Powder Technology 223 (2012) 83–91

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DEM investigation of energy distribution and particle breakage in tumbling ball mills

M.H. Wang, R.Y. Yang ⁎, A.B. YuLaboratory for Simulation and Modelling of Particulate Systems, School of Materials Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia

⁎ Corresponding author.E-mail address: r.yang@unsw.edu.au (R.Y. Yang).

0032-5910/$ – see front matter © 2011 Elsevier B.V. Aldoi:10.1016/j.powtec.2011.07.024

a b s t r a c t

a r t i c l e i n f o

Available online 23 July 2011

Keywords:Tumbling ball millsPopulation balance modelsDiscrete element methodEnergy distributionGrinding

This work investigates the grinding process in tumbling ball mills using a discrete element method (DEM)based model. Binary particles are used to represent grinding balls and ground powders, respectively. Threeforms of energy inside mills, collision energy, dissipated energy and maximum impact energy, areinvestigated and linked to particle breakage. By linking the energy information with population balancemodels, the evolution of product size with grinding time is predicted and the results are compared with theexperimental data from the literature. The results indicate that the collision energy is more directly linkedwith particle breakage and can predict the product size reasonably well without adjustment. Other twoenergies can also be used for size prediction but the selection function needs to be carefully calibrated. Theeffects of operation conditions such as rotation speed and loadings of grinding balls and ground material arealso investigated.

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© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Grinding is an important unit operation for many industries, suchas mineral, food and pharmaceutical industries [1]. It is a capital- andenergy-intensive process and may account for up to 50% of the directoperating cost in a mineral processing plant due to its low powerefficiency (4–8%) [2]. Therefore, it is important that the grindingprocess is properly designed and grinding devices are operated atoptimum operating conditions. Any small improvement in theefficiency of mill operation provides an opportunity of immenseeconomic benefit to the industries. To achieve this aim, reasonablyaccurate models based on the basic physics and mechanics arerequired to understand the dynamic characteristics of grinding.

The phenomenological population balance model (PBM) is oftenused to analyse the grinding process at the process length scale. Thegeneral form of PBM for a batch grinding process is given by [3]:

dmi tð Þdt

= −Simi tð Þ + ∑i−1

j=1bijSjmj tð Þ ð1Þ

wheremi is the mass fraction of particles in the ith size interval, Si thebreakage rate or the selection function and bij the breakage function.The selection function Si describes the fractional rate at which aparticle is broken out of the ith size interval, whereas the breakagefunction bij represents the fraction of the primary breakage product ofmaterial in the jth size interval which falls inside ith size intervalfollowing a breakage event. While PBM has been demonstrated toprovide a better prediction of grinding performance than the

empirical Bond work index models [4], a priori knowledge of S andb has to be obtained. The two functions are material properties andclosely related to the energy distribution inside mills [5] which isquite difficult to obtain from experiments. Therefore various forms ofselection and breakage functions have been proposed and theparameters in the functions were largely fitted with experimentaldata [6–8]. As these methods are basically the data fitting process, theparameters in one mill cannot be used to predict the grindingperformance under another grinding condition.

Grinding is a complex process, but the complexity at the bulk levelis the collected outcome of the interactions among grinding media,ground materials and grinding devices. While such information isdifficult to obtain from experiments, numerical modelling based on thediscrete element method (DEM) [9] can readily determine the energydistribution based on the well established contact mechanics. Mishraand Rajamani [10] were the first to use DEM to simulate the chargemotion in a tumblingmill. Since then, DEMhas been used for modellinga wide range of grinding devices including tumbling mills, planetarymills and stirred mills, as reviewed recently by Zhu et al. [11].

While the DEM studies have significantly improved our knowledgeof particle flow inside mills, a few obstacles need to be overcome inorder to link the microscopic information from DEM with PBM topredict product sizes. Firstly, different forms of energy, such ascollision and dissipated energy, can be obtained from DEM and theyall have been used in the literature to characterise the particlebreakage [12,13]. For example, Datta and Rajamani [12] proposed anapproach to link the dissipated energy fromDEM simulationswith theselection and breakage functions of particles determined fromexperiments. PBM was then solved to predict the particle sizes. Theirwork, however, was based 2D DEM simulations and a calibrationparameter was required in order to match the PBM prediction with

Table 1Parameters and their values in the simulations.

