Minerals Engineering 43–44 (2013) 91–101

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Minerals Engineering

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Predicting the effect of operating and design variables on breakage ratesusing the mechanistic ball mill model

Rodrigo M. de Carvalho, Luís Marcelo Tavares ⇑Department of Metallurgical and Materials Engineering, COPPE, Universidade Federal do Rio de Janeiro, UFRJ, Cx. Postal 68505, CEP 21941-972, Rio de Janeiro, RJ, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:Available online 6 November 2012

Keywords:GrindingModelingDiscrete element methodBall mill

0892-6875/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.mineng.2012.09.008

⇑ Corresponding author. Tel.: +55 21 2562 8538.E-mail address: tavares@ufrj.br (L.M. Tavares).

Batch grinding tests have been a very good tool to aid in understanding the effect of design and operatingvariables in ball milling, as well as in providing data for a couple of successful scale-up methods. Recently,a mechanistic model of the ball mill has been proposed, and the present paper describes its application inthe simulation of batch mills operating under a range conditions. First-order breakage rates have beenestimated using data from these simulations, and used to investigate the effect of operating and designvariables in milling. Predictions using the mechanistic model are then compared to those using the scale-up relationships proposed by Austin and collaborators and Herbst and Fuerstenau. The trends predictedusing the mechanistic model are in general agreement with the empirical models. Good correlation hasalso been observed between the simulated specific breakage rates and the specific mill power, which is inagreement with the scale-up method proposed by Herbst and Fuerstenau.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The traditional population balance model has been widely andsuccessfully used to describe comminution in tumbling mills(Austin et al., 1984; King, 2001). Although it can describe effectivelythe process, by accounting for the selection, breakage, internal clas-sification and transport in mills, the model itself has no intrinsiccapabilities to allow simulating the process under conditions thatare different from those that were used to fit its parameters. Thereason is that the population balance model is essentially a massbalance over a range of sizes, having no direct information on theunderlying physics of the process. In fact, a large number of papershas been published in this last 40 years or so to study, from tests inbatch mills, how ball mills respond to changes in a number of designand operating variables, including mill diameter, mill speed, ballfilling, ball size, ball density, and powder filling.

Perhaps the greatest relevance of understanding relationshipsbetween design and operating variables and mill performance isin developing scale-up rules, from which the performance of indus-trial mills can be predicted using data collected from tests con-ducted either in batch laboratory or continuous pilot-scale mills.Two of the most well-known ball mill scale-up procedures thatare based on the traditional population balance model were pro-posed by Herbst and Fuerstenau (1980) and Austin et al. (1984).Both consider that the breakage distribution function is mostlymaterial-dependent whereas the breakage rate (selection) function

ll rights reserved.

is influenced by both material and grinding environment. Herbstand Fuerstenau (1980) proposed a scale-up procedure in whichthe specific rate of breakage varies with the specific power inputto the mill. Austin et al. (1984), on the other hand, proposed empir-ical expressions to account for the effect of design and operatingvariables on the specific breakage rates.

With the aim of overcoming a number of limitations of the tra-ditional population balance model and of the scale-up approachesa mechanistic model of the ball mill has been proposed (Tavaresand Carvalho, 2009; Tavares and Carvalho, 2010). While maintain-ing the mass balancing capabilities of the population balance mod-el, the mechanistic model allows describing in much greater detailthe effect of operating and design variables since it decouplesmaterial from mill contributions in the process. This is accom-plished by both characterizing and modeling in detail each of thebreakage mechanisms (body and surface) and using the DiscreteElement Method to describe the mechanical environment withinthe mill, using the population balance model as the framework.

The present paper investigates the influence of selected operat-ing and design variables in grinding, by simulating batch ball millsusing the mechanistic model and calculating the equivalent first-order breakage rates. Simulation results are discussed and trendsobserved are compared to predictions using the scale-up proceduresproposed by Herbst and Fuerstenau (1980) and Austin et al. (1984).

2. Model description

The model considers that when particles suffer an impact, someof them undergo catastrophic breakage and some do not (Fig. 1). If

92 R.M. de Carvalho, L.M. Tavares / Minerals Engineering 43–44 (2013) 91–101

stresses are insufficient to cause catastrophic (body) fracture, theywill undergo surface breakage (abrasion/chipping) and will alsobecome progressively weaker.

The batch grinding process equation can be derived from a moregeneral formulation of the traditional population balance modelapplied to the microscale description of size reduction processes(Carvalho and Tavares, 2009). The equation that describes the ratesof changes in mass of material contained in size class i is

dwiðtÞdt

¼ xM½�Di;bðtÞ � Di;sðtÞ þ Ai;bðtÞ þ Ai;sðtÞ� ð1Þ

where M is the material load, also called mill hold-up. wi(t) is themass fraction of particles contained in size class i in the mill andx is the frequency of stressing events in the comminution machine.Functions A and D represent the rate of appearance and disappear-ance of material in class i due to fracture, being defined in Table 1,whereas subscripts b and s stand for body and surface breakagemechanisms, respectively.

In the equations listed in Table 1, p(E) is the distribution ofstressing energies E in the mill, mj is the mass of particles con-tained in size class j captured in each stressing event and p(e) isthe function that represents the energy split among particles. aij

and bij are the breakage functions in density form, correspondingto the mechanisms of surface and body breakage, respectively,the later dependent on stressing energy (Tavares and Carvalho,2009). ji is the first order surface breakage rate. The density distri-bution of stressing energies in the mill p(E) is calculated from thecumulative distribution, from P(E) = dp(E)/dE. The product eE isthe fraction of the impact energy that is absorbed by each particlecaptured in an impact event.

