Minerals Engineering 63 (2014) 100–109

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Minerals Engineering

journal homepage: www.elsevier .com/ locate/mineng

Scale-up of batch grinding data for simulation of industrial millingof platinum group minerals ore

http://dx.doi.org/10.1016/j.mineng.2014.01.0230892-6875/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +27 73 183 8174.E-mail address: ngodzazw@gmail.com (N. Chimwani).

1 The COMPS unit has now relocated to The University of South Africa, where it iscontinuing its work under the name of Material and Process Synthesis.

Ngonidzashe Chimwani a,⇑, François K. Mulenga b, Diane Hildebrandt b, David Glasser b,Murray M. Bwalya a

a School of Chemical and Metallurgical Engineering, University of the Witwatersrand, Private Bag X3, Wits 2050, Johannesburg, South Africab Department of Electrical and Mining Engineering, University of South Africa (UNISA), Florida Campus, Private Bag X6, Johannesburg 1710, South Africa

a r t i c l e i n f o

Article history:Available online 7 February 2014

Keywords:Attainable regionScale-up procedurePopulation balance modelSelection functionBreakage functionMilling parameters

a b s t r a c t

The attainable region analysis of batch grinding is a graphical method of establishing limits of perfor-mance of possible outcomes of any defined process. When applied to batch grinding, the results haveoften come into conflict with traditionally acceptable milling practice under which most concentratorsoperate (Metzger et al., 2011). This novel technique has not been tested on an industrial scale and thus,application of the scheme to full scale industrial mill is naturally the next step.

This research entailed scaling-up data obtained from laboratory batch milling of a platinum ore usingempirical models. Using the parameters obtained in the laboratory tests, the authors applied a schemedeveloped by Austin et al. (1984) to predict the selection function and breakage function parametersfor an operational industrial mill on which some plant survey had been conducted. It was found thatthe simulated product size distributions based on this scale up-procedure displayed a close match withthe actual obtained from an operational industrial mill.

The attainable region plots from the scaled-up data showed that a finer product is achieved by usingsmall balls. This is in agreement with initial findings based on laboratory batch tests only. It is also antic-ipated that pilot tests, industrial tests or simulations should be the next step in the quest for bridging thegap between the attainable region methodology and industrial experience. It has also been validated onindustrial scale that less powder and grinding balls are needed to achieve finer grinding. However, it wasinteresting to note that the factors that produced a coarser product when analysed from a particle pointof view were the same as those that yield the greatest amount of the desired size class when viewed froman attainable region perspective.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction proportion of the costs involved in operating processing plants.

The work described in this paper was part of a Minerals Process-ing project that has been ongoing in the internationally-recognisedCentre of Material Process and Synthesis (COMPS), University ofthe Witwatersrand, for many years. Many of the innovations sug-gested by researchers at COMPS as a result of their own investiga-tions have built on the previous contributions of their colleagues inthis ever-expanding field, which explains why many of the refer-ences to previous publications mention names associated withthe Centre1.

In industrial milling, size reduction is commonly known tobe a highly energy-intensive process that accounts for a major

This explains why engineers designing concentrators strive tooperate the grinding systems and circuits in the most energy-efficientway. To identify optimal design configurations, researchers haveresorted to laboratory batch tests to establish selection and break-age function parameters that enable them to identify optimaldesign configurations (Herbst and Fuerstenau, 1980; Herbst et al.,1981; Austin et al., 1984; Rajamani, 1991; King, 2001; Datta andRajamani, 2002).

The Attainable Region (AR) approach, initially proposed for theanalysis of chemical engineering systems, has now been extendedto comminution processes and has already been applied success-fully to optimise batch milling of different types of ore (Mulengaand Chimwani, 2013). Research carried out on laboratory millshas demonstrated the capacity of the AR technique to determinethe set of all the achievable size distributions under different pro-cess conditions (Metzger et al., 2011). This information providesthe engineer with accurate data on the operating conditions

N. Chimwani et al. / Minerals Engineering 63 (2014) 100–109 101

required to achieve a specific objective function, such as thosementioned in the previous paragraph (Khumalo et al., 2006).

As an illustration of the above claim, the investigation carriedout by Khumalo et al. (2006, 2007, 2008) used the AR techniqueto obtain a desired product using a minimum amount of energy.The results from his undertaking showed how the optimisationproblem of the level of specific energy to be used in a given equip-ment, to achieve a certain objective function can be answered.Most importantly, the authors pinpointed the stage at which theenergy intensity in a comminution process can be controlled in or-der to develop an optimal energy regime. On similar lines, Metzgeret al. (2009) looked at means of minimizing the total time requiredfor operation. Their results showed how useful AR is in determin-ing optimal policies to reduce milling processing times. Katubilwaet al. (2011) analysed the effect of ball size on milling, and demon-strated clearly the advantage of mixing the sizes of the grinding ballsto produce a maximum amount of material in a target size range.

The question being addressed in this work is whether the ARtechnique can be successfully applied to industrial mills. The pres-ent work also seeks to demonstrate how the tool can explore out-puts that can assist in choosing optimal operating conditions. It isenvisaged that this technique will serve as a complimentary ana-lytical tool for the optimisation of milling circuits.

