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.     

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.     

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。     

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.     

Discrete thermal element modelling of heat conduction in particle systems: Pipe-network model and transient analysis

In our recent work [Y.T. Feng, K. Han, C.F. Li, D.R.J. Owen. Discrete thermal element modelling of heat conduction in particle systems: basic formulations. Journal of Computational Physics. 227: 5072-5089, 2008], a novel numerical methodology, termed the discrete thermal element method (DTEM), is proposed for the modelling of heat conduction in systems involving a large number of circular particles in 2D cases. The method cannot be easily extended to transient analysis, which causes difficulties in combining the DTEM with the conventional discrete element method for modelling thermal/mechanical coupling problems in particle systems. This paper presents a simplified version of the DTEM, termed the pipe-network model, in which each particle is replaced by a simple thermal pipe-network connecting the particle centre with each contact zone associated with the particle. The model essentially neglects the direct heat transfer between the contact zones and thus significantly simplifies the solution procedure of the original DTEM. With this feature, transient heat conduction analysis can now be performed in a straightforward manner. In addition, the entire algorithmic structure of the pipe-network model is compatible with the discrete element method, leading to an effective scheme for simulating thermal-mechanical coupling problems. Numerical experiments are conducted to establish the solution accuracy of the proposed model.     

Large-scale numerical investigation of solids mixing in a V-blender using the discrete element method

Over the past few years, the discrete element method (DEM) has been used in models for the simulation of granular flows in various mixing applications. If these models have shown rather efficient, they have so far been applied to predict the behavior of small numbers of particles over limited spans of time. The objective of this work is to show that DEM-based models can be used to predict the flow behavior of large numbers of particles over large spans of time and, more particularly, mixing phenomena that take time to manifest in such systems. To this end, several large-scale DEM-based numerical investigations of the flow of monodisperse and bidisperse blends of up to 225 000 particles over a span of 120 s in a V-blender will be discussed using entities such as the particle velocity and granular temperature, the torque of the mixing system, RSD curves and mixing times.     

A soft-sensor approach to flow regime detection for milling processes

Due to the many industrial applications of rotating drums, a wide range of operating conditions, including different particle flow regimes, are used. Knowledge of the flow regimes inside a drum is beneficial for process optimisation and control. This paper shows how the unique insights provided by a discrete element method (DEM) model of a rotating drum can be used to create soft-sensor models that detect flow regime. Impacts between particles and the drum wall are simulated, from which the feature variables are extracted. A soft-sensor model which links these feature variables to flow regime is constructed using the multivariate statistical technique of Fisher discriminant analysis (FDA). This model is able to successfully classify new testing data, which are not used in soft-sensor model training, as belonging to rolling, cascading and cataracting flow regimes.     

An analytical solution of different configurations of the linear viscoelastic normal and frictional-elastic tangential contact model

In the current study an analytical solution describing the impact of a spherical particle on a rigid wall is derived. The contact is linear viscoelastic in normal and frictional-elastic in tangential direction. Due to its simplicity, the model combination considered is one of the most common in the framework of the Discrete Element Method especially for large-scale simulations. The linear viscoelastic normal model is realized according to Zhang and Whiten [1996. The calculation of contact forces between particles using spring and damping models. Powder Technology 88, 59-64.] assuming a contact to be ceased when the normal force attains a value of zero. In literature the frictional elastic tangential model is employed in three different configurations following 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] and 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. Kluwer Academic Publishers, Dordrecht, 1998, p. 313]. The differences among these configurations are briefly explained, whereas the focus is set on the two most accurate model formulations given in the first two papers. All important final collision properties as positions and velocities are derived in an analytical form. Based on these results a comparison with experimental data of particle/wall and particle/particle collisions 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] and Gorham and Kharaz [2000. The measurement of particle rebound characteristics. Powder Technology 112(3), 193-202] is performed, showing very good agreement especially for the model configuration of Di Maio and Di Renzo. The analytical solution as derived here helps understanding the complex collision process and can be applied for the evaluation of integration methods or in the context of an event-driven discrete element method.     

Utilization of a Radial Temperature Model for Heat Conduction within Spherical Particles to Model Granular Assemblies

A heat transfer (DEM) model for application in the particle based discrete element simulation method is presented. It utilizes an analytical solution of the heat diffusion equation for a solid spherical particle to obtain temporal and radial solutions of the temperature distributions within the particles. This radial temperature model avoids the shortcomings of the usual assumption of spatially uniform temperature profiles in particles. The concept is designed to minimize computing power and memory requirement in order to allow the computation of granular assemblies consisting of a large number of particles. Results obtained for a particle subject to transient convective boundary conditions are compared with a Crank‐Nicholson implicit scheme as numerical reference solution. A first implementation of the radial temperature model in a discrete element code reveals the additional computational cost as negligible compared to the demands of contact identification and force calculation.     

