to become unpractically small. The values for kn and k, are thus mainly determined by computational efficiency and not by the material properties. More on this point is given in the Section III.B.7 on efficiency issues. So, finally we are left with three collision parameters e, et, and which are typical for the type of particle to be modeled. 

Exercise. Show that in equilibrium the last three lines of (5.9) vanish, provided that w P, Pi P3iPa) = w P2 P4. P >P2 which means that to each collision corresponds an inverse or restituting collision with equal cross-section. ... 

In order to determine the unknown velocities after collision uniquely, two more relationships for impact velocities need to be specified. For certain values of the coefficient of friction and the coefficient of restitution, simple expressions of impact velocities can be obtained. In a collision of two completely rough and inelastic spheres, where the coefficient of friction / reaches infinity and the coefficient of restitution e is equal to zero, the relative velocities must vanish. Therefore, we have... 

Any solid material has its own upper limit of elastic deformation under either normal or tangential stresses. Once the stresses exceed this limit, plastic deformation will occur. In this section, collisions of inelastic spheres are presented. The degree of inelastic deformation is characterized by the restitution coefficient. 

The recoverability or restitution of the kinetic energy during a normal collision between two objects can be represented by the coefficient of restitution defined by Eq. (2.3). Note that the coefficient of restitution cannot be used as a criterion to judge whether a collision is elastic or not unless the collision is solely considered as a normal collision. For example, the sliding at contact for the collision between two elastic spheres will make the collision inelastic while the value of the coefficient of restitution in this case is equal to 1. 

So far, in this model, we assume that all the particles are elastic and the collision is of specular reflection on a frictionless smooth surface. For inelastic particles, we may introduce the restitution coefficient e, which is defined as the ratio of the rebound speed to the incoming speed in a normal collision. Therefore, for a collision of an inelastic particle with a frictionless surface as shown in Fig. 5.9, we have... 

The effect of the particle movement on the fluid phase, and interactions between particles are neglected. This is justified because the number of particles in the freeboard is low. Particles are allowed to bounce off walls, where a coefficient of restitution of 0.8 is applied, which is based on the elasticity of sand and metal collisions but can be varied considerably (from 0.5 to 0.9) without affecting the entrainment results noticeably. 

When simulating the trajectories of dispersed phase particles, appropriate boundary conditions need to be specified. Inlet or outlet boundary conditions require no special attention. At impermeable walls, however, it is necessary to represent collisions between particles and wall. Particles can reflect from the wall via elastic or inelastic collisions. Suitable coefficients of restitution representing the fraction of momentum retained by a particle after a collision need to be specified at all the wall boundaries. In some cases, particles may stick to the wall or may remain very close to the wall after they collide with the wall. Special boundary conditions need to be developed to model these situations (see, for example, the schematic shown in Fig. 4.5). 

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). 

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 fluidization 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... 

It is noticed that in-elastic collisions are characterized by the degree to which the relative speed is no longer conserved. For this reason the coefficient of restitution e in a collision is defined (i.e., with basis in (2.120)) as the ratio of the relative velocity after collision, divided by the relative velocity of approach [45] [69] ... 

For these inelastic particles it is required that the relative velocity component normal to the plane of contact, g2i k (before collision) and C21 k (after collision) satisfy the empirical relation (2.123) [31]. If the restitution coefficient therein is equal to one, the collision is elastic, which means that there is no energy loss during collision. Otherwise the collision is inelastic, which means that there is energy dissipation during collision. 

The theoretical studies of rapid granular flows are generally based on the assumption that the energy dissipation in a binary particle collision is determined by a constant coefficient of restitution e, the ratio of the relative approach to recoil velocities normal to the point of impact on the particle. However, measurements show that the coefficient of restitution is a strong function of the relative impact velocity [10]. Physically, the energy dissipation relates to the plastic deformation of the particle s surface. Thus, a realistic microscopic model should include the deformation history of the particle s surface. However, such a model might become computationally demanding and thus not feasible. 

The coefficient of restitution depends on the mechanical properties of the particle and surface. For perfectly elastic collisions, e — 1 and the particle energy is conserved after collision. Deviations from unity result from dissipative processes, including internal friction, that lead to the generation of heat and the radiation of compressive waves into the. surface material. 

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