Simulation Of Granular Flows Research Articles

Using the Web to increase the availability of DEM-based mill modelling

Discrete element method (DEM) simulation of granular flows has attained a degree of maturity that is consistent with application to the modelling of commercial-scale processes. Modelling of tumbling mills (SAG/AG or ball) was one of the first industrial applications of DEM and continues to be a major focus of technology developers. Despite this, industry makes limited use of mill modelling. This is partly due to issues of confidence in the technology and barriers to access resulting from the often prohibitive cost of obtaining and running DEM simulation packages. In an attempt to overcome some of these problems, a Web-based interface to CSIRO’s DEM capabilities has been developed. This interface, referred to as webGF- Mill ®, allows users to specify mill simulations that are then performed promptly using CSIRO software and computing facilities. Results are provided to users electronically and can be employed to better inform decisions about new designs or how best to operate an existing mill. Apart from making the benefits of DEM mill modelling more readily available to industry, it is hoped that webGF- Mill ® will play a significant role in fostering interaction between the users and developers of DEM technology, and consequently increase its value to commercial mill operators.

Modeling and direct simulation of near-field granular flows

A wide range of modern applications have emerged where a successful analysis requires the simulation of flowing granular media which simultaneously incorporates near-field interaction between charged particles and intergranular contact. Such systems naturally arise in powder processing, micro- and nanotechnology, as well as in applications arising from the study of aerosols, epitaxy and astro- and geophysics. The focus of this communication is to develop models and physically-based high-performance solution strategies for the direct rapid simulation of such flowing granular media.

Partitioned Distinct Element Method Simulation of Granular Flow within Industrial Silos

A method for performing discrete element simulations of granular flow and pressures within industrial silos is presented. Special attention is devoted to complex problems involving large numbers of particles and sophisticated boundary conditions due to the presence of inserts. The proposed method consists of partitioning the silo into layers that are analyzed sequentially, and in determining stresses and velocities at the virtual interlayer boundaries. The method is first validated by simulating the discharge of a single insert hopper containing 20,000 particles, performing both a simulation of the whole silo and a multilayer partition. The results show a small discrepancy in the displacement fields produced by the two simulations. Then the discharge of an industrial silo containing 170,000 particles with several inserts of different size and shape is simulated. The relevance of the stress and velocity fields obtained confirms the feasibility and the efficiency of the procedure. The method allows for managing huge numbers of particles with a limited memory capacity and a gain of computational time that may be significant depending on each particular case.

Strategies for contact resolution of level surfaces

Particles in granular matter can have very different and irregular shapes. The computational treatment of nonspherical objects is a major difficulty in the simulation of granular flows. In this paper, two basic strategies for contact resolution between objects described by level surfaces are presented and analyzed. They are based on the iterative solution of systems of nonlinear equations. The major difficulties are pinpointed and necessary steps toward a generic algorithm are proposed. A test case of colliding cardioids in two dimensions is used to demonstrate the algorithms and illustrate common pitfalls.

An accurate tangential force–displacement model for granular-flow simulations: Contacting spheres with plastic deformation, force-driven formulation

An elasto-plastic frictional tangential force–displacement (TFD) model for spheres in contact for accurate and efficient granular-flow simulations is presented in this paper; the present TFD is consistent with the elasto-plastic normal force–displacement (NFD) model presented in [ASME Journal of Applied Mechanics 67 (2) (2000) 363; Proceedings of the Royal Society of London, Series A 455 (1991) (1999) 4013]. The proposed elasto-plastic frictional TFD model is accurate, and is validated against non-linear finite-element analyses involving plastic flows under both loading and unloading conditions. The novelty of the present TFD model lies in (i) the additive decomposition of the elasto-plastic contact area radius into an elastic part and a plastic part, (ii) the correction of the particles' radii at the contact point, and (iii) the correction of the particles' elastic moduli. The correction of the contact-area radius represents an effect of plastic deformation in colliding particles; the correction of the radius of curvature represents a permanent indentation after impact; the correction of the elastic moduli represents a softening of the material due to plastic flow. The construction of both the present elasto-plastic frictional TFD model and its consistent companion, the elasto-plastic NFD model, parallels the formalism of the continuum theory of elasto-plasticity. Both NFD and TFD models form a coherent set of force–displacement (FD) models not available hitherto for granular-flow simulations, and are consistent with the Hertz, Cattaneo, Mindlin, Deresiewicz contact mechanics theory. Together, these FD models will allow for efficient simulations of granular flows (or granular gases) involving a large number of particles.