Simulation of flow and drying characteristics of high-moisture particles in an impinging stream dryer via CFD-DEM

Impinging stream dryer (ISD) is an alternative for drying high-moisture particulate materials. Due to the complex multiphase transport phenomena that take place within an ISD, use of a reliable computational model instead of a tedious experimental route to aid the design of the dryer is desirable. In the present study, computational fluid dynamics were used in combination with the discrete element method (CFD-DEM) to predict, for the first time, the multiphase transport phenomena within a coaxial ISD; results from a model that does not consider particle-particle interactions (CFD) were also obtained and compared with those from the CFD-DEM model. In all cases, high-moisture particles having negligible internal transport resistance were assumed. Both models were used to simulate the gas-particle motion behavior, particle mean moisture content, particle mean residence time, and particle residence time distribution. The simulated results from both models were compared with the experimental data whenever possible. The results showed that the CFD-DEM model could be utilized to predict the particle motion behavior and led to more physically realistic results than the CFD model. The CFD-DEM model also gave predictions that were in better agreement with the experimental mean particle residence time and moisture content data.     

Experimental validation of polyhedral discrete element model

The flow of polyhedral granular particles in a small 3D slice hopper is studied experimentally and computationally by applying the discrete element method (DEM). A high speed camera was used to obtain the experimental results. The experimental packing structure, flow behaviour, arching and discharging in the hopper are analysed and compared with the DEM results for three hopper half angles. Reasonable agreement is shown on the static packing, flow behaviour and hopper discharge rates. The critical orifice length at which flow ceases to be smooth is investigated and arching of the material around the orifice is demonstrated experimentally and computationally. Spherical particles of nearly identical volume and density to the average of the polyhedral particles are also tested and compared to the polyhedra. The DEM is shown to be reasonably adept at modelling the interactions between polyhedral particles in a system in which there are very many possible particle geometrical interactions. Further work should consider the cohesion between the particles and the particle and the wall. Simulations of a greater number of particles in different hopper geometries should also be explored.     

DEM simulations of the particle flow on a centrifugal fertilizer spreader

Usually, the performance of centrifugal spreaders must be evaluated in large halls by capturing the fertilizer distribution patterns in standardized tests, often carrying a big cost to the manufacturers. In contrast, this paper proposes a first attempt to model a particle flow subjected to a spinning disc using the Discrete Element Method (DEM) starting from the particle outflow of a bin, using flat as well as inclined discs. The model is validated by experiments in two different ways. The first manner is the measurement of the cylindrical mass distribution along the edge of the disc by a device that collects the fertilizer particles in a tray of baskets around the disc. A second method consists of collecting the particles on the ground after their ballistic flight through the air. Both validation methods are relatively cheap and fit into the present statistical or qualitative interpretation of DEM simulations. Additionally, a number of rotational disc speeds is chosen (300-650 rpm) to incorporate velocity dependent effects of the particle flow. It was found that the DEM simulations show a good qualitative and considerable quantitative agreement with the experiments. The deviations between the simulations and experiments are profound at high disc rotational speeds (500-650 rpm) and can be identified as (1) an underestimation of the simulated particle velocities at the edge of the disc and (2) a too low dispersion on the (vertical) simulated particle velocities at the edge of the disc. A parameter study revealed that (1) can be resolved by introducing a velocity dependent friction coefficient, in agreement with literature. The influence of other model parameters such as particle damping and stiffness appears to be small, while the introduction of a rolling friction coefficient to mimic rolling resistance or particle shape does not provide any answer either, and hence reason (2) at this moment must be addressed to unknown external factors such as disc plane vibrations appearing at higher disc speeds.     

Comparison of discrete elemental modelling to experimental data regarding mixing of solids in the transverse direction of a rotating kiln

Finnie et al. [Finnie, G.J., Kruyt, N.P., Ye M., Zeilstra, C., Kuipers, J.A.M., 2005. Longitudinal and transverse mixing in rotary kilns: a discrete element method approach. Chemical Engineering Science 60, 4083-4091] developed a discrete elemental model to simulate the mixing of solids in a rotary kiln. It is interesting to analyse and compare their results to those obtained through experimental observations such as those by Henein et al. [Henein, H., Brimacombe, J.K., Watkinson, A.P., 1983. Experimental study of transverse bed motion in rotary kilns. Metallurgical Transactions B 14B, 191-205] Van Puyvelde et al. [Van Puyvelde, D.R., Young, B.R., Wilson, M.A., Schmitd, S.J., 1999. Experimental determination of transverse mixing kinetics in a rolling drum by image analysis. Powder Technology 106, 183-191; 2000. Modelling transverse mixing in a rolling drum. Canadian Journal of Chemical Engineering 78, 635-642] and Mellmann [Mellmam, J., 2001. The transverse motion of solids in rotating cylinders—forms of motion and transition behavior. Powder Technology 118, 251-270].     

Derivation and validation of a binary multi-fluid Eulerian model for fluidized beds

The paper presents a multi-fluid Eulerian model derived from binary kinetic theory of granular flows, free path theory and an empirical friction theory. The effects of the inter- and inner-particle collisions, particle translational motions and particle–particle friction are included. As the effects due to fluiddynamic particle velocity differences and particle–particle friction are considered, some unconventional terms are produced compared with the previous models. Model validation using the data from Mathiesen et al. (2000) shows that the coupling terms give a stronger and more realistic particle–particle coupling because the effects due to the fluiddynamic velocity differences are considered. The model gives reasonable predictions of the particle volume fraction, particle velocities and velocity fluctuations. The model analysis reveals that the basic particle velocity fluctuations constitute 2 terms: the velocity fluctuations of the discrete particles, and the velocity fluctuations of the continuous fluid flow. Furthermore, the simulation results show that the velocity fluctuations of the continuous fluid flow are dominant in a binary riser flow.