Powder Technology 214 (2011) 431–442

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Powder Technology

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Experimental validation of polyhedral discrete element model

Stuart Mack a, Paul Langston b,⁎, Colin Webb a, Trevor York c

a School of Chemical Engineering and Analytical Science, University of Manchester, United Kingdomb Faculty of Engineering, Process & Environmental Division, University of Nottingham, United Kingdomc School of Electrical and Electronic Engineering, University of Manchester, United Kingdom

⁎ Corresponding author. Tel.: +44 115 951 4177; faxE-mail address: P.Langston@nottingham.ac.uk (P. La

0032-5910/$ – see front matter © 2011 Elsevier B.V. Alldoi:10.1016/j.powtec.2011.08.043

a b s t r a c t

a r t i c l e i n f o

Article history:Received 14 October 2010Received in revised form 24 May 2011Accepted 28 August 2011Available online 3 September 2011

Keywords:Non-spherical particleDEMPolyhedraComputationGranular materialsHoppers

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

: +44 115 951 4115.ngston).

rights reserved.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

1.1. Background

Granular materials are present in many forms, from large coalnuggets to tiny flour grains. The efficient handling and storage ofgranular materials have concerned engineers for many years. Hopperstorage and flow can be considered as an archetypal process for gran-ular materials; if this can be predicted then our general understand-ing of granular behaviour will be well established. Much of the earlywork considered the mass flow rate (Fowler and Glastonbury [1],Brown and Richards [2], Beverloo [3] and later Al-Din and Gunn [4],Humby et al. [5]) and theoretical predictions of the wall and internalstress during the filling and discharge from hoppers (Walker [6],Jenike [7], Walters [8,9], Enstad [10]). The correlations derived byJenike from shear tester methods were found to suggest larger hop-per outlets than are actually required (Enstad [9]) and most practicaldesign methods are based on theoretical-empirical approaches. Theincrease in computing power since the late 1980s has resulted inmany natural phenomena being simulated on computers, includinggranular material flow. The effect of particle shape on hopper dis-charge and stress patterns has been shown to be significant (Clearyand Sawly [11], Langston et al. [12]). The mass flow rate has been pre-dicted using the DEM (Anand [13]) and models have also been

proposed to predict the mass flow rate of non-spherical particles(Al-Din and Gunn [4]) and binary mixtures (Humby et al. [5]).

1.2. DEM

The discrete element method DEM describes a suite of numericalmethods that consider particles as discrete finite entities. This is incontrast to continuum methods such as Plasticity Theory in which in-dividual particle properties are averaged over a continuum. Cundalland Strack [14] applied the DEM to granular systems and a consider-able number of applications have resulted mainly utilising sphericalparticles in the DEM, for example in vibration induced flow in hoppers(Langston et al. [15]). The method has been considerably extended,tested and applied to various particle shapes including convex rectan-gles (Kohring et al. [16]), super-quadrics (Clearly and Sawly [11]),Sphero-discs (Li et al. [17]), cylinders with spherical ends (Langstonet al. [12]), cubes (Fraige et al. [18]), corn shaped particles (Tao et al.[19]), tetrahedra and octahedra (Wang et al. [20]). These studiesshow that particle shape can be very significant.

Latham et al. [21] describes how non-spherical particles can berepresented for ‘soft contact’ modelling using either combined finite-discrete element (FEM/DEM)methods, or non-spherical DEMmethodssuch as multi-sphere approximations of irregular geometry. O'Connoret al. [22] use the Discrete Function Representation (DFR) systemto determine contacts between irregularly shaped particles in 2Dusing an algorithm that processes information at successive increasingdifferent levels of detail: bounding sphere, bounding box, cellular re-gions, and surface facets. This leads to an efficient algorithm. The

Fig. 1. Polyhedral dice used in practical experiments. From top left to bottom right downeach column: a) two types of hexahedron; b) octahedron, c) icosahedron, d) pentagonaltrapezahedron, e) tetrahedron.

Table 1Polyhedra dimensions.

