Motion Of Granular Material Research Articles

Mixing in a vibrated granular bed: Diffusive and convective effects

In this study, Discrete Elementary Method (DEM) is employed to simulate the motion and mixing behavior of granular materials in a three-dimensional vibrated bed, which is energized by vertical sinusoidal oscillations under different vibrating conditions. With frictional sidewalls, the convection flow is a very important phenomenon in the vibrated granular bed. The influence of vibrating conditions, including vibration acceleration and frequency, on the formation of symmetric convection flow is investigated in this study. In order to characterize the convective flow and the diffusive motion of granular materials, the dimensionless convection flow rate, J conv, and the vertical self-diffusion coefficient, D yy , are defined, respectively. Péclet number, Pe, is employed to characterize the ratio of the convective flow to the diffusive motion in vertical direction. The role of Pe in the formation of symmetric convection flow is discussed in detail. Moreover, the top-bottom initial loading pattern of two groups of glass beads with different colors is employed to investigate mixing behavior of granular materials. The well-known Lacey index is employed as the mixing degree, M, to quantify the mixing quality. The mixing rate is calculated from a least-square fit using the time evolution of M. The simulation results demonstrate that the mixing rates increase with increasing J conv and D yy in exponential relations.

Capturing blast waves in granular flow

In this paper we continue the analysis of compressible Euler equations for inelastic granular gases described by a granular equation of state due to Goldshtein and Shapiro [Goldshtein A, Shapiro M. Mechanics of collisional motion of granular materials. Part 1: General hydrodynamic equations. J Fluid Mech 1995;282:75–114], and an energy loss term accounting for inelastic collisions. We study the hydrodynamics of blast waves in granular gases by means of a fifth-order accurate scheme that resolves the evolution under different restitution coefficients. We have observed and analyzed the formation of a cluster region near the contact wave using the one-dimensional and two-dimensional versions of our code.

Simulating Motion of Toner Using the Discrete Element Method

The Discrete Element Method (DEM) is one of the most popular and powerful tools for simulating the motion of granular materials. However, the traditional DEM focuses only on cohesionless granular materials; that is, materials whose motion is dominated by extra forces without significant influence of particle to particle attractive forces. For this reason the DEM approach has seen limited application in the study of toner. This study, however, proposes a numerical DEM model that incorporates cohesive forces, namely Van der Waals' and electrostatic forces, in the study of toner movement. Van der Waals' forces are those which exist due to the polarization of molecules in the toner particles into dipoles of positive and negative charge. Electrostatic forces exist between charged particles such as toner and are directly proportional to the magnitudes of the charges and inversely proportional to the square of the distance between them. The DEM model was decomposed into three sub-models, and these decomposed sub-models were validated separately by comparing the simulation results with experimental results. Finally, the model was used to simulate the motion of toner in the developer nip of an electrophotographic imaging process. The simulation results were compared with experimental results, and the simulation results fit very well with experimental results. This work shows that it is possible to achieve quantitative agreement between DEM simulations using cohesive forces and experimental results.

446 粒状体ダンパの減衰メカニズムに関する研究 : 粒状体の挙動観察と減衰特性

446 粒状体ダンパの減衰メカニズムに関する研究 : 粒状体の挙動観察と減衰特性

Silo music: Sound emission during the flow of granular materials through tubes

The present work deals with experiments and a model related to the emission of sound during the flow of granular materials through tubes. Experiments have been conducted using tubes fitted either with pistons or orifice plates at the ends. Earlier workers have observed that the height H of the granular material above the piston or orifice must exceed a critical value H c for the music to occur. The critical height H c was determined using the recorded sound signal for flow velocities in the range 0.4–1.8 mm/s (piston experiments) and 3–36.2 mm/s (orifice experiments). A simple model is constructed by treating the powder column as a spring and slider in series. The slider interacts with the tube wall via a rate and state variable friction law. The model predicts that the material either moves with a constant velocity (steady sliding) or in a jerky manner (stick-slip), depending on the parameter values used. As suggested by earlier workers, we postulate that the silo music is generated by the stick-slip motion of granular material. For H < H c, it is postulated that steady sliding occurs, and this is shown to be consistent with the model predictions. For H > H c, the model predicts that steady sliding is unstable with respect to small perturbations. The velocity of the slider exhibits sustained oscillations, which is the expected behavior of a system undergoing stick-slip motion. The model parameters have been estimated by fitting data on the variation of H c with the flow velocity. The predicted acceleration of the slider is similar in some respects to the data of Muite et al. (Powder Technol. 145 (2004) 190–202) for the flow of glass beads in a tube fitted with an orifice. Muite et al. found that the dominant frequency of the sound was given by c s / (4 L a), where c s is speed of sound in air and L a is the length of the air column above the granular material. This expression also fits the present data for the piston experiments.

