Elastohydrodynamic theory for wet oblique collisions

This paper presents a model for oblique collisions of spherical particles with a plane surface covered with a thin liquid layer. Elastohydrodynamic theory developed previously for fully immersed collisions [Davis, Serayssol and Hinch 1986 JFM 63 479-497] is modified for the normal component of motion to account for the finite thickness of the liquid layer. The resulting time evolution of the film thickness profile is then used along with sliding lubrication to determine the tangential component of motion. The critical Stokes number (dimensionless ratio of particle inertia and viscous forces), below which no rebound is seen, is predicted in terms of the physical properties of the materials involved in the collision, as described by a compliance parameter representing a dimensionless measure of elastic deformation due to viscous forces. Beyond the critical Stokes number, the normal restitution coefficient is found to increase with the Stokes number and the compliance parameter, asymptoting to the dry restitution coefficient at high Stokes numbers. The lubrication suction resistance during rebound is limited by cavitation. The tangential restitution is independent of the impact angle and is linearly dependent on the ratio of the fluid layer thickness to the sphere radius, in addition to depending on the Stokes number and compliance parameter. The tangential restitution is found to be close to unity and is generally higher for a larger value of the compliance parameter. Moreover, the tangential restitution is seen to increase with the Stokes number at small compliance and decrease with the Stokes number at large compliance. The change in rotational velocity exhibits trends that are the reverse of the tangential restitution. Finally, closed-form expressions have been developed for describing the restitution coefficients and dimensionless change in rotational velocity.     

A bubbling fluidization model using kinetic theory of rough spheres

Collisional motion of inelastic rough spheres is analyzed on the basis of the kinetic theory for flow of dense, slightly inelastic, slightly rough sphere with the consideration of gas–solid interactions. The fluctuation kinetic energy of particles is introduced to characterize the random motion of particles as a measure of the translational and rotational velocities fluctuations. The kinetic energy transport equation is proposed with the consideration of the redistribution of particle kinetic energy between the rotational and translational modes and kinetic energy dissipation by collisions. The solid pressure and viscosity are obtained in terms of the particle roughness and restitution coefficient. The partition of the random‐motion kinetic energy of inelastic rough particles between rotational and translational modes is shown to be strongly affected by the particle restitution coefficient and roughness. Hydrodynamics of gas–solid bubbling fluidized beds are numerically simulated on the basis of the kinetic theory for flow of rough spheres. Computed profiles of particles are in agreement with the experimental measurements in a bubbling fluidized bed. The effect of roughness on the distribution of energy dissipation, kinetic energy, and viscosity of particles is analyzed. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

Collisions of spheres with wet and dry porous layers on a solid wall

A theory that combines Darcy's law for flow in porous media with inelastic solid mechanics, to model collisions of solid spheres with wet or dry porous layers placed on a solid wall, is found to closely describe the trends in data collected from particle-collision experiments. An exponential-hardening, stress-strain model is used for the porous layer, validated with dynamic mechanical analyzer measurements. Low-velocity collisions were performed in the low-gravity environment afforded during parabolic flight of a KC-135 aircraft, and also under normal gravity with a pendulum-based setup. Both theory and experiments show a decrease in the dry restitution coefficient with an increase in impact velocity, mainly due to increased inelastic losses in the porous material. The wet restitution coefficient is also found to decrease with an increase in the impact velocity, in contrast to the wet restitution coefficient for collisions of a solid sphere with a wet wall without a porous layer. Moreover, a critical impact velocity (below which no rebound occurs) is observed for wet collisions without a porous layer but not with a porous layer. The wet restitution coefficient is always found to be lower than the dry restitution coefficient, due to the viscous losses associated with fluid flow in addition to the inelastic losses associated with the porous layer.     

Einfluss der Plattendicke auf die Kontaktzeit beim elastischen Stoßvorgang

Using a free fall apparatus, the coefficient of restitution and the contact time of steel spheres at impact on thin gold‐coated glass plates were measured experimentally. The influence of the impact velocity, the particle size, and the plate thickness has been investigated. The measurements were evaluated using the Zener model. However, since the model merely considers energy dissipation due to elastic flexural waves and ignores additional energy dissipation by friction and viscous damping, the experimental values remain slightly below the theoretical Zener curve.     

