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Contributions to theexperimental validation of the

discrete element methodapplied to tumbling mills

Andrew McBride, Indresan Govender, Malcolm Powelland Trevor Cloete

Department of Mechanical Engineering, University of Cape Town,Cape Town, South Africa

Keywords Discrete manufacturing, Experimentation, Simulation

Abstract Accurate 3D experimental particle trajectory data, acquired from a laboratorytumbling mill using bi-planar X-ray filming, are used to validate the discrete element method(DEM). Novel numerical characterisation techniques are presented that provide a basis forcomparing the experimental and simulated charge behaviour. These techniques are based onfundamental conservation principles, and provide robust, new interpretations of charge behaviourthat are free of operator bias. Two- and three-dimensional DEM simulations of the experimentaltumbling mill are performed, and the relative merits of each discussed. The results indicate that inits current form DEM can simulate some of the salient features of the tumbling mill charge,however, comparison with the experiment indicate that the technique requires refinement toadequately simulate all aspects of the system.

IntroductionSemi-autogenous and autogenous milling have become an integral componentof modern mining operations. These processes allow large volumes of raw rockfeed to be processed efficiently and cost-effectively. Current semi-empiricalmethods for the design of mills are based largely on data obtained from pilotand full scale plant operations. While these methods are highly successfuland currently indispensable, they provide little insight into the mechanics ofthe charge motion and scale poorly as one moves away from the window ofoperating conditions in which they were formulated. The discrete elementmethod (DEM) (Cundall and Strack, 1979) is a promising numerical tool capableof simulating the complex dynamic particle motion and interactions withintumbling mills. It is envisioned that DEM will be eventually used inconjunction with empirical methods to better optimise the mill design process.

Prior to this occurring, however, DEM must be rigorously validated from itsmost fundamental level upwards. This paper presents and applies several

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This work is part of the AMIRA P9M project. Professor Doubell and the radiographers atTygerberg hospital are thanked for their assistance.

Contributions tothe experimental

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Received February 2003Revised July 2003

Accepted July 2003

Engineering ComputationsVol. 21 No. 2/3/4, 2004

pp. 119-136q Emerald Group Publishing Limited

0264-4401DOI 10.1108/02644400410519703

techniques for such a validation. Three-dimensional particle trajectory dataobtained from a laboratory mill are used to validate DEM and a series of robustalgorithms are developed to characterise the charge, allowing meaningfulcomparisons to be made between the numerical and simulated data.

The ability of DEM to accurately simulate the behaviour of experimentaltumbling mills has been investigated by several other researchers, withpromising results. The following is a brief, but by no means complete, overviewof the DEM validation to date in the field of milling: Cleary and Hoyer (2000)demonstrated good agreement (in terms of power draw and visual snapshotcomparisons of the charge motion) between the experimental and 2D DEMsimulations of a centrifugal mill. Agrawala et al. (1997) and Rajmani et al.(2000) presented similar comparisons between the experimental tumbling millsand DEM simulations thereof. The ability of 2D DEM to model the dynamics ofan instrumented experimental mill at start-up was established by Monamaand Moys (2002) by comparing the power drawn in both experimental andsimulated mill. The position of the centre of circulation (CoC) (Powell andNurick, 1996) was used by Govender et al. (2001a) to compare 2D DEMsimulations with the experimental particle trajectory data from a scaletumbling mill. More recent comparisons of 2D and 3D DEM with anexperimental SAG mill using the position of the shoulder and toe of the chargeas well as the CoC were presented by Cleary et al. (2003). While the powerdrawn by a mill provides a consistent measure of the mill behaviour, visualcomparisons of charge motion will always introduce some measure ofsubjectivity.

None of the validation techniques presented in the literature are sufficientlyrigorous to claim that DEM can simulate all aspects of the system correctly.Validating DEM against one aspect of a complex system, such as a mill, doesnot imply that the model adequately describes the full system. The hypothesisof the authors is that a series of validation techniques (a validation toolbox) andobjective comparisons are required to ensure that all features of the system areadequately simulated. This work contributes several techniques towards sucha validation toolbox.