Wet granular materials in sheared flows

The transport properties of wet granular materials in a shear cell apparatus have been studied. If the particles are wet, the flow becomes more viscous forming liquid bridges between particles. The dynamic liquid bridge forces are considered as the cohesive forces between particles to restrict their movements. The cohesive forces make the particles stick tighter with each other and hamper the movement of particles. The mixing and transport properties are influenced seriously by the amount of moisture added in the flow. This paper discusses a series of experiments performed in a shear cell device with five different moisture contents using 3-mm glass spheres as the granular materials. The motion of granular materials was recorded by a high-speed camera. Using the image processing technology and particle tracking method, the average and fluctuation velocities in the streamwise and transverse directions could be measured. The self-diffusion coefficient could be found from the history of the particle displacements. The self-diffusion coefficients and fluctuations in the streamwise direction were much larger than those in the transverse direction. Three bi-directional stress gages were installed to the upper wall to measure the normal and shear stresses of the granular materials along the upper wall. For wetter granular material flows, the fluctuation velocities and the self-diffusion coefficients were smaller.

Numerical predictions on the influences of the air blast velocity, initial bed porosity and bed height on the shape and size of raceway zone in a blast furnace

A numerical model has been developed to predict the shape and size of the raceway zone (a void space) created by the force of the blast air injected through the tuyeres in the packed coke bed of a blast furnace. The model is based on the solution of conservation equations of both gas and solid phases as interpenetrating continua on a Eulerian–Eulerian frame. A modified k–ε model has been adopted for gas phase turbulence. The solid phase constitutive equation is characterized by the solid pressure, bulk viscosity and shear viscosity, which are evaluated from the kinetic theory of random motions of granular materials in a fluid flow. The influences of the air blast velocity, initial porosity of the coke bed and the bed height on the shape and size of the raceway zone have been predicted.

An Interactive Deformation System for Granular Material

Abstract Computer Graphics (CG) animations of natural phenomena are currently widely used for movies and in video games. Granular materials occur widely in nature, and therefore it is necessary that CG animations represent ground surfaces composed of a granular material as well as model deformations when the granular material comes into contact with other physical rigid objects (called solid objects). In this paper, we propose a deformation algorithm for ground surfaces composed of granular material. The deformation algorithm is divided into three steps: (1) detection of the collision between a solid object and the ground surface, (2) displacement of the granular material and (3) erosion of the material at steep slopes. The proposed algorithm can handle solid objects of various shapes, including concave polyhedra by additionally using a layered data structure called the Height Span Map. Furthermore, a texture sliding technique is presented to render the motion of granular materials. In addition, our implementation of the deformation algorithm can be used at interactive frame rates.

Numerical simulation of the motion of granular material using object-oriented techniques

The objective of this contribution is to present a numerical simulation method to model the motion of a packed bed on a moving grate or in a rotary kiln using object-oriented techniques. The packed bed can be described as granular material consisting of a large number of particles. The method chosen is the Lagrangian time-driven method and it uses the position, the orientation, the velocity and the angular velocity of particles as independent variables. These are obtained by time integration of the three-dimensional dynamics equations which were derived from the classical Newtonian mechanics approach based on the second law of Newton for the translation and rotation of each particle in the granular material. This includes keeping track of all forces and moments acting on each particle at every time-step. Particles are treated as contacting visco-elastic bodies which can overlap each other. Contact forces depend on the overlap geometry, material properties and dynamics of particles and include normal and tangential components of repulsion force with visco-elastic models for energy dissipation through internal and surface friction. The resulting equations of particle motion are solved by the Gear predictor–corrector scheme of fifth-order accuracy. The simulation method is based on object-oriented methodologies and programmed in the programming language C++. This approach supports objects which can be used for three-dimensional particles of various shapes and sizes and for walls as boundaries. The programming modules are implemented in the TOSCA (tools of object-oriented software for continuum mechanic applications) software package which allows for a high degree of flexibility and for shortening the duration of the software development process. As methods for particle motion may deal with particles of different sizes and materials, the approach allows to describe transport processes in technical applications.

