Ž .Powder Technology 105 1999 250–256www.elsevier.comrlocaterpowtec

The discrete element method for fine grinding scale-up in Hicom mills

David I. Hoyer )

Hicom International, 8 Khartoum Road, North Ryde, Sydney, NSW 2113, Australia

Abstract

The Hicom mill is a high-intensity grinding mill in which the grinding media tumbles in a strong centrifugal acceleration field. Typicalapplications include wet or dry fine grinding, liberation and concentration of diamonds, and breaking up mineral agglomerates and clayballs. When fine grinding, the intense grinding environment results in a rapid production of very fine material from most mineral ores,down to below 10 mm. However, scale-up from laboratory size data to industrial sized mills becomes increasingly difficult as the product

Ž .size becomes finer. The discrete element method DEM is a computation technique which models the movement of collections ofseparate particles. DEM can be used to track individual collisions as particles tumble over each other, and to calculate the energyassociated with each collision. It is shown how DEM techniques can be applied to the grinding media in a Hicom mill to producefrequency distribution plots of collision energies under different mill operating conditions, including ball size and density, mill speed andmill diameter. These results are shown to be well-correlated with conventional experimental and theoretical results for Hicom mills. Theefficiency of fine grinding in the Hicom mill can be correlated with collision energies calculated by DEM. This has resulted in improvedscale-up methods for fine grinding in the Hicom mill. q 1999 Elsevier Science S.A. All rights reserved.

Keywords: Comminution; Discrete element method; Fine grinding; Scale-up; Nutating mill

1. Introduction

w xThis paper extends the work of a 1994 publication 1which described a scale-up method for fine grinding in theHicom mill. It proposes using the discrete element methodŽ .DEM as a means of determining collision or impactenergies in the Hicom mill, and it is part of an on-goingprogram for improving fine grinding performance andscale-up procedures for the Hicom mill.

w xThe Hicom mill 1–6 is a high-intensity grinding millin which the grinding media tumbles in a strong centrifu-gal acceleration field. Fine grinding of hard minerals toless than 10 mm has been shown to be energy-efficient inbatch tests in small Hicom mills. However, it has also been

w xshown 1 that the finer the grind, the more sensitive theefficiency of grinding is to mill operating conditions suchas mill speed, size, and ball diameter. This has also meantthat conventional scale-up methods based on for example

w xpopulation balance models 2 are often not adequate forreliable scale-up from small batch Hicom mills to pilotplant and full scale situations. A method for relatinggrinding efficiency to impact energies was proposed in

) Tel.: q61-2-9919-1203; fax: q61-2-9887-3034; E-mail:dhoyer@chwarman.com

1994, but this relied on assumptions regarding the way inwhich impact energies vary with operating conditions in aHicom mill.

The DEM is a computational method for modeling themotion of collections of particles. It was developed origi-

w xnally by Cundall and Strack 10 , and extended by Mishraw x w xand Rajamani 11 and Rajamani et al. 12 into a model of

the motion of spherical balls in tumbling mills. The codewas further adapted by the author to model the motion ofballs in centrifugal and nutating mills.

It was proposed to use the DEM to try to find aworkable method for improving scale-up procedures forthe Hicom mill. Indeed, the methods proposed here mightalso be suitable for improving scale-up procedures in othertumbling mills.

2. The Hicom mill

w xThe Hicom mill 1–6 is an adaptation of the concept ofw xthe centrifugal mill 7–9 in an arrangement that over-

comes the discharging difficulties experienced with cen-trifugal mills at high throughput rates. Very rapid breakagerates are attainable per unit mill volume in the Hicom mill,of the order of 50 to 100 times greater than in conventional

0032-5910r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved.Ž .PII: S0032-5910 99 00145-X

( )D.I. HoyerrPowder Technology 105 1999 250–256 251

tumbling or ball mills. The small physical size and intensegrinding action of these mills make them suitable for awide range of industrial applications. Some typical applica-tions include fine and ultrafine grinding of hard and soft

w xmaterials 1 , the liberation of diamonds from sea shellsw xand from kimberlite 5,6 , the production of exotic alloys

w xby mechanical activation 5 , breaking up mineral agglo-merates and clay balls, and the reduction of critical size

w xpebbles extracted from autogenous mills 4,5 .The Hicom mill uses strong acceleration fields to impart

an intense tumbling and agitation motion to steel, ceramicor rock particles, producing very rapid breakage rates ofmineral ores. In operation, it can be thought of as ahigh-intensity ball or autogenous mill with a small, rapidlymoving grinding chamber. Fig. 1 shows a commercialimplementation of the Hicom mill. The grinding chamberis a truncated cone with a rounded base, open at the topand with a grate discharge at the bottom. An especiallydeveloped rolling bearing and mechanical drive arrange-ment cause the mill axis to nutate about a fixed nutationpoint at a nutation angle of 4.758. The motion is similar toswirling a conical flask in the wrist: the top wobbles andthe bottom describes a circle. The eccentricity ´ is theradius of this circle at any position down the axis, and