Laboratory ball millThe 3D trajectory history of a single particle within the bulk charge of alaboratory mill is recorded using an automated tracking technique andbi-planar X-ray filming (Govender et al., 2001b). The 142 mm diameter, 12 lifterPerspex mill is filled with approximately 4,000, 6.1 mm diameter plasticspheres constituting the charge (Table I) for the measured experimental millspecifications and particle properties. A medical diagnostic tool, the bi-planarangiograph, is used to digitally film the mill with X-rays in two planessimultaneously at a sampling rate of 50 frames/s and a shutter speed of1/3,000 s for a duration of 67 s per condition (Figure 1). The tracked particle isone of the bulk charge particles coated with a thin layer of silver paint, causing

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it to attenuate more radiation and thus appear darker on the X-ray images.The digital images are processed using a fully automated imaging technique,which locates the 3D coordinates of the marked particle to within 0.2 mm. Theuncertainty of 0.2 mm is guaranteed for the mill speeds used in this work, i.e. upto a maximum measured particle velocity of 1.06 m/s. The 3D coordinates of thetracked particle provide the statistically significant and accurate data requiredfor the rigorous validation of DEM.

The behaviour of a single isolated particle within a tumbling mill has beeninvestigated by Dong and Moys (2002) using multi-exposed photographs witha stroboscope as the light source. The behaviour of the single isolated particleallows the value of the coefficient of restitution and friction to be determined forsubsequent incorporation into a DEM simulation. Dong and Moys’s work islimited to 2D, and requires the user to discern the particle trajectory path froma photograph thereby limiting the amount of the experimental data that can begathered from an experiment, and would not be able to capture the behaviourof a particle within the bulk charge.

The non-invasive positron emission particle tracking (PEPT) technique(Parker et al., 1997) has similar capabilities to the bi-planar X-ray filming

Figure 1.Experimental mill

within the bi-planarangiographic equipment

Mill length (mm) 142Mill internal diameter (mm) 142Lifter height (mm) 9Lifter width (mm) 9Lifter angle (8) 60Mill filling ( per cent volume) 40Particle density (kg/m3) 780Mean particle diameter (mm) 6.1Mill speed (rpm) 61 (test series 1)

71 (test series 2)

Table I.Measured experimentalmill specifications and

particle properties


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method employed in this work in that the trajectory path of a single positronemitting tracer particle within the bulk charge can be determined. PEPT isbased on the detection of nearly collinear gamma rays emitted during theprocess of positron decay and subsequent annihilation of the positrons withelectrons within the tracer particle or surrounding material. The back-to-backgamma rays are detected using a positron camera consisting of two gamma raydetectors and the position of the annihilation event determined usingtriangulation. In practice, the position of the annihilation event is determinedusing multiple gamma ray pairs. The uncertainties in the positional location ofthe tracer particle and the detection frequency are dependent on variousfactors, including the velocity of the tracer. A tracer moving at 1 m/s can belocated within 5 mm 250 times per second while a tracer moving at 0.1 m/s canbe located within 2 mm 25 times per second (Parker et al., 1997). The maximummeasured particle velocity in the current test series is 1.06 m/s. The accuracy ofthe X-ray filming method employed in this work is therefore approximately 25fold greater than what would be obtainable using PEPT. An advantage of thePEPT method compared to the bi-planar X-ray filming is the extended durationover which the tracer is tracked; approximately 1-2 h compared to 67 s in thiswork. The PEPT method has been used by various researchers for the purposeof validating DEM (Stewart et al., 2001; Yang et al., 2003).

Validation algorithmsA series of validation algorithms were developed to allow meaningfulcomparisons to be made between the trajectory history of the tracked particlewithin the experimental mill and the trajectory histories of ten randomlyselected particles within the DEM simulations. The validation algorithms allowthe behaviour of a system of near identical particles to be inferred from that ofa single particle or multiple particles. The duration over which the markedparticle is tracked must be sufficient to allow the particle to pass through allregions of the charge and thereby provide a representation of the bulk chargebehaviour.

Bin algorithmsTo provide a statistically meaningful means of comparison between theexperiment and simulation, a probability distribution function of particleposition within the mill is generated using a binning algorithm (Govender et al.,2001a).

