A theoretical analysis of the force models in discrete element method

This paper presents an investigation of the equilibrium, stability and pure rolling problems of a sphere moving on a flat plane with special reference to a few force models which are commonly used in the discrete element method (DEM) and here categorized into two types: with and without rolling friction. It is obtained that according to the models without rolling friction, the set of equilibrium states of the system is not asymptotically stable, which does not agree with the fact that the sphere with small initial tangential velocity, angular velocity and tangential displacement should eventually stop. The models also display that the sphere can roll on the plane without sliding only when both the tangential force and torque acting on the sphere are zero, which is not reasonable for a viscoelastic sphere moving on a hard plane. On the other hand, the models with rolling friction cannot describe the pure rolling motion of the sphere with any material properties. The results highlight the theoretical deficiency associated with the force models in DEM. Based on the findings, modified models are proposed to overcome the above problems.     

Effects of particle size distribution in a three-dimensional micropolar continuum model of granular media

A particle size distribution is incorporated into a three-dimensional homogenisation scheme, devised on the scale of a particle and its immediate (or first ring) of neighbours. Based on this scheme, micropolar continuum models for polydisperse, dry, and densely packed granular assemblies of spherical particles undergoing quasi-static deformation are developed for various particle size distributions. Three different cases are considered: (1) a monodisperse assembly, (2) a defect particle in an otherwise monodisperse assembly, and (3) an assembly of a given particle size distribution. In Case 1, an additional dependence on particle radius is found in 3D systems, compared with previous 2D constitutive laws. In Case 2, it is found that a small (large) particle in an otherwise monodisperse system increases (decreases) the stress compared to a purely monodisperse assembly, but the couple stress may increase or decrease depending on the relative size of the rolling resistance compared with the tangential stiffness coefficients. On the other hand, if the defect particle is substantially smaller or larger than the monodisperse particle size, the stress and couple stress are always increased. In Case 3, three different distributions are examined, i.e. square, normal and a lognormal distribution. For Cases 2 and 3, both the stress and the couple stress increased with the degree of dispersity, from the lower bound value corresponding to the monodisperse system considered in Case 1. Finally, the paper highlights areas that will need to be addressed to enable the future advancement of micromechanical continuum models.     

A study on tangential force laws applicable to the discrete element method (DEM) for materials with viscoelastic or plastic behavior

The discrete element method is a widely used particle orientated simulation approach for modeling granular systems. It is based on tracking each particle's movement and its interactions with the surroundings over time. The motion of a particle is given by a system of coupled ordinary differential equations which are solved numerically. Therefore, models for the forces acting between particles in contact need to be specified. In the past, detailed investigations dealing with the accuracy of tangential force-displacement models have been very limited, with sparse experimental data considered and the frequent restriction of including only fully elastic materials. In large scale discrete element simulations, on the other hand, viscoelastic or plastic material behavior is often assumed for normal contacts and combined with arbitrary tangential models. To address this situation a number of tangential force-displacement models are reviewed including linear models by Cundall and Strack [1979. A discrete numerical model for granular assemblies, Geotechnique 29, 47-65], Di Maio and Di Renzo [2004. Analytical solution for the problem of frictional-elastic collisions of spherical particles using the linear model. Chemical Engineering Science 59(16), 3461-3475], Brendel and Dippel [1998. Lasting contacts in molecular dynamics simulations. In: Herrmann, H.J., Hovi, J.-P., Luding, S. (Eds.), Physics of Dry Granular Media, Dordrecht. Kluwer Academic Publishers, pp. 313], Walton and Braun [1986. Viscosity, granular temperature and stress calculations for shearing assemblies of inelastic, frictional disks. Journal of Rheology 30, 949] and simple non-linear models by Brilliantov et al. [1996. Model for collisions in granular gases. Physical Review E 53(5), 5382-5392], Tsuji et al. [1992. Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technology 71, 239-250] and Di Renzo and Di Maio [2005. An improved integral non-linear model for the contact of particles in distinct element simulations. Chemical Engineering Science 60(5), 1303-1312]. Whereas for fully elastic materials the parameters of the tangential force-displacement models can be derived directly from mechanical properties a scaling approach is proposed for the estimation of the parameters in the non-elastic case. The effect of different normal force-displacement models is analyzed. For all model combinations macroscopic final collision properties are derived and compared to experimental results by Foerster et al. [1994. Measurements of the collision properties of small spheres. Physics of Fluids 6(3), 1108-1115], Lorenz et al. [1997. Measurements of impact properties of small, nearly spherical particles. Experimental Mechanics 37(3), 292-298], Gorham and Kharaz [2000. The measurement of particle rebound characteristics. Powder Technology 112(3), 193-202] and Dong and Moys [2003. Measurement of impact behaviour between balls and walls in grinding mills. Minerals Engineering 16(6), 543-550; 2006. Experimental study of oblique impacts with initial spin. Powder Technology 161(1), 22-31].     