基于颗粒流运动论的喷动流化床气固流动行为三维模拟

1 INTRODUCTION Spout-fluid beds have been of increasing interest in the petrochemical, chemical and metallurgic indus-tries since spout-fluid beds can reduce some of the limitations of both spouting and fluidization by su-perimposing the two type of systems[1―4]. In recent years, spout-fluid beds have become an alternative for gas/solid contactors in coal gasification. Spout-fluid bed coal gasifiers have been adopted for APFBC-CC (advanced pressurized fluidized bed combus-tion-combined…     

Influence of Particle Shape on Filtration Processes

The influence of particle shape on filtration processes was investigated. Two types of particles, including spherical polystyrene latex (PSL) and iron oxide, and perfect cubes of magnesium oxide, were examined. It was found that the removal efficiency of spherical particles on fibrous filters is very similar for corresponding sizes within the range of 50–300 nm, regardless of the fact that the densities of PSL and iron oxide differ by a factor of five. On the other hand, the removal efficiency of magnesium oxide cubic particles was measured, and found to be much lower than the removal efficiency for the aerodynamically similar spheres. Such disparity was ascribed to the different nature of the motion of the spherical and cubic particles along the fiber surface, following the initial collision. After touching the fiber surface and before coming to rest, the spherical particles could either slide or roll compared to the cubic ones, which could either slide or tumble. During tumbling, the area of contact between the particle and the fiber changes significantly, thus affecting the bounce probability, whilst for the spheres, the area of contact remains the same for any point of the particle trajectory. The extra probability of particle bounce by the cubes was derived from the experimental data. The particle kinetic energy was proposed to be responsible for the difference in removal efficiency of particles with alternative shapes, if all other process parameters remain the same. The increase in kinetic energy is shown to favor the increase of the bounce probability.     

A Study of Particle Rebounding Characteristics of a Gas–Particle Flow over a Curved Wall Surface

The particle rebounding characteristics of a gas–particle flow over a cylindrical body is investigated. With the aid of both computational and experimental approaches, the mean particle flow patterns, comprising both incident and rebound particles resulting from the impact of particles on a curved wall surface, are examined. In the experimental investigation, a two-dimensional Laser Doppler Anemometry (LDA) technique is used in the immediate vicinity of the body surface to measure the instantaneous incident and rebound particle velocities. The Reynolds-Averaging Navier-Stokes equations are solved for the continuum gas phase, and the results are used in conjunction with a Lagrangian trajectory model to predict the particle-rebound behavior in the immediate vicinity of the cylindrical wall. The computational observations, also confirmed through experiments, reveal a particle rebound zone where the mean particle flow pattern is significantly modified due to the contribution of the rebound particles during the process of particle–wall impact interaction. This particle rebound zone is found to be a function of mainly the Stokes number (particle inertia), and to a lesser extent on the fluid Reynolds number (gas flow condition), except for high gas flow velocities and restitution coefficients (particle-wall impact characteristics). Analysis of the effect of the above-mentioned parameters on the rebounding particle flow characteristics and their interrelationship has provided a better understanding of the behavior of particle flow impinging on a solid wall body. The beneficial contributions of the experimental and computational approaches in their ability to better quantify the particle–wall impact interaction phenomena present additional foundational investigations that could be further undertaken to better comprehend the particle behavior in curved wall surfaces. Such invaluable information has direct applications to industrial devices such as commercial heat exchangers and inertial impactors.     

基于粗糙颗粒动理学流化床内颗粒与幂律流体两相流动特性的数值模拟研究

在颗粒动理学理论（KTGF）的基础上,通过引入表征粗糙颗粒摩擦和切向非弹性的切向弹性恢复系数β,以及综合反映颗粒平动和旋转运动脉动强度的颗粒拟总温e₀,结合输运理论建立了考虑颗粒旋转作用的颗粒相质量、动量和颗粒拟总温守恒方程。并在求解了同时具有平动和旋转运动的能量耗散和颗粒相应力等参数的前提下提出了颗粒相压力、剪切黏度和能量耗散等本构关系式以及边界条件,最终得出了粗糙颗粒动理学理论（KTRS）。通过改变液相的流变特性,分析了幂律流变模型中流动指数n和稠度系数K_l对流化床内流固两相流动特性的影响,模拟结果表明：随着流动指数和稠度系数的增大,液相湍动能耗散率逐渐增大,而颗粒相压力逐渐减小,颗粒旋转先增大后减小。     