Parameters Steel balls Limestone powders

Density, ρ (kg/m3) 7.8×103 1.5×103

Young's modulus, Y (N/m2) 1×107 1×107

Poisson's ratio, σ 0.27a 0.4b

Sliding friction coefficient, μ 0.3c 0.75c

Rolling friction coefficient, μr 0.02 0.02Normal damping coefficient, γn (s−1) 3×10−5 3×10−5

a Steel A. Product data bulletin: 17-4 PH Stainless Steel. West Chester, OH: AK SteelCorporation 2007.

b Gere JM, Timoshenko SP. Mechanics of Materials. 4th ed: Pws Pub Co 1997.c http://www.supercivilcd.com/FRICTION.htm.

Table 2Mill configurations and operation conditions.

Parameters Mill-A Mill-B Mill-C

Mill diameter (mm) 254 381 900Mill length (mm) 292 292 500Lift number 8 10 6Lift width (mm) 19 25 25Lift length (mm) 5 9 25Mill rotation speed (rpm) 55 41 22–67Grinding ball size (mm) 50.8 50.8 50.8Grinding ball loading 35% 35% 10%–50%Ground material size (mm) 15.2 15.2 15.2Ground material loading 15% 15% 4–24%

0.2

0.4

0.6

0.8

1

1.2

1.4

Ave

rage

vel

ocity

, v (

m/s

)

Mill-AMill-B

a

84 M.H. Wang et al. / Powder Technology 223 (2012) 83–91

experimental results. So far, it is not clear which form of energy is moresuitable. Secondly, most of the previous studies mainly focused on themotion of grinding media without paying much attention to groundmaterials. It is still unclear how the presence of ground particles in thesimulationswill affect theflowpattern and internal energy distribution.More recently, Cleary andMorrison [14] investigated the particleflow ina mill involving both grinding media and ore particles. No attempt,however, was given in their work to predict particle sizes.

The presentwork is therefore to investigate the grinding process in atumbling ball mill involving both grinding media and ground powders.The simulations are based on our previously developed DEM modelwhich successfully simulated both dry andwet particle flows in rotatingdrums [15–17]. Different forms of energy will be analysed and theresults are then linked to PBM to predict product sizes. The effects ofoperation conditions on the grinding process will also be investigated.

2. Numerical model and simulation conditions

An in-house developed DEMmodel is used in the present study. InDEM simulations, the motions of a particle of radius Ri and mass mi isdescribed by [18]

midvi

dt= ∑

jFnij + Fs

ij

� �+ mig ð2Þ

Iidωi

dt= ∑

jRi × Fs

ij−μrRijFnijjω̂i

� �ð3Þ

where vi, ωi and Ii are, respectively, the translational and angularvelocities, and themoment of inertia of the particle. Ri is a vector runningfrom the centre of the particle to the contact point with its magnitudeequal to particle radius, and μr is the coefficient of rolling friction. Fn

ij andFsij represent, respectively, the normal contact and tangential contact

forces imposed on particle i by particle j (Fig. 1), given by [19,20]:

Fnij =

23E

ffiffiffiffiR

pξ

32n−γnE

ffiffiffiffiR

p ffiffiffiffiffiξn

qvij·n̂ij

� �� �n̂ij ð4Þ

Fsij = −sgn ξsð Þμ jFn

ij j 1− 1−min ξs; ξs;max

� �ξs;max

0@

1A

32

24

35 ð5Þ

where E = Y1−σ̃ 2 and Y and σ̃ are, respectively, Young's modulus and

Poisson's ratio;ξn is theoverlapbetweenparticles iand j,n̂ij is aunit vector

Fig. 1. Schematic illustration of the forces acting on particle i from contacting particle j.

running from the centre of particle j to the centre of particle i,R = RiRj = Ri + Rj

� �. The normal damping constant γn is directly linked

to the restitution coefficient of particles. ξs and ξs, max are, respectively, thetotal and maximum tangential displacements of the particles.

b

0 2 4 6 80

Time, t (s)

0 2 4 6 80

100

200

300

400

500

Time, t (s)

Pow

er d

raw

, P (

W)

Mill-AMill-B

Fig. 2. Temporal variations of (a) mean particle velocity; and (b) power draw for Mill-Aat 55 rpm and Mill-B at 41 rpm.

