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
Received February 2003Revised July 2003
Accepted July 2003
Engineering ComputationsVol. 21 No. 2/3/4, 2004
pp. 119-136q Emerald Group Publishing Limited
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
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 Moyss 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
Contributions tothe experimental
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
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
Contributions tothe experimental
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 particles 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 particles 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 sgn ~d~n~n ~nz
Figure 3.Schematic sectioning ofthe mill using variouscontrol surfaces andthe mass flux countperformed along aselected control surfaceto determine the positionof the mass fluxequilibrium point
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 vector
orthogonal 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
d l 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 b...