Restitution coefficients

Once the relative impact velocity between two colliding spheres is higher than the critical yield velocity, plastic deformation must occur. Heat loss is another phenomenon often coupled with such collisions. Collisions with plastic deformation are referred to as inelastic collisions. All the energy transfer in the form of plastic deformation and heat loss in an inelastic collision is considered as a kinetic energy loss. 

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. 

A complete course of collision consists of the compression and rebound processes. The total work done during the compression process without heat loss can be calculated as 

Prediction of the restitution coefficient has been a challenging research topic for decades. Unfortunately, no reliable and accurate prediction method has been found so far. However, some useful simplified models with certain limits have been developed. One of them is the elastic-plastic impact model in which the compression process is assumed to be plastic with part of the kinetic energy stored for later elastic rebounding, with the rebound process considered to be completely elastic [Johnson, 1985]. In this model, it is postulated that (1) during the plastic compression process, a — r3/2a (2) during the compression process, the averaged contact pressure pm is constant and is equal to 3 Y and (3) the elastic rebound process starts when maximum deformation is reached. Therefore, the compressional force is 

for both elastic and plastic deformation, we have 

---

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). 

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 elastic-plastic model reveals that the restitution coefficient depends not only on the material properties but also on the relative impact velocity. Equation (2.166) also indicates that the restitution coefficient decreases with increasing impact velocity by an exponent of 1/4, which is supported by experimental findings, as shown in Fig. 2.18. For high relative impact velocities, the model prediction is reasonably good. However, for low relative impact velocities, the prediction may be poor because the deformation may not be in a fully plastic range as presumed. 

Eliminating the yield stress in Eq. (2.166) by substitution of the critical yield velocity Ui2Y from Eq. (2.157), the restitution coefficient becomes... 

Figure 2.18. Measurements of the restitution coefficient of a steel ball on blocks of various materials (from Goldsmith, 1960).

Example 2.4 A copper ball of 1 cm diameter normally collides with a stainless steel wall with an impact velocity of 0.5 m/s. Estimate the restitution coefficient using the elastic-plastic model. What is the rebound velocity of the ball The yield strength of copper is 2.5 x 108 N/m2. It can be assumed that the yield strength of the stainles steel is higher than that of copper. 

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... 

Dj Fickian diffusion coefficient of species i, m /s e Restitution coefficient... 

Higa et al. [13,14] studied the restitution coefficient of an ice ball of radius in the range 0.14 < r-p < 3.6 cm at impact velocities Vi in the range 1 < Vi < 700 ctn/s and temperatures T in the range 113 < T < 269 K. Figure 9.4 shows the result obtained for ice balls with a smooth surface. They found a relationship between restitution coefficient and ice strength. Their results show that the restitution coefficient drops suddenly at a critical velocity for each temperature and impactor size. Beyond the critical velocity, Vc the coefficient 8 can be fit by the empirical equation. 

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... 

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, e, . 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... 

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... 

Turbulent kinetic energy contained in wave number range of k iok+dk Restitution coefficient Mixture fraction, Eq. (5.19)... 

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. 

---