The body breakage function is described on the basis of expres-sions that account for the effect of particle size and stressingenergy on the size distribution of the progeny fragments. Thereduction in size of the progeny with increasing applied energyhas been described very successfully using the t10-procedure, orig-inally proposed by Narayanan and Whiten (1988). The size distri-bution from impacts at a specific energy Em on particlescontained in size class j is calculated by first estimating the fractionof material passing 1/10th of the original particle size, called t10,using the expression (Tavares, 2009)

t10;j ¼ A 1� exp �b0Em

Em50b;j

� �� �ð2Þ

where A and b0 are model parameters and Em50b,j is the medianmass-specific particle fracture energy of the particles that are

Not captured

Captured Not fractured

Fractured Remained in original size

Left original size

Fig. 1. Schematic diagram of the structure of the mechanisti

broken as a result of the impact of magnitude Em. When the colli-sion energy is higher than the fracture energy of the toughest par-ticle contained in size class j, then Em50b,j = Em50,j, otherwise itshould be calculated numerically (Tavares and Carvalho, 2009).The validity of Eq. (2) in describing single particle breakage datahas been demonstrated in other publications (Tavares, 2007).

From the t10 value calculated using Eq. (2), it is then possible toestimate the proportions passing (tn values) of different fractions ofthe original parent size for each stressing energy and particle sizeinvestigated. This is done with the aid of a model that is based onthe incomplete beta function, given by

tnðt10Þ ¼100R 1

0 xan�1ð1� xÞbn�1dx

Z t10=100

0xan�1ð1� xÞbn�1 dx ð3Þ

where each tn is the percent passing in a dj/n size, in which dj is theoriginal size of the particle. an and bn are parameters that are fittedto the experimental data. The size-normalisable and energy-specificbreakage function is then obtained by interpolating the various tns,so that the elements of the breakage matrix are calculated byBijðEmÞ ¼ interpðt10; tnÞ. The breakage function in density form maythen be calculated by

biiðEmÞ ¼ 1� BiiðEmÞbijðEmÞ ¼ Bi�1;jðEmÞ � BijðEmÞ

ð4Þ

Finally, surface breakage is modeled using data from the JKMRCtumbling test (Napier-Munn et al., 1996) assuming that breakagefollows first-order kinetics and a size-independent surface break-age rate (ji = j), being independent of collision energy. Its break-age function in the cumulative form is given by Aij ¼ ðdi=dAÞk,where dA and k are model parameters.

Eq. (1) should be solved simultaneously with the equation thatdescribes how the fracture probability distribution of each compo-nent varies with time, which may be calculated by

FiðE; t þ DtÞ ¼ GiF�i ðE; t þ DtÞ þ HiFiðE;0Þ þ IiFiðE; tÞ

Gi þ Hi þ Iið5Þ

where the fracture probability distribution of the original materialFi(E, 0) can be, generally, described well using the upper-truncatedlognormal distribution, given by (Tavares and King, 1998)

FiðE;0Þ ¼12

1þ erfln E� � ln E50;iffiffiffiffiffiffiffiffiffi

2r2i

q0B@

1CA

264

375 ð6Þ

None

Body (energy-specific)

Surface

BreakageFractureenergies

Unchanged

Reduced by damage

Body (energy-specific)

Unchanged

c model at the microscale (Tavares and Carvalho, 2009).

Table 1Definition of terms in Eq. (1).

Breakagemode

Rate of appearance (A)

Body (b) Ai;bðtÞ ¼Pi�1

j¼1wjðtÞR1

0 mjðEÞpðEÞR 1

0 bijðeE; tÞFjðeE; tÞpðeÞdedE

Surface (s) Ai;sðtÞ ¼Pi�1

j¼1wjðtÞjjR1

0 mjðEÞpðEÞR 1

0 aijðeE; tÞ½1� FjðeE; tÞ�pðeÞdedE

Rate of disappearance (D)Body (b) Di;bðtÞ ¼ wiðtÞ

R10 miðEÞpðEÞ

R 10 ½1� biiðeE; tÞ�FiðeE; tÞpðeÞdedE

Surface (s) Di;sðtÞ ¼ wiðtÞjiR1

0 miðEÞpðEÞR 1

0 ½1� FiðeE; tÞ�pðeÞdedE

R.M. de Carvalho, L.M. Tavares / Minerals Engineering 43–44 (2013) 91–101 93

where

E� ¼ Emax;iEEmax;i � E

ð7Þ

where E50,i is the median particle fracture energy, r2i is the variance

and Emax,i is the upper truncation value of the distribution.Typically, as particles become finer, they become tougher and

their specific fracture energy increases. An expression that can beused to describe that variation is given by

E50;i ¼ E1 1þ do

di

� �/" #

�mp;i ð8Þ

where di is the representative size of particles contained in the ithclass, E1, do and / are model parameters that should be fitted toexperimental data. The mean weight of particles contained in sizeclass i may be estimated by

�mp;i ¼ qbd3i ð9Þ

where q is the material specific gravity and b the volume shapefactor.