Particle No. Volume cm3 Density g cm−3 Side length cm

Pentagonal trapezohedra 90 2.73 1.65Octahedra 31 2.57 1.56 1.76Tetrahedra 29 1.05 2.19 2.08Icosahedra 81 5.02 1.27 1.32Hexahedra 91 3.83 1.17 1.56

The radius of curvature of the edges of the shapes was estimated to be 0.1 cm.

432 S. Mack et al. / Powder Technology 214 (2011) 431–442

Common Plane Method defines a virtual common plane that dissectsthe space between two particles, whereby the particle–particle contactdetection problem is simplified to an easy particle-plane contact detec-tion problem. Generally CPM needs a large number of iterations to ob-tain the common plane. Nezami et al. [23] proposed an improvedversion of CPM called fast common plane method (FCP). This reducedthe number of iterations significantly by limiting the search space ofthe common plane to a few candidates. The shortest link method(SLM) was developed by Nezami et al. [24]. Beyabanaki et al. [25] de-scribe an algorithm for determining contacts between two polyhedrawith planar boundaries. All six types of contact are transformed intothe form of point-to-face contacts. Relevant contact force algorithmsare also derived.

DEM is based on the principle that during a small time step, forcesand accelerations can be assumed to be constant, while the velocityvaries linearly. It is also assumed that within one time step, distur-bances cannot propagate from any particle further than its immediateneighbours. Choosing an appropriate time step is therefore crucial inmaintaining stability and accuracy in DEM simulations as well as forefficiency.

1.3. Objectives

This paper validates the application of a 3D DEM for modellingflow of mixtures of polyhedral particles in rectangular hoppers bycomparing computer simulations with controlled lab experimentsusing “real” particles. It also provides an insight into the underlyingphysics of polyhedral particle flow. The particles are modelled asnon-cohesive rounded polyhedra. Comparisons are made betweenthe experimental and simulated values of the flow rate, packingstructure and flow patterns at different hopper half angles formodel validation and to enhance our understanding of the underlyingphysics of polyhedral particle flow. The study also compares the poly-hedra flowwith the flow of spherical particles of a similar volume anddensity. Many researchers have applied DEM to simulate the flow ofspherical particles, hence the inclusion of spherical particles in thisstudy is to facilitate comparison with previous work. The critical ori-fice width at which arching occurs is investigated for the polyhedra.The experiments were recorded using a high speed video camera.

Section 2 details the equipment and the methods employed in thepractical experiments. The methods used in the simple experimentsto determine the input parameters for the DEM simulations are alsodescribed. In Section 3 a summary of the spherical and polyhedralDEM programs is described. In Section 4 the principal data used inthe DEM simulations are given and in Section 5 the results from theexperiments and computer simulations are presented and discussed.Section 6 provides a conclusion to the study and suggestions for fur-ther work.

2. Experiments

2.1. Test particles

Polyhedral dice, obtained from www.trollandtoad.com, were cho-sen to represent the polyhedral particles as shown in Fig. 1. A batchof 322 particles comprising a mix of five polyhedra types were usedto observe the main features of the particle flow and allow compari-sons to be made between simulation and experiment. These relativelylarge “granular” particles allow good visualisation and a reasonablylarge bench scale system for detailed DEM validation.

The dimensions of each polyhedral type are given in Table 1 andfurther dimensions of the pentagonal trapezohedron are shown inFig. 2. The mass of the particles was obtained by weighing ten of theparticles to an accuracy of 0.001%. The volumes of the polyhedrawere calculated from formulae by substituting in the length of one

of the sides. The calculated values were also tested experimentallyby water displacement.

Three hundred and twenty two “bubble gum balls” were used forthe comparisons of flow behaviour with the polyhedra in the samehopper. These particles, which will be referred to as spheres, were se-lected because they have a similar volume and density to the averagevolume and density of the 322 polyhedral particles and a uniformspherical shape. They also have very similar friction and dampingproperties to the polyhedral particles.

