Coefficient Of Normal Restitution Research Articles

A model for oblique impacts on material surfaces

Many practical situations of material damage, wear, and erosion involve collisions between small particles and surfaces at inclined angles. While there are many well-validated models of normal incidence impact situations, elastic-plastic models for oblique incidence impact events are lacking. Here the finite element method is used to predict the normal and tangential coefficient of restitution in oblique impacts for hard, elastic spheres impacting an elastic-perfectly plastic material surface. The proposed model covers various impact angles ranging from 0° to 45°, within a limiting impact velocity below which the effects of heating are negligible. The normal coefficient of restitution follows power-laws with respect to normalized values of the impact velocity. Interestingly, the tangential coefficient of restitution follows a linear relationship with impact velocity. Together, these results provide a semi-empirical set of equations predicting oblique impact rebounds (both velocity and trajectory) for a wide range of conditions and material properties, with which experimental results can be rapidly interpreted. Laser-Induced Particle Impact Test (LIPIT) data are also presented for aluminum particles impacting aluminum substrates, at impact angles of 25° and 40°; the results compare favorably with the model and validate the general use of such models for the analysis of experimental data.

A novel variable restitution coefficient model for sphere–substrate elastoplastic contact/impact process

The restitution coefficient serves as a critical parameter to evaluate the energy loss during the contact/impact process. Its in–depth researches are beneficial to accurately describing the contact/impact phenomenon. Given the limitations of existing restitution coefficient models, a novel variable restitution coefficient model, which considers the material properties and the initial relative contacting velocity between two colliding bodies, is proposed in this work. Firstly, the function relationship between the restitution coefficient and the equivalent plastic strain can be obtained based on the energy equivalence principle. After that, multi–condition FEM numerical simulation cases are conducted to explore the mapping relationship among the equivalent plastic strain, material properties and initial relative contacting velocity, with which the new normal variable restitution coefficient model can thus be established. Finally, various comparisons with several existing restitution coefficient models are conducted to showcase its superior performance, with the low–speed experimental data and high–speed FEM results acting as reference values. Furthermore, the new restitution coefficient model is extended to describe interaction process between two spheres, and also employed for the establishment of a new continuous contact force model.

Transition from bouncing to rolling on a horizontal surface

If a ball is incident obliquely on a horizontal surface and is allowed to bounce more than once, then it is likely to bounce many times before it starts rolling along the surface. The number of bounces before rolling commences depends on the initial vertical speed and the normal coefficient of restitution. The transition from bouncing to rolling is examined using a simple theoretical model and is compared with experimental data obtained by filming the process with a video camera. We find that the final rolling speed is proportional to the initial horizontal speed of the ball and depends on the initial ball spin, but is independent of the tangential coefficient of restitution. Representative videos for different balls are included as supplementary material, including a superball thrown with a backspin that creates a back and forth motion. Instructors could use the experiment and/or analysis for an advanced undergraduate lab or use a simplified observational exercise for non-majors.

The influence of self-assembled particle layer on particle collision properties

Interactions between particle and particle layers are fundamental in modeling deposition processes. The complex mechanical behaviors of these layers pose significant challenges in precisely predicting the rebound characteristics of incident particle, especially at the micron and nanoscale. Very few related experimental studies have been reported in this area. To better control the structure of particle layers and offer a path to more systematic research, this study introduces the particle self-assembly method, combined with high-speed imaging technology, to investigate the influence of particle monolayer, their diameters, and impact velocity on critical capture velocity (vcr), normal coefficient of restitution (COR), and angle deflection of incident particles. Results indicate that self-assembled particle layers remain stable within an impact velocity range and experimental data within this stable range exhibit good repeatability. These layers significantly enhance kinetic energy dissipation. A semi-logarithmic plot revealed a negative linear relationship between vcr and particle size ratio. The effect of particle layer on COR depends on particle layer diameter and impact velocity. Specifically, the dissipation of particle layer diminishes and approaches zero for smaller layer particles (0.5 μm), but increases and stabilizes for larger particles (2 μm). Moreover, statistical results indicate that angle deflection decreases as particle size ratio increases, aligning with a power-law relationship predicted by Monte Carlo simulations. Overall, this study presents a novel method for understanding micro and nano particle layers' influence on incident particle dynamics.