In analogy to Eq. (8), the standard deviation of the lognormaldistribution of particle fracture energies can vary with size, oftenincreasing as particle sizes decrease. An expression that can beused to describe this variation is

ri ¼ r1 1þ d0

di

� �h" #

ð10Þ

where d0and h are fitting parameters.In Eq. (5) Fi(E, t) is the distribution of fracture energies of the

material contained in size class i that did not suffer any impactevent during the time interval, whereas F�i ðE; t þ DtÞ is the distribu-tion of fracture energies of the particles in size class i that werecaptured in an impact event, but which did not fracture, being gi-ven by

F�i ðE; t þ DtÞ ¼

R E�i0 pðEkÞ

R 10

Fi ½E=ð1�DÞ;t��FiðeEk ;tÞ1�FiðeEk ;tÞ

h ipðeÞ dedEkR E�i

0 pðEkÞdEk

ð11Þ

and

D ¼ 2cð1� DÞð2c� 5Dþ 5Þ

eEk

E

� �2c5

ð12Þ

where E�i is the maximum fracture energy of particles contained inclass i, which is equal to Emax in the first time interval and FiðE�i Þ ¼ 1as comminution progresses (Tavares and Carvalho, 2009). The mod-el is able to account for the fact that particles that are stressed butdo not fracture in a stressing event, may become progressivelyweaker. The various terms in Eq. (5) are then given by

GiðtÞ ¼xDtMðtÞwiðtÞð1� jiÞ

Z 1

0miðEÞpðEÞ

Z 1

0½1� FiðeE; tÞ�pðeÞdedE

ð13Þ

which is the fraction of material in the class that has been damagedbut remained in the original size range, and

Hi ¼xDtMðtÞ

Xi

j¼1

wjðtÞZ 1

0mjðEÞpðEÞ

Z 1

0bijðeE;tÞFjðeE;tÞpðeÞdedEþAi;s

" #

ð14Þ

which is the fraction of material that appeared due to body and sur-face breakage, and

IiðtÞ ¼wiðtÞMðtÞ MðtÞ �xDt

Z 1

0miðEÞpðEÞdE

� �ð15Þ

which is the fraction of material that was not captured in the timeinterval.

The mass of material captured between two grinding media ele-ments (mi) may be estimated from the product of the number ofparticles caught between the media elements (Ncap,i) and the aver-age weight of the particles ð �mp;iÞ,

mi ¼ �mp;i Ncap;i ð16Þ

Given the evidence from earlier work (Höfler, 1990), that parti-cles in an unconfined bed normally only break when they are typ-ically squeezed down to a monolayer between grinding media, it isassumed that particles are modeled as a packed monolayer bed. Ifparticles had spherical shapes, and if they were arranged accordingto a dense hexagonal packing, then the number of particles cap-tured as a function of radius could be estimated by (Barrios et al.,2011)

Ncap;i ¼ 14þ 3

42rcdi

� �2for rc P di=2

¼ 1 for rc < di=2ð17Þ

where rc is the radius of the bed, also called radius of capture, and di

is the mean size of the particles caught by the colliding steel ball,estimated from the geometric mean of sieves containing the narrowsize fraction.

Barrios et al. (2011) proposed that the radius of capture can becalculated by adding the radius of contact due to geometry (rg) andthe radius of contact due to the elastic deformation in the vicinityof contact (re), giving

rc ¼ re þ rg ð18Þ

The radius of the elastic deformation zone, resulting from thecollision of two bodies, is given by Hertz contact theory as

re ¼15E8Ke

� �1=5

K2=5g ð19Þ

where Kg is the geometric constant of the contact, given by

Kg ¼rc1rc2

rc1 þ rc2ð20Þ

and the elastic constant of the contact is given by (Tavares and King,2004)

Ke ¼K1K2

K1 þ K2ð21Þ

where rci and Ki are the radius of curvature and the elastic stiffnessof each of the bodies in contact, respectively.

It is then necessary to account for the fact that the radius of thezone in which particles are captured increases, due to the influenceof their height and to the degree of penetration of the striking ballin the bed resulting from breakage. Barrios et al. (2011) proposed amodel in which the radius of contact due to geometry is calculated by

rg ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2

g � Kg � hDh

� �� �2s

ð22Þ

in which D is the maximum deformation of the bed during impact.In the case of a monolayer bed, the initial bed height (h) may be

0.0

0.2

0.4

0.6

0.8

1.0

0.1 1 10 100

Cum

ulat

ive

dist

ribut

ion

Normal collision energy - E (mJ)

0.62

1.10

Fig. 3. Normal collision energy spectra for DEM simulations presented in Fig. 2.Collision frequencies were 479.957 and 340.249 s�1 and median collision energieswere 0.62 mJ and 1.10 mJ for the mill with smooth and square lifters, respectively.

94 R.M. de Carvalho, L.M. Tavares / Minerals Engineering 43–44 (2013) 91–101

considered equal to di, although it is certainly influenced by particleshape.

The term D/h, called ‘‘maximum relative deformation of thebed’’ has been modeled by Barrios et al. (2011) using the empiricalexpression

Dh¼ ac 1� exp �bc

EE50b;i

� �c� �� �ð23Þ

in which ac, bc and c are constants that must be fitted to the data.Data from impacts involving different ball diameters, particle sizesand materials (copper ore and limestone) were fitted to Eq. (23)(Barrios et al., 2011), resulting in the constants ac = 0.3621,bc = 0.1517, and c = 0.4125, which were found to be reasonably validfor both materials and all size ranges studied.

In the description of the energy split function in the bed p(e), itis considered that the stressing energy is split equally among par-ticles positioned within the active breakage zone in the mill. In thiscase the energy distribution function may be given by

pðeÞ ¼ dðe� 1=Ncap;iÞ ð24Þ

where d is the Dirac delta function.Some of the assumptions in the mechanistic model of the ball

mill include:

(i) the mechanical collision environment is determined exclu-sively by the motion of grinding media and the energy avail-able for breakage is given by the energy loss in the collision;

(ii) breakage results only from the normal component of thecollisions;

(iii) balls and powder are perfectly mixed within mill charge;(iv) powder particles are involved in every collision event within

the mill, and particles are stressed as packed monolayer par-ticle beds; and

(v) the fracture strength of a particle does not vary if it was pro-duced as a progeny of a low-energy or a high-energy stress-ing event.

