Tangential Coefficient Of Restitution Research Articles

Moderately dense granular gas of inelastic rough spheres

A kinetic theory for moderately dense gases of inelastic and rough spherical molecules is developed from the Enskog equation where a macroscopic state is characterised by 29 scalar fields which correspond to the moments of the distribution function: mass density, hydrodynamic velocity, pressure tensor, absolute temperature, translational and rotational heat fluxes, hydrodynamic angular velocity and angular velocity flux. The balance equations for the 29 scalar fields are obtained from a transfer equation derived from the Enskog equation where the kinetic and potential parts of the new moments of the distribution function and production terms are calculated from Grad’s distribution function for the basic fields. The transition from the 29 field theory to an eight field theory—with mass density, hydrodynamic velocity, absolute temperature and hydrodynamic angular velocity—leads to the determination of the transport coefficients of the Navier–Stokes and Fourier laws. The transport coefficients are functions of the normal and tangential restitution coefficients and of the local equilibrium radial distribution function. The transport coefficients in the limiting case of elastic rough spheres is also determined.

Could the Rebound Characteristics of Oblique Impact for SiO2 Particles Represent the Ash Particles?

Collisions between particles or with a surface have been widely applied, in which the restitution coefficients are the important parameter to describe the particle rebound behavior. SiO2 particles are often used instead of ash particles in theoretical analyses; however, whether this is justifiable has not been confirmed. This paper compares the rebound characteristics of oblique impact for SiO2 particles and ash particles by experimental and theoretical analyses. Based on the rigid-body theory, the tangential restitution coefficients, rebound angle-particle center, and reflection angle-contact path predicted by SiO2 particles are basically in agreement with the experimental results for ash particles, especially at large impact angles. However, there is a slight error at 2.2 m/s as the velocity approaches the critical capture velocity.

Coefficient of tangential restitution for non-spherical particles

In various industrial and naturally occurring multiphase flows, whether dilute or dense, particle interaction plays a crucial role. In most cases, the particles are non-spherical, which poses a computational challenge in terms of particle motion and particle interaction, i.e. both particle–particle and particle–wall collisions. In this study, we present a novel frictional particle collision model to be used in general fluid flows. The model is derived for superellipsoidal particle shapes and thus allows consideration of frictional collisions of a wide variety of particle geometries. In this context, we derive an expression for the tangential coefficient of restitution applicable to arbitrarily shaped particles. Furthermore, we present the performance of the novel model by applying it to demonstrative examples ranging from two- to multi-particle systems.

Numerical simulation of collision dynamics between a dry particle and a liquid-coated wet particle

A comprehensive understanding of wet particle collisions in the presence of a liquid film is essential for industrial processes. In this study, we conduct a numerical investigation into the dynamics of both normal and oblique collisions between a dry particle and a liquid-coated wet particle. In normal collisions, the gas flow velocity near the stretching liquid bridge increases significantly, while oblique collisions exhibit no such increase. The wetting area of particles reaches its maximum when solid particles come into contact, at which point the viscous and capillary forces acting on the particles also reach their peaks. The capillary force causes kinetic energy dissipation to be one magnitude greater than viscous energy dissipation. The normal wet restitution coefficient rises with the logarithm of the modified Stokes number, reflecting energy dissipation effects. As the collision angle increases, the normal wet restitution coefficient gradually increases and surpasses the tangential wet restitution coefficient.

Experimental study of particle collision and rebound with moving walls

Particle–wall collision always occurs in transportation systems with multi-phase flows containing particles in pneumatic conveying, coal liquefaction production, and wastewater treatment. Especially within rotating fluid machinery, collision and rebound between particles and moving walls occur frequently. It is necessary to conduct research, as it can provide theoretical support for controlling particle trajectories and reducing particle erosion. In this study, experimental investigations are carried out to analyze the trajectories and restitution coefficients of particle collision and rebound at different velocities of the moving wall, wall roughnesses, and collision angles. Particle trajectories and the influence of particle rotation on restitution coefficients in normal collisions are investigated, and particle motion behavior, tangential and normal restitution coefficients of the particle collision and rebound, and effective rebound angles in oblique collisions are analyzed and discussed at different operating conditions. The differences in collision and rebound characteristics between normal and oblique particle impacts are identified.