Atomistic simulations of oblique collisions between crystalline and amorphous SiC nanoparticles: Mechanisms of deformation and scaling coefficients of restitutions

Systematic atomistic simulations of oblique collisions between spherical crystalline, 6H and 3C, and amorphous SiC nanoparticles (NPs) are performed in the bouncing and fragmentation regimes for the particles with diameters from 8 nm to 30 nm, impact velocities from 100 ms−1 to 5000 ms−1, and in the whole range of the geometric impact parameter. The critical sticking velocity of 6H-SiC NPs reduces from ∼490 ms−1 to ∼150 ms−1 when the NP diameter increases from 10 nm to 30 nm. Below the melting threshold, the collision of crystalline NPs induces the formation of localized zones of densified material, while the remaining parts of NPs are not affected by plastic deformations. The impact of amorphous NPs results in the plastic deformation of significant parts of the NP material and much larger irreversible losses of mechanical energy. The post-collision translational and rotational velocities of crystalline NPs as well as coefficients of restitution (CORs) demonstrate weak dependence on the NP size if the NP diameter is greater than 20 nm. At constant impact velocity, the normal COR only marginally varies for head-on and oblique collisions, when the collision angle is smaller than ∼53°. The tangential center-of-mass and point-of-contact CORs, as functions of the impact parameter, demonstrate extremal behavior with the minima at collision angles from ∼27° to ∼37°. The normal COR exhibits a linear decay as a function of the impact velocity where the slope coefficient is different below and above the melting threshold. The tangential CORs attain the minimum values at the impact velocity of ∼300 ms−1. The obtained results are summarized in the form of fitting equations for CORs as functions of the impact parameter and velocity, which can be readily used to predict the post-collision translational and angular velocities of NPs in large-scale simulations of granular gas flows.

Ash fouling characteristic analysis and prediction for pillow plate heat exchanger in waste heat recovery based on attentive-feature decision algorithm

Ash fouling on heat exchanger surfaces in waste heat recovery and utilization is an inevitable result of solid fuel combustion and particle deposition, affecting the efficiency, availability and operating cost of an energy utilization system. Fouling prediction plays a critical role in reasonable design and efficient operation of heat exchangers. However, credible monitoring and accurate prognostic towards fouling condition always remain a challenge, largely due to the complex flue gas component and abominable service environment of heat exchangers. In this paper, a novel ash fouling prediction method combined with Effective Particle Size Filtering and Multistep Attentive-Feature Decision algorithm (EF-MAFD) for Pillow Plate Heat Exchanger (PPHE) is proposed. First, a numerical fouling model with Discrete Phase Model (DPM) combined particle deposition and removal model is developed. Validation work is conducted on Nusselt number, pressure drop coefficient and normal restitution coefficient with existing experimental data. The effect of geometrical configuration of PPHE and condition on the fouling characteristic is then evaluated with this model. Next, the EF-MAFD ash fouling prediction model is established, in which the initial particle size distribution is transformed into an effective diameter innovatively and two indicators evaluating the fouling state, namely critical fouling resistance and critical fouling time are defined and predicted. The results indicate that fouling mainly occurs downstream of the welding spots and shows an asymptotic growing trend over time. PPHE with smaller diameter of welding spot, channel height and pitch ratio possess anti-fouling potential. In comparison to existing methods, the newly developed EF-MAFD method has the capacity to deal effectively with the complex particle distribution and provides the most satisfying prediction ability with coefficient of determination (R2) of 0.93 and mean absolute prediction error (MAPE) of 7.8 %. The proposed method could be utilized as a reliable tool to support fouling condition-based maintenance and design for various types of heat exchangers operating in fly ash conditions.