The cross section of the mill is uniformly divided into a fine grid (50 £ 50cells) where each cell represents a bin. All data points within the individualbins are grouped together. The normalised count of experimental points fallingwithin each bin represents the probability function of particle position withinthe mill. The bin algorithm allows the frequency of any measured variable tobe expressed as a function of position within the mill. The velocity andacceleration of the tracked particle is determined from the experimental

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trajectory data using a second order Lagrange interpolation polynomial, finitedifferencing scheme (Chapra and Canale, 1989). The same finite differencingscheme is used to determine the accelerations of the tracked particles from theirsimulated values of velocity in the DEM simulation.

Bin plots of a selected variable in both experiment and DEM simulationscan be subtracted from one another to give an indication of the relative error;a bin difference matrix containing only zero values would indicate a perfectmatch.

Locating key features of the bulk chargeThe identification of unique features of the charge allows direct comparisons tobe made between the experimental and DEM data. Two such features are theCoC and the equilibrium surface (Powell and Nurick, 1996). Powell and Nurickdefined the CoC as the point about which all the charge in the mill circulatesand the equilibrium surface as the surface dividing the ascending, en massecharge from the descending charge. The process of identifying the CoC and theequilibrium surface was via visual inspection of photographs of the mill(Figure 2) and X-ray trajectory plots of a single tracked particle, and thereforeintroduced operator bias. Cleary et al. (2003) used the concept of the CoC(termed the vortex centre or centre of recirculation) and the positions of theshoulder and toe of the charge to compare the charge motion in a scale mill withDEM simulations. The position of the CoC (determined via visual inspection ofstreak images) was shown to provide the most sensitive measure of theaccuracy of the DEM simulation.

Figure 2.Photograph of anexperimental mill

showing the CoC and theequilibrium surface


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This work provides rigorous definitions for the CoC and equilibrium surfaceusing basic conservation principles, allowing their positions to be evaluatedobjectively using automated numerical techniques.

The principle of conservation of mass flow within a closed system is used toidentify the CoC and equilibrium surface. Consider the cross section of the millshown in Figure 3 with the tracked particle’s trajectory data superimposed.The cross section of the mill is sectioned using horizontal, vertical and radialplanes, termed control surfaces, whose normals lie in the plane of the crosssection. The mill is currently analysed as if it were a 2D problem in the X-Yplane bisecting the length of the mill. End effects due to the interaction of theparticles with the front and back ends of the mill and other possiblelongitudinal position dependent effects are not quantified in this work.The number of particle trajectories crossing each control surface, the location ofthe intersection point (defined as the point where a particle’s trajectory pathintersects the control surface) and the relative sense of the crossing s arerecorded for the duration of the analysis. The relative sense of crossing isdefined as follows:

s ¼ sgnð ~d†~nÞ

~n ¼ ~nz £$

AB

Figure 3.Schematic sectioning ofthe mill using variouscontrol surfaces andthe mass flux countperformed along aselected control surfaceto determine the positionof the mass fluxequilibrium point

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where ~d is the particle trajectory vector, ~n the normal vector to the controlsurface, ~nz a vector defining the mills axis of rotation, and

$

AB a vectororthogonal to ~n in the XY plane (Figure 3). The definition of the normal vectorto the control surface ~n is dependent on whether the mass flux count isperformed from A to B or from B to A.

The cumulative count of particles crossing the control surface (representingthe mass flux f along the control surface) is performed for each control surfacefrom both A to B and from B to A. The point of maximum mass flux along acontrol surface, termed the mass flux equilibrium point or the balance point,occurs at the intersection of the mass flux count performed from A to B andfrom B to A. The equilibrium point represents the position along a controlsurface where the amount of charge moving in a positive sense relative to thecontrol surface is in equilibrium with the amount of charge moving in anegative sense, i.e. where

df

d l¼ 0

and l is the distance along the control surface.The surface formed by linking successive mass flux equilibrium points

within a family of control surfaces is termed as the mass flux equilibriumsurface. The position and physical interpretation of the mass flux equilibriumsurface is governed by the orientation of the control surfaces over which themass flux summation was performed and is apparent when the equilibriumsurfaces are superimposed on selected velocity component bin plots, as shownin Figure 4. The equilibrium surface generated by sectioning the mill with afamily of control surfaces whose normals are parallel to the vertical Y axis istermed the vertical mass flux equilibrium surface. The vertical mass fluxequilibrium surface separates the ascending, i.e. travelling in the positive Ydirection, en masse charge from the descending charge. The horizontal massflux equilibrium surface, formed by sectioning the mill using a family of controlsurfaces whose normals are parallel to the horizontal X axis, separates theen masse charge moving in a positive sense in the X direction from that movingin a negative sense. The radial mass flux equilibrium surface is formed by