Experimental study of particle deposition characteristics in a gasifier: Effect of slag wall temperature

Based on the visualization system, the particle tracking method was used to study the influence of slag wall temperature on the deposition characteristics of coal particles under gasification conditions. The results show that particle deposition can be divided into five types: rebound, rebound after blocking for a distance, crushing, rolling, and adhesion. When slag wall temperature is lower than the deformation temperature, the probability of rebound is over 70%. With the increase of slag wall temperature, the global coefficient of restitution remains constant and then decreases with 1100 °C as the dividing line and the tangential coefficient of restitution concentrates around 0.65. It is proved that the fusion characteristics of slag wall are the main cause of the change in particle deposition on the wall. The collision theory is introducted in the study of deposition characteristics of particles. The collision process is classified into “elastic-plastic collision” and “wet collision”.

Occurrence conditions of the reverse rotation of rockfall and its influence on the restitution coefficient

When rockfall occurs along dense rock slopes, the rotation direction of rockfall is not always downhill. Specifically, the rockfall may obtain a reverse rotation speed (RRS) after impact under certain conditions, the effect of which on the restitution coefficient (RC) cannot be ignored. According to the statistical results of the reverse rotation (RR) phenomena of blocks obtained from previous experiments, the occurrence of RR is correlated to the block shape, incident angle, and contact attitude. In this study, considering a typically shaped cubic block, the critical condition for the RR is preliminarily deduced. Based on the results, the influence of the RRS on the RC for four typically shaped blocks is examined using a customized device. Results show that the tangential RC (Rt) values of each block are not sensitive to the change in the RRS, the distribution is relatively concentrated and the values are high. Moreover, the normal RC (Rn) values are not sensitive to the RRS, and the distribution is relatively discrete. The RRS influences Rn; however, it is difficult to directly establish the relationship between them. To this end, considering the contact attitude and shape of the block, an integral variable, the impact coefficient (Ic), is proposed to determine the influence of RRS on Rn. Moreover, the impact-bounce behaviours of the block are categorized and analysed. For the block rebound following a single impact, Ic and Rn are positively and negatively correlated when the mass centre of the block (MC) is in front and behind the contact point (CP), respectively. For the block rebound following two successive impacts, with the increase in Ic, the Rn increases. These conclusions help clarify the mechanism of the influence of the RRS on RC and provide vital information and ideas for the development and optimization of a program to accurately predict rockfall trajectories.

Modeling Ellipsoidal Block Impacts by an Advanced Rheological Model

In this paper, an advanced rheological model for impacts of ellipsoidal blocks on deformable ground surfaces, introducing the effects of block eccentricity and orientation at impact, is presented. This allows us to assess impact penetration and force, restitution coefficients, and block trajectories. A parametric analysis was carried out by considering different block aspect ratios, impact angles and initial block orientations at impact. The results are presented in terms of restitution coefficients, penetration and force time histories, maximum penetration depth, maximum force and rotational/total kinetic ratios. Impacts along the major block axis, versus those along minor axis, are characterized by larger penetrations (ranging from 3.3 to 50%), shorter impact durations (ca 50%) and very slightly larger vertical forces (ranging from 0.3 to 60%) according to the model parameter used. In contrast, the impact angle is shown to strongly affect maximum penetration and force values, and markedly increase rotation at impact. Analogously, normal restitution coefficient is severely dependent on impact angle, with a variation of more than two orders of magnitude. A mathematical expression for computing the energetic restitution coefficient from the normal and tangential apparent restitution coefficients and the ratio between the rotation and total kinetic energy is proposed. This overcomes the drawback of classical restitution coefficients greater than one when a change in block rotation occurs allowing us to bracket the coefficient of restitutions values to support and improve classical rock fall simulations also highlighting their intrinsic limitations. Finally, the effects of block geometry and initial angular velocity on rockfall simulations were analyzed by implementing the approach in the HyStone simulation code. The simulated frequencies of the maximum height during each ballistic trajectory follow an exponential distribution, whereas those for normal and tangential apparent restitution coefficients follow normal distributions.