Fig. 1. A commercial implementation of the Hicom nutating mill.

increases linearly from the top down to the base of thegrinding chamber. Both the mill chamber diameter D andthe eccentricity ´ increase down the mill axis. Hence, thecentrifugal acceleration also increases down the nutationaxis.

The motion of the Hicom mill is similar to that of acentrifugal or planetary mill, except that the diameter andacceleration intensity vary down the length of the mill.There is no critical speed in these mills — they can beoperated at any speed, limited only by the mechanicalstrength of the mill. The magnitude of the accelerationfield in which the mill contents tumble is equal to theproduct of the eccentricity and the square of the millspeed:

Asv 2´ 1Ž .

where v is the mill speed and ´ is the eccentricity orradius of nutation at any point along the axis. It has been

w xshown 3 that the power consumption for nutating millsincreases with the cube of the mill speed. At constantacceleration intensity and mill chamber geometry the powerincreases with the mill diameter raised to the power 3.5.

3. 2D-DEM

3.1. Background

The DEM is a computational method for modeling themotion of collections of particles. It was developed origi-

w xnally by Cundall and Strack 10 , and extended by Mishraw x w xand Rajamani 11 and Rajamani et al. 12 into a model of

the motion of spherical balls in tumbling mills. Mishra andRajamani’s code was made available to the author andfurther adapted to model the motion of balls in centrifugaland nutating mills. At every contact the forces developingdue to collisions are modeled by a pair of normal and

w xtangential spring-dashpots. Mishra and Rajamani 11 foundthat the following parameter values were realistic in ballmill simulations in the presence of powder: normal stiff-nesss400 kNrm, shear stiffnesss300 kNrm, dampingcoefficients0.5, coefficient of restitutions0.4 to 0.6 de-pending on contact conditions. These were the values usedin the Hicom simulations.

The model not only yields the motion of individualballs in the mill, but also keeps a log of every collision,and the energy associated with each collision. Dynamiccomputer-generated movies of the mill load motion can beconstructed. By simulating a sufficient number of colli-sions it is possible to construct a plot of the magnitude vs.the frequency of collisions in a tumbling mill, and this plotserves as a comprehensive description of the grindingenvironment.

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Fig. 2. Photos of mill load and DEM simulations.

3.2. Photographic comparisons

Photographic images of the load in a centrifugal millw xwere published in 1984 9 along with an analytical method

for determining the shape of the free surface of the millload. Fig. 2 shows several DEM simulations comparedwith the photographic images at the same mill conditions.The images are for a centrifugal mill with an eccentricity

ratio of ´rDs0.2, and mill fillings of Js0.25, 0.5 and0.75. Clearly, the DEM results are in close agreement withthe photographic images. They show a stable cascadingload at mill fillings of Js0.75 and 0.5, and they alsoshow the degeneration into an unstable regime at Js0.25.These results demonstrate that the DEM simulations pro-vide a good model of the motion of balls in centrifugalmills.

Fig. 3. The distribution of collision energies from the simulation in Fig. 2 at mill filling of Js0.5.

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3.3. Distribution of collision energies

During each DEM simulation a log is kept of the energyassociated with each collision between balls and againstthe mill liners. Individual collision energies vary widely inmagnitude and can be presented as a frequency distributionplot such as that shown in Fig. 3. This shows the range ofcollision energies logged during the DEM simulation whichwas used to produce the image shown in Fig. 2, at a millfilling of Js0.5. The DEM simulation was allowed toproceed until a total of 50 000 separate collisions had beenlogged. The collision energies were divided into bands,and the figure shows the number of collisions in eachenergy band. Plots such as these represent a precise charac-terisation of the grinding environment at the conditions ofeach simulation. A five-parameter skewed Gaussian curvewas fitted to the bell-shaped curve, and it was reasonedthat these five parameters form a convenient summary ofthe grinding environment in the mill. The functional form

Ž .of the skewed Gaussian curve is given by Eq. 2 below.Ž .p4qp5 ln x yp2Ž .Ž .yabs ln x yp2Ž .Ž .

f x sp1 exp .Ž .p3

2Ž .