Integration schemes and damping algorithms in distinct element models

Abstract Computational run times in distinct element method (DEM) simulations of granular flow can be large and limit the size of the system being modeled. Hence, it is important to use efficient numerical integration schemes. This paper investigates some numerical integration schemes for their accuracy, stability and computational efficiency. It also investigates the effect of different particle contact damping algorithms on the model mathematical accuracy and stability. It is shown that the half-step leapfrog Verlet algorithm is the best integration scheme, while Euler is poor in terms of accuracy. Non-linear damping has been shown in the literature to be more realistic in terms of experimental data on particle impact coefficient of restitution. This was reproduced here. This paper also shows that no n-linear damping reduces the mathematical error in the integration scheme because the force change is less discontinuous. However, in particle assembly simulations, filling a hopper, the non-linear damping model was less stable, probably because less energy is dissipated at low velocities.

Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes

The distinct element method (DEM) has proven to be reliable and effective in characterizing the behavior of particles in granular flow simulations. However, in the past, the influence of different force–displacement models on the accuracy of the simulated collision process has not been well investigated. In this work, three contact force models are applied to the elementary case of an elastic collision of a sphere with a flat wall. The results are compared, on a macroscopic scale, with the data provided by the experiments of Kharaz et al. (Powder Technol. 120 (2001) 281) and, on a microscopic scale, with the approximated analytical solution derived by Maw et al. (Wear 38 (1976) 101. The force–displacement models considered are: a linear model, based on a Hooke-type relation; a non-linear model, based on the Hertz theory (J. Reine Angew. Math. 92 (1882) 156) for the normal direction and the no-slip solution of the theory developed by Mindlin and Deresiewicz (Trans. ASME. Ser. E, J. Appl. Mech. 20 (1953) 327) for the tangential direction; a non-linear model with hysteresis, based on the complete theory of Hertz and Mindlin and Deresiewicz for elastic frictional collisions. All the models are presented in fully displacement-driven formulation in order to allow a direct inclusion in DEM-based codes. The results show that, regarding the values of the velocities at the end of collision, no significant improvements can be attained using complex models. Instead, the linear model gives even better results than the no-slip model and often it is equivalent to the complete Mindlin and Deresiewicz model. Also in the microscopic scale, the time evolution of the tangential forces, velocities and displacements predicted by the linear model shows better agreement with the theoretical solution than the no-slip solution. However, this only happens if the parameters of the linear model are precisely evaluated. The examination of the evolution of the forces, velocities and displacements during the collision emphasizes the importance of correct accounting for non-linearity in the contact model and micro-slip effects. It also demonstrates how these phenomena need to be considered into the model in order to perform deeper analyses on granular material in motion and, in general, for systems sensitive to the actual force or displacement. For these cases, more accurate models such as the complete Mindlin and Deresiewicz model should be addressed.

Stability Analysis of Collisional Granular Flow on a Slope

We analyze the linear stability of a collisional granular flow on a slope under gravity using hydrodynamical equations based on kinetic theory of inelastic particles. It is shown that the steady, uniform flow is unstable against longitudinal long-wavelength perturbations in lower density region. The results are compared with the instabilities found in numerical simulations of granular flows.

Order parameter description of stationary partially fluidized shear granular flows.

We carry out a detailed comparison of soft-particle molecular dynamics simulations with the theory of partially fluidized shear granular flows. We verify by direct simulations a constitutive relation based on the separation of the shear stress tensor into a fluid part proportional to the strain rate tensor, and a remaining solid part. The ratio of these two components is determined by the order parameter. Based on results of the simulations we construct the "free energy" function for the order parameter. We also present the simulations of the stationary deep 2D granular flows driven by an upper wall and compare it with the continuum theory.