2.2. Experimental apparatus

A small 3D rectangular wedge shaped hopper shown in Fig. 3 wasused to conduct the experiments. This consisted of two parallel trans-parent sheets (101 cm×100 cm in size) that formed the front andback walls and provided a clear view of the particles. Four perspexstrips were inserted between the transparent sheets to form the hop-per and orifice and the vertical sides of the silo for a flat-bottomedhopper. The Perspex strips could be altered to adjust the hopperhalf angle from the vertical. The gap between the transparent boardswas set to 2.5 cm (hopper thickness) and when the hopper half anglewas set to 90° the vertical sides of the silo were set 45 cm apart. Twofurther Perspex strips were fixed at the bottom of the hopper to se-cure the transparent sheets and to provide support for the other Per-spex strips. The fixed strips were moved along a screw thread so thatthe orifice length could be adjusted. The orifice length was varied in1 cm intervals between 12 cm and 6 cm to obtain the critical orificeopening. It should be noted that the hopper thickness was just largerthan the largest polyhedra dimension. It was truly 3D in that the par-ticles could rotate about all axes and translate in all axes, albeit asmall distance in one. All the particles were clearly visible. This is use-ful for DEM validation, but of course not an ideal design for an indus-trial hopper. The particle-wall interaction is quite significant here, soparticle-wall as well as particle-particle system properties were care-fully measured.

1.67cm

1.32cm

2.21cm

This is total height from apex to apex

Fig. 2. Dimensions of the pentagonal trapezahedron.

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2.3. Coefficients of friction

The coefficient of sliding friction between the polyhedra was de-termined by placing a combination of ten different polyhedra inpairs on top of each other on a flat strip and inclining the surfaceuntil they started to slide. The sliding friction between the sphereswas determined by sticking ten combinations of four spheres to a sur-face and then placing another set of four spheres, that were attachedwith tape, on top of them and similarly inclining the surface until thetop layer of spheres started to slide. The particle-particle friction wasdetermined from Eq. (1). With the general level of assumptions usedin DEM it is not considered necessary here to distinguish betweenstatic and dynamic friction.

μ ¼ tan θs ð1Þ

The coefficient of sliding friction between the polyhedra and thewall was obtained by placing ten different particles in turn onto aflat strip of perspex and inclining them until they started to slide.The coefficient of sliding friction between the spheres and the wallwas determined by sticking ten combinations of four spheres together,placing them on a perspex surface and inclining the surface until theystarted to slide.

Rolling friction is modelled for the spheres. It is assumed that thisphenomenon is not significant for the polyhedra. The coefficient ofsphere-wall rolling friction was determined by applying a methodgiven by Ketterhagen et al. [26]. The spheres were placed at the topof a ramp and allowed to roll down the ramp onto a horizontal

Fig. 3. Experimental rig.

plane. The sphere-wall rolling friction was then determined usingthe equation

μ r;pw ¼ hD

ð2Þ

where h is the initial height of the sphere contact on the ramp and D isthe distance travelled by the sphere on the horizontal plane beforecoming to rest. The sphere–sphere rolling friction was determinedby applying the correlation given by Zhou et al. [27] as follows

θ0 ¼ 68:61� μ0:27s;pp � μ0:22

s;pw � μ0:06r;pp � μ0:12

r;pw � d−0:2 ð3Þ

θ ¼ θ0 1þαe−kw� �

ð4Þ

where μr,pp and μr,pw are the sphere–sphere and sphere–wall rollingcoefficients, μs,pp and μs,pw are the sphere–sphere and sphere–wallsliding coefficients, θ0 is the angle of repose of a heap of materialand d is the particle diameter (mm), w is the container thickness(mm) and θ is the angle of repose of the material between the con-tainer sides. Zhou et al. [27] deduced that α=1 and k=0.18/d.Zhou et al. [27] discuss the prospect that the rolling friction coefficientmay be proportional to particle diameter, so that thepower of d becomes−0.02. Applying the method developed by Ketterhagan [26] et al. dis-cussed earlier, the sphere–wall rolling friction coefficient of four differ-ent sizes of ball bearing was measured. Results (not shown here) fromsimple experiments conducted with ball bearings seem to suggest thatthe rolling friction coefficientmay indeed be proportional to the particlediameter. The angle of repose, θ, of the particles placed in the 3D hopperslicewas obtained by setting up the rig tomodel a flat bottomed hopper.The rig was allowed to drain and the angle from the horizontal to the in-cline of the particles left in the corners of the rig was measured. Thesphere–sphere rolling friction was obtained from the equation