Fig. 4. Particle flows at steady-state: (a) Mill-A at 55 rpm; and (b) Mill-B at 41 rpm.

85M.H. Wang et al. / Powder Technology 223 (2012) 83–91

Tumbling ball mills partially filled by binary sized sphericalparticles were simulated. The large particles (d=50.8mm) representthe grinding media (steel balls) while the small ones (d=15.2mm)are the ground materials (limestone powders). Note in practice, theground powders are much smaller. Limited by computing capacity,the current simulations were restricted to the grinding ball/groundpowder size ratio of 1:0.3. It was assumed that the relatively largeground powder size has no significant effect on the overall flowpattern and the impact energy acting on the ground powders. Thiswill be further discussed later. Table 1 shows the properties of twomaterials which were selected to match those of steel balls andlimestone powders.

A simulation started with the random generation of all particleswith no overlap inside a mill. The particles started to fall down undergravity and collided with others. The packing process proceeded untilall particles reached their stable positions with essentially zerovelocities as a result of the damping effect for energy dissipation. Aftera packed bed was formed, the mill was then rotated at differentspeeds. After the flow reached the steady state (determined from flowvelocity and power draw as shown later in Fig. 3), the flowinformation was then collected and analysed.

Three mills were simulated in the present work. Two mills (Mill-Aand Mill-B) were selected to match those used in Datta andRajamani's study [12] so the present results can be compared withtheir experimental data. The third mill (Mill-C) was much larger thanother twomills so the grinding performance under different operationconditions can be investigated. Table 2 lists the grinding conditions ofthree mills.

b

0 0.5 1 1.5 20

0.5

1

1.5

2

Particle velocity, v (m/s)

Pro

babi

lity

dens

ity fu

nctio

n,P

(v)

2s4s6s

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

Particle velocity, v (m/s)

Pro

babi

lity

dens

ity fu

nctio

n,P

(v) 2s

4s6s

a

Fig. 3. Distributions of particle velocity at different times for (a) Mill-A at 55 rpm; and(b) Mill-B at 41 rpm.

b

Ave

rage

Col

lisio

n E

nerg

y (J

)

Mill-A Mill-B10-4

10-3

10-2

10-1

100

101

Small-SmallLarge-SmallLarge-Large

Tota

l Col

lisio

n E

nerg

y (J

)

Mill-A Mill-B100

101

102

103

Small-SmallLarge-Large

Large-Small

a

Fig. 5. Collision energy among different sized particles: (a) average collision energy;and (b) total collision energy. Note the logarithmic scale for Y axis.

a

b

10-4 10-3 10-2 10-1 10010-1

100

101

102

103

104

10-1

100

101

102

103

104

Energy, E (J)

10-4 10-3 10-2 10-1 100

Energy, E (J)

Pro

babi

lity

dens

ity fu

nctio

n, P

(E

) Collision energyDissipated energyMaximum impact energy

Pro

babi

lity

dens

ity fu

nctio

n, P

(E)

collision energydissipation energyimpact energy

Fig. 6. Distributions of three types of energy in (a)Mill-A at 55 rpm; and (b)Mill-B at 41 rpm.

86 M.H. Wang et al. / Powder Technology 223 (2012) 83–91

3. Result and discussion

3.1. Dynamics of particle flow

This section investigates the flow properties of Mill-A and Mill-B.Flow velocity and mill power draw are two important properties tocharacterise the dynamics of flow, and thus are analysed first andcompared with the previous results. In the present study, power drawis the product of mill rotation speed and total torque acting on themill. The mills are first loaded with steel balls only and rotated at the

Fig. 7. Variation of collision energy distribution in Mill-A with ground particle size.