Figure 4.Physical interpretation

of the mass fluxequilibrium surfaces


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sectioning the mill using a family of control surfaces passing through the centreof the mill’s cross section and whose normals are orthogonal to the mill’s axis ofrotation. The radial equilibrium surface separates en masse charge possessinga positive radial velocity from that possessing a negative radial velocity.

As a direct consequence of the principle of mass conservation, the mass fluxalong any control surface passing through the CoC will be maximum andindependent of the control surface’s orientation. This allows the CoC to beidentified as follows (Figure 5): the mass flux equilibrium point (the point ofmaximum mass flux along a single control surface within the family) on thecontrol surface with the greatest maximum mass flux is identified andrepresents the CoC for that specific equilibrium surface.

It is possible for multiple CoCs to be identified using the aforementionedprocess. These multiple maxima bound the region in which the instantaneousCoC moves as a natural consequence of fluctuations in the motion of thetumbling charge (Figure 6). The CoC along the respective control surface isassumed to be that of the position of the mean of the multiple maxima. Theprocedure described to determine the CoC was then applied to all three familiesof control surfaces (horizontal, vertical and radial) and their CoCs arecalculated. The averaged position of the CoC, used to compare the experimentaland simulated data, is defined as the centroid of the triangle whose vertices arethe CoCs of the three different equilibrium surfaces.

An alternative approach to determine the position of the CoC would be tocalculate the intersection of the mass flow equilibrium surfaces generated fromthe three different families of control surfaces.

Powell and Nurick’s (1996) original definition of the equilibrium surface asthe surface separating the ascending en masse charge from the descendingcharge, as shown in Figure 2, is the combination of the horizontal equilibriumsurface below the CoC and the vertical equilibrium surface above the CoC.

Previous attempts to locate the CoC and the equilibrium surfaces usingnumerical techniques by identifying the particle trajectory turning points

Figure 5.Relating the points ofmaximum mass fluxalong each equilibriumsurface to the positionof the CoC

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(the points where the horizontal or vertical component of velocity of a particlechanges sign) and fitting either an ellipse (Govender et al., 2001a) or a leastsquares cubic spline to the data have several disadvantages when compared tothe mass flux method presented. The trajectory path of a particle within thebulk of the charge is not smooth. Deviations from a smoothed trajectory pathoccur due to the numerous particle interactions within the charge. Thesedeviations cause significant scatter in the particle turning point data andnecessitated the development of techniques requiring operator input to identifythe actual turning points, thereby introducing some level of potential user bias.Spurious turning points are dealt with naturally using the mass flux balancemethod as deviations from the smoothed trajectory path are cancelled uponintegration. Inherent error estimates, based on parameters such as the massflux across any control surface passing through the CoC being a maximum andindependent of the orientation of the control surface and the net mass fluxacross an equilibrium surface being zero, add to the robustness of the method.

DEM simulationsDEM modelThe DEM simulations of the experimental mill were performed using both 2Dand 3D DEM software package particle flow code (PFC) from Itasca. Contactbetween particles is modelled using the soft contact approach, wherebycontacting objects are allowed to overlap. The contact force law, used to relatethe overlap between contacting particles to a contact force, is the standardlinear spring and dashpot model. A thorough overview of the details of theDEM and the contact model used in this work is omitted for the sake of brevityand the interested reader is referred to Itasca (1999) and Mishra (1991).

