Design and Engineering
Application of the Discrete Element Method in Materials Handling Part 1: Basics and Calibration T. Gröger and A. Katterfeld, Germany The potentials of the Discrete Element Method (DEM) to further optimise not only the design of bulk materials handling equipment but also to vastly improve the engineering process itself are considered to be enormous. This paper, the first of a series of four papers in total, desribes the basics of the DEM and is focused on determination of the material properties. The goal of this series is to describe the current potentials and limitations of the DEM.
I
n the last years an increasing interest of materials handling industries in the numerical simulation of bulk solids could be noticed. In this process the comparatively new methods for the simulation of particles of bulk materials receive particular interest. The available computational power allows to run numerically experiments using these methods on PCs. Such simulations can be used to optimise and design machines and apparatuses and to perform research work on the basics of particle processes. Among the direct particle methods the Discrete Element Method (DEM) belongs to those, which come very close to the nature of bulk solids. Because of the large potential for the modelling of processes in bulk solid industries Discrete Element Simulations became popular in the past years. This quadripartite series of papers tries to communicate the potentials and the limitations of DEM. The following paper explains the basics of the Discrete Element Method and concentrates on the determination of material properties. Three more articles in this journal talking about screw conveyors, bulk material transfer
stations and about bucket elevators and scraper conveyors will demonstrate the usage and the current limitations of DEM for the application in materials handling.
1
Introduction
A homogeneous bulk material seen from some distance appears to be a continuum with properties showing similarities to solids, liquids and gases. For instance, bulk solids that are stored in a silo or conveyed on a conveyor belt are subjected to small deformations and small shear rates. Under these conditions the material exhibits a comparatively high density and the shear stresses are proportional to the normal stresses. For this reason numerous continuum mechanical models for bulk solids are based on the theories of elasticity and plasticity. Discharging bulk solids from the silo or off the belt conveyor is accompanied by a loosening process reducing the density of the ma-
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terial. When this material is guided or deflected by chutes or walls it will deform under high shear rates, which invalidate the continuum mechanical models based on the solid-like state. Since, under these conditions, the shear stresses become proportional to the shear rates it is possible to derive models from the fluid mechanics and from the kinetic gas theory. Fig. 1:
Fig. 2: Example of a contact model for spherical particles; spring: elastic force-displacement-law, dashpot: viscous damping law, frictional element: Coulomb friction, meniscus: liquid bridge (attractive force)
Rotating drum with a consolidated, solid-like range (range radially outwards from blue particles to the drum) and a fast flowing, liquid-like still (red)
For free-flowing materials the transition between these states can occur spontaneously, e.g. by applied oscillations. Unfortunately, currently no continuum mechanical model is known that describes all states of granular matter with the same quality. However, there are numerous machines and processes at which two or three of these states occur simultaneously. Representatively, Fig. 1 shows the behaviour of a particle system in the rotating drum. In contrast to the above described behaviour cohesive materials do not tend to fluidise but usually they exhibit shear localisations characterised by large shear gradients, which also cause problems for continuum mechanical models. Looking somewhat closer the causes of the peculiarities of bulk solid materials can be recognized. Bulk solids are systems of particles linked by short ranged potentials, i.e. a particle is only influenced by its neighbouring particles or by walls. Because of the comparatively small bonding forces particles can easily be detached, which interrupts and changes the flux of forces. On the microscopic scale fluidisation and shear localisation cannot be recognized directly but the macroscopic states of granular matter manifest themselves by the contact duration. Obviously, the complex behaviour that bulk solids exhibit on the macroscopic scale is caused by comparatively simple but repeatedly applied microscopic interaction laws, which remain constant if the atomic conditions do not change quantitatively. This fundamental property can be used by the methods for direct particle simulations, among which the Discrete Element Method is the numerically most expensive but also the most realistic representative. Subsequently this method is explained in principle. For a more detailed description it is referred to further literature [1, 2].