speeds of 55 rpm and 41 rpm, corresponding to 65% of their criticalspeeds, respectively. Fig. 2 plots the variations of the average flowvelocity andmill power drawwith time. Mill-B has a larger mean flowvelocity due to its relatively larger size even it rotates slower thanMill-A, generating larger kinetic energy for the particles. For the samereason, the power draw of Mill-B is 160 W, almost 3 times larger thanthat of Mill-A (56 W). The strong fluctuations of power draw are dueto the inherently dynamic interactions of particles with the millsurfaces. The mean results of power draw from the presentsimulations are comparable with experimental measurements(60 W and 165 W) and 2D simulation results (58 W and 155 W)[12], indicating the validity of the model. Fig. 2 also indicates that theparticle flows in both mills reach the steady states within a very shorttime (less than 2s). This can be further demonstrated from thedistribution of particle velocity at different times (Fig. 3) which showsno significant change. All analyses below are carried out when theparticle flows are in the steady states.

Fig. 4 shows the flow patterns when limestone powders areincluded. The colour represents the speeds of the particles. The flowsare in the cascading/cataracting regimes which are similar to the flowpatterns with grinding media only and are also comparable with the2D flow patterns obtained by Datta and Rajamani [12]. This indicatesthat the inclusion of ground materials has not significantly alerted theparticle flows.

The energy distribution in the mills is investigated in terms of threetypes of energy: collision energy, dissipated energy and maximumimpact energy. The collision energy between two particles is calculatedby 1/2mvij

2, where vij=vi−vj is the relative normal velocity of twoparticles at the collision and m is average mass m = mi + mj

2

� �for particle–particle collision or the mass of the particle for particle-mill collision. While the reduced mass (m=2mimj/(mi+mj)) isoften used [21] to calculate the collision energy, its use results astrong dependence of the collision energy on the ground particlesize, which is very difficult to quantify as the ground particlesare continuously broken out in grinding. So the average mass isadopted in the present study to minimise the effect of the change ofground particle size. The dissipated energy, caused by the inelasticcontact of two particles (e.g. plastic deformation) [12], is calculated asthe integral of damping forces with respect to displacements over awhole contact period (E = ∫ tcontact

0 Fdndξn + Fsdξs� �

). The maximumimpact energy, characterising the maximum stress experienced by aparticle, is the integral of normal contact force and normal displacementwhen the overlap between two particles reaches the maximum(E = ∫ ξn; max

0 Fnijdξn). In the following discussion, the results are the

average of 25 samples with each sample being collected over 0.2 s atdifferent grinding times.

Fig. 5 shows the average and total collision energy for differenttypes of collisions, plotted in a logarithmic scale. As shown in Fig. 5a,Mill-B generates larger average collision energy comparing with Mill-A. Overall the average collision energy of the large–large particles isonly slightly larger than that of the large–small particles using thecurrent method. Meanwhile, the average collision energy for thesmall–small particles is significantly smaller (0.0006 J and 0.001 J forMill-A and Mill-B, respectively). Therefore their contributions toparticle breakage can be reasonably ignored. On the other hand, thetotal collision energy (Fig. 5b) due to the large-small particlescontributes more than 80% of the total collision energy due to thelarge number of collisions between the large and small particles.

Table 3Fitting values of α for different types of energy.

Type of energy α Squared residuals R2

Collision energy 1 0.185 0.895Dissipated energy 0.57 0.065 0.963Maximum impact energy 5.8 0.230 0.870

bParticle size, (mm)

Sel

ectio

n fu

nctio

n, S

(min

-1)

0.5 1.0 1.5 2.0 2.5

0.4

0.6

0.8

1.0collision energy

maximum impact energy

dissipated energy

Particle size, (mm)

Sel

ectio

n fu

nctio

n, S

(min

-1)

0.5 1.0 1.5 2.0 2.50.2

0.4

0.6

0.8collision energy

maximum impact energy

dissipated energy

a

Fig. 8. Predicted selection functions using different kinds of energy: (a) Mill-A at55 rpm; and (b) Mill-B at 41 rpm.

bParticle size, (mm)

Cum

ulat

ive

unde

rsiz

e

0.5 1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

0.8

1.0

0.5min

4min

Particle size, (mm)

Cum

ulat

ive

unde

rsiz

e

0.5 1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

0.8

1.0

4min

0.5min

a

Fig. 9. Prediction of product sizes at different grinding times for (a) Mill-A at 55 rpm;and (b) Mill-B at 41 rpm using different forms of energies: collision energy ( ),dissipated energy ( ) and maximum impact energy ( ) in comparison withexperimental results (■) [12].