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.

The normal restitution coefficient and critical sticking velocity of disk-shaped adhesive particles

The particle-wall collision is an important phenomenon in gas-solid flows. For non-spherical particles like disk-shaped particles, the classic models based on quasi-spherical assumptions cannot be directly applied. It is still challenging to evaluate the loss of kinetic energy by resolving the adhesion and viscoelastic damping because of their strong coupling. To address this issue, we employ the finite element method (FEM) to conduct a numerical investigation of the particle-wall collision dynamics of adhesive, disk-shaped particles. The viscoelastic model and adhesion are coupled to naturally reproduce the energy dissipation. Our findings reveal that the non-spherical nature of the particles influences energy dissipation through two distinct mechanisms: viscoelastic dissipation by changing the collision time and adhesion by changing the contact area. Explicit expressions of the restitution and the critical sticking velocity of the disk-shaped particles are given, which can be directly used in the numerical simulation under the Euler-Lagrange framework.

Experimental Study on Coefficient of Restitution of Small-Sized Spherical Particles during Low-Speed Impact

This study presents experimental investigations on the normal restitution coefficients of a titanium bead (Ti), zirconia bead (ZrO2), and amorphous zirconium alloy sphere (Amor). The research explores the influence of particle diameter and collision velocity on the normal restitution coefficient between two independent, identical spherical particles of different materials. The experimental findings demonstrate that increasing the particle diameter results in more effective plastic deformation, leading to higher energy losses and, subsequently, smaller coefficients of restitution. Similarly, higher particle velocities cause more energy dissipation during collisions, resulting in smaller restitution coefficients. Comparing particles of different materials, those with larger yield strengths exhibit more elastic behavior, experience less initial energy loss due to deformation, and reach the maximum restitution coefficient (elastic state) with fewer collisions. This finding suggests that material properties significantly influence the overall energy dissipation and elastic response in the particles. To validate the experimental results, existing models are compared and discussed. Furthermore, potential physical mechanisms responsible for the observed behavior are explored, providing valuable insights into the collision dynamics in spherical particle interactions. Overall, this study contributes to a better understanding of the factors affecting the normal restitution coefficient in particle collisions, enabling the design and optimization of particle systems for diverse applications in condensed matter and related fields.

Measurement and analysis of impact dynamic parameters of micron-sized single particles using particle shadow velocimetry

The impact dynamics of micro-sized particles is still a fascinating topic of research, which can contribute to the understanding of the mechanisms of adhesion and rebound of these particles. A high-precision measurement system was developed that combines hardware, software, and numerical simulation methods to determine the impact dynamics parameters of micro-sized single particles under high temperature conditions by using Particle Shadow Velocimetry (PSV). The impact dynamic parameters of three standard particles (impact velocity, impact angle, coefficient of restitution, critical sticking velocity, etc.) were measured as examples to validate the reliability of this measurement system. The results demonstrated the reliability of the developed particle image processing algorithm. Corrections on the impact velocity and rebound velocity of small particles at low impact velocities were necessary. Particle size had a significant influence on the variation characteristics of the normal coefficient of restitution in particle impacts. A power function relationship between the critical sticking velocity and particle size was obtained. This paper provides a strong technical guidance for the future research on the impact dynamics of fly ash particles produced in the boiler field or other related fields.