3. Model parameters

First, DEM contact parameters have been fitted from compari-sons of estimates of charge toe and shoulder position from imagescaptured of a 30.5 � 30.5 cm lab mill fitted with an acrylic cover tosimulations conducted using EDEM� software (Ramos et al., 2011).Besides that, comparisons have also been made between the shaftpower measured using a torque sensor to estimates of the total en-ergy loss in normal collision events per unit time. The optimum

Fig. 2. DEM simulations on the effect of liner design in a 0.6 m diameter mill op

contact parameters found were 0.35 for the coefficient of restitu-tion of ball–ball and ball–wall contacts, 0.34 for the coefficient ofstatic friction for ball–ball and ball–wall contacts, and 0.28 forthe coefficient of rolling friction of both contact types. The Pois-son’s ratio used for steel was 0.3, the shear coefficient, 10 GPa,and the specific gravity, 7800 kg/m3. All these values were main-tained constant for all simulated mills and conditions studied.

It is worth noting that DEM simulations have included only thegrinding media, not the particles, whereas physical tests have beenconducted with the mill fed also with powder. As such, the contactparameters estimated are representative of ball–ball collisions inwhich the contacts are accounting for the presence of particles,not steel-on-steel collisions.

Typical results from DEM simulations are shown in Fig. 2 for dif-ferent types of mill liners, showing that the effect can be signifi-cant, which is consistent with simulations conducted by otherauthors (Cleary, 2001). Collision energy spectra from these simula-tions have been compiled, by logging the normal components ofthe energy loss over the collision events that happen in the mill(Fig. 3). These have been truncated by eliminating collisions withmagnitude below 10�4 J (0.1 mJ), which were simply regarded asan artifact of DEM in tracking down the contacts (Powell et al.,2008). This allows increasing computation efficiency without los-ing relevant information on the collision events.

In addition to parameters describing the mechanical environ-ment, the mechanistic model of the ball mill also requires a

erating at 68% of critical speed, with 30% ball filling and 25 mm steel balls.

Table 2Summary of particle breakage parameters.

Copper ore Granulite Limestone #1 Limestone #2 Iron ore #1 Iron ore #2

E1 (J/kg) 60.0 130.7 7.0 150.0 44.9 16.8do (mm) 400 1.1 100 0.79 4.3 20.1/ (–) 0.45 1.99 0.80 1.30 1.28 0.84r1 0.40 0.90 0.39 0.30 0.46 0.40r0 (mm) 1 0.173 10.47 1 1 1r� 0 1.65 0.08 0 0 0a1.2/b1.2 0.51/11.95 0.43/10.26 0.19/7.78 0.08/8.76 0.98/5.99 1.78/22.03a1.5/b1.5 1.07/13.87 0.92/10.74 0.56/7.51 0.56/7.48 1.01/5.01 2.45/20.67a2/b2 1.01/8.09 1.31/9.15 0.78/5.55 1.31/7.57 1.36/3.80 1.53/8.26a4/b4 1.08/3.03 1.18/2.97 1.12/3.01 1.21/3.03 1.22/2.17 1.50/3.64a25/b25 1.01/0.53 0.93/0.49 1.17/0.54 0.98/0.50 0.95/0.67 0.82/0.42a50/b50 1.03/0.36 0.92/0.39 1.43/0.40 0.98/0.31 0.89/0.48 0.76/0.24a75/b75 1.03/0.30 0.90/0.31 1.92/0.42 0.95/0.22 0.92/0.41 0.70/0.16c 5.0 5.4 5.4 5.0 3.0 5.0q (g/cm3) 2.93 2.79 2.71 2.98 4.35 2.8b (–) 0.62 0.64 0.60 0.62 0.62 0.62A (%) 67.7 47.5 53.3 63.4 60.4 44.2b0 (–) 0.029 0.027 0.033 0.033 0.051 0.029ta (%) 6.7 6.7 1.6 5.0 1.6 0.2dA (mm) 0.25 0.25 0.21 0.21 0.21 0.30k (–) 0.3 0.3 0.3 0.3 0.3 0.3

20

40

60

80

100

Pass

ing

(%)

Initial

0.5 min

1 min

4 min

8 min

R.M. de Carvalho, L.M. Tavares / Minerals Engineering 43–44 (2013) 91–101 95

description of parameters characterizing the breakage response ofmaterials. A summary of these parameters for materials simulated,with widely different characteristics, is shown in Table 2. Whereasthe softest material (limestone #1) is characterized by an exceed-ingly low breakage strength, evident from the A�b in the JKDropWeight Test of 632, ta (abrasion index) of 3.1 in JKMRC standardtumbling test (Napier-Munn et al., 1996) and a Bond ball mill workindex (BWi) of 6.2 kW h/t, the toughest material, which corre-sponds to the copper ore, has comparably high strength, with A�bof 32, ta of 0.45 and BWi of 22.1 kW h/t. Such significant differencesin strength are illustrated in Fig. 4, which shows the effect of par-ticle size on the median particle fracture energies of the materialsstudied.

00.1 1 10

Size (mm)

Fig. 5. Typical results from simulations of batch grinding of limestone #1 (30% ballload, 25 mm ball size and 68% of critical speed in a 0.3 m diameter mill).