87M.H. Wang et al. / Powder Technology 223 (2012) 83–91

Other two types of energy (the dissipated energy and maximumimpact energy) also show similar trends (not shown here). Since thecollisions of the small-small particles and large-large particles do notcontribute to particle breakage, the following discussion only focuseson the interactions between large and small particles.

Fig. 6 shows the distributions of three types of energy for the large-small particle interactions. Similar distributions are observed for bothmills, showing that the large number of interactions occur at smallcollision energy with the number of interactions of larger energydecaying rapidly. The results are comparable with the previousfindings in the flow in rotating drums [16]. Among different types ofenergy, the collision energy is the largest, followed by the dissipatedenergy while the maximum impact energy is the smallest.

In the grinding process, the ground powders experience contin-uously breakage. It therefore would be interesting to investigate theeffect of the change of powder size on the energy distribution insidemills. Three sized of ground powders, 15.2 mm, 11.2 mm and 8.6 mm,are selected while the grinding balls are fixed at 50.8 mm, making thesize ratio of grinding balls and ground particles 3.3:1, 4.5:1 and 6:1respectively. The energy distributions for different sized groundparticles in Mill-A at the rotation speed of 55 rpm are plotted in Fig. 7.It is observed that the distribution has no significant change as theground particle size decreases. While some differences can beobserved at small (b0.1 J) and very large (N0.715 J) collision energyranges, their contributions to particle breakage are not importantbecause either the collision energy is too small to break particles orthe number of collisions is too rare to affect the overall breakage. Inthe energy region which contributes the most to particle breakage(0.115 J–0.465 J), the distributions are almost the same. So it isreasonable to assume that the energy distribution inside mill does notchange with grinding time.

In the following, the energy distribution will be linked to PBM topredict particle sizes with grinding time.

3.2. Prediction of particle size

To predict particle size, the breakage behaviour of ground powders(limestone) should be known, which can only be determined fromexperiments. In the present work, the experimental data obtained byDatta and Rajamani [12] is adopted (Figs. 3 and 5 in [12]). In theirexperiments, the breakage behaviour of particles is determined by theball-drop test: a steel ball of 5.08cm in diameter was dropped fromdifferent heights on a packed bed of four layers of ground particles. Afterthe test, the broken mass was sieved to determine the particle sizedistribution. Thebreakageand selections functionswere thendeterminedby averaging the results from 40 to 50 test samples. It was observed thatthe size distributions of broken particles, when normalised with respectto the original particle size, are same, indicating that the breakagefunction does not change with grinding time. The selection function, onthehand,wasdependent onboth impact energy andoriginal particle size.

As proposed by Datta and Rajamani [12], the selection andbreakage functions of the flow system under a given grindingconditions are given by:

Si = α ∑N

k=1

λkmi;k

Hð6Þ

bij = ∑N

k=1

λkmj;kbij;k

∑Nk=1λkmj;k

ð7Þ

Fig. 10. Particle flow pattern in Mill-C at different rotation speeds.

0.4 0.8 1.2 1.6 2.0 2.4

10

20

30

40

50

60

70

80

90

100

Particle size, (mm)

Mas

s fr

actio

n le

ss th

an s

ize,

(%

)

4 min3 min2 min1 min

a

88 M.H. Wang et al. / Powder Technology 223 (2012) 83–91

where λk is the number of collisions per second under collision energyek, mj,k represents the mass of particles broken in the jth size intervalunder ek and bij,k is the breakage function of each ek. H is the totalmass of ground materials. To compare the three different kinds ofenergy, a parameter α is introduced in Eq. (6), which was adjusted tomatch the experimental measurements [12]. The following procedurewas adopted to determine the values of α:

• The broken mass mj,k.and breakage function bij,k were determinedfrom the experimental data;

• By substituting the broken mass, the breakage function and theenergy distributions obtained from the simulations into Eqs. (6) and(7), the equations were solved with α the only unknown variable;

• By substituting Eqs. (6) and (7) into Eq. (1), the evolutions ofparticle sizes was predicted. The value of α was varied and selectedwhen the experimental data were best fitted with the least squaremethod.