Energy monitoring in distinct element models of particle systems

This paper describes techniques for monitoring energy in discrete element method (DEM) simulations of granular flow. They have been applied to DMX a three-dimensional model of polydisperse cohesive spheres flowing into a cylindrical vessel, settling and then subject to vibration. The model takes account of gravitational potential energy, linear and tangential ‘particle spring’ potential energies and net work done by the particles, normal and angular kinetic energies, dissipated energies due to linear and tangential damping and friction, and the work done by the vibrating vessel on the particle system. Energy monitoring enhances understanding of the physics and further validates the program code. It was found that the numerical technique inherently introduces artificial energy components, but that these can be explicitly monitored. Energy conservation was thus verified and the artificial components explained. Simulations of various particle types and sizes were performed monitoring all the energy components with time. The results show that explicit dissipated energy calculation is required and cannot be simplified as the remainder term of total minus potential and kinetic energies, and that energy is dissipated mainly in normal damping and gross sliding. Total energy dissipation is not sensitive to particle stiffness, but moderately sensitive to damping and friction. However, the maximum rate of energy dissipation is significantly affected by the damping coefficient and the particle stiffness, and only negligibly by the friction coefficient. Initial studies showed that in some low energy vibration the artificial energy component is not negligible and its effect must be considered in some DEM applications.

A method to extract the mechanical properties of particles in collision based on a new elasto-plastic normal force–displacement model

We present a general method to extract the elasto-plastic mechanical properties such as the coefficient of restitution, Young’s modulus, Poisson’s ratio, and the yield stress of granular materials from simple experimental measurements. These mechanical properties are important in the simulation of granular flows, but are not readily available and/or cannot be measured through direct experiments. The method developed here is based on the new elasto-plastic normal force–displacement (NFD) model [presented in Proceedings of the Royal Society of London, Series A 455 (1999b) 4013]. Using the Vu-Quoc and Zhang NFD model to simulate both the dropping test and the compression test of a particle, the mechanical properties of the particle can be extracted by minimizing, in the least-square sense, the difference between the experimental and simulated force–displacement relation. An application of the proposed method to soybeans is presented.

DEM modelling of industrial granular flows: 3D case studies and the effect of particle shape on hopper discharge

While the discrete element method (DEM) is attracting increasing interest for the simulation of industrial granular flow, much of the previous DEM modelling has considered two-dimensional (2D) flows and used circular particles. The inclusion of particle shape into DEM models is very important and allows many flow features, particularly in hoppers, to be more accurately reproduced than was possible when using only circular particles. Elongated particles are shown here to produce flow rates up to 30% lower than for circular particles and give flow patterns that are quite different. The yielding of the particle microstructure resembles more the tearing of a continuum solid, with large-scale quasi-stable voids being formed and large groups of particles moving together. The flow becomes increasingly concentrated in a relatively narrow funnel above the hopper opening. This encourages the hope that DEM may be able to predict important problems such as bridging and rat-holing. Increasing the blockiness or angularity of the particles is also shown to increase resistance to flow and reduces the flow rates by up to 28%, but without having perceptible effect on the nature of the flow. We also describe our methodology for constructing and modelling geometrically complex industrial applications in three dimensions and present a series of industrially important three-dimensional (3D) case studies. The charge motion in a 5 m diameter ball mill and in a Hicom nutating mill, discharge from single- and four-port cylindrical hoppers, and particle size separation by a vibrating screen are demonstrated. For each case, plausible particle size distributions (PSDs) have been used. The results obtained indicate that DEM modelling is now sufficiently advanced that it can make useful contributions to process optimisation and equipment design. Finally the parallelisation of such a DEM code is described and benchmark performance results for a large-scale 2D hopper flow are presented.

Numerical Aspects of FE Simulations of Granular Flow in Silos

Some important numerical aspects concerning the finite-element simulation of granular flow are presented. The main topic of this paper is the so-called “switch” phenomenon, which is a stress peak at the junction of the vertical and the inclined part of a mass flow silo during its discharge. Case studies prove that the FE simulation of silos and simplified systems should be handled with great care, as many wall pressure peaks occurring at places where the shape of the silo is changing may well be caused by numerical corner singularities. Other numerical aspects lead to consequences for the filling process and the opening phase of silo emptying.

Normal and tangential force–displacement relations for frictional elasto-plastic contact of spheres

In this paper, we present the data obtained from numerical experiments that are at the foundation of our successful elasto-plastic force–displacement (FD) models in both the normal direction and tangential direction of spheres in collision. We also present in detail how these data were obtained. Our numerical experiments are in the form of frictional elasto-plastic finite-element analysis of spherical particles in contact. We use the numerical experiments to observe the behavior of various quantities involved in the contact (e.g. contact-area radius, contact curvature, stresses, FD relations, etc.). This observation has helped us to devise an accurate and consistent model for frictional elasto-plastic contact of spherical particles that are used in granular-flow simulations that may involve tens of thousands of these particles.