Particle Collision Study Based on a Rotational Boundary Condition

The main engineering machinery for the hydrodynamic lifting of seafloor mineral particles is rotor machinery with rotating impeller motion. It is important to study the rebound mechanism of collisions between particles and rotating walls to improve the accuracy of numerical simulation of rotor machinery. In this study, the law of motion change after collisions between particles and rotating walls is investigated using an experimental research method. The results show that the deflection angle of the particles after collision decreases with increases in the rotational speed of the wall, and the spin angular velocity increases with increases in the rotational speed of the wall. The normal velocity coefficient of restitution under the rotating wall is not affected by the rotational speed of the wall. The tangential coefficient of restitution under rotational boundary condition is smaller than the tangential coefficient of restitution under the stationary wall, and the higher the rotational speed, the closer it is to the coefficient of restitution under the stationary wall. During collision in the experiment, the main mode of contact between the particle and the rotating wall is sliding contact. Sliding friction between the particle and the rotating wall results in energy loss in the tangential velocity of the particle, and also provides energy for deflection of the particle’s trajectory and increased kinetic energy from the spin angular velocity; sliding friction loss is affected by the speed of the wall.

Energy loss in oblique collisions

The decrease in kinetic energy of a ball incident vertically on a rigid horizontal surface depends on the normal coefficient of restitution, [Formula: see text]. It is shown in the present paper that if the ball is incident obliquely on the surface then the total decrease in kinetic energy has two independent contributions, one depending on [Formula: see text], the other depending on the tangential coefficient of restitution, [Formula: see text].

Study of the process of metal droplets with high surface tension impinging on wall

The behavior of aluminum droplets impacting a wall critically affects the operation of solid rocket motors. In the present study, the flow and rebound of aluminum droplets with a high surface tension are studied experimentally and numerically. In addition, the impact of aluminum droplets at different inclined angles is monitored experimentally. When the droplet is about to rebound away from the wall, it is stretched to its maximum length. An empirical correlation formula is proposed to predict the maximum length when a droplet is about to bounce off a flat plane. The velocity and pressure distributions of a droplet flowing over a flat plane and an inclined plane are compared by using the volume of fluid method. Furthermore, the restitution coefficient of the droplet is discussed in detail. When normal Weber number Wen ranges from 0 to 20, normal restitution coefficient en ranges from 0.3 to 0.6. When tangential Weber number Wet <10, tangential restitution coefficient et is subject to great uncertainty. When 10 < Wet < 80, et is maintained at 0.75. The formulas for the total restitution coefficient, normal restitution coefficient, and tangential restitution coefficient of the aluminum droplets are also presented. Based on the principle of conservation of energy, we calculate the remaining energy of the aluminum droplets impinging on a wall and provide the relationship between the restitution coefficient and the dissipated energy. The results elucidate the mechanisms at work when aluminum droplets collide with a wall.

Granular Gas of Inelastic and Rough Maxwell Particles

The most widely used model for granular gases is perhaps the inelastic hard-sphere model (IHSM), where the grains are assumed to be perfectly smooth spheres colliding with a constant coefficient of normal restitution. A much more tractable model is the inelastic Maxwell model (IMM), in which the velocity-dependent collision rate is replaced by an effective mean-field constant. This simplification has been taken advantage of by many researchers to find a number of exact results within the IMM. On the other hand, both the IHSM and IMM neglect the impact of roughness—generally present in real grains—on the dynamic properties of a granular gas. This is remedied by the inelastic rough hard-sphere model (IRHSM), where, apart from the coefficient of normal restitution, a constant coefficient of tangential restitution is introduced. In parallel to the simplification carried out when going from the IHSM to the IMM, we propose in this paper an inelastic rough Maxwell model (IRMM) as a simplification of the IRHSM. The tractability of the proposed model is illustrated by the exact evaluation of the collisional moments of first and second degree, and the most relevant ones of third and fourth degree. The results are applied to the evaluation of the rotational-to-translational temperature ratio and the velocity cumulants in the homogeneous cooling state.