Table 3 show the fitting values of α, the fitting residuals and fittingconfidence (coefficient of determination) for different types of energy.Different values of α are obtained for different types of energy. It isobserved that no adjustment is required (α=1) when the collisionenergy is used to describe the particle breakage. When the dissipatedenergy is used,α needs to be adjusted to 0.57 for both mills. The valueis close to 0.8 adopted in literature [12]. However the value of α forthe maximum impact energy is 5.8 for the two mills because themaximum impact energy is much smaller than other two types ofenergy. Fig. 8 shows the selection functions of different sized particlespredicted with different forms of energy when the mills rotate at55 rpm and 41 rpm, respectively. Clearly, the smaller particles are

20 30 40 50 60 700

10

20

30

40

50

60

70

Mill rotation speed, ω (rpm)

Ave

rage

col

lisio

n en

ergy

, E (

mJ)

Fig. 11. Variation of average collision energy in Mill-C with rotation speeds.

more difficult to break. The results also indicate that while thecollision energy is more directly linked with particle breakage, othertwo can also generate similar selection functions if α is carefullyselected.

Fig. 9 shows the cumulated product size distributions aftergrinding for 0.5min and 4min in the two mills. Experimental dataare also plotted for comparison. For Mill-A, the predictions aftergrinding for 0.5 min based on all three energies are quite comparablewith the experimental results. After grinding for 4 min, the pre-dictions are still close to the measured results although the predictionbased on the collision energy slightly underestimates the product sizeat the finest size. Meanwhile, all predictions for mill-B are comparable

b

1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Grinding time, t (min)

t 50

22rpm31rpm36rpm45rpm67rpm

Fig. 12. Variations of (a) the cumulative size distribution of the ground samples withtime at the rotation speed of 36 rpm; and (b)mass fraction of particles smaller than 50%of original particle size with time for different rotation speeds.

Fig. 13. Particle flow in Mill-C at different loading of grinding balls. ω=36 rpm.

89M.H. Wang et al. / Powder Technology 223 (2012) 83–91

to the experimental data after 0.5 min of grinding, but slightlyunderestimate the amount of fine particles after 4 min of grinding forthe maximum impact energy. Again, the predictions based on thedifferent energies are similar if α is properly calibrated. In thefollowing, the collision energy will be used to investigate how theoperation conditions of mills affect the grinding performance.

3.3. Effects of operation conditions

In this section, the effects of operation conditions such as rotationspeeds, the loading of grinding balls and ground materials areinvestigated. Here the mill loading is defined as the ratio of millvolume occupied by particles to the total volume of the mill. The

b

10 20 30 40 500

10

20

30

40

50

Loading of grinding ball, JB (%)

Ave

rage

col

lisio

n en

ergy

, E (

mJ)

1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Grinding time, t (min)

t 50

10%20%30%40%50%

Loading of grinding ball

a

Fig. 14. (a) Average collision energy at different grinding ball loadings; and (b)variation of mass fraction of particles smaller than half of original particle size with timefor different grinding ball loadings. ω=36 rpm.

grinding process in Mill-C is simulated so the data can be comparedwith the results in the previous section.

Rotation speed is one of the critical operation conditions affectingthe grinding process. Austin et al. [22] stated that the optimalperformance is usually obtained when mills rotate between 70% and85% of the critical speed. Napier-Munn et al. [23] also pointed out that80% of the critical speed is best for grinding in mills of relatively highloadings. Fig. 10 shows the flow patterns at different rotation speeds.The mill is 40% loaded with grinding balls (580 particles) and 20%loaded with ground materials (10800 particles). As the rotation speedincreases from 22 rpm to 45 rpm, more particles are lifted up and theparticle velocity increases significantly. The shoulder of the flow goesup and the flow moves from the cascading regime to the cataractingregime. The change in the flow pattern also results in the increasingcollision energy as shown in Fig. 11. The average collision energyincreases almost linearly from 22 rpm to the critical speed of 45 rpm,but drops sharply when the mill rotates at the speed of 67 rpm.