100

size

cla

ss

4. Simulations using the mechanistic model

4.1. Estimation of breakage rates

Simulations of grinding using the mechanistic model have beencarried out by imitating batch tests with narrow size samples, suchas those suggested as part of the scale-up method proposed byAustin et al. (1984) to determine breakage distribution and break-age rate functions. Particle sizes tested ranged from 75–106 lm to5.6–7.9 mm, covering a total of 13 size intervals. Typical results

10

100

1000

10000

1001010.1Med

ian

spec

ific

parti

cle

fract

ure

ener

gy -

E m50

(J/k

g)

Particle size (mm)

Copper oreGranuliteIron ore #1Iron ore #2Limestone #1Limestone #2

Fig. 4. Effect of particle size on the median particle fracture energies for thematerials studied.

1

10

0 5 10 15 20 25 30% m

ater

ial r

emai

ning

in to

p

Time (min)

0.21mm-0.15mm0.43mm-0.30mm1.67mm-1.18mm7.92mm-5.60mm

Fig. 6. Simulated disappearance plots of batch grinding tests for granulite (30% ballload, 25 mm ball size and 68% of critical speed in a 0.3 m diameter mill).

from simulations are shown in Fig. 5. Previous studies have dem-onstrated the good agreement between experiments and simula-tions using the mechanistic ball mill model (Tavares and

96 R.M. de Carvalho, L.M. Tavares / Minerals Engineering 43–44 (2013) 91–101

Carvalho, 2009; Tavares and Carvalho, 2010) but a detailed valida-tion of the model is the subject of a future publication.

Disappearance plots have been prepared as a function of parti-cle size and results are shown in Fig. 6. From these plots it is pos-sible, in analogy to the analyses of experimental data from batchgrinding tests, to estimate the equivalent first-order breakagerates. These are often estimated assuming the validity of theexpression (Austin et al., 1984)

dw1ðtÞdt

¼ �k1w1ðtÞ ð25Þ

which has the solution

lnw1ðtÞw1ð0Þ

� �¼ �k1t ð26Þ

where k1 are the specific breakage rates, given by the slope of thebest-fit line. The least squares best-fit values of the breakage rateshave been estimated from the simulated data up to a grinding timeof 30 min or until no more than 10% of the original materialremains.

Fig. 6 shows a general agreement with the hypothesis of first-order breakage rates, which forms the basis of the linear popula-tion balance model of ball mills (Herbst and Fuerstenau, 1980;Austin et al. 1984). However, it demonstrates that the model pre-dicts some deviations from straight lines, which represent first-order breakage rates. This is observed especially for coarser sizes,being consistent with observations from an earlier work by theauthors (Tavares and Carvalho, 2009), and is identified by Austinand co-workers as ‘abnormal breakage region’ (Austin et al., 1982).

0.01

0.1

1

10

1010.1

Spec

ific

brea

kage

rate

- k i (

1/m

in)

Particle size (mm)

Copper oreGranuliteLimestone #1Limestone #2Iron ore #1Iron ore #2

Fig. 7. Simulations on the effect of particle size on breakage rates of selectedmaterials (30% ball load, 25 mm ball size and 68% of critical speed in a 0.3 mdiameter mill).

Fig. 8. Simulations on the effect of mill speed: breakage rates as a function of particle sizfor Copper ore (30% ball load, 25 mm ball size in a 0.6 m diameter mill).

The influence of a number of variables in batch grinding is ana-lyzed as follows, including the effect of particle and ball size, milldiameter, speed, mill filling and liner configuration. The investiga-tion of the effect of powder filling (U) has not been conducted,since assumption (iv) in the current version of the model wouldmake predictions unrealistic. As such, all simulations in the presentwork have been carried out considering 100% powder (interstitial)filling.

4.2. Effect of particle size on specific breakage rates

The effect of particle size on the breakage rates of a number ofmaterials is illustrated in Fig. 7. The coarsest particle size simulatedwas limited to 7.9 mm, since the validity of the simulations usingthe mechanistic ball mill model becomes questionable as particlesize increases, given assumption (i) in the model. Fig. 7 reproducesthe well-documented trend of breakage rates with particle size,with an increase in breakage rates up to a maximum, at a particlesize of about 1–4 mm, following by a drop at coarser sizes for thedifferent materials. Simulated results, represented using symbols,were used in fitting parameters of the equation proposed by Austinet al. (1984):

ki ¼S1da

i

1þ ðdi=lÞKð27Þ

where S1, a, l and K are model parameters that must be fitted todata, with di and l given in millimeters. Parameter S1 approachesthe specific breakage rate of 1 mm particles as l increases.

4.3. Effect of mill speed on specific breakage rates

The effect of mill speed on the breakage rates is illustrated inFig. 8, which also shows its effect on the frequency of collisionsand the median collision energy loss. It shows that breakage ratesincrease significantly with fraction of critical speed, especiallyabove 20%, dropping above the critical speed. This is associatedwith the increase in median specific collision energies, and thereduction in their specific frequency (number of collisions dividedby ball load) for mill speeds above, about, 50%. It is important tostate, however, that the assumption of perfect mixing (iii) of ballsand powder charge becomes questionable as mill frequencies ap-proach and surpass the critical speed, since, in these conditions,particles are more amenable to centrifuge than the balls, giventheir finer sizes.