μ r;pp ¼ θ

1þ e−0:18w

d

� �68:61� μ0:27

s;pp � μ0:22s;pw � μ0:12

r;pw � d−0:2

0@

1A−0:06

: ð7Þ

2.4. Coefficient of restitution

The particle–wall coefficient of restitution ewas obtained by drop-ping particles from an initial height h0, recording the rebound heighth1, and substituting the values into Eq. (8).

e ¼ffiffiffiffiffih1h0

sð8Þ

Ten different spheres and ten different hexahedra were droppedfrom a height of 50 cm. Only the hexahedra were chosen, because itwas difficult to drop the other shapes onto their faces, so that theywould rebound vertically. It is recognised that this is somewhat approx-imate for the polyhedra, but the results here are considered sufficientlyaccurate to test the DEMwith its inherent assumptions. Indeed it is con-sidered that the discharge phase is not sensitive to the contact damping.

2.5. Experimental procedure

Three hundred and twenty two particles consisting of six differenttypes of polyhedra were randomly poured into the hopper from thetop and allowed to free fall and come to rest on the bottom of thehopper as shown in Fig. 3. When the hopper filling was complete ahigh speed video camera, Phantom V710 was switched on. The orificeoutlet was opened and the particles started to flow out of the hopper.The video camera recorded images at 1000 frames/s. The same proce-dure was conducted with the 322 spherical particles.

434 S. Mack et al. / Powder Technology 214 (2011) 431–442

3. DEM models

3.1. Spherical particle flow model

The DEM technique uses an explicit time stepping approach to nu-merically integrate themotion of each particle from the resulting forcesacting on them at each timestep. The particle flowmodel follows a fairlystandard DEM approach for spheres. The inter-particle and particle wallcontacts aremodelled using the spring–dashpot–slider analogy. Contactforces are modelled in the normal and tangential directions with re-spect to the line connecting the particles centres. Particle cohesion canalso be modelled but this is not used in this study.

Rolling friction is modelled as an angular torque arising from theelastic hysteresis loss or viscous dissipation. The value of the particle–particle μpp,r and particle–wall μpw,r rolling friction coefficients is usuallydifferent. It has been shown that rolling friction is necessary in achiev-ing a simulation that accurately models the kinematics of spherical par-ticles (Zhou [27]).

The translational and rotational motion of each particle is mod-elled using a half step leap-frog Verlet numerical integration schemeto update particle positions and velocities. A zoning method is used toincrease the computational efficiency of determining particle contacts(Asmar et al. [28]).

3.2. Polyhedron 3D model

Fraige et al. [18] describe details of a DEM model for polygons. Thiswas further developed to model convex polyhedron shaped particlesin 3D (Wang et al. [20]). A summary is given here. The definition ofthe polyhedron particle modelled here is shown in Fig. 4. It should benoted that the vertices are “rounded”, and hence the edges are corre-spondingly rounded. This is to facilitate the model and to acknowledgethat real granular particles do not contain perfect vertices. The radius ofthe “vertex-spheres” can be selected by the user. This may also have aneffect on the timestep required for the simulations. The volume,