Granular flow in partially filled slowly rotating drums

In many industrial processes granular materials are mixed together in partially filled slowly rotating drums. In this paper a general theoretical framework is developed for the quasi-two-dimensional motion of granular material in a rotating drum. The key assumption is that the body can be divided into a fluid-like and a solid-like region, that are separated by a non-material singular surface at which discontinuities occur. Experiments show that close to the free surface there is a thin rapidly moving fluid-like avalanche that flows downslope, and beneath it there is a large region of slowly rotating solid-like material. The solid region provides a net transport of material upslope and there is strong mass transfer between the two regions. In the theory the avalanche is treated as a shallow incompressible Mohr–Coulomb or inviscid material sliding on a moving bed at which there is erosion and deposition. The solid is treated as a rigid rotating body, and the two regions are coupled together using a mass jump condition. The theory has the potential to model time-dependent intermittent flow with shock waves, as well as steady-state continuous flow. An exact solution for the case of steady continuous flow is presented. This demonstrates that when the base of the avalanche lies above the axis of revolution a solid core develops in the centre of the drum. Experiments are presented to show how a mono-disperse granular material mixes in the drum, and the results are compared with the predictions using the exact solution.

Arching phenomena in a vibrated granular bed

The motions of granular materials in a vertical shaker are complicated. According to the levels of the amplitude of the vibration acceleration, the particle motion phenomena may include the following convective types: heaping; fluidization; segregation; surface wave; and arching. This paper studied the arching phenomena of granular materials in a vertical shaker. The arching motions were investigated by high speed photography and image processing technology. The velocity fields of the granular materials during vibration were calculated from the recorded images. The ensemble average velocities in the horizontal and vertical directions were close to 0. The velocity fluctuations were also measured. The ensemble average granular temperature increases with the amplitude of the vibration acceleration, however, the variation of granular temperature with the acceleration in the horizontal direction is not as significant due to the existing side walls.

Mechanics of collisional motion of granular materials. Part 4. Expansion wave

The problem of expansion into a vacuum of a semi-infinite layer composed of chaotically moving inelastic rough spherical particles is solved analytically. A variant of the matched asymptotic expansion scheme is used to obtain a matched composite solution, which is valid for both small and large times in the wave head, wave tail and intermediate domains of the disturbed part of the layer.The effects of granular initial energy and particle collisonal properties on their hydrodynamic velocity, temperature, density and pressure are studied in the limit of low initial density of the granular gas. The total granular mass, M, within the disturbed region was found to change with time as log t. This is in contrast with the comparable classical result M ∼ t obtained for conservative (molecular) gases. This logarithmic dependence stems from the influence of kinetic energy losses, which reduce the granular temperature and speed of sound in the wave head region.The ultimate escape energy and momentum (i.e. those achieved for long times by the expanding part of the layer) are shown to be finite quantities, dependent on the particle restitution coefficient, roughness and the initial granular temperature. The estimated mass of the escaping part of the layer, calculated here for dilute gases, serves as an upper bound on this quantity for all (also dense) comparable granular gases. This mass is determined by the collisional losses, as embodied within the particle restitution and roughness coefficients.

Mechanics of collisional motion of granular materials. Part 3. Self-similar shock wave propagation

Shock wave propagation arising from steady one-dimensional motion of a piston in a granular gas composed of inelastically colliding particles is treated theoretically. A self-similar long-time solution is obtained in the strong shock wave approximation for all values of the upstream gas volumetric concentration v0. Closed form expressions for the long-time shock wave speed and the granular pressure on the piston are obtained. These quantities are shown to be independent of the particle collisional properties, provided their impacts are accompanied by kinetic energy losses. The shock wave speed of such non-conservative gases is shown to be less than that for molecular gases by a factor of about 2.The effect of particle kinetic energy dissipation is to form a stagnant layer (solid block), on the surface of the moving piston, with density equal to the maximal packing density, vM. The thickness of this densely packed layer increases indefinitely with time. The layer is separated from the shock front by a fluidized region of agitated (chaotically moving) particles. The (long-time, constant) thickness of this layer, as well as the kinetic energy (granular temperature) distribution within it are calculated for various values of particle restitution and surface roughness coefficients. The asymptotic cases of dilute (v0 [Lt ] 1) and dense (v0 ∼ vM) granular gases are treated analytically, using the corresponding expressions for the equilibrium radial distribution functions and the pertinent equations of state. The thickness of the fluidized region is shown to be independent of the piston velocity.The calculated results are discussed in relation to the problem of vibrofluidized granular layers, wherein shock and expansion waves were registered. The average granular kinetic energy in the fluidized region behind the shock front calculated here compared favourably with that measured and calculated (Goldshtein et al. 1995) for vibrofluidized layers of spherical granules.