Figure 6.DEM trajectory data

showing the region inwhich the instantaneous

CoC moves


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The selection of the input parameters for the DEM simulation can have asignificant influence on the behaviour of the simulation (Cleary et al., 2003;Dong and Moys, 2002). The experimental measurement of these inputparameters (in particular, the coefficient of normal and tangential restitution,and the coefficient of friction) needs to be performed under conditions thatreproduce those found in the experimental system, as they may be dependenton the relative contact velocity (Lorenz et al., 1995). Work is currentlyunderway to construct an experimental device to measure the particleinteraction properties and the coefficient of friction under conditions similar tothose experienced in the experimental mill. The behaviour of a single particlewithin the mill will also be investigated to determine the coefficients of normalrestitution and interface friction. As this work is not yet complete, particleinteraction properties from similar tests were used as estimates (Lorenz et al.,1995) and a series of 2D DEM simulations performed to adjust their values toaccount for the specific conditions found in the experimental mill (Govenderet al., 2001a). The interaction properties were adjusted within a realistic rangeto find the best agreement between the simulated and experimental position ofthe CoC. Bin difference plots of particle position, velocity and acceleration werealso used to assess the agreement between the experimental and simulatedsystems. The parameters used in the DEM simulations are listed in Table II.

Discrepancies between the experimental beads and the numericalrepresentation thereof as perfect spheres or discs arise as the plastic beadsare not perfectly spherical and contain a hole through their centre.

The mill shell is represented as a cylindrical geometric primitive in the 3DDEM simulation and as a series of line segments in the 2D simulation. In the 2DDEM simulation ten equal length line segments are used to approximate theshell between consecutive lifters. Discrepancies between the experimental millshell and DEM representation therefore exist and contribute to differences inthe experimental and simulated data.

The DEM simulation was broken into two stages; namely, settlement andanalysis. The particles were generated within the mill and allowed to settleunder gravity during the settlement stage. A representative sample of theplastic beads that constitute the bulk charge diameters were measured and the

Fraction of critical damping in the normal direction cn* (2 ) 0.2Fraction of critical damping in the shear direction cs* (2 ) 0.1Normal contact spring stiffness (N/m) 1 £ 106

Shear contact spring stiffness (N/m) 1£ 106

Coefficient of friction (2 ) 0.23Maximum permissible time-step duration (s) ** 1 £ 1025

Notes: *The coefficient of restitution is related to the fraction of critical damping via anexpression described by Rajmani et al. (2000); **the actual time-step duration could be below thisvalue to satisfy stability criteria (Itasca, 1999).

Table II.Parameters used inthe DEM simulations.The particles and themill boundaries wereassumed to have thesame properties

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standard deviation thereof is included in the numerical model. An estimatefor the number of particles to be generated was obtained by assuming an85 per cent mill filling by volume and a void ratio of 0.3. By overfilling the mill,subsequent analyses can be performed without needing to regenerate theparticle system. The “random” filling of an arbitrary region in space withparticles conforming to a size distribution is an area of immense importance forthe development of DEM and the seminal work of Feng et al. (2001), and morerecently Feng et al. (2002), deserve mention in this regard. The initial particlepositions within the simulated mill were obtained using the random generationof particle positions, subject to the criteria that generated particles could notoverlap. The initial radii of the generated particles were reduced by a factor of30 per cent to expedite this process. The particles’ radii were restored to theirinitial values once all particles had been generated. The process of expandingthe particle radii can cause considerable overlap to occur between particles,resulting in excessive contact forces, and in turn excessive particle velocities.The particles’ velocities were therefore set to zero every 100 time-steps duringthe first quarter of settlement phase. Then the particles settled normally undergravity for the remainder of the settlement phase.

Particles with upper extents greater than the required filling were removedat the beginning of the analysis phase. The mill was then rotated at full speedfor two revolutions and the power draw was monitored to assess the systemstate. It was found that approximately one mill revolution was required tostabilise the power draw, indicating that a steady state had been achieved. Thetrajectory histories of ten randomly selected particles within the bulk chargewere then monitored for a specified number of revolutions (15 revolutions in the3D simulation and 45 in the 2D).