2
The Principle of the Discrete Element Method
The Discrete Element Method was developed by C (1979) and a lot of detailed descriptions have been published ever since. Therefore only a brief survey will be given here. For algebraic modelling, the particles of bulk solids need to be represented by well defined geometrical objects. For performance reasons,
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spheres or sphere conglomerates are preferred. The particles themselves are assumed to be rigid however they are allowed to overlap. These overlaps are regarded as contact deformation from which an elastic contact force arises. Dependent on the applied contact model (Fig. 2) other types of contact forces can contribute to the total contact force. Accumulating all contact forces on a particle delivers the resulting force and moment for this particle. With the mass and the momentum of inertia the Newtonian equation can be integrated for a very short time step. This places a particle onto its new position and hence a new contact detection has to be performed as existing contacts may have vanished or new contacts may have formed. The described cycle needs to be executed in a loop until the desired process time is reached.
3
Determination of Particle and Contact Properties
3.1
Elastic Contact Properties
In the simplest case the elastic contact deformations can be modelled by a linear spring law. However, for spherical particles a Hertzian law is more appropriate. Only in very rare cases where the real particles exhibit a spherical shape Young’s modulus and Poisson’s ratio of the solid material can be used directly. If more complex particles are modelled by spheres this simplification needs to be compensated by a calibration of the contact law. For geo-mechanical applications the particle stiffness is adjusted by means of numerical triaxial tests with the goal to fit a measured macroscopic stress-strain-curve. With models of very coarse geo-materials numerically stable simulations can be achieved with the realistic stiffness and the realistic masses. For quasi-static processes it is often applicable to up-scale volumes and/or masses in order to achieve numerical stability. Unfortunately, the majority of processes from the field of materials handling and process engineering exhibits both fast flow regimes and comparatively small particles, which do not allow a mass or volume scaling. In order to obtain numerical stable time steps that enable a reasonable computing time (less than a month for most consulting jobs), though, the particle stiffness needs to be reduced. For instance, large scale simulations on high-end PC’s require the stiffness of minerals to be decreased by a factor of 100 or higher. Therefore, it is currently not possible to calibrate the particle stiffness for the majority of applications
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from process engineering and materials handling. It is recommended to choose the particle stiffness as high as the overall computational time allows it.
3.2
Damping
Very often the size of the simulated particles is large enough that global damping effects of the surrounding medium can be neglected. For fine particles or surrounding fluids an appropriate damping law can be applied if needed. However, it is essential for most cases of handling and processing of bulk solids to consider the contact damping. Usually, contact damping is modelled in dependency on the relative velocity of the contact partners and occasionally dependent on the contact deformation. Except for nearly spherical particles that enable the measurement of the rebound height of a dropped particle no experiments are known that could be used for a calibration procedure. Practically relatively high contact damping coefficients are required. It is noted that higher damping forces can be achieved for a larger contact stiffness.
3.3
Fig. 3:
Yield loci obtained from simulated shear tests; the inclination is a measure for the macroscopic friction; (intersection with the ordinate) was caused by liquid bridges
Fig. 4:
Examples for an offset of the contact force from the centre of mass
Coulomb Friction and Rolling Friction
In process engineering and materials handling the macroscopic friction angle of bulk solids is of particular importance. Besides cohesion, friction determines the flow properties of a particulate material significantly. Simultaneously, it is one of the most complex parameters since macroscopic friction is the result off particle friction and rolling friction on the microscopic level as well as the particle shape, the standard deviation of the particle size distribution, the packing structure and the packing density. In general, shear tests are performed numerically and experimentally in order to compare the inclination of the yield loci, which is a measure of the macroscopic friction. Fig. 3 shows examples of simulated yield loci. It is evident that the particle shape has a considerable influence on the macroscopic friction angle. Unfortunately, the depicted particles composed of a number of spheres demonstrate two disadvantages. Firstly, with an increasing number of primary spheres the computational effort increases, too, and secondly in sections the particles can roll without any resistance. Therefore, it can be of advantage to introduce a rolling resistance (moment) that arises from an offset of the contact force from the centre of mass as depicted in Fig. 4. Fig. 6:
Different experimental methods for the investigation of the angle of repose and their numerical representation
Fig. 5:
Simulated macroscopic friction angle [°] dependent on the particle (Coulomb) friction μ [-] and rolling friction μr [-] for spheres with diameter d = 2.3 mm to 2.6 mm in a shear tester
There are a number of factors that can be responsible for the force offset, such as the deformation due to rolling (Fig. 4 left), the particle shape (Fig. 4 center) and asperities on the surface of the particles (Fig 4 right), as well. These effects can all be covered with the coefficient of rolling friction, which is multiplied with the particle radius to obtain the amount of the offset (lever of the force).