Computational studies of granular mixing

Particulate systems have proven difficult to probe experimentally in many instances. Simulations of granular flows, and mixing flows in particular, provide a useful means of studying particulate behavior. Mixing flows generate large scale patterns and structures which can be easily visualized. Thus, mixing studies provide a means of indirectly examining granular flows. In this paper we review recent computational studies of tumbler mixing, focusing on two very different, yet complementary, techniques: Particle Dynamics and Lagrangian Simulation. We discuss mixing in different tumbler geometries, as well as segregation and cohesive effects.

An elastoplastic contact force–displacement model in the normal direction: displacement–driven version

We present an elastoplastic normal forcedisplacement (NFD) model for spheres in collision. Among several possible applications of this model are the granularflow simulations using a discreteelement...

Comparison of Kinetic Theory and Discrete Element Schemes for Modelling Granular Couette Flows

Discrete element based simulations of granular flow in a 2d velocity space are compared with a particle code that solves kinetic granular flow equations in two and three dimensions. The binary collisions of the latter are governed by the same forces as those governing the discrete elements. Both methods are applied to a granular shear flow of equally sized discs and spheres. The two dimensional implementation of the kinetic approach shows excellent agreement with the results of the discrete element simulations. During change to a three dimensional velocity space, the qualitative features of the flow are maintained. However, some flow properties change quantitatively.

A Normal Force-Displacement Model for Contacting Spheres Accounting for Plastic Deformation: Force-Driven Formulation

In this paper, we present a simple and accurate model for the normal force-displacement (NFD) relation for contacting spherical particles, accounting for the effects of plastic deformation. This NFD model, based on the formalism of the continuum theory of elastoplasticity, is to be used in granular flow simulations involving thousands of particles; the efficiency of the model is thus a crucial property. The accuracy of the model allows for an accurate prediction of the contact force level in the plastic regime. In addition to being more accurate than previously proposed NFD models, the proposed NFD model also leads to more accurate coefficient of restitution that is a function of the approaching velocity of two particles in collision. The novelty of the present NFD model is the additive decomposition of the contact-area radius, and the correction of the curvature of the particles at the contact point due to plastic flow. The accuracy of the proposed model is validated against nonlinear finite element results involving plastic flow in both loading and unloading conditions. [S0021-8936(00)03102-0]

Stability of the three‐dimensional Coulomb friction law

It is known that problems arise over existence and uniqueness of solution in quasi–static contact problems involving large coefficients of Coulomb friction. This paper considers the behaviour of an elastically supported mass with three translational degrees of freedom that can make contact with a rigid Coulomb friction support. A critical coefficient of friction is identified, above which the quasi–static solution can be non–unique. A numerical solution of the corresponding dynamic problem shows that the state realized then depends on the initial conditions. Even below the critical coefficient of friction, the dynamic solution can deviate from the quasi–static even for arbitrarily small loading rates. Typical dynamic responses include limit–cycle oscillations in velocity, oscillations involving a brief period of zero velocity (stick) and ‘stick–slip’ motion in which the mass spends significant periods in a state of stick, interspersed with short rapid–slip periods. All these non–steady motions involve non–rectilinear motion, even in cases where the time derivative of the applied load is constant in direction. A perturbation analysis is performed on the quasi–static frictional slip solution and predicts instability for certain slip directions, depending on functions of the off–diagonal stiffness components and the coefficient of friction. These results are presented in dimensionless form in stability diagrams. It is also shown that quasi–static slip that is stable when there is no damping can be destabilized if the damping matrix has sufficiently large off–diagonal components.

The importance of computer simulations of granular flow

The importance of the technological applications of granular media in silos or conveyer belts, in ball mills or under bulldozers, demands more efficient simulation techniques to reproduce entire industrial processes. Although several commercial programs for the simulation of granular material exist, their use is not nearly as common as in fluid or solid mechanics. This is mainly due to historical reasons and to limitations in the number of simulated particles. The advancement of computer hardware and the rapid advance in simulation algorithms ensure that particle simulations will become much more commonplace in power technology. Other areas will certainly profit from this process in computational granular materials, such as geosciences and astrophysics. One beautiful example is the simulation of the transport of particles under the action of wind, and the subsequent formation of dunes.