x y

z2vcr

r distance from edge of node to centre ref O

O

Fig. 4. Polyhedron particle definition. For each type of polyhedron the user specifies thenumber of vertex nodes in the particle (e.g. 4 for a tetrahedron) and vertex radius vcr.For each node the user specifies a radius r, azimuth angle α and elevation angle β fromthe particle reference centre at O. The angles are defined in a local axis system. The pro-gramme calculates the vertex node centre local co-ordinates: starting at (r–vcr, 0, 0) rotateabout y by −β then about z by +α. The user specifies the number of triangles whichdefines the particle surface (e.g. 4 for a tetrahedron) and identifies which 3 vertexnodes define each triangle. (The particle can be imagined as a series of triangles withcylindrical edges and spherical nodes.) The particle must be convex, that is the outerangle at each edge is always greater than 180°. A cohesion distance is defined surroundingthe outer surface (not shown). The material density is specified and the programmecalculates numerically the mass, centre of gravity (not necessarily at O) and momentof inertia tensor. For numerical efficiency in the particle–particle interaction model,the radius of a sphere centred on O which just encloses the particle and its cohesiveregion is calculated. In the current model the polyhedron does not have to be regularbut the local axesmust be principal axes of rotation (off diagonal elements of themomentof inertia tensor are zero).

moment of inertia tensor and centre of gravity are calculated numeri-cally by summation over small elements of the particle.

A significant difference herewith spherical particles is that: (a) therecan be “multiple contact points” detected in a particle–particle interac-tion; (b) the combined length and area of contact could be significantlygreater. This is illustrated in Fig. 5. For each contact point (red) themodel calculates the forces using the same equations as for the spheri-cal particles. Friction force is calculated at the contact point which hasthe largest solid overlap. That is, there is one friction force modelledper particle–particle interaction. This is a simplification but is consistentwith general simplifications in DEM. Rolling friction is not included;theremay be some rolling on the vertices, but this is not considered sig-nificant. There is no cohesion force in this study.

4. Modelling data

The principal simulation data are given in Table 2. These were cho-sen by obtaining an average value of measurements as described inSection 2. No parameters were selected to fit the subsequent results.The hopper geometry was set to be the same as in the experiment.The particles were dropped into the hopper in the simulation asshown in Fig. 6 and allowed to settle before the orifice was openedand discharge commenced. The filling method was chosen to be asclose as possible to the experiments, i.e. random particle sequence.

5. Results analysis and comparison

5.1. Comparison of DEM and experimental static packing

Fig. 7 shows a visual comparison between the DEM and experimentfor the static packing of the polyhedral particles in the hopper at hopperhalf angles α=30°, 60° and 90°. There is good agreement betweenpacking height. A range of particle orientations is observed in theDEM and experimental packings which qualitatively appear similar.Fig. 8 shows a comparison for the spheres. The spheres are shown topack in a uniform crystalline structure. This crystalline structure isshown to breakdownat the sides of the hopper in both theDEMand ex-periment as shown in Fig. 8c. It iswell known that a small polydispersity

Fig. 5. Multiple contact points. This illustrates the scenario where two polyhedra (blueand green) contact over a regionwhich cannot be described as a point contact. First figureis fromside inplane of contact, secondfigure shows plan viewof plane of contact. The areaof contact is estimated by calculating the average co-ordinates of the four contact pointsshown in red.

Table 2Principal hopper and particle data.

Parameter Value

Silo dimensions: height, width, thickness H=60, D=45, W=2.5 cmOrifice length B=12 cmOrifice width As hopperHopper half angle α=30°,60°,90°Number of particles N=322Particle dimensions a) For polyhedra see Table 1

b) r=1 cmParticle density a) For polyhedra see Table 1

b) 1.34 g cm−3

Coefficient of restitution a) 0.54, b) 0.59Time step 1.0×10−5 sFriction coefficientsSliding a) and b) 0.28Rolling(spheres) particle–wall 0.019particle–particle 0.0005

a) Polyhedra and b) Spheres.A “softened” particle stiffness (normal and tangential) of 5×107 dyn cm−1 was used—

see Section 6.2.

435S. Mack et al. / Powder Technology 214 (2011) 431–442

can disorder the structure, but since the primary objective here is ex-perimental comparison, the mono-sized particles were used.