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the determination of the two unknowns. Therefore the Institute of Logistics and Materials Handling (ILM) investigates further methods of measuring the macroscopic friction. Currently, the angle of repose formed in a rotating drum as well as formed by a vertical cylinder is being investigated (Fig. 6). Although no results can be presented, yet, it is reasonable to assume that in the process of forming the angel of repose the coefficients of friction and rolling friction have a differently weighted influence compared to shearing a consolidated system. This will lead to diagrams of the same type shown in Fig. 5. However different gradients are expected to be apparent. Hence, overlaying two of these diagrams should deliver an intersection at the desired macroscopic friction coefficient that delivers the pair of frictional coefficients that is representative for the majority of flow conditions. The described procedure is numerically expensive and further research is needed to find short cuts for the calibration process.
3.4
Fig. 7:
Influence of micromechanical effects on the macroscopic tensil strengths and the particle size
Fig. 5 shows the influence of the particle friction coefficient and the rolling friction coefficient on the macroscopic friction of a particulate system that is subjected to direct shearing in a Jenike shear cell. Obviously, the same macroscopic friction can be obtained from different combinations of rolling friction and particle friction (e.g. along the lines between two colored areas). Since it is desirable to find the pair of coefficients that is valid for all flow conditions, regardless if it is a slow shearing or a fast flowing material, a single type of experiment seems to be insufficient for Fig. 8:
Model of the pendular liquid bridge between spherical particles
Cohesion
Macroscopic cohesion may arise from a number of microscopic causes, such as Van-der-Waals-Forces and liquid bridges (Fig. 7). The attractive forces on the microscopic level are comparatively well investigated and several mathematical models exist, which can be embedded in a contact model used for DEM-simulations. Considering the relatively high tensile strength of wet bulk solids, moisture is one of the most important factors causing cohesion. Even for small degrees of saturation of less than S = 5 % pendular liquid bridges will emerge that cause a cohesion in the order of 10 kPa for a particle size of d < 100 µm. Below the contact law for modelling liquid bridges and its application to numerical shear tests is presented. The tensile force resulting from liquid bridges can be computed at the gorge of the bridge by: FLB = π · R22 · Δp + 2 · π · R2 · γ
(1)
With the capillary pressure
(
1 − __ 1 Δp = γ · __ R1 R2
)
(2)
and the principal radii R1, R2 (Fig. 8) as well as the surface tension γ, Eq. (1) can be rewritten to:
(
R2 FLB = π · γ · R2 · __ +1 R 1
)
(3)
The principal radii R1, R2 dependent on the separation distance a and the filling angle β and need to be computed iteratively during the simulation. The failure limit of the approach can be expressed by the critical separation distance ac. L et al. [4] have found the relation 3 ac = ( 1 + _12_ · δ ) · VLB _1_
(4)
by solving the exact equation of a liquid bridge numerically. In this equation δ means the wetting angle at the three face boundary and VLB as the volume of the liquid bridges. In order to calibrate and test the contact model direct shear tests can
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be performed. By means of yield loci (shear stress vs normal stress) the simulation results can be compared with the results of experiments on Jenike-shear-testers or Ring-sheartesters. Fig. 9 shows a suitable model to simulate direct shear tests. The shear plane is limited by periodic boundaries, which enable the simulation of endless particle assemblies. For this particles leaving the model on one side will be reinserted on the opposite side with the same velocity. In contrast to real shear testers there are no side walls influencing the results. The normal load is applied by two plane walls which are controlled by a servo algorithm. The actual shear process is conducted by moving the walls in opposite direction with a constant velocity. From such a simulation the macroscopic stress tensor σij can be calculated by means of