Fig. 12a shows the variation of the cumulative size distribution ofthe ground samples with time at the rotation speed of 36 rpm. Similardistributions can also be obtained for other speeds. The distributioncurves shift continuously towards the finer side as grinding pro-gresses. To quantitatively investigate the grinding performance underdifferent conditions, t50, defined as the mass fraction of particlessmaller than 50% of original particles [14], is plotted in Fig. 12b as afunction of grinding time. It shows that t50 increases linearly withtime for all cases although different slopes are observed for differentrotation speeds. The breakage rate (i.e. slope) increases as the rotationspeed increases from 22 rpm to 45 rpm, but decreases when therotation speed reaches 67 rpm, indicating that the optimal rotationspeed for grinding is probably between 36 rpm and 67 rpm. However,more work is required to quantitatively determine such value.

Mill loading is another critical parameter affecting the grindingprocess. Fig. 13 shows the flow patterns at the rotation speed of36 rpmwhen the loading of grinding balls varies from 10% to 50% andthe loading of ground materials is kept at 20%. The flow patterns arequite similar, showing the particles are lifted up to the shoulder. Theheights of the flow shoulders are also similar for all the cases.However, more particles are lifted up as the loading of grinding ballsincreases. Fig. 14 plots the variation of the average collision energywith loading and the evolution of particle size with time. The averageimpact energy does not change significantly, fluctuating between28 mJ and 42 mJ (Fig. 14a). This is because the flows show similarpatterns within the range investigated in this work. However, asshown in Fig. 14b, the grinding performance has increased signifi-cantly with increasing loading up to 40% beyond which the grindingperformance drops slightly at the 50% loading.

To investigate the effect the ground material loading, the loadingof ground materials varies from 4% to 24% while keeping the loadingof grinding balls at 40%. As shown in Fig. 15, more ground materials

Fig. 15. Particle flow at different loadings of ground materials. ω=36 rpm.

90 M.H. Wang et al. / Powder Technology 223 (2012) 83–91

are lifted up and then flow down to the free space of the mill. Fig. 16ashows the average collision energy decreases quickly first from 4% to16% of the loading due to the decrease in the free space for the particlecollisions. As the loading increases, the average impact energy variesslightly around 38mJ. Fig. 16b shows the variation of t50 with grindingtime at different ground material loadings at the rotation speed of36 rpm. With the constant grinding ball loading, less ground materialgives better grinding performance. However, the grinding efficiencyshould also be considered here to determine the optimal loading ofground materials.

b

4 8 12 16 20 240

50

100

150

200

Loading of ground materials, fc (%)

Ave

rage

col

lisio

n en

ergy

, E(m

J)

0 1 2 3 40

0.2

0.4

0.6

0.8

1

Grinding time, t (min)

t 50

8%12%16%20%24%

Loading of ground materials

a

Fig. 16. (a) Average collision energy at different loadings of ground powders; and (b)variation of mass fraction of particles smaller than half of original particle size with timefor different ground powder loadings. ω=36 rpm.

4. Conclusion

DEM simulations of binary particles in tumbling ball mills werecarried out. Different forms of energy, collision energy, dissipationenergy and maximum impact energy, were analysed. By linking theenergy on ground powders with PBM, the product size distributionsunder different conditions were predicted. It was observed that thecollision energy between the large-small particles can be used directlyto predict product sizes without a need of data fitting. Other twoenergies, on the other hand, require an adjustable parameter whichneeds to be calibrated. Based on the information of collision energy,the effects of mill rotation speed, loadings of grinding balls and theground materials on grinding performance were investigated.

Acknowledgement

Authors are grateful to the Australia Research Council (ARC) forthe financial support for this work.