Elimination method of sliding blockage effect in discrete element simulation of rolling stone based on discrete fracture network and damping joint model

The particle essence of the discrete element method could prevent the rock block from sliding on the slope surface due to the stress concentration between the convex particles of the sliding block and the slope. To eliminate this sliding blockage effect and simulate more realistic rockfall, slope surface was transformed into a series of successive joints by the discrete fracture network technique, and a damping joint model was developed by adding a normal dashpot and a shear dashpot on the smooth joint model that does not have damping properties. The motion processes of sliding, rolling, and collision were simulated by the proposed method, and the damping and movement performances were investigated. The macroscopic sliding and rolling friction coefficients significantly increase with the microscopic friction coefficient and the shear damping ratio. The normal restitution coefficient of collision decreases with the normal damping ratio, while the shear restitution coefficient of collision decreases with both the shear damping ratio and the friction coefficient. Furthermore, the simulation of a rockfall case demonstrates the proposed method represented a more realistic movement process and more plentiful characteristics of collision, sliding and rolling, compared with the simulated results of Rocfall software.

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.

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.

Independent friction-restitution description of billiard ball collisions

The oblique impact of a cue ball moving on a horizontal surface with an arbitrary spin on an object ball at rest is described using the independent friction-restitution modeling of impact. The model yields theoretical expressions for the post-collision linear and angular velocities in terms of ‘constant’ coefficients of normal restitution, tangential restitution, and friction for both stick and slip regimes of impact. The model also predicts the post-impact velocities reached by the balls when, as a result of friction with the supporting surface, the pure rolling motion is established. Theoretical predictions are in satisfactory agreement with experimental data for collisions between regulation billiard balls, steel, brass, and rubber spheres.

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].

Assessment of kinetic theories for moderately dense granular binary mixtures: Shear viscosity coefficient

Two different kinetic theories [J. Solsvik and E. Manger (SM), Phys. Fluids 33, 043321 (2021) and Garzó et al. (GDH), Phys. Rev. E 76, 031303 (2007)] are considered to determine the shear viscosity η for a moderately dense granular binary mixture of smooth hard spheres. The mixture is subjected to a simple shear flow and heated by the action of an external driving force (Gaussian thermostat) that exactly compensates the energy dissipated in collisions. The set of Enskog kinetic equations is the starting point to obtain the dependence of η on the control parameters of the mixture: solid fraction, concentration, mass and diameter ratios, and coefficients of normal restitution. While the expression of η found in the SM-theory is based on the assumption of Maxwellian distributions for the velocity distribution functions of each species, the GDH-theory solves the Enskog equation by means of the Chapman–Enskog method to first order in the shear rate. To assess the accuracy of both kinetic theories, the Enskog equation is numerically solved by means of the direct simulation Monte Carlo method. The simulation is carried out for a mixture under simple shear flow, using the thermostat to control the cooling effects. Given that the SM-theory predicts a vanishing kinetic contribution to the shear viscosity, the comparison between theory and simulations is essentially made at the level of the collisional contribution ηc to the shear viscosity. The results clearly show that the GDH-theory compares with simulations much better than the SM-theory over a wide range of values of the coefficients of restitution, the volume fraction, and the parameters of the mixture (masses, diameters, and concentration).

High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States

The Boltzmann equation for d-dimensional inelastic Maxwell models is considered to determine the collisional moments of the second, third and fourth degree in a granular binary mixture. These collisional moments are exactly evaluated in terms of the velocity moments of the distribution function of each species when diffusion is absent (mass flux of each species vanishes). The corresponding associated eigenvalues as well as cross coefficients are obtained as functions of the coefficients of normal restitution and the parameters of the mixture (masses, diameters and composition). The results are applied to the analysis of the time evolution of the moments (scaled with a thermal speed) in two different nonequilibrium situations: the homogeneous cooling state (HCS) and the uniform (or simple) shear flow (USF) state. In the case of the HCS, in contrast to what happens for simple granular gases, it is demonstrated that the third and fourth degree moments could diverge in time for given values of the parameters of the system. An exhaustive study on the influence of the parameter space of the mixture on the time behavior of these moments is carried out. Then, the time evolution of the second- and third-degree velocity moments in the USF is studied in the tracer limit (namely, when the concentration of one of the species is negligible). As expected, while the second-degree moments are always convergent, the third-degree moments of the tracer species can be also divergent in the long time limit.