the size of the liquid bridges was deduced from the water content determined by drying experiments. For the elastic contact law the parameters of glass, namely Young’s modulus and the Poisson ratio were taken from tables. The surface roughness of the particles and the frictional coefficient of particle-particle contacts are calibrated by means of the yield loci. Fig. 11 shows that the simulated yield loci match well with the yield loci obtained from experiments. The straighter curve produced by the simulation can be explained by the ideal sphericity of the particle model compared to a slight nonsphericity of the real glass beads. Fig. 3 shows that bended yield loci can be simulated in principle by using non-spherical particles.
From Fig. 7 it can be deduced that moisture, which is the main cause for caking of Fig. 9: DEM-model for the simulation bulk solids in conveyors and apparatuses, of direct shear tests becomes relevant for particle sizes below where M is the number of contacts, V is the 1 mm. If this particle size were considered considered volume, di is the vector connecting for the simulation of real industrial applications, the number of particles and hence the computational the particle centres and Fi is the contact force. time would grow to an extent that is not of interest for practical As an example Fig. 10 shows the simulated curve of the normal purposes. In order to solve such problems it is conceivable to stresses and the shear stresses as well as the volumetric dilatadevelop hybrid models that allow, to describe the caked matetion dependent on the shear strain. Because of the over-consolirial by continuum mechanical assumptions. dated conditions of the sample the shear stress reaches a maximum at which the plastic deformation starts. By means of the dilatation curve it can be recognized that the particle system Fig. 10: Curves of the normal and shear stresses as well as the volumetric dilatation for an over-consolidated sample loosens during the shear process. This reduces the shear resistance of the particle system which causes the shear stress to reach the steady value. M · d ·F +d ·F σij = ____ j i) 2·V ( i j (5)
The maximum of the shear stress in the related normal stress deliver one point on the yield locus. In order to obtain further points of the same yield locus shear tests with the same consolidation state at different normal loads need to be performed. For each experiment this requires a new sample preparation and a new pre-shear test. In contrast to this the same dataset of the initial model can be used with varied normal loads for the DEMsimulations. In Fig. 3 the simulated yield loci are shown for two wetted particles systems with different particle shapes. For the system consisting of 3-atomic particles the yield locus for dry conditions is also depicted. Referring to this yield locus the curve of the identical, wetted system is shifted by the amount of the cohesion towards higher shear stresses. Other than in real shear tests the numerical model can easily be subjected to tensile forces. The yield loci reveal that the curvature can be extended to the tensile stress range, initially. Shortly before the failure limit is reached the curves bend sharply in order to meet the point of the Tensile strength. For a quantitative assessment of the shear simulations real shear tests have been performed with wet glass beads (d = 684 µm, see appendix) by means of a ring shear tester [3]. For the simulation
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been discussed and their influence of the flow behaviour was explained. Currently, not all parameters can be calibrated to represent the properties of particulate systems realistically. In case of the elastic properties this is caused by the limitations of the available computational power. In other cases, such as contact damping and friction, fundamental experimental methods for the determination of these properties are still to be developed.
References Fig. 11:
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[1]
Cundall, P.A. and Strack, O. D. L.: A discrete numerical model for granular assemblies. Geotechnique, 29 (1979) 1, pp. 47-65.