References

[1] C.L. Prasher, Crushing and Grinding Process Handbook, Wiley, Chichester, 1987.[2] B.A. Wills, Mineral Processing Technology, Pergamon Press, Oxford, 1992.[3] L.G. Austin, A Review–Introduction to the mathematical description of grinding as

a rate process, Powder Technology 5 (1971/1972) 1–17.[4] K.J. Reid, A solution to the batch grinding equation, Chemical Engineering and

Processing 20 (1965) 953–963.[5] J.A. Herbst, D.W. Fuerstenau, Scale-up procedure for continuous grinding mill

design using population balance models, International Journal of MineralProcessing 7 (1980) 1–31.

[6] R.R. Klimpel, L.G. Austin, The back-calculation of specific rates of breakage andnon-normalized breakage distribution parameters from batch grinding data,International Journal of Mineral Processing 4 (1977) 7–32.

[7] J.A. Herbst, D.W. Fuerstenau, Zero-order production of fine sizes in comminutinonand its implications in simulation, Transaction AIME 241 (1968) 538–548.

[8] K. Rajamani, J.A. Herbst, Simultaneous estimation of selection and breakagefunctions from batch and continuous grinding data, Transactions of the Institutionof Mining and Metallurgy 93 (1984) C74–C85.

[9] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies,Geotechnique 29 (1979) 47–65.

[10] B.K. Mishra, R.K. Rajamani, The discrete element method for the simulation of ballmills, Applied Mathematical Modelling 16 (1992) 598–604.

[11] H.P. Zhu, Z.Y. Zhou, R.Y. Yang, A.B. Yu, Discrete particle simulation of particulatesystems: a review of major applications and findings, Chemical EngineeringScience 63 (2008) 5728–5770.

[12] A. Datta, R.K. Rajamani, A direct approach of modeling batch grinding in ball millsusing population balance principles and impact energy distribution, InternationalJournal of Mineral Processing 64 (2002) 181–200.

[13] H. Lee, H. Cho, J. Kwon, Using the discrete element method to analyze thebreakage rate in a centrifugal/vibration mill, Powder Technology 198 (2010)364–372.

[14] P.W. Cleary, R.D. Morrison, Understanding fine ore breakage in a laboratory scaleball mill using DEM, Minerals Engineering 24 (2011) 352–366.

[15] R.Y. Yang, R.P. Zou, A.B. Yu, Microdynamic analysis of particle flow in a horizontalrotating drum, Powder Technology 130 (2003) 138–146.

[16] R.Y. Yang, A.B. Yu, L. McElroy, J. Bao, Numerical simulation of particle dynamics indifferent flow regimes in a rotating drum, Powder Technology 188 (2008)170–177.

91M.H. Wang et al. / Powder Technology 223 (2012) 83–91

[17] P.Y. Liu, R.Y. Yang, and A.B. Yu, Dynamics of wet particles in rotating drums: Effectof liquid surface tension. Physics of Fluids. 23 013304–9.

[18] H.P. Zhu, Z.Y. Zhou, R.Y. Yang, A.B. Yu, Discrete particle simulation of particulatesystems: theoretical developments, Chemical Engineering Science 62 (2007)3378–3396.

[19] N.V. Brilliantov, F. Spahn, J.M. Hertzsch, T. Poschel, Model for collisions in granulargases, Physical Review E 53 (1996) 5382–5392.

[20] P.A. Langston, U. Tuzun, D.M. Heyes, Discrete element simulation in granular flowin 2D and 3D Hoppers:dependence of discharge rate and wall stress on particleinteractions, Chemical Engineering Science (1995) 967–987.

[21] C.C. Kwan, H. Mio, Y. Qi Chen, Y. Long Ding, F. Saito, D.G. Papadopoulos, A. CraigBentham, M. Ghadiri, Analysis of the milling rate of pharmaceutical powdersusing the Distinct Element Method (DEM), Chemical Engineering Science 60(2005) 1441–1448.

[22] L.G. Austin, R.R. Klimpel, P.T. Luckie, Process Engineering of Size Reduction, SME,New York, 1984.

[23] T.J. Napier-Munn, S. Morrell, R.D. Morrison, T. Kojovic, Mineral comminutioncircuits: their operation and optimisation, in: T.J. Napier-Munn (Ed.), JKMRCMonograph Series in Mining and Mineral Processing, Julius Kruttschnitt MineralResearch Centre, Brisbane, 1996.