A cluster-based drag model of rough sphere for the simulation of fast fluidization

The non-uniform flow caused by cluster greatly influences the performance in fast fluidization systems. Rough particles result in extra energy dissipation by particle rotation, which finally affects the cluster formation and evolution. In this study, the kinetic theory of rough sphere (KTRS) is employed and coupled with the cluster structure-dependent (CSD) drag model to consider the rotation effect on cluster formation, where the total granular temperature is introduced and the cluster-scale effect on the collision is incorporated. The effect of particle roughness on energy consumption is also discussed. The result demonstrates that the cluster-scale pressure plays an important role in the collision. The energy consumption is dependent on the particle roughness or tangential restitution coefficient. Meanwhile, the change of particle roughness also alters mesoscale parameters like cluster size and cluster solid concentration.

Bouncing and spinning of amorphous Lennard-Jones nanoparticles under oblique collisions

Collisions of Lennard-Jones nanoparticles (NPs) may be used to study the generic collision behavior of NPs. We study the collision dynamics of amorphous NPs for oblique collisions using molecular dynamics simulation as a function of collision velocity and impact parameter. In order to allow for NP bouncing, the attraction between atoms originating from differing NPs is reduced. For near-central collisions, a finite region of velocities – a ‘bouncing window’ – exists where the 2 NPs bounce from each other. At smaller velocities, energy dissipation and – at larger velocities – also NP deformation do not allow the NPs to surpass the attractive forces such that they stick to each other. Oblique collisions of non-rotating NPs convert angular momentum into NP spin. For low velocities, the NP spin is well described by assuming the NPs to come momentarily to a complete stop at the contact point (‘grip’), such that orbital and spin angular momentum share the pre-collision angular momentum in a ratio of 5:2. The normal coefficient of restitution increases with impact parameter for small velocities, but changes sign for larger velocities where the 2 NPs do not repel but their motion direction persists. The tangential coefficient of restitution is fixed in the ‘grip’ regime to a value of 5/7, but increases towards 1 for high-velocity collisions at not too small impact parameters, where the 2 NPs slide along each other.

Effects of particle size and liquid film viscosity on characteristics of particles colliding with dry and wet walls

Particle-wall collisions frequently occur in fluidized beds, granulation processes, and the transportation of gas–solid two-phase flows in pipelines. Exploring the rebound characteristics of particles after colliding with the wall will enable more accurate calculation models of collision rebound, enhanced control of the collision–rebound process, and improved efficiency of industrial production. This paper studies the relationship between the energy consumed by the liquid film and the energy consumed in the collision and investigates the effects of different particle sizes and liquid film viscosities on the liquid bridge morphology, coefficient of restitution (COR), and critical capture angle (CCA). The results show that the normal COR is affected by particle size, incident velocity, and viscosity, while the tangential COR is only related to the existence of the liquid film. Additionally, the maximum size of the liquid bridge and the growth rate of COR change suddenly in a transition region. The particle size and liquid film viscosity have significant effects on the CCA of the particle. The numerical results enable us to propose a general form of the normal wet collision model.

Oblique impact of ice hockey and plastic pucks with a rigid surface

The collision of a disk with a rigid surface is analysed in this paper assuming that the disk slides throughout the collision at glancing angles or grips the surface at other angles of incidence. Experimental results are presented for an ice hockey puck and a plastic disk, showing that there is no rolling involved, as assumed in previous studies. Measurements are presented of the outgoing speed, angle and spin as a function of the angle of incidence, and the results are described in terms of the normal and tangential coefficients of restitution plus the coefficient of sliding friction. The experiment would be suitable for use in a student laboratory.