5.2. Experimental flowrate

Particle flowrate at the orifice is analysed by examining therecorded images of the particles discharging from the three anglesα=30°, 60° and 90° for both the polyhedra and the spheres. Experi-mental tests of the polyhedra and spheres were conducted threetimes at each of the hopper half angles. Fig. 9 shows the fraction ofpolyhedral particles discharged from the hopper against time t.Apart from the case at angle α=30°, where arching and flow block-age of the polyhedra occurred, particles discharge smoothly. The par-ticles are shown to accelerate to a steady ratewithin the first 0.1 s whenthe orifice is opened. Fig. 10 shows the fraction of spheres dischargedfrom the hopper against t. The spheres are shown to flow out of the

a

b

Fig. 6. DEM simulation of a) hopper filling an

hopper with a uniform velocity in the 30° and 60° hopper angle cases.In the 60° hopper angle case the flow rate slightly decreases in the last0.2 s before the hopper is empty. In the 90° hopper angle case theflow rate is uniform for the first 1.5 s and then decreases as the particlesin the corners start to flow out. The flow rate of the polyhedra in the 30°hopper is relatively linear. The video shows them “jostling for position”in the hopper as they flowout. In the 60° and 90° hoppers the polyhedraand spheres are observed to exhibit “core flow” in which a central fun-nel of polyhedra are initially shown to flow out of the hopper. Polyhedraand spheres either side of the funnel collapse and join the flow in thecentral core. In the 30° case the polyhedra and spheres exhibit “massflow”. There is a greater variation in the results from the three trialsof the polyhedral flow than the flow of the spheres.

Averaged discharge of the three trials of the polyhedra and spheres isshown in Fig. 11. In the 90° hopper the polyhedra discharge at a very sim-ilar rate to the spheres and more of the particles accumulate in the cor-ners. In the 60° hopper the polyhedra flow 20% faster than the spheres.The 30° hopper case ismore complex; the flowrates are similar, howevera blockage occurs during one of the polyhedral particle flow cases.

5.3. Comparison of DEM and experimental flow behaviour

Fig. 12 shows a comparison of the DEM and experimental flowrate for the polyhedra averaged over 3 cases for each. Very goodagreement is shown for the 90° and 60° cases. The 30° cases are dif-ferent due to the blockage that occurred in one of the experiments.Blockages are considered later. There are some slight differences be-tween the DEM and experimental results, this is most likely due tothe non-linear and stochastic nature of the contacts and dischargepatterns, resulting from the numerous combinations of particle pack-ings that are available with the six different types of 322 dice.

Simulation and video images of the particle flow patterns are shownin Figs. 13–15 for the 30°, 60° and 90° cases respectively. In the case ofangle α=30° the packing top profile is different after 0.5 s, as it wasprior to discharge, but after 0.8 and 1.0 s there is a close similarity be-tween the experiment and simulation. Similarly the 60° hopper shows

d b) discharging:3D and 2D projections.

a

b

c

Fig. 7. Hopper filled with polyhedral particles, on the left DEM simulation and on the right practical experiment, for hopper half angle a) 30°, b) 60°, and c) 90°.

a

b

c

Fig. 8. Hopper filled with slightly polydisperse spheres, on the left DEM simulation and on the right experiment, for hopper half angle a) 30°, b) 60°, and c) 90°.

436 S. Mack et al. / Powder Technology 214 (2011) 431–442

Fig. 9. Experimental discharge dynamics of the polyhedra.

437S. Mack et al. / Powder Technology 214 (2011) 431–442

differing top profiles at start and after 0.5 s and 0.8 s (not shown here)but a similar profile after 1.0 s. The 90° hopper case shows a slightly fas-ter flow in the experiment compared to the simulation and a greaternumber of particles in the experiment piling up at the corners. After1.0 s experiment and simulation look similar.

A comparison of the DEM and experimental flow rate for thespheres is shown in Fig. 16. These show good agreement as per previ-ous studies in the literature.