It is now worthwhile to compare the predictions using themechanistic model to estimates using the scale-up approaches byAustin et al. (1984) and Herbst and Fuerstenau (1980). In the

ϕ

e (left) and median impact energy and specific collision frequency from DEM (right)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 20 40 60 80 100 120

S 1/S

1 re

f

% mill speed - ϕc

Copper ore

Limestone #2

Austin et al. (1984)

Fig. 9. Simulated normalized breakage rates predicted by the mechanistic mode forselected materials and predictions using the model by Austin et al. (1984) andHerbst and Fuerstenau (1980) (30% ball load, 25 mm ball size in a 0.6 m diametermill).

0.0

0.5

1.0

1.5

0 10 20 30 40 50 60

S 1/S

1ref

Ball load (%)

Granulite

Copper ore reference condition

Fig. 11. Simulated normalized breakage rates predicted by the mechanistic modelfor selected materials, and predictions using the model by Austin et al. (1984) andHerbst and Fuerstenau (1980) (25 mm ball size and 68% of critical speed in a 0.9 mdiameter mill).

R.M. de Carvalho, L.M. Tavares / Minerals Engineering 43–44 (2013) 91–101 97

approach proposed by Austin and co-workers, mill frequency influ-ences the scale-up parameter S1 according to the relationship(Austin and Concha, 1993)

S1 / ðuc � 0:1Þ 11þ exp½15:7ðuc � 0:94Þ�

� �;

for 0:4 < uc < 0:9 ð28Þ

where uc is the fraction of critical speed.Herbst and Fuerstenau (1980) proposed that the specific break-

age rates vary in proportion to the ratio between the net millpower and the mill load (P/M). Different models have been pro-posed in the literature to describe the variation of mill power withmill design and operating variables (King, 2001) and, thus, predic-tions will vary with the chosen model. If the expression proposedby Rowland and Kjos (1978) is used, and assuming that the param-eter S1 from Eq. (27) can be used to describe the variation of thebreakage rates with particle size within the Herbst and Fuerstenau(1980) scale-up approach, then

S1 / uc 1� 0:1

29�10uc

� �ð29Þ

In order to conveniently compare values of the parameter S1

estimated from predictions using the mechanistic model to thosepredicted by the scale-up rules, results were represented as the ra-tio between S1 values estimated for the different percentages ofcritical speed and a reference value, arbitrarily taken as 68% of

0.1

1

10

0.1 1 10

Spec

ific

brea

kage

rate

- k i (

1/m

in)

Particle size (mm)

10%20%30%40%50%

Ball load

Fig. 10. Simulations on the effect of mill filling: breakage rates as a function of particle si(25 mm ball size and 68% of critical speed in a 0.9 m diameter mill).

critical speed. Fig. 9 shows that the different approaches provideconsistent predictions, although not identical. The mechanisticmodel predictions were positioned within the range of values esti-mated using Eqs. (28) and (29). It is important to realize, however,that the Herbst and Fuerstenau (1980) and Austin et al. (1984) pre-dictions are independent of feed material when plotted in a graphsuch as the one shown in Fig. 9, whereas the mechanistic model ac-counts for material effects. Further, some studies in the literature(Shoji et al., 1982) have shown that the relationship betweenpower and percent of critical speed does not remain invariant withother variables, such as mill diameter, filling and liner configura-tion, which further limits the validity of Eqs. (28) and (29), inparticular for small diameter mills.

4.4. Effect of mill filling on specific breakage rates

Predictions on the effect of mill filling on the breakage rates areillustrated in Fig. 10. It shows that mill filling has a very limited ef-fect on the breakage rates over a wide range of simulated condi-tions. Fig. 10 also shows that the magnitude of the collisionsdecrease with an increase in ball load, while their specific fre-quency first increases significantly, then drops for mill fillingsabove 20%.

Results such as those from Fig. 10 can then be compared to pre-dictions using the scale-up relationships proposed by Austin et al.(1984) and Herbst and Fuerstenau (1980). Austin et al. (1984)

1

10

0

1

2

3

4

5

0 20 40 60

Median collision energy -E

(mJ)

Spec

ific

collis

ion

frequ

ency

-/M

b

(1/t.

s)

Ball load (%)

ze (left) and median collision energy and specific frequency (right) for limestone #2

0.01

0.1

1

10

0.1 1 10

Spec

ific

brea

kage

rate

-k i

(1/m

in)

Particle size (mm)

0.3 m0.6 m0.9 m1.8 m

Mill diameter

0.1

1

10

0 0.5 1 1.5 20

1

2

3

4

5 Median collision energy -E

(mJ)

Spec

ific

collis

ion

frequ

ency

-ω

/Mb

(1/t.

s)

Mill diameter (m)

Fig. 12. Simulations on the effect of mill diameter: specific breakage rates as a function of particle size (left) and median collision energy and specific frequency (right) forCopper ore (25 mm ball size, 30% mill filling and 68% of critical speed).

98 R.M. de Carvalho, L.M. Tavares / Minerals Engineering 43–44 (2013) 91–101

proposed that the relationship between parameter S1 and the frac-tional mill filling (J) could be described by

S1 /1

1þ 6:6J2:3 ; for 0:2 < J < 0:6 ð30Þ

whereas in the approach proposed by Herbst and Fuerstenau(1980), considering Rowland and Kjos (1978) mill power model,the relationship would be (Austin and Concha, 1993)

S1 / Jð1� 0:937JÞ ð31Þ

Fig. 11 compares the results and shows that the mechanisticmodel is in reasonable agreement with predictions from thescale-up methods within the range of mill loads from 20% to50%. The figure once again shows the material-specific effect cap-tured by the mechanistic model, which is not taken into accountusing either scale-up method.