The physical meanings associated with each parameterare listed below.p1 The height of the peak.p2 The mode, which is the position of the peak on the

x-axis.p3 The standard deviation of the curve.p4 The kurtosis of the curve. A normal Gaussian curve

has a kurtosis of 2. The curve peaks more sharply forlower kurtosis values, and has a flatter top for higherkurtosis values.

p5 The skewness of the curve. A normal Gaussian curvehas a skewness of 0. The curve leans right forpositive skewness, and left for negative skewness.

3.4. Characterising the grinding enÕironment

Fig. 3 showed the range of collision energies at a millfilling of Js0.5. Additional DEM simulations were doneat a number of mill filling levels ranging from Js0.2 to0.8, and the resulting five parameters p1 to p5 are shownplotted against mill filling in Fig. 4. There are systematicchanges in parameters p1 to p3, representing the positionand height of the peak, and the spread or standard devia-tion. There is not a strong effect on the skewness and

Fig. 4. DEM simulation results showing the effect of mill filling levels on fitted parameters p1 to p5.

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kurtosis parameters — they show more scatter, and aregrouped over a relatively narrow range of values. A largenumber of similar collision energy plots have been con-structed for a wide range of operating conditions, includingvariations in mill size and speed, mill filling, and thediameter and specific gravity of the ball load. Mixtures ofdifferent ball sizes and specific gravities can also be done.In most cases it has been found that the variations in theskewness and kurtosis parameters are not large, and thatthey generally do not follow simple trends.

For a constant total number of collisions per simulationthe peak height p1 is closely correlated with the standarddeviation p3 when the skewness and kurtosis do not varymuch. The most important characterising parameter is p2,the mode of the distribution, which can be interpretedŽ .loosely as being related to the average collision energy,and hence to the intensity of the grinding environment.The remaining parameters describe the spread of collisionenergies around this average value. For the sake of sim-plicity in this investigation it was decided to concentrate

Ž .on the variation in the mode parameter p2 , and leave amore detailed investigation of the other parameters for afuture study.

3.5. The effects of operating conditions

Fig. 5a shows how the mode p2 varies with milldiameter. It shows in effect that the average collisionenergy increases with mill diameter, as one would expect:a larger mill diameter results in greater tumbling distancesand larger forces within the mill load. The other millconditions for these simulations were mill speeds1160rpm, ball diameters8 mm, mill filling Js0.5, eccentri-city ratio ´rDs0.16, and media rs7.8 for steel balls.An inspection of the figure shows that the collision energyincreases approximately with mill diameter to the power1.3, which is lower than the previous assumed value of 3w x1 . This value 1.3 is the slope of the curve after taking

logs on the x-axis values; the y-axis values are alreadylogarithmic. It is important to note that if the mill speed,filling, ´rD ratio and ball size and density were changedand the simulations for the effect of mill diameter wererepeated, the resulting curve may differ from that shown inFig. 5a. The way in which the distribution of collisionenergies changes with mill conditions is complex andgenerally does not follow simple geometrical relationships— they need to be calculated individually for each case.

ŽFig. 5b shows how the mode p2 loosely, the average.collision energy increases rapidly at small eccentricity

values, and more slowly at larger values. Fig. 5c showshow the average collision energy increases steadily withthe square of mill speed, and this is in agreement with

w xearlier analyses 1 . Fig. 5d–f show how the averagecollision energy varies with ball size, mill filling, andmedia density. The somewhat chaotic variation at smallvalues of mill filling J reflects the chaotic nature of theload at low filling levels — see Fig. 2 for example.

4. Extension to 3D for the Hicom mill

At the time of writing this paper, the DEM code fortumbling mills was available only in a 2D version, whichmodels a 2D cross-section of a mill. The previous analyseshave been applied to centrifugal mills, which can beadequately described by a cylindrical mill section. How-ever, the Hicom mill grinding chamber is approximatelyconical in shape and has a nutating motion with varyingacceleration intensity down its length. This has been mo-deled successfully in the past by treating it as a series of

w xshort cylindrical sections 2 of increasing diameter andacceleration intensity, as noted earlier. It was also shownthat the ball load in a Hicom mill tends to segregate intosize order, with small balls located near the base and largerballs near the top of the grinding chamber. To get collisionenergy distributions for Hicom mills, the Hicom mill was

Ž .Fig. 5. DEM simulation results showing the effect of various mill operating conditions on ln p2 .