5.4. Comparison of sphere and polyhedra flow from DEM data

Fig. 17 shows a comparison of the results from the DEM simula-tions of the flow of the polyhedra and volume equivalent sphereswith identical friction and damping and rolling friction. A comparisonof polyhedra and equivalent sphere flow is complex here. Generallythe polyhedra flow slightly faster than the equivalent spheres. This

Fig. 10. Experimental discharge dynamics of t

is probably due to inter-planar surface sliding. However, for the 90°hopper the flow is slower towards the end and more particles areretained in the corners as expected. Polyhedra also have a muchgreater tendency to blockage as discussed later.

5.5. Sensitivity of polyhedra flow to friction

The coefficient of friction between the polyhedra was assumed tobe identical in the DEM simulations for all combinations of particlecontacts. It is considered that this parameter has the greatest uncer-tainty (5%), hence some simulations were undertaken to test the sen-sitivity of the results to friction. Fig. 18 shows the variation in the flowrate of the polyhedra at five different coefficients of friction. The coef-ficient of friction for the dice was estimated to be 0.28 as describedbefore. Reducing this to 0.18 increases the discharge rate by about20%, but increasing to 0.38 makes little difference. (The higher friction

hree experimental trials for the spheres.

Fig. 11. Comparison of average experimental discharge dynamics between the polyhedra and the spheres.

438 S. Mack et al. / Powder Technology 214 (2011) 431–442

gives a very small increase in discharge at some points — this is prob-ably stochastic.) Reducing to 0.08 or increasing to 0.48 changes theflowrate by about a further 10%. It is considered that differences infriction could account for some of the small differences observed be-tween experiment and simulation. If DEM is to be used in a predictivecapacity friction should be carefully considered.

5.6. Critical orifice diameter

Blockages occurred in the polyhedra experiments in the hop-per angle α=30°. The orifice length was set to 12 cm in theDEM and experimental comparisons so that a relatively unim-peded flow could be established in the hopper. Experimental

Fig. 12. Comparison of DEM and average ex

tests were conducted at angles α=30°, 60° and 90° to deducea critical orifice width at which hopper discharge was unlikely.DEM simulations were also conducted at different values of theorifice length. Fig. 19 shows the blockage that occurred in a hop-per with angle α=30° and orifice length 10 cm for experimentand DEM. The arching structure is similar in these cases.Table 3 shows the orifice lengths estimated from these experi-ments and simulations:

• B0 at which it can be reasonably certain that complete dischargewill NOT occur

• B100 at which it can be reasonably certain that complete dischargewill occur

perimental discharge for the polyhedra.

a

b

c

Fig. 13. Polyhedra discharging at hopper half angle 30° a) t=0.5 s, b) t=0.8 s, and c) t=1.0 s.

439S. Mack et al. / Powder Technology 214 (2011) 431–442

The experimental results are similar in magnitude and trend to thesimulation, but they extend over a slightly wider range. The criticalorifice length generally increases as the hopper half-angle decreases.From these simulations and experiments it appears that many repli-cations are needed to obtain accurate statistical probabilities of com-plete discharge for intermediate orifice sizes which could be thesubject of a further study.

6. Conclusions

6.1. DEM validation

The primary objective of the paper is to help validate the recentpolyhedral DEM model by close comparisons with carefully con-trolled experiments. Previous studies have validated DEM with

Fig. 14. Polyhedra discharging at h

spheres and some other shapes. Comparisons have been made herebetween DEM simulations and practical laboratory scale experimentsof 322 non-cohesive large particles in a small 3D hopper at anglesα=30°, 60° and 90°. Polyhedral particles and spheres have been ex-amined. Individual particle properties were measured prior to thesimulations. No model parameters we subsequently changed to fitthe experiment. The static packing height, flow rate and critical orificelength were compared and visual inspections were made of the pack-ing structure, flow patterns and blocked arch structure. In all casesDEM simulations showed good general agreement with experiment.This gives increased confidence in use of such a model for predictingthe behaviour of granular materials which are polyhedral-like inshape. There are some differences between the top surface profilesin the DEM simulations and experiments. This is mostly likely dueto the inability to exactly replicate the filling method between the

opper half angle 60° t=1.0 s.