A novel coalescence model for binary collision of identical wet particles

A coalescence model that incorporates both capillary and viscous contributions of the liquid binder, along with the liquid bridge volume effects, has been developed. The objective is to predict the coalescence of two spherical particles coated with a thin liquid film approaching with equal initial velocities from opposite directions. The model is based on an overall coefficient of restitution that is determined with the aid of the approximate analytical values of the maximum possible energy dissipation and a critical value that depends on the total initial kinetic energy of particles. The maximum possible energy dissipation accounts for the energy loss due to the viscous and capillary effects and inelastic collision. The proposed simplified method to determine the critical velocity has been compared with the numerical solution of the general equation of motion, and an excellent agreement has been found. The coalescence model has been investigated at the limiting conditions by neglecting either the capillary effect or the viscous effect, respectively. Finally, comparisons have been made with experimental data, and a reasonable agreement has been found.     

Granular pressure and particle velocity fluctuations prediction in liquid fluidized beds

A numerical modelling approach for the dynamic simulation of solid-liquid fluidized beds is evaluated. This approach is based on an Eulerian two-fluid formulation of the transport equations for mass, momentum and fluctuating kinetic energy. The solid-phase fluctuating motion model is derived in the frame of granular medium kinetic theory accounting for the viscous drag influence of the interstitial liquid phase. Solid-liquid fluidized bed two-dimensional simulations were performed for flow configurations taken from the experimental work of Zenit et al. [1997. Collisional particle pressure measurements in solid-liquid flows. Journal of Fluid Mechanics 353, 261-283], for three types of solid particles of contrasted inertia in water at high particle Reynolds number (nylon, glass and steel beads). Experimental and numerical granular pressures exhibit a satisfactory agreement with both low and high inertia particles, the best level of prediction being observed with the most inertial particles. Sensitivity of the predictions to the phenomenological laws used in the model is also presented and it appears that, due to non-linear correlations, the average granular pressure in the bed is a less sensitive variable than the fluctuating kinetic energy (or granular temperature). The transport mechanisms of the mean granular temperature in the bed are therefore analyzed as a function of the solid fraction and the particle inertia. At low and moderate Stokes number (nylon and glass beads) and in all range of solid-phase fraction, the dominant production mechanism of fluctuating kinetic energy is due to the mean velocity gradient, whereas the main dissipation term is that induced by the viscous drag. At higher Stokes number (steel beads) and concentration, the production of the granular temperature is controlled by the compressibility effects via the granular pressure. In this case, the dissipation is mainly provided by inter-particle collisions.     

颗粒特性对撞击分离器性能影响的实验与数值研究

颗粒-壁面撞击行为和气固相间作用对撞击分离器的性能具有重要影响。基于玻璃珠及煤粉的单颗粒撞击实验数据建立平均撞击恢复系数模型,采用非球形颗粒曳力模型对平板式撞击分离器的分离性能和气固流动开展数值研究。结果表明,采用基于实验的平均恢复系数模型以及考虑颗粒形状的曳力模型,能够准确地预测撞击式分离器的总分离效率和分级分离效率。颗粒分离过程中,Stokes数较大的颗粒对颗粒-壁面撞击模型比较敏感,Stokes数较小的颗粒对气固曳力模型比较敏感。     

Flow of particles suspended in a sheared viscous fluid: Effects of finite inertia and inelastic collisions

We investigate in this article the macroscopic behavior of sheared suspensions of spherical particles. The effects of the fluid inertia, the Brownian diffusion, and the gravity are neglected. We highlight the influence of the solid‐phase inertia on the macroscopic behavior of the suspension, considering moderate to high Stokes numbers. Typically, this study is concerned with solid particles O (100 μm) suspended in a gas with a concentration varying from 5% to 30%. A hard‐sphere collision model (with elastic or inelasic rebounds) coupled with the particle Lagrangian tracking is used to simulate the suspension dynamics in an unbounded periodic domain. We first consider the behavior of the suspension with perfect elastic collisions. The suspension properties reveal a strong dependence on the particle inertia and concentration. Increasing the Stokes number from 1 to 10 induces an enhancement of the particle agitation by three orders of magnitude and an evolution of the probability density function of the fluctuating velocity from a highly peaked (close to the Dirac function) to a Maxwellian shape. This sharp transition in the velocity distribution function is related to the time scale which controls the overall dynamics of the suspension flow. The particle relaxation (resp. collision) time scale dominates the particulate phase behavior in the weakly (resp. highly) agitated suspensions. The numerical results are compared with the prediction of two statistical models based on the kinetic theory for granular flows adapted to moderately inertial regimes. The suspensions have a Newtonian behavior when they are highly agitated similarly to rapid granular flows. However, the stress tensors are highly anisotropic in weakly agitated suspensions as a difference of normal stresses arises. Finally, we discuss the effect of energy dissipation due to inelastic collisions on the statistical quantities. We also tested the influence of a simple modeling of local hydrodynamic interactions during the collision by using a restitution coefficient which depends on the local impact velocities. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