4.5. Effect of mill diameter on specific breakage rates

The effect of mill diameter on the breakage rates is illustrated inFig. 12, which shows that it is significant. The increase in milldiameter did not only increased the specific breakage rates forthe different sizes simulated, but also shifted the maximum in eachcurve to coarser sizes, which is an effect that has been observed byother researchers (Austin et al., 1984). This shift is associated to theincrease in the magnitude of the impacts, which is evident in the

0.1

1

10

0 0.5 1 1.5 2

S1/

S1r

ef

Mill diameter - Dm (m)

Copper ore

Granulite

reference condition

Fig. 13. Simulated normalized breakage rates predicted by the mechanistic modelfor selected materials and predictions using the model by Austin et al. (1984) andHerbst and Fuerstenau (1980) (25 mm ball size, 30% mill filling and 68% of criticalspeed).

variation of the median collision energy with mill diameter shownin Fig. 12.

Austin and Concha (1993) observed that there are very few di-rect measurements of breakage rate and breakage distributionfunctions of large-diameter mills and that back-calculating thesefunctions from continuous grinding data is subject to significanterrors. As such, the scale-up rules have been mostly developedfor smaller diameter mills, being extrapolated to larger mills. Theeffect of mill diameter on the scale-up methods can be describedby (King, 2001)

S1 / Dsm ð32Þ

Austin et al (1984) proposed using s = 0.5, whereas Rowlandand Kjos (1978) proposed 0.3 for the relationship between the ratioof power and mill load and mill diameter.

Fig. 13 compares predictions using the mechanistic model onthe effect of mill diameter to results from Eq. (32). All approachesshow an increase in specific breakage rates with mill diameter, butthe mechanistic model predicts a much more significant effect.Recognizing the importance of mill power on breakage rates, thepowers calculated by DEM were compared to those calculatedusing the expression proposed by Rowland and Kjos (1978), andit was found that values matched for the 0.3 m diameter mill,but differ as mill diameter increased, being three times higher forDEM for the 1.8 m. This demonstrates that the assumption consid-ered in the present work that contact parameters used in DEM sim-ulations could be maintained the same from mills with differentdimensions was not valid.

4.6. Effect of ball size on specific breakage rates

The effect of ball size on specific breakage rates predicted usingthe mechanistic model is illustrated in Fig. 14 for selected materi-als, and shows that the model is capable of accounting for the shiftin the size corresponding to the maximum breakage rate with achange in ball size, which has been widely documented in the lit-erature (Austin et al., 1984; Austin and Concha, 1993; Erdem andErgun, 2009; Katubilwa and Moys, 2009). Smaller ball sizes pro-duce less energetic impacts (Fig. 15) and each impact captures few-er particles (Barrios et al., 2011), being also less efficient at nippingcoarser particles. Offsetting these effects that tend to decrease thespecific rate of breakage as ball size decreases is the increased fre-quency of impacts that results from the larger number of smallerballs in the mill (Fig. 15).

Fig. 14 also shows that the slope of the line at finer sizes, repre-sented by the fitting parameter a is Eq. (27), may or not vary withball diameter. In several studies reported in the literature it was

0.01

0.1

1

Spec

ific

brea

kage

rate

-k i

(1/m

in)

Particle size (mm)

15 mm25 mm40 mm60 mm

Ball size

0.1

1

10

0.1 1 10 0.1 1 10

Spec

ific

brea

kage

rate

-k i

(1/m

in)

Particle size (mm)

15 mm25 mm40 mm60 mm

Ball size

Fig. 14. Simulations on the effect of ball size: breakage rates as a function of particle size for a lab scale ball mill for two materials, Copper ore (left) and limestone #2 (right)(30% mill filling and 68% of critical speed in a 0.3 m diameter mill).

0.1

1

10

0.0

0.1

0.2

0.3

0.4

0.5

0 20 40 60 80

Median collision energy (m

J)

Spec

ific

collis

ion

frequ

ency

-ω

/Mb

(1/t.

s)

Ball diameter (mm)

Fig. 15. DEM median impact energy and collision frequency as function of ball size(30% mill filling and 68% of critical speed in a 0.3 m diameter mill).

S(1

/min

)S

1(1

/min

)

0.0

0.

1

01

.1

1

0

1

CoGrIroIroLimLim

oppranuon oon omesmes

per ouliteore ore stonston

oree#1#2ne #ne #

#1#2

10Speeciffic ppowwer - PP/MM (kW

100W/t))

1

10000

Fig. 17. Specific breakage rate parameter S1 estimated from predictions using themechanistic model as a function of mill specific power.

R.M. de Carvalho, L.M. Tavares / Minerals Engineering 43–44 (2013) 91–101 99

assumed that this slope remained constant as ball sizes varied,although it is recognized that this assumption was less due toexperimental evidence and more because of lack of precise infor-mation for finer sizes (Austin et al., 1984; Katubilwa and Moys,2009). Indeed, variations in the slope of the specific breakage ratefunctions with ball sizes have been observed by a number ofresearchers (Herbst and Lo, 1992; Martinovic et al., 1990). The sim-ulated results from Fig. 14 demonstrate that these two situationscan occur, depending on material breakage properties.