Fig. 15. Polyhedra discharging at hopper half angle 90° t=1.0 s.

440 S. Mack et al. / Powder Technology 214 (2011) 431–442

experiments and the simulations, however this has not greatly affect-ed the flow rate.

6.2. Effect of particle parameters

A contentious point in DEM is how accurately the normal force ofparticle interaction should be modelled. The use of the “soft” linearspring to model the particle contacts is shown to be accurate in pre-dicting the flowrate and flow patterns of polyhedral particles andfor spheres. Similarly this study shows that other approximationsfor the polyhedral model described in the text are not critical fornon-cohesive particles. It is considered that particle friction is themost important particle surface property here. Some simulationsshowed that the discharge rate of the polyhedra is moderately sensi-tive to the friction coefficient.

6.3. Effect of particle shape

Polyhedra flow is generally slightly faster than the spheres. Thismay be because the polyhedra have many flat surfaces which enablethem to slide past each other more easily. Slow motion videos of theflowing polyhedral particles show their ability to pack more tightlyand slide past each other more easily than the spheres. However,polyhedra have a greater tendency for blockage. The likelihood of ablockage occurring is greater for the hopper with angle α=30°,

Fig. 16. Comparison of DEM and average e

however more polyhedral particles, than spheres, pile up in the cor-ners in the 90° hopper half angle case.

6.4. Further work

Further work should consider the affect of cohesion between theparticles. Ideally various coatings should be experimentally testedand the results compared with DEM simulations. This could also beused to change the friction properties. This would enhance our under-standing of some important physics for granular flow and give a morerigorous test of the DEM technique. Further modelling techniquesshould be employed to improve efficiency. Currently the full particle–particle interaction algorithm is employed at each time step. Utilisingparallel programming techniques should also enable simulation of alarger number of particles. All these tasks would help in the generaldevelopment of DEM for “real” particles alongside other techniquesbeing developed as summarised in Section 1. A key question to con-sider is what level of detail is required to reproduce the importantphysics.

NotationB orifice lengthd particle diameterD distance travelled by sphere on horizontal plane in Eq. (2)e coefficient of restitution

xperimental discharge for the spheres.

Fig. 17. Comparison of DEM polyhedra and DEM sphere discharging. Total volume and mass of spheres is the same as for the polyhedra.

441S. Mack et al. / Powder Technology 214 (2011) 431–442

h height of sphere contact on ramp in Eq. (2)h0 initial drop heighth1 rebound heightr radius of spherical particlesw container thickness

Greek letters

α hopper half angle

Fig. 18. Comparison of DEM polyhedra simulations at hopper angle 60° for v

θ angle of repose of material between container sidesθ0 angle of repose of a heap of materialθs angle of inclined surface in Eq. (1)μ coefficient of sliding friction in Eq. (1)μr particle–particle or particle–wall rolling frictionμr,pp particle–particle rolling frictionμr,pw particle–wall rolling frictionμs particle–particle or particle–wall sliding frictionμs,pp particle–particle sliding frictionμs,pw particle–wall sliding friction

Acknowledgements

The authors greatly acknowledge the support of the Engineeringand Physical Sciences Research Council for funding the project andto the EPSRC instrument pool for the loan of the high speed camera.

ariation of the friction coefficient (particle–particle and particle–wall).

Fig. 19. Polyhedral dice blocked in hopper with half angle 30° and orifice length 10 cm. a) Experiment, b) DEM.

Table 3Estimates of critical orifice lengths for flow.

α (deg) B0 (cm) B100 (cm)

Experiment 30 8.0 13.0⁎

60 7.0 12.090 6.0 11.5

DEM 30 8.5 11.560 8.5 11.090 7.0 10.5

B0 to prevent complete discharge.B100 for complete discharge (90° ignore particles left in corners).⁎ Extrapolated.

442 S. Mack et al. / Powder Technology 214 (2011) 431–442

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