[2]
Gröger, T.: Schüttgutmechanische Untersuchungen zur senkrechten Schlauchgurtförderung. Logisch GmbH, Magdeburg, 1999, ISBN3-930385-21-X.
[3]
Gröger, T., Tüzün, U. and Heyes, D.M.: Modelling and measuring of cohesion in wet granular materials. Elsevier, Powder Tech. 133, (2003), pp. 203-215.
[4]
Lian, G., Thornton, C. and Adams, M.J.: A theoretical study of the liquid bridge forces between two rigid spherical bodies. J. Colloid Interface Sci. 161 (1993), pp. 138–147. ■
Comparison of yield loci obtained from simulations and experiments
Conclusion
Discrete element simulations are an interesting tool for consulting and research concerned with bulk solid materials. The development of realistic models for the simulation of bulk solids requires the calibration of the model parameters by means of experimental data. Several microscopic parameters used for the direct simulation of particulate systems, such as powders and bulk solids have Table A1: Parameters of the simulations
poly-disperse spheres
3-/4- atomic particles
Particle size
dm = 684 μm, σ = 93 μm
d = 30 … 40 μm
Coefficient of friction
m = 0.05
μ = 0.3
Diameter of primery particles
–
da = 20 μm
Density of glass
ρs = 2.5 g/cm3
ρs = 2.5 g/cm3
Particle mass
mm = 0.42 mg
m = 30 … 40 mg
Young’s modulus of glass
E = 70 GPa
E = 70 GPa
Poisson’s ratio of glass
υ = 0.2
υ = 0.2
Volume of liquid bridges
V = 5.6 · E-4 mm3
V = 10 μm3
Wetting angle
δ=0
δ=0
Surface tension of water
γ = 72 mN/m
γ = 72 mN/m
Time step
Δt = 10 ms
Δt = 1 μs
Factor of mass scaling
103
105
Particle shape
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About the Author
Dr. Torsten Gröger Dr. Torsten Gröger studied Mechanical Engineering at the Otto-von-Guericke University, Magdeburg, Germany, and specialised in continuous conveying of bulk materials under guidance of Prof. Friedrich Krause. He received his Ph.D. with his work on DEM simulations of vertical pipe conveying (Discrtete-Elemente Simulationen zur senkrechten Schlauchgurtförderung). For one year, Dr. Gröger worked as a Post-Doc with Prof. Jürgen Tomas (Institute for Mechanical Processing, Magdeburg) and for three years with Prof. U. Tüzün (Institute for Chemical and Process Engineering, University of Surrey, United Kingdom). During his work for Itasca Consultants GmbH he gained imporant experience in the execution of industry projects on basis of DEM simulations. Dr. Gröger is the Managing Director and co-founder of CeParTec GmbH. Contact: CeParTec GmbH Dr.-Ing. Torsten Gröger Uhlandstrasse 4, 39108 Magdeburg, Germany Tel.: ++49 (0) 2327 96 56 48 Fax: ++49 (0) 2327 96 56 49 E-Mail:
[email protected]
About the Author
Dr. André Katterfeld Dr. André Katterfeld currently works as a Scientific Assistant and Project Manager at the Institute for Conveying- and Construction Machinery Technology, Steel Construction and Logistics at the Otto-von-Guericke University, Magdeburg, Germany. After receiving his Ph.D. with his work on the Functional Analysis of Pipe Chain Conveyors (Funtionsanalyse von Rohrkettenförderern), he took part in numerous research projects about the application of computer simulations on basis of the Discrete Element Method in the area of bulk material conveying technology. Dr. Katterfeld is co-founder of the CeParTec GmbH. Contact: Otto-von-Guericke Universität Magdeburg Dr.-Ing. André Katterfeld Institut für Logistik und Materialflusstechnik Universitätsplatz 2, 39106 Magdeburg, Germany Tel.: ++49 (0) 391 67 12245 Fax: ++49 (0) 391 67 12518 E-Mail:
[email protected]
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