It is important to realize that the different scale-up methodsrecognize the effect of ball size not only on the specific breakagerates, but also on the breakage distribution function. Herbst andFuerstenau (1980) suggested that batch tests should be conducted

0.01

0.1

1

Spec

ific

brea

kage

rate

-k i

(1/m

in)

Particle size (mm)

Square, Dm=0.3m

Square, Dm=0.6m

Smooth,Dm=0.3m

Smooth, Dm=0.6m

0.1 1 10

Fig. 16. Effect of liner profile, depicted in Fig. 2, on breakage rates of Copper o

with the ball size distribution that would be used in the industrialmill, primarily with the aim of describing the breakage distributionfunction. On the other hand, Austin et al. (1984) suggested con-ducting batch grinding tests with different ball diameters and thencalculating the weighed-average breakage function for the actualball size distribution that would be used in the full-scale mill. In-deed, these alternatives have been successfully used in the past,but a challenge remains when larger ball sizes are to be used inan industrial mill. Tests in small diameter batch mills should notbe conducted using large ball sizes, since the motion of balls ofsuch diameters will bear very limited resemblance to what hap-pens in the operation of an industrial mill (Austin et al., 1984).As such, the mechanistic model has a significant potential benefit

0.1

1

10

0.1 1 10Spec

ific

brea

kage

rate

-k i

(1/m

in)

Particle size (mm)

Square, Dm=0.3mSquare, Dm=0.6mSmooth,Dm=0.3mSmooth, Dm=0.6m

re (left) and Granulite (right) (30% mill filling and 68% of critical speed).

0

20

40

60

80

100

0.01 0.1 1 10

Pas

sing

(%

)

Particle size (mm)

90% / 30%

68% / 40%

% mill speed / Ball load

Fig. 18. Product size distributions predicted using the mechanistic model fromgrinding the Copper ore in a 1.8 m diameter mill operating at two differentconditions, which resulted in the same specific power (186 kW/t) but different sizedistributions in a batch mill from a constant feed size distribution. t1 and t2

represent two different grinding times.

100 R.M. de Carvalho, L.M. Tavares / Minerals Engineering 43–44 (2013) 91–101

over the other scale-up approaches when dealing with predictionsof mills that use large ball sizes. This is particularly relevant, giventhe current trend in the minerals industry of using mills of largerdiameters, which are being progressively used to process increas-ingly coarser feeds and, thus, requiring larger diameter make-upballs.

4.7. Effect of mill liner configuration

The effect of mill liner on the specific breakage rates predictedusing the mechanistic model is illustrated in Fig. 16, which showsthat it has a significant effect in the mills studied. When taller and,thus, more aggressive liners (Fig. 2) are used, the specific breakagerates of particles over the range of sizes also increase when com-pared to the nearly smooth liners. This is a variable which hasnot been incorporated in any of the empirical scale-up methodsin the literature and its effect in grinding can be assessed usingDEM-based scale-up models, such as the mechanistic ball millmodel.

4.8. Specific power and breakage rates predicted using the mechanisticmodel

In the scale-up approach proposed by Herbst and Fuerstenau(1980), specific breakage rates vary directly with specific millpower, that is, the ratio between the net mill power and the millload. This is in qualitative agreement with Bond’s method ofdesigning ball mills, in which the fineness of a ball mill productis dictated primarily by specific energy and ore grindability (Austinand Concha, 1993; King, 2001). Although not explicitly, the scale-up approach proposed by Austin et al. (1984) also incorporatesthe influence of operating and design variables in grinding as theyaffect mill power through functional forms that are similar to thosegiven, for instance, by Eqs. (28) and (30). In the mechanistic ballmill model, the frequency and the distribution of collision energies,rather than the mill power, is used, besides the material breakageproperties, as input to the simulations. Since these pieces of infor-mation are highly convoluted in the mechanistic model (Table 1), itis worthwhile analyzing if the predicted specific breakage rates arerelated to mill power.

Fig. 17 compares for all materials and grinding conditions sim-ulated the relationship between specific net mill power, given bythe ratio of mill power calculated by DEM and mill load, and thespecific breakage rate parameter S1. Condensing a total of over fivethousand simulations that included six materials over 13 size clas-ses and conducted with a range of conditions, including mill andball diameters, liner designs, mill fillings and speeds, Fig. 17 shows

that a material-specific signature appears. Indeed, the approxi-mately linear relationship between S1 and the net specific millpower is consistent with the scale-up approach proposed byHerbst and Fuerstenau (1980), which is based on the definitionof the material and energy-specific breakage rates in ball milling.

It is important to realize, however, that although the mechanis-tic model recognizes the dominant effect of specific power onbreakage rates, the scatter around the trend line suggests thatother effects can have a smaller, but important role. This is illus-trated in Fig. 18, which compares predictions from batch grindingat the same specific power for the same mill but operating at dif-ferent conditions such as mill speed and ball load.

5. Conclusions

Simulations of grinding narrow size samples using the mecha-nistic ball mill model allowed estimating first-order breakage ratesas a function of a number of variables for selected materials. Thepredicted influence of particle and ball size, mill filling, speedand frequency was in general qualitative agreement with evidencefrom the literature, as well from predictions using the traditionalscale-up methods of Herbst and Fuerstenau (1980) and Austinet al. (1984).

Unlike the traditional scale-up methods, the mechanistic ballmill model is able to predict synergic effects between materialand milling variables.

For each given material, specific breakage rates predicted usingthe mechanistic model were found to correlate with the specificpower in comminution. Nevertheless, different combinations ofmilling variables that resulted in the same net specific mill poweralso were found to yield different breakage rates and size distribu-tions. This demonstrates that, similarly to the traditional scale-upmethods, the mechanistic ball mill model responds primarily tospecific power. Nevertheless, the individual milling variables alsoplay a role that is described in the model.

Acknowledgements

The authors would like to thank the financial support from theBrazilian research agencies CNPq and CAPES, as well as from VALEand AMIRA, through the P9O project. The authors also thank DEMSolutions for providing the software EDEM� through the AcademicProgramme.

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