Numerical investigation of a droplet impacting obliquely on a horizontal solid surface

A droplet will be reflected like a light beam when it impacts obliquely on a horizontal solid surface. In the present work, whether or not the droplet obeys the so-called `reflection law' like a light beam is numerically explored. It is found that the reflection angle is always larger than the incident angle as the magnitudes of the normal and tangential velocities reduce disproportionately after reflection. Specifically, the normal restitution coefficient (${ϵ}^{n}$) satisfies the general scaling relationship of ${ϵ}^{n}$ ~ We${}^{\ensuremath{-}1/4}$ for moderate normal Weber number We while the tangential restitution coefficient (${ϵ}^{t}$) scatters around a constant value.

Experimental study of the gear effect

The gear effect arises during the collision of a ball and an extended object due to rotation of the object across the surface of the ball. The effect is well known in golf, drivers being designed with a convex face to help reduce the problem of a mis-hit. An experiment is described to examine the gear effect for two different balls, one with a high tangential coefficient of restitution and one with a low coefficient. The outgoing ball spins were different, indicating that a driver designed for one type of golf ball is unlikely to work well with a different type of golf ball.

Laboratory investigation of the effect of the rotational speed on the coefficient of restitution

In the process of impact-rebound, rockfalls with an initial rotational speed (RS) often undergo a violent transformation between rotational energy and translational energy, which may affect the coefficient of restitution (COR). However, owing to the complexity of experimental simulations and the influence mechanism, this issue is often neglected during research. Therefore, in this study, four representative blocks (ellipsoid, cylinder, cone with a spherical base, and disc) were selected, and impact tests of each block rotating around different axes with various RSs were conducted using a specially developed device. Based on the impact dynamics theory, the RS has no significant effect on the tangential COR (Rt), while it may exhibit a positive correlation with the Rt when the size of the block size is large enough. When the block rotates around the X-axis (approximately normal impact), there is no obvious relationship between the RS and the normal COR (Rn). Conversely, when the block rotates around the Y-axis (often non-normal impact), the RS influences the Rn, but it is difficult to determine in what manner and to what extent. Thus, to clarify the influence mechanism of RS on the Rn for non-normal impacts, a comprehensive variable, the impact posture coefficient (IPC), which comprehensively considers the RS, block shape, and impact posture, was introduced, and the impact-rebound characteristics of the blocks were classified and discussed. Under the condition of bouncing after only one impact, the IPC and Rn values exhibited a positive linear correlation when the block mass centre (C) is behind the contact point (CP), whereas the opposite behaviour was observed when the C is in front of the CP. Meanwhile, under the condition of bouncing after two impacts, the Rn and IPC values had a positive linear correlation. These findings explain the influence mechanism of RS on the COR and provide a new idea for the accurate prediction of the impact-rebound processes of rotating irregular rockfalls.

Analysis of normal and tangential restitution coefficients in car collisions based on finite element method

In momentum-based impact models used in accident reconstructions, the restitution coefficient is a key role to determine impulses and delta-V accurately in car-to-car collisions. The restitution coefficients can be defined as the normal and the tangential direction in the contact plane, and the same value of the normal and tangential coefficients is employed in a two-dimensional PC-Crash model. Ishikawa (1993) proposed an impact model with the normal and tangential restitution coefficient, however, to the best of the author's knowledge, few studies have further investigated tangential restitution coefficients in detail. In this study, the normal and tangential restitution coefficients of the impact model were investigated based on the finite element (FE) analysis. FE simulations of angled barrier crashes with varying barrier angles as well as 16 car-to-car collisions were conducted. As the contact plane slides, the ratio of the tangential impulse to the normal impulse becomes the same value with the maximum friction coefficient. The tangential restitution coefficient is related to whether sliding occurs on the contact plane, and can be expressed as a function of the impulse ratio. The tangential restitution coefficient decreased with increasing the impulse ratio and is positive in non-sliding collisions and negative in sliding collisions. An approximate expression for the tangential restitution coefficient was formulated as a function of the generalized impulse ratio. The calculated impulses of the impact model with normal and tangential restitution coefficients were closer to the FE analysis than the PC-Crash model which uses a constant restitution coefficient of 0.1.