Comparison of DEM and experimental dataThe trajectories of the monitored particles in the experimental and simulatedsystems are superimposed upon one another in Figure 7. The position of thetoe, the region where the outermost layer of cateracting or cascading particlesimpact the mill liner or other particles, prior to being drawn into the bulk ofthe charge, is exaggerated in the 2D simulation (see Figure 2 for a schematicdefinition of the terminology). This exaggeration is due to the void ratio of a 2Dsystem of disks being greater than a similar 3D system of spheres, i.e. thepacking efficiency is greater in a 3D structure than in a 2D one. This limitationof 2D simulations was also noted by Cleary et al. (2003) when comparing 2DDEM simulations with the experimental mill trajectory data. The position ofthe shoulder of the charge, the region where particles leave the liner and enterinto projectile motion or cascade down the surface of the charge, is exaggeratedin both 2D and 3D simulations. The simulated trajectory paths cross theexperimental ones in this region, indicating that the motion is not identical.The reason for this discrepancy is not verified at present, but it is most likelydue to the estimated coefficient of friction being incorrect, and using only one


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coefficient of friction as opposed to different static and kinetic coefficients offriction, which was identified by Powell (1991) as being critical in obtaining thecorrect particle trajectory as it is projected off a lifter bar.

Experimental data appears to be lacking in certain regions, most notably inthe region bounded by the cateracting and cascading trajectory paths. If noexperimental data occur within a certain region it becomes rather subjective asto whether the experimental sampling duration was insufficient to capturebehaviour in this region or if instead it is highly improbable that a particle willever enter the region. Only additional experiments will resolve this issue.

Figure 7.Comparison of theDEM and experimentaltrajectory data

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One major component of the charge motion that obviously cannot besimulated in the 2D DEM is the longitudinal particle drift in the mill. Figure 8shows the trajectory path of a single tracked particle in both experimental andsimulated mills. The effect of the longitudinal charge motion becomessignificant when flow through the mill, classification by a grate, and end effectsare to be considered. The amount of longitudinal drift is significant in bothexperimental mill and DEM simulation (note, the duration over which theexperimental data was sampled is longer than the DEM simulation period andthus more longitudinal drift is present in the experimental system). No attemptwas made to quantify the longitudinal drift in the current work.

DEM must be able to estimate accurately the energy absorbed by particlesduring contact events if it is to a mill design tool. It was found that, on anaverage, the number of contact events per particle per mill revolution in the 3Dsimulation was 15 per cent higher than in the corresponding 2D simulation.The energy absorbed during contact events needs to be artificially scaled inthe 2D simulations to account for the reduced number of contacts that eachparticle experiences. This introduces errors when attempting to model particlebreakage within the mill.

The experimental and simulated bin plots of the X, Y and radial componentsof velocity for the first test series (mill speed of 61 rpm) are shown in Figure 9with the mass flux equilibrium surfaces superimposed. The smoothness of thebin plots for the simulated data relative to the experimental data highlights theneed to perform additional experiments. The upper bound of the Y and radialcomponents of velocity in the experimental and simulated systems are in goodagreement. The upper bound of the X-component of velocity in the experimentaldata is approximately twice that of the simulated systems. Closer analysisrevealed that this was probably due to linking the particle trajectory data fromseparate experimental runs or due to missing experimental data. The absolutevelocity of the particles in contact with the liner within the bulk of the chargecorresponds exactly, as expected, with the rotational velocity of the mill(0.46 m/s). Work is currently underway to objectively define other key regions

Figure 8.Comparison of the DEM

and experimentaltrajectory data along the

length of the mil


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of the charge, such as the shoulder and toe, using the bin plots, therebyproviding physically correct definitions that limit possible user biasing.

The bin difference plots of normalised frequency (i.e. the normalisedprobability distribution plots of particle position) and absolute velocitybetween the experimental and simulated mill system for the first test series(mill speed of 61 rpm) are shown in Figure 10; the scale indicates the percentageerror between the experimental and simulated data sets with a zero bin valuebeing a perfect correlation. The normalised probability distribution plots ofparticle position were calculated by dividing the value in each cell of the binplot of particle position by the cumulative bin count and then normalising tounity. The error E is calculated for each cell in the bin plot as follows:

E ¼ðgE 2 gDEMÞ

gE£ 100 per cent

where gE the bin value of the measured parameter in the experimental systemand gDEM is the bin value of the parameter in the simulated system. The error isundefined in regions where the experimental data are not present, and bydefinition is set to zero in these regions. Figure 11 shows the bin differenceplots for the 2D and 3D DEM simulations. The gE and the gDEM terms in theerror expression now represents the 2D and 3D data, respectively.