Understanding factors leading to bias for falling-stream cutters using discrete element modelling with non-spherical particles

Previous studies of the possible bias of falling-stream sample cutters have used physical experiments, two-dimensional and three-dimensional DEM (discrete element modelling) models with spherical particles. The present study uses super-quadric particles describing particles with variable aspect ratios and variable blockiness, and is more realistic than previous modelling studies. Our results support the commonly used rule that the cutter aperture should be at least three times the nominal top size of the material being sampled, but do not support a commonly used rule relating acceptable cutter aperture to cutter speed and suggest that particle shape, belt load and distance of cutter below the head pulley are also key factors which can influence cutter bias.     

颗粒移动床内不稳定运动的计算颗粒动力学模拟

采用考虑滚动摩擦的三方程离散单元法（DEM）模型对侧开孔的移动床中的颗粒流动进行了数值研究。结果表明，计算颗粒动力学(CGD)方法可对复杂颗粒系统内颗粒的运动行为进行准确的预测，包括时均速度场和脉动速度场。讨论了模型中颗粒摩擦参数的重要影响，并对颗粒流动表现出的间歇现象进行了分析。颗粒流动与流体流动有相似之处，都存在随机的脉动，但颗粒流的随机脉动机理与流体中的湍流机理有很大不同，颗粒流动会表现出很强的不连续性。     

Numerical simulation of liquid transfer between particles

Liquid transfer between particles plays a central role in the operation of a variety of particle processing equipment, including flotation, spray-coating, flocculation, granulation, and drying. In each of these applications, the local liquid concentration within the bed dramatically affects the flow behavior of the system and can strongly impact performance. In this work, we introduce a dynamic liquid transfer model for use in discrete element modeling (DEM) of heterogeneous particle systems. We explicitly track moisture levels on individual particles and utilize an experimentally validated rule-set for liquid transfer upon forming/breaking contacts. As a test of this new model we present results from the simulation of a rotary drum spray-coating system, but expect that this liquid transfer-modified DEM is general and would be applicable to wide range of processing operations.     

Dynamics of gas–solid fluidised beds with non-spherical particle geometry

Fluidised beds play an important role in physical and chemical engineering processing. Understanding the granular motion within these beds is essential for design, optimisation and control of such processes. Motion on the particle scale is difficult to measure experimentally, making computational simulations invaluable for determining the dynamics within such systems. Computational models which have had the greatest success at capturing the full range of dynamics are coupled discrete element model and Navier–Stokes solvers, based on a pressure-gradient-force formulation. However, most discrete element models assume spherical geometry for the particles. Particle shape in many important industrial processes, such as catalysis and pyrolysis, is often non-spherical. We present a re-formulation of the pressure-gradient force model, based on a modified pressure correction method, coupled to a discrete element model with non-spherical grains. The drag relations for the coupling are modified to take into account the grain shape and cross-sectional area relative to the local gas flow. We show that grain shape has a significant effect on the dynamics of the fluidised bed, including increased pressure gradients within the bed and lower fluidisation velocities when compared to beds of spherical particles. A model is presented to explain these effects, showing that they are due to both decreased porosity within the bed as well as the relative particle cross-sectional area creating a greater net drag over the bed. Our findings will be of interest from an applied standpoint as well as showing fundamental effects of particle shape on coupled fluid and granular flow.     

Axial dispersion of granular material in horizontal rotating cylinders

The discrete element method is used to calculate axial dispersion coefficients for approximately monosized particles in a rotating horizontal cylinder. Axial dispersion within the cylinder is shown to follow Fick's second law, in that the mean squared deviation of axial position of a pulse of particles is proportional to time. The axial dispersion coefficient is found to depend on the particle size, gravity and drum rotation speed, allowing a dimensionless group to be formed using these four quantities. For sufficiently large cylinders, the axial dispersion coefficient is found to be independent of drum diameter. A general argument is given which suggests that axial dispersion in physical beds of approximately monosized particles should follow Fick's second law.     

Numerical simulation of particle dynamics in different flow regimes in a rotating drum

Flow regimes in a horizontal rotating drum are important to industrial applications but the underlying mechanisms are not clear. This paper investigated the granular flow dynamics in different regimes using the discrete element method. By varying the rotation speed and particle-wall sliding friction over a wide range, six flow regimes were produced. The macroscopic and microscopic behaviour of the particle flow were systematically analysed. The results showed that the angle of repose of the moving particle bed had a weak dependence on the rotation speed in the slumping and rolling regimes, and increased significantly as the flow transited to the cascading and cataracting regimes. The mean flow velocity increased with the rotation speed, but the normalised velocity against the drum speed in the continuous regimes collapsed into a single curve, which can be well described by a log-normal distribution. The particle bed at low rotation speed had a similar density to those of the random loose packing, and became more dilated with the increase of the rotation speed. Similarly, the mean coordination number showed linear dependence on the drum speed. Both the collision energy and collision frequency increased with the rotation speed. However, the normalised collision energy in different regimes can be fitted with a simple scaling law.