A novel approach to determine wet restitution coefficients through a unified correlation and energy analysis

Wet particle interactions are observed in many applications, for example, pharmaceutical, food, agricultural, polymerization, agglomeration, and coating, in which an accurate evaluation of the wet restitution coefficient (e_wet) is crucial to understand the particle flowability, operating conditions and product size distribution. Experiments were performed to measure the wet restitution coefficient by impacting a spherical particle on a stationary plate covered with a thin liquid layer of water or glycerol solution in this work. Furthermore, novel approaches for estimation of e_wet were developed using dimensional analysis (using the Buckingham π theorem and regression analysis) in combination with energy budget analysis. In the correlation development, the dominant physical properties of solid and liquid, particle impact velocity and liquid layer thickness are grouped into well‐known dimensionless numbers viz. Reynolds, Weber and Stokes. Whereas in the energy analysis, the energy dissipation rates were determined for five distinct collision phases, that is, dipping, dry collision, undipping, formation and breakage of the liquid bridge, and added mass. The efficacy of the developed approaches was analyzed by comparing obtained results with experiments and an elastohydrodynamic model, and a modified elastohydrodynamic model. © 2014 American Institute of Chemical Engineers AIChE J, 61: 769–779, 2015     

Using the discrete element method to predict collision-scale behavior: A sensitivity analysis

Discrete element method (DEM) simulations have recently been used to investigate collision-scale measurements such as collision frequency and impact velocity distributions. These simulations are typically validated against particle velocity fields using experimental techniques such as particle image velocimetry or positron emission particle tracking. An important question that has not been addressed is whether validation of a macroscopic velocity field or solid fraction field also implies a validation of collision-scale measurements such as collision frequency. In this study, DEM measurements of solid fraction, shear rate, collision frequency, and impact velocity are made in a small region just beneath the free surface in a rotating drum. The effects of periodic drum length, particle stiffness, coefficient of restitution, and particle size are investigated. The solid fraction and shear rate do not vary with particle stiffness or coefficient of restitution over the range of values studied. However, the collision rate increases with increasing particle stiffness and coefficient of restitution. In addition, the average collision speed decreases as particles become stiffer or less elastic. The shear rate varies with particle size, but the average collision velocity remains constant. These findings indicate that validation against particle velocity and solid fraction fields does not necessarily imply validation of collision frequency and impact velocity. Indeed, the velocity and solid fraction fields were found to be relatively insensitive to a range of DEM contact stiffnesses and coefficients of restitution while the collision distributions were sensitive.     

Shape Effects on Microbody Impacts against a Flat Surface

A general numerical model is developed to simulate the impact of an elongated microbody with a spherical tip against a flat surface. Experimental data of sphere impacts are used to determine the parameters used in the simulations. Two kinds of microbodies are considered: a rod with spherical ends and 2 hemispheres connected by a thin rigid rod. The results show that under the same incident velocity and orientation angle, the impacts are affected by the microbody shape. Angular velocity changes are quite sensitive to the length and orientation of the rod. Rotational energy balance plays an important role for long microbodies. The rebound velocity at the contact tip and the center of mass is different and can lead to secondary impacts. In contrast to spheres, tangential (friction) and normal forces are coupled for elongated microbodies. Because the tangential and normal forces act over the contact area at the end of the rod, a moment about the mass center is produced. The rotation of the rod is driven by this moment, which, in turn, changes the relative velocity and contact forces over the contact area. Thus the coefficient of restitution at the contact tip is also effected by and becomes a function of the geometry and orientation of the microbody. The simulation results support that three-dimensional (3D) microbody impact response is determined not only by the material and incident velocities but also by the geometry and orientation of the principal axis of the microbody.     