Numerical simulation of granular-material flow on a slope

Particle/particle interaction is one of the most important factors in the transport process of granular material. In this study, the motion of granular material flowing on a slope is simulated by using the distinct element method, in which the motion of the individual particle is traced directly. The characteristics of the velocity profile of particles, or the transition of the type of the velocity profile with the increase of the inclination angle of a slope-from an upward convex curve to an upward concave curve via a straight line - found in the previous experiments is reproduced by the present simulation. The physics of such transition is investigated through the simulation data of individual particle motion.

Mechanics of collisional motion of granular materials. Part 2. Wave propagation through vibrofluidized granular layers

According to numerous experimental observations and theoretical models vibrated layers composed of large granules behave like a solid plastic body. In contrast, in this study experimental data are presented that reveal that, for constant vibration amplitudes A ≥ 1 cm with the frequency ω increasing from zero, all layers pass through three vibrational states, with the respective behaviours being as of (i) a solid plastic body, (ii) a liquid, (iii) a gas. In the liquid-like vibrational state transverse waves propagating along the layer width were observed. These waves were shown to be gravitational resonance waves, with the corresponding frequencies well correlated by the known formula for incompressible liquids. In the gas-like vibrational state compression (shock) and expansion waves propagating across the layer height, were observed.A theoretical model for time-periodic collisional vibrational regimes was developed on the basis of the Euler-like equations of a granular gas composed of inelastic spheres. The model shows that the vibrational granular state (bed porosity, shock wave speed, granular pressure and kinetic energy) is inter alia governed by the dimensionless parameter V = (Aω)/(hmg)1/2, with g, hm being the gravitational acceleration and the height of the resting layer, respectively. This is in contrast with the previous studies, where the behaviour of vibrated granular layers was interpreted in terms of the dimensionless acceleration Δ = (Aω2)/g. The proposed model was tested by processing the data obtained from photographs of the particle distribution within vibrated layers. Theoretical predictions of the particle average concentration compared favourably with the experimental data.Other phenomena observed in vibrated granular layers include the formation of caverns, circulatory motion of granules and synchronized periodic motion of two adjacent vibrated layers of different widths. The importance of the observed phenomena in relation to various technological processes involving bulk materials (vibromixing, vibroseparation, etc.) is discussed.

Mechanics of collisional motion of granular materials. Part 1. General hydrodynamic equations

Collisional motion of a granular material composed of rough inelastic spheres is analysed on the basis of the kinetic Boltzmann–Enskog equation. The Chapman–Enskog method for gas kinetic theory is modified to derive the Euler-like hydrodynamic equations for a system of moving spheres, possessing constant roughness and inelasticity. The solution is obtained by employing a general isotropic expression for the singlet distribution function, dependent upon the spatial gradients of averaged hydrodynamic properties. This solution form is shown to be appropriate for description of rapid shearless motions of granular materials, in particular vibrofluidized regimes induced by external vibrations.The existence of the hydrodynamic state of evolution of a granular medium, where the Euler-like equations are valid, is delineated in terms of the particle roughness, β, and restitution, e, coefficients. For perfectly elastic spheres this state is shown to exist for all values of particle roughness, i.e. − 1≤β≤1. However, for inelastically colliding granules the hydrodynamic state exists only when the particle restitution coefficient exceeds a certain value em(β)&lt; 1.In contrast with the previous results obtained by approximate moment methods, 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. The effect of increasing inelasticity of particle collisions is to redistribute the kinetic energy of their random motion in favour of the rotational mode. This is shown to significantly affect the energy partition law, with respect to the one prevailing in a gas composed of perfectly elastic spheres of arbitrary roughness. In particular, the translational specific heat of a gas composed of inelastically colliding (e = 0.6) granules differs from its value for elastic particles by as much as 55 %.It is shown that the hydrodynamic Euler-like equation, describing the transport and evolution of the kinetic energy of particle random motion, contains energy sink terms of two types (both, however, stemming from the non-conservative nature of particle collisions) : (i) the term describing energy losses in incompressibly flowing gas; (ii) the terms accounting for kinetic energy loss (or gain) associated with the work of pressure forces, leading to gas compression (or expansion). The approximate moment methods are shown to yield the Euler-like energy equation with an incorrect energy sink term of type (ii), associated with the ‘dense gas effect’. Another sink term of the same type, but associated with the energy relaxation process occurring within compressed granular gases, was overlooked in all previous studies.The speed of sound waves propagating in a granular gas is analysed in the limits of low and high granular gas densities. It is shown that the particle collisional properties strongly affect the speed of sound in dense granular media. This dependence is manifested via the kinetic energy sink terms arising from gas compression. Omission of the latter terms in the evaluation of the speed of sound results in an error, which in the dense granular gas limit is shown to amount to a several-fold factor.