Figure 9.Experimental and DEMbin plots of selectedcomponents of particlevelocity with the massflow equilibriumsurfaces superimposed

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The comparison of the 2D and 3D simulations in Figure 11 shows furthershortcomings of the 2D DEM. The bin difference plot of normalised particleposition highlights the fundamentally incorrect manner in which the particlespack in 2D. The trajectory of a particle within the bulk of the charge in a 2Dsimulation is confined to set paths that appear as concentric circles centred onthe CoC. As mentioned earlier, the 2D simulation exaggerates the position ofthe toe; reflected by the near 100 per cent error in the bin difference plot. The bin

Figure 10.Bin difference plots

(expressed as per centdifference) of frequencyand absolute magnitudeof velocity between the

experimental andsimulated systems

Figure 11.Bin difference plots

(expressed as per centdifference) of frequencyand absolute magnitudeof velocity between the2D and 3D simulations


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difference plots of absolute velocity indicate that in general the 2D and 3Dsimulations are fairly similar. Significant discrepancies however exist in theregion above the CoC and appear to adhere to a pattern. The reason for thesediscrepancies is presently unclear, but could well be due to the artificialpacking structures that develop in the 2D DEM simulations.

The position of the CoC can be used to compare the experimental andsimulated data. The position of the CoC would be invariant under stableequilibrium conditions, but the location of the charge in a mill fluctuates withtime and a detailed investigation showed that the CoC moves within a boundedregion between successive mill revolutions, as shown for the simulated 3Dtrajectory data in Figure 12 (the analysis duration of 15 revolutions wasdivided into five subsets of three revolutions and the CoC was calculated foreach subset). This motion can be attributed to the surging and slumping of thecharge, a phenomenon observed in large scale mills (Vermeulen et al., 1984) andDEM simulations of an experimental tumbling mill (Agrawala et al., 1997). TheCoC is therefore the centroid of the region of maximum mass flux that boundsthe instantaneous CoC.

The position of the centre of mass of the charge also moves in a similarmanner to the CoC causing fluctuations in the power drawn by the mill, asevident in the simulated system. The time averaged position of the CoC,summarised in Table III, provides an unique means to characterise the charge,provided that the duration of the experiment is sufficient. As the mill speedincreases from test 1 to test 2, the CoC moves upwards towards the shoulderand outwards towards the mill shell.

The 2D simulations appear to give better correlation to the experimentaldata than the 3D, but this is misleading. The exaggerated position of the toe inthe 2D simulation draws the position of the CoC towards the toe and in turntowards the position of the experimental CoC, thus improving the simulatedresults. This emphasises the fact that the validation of DEM needs to beperformed using a range of different comparative techniques. The simulatedresults differ by not more than approximately one and a half times the meanparticle diameter from the experimental CoC. Future tests and additionalexperimental data will allow the surface traced out by the movement of the CoCto be used to characterise the charge behaviour over a range of mill speeds,such as that shown in Figure 12.

ConclusionsThe DEM method has the potential to simulate the bulk behaviour of the millcharge, but in the current simulations discrepancies occur in the regions of theshoulder and toe. Charge velocity comparisons show good general agreement,except in regions where the experimental data are insufficient. The 2D and 3Dsimulations give similar results, but the 3D simulations are superior due totheir ability to simulate realistic particle packing, longitudinal particle motionand end effects.

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The validation techniques presented characterise the mass behaviour of thecharge based on the observations of a limited number of particles. They arerobust and noise free due to the inherent insensitivity to minor fluctuations incharge motion (i.e. spurious turning points) and the noise suppressing natureof the integration of experimental data. They also provide inherent errorestimates. These methods have scope beyond experimental tumbling mills andcan be used to characterise the motion of many closed systems.

References

Agrawala, S., Rajmani, R.K., Songfack, P. and Mishra, B.K. (1997), “Mechanics of media motionin tumbling mills with 3D discrete element method”, Minerals Processing, Vol. 10 No. 12,pp. 215-27.