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treated as a series of short centrifugal mill slices, each slicehaving its own diameter, acceleration intensity, and ballsize. It was assumed that the mill filling remained constantacross all slices of the grinding chamber.

The collision energy distribution plot for any Hicommill configuration can therefore be obtained by integratingacross the mill slices. In practice this meant doing aseparate DEM simulation for each slice of the Hicom mill,then averaging the parameter values after weighting themin proportion to the square of the diameter of each millslice.

Using this method the collision energies for any Hicomnutating mill configuration can be determined. This willnow be used to review a previously proposed method forrelating grinding efficiencies to collision energies.

4.1. Effect of collision energies on grinding efficiency

w xIt has been proposed previously 1 that the efficiencyof grinding in the Hicom mill is a simple function of theimpact or collision energy density in the mill, and thattypical collision energy densities in the Hicom mill could

Ž .be expected to vary approximately by Eq. 3 below. Notethat this is not a formal energy density in the sense of amechanical stress, but it relates to the energy imparted toeach particle during a collision, taking into account beddepth and the amount of powder associated with each

w xcollision 1 .

E sr dv 2D3. 3Ž .n

Furthermore, it was shown that when comparing grind-ing results at different mill operating conditions, the grind-ing efficiency in kW hrt to reach some specified product

size would be approximately constant when the collisionenergy density was constant. For example, grinding withsmall balls at high speed has a similar energy efficiency togrinding with larger balls at a lower speed, if the combina-tion of speed and ball size was chosen to give a constantcollision energy density.

Ž .Eq. 3 was derived from a consideration of the forcesin the Hicom mill. The DEM simulations described earlierprovide an alternate means for determining these typicalcollision energies. Specifically, the mode p2, when inte-grated down the length of a Hicom grinding chamber,gives a value related to the average collision energy for themill. For fine grinding applications a number of particleswill be subjected to comminution in every collision, so thecollision energy density is given by:

Collision energy density by DEM E sp2rd2 . 4Ž .DEM

Fig. 6a shows collision energy densities for two seriesof grinding tests at constant grinding efficiencies, demon-strating that when the collision energy is constant thegrinding efficiency is also constant, or approximately so.

w x Ž .The data were published previously 1 using Eq. 3 toŽ .determine the E collision energy density values. Fig. 6bn

shows the same data, but using average collision energyŽ .densities determined by the DEM simulations and Eq. 3 .

The results are quite similar, indicating that the DEM canprovide a useful and practical method for scale-up. Themethod does not have a predictive capability in the senseof finding optimum grinding efficiencies, but it does meanthat if grinding efficiency is optimised in a small labora-tory mill then the method will help to duplicate the opti-

Fig. 6. DEM vs. previous method for relating grinding efficiency to collision energies.

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mised grinding efficiency in a larger mill by finding theoperating conditions that give rise to the same collisionenergy density.

5. Conclusions

The DEM is a powerful analytical tool for characteris-ing the grinding environment in the Hicom mill as afunction of operating variables such as mill size and speed,and grinding media size and density. The method enablesthe calculation of average collision energy values in anyHicom mill grinding chamber. This method shows promiseas an alternative to previous work which related the effi-ciency of fine grinding to collision energies estimated by aformula.

There is scope for further development of the methodby considering not only the average collision energy in anyparticular mill, but the way in which the spread of collisionenergy values is affected by the milling conditions. Thegeneral method of analysis can probably be applied suc-cessfully to ball mills and other grinding mills as well.

6. Nomenclature

Ž 2 . Ž Ž ..A Acceleration intensity mrs Eq. 1Ž .d Ball diameter mmŽ .D Mill diameter mm

E Collision energy intensity by formulan

E Collision energy intensity by DEMDEM

J Fractional filling of mill with grinding mediap1 to p5 Parameters fitted to the DEM collision energy

curvesŽ .´ Eccentricity of mill axis mm . This is the dis-

tance between the mill axis and the nutationaxis, measured perpendicularly from the nuta-tion axis

Ž .v Mill speed radrsr Specific gravity of grinding media

Acknowledgements

This work was sponsored by Hicom International. Thepermission granted by that company to publish this paperis gratefully acknowledged. The author thanks Raj K.

ŽRajamani The Utah Comminution Centre on the campusof the University of Utah, Salt Lake City, UT 84112,

.USA for providing the discrete element method code fortumbling mill simulation, which was adapted for centrifu-gal mill simulations.

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