Three-dimensional numerical simulation of the motion of particles discharging from a rectangular hopper using distinct element method and comparison with experimental data (effects of time steps and material properties)

The distinct element method (DEM) is a simple numerical model which is capable of describing the motion of granular materials. In this study, the motion of particles discharging from a rectangular hopper has been numerically simulated by the DEM whose time steps are Δ t = 10 −6, 10 −5 and 10 −4s, which correspond to stiffness k n = 7.0 × 10 7, 7.0 × 10 5 and 7.0 × 10 3 N/m, respectively. The calculated results have been compared with the experimental data. All of the calculated results not only qualitatively describe the experimental results, but also approximately describe the quantitative characteristics of the measured data, although the stiffness of the calculations is greatly different from the real stiffness of the materials. However, the calculated result of the DEM using the smallest time step (Δ t = 10 −6s) could not give the best description of the experimental data. This suggests that the calculated results of the DEM could not accurately describe the real granular flow even if the time step is small enough to use the real stiffness of the materials. The most significant approximation of the DEM is that distinct particles do not interact with any other particles and displace independently from one another during one calculation time step. The discrepancy between the model based on this approximation and the real granular flow would not decrease as the time step is decreased.

Rapid flow of granular materials with density and fluctuation energy gradients

A constitutive relationship for the stress tensor of granular materials during the rapid flow regime that includes the effects of density gradient and fluctuating velocity is proposed. Explicit expressions for various stress components are obtained and simple examples are presented. It is shown that this continuum-based model is capable of predicting normal stress differences. The model also contains many earlier models as its special cases. The results also show that the model may be suitable for analyzing rapid, as well as slow, motions of granular materials.

A comparison of the solutions of some proposed equations of motion of granular materials for fully developed flow down inclined planes

In the past few years kinetic theory has been used to derive equations of motion for rapidly shearing granular materials, and there have been empirical extensions of these to take into account stress transmitted by sustained sliding and rolling contacts between particles. The equations are complicated and solutions have been generated only for very simple flows. In this paper three forms for the equations of motion are considered; one representing interaction by collisions only, one which is a high-density asymptotic form of this, and a third which includes terms representing the ‘frictional’ stresses associated with the sustained contacts referred to above. Solutions are found for fully developed flow under gravity down an inclined plane, and it is shown that the relation between the flow rate and the depth of the flowing layer predicted by the first two sets of equations is not in accord with observations. The third form appears to eliminate much of the discrepancy, but its predictions have not been explored over the whole parameter space. It is emphasized that the form of the solutions should be studied over a wide range of operating conditions in order to assess the usefulness of proposed equations.

The flow of granular material

The motion of granular material is not readily predicted by the techniques developed for either fluid or solid mechanics. Although there are many industrial applications (i.e. transport of coal, grain, powder, salt etc.), this field still requires more rigorous scientific analysis. Experiments conducted with the flow of several (non-cohesive) granular materials through a slot indicate the existence of three flow regions. These regions may be catalogued into domains described by: (I) soil mechanics; (II) free-particle flow; and (III) frictional flow. It is also observed that a choking phenomenon occurs in which the slot velocity attains a critical maximum value independent of the slot area or height of the material. An analysis of this interdisciplinary topic and an explanation of the experimental observations is presented.