Chapra, S.C. and Canale, R.P. (1989), Numerical Methods for Engineers, 2nd ed., McGraw andHill, New York, NY.

Cleary, P.W. and Hoyer, D. (2000), “Centrifugal mill charge motion and power draw: comparisonof DEM predictions with experiment”, International Journal of Minerals Processing,Vol. 59, pp. 131-48.

Cleary, P.W., Morrison, R. and Morrell, S. (2003), “Comparison of DEM and experiment for a scalemodel SAG mill”, International Journal of Minerals Processing, Vol. 68, pp. 129-65.

Figure 12.Variation in the position

of the CoC and acomparison of simulated

and experimentalequilibrium surfaces

(mill speed of 61 rpm)

COC Test 1 COC Test 2

Speed 61.34 rpm Speed 70.88 rpm

Position (mm) Deviation (mm) Position (mm) Deviation (mm)

2D DEM X ¼ 26 and Y ¼ 230 3 X ¼ 28 and Y ¼ 227 43D DEM X ¼ 29 and Y ¼ 222 10 X ¼ 29 and Y ¼ 223 6Experiment X ¼ 27 and Y ¼ 232 X ¼ 32 and Y ¼ 228

Table III.Experimental andsimulated position

of the CoC


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Cundall, P.A. and Strack, O.D.L. (1979), “A discrete numerical model for granular assemblies”,Geotechnique, Vol. 29 No. 1, pp. 47-65.

Dong, H. and Moys, M. (2002), “Assessment of discrete element method for one ball bouncing in agrinding mill”, International Journal of Mineral Processing, Vol. 65 Nos. 3-4, pp. 213-26.

Feng, Y.T., Han, K. and Owen, D.R.J. (2001), “Filling domains with discs”, Proceedings ofthe 4th International Conference on Analysis of Discontinuous Deformation (ICADD-4),6-8 June 2001, Glasgow, UK.

Feng, Y.T., Han, K. and Owen, D.R.J. (2002), “An advancing front packing of polygons, ellipsesand spheres”, Proceedings of the 3rd International Conference on Discrete ElementMethods, 23-25 September 2002, Santa Fe, New Mexico, USA.

Govender, I., Powell, M.S. and Nurick, G.N. (2001b), “3D particle tracking: a rigorous techniquefor verifying DEM”, Minerals Engineering, Vol. 14 No. 10, pp. 1329-40.

Govender, I., Balden, V., Powell, M. and Nurick, G. (2001a), “Validated DEM-potential majorimprovements to SAG mill modelling”, in Barratt, et al. (Eds), Proceedings InternationalAutogenous and Semiautogenous Grinding Technology, 30 September- 3 October, Vol. IV,CIM, pp. 101-14.

Itasca (1999), PFC User Manual, Itasca Consulting Group Inc., Minneapolis, USA.

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Mishra, B.K. (1991), “Study of media mechanics in tumbling mills by the discrete elementmethod”, PhD thesis, Department of Metallurgical Engineering, University of Utah.

Monama, G.M. and Moys, M.H. (2002), “DEM modelling of the dynamics of mill startup”,Minerals Engineering, Vol. 15, pp. 487-92.

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Rajmani, R.K., Mishra, B.K., Venugopal, R. and Datta, A. (2000), “Discrete element analysis oftumbling mills”, Powder Technology, Vol. 109, pp. 105-12.

Stewart, R.L., Bridgwater, J., Zhou, Y.C. and Yu, A.B. (2001), “Simulated and measured flow ofgranules in a bladed mixer – a detailed comparison”, Chemical Engineering Science, No. 56,pp. 5457-71.

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Further reading

Govender, I., Powell, M. and Nurick, G. (2002), “Automated imaging to track the 3D motion ofparticles”, Journal of Experimental Mechanics, Vol. 42 No. 2, pp. 153-60.

Mishra, B.K., Thornton, C. and Bhimji, J. (2002), “A preliminary numerical investigationof agglomeration in a rotary drum”, Minerals Engineering, Vol. 15 No. 1-2, pp. 27-33.

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