Rebound coefficient

The coefficient of restitution is defined as the ratio of the vertical components of the impact and rebound velocities resulting when a bad is dropped or thrown onto a playing surface. The velocities or related rebound heights may be measured photographically. Criteria such as bad inflation pressure, air temperature, and other detads must be specified. 

By experiment Langmuir showed that the reflexion of a gas striking a solid, i.e. the number of molecules which strike a surface and rebound without condensation, is generally very small, the coefficient of accommodation a being nearly unity. 

The rebounding velocities of the colliding spheres can be expressed in terms of the coefficient of restitution as... 

Equation (2.3) also provides a basis for the experimental determination of the coefficient of restitution. Consider the case where a ball at rest is dropped from a height h to a horizontal stationary massive rigid surface, rebounding back to a height of h . If we label the ball with the subscript 1 and the massive plane with 2, Eq. (2.3) can be rearranged to... 

Example 2.2 Consider an impact between a polyethylene particle (dp = 1 cm) and a copper wall. The incident velocity is 2 m/s, and the incident angle is 30°. The friction coefficient of the interface is 0.2. The densities of polyethylene and copper are 950 and 8,900 kg/m3, respectively. What is the contact time duration for the collision Estimate the rebound velocity of the particle. Repeat the problem for a copper particle colliding with a polyethylene wall. 

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... 

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... 

Fig. 5 Adiabatic and non-adiabatic ET processes. In the adiabatic process (Fig. 5a), Vel > 200 cm and the large majority of reaction trajectories (depicted as solid arrows) which reach the avoided crossing region remain on the lower energy surface and lead to ET and to the formation of product (i.e., the electronic transmission coefficient is unity). In contrast, non-adiabatic ET is associated with Vel values <200 cm-1, in which case the majority of reaction trajectories which reach the avoided crossing region undergo non-adiabatic transitions (surface hops) to the upper surface. These trajectories rebound off the right-hand wall of the upper surface, enter the avoided crossing region where they are likely to undergo a non-adiabatic quantum transition to the lower surface. However, the conservation of momentum dictates that these trajectories will re-enter the reactant well, rather than the product well. Non-adiabatic ET is therefore associated with an electronic transmission coefficient which is less than unity.

M. BOUDART Let us take a simple case that of a gas on a liquid surface the principle is the same for a solid surface. Consider now a molecule impinging on the liquid surface, will it stick or rebound Knudsen would say that it would stick. For a simple molecule condensing on a simple liquid, we postulate that in the transition state between the gas and the surface layer the molecule has two-dimensional translational freedom and is still free to rotate. Then the sticking probability would be unity. If, however, the activated complex has no rotational freedom, the sticking coefficient is much less than unity in accordance with measurements. Various values of the activation entropy will be obtained for other systems depending on the number of translational, rotational, and vibrational degrees of freedom involved. 

The critical velocity was found to depend on the temperature and the impactor size it increases with decreasing temperature and impactor size. These dependencies are analyzed using a theoiy of contact mechanics that includes a dynamic effect [15]. They observed that the rebound velocity dropped suddenly at the impact velocity when macroscopic cracks began to appear at the impact point, so they thought that the impact condition for crack initiation was related to the critical velocity. Their analysis concluded that the critical velocity was determined by the ice strength, which was known to depend on temperature and strain rate. The unified equation of restitution coefficient that they derived is... 

Let us emphasise that the non-linearity in the particle drag coefficient is caused by the particle movement relative to the liquid. Its manifestation in the condition of particle rebound is weak and can be neglected. Let us estimate the effect for the onset of the recoil of the particle... 

Since C, the capture coefficient is defined as the fraction of the molecules that condense on collision with the surface, then 1 — C equals the fraction that rebound or escape,... 

Comment by D. J. Santeler, Aero Vac Corporation The interesting analysis of the molecular rebound from a tube causing an increase in sticking coefficient from 0.90 to 0.997 is equally applicable to the thermal adsorption problem of honeycomb structures. Experimental tests on flat panel vs, honeycomb have given the following improvements 0.70 to 0.91, 0.90 to 0.98, and 0.95 to 0.99, all in good agreement with the molecular equations presented. 

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). 

The critical particle velocity that enables elastic rebound is a function of adhesion energy (Pad), particle mass (m), and the coefficient of restitution (e) ... 

In nanometric machining the microstructure of the workpiece material will play an important role in affecting the machining accuracy and machined surface quality. For example, when machining polycrystalline materials the difference in the elastic coefficients at the grain boundary and interior of the grain causes small steps to form on the cut surface, since the respective elastic rebound varies [59]. The study by Lee... 

As already noted (see Fig. IX.l), part of the particles will rebound from the walls of the duct. By means of the coefficient Xq, we can evaluate the fraction of the particles rebounding on the duct surface. For Reynolds numbers in the range of 10 and 10 and a stream particle concentration of 1.2 10 g/cm, the coefficient Kq varies with particle size as follows [252] ... 

In the present case, r o is the ratio of the number of adherent particles to the total number of particles passing across the midsection of the obstacle. The amount of adherent dust and the value of the capture coefficient will depend on the conditions of flow around the obstacle by the dust-laden stream, on the possibility of particle rebound from the surface, and on the adhesive forces capable of holding these particles. The capture coefficient will have a value less than unity. 

The coefficient Kq, in contrast to the capture coefficient 170, determines only the particle rebound it does not take any account of the conditions of flow around the surfaces. Hence the capture coefficient gives a more nearly complete characterization of particle adhesion for the flow around different obstacles. 

In accordance with the value of the Stokes number, particle deposition takes place on the front side (see Fig. IX.5). An analysis of the experimental data shows that in flow around cylindrical and spherical surfaces, the number of solid particles held on the surface will always be less than the number of particles in the impinging stream this is a result of rebound, the probability of which increases with increasing particle velocity. Hence, the Stokes number can be used to characterize particle adhesion only on the front side of the object and only with relatively low flow velocities. Moreover, the relationship between the capture coefficient and the Stokes number has thus far been expressed only qualitatively. 

---