The influence of DEM simulation parameters on the particle behaviour in a V-mixer

Results are described of simulations based on the discrete element method (DEM) using a code developed by Tsuji, Kawaguchi, and Tanaka (Discrete particles simulation of 2-dimensional fluidized bed. Powder Technology 77 (1993) 79-87). The mechanical interactions between particles and also between particles and the walls in granular flows are modelled by linear springs, dash-pots and friction sliders. The simulation parameters are the restitution coefficient, normal stiffness, friction coefficient between particles and between particles and the walls, and two ratios which relate the normal and tangential stiffness and damping coefficients. Their influence on particle motion in a V-mixer has been evaluated and compared with radioactive tracer measurements of particle motion. A number of quantitative methods for comparing DEM and experimental data were developed. Given the simplified nature of the modelled interactions, the agreement between the predicted and measured data is remarkably close for restitution coefficient values of 0.7 and 0.9, internal friction coefficient values of 0.3 and 0.6 and wall friction coefficient values of 0 and 0.3. The internal and wall friction coefficients are important in determining the initiation of particle movement, while the value of the restitution coefficient has a larger influence on particles in a dynamic state. The simulation of the fully elastic case (coefficient of restitution =1.0) with zero internal and wall friction, gives results that are very different from the experiment data.     

Motion of a sphere parallel to plane walls in a Poiseuille flow. Application to field-flow fractionation and hydrodynamic chromatography

The motion of a solid spherical particle entrained in a Poiseuille flow between parallel plane walls has various applications to separation methods, like field-flow fractionation and hydrodynamic chromatography. Various handy formulae are presented here to describe the particle motion, with these applications in mind. Based on the assumption of a low Reynolds number, the multipole expansion method coupled to a Cartesian representation is applied to provide accurate results for various friction factors in the motion of a solid spherical particle embedded in a viscous fluid between parallel planes. Accurate results for the velocity of a freely moving solid spherical particle are then obtained. These data are fitted so as to obtain handy formulae, providing e.g. the velocity of the freely moving sphere with a 1% error. For cases where the interaction with a single wall is sufficient, simpler fitting formulae are proposed, based on earlier results using the bispherical coordinates method. It appears that the formulae considering only the interaction with a nearest wall are applicable for a surprisingly wide range of particle positions and channel widths. As an example of application, it is shown how in hydrodynamic chromatography earlier models ignoring the particle-wall hydrodynamic interactions fail to predict the proper choice of channel width for a selective separation. The presented formulae may also be used for modeling the transport of macromolecular or colloidal objects in microfluidic systems.     

Hydrodynamic modeling of particle rotation for segregation in bubbling gas-fluidized beds

A multi-fluid Eulerian model has been improved by incorporating particle rotation using kinetic theory for rapid granular flow of slightly frictional spheres. A simplified model was implemented without changing the current kinetic theory framework by introducing an effective coefficient of restitution to account for additional energy dissipation due to frictional collisions. Simulations without and with particle rotation were performed to study the bubble dynamics and bed expansion in a monodispersed bubbling gas-fluidized bed and the segregation phenomena in a bidispersed bubbling gas-fluidized bed. Results were compared between simulations without and with particle rotation and with corresponding experimental results. It was found that the multi-fluid model with particle rotation better captures the bubble dynamics and time-averaged bed behavior. The model predictions of segregation percentages agreed with experimental data in the fluidization regime where kinetic theory is valid to describe segregation and mixing.     

Numerical simulations of the normal impact of adhesive microparticles with a rigid substrate

The rebound behavior of elastic and elastoplastic microspheres impacting normally with a rigid wall is studied using the finite element method. The interfacial adhesion forces are introduced by adding piecewise-linear spring elements with a particular constitutive relation characterizing the adhesion property. The effect of adhesion hysteresis is taken into account by assuming that the adhesion work during the incident stage is smaller than that during rebounding. The influences of the interfacial adhesion parameters, the constitutive relations, size, and incident velocity of the particle on the coefficient of restitution (COR) are all examined. We also analyze the changing tendency of the kinetic energy, elastic strain energy, adhesion work, and their interchange during impact. It is found that besides interfacial adhesion and plastic dissipation, the residual stress field caused by incompatible plastic deformation has a considerable influence on the impact behavior of the sphere as well. For smaller impact velocities, interfacial adhesion plays a dominant role in the impact process, while for higher incident velocities, the COR depends mainly on plastic deformation. In addition, the COR shows a distinct dependence on the particle size. Finally, our numerical results are compared with the relevant experimental results.