Coefficient of normal restitution

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. 

According to the definition, the coefficient of normal restitution is given by... 

In previous work, we have mainly used the DPM model to investigate the effects of the coefficient of normal restitution and the drag force on the formation of bubbles in fluidized beds (Hoomans et al., 1996 Li and Kuipers, 2003, 2005 Bokkers et al., 2004 Van der Floef et al., 2004), and not so much to obtain information on the constitutive relations that are used in the TFMs. In this section, however, we want to present some recent results from the DPM model on the excess compressibility of the solids phase, which is a key quantity in the constitutive equations as derived from the KTGF (see Section IV.D.). The excess compressibility y can be obtained from the simulation by use of the virial theorem (Allen and Tildesley, 1990). 

Fig. 19. Simulation results for both the soft-sphere model (squares) and the hard-sphere model (the crosses), compared with the Carnahan-Starling equation (solid-line). At the start of the simulation, the particles are arranged in a FCC configuration. Spring stiffness is K = 70,000, granular temperature is 9 = 1.0, and coefficient of normal restitution is e = 1.0. The system is driven by rescaling.

Next, we consider a system of inelastic spheres (ISs). As can be seen from Eq. (81), the KTGF predicts that the excess compressibility yls of ISs is a linear function of the coefficient of normal restitution e,... 

In Fig. 21, the excess compressibility is shown as a function of the solid fraction for different coefficients of normal restitution e. These results are compared with the Eq. (54), where the excess compressibility yES is taken from either the Ma-Ahmadi correlation (Ma and Ahmadi, 1986) or the Carnahan-Starling correlation. As can be seen, the excess compressibility agrees well with both correlations for a solid fraction ss up to 0.55. For extremely dense systems, i.e., es>0.55, the Ma-Ahmadi correlation presents a much better estimate of the excess compressibility, which is also the case for purely elastic particles (see Fig. 23). 

Fig. 20. Excess compressibility yIS for a system of inelastic hard spheres, as function of the coefficient of normal restitution, for one solid fraction (as = 0.05). The excess compressibility has been normalized by the excess compressibility y is of the elastic hard spheres system. Other simulation parameters are as in Fig. 19.

Fig. 21. The excess compressibility from soft-sphere simulations, with random initial particle positions, for different coefficients of normal restitution e (a) e = 1.0 (top-right) (b) e = 0.95 (top-left) (c) e = 0.90 (bottom-right) (d) e = 0.80 (bottom-left). The simulation results (symbols) are compared with Eq. (54) based on the Ma-Ahmadi correlation (solid line) or the Camahan-Starling correlation (dashed line). The spring stiffness is set to k = 70,000.

Fig. 22. The effect of the cohesive force on the excess compressibility. The coefficient of normal restitution is e = 1.0, and granular temperature is T = 1.0. The Hamaker constant is A = 3.0 x 10-12 (circles) and 3.0 x 1CT10 (crosses).

Consider a simply sheared stationary monodisperse granular system, whose collisions are characterized by a fixed coefficient of normal restitution, e. Let the macroscopic velocity field, V, be given by V = 7J/x (a well known solution of the pertinent equations of motion), where 7 is the shear rate, y is the spanwise coordinate and x is a unit vector in the streamwise direction. Both phenomenological [2,9] (even dimensional) and kinetic theoretical calculations show that the granular temperature in this system approaches a steady state value given by [4] ... 

The coefficients of normal restitution of the sugar pellets was obtained at an impact velocity of 1.28 0.03m/s for particle-steel apparatus wall impact with Cp w = 0.54 0.08 and for particle- lass wall impact with... 

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. 

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. 

For the normal components of particle velocities, the boundary conditions may be expressed by using the coefficient of restitution as... 

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. 

Here, n is the normal unit vector and v>j is the relative velocity at the contact point. The elastic part of the contact force is represented by a non-linear spring, assumed proportional to the spring stiffness and to s- (sn displacement). Additionally, to account for viscoelastic material properties that cause energy dissipation, a damping factor Pn related to the coefficient of restitution is included in the model ... 

The normal coefficient of restitution Cn is defined as the ratio of the rebound velocity to the impact velocity... 

In these equations, (-) and (+) refer again to the quantities before and after impact respectively, N is a composite normal vector for all coordinates, e the coefficient of restitution, 4 the velocity vector, x the impulse due to the contact force, a the Lagrange mul liers associated with velocity constraints, M the system mass matrix, D = [ Ir, ], D the constraints Jacobian matrix, and p the vector of system momenta. 

With T given by (4.19), the normal stress and shear stress vary with the square of the shear rate. This is what is seen in the experiments (Bagnold 1954, Savage 1972, Savage and McKeown 1983, Savage and Sayed 1984, Hanes and Inman 1985). The results of the numerical simulations of Walton and Braun (1986) are in excellent agreement with the values of the predicted stress, provided that the coefficient of restitution is greater than 0.8. 

In Figure 4.3, we show the profiles of normalized fluctuation velocity, average velocity, and volume fraction in half a shear cell with a thickness L = 14.44a, a wall bumpiness 0 = tc/5.5, and a wall volume fraction Vi = 0.25, for three values of the coefficient of restitution, e = 0.80,0.85, and 0.90. The corresponding values of the stress ratio and pressure, made dimensionless by the product pgU, are 0.3470,0.3094,0.2609, and 0.0141,0.0144,0.0153, respectively. The values of w/U and u/U at a given height increase with e while those of v decrease. The assumption of moderate volume fractions is abused at both the wall and the centerline of the flow. At the wall, the volume fraction is somewhat low at the centerline, the volume fraction for the more dissipative spheres is too high. Near the centerline, a theory for denser shearing flow is required for the more dissipative spheres. 

Figure 4.3 Profiles of normalized fluctuation velocity, average velocity, and volume fraction in half a symmetric shear cell for spheres with coefficients of restitution e = 0.80,0.85,0.90 and Vi = 0.25.

A requirement of the DSMC method is that expressions are available for the postcoUisional velocities of two particles, given their precolUsional positions and velocities. For solid particles, there are well-known expressions in terms of normal and tangential coefficients of restitution and fiiction coefficients (Van der Hoef et al, 2008). However, for droplets and bubbles (as well as wet particles), additional correlations are needed detailing the outcome of a binary collision, such as bouncing, coalescence, or breakup. In the topical sections, we will discuss these correlations for some representative systems. 

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. 

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... 

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. 

As already mentioned, in the present study all the collision interactions between the droplets and particles are disregarded. Although two cases of particle-wall interaction are investigated (a) particles hitting walls are escaped from the computational domain, that is, the trajectories of drop-lets/particles are terminated if striking against the chamber walls, and (b) particles can rebound from the walls with restitution coefficients 0.9 (normal) and 0.5 (tangential). 

When two particles are in contact, a collision analysis can be conducted to obtain the velocities of the particles after the collision. To simplify the analysis, it can be assumed that the tangential traction and the resulting displacements have no effect on the normal collision. For the collision between particles a and b, the normal components after collision can be obtained by solving the equations for the restitution coefficient and the conservation of momentum. 

The relative velocities of the centers of the spheres immediately before and after a collision are still given by (2.111). For these inelastic particle collisions it is required that the relative particle velocity component normal to the plane of contact, g2i k (before collision) and k (after collision) satisfy the empirical relation (2.110) [68], 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. It is required that the component of the relative velocity perpendicular to the apse line should be unchanged in a collision, thus the impulse J12 must act entirely in the k direction. On this demand, J12 can be determined from (4.47), (4.48) and (2.110). 

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