Initially, standard laboratory batch experiments on a Platinumore to determine ore breakage characteristics were conductedand an empirical scale-up procedure was carried for full industrialmill application. After validating the scaled-up parameters withindustrial survey data, the attainable region scheme was used toinvestigate the extent to which variations in the selection functionparameters (l, a, a and K, as symbolically presented by Austinet al. (1984)) affect the final product distributions in an industrialset-up. The results demonstrate that the AR technique can be usedas a good tool for the design and analysis of mineral processingcircuits.

2. Literature review

2.1. Population balance models

Population balance modelling has found widespread use in thesimulation, control and optimization of various particulate pro-cesses. It is one of the most comprehensive tools for analysingand tailoring the particle size distribution (PSD) resulting from asize reduction process; investigating the grinding process; andstudying the breakage mechanisms involved. The latter includemassive fracture, cleavage and attrition (Bilgili et al., 2005). Thepopulation balance modelling approach requires an understandingof two concepts: the selection and the breakage functions. Both ofthese functions can be applied to predict grinding results once alltheir parameters are known.

As far as the selection function (Si) is concerned, Austin et al.(1984) have proposed an empirical relationship that has gainedwide acceptance by many researchers (King, 2001; Yekeler, 2007;Tavares and Carvalho, 2013), which is expressed as follows:

Si ¼ a � xai � Q i ¼ a � xa

i1

1þ xil

� �K ; on condition that K P 0 ð1Þ

where xi is the upper limit in the screen size interval i in mm, K anda are characteristic constants which are dependent on materialproperties, a is a characteristic constant dependent on mill condi-tions and can also dependent on material properties since it implieshow fast grinding occurs (Makokha and Moys, 2006), l is a param-eter dependent on mill conditions, Qi is the correction factor (takenas unity for lower size classes, i.e. xi� dball), dball is the diameter ofthe spherical grinding media.

Eq. (1) contains four parameters, two of which are dependenton mill conditions (a and l) while the other two (K and a) arematerial-dependent.

Austin et al. (1984) also proposed an empirical model relatingthe cumulative breakage function Bi,j to particle size xi. The modelhas three parameters (c, U and b) which have been found to de-pend on the properties of the material being ground. For a normal-isable material, that is, a material that breaks following the samepattern regardless of relative size, the cumulative breakage func-tion Bi,j is given by:

Bi;j ¼ Ujxi�1

xj

� �c

þ ð1�UjÞxi�1

xj

� �b

; on condition that n P i

P j P 1 ð2Þ

where b is a parameter characteristic of the material used, the val-ues of which are generally greater than 2.5, c is a material depen-dent parameter, the values of which are typically found to begreater than 0.6, Uj represents the fraction of fines that are pro-duced in a single fracture step, dependent on the material usedand ranges from 0 to 1.

With the selection function and breakage function known, amass balance in each size interval can be performed. Ultimately,all the amount of material in each individual size class can be cal-culated using the population balance model (PBM) of milling. Inthe case of batch milling, the model reduces to the batch grindingequation (Reid, 1965):

dwiðtÞdt

¼ �SiwiðtÞ þXi�1

j¼1i>1

bi;jSjwjðtÞ; on condition that n P i

P j P 1 ð3Þ

where wi(t) and wj(t) are the mass fractions of size i and j materialpresent in a mill at time t respectively, bi,j is the discretized break-age function which gives the proportion of the particles of j that re-ports to size fraction i after one breakage event: bi;j ¼ ðBi�1;j � Bi; jÞ,Si is the selection function of the material considered of size i.

Eq. (3) describes the evolution of mass fractions within eachsize class interval as a function of time while taking into accountall sub-processes of breakage such as the birth and death of parti-cles (King, 2001).

2.2. Factors affecting the breakage rate

In tumbling ball mills, the rate of breakage (Si) and overall millperformance are affected by fractional ball filling (J), fraction of themill volume filled by powder (fc), powder filling (U), fraction of crit-ical speed (/c), and ball diameter. These are discussed below.

2.2.1. Ball fillingFractional ball filling (J) is conventionally expressed as the frac-

tion of the mill volume filled by the ball bed at rest, assuming a for-mal bed porosity of 0.4. It can be calculated as follows:

J ¼

mass of ballsball density

� �

mill volume� 1:0ð1� 0:4Þ ð4Þ

The rate of breakage has been found to depend primarily onhow much of the mill volume is filled with grinding balls, i.e. ballfilling (J). Indeed, as the mill rotates, the grinding media reach apoint where the balls will either be thrown off or roll off the linersand the charge separates from the shell, forming the shoulder ofthe charge. In the former case, the motion is referred to as cataract-ing whereas the latter as cascading. In general, the mill load willexperience a combination of both. Fortsch (2006) reported that

102 N. Chimwani et al. / Minerals Engineering 63 (2014) 100–109

operating a ball mill at 75% of critical speed and 22% charge levelhas a higher cataracting region than at 75% critical speed and40% charge level. These researchers further observed that cascad-ing action provides the highest grinding efficiency because of thehigh total surface area in contact. Finally, they argued that as ballfilling is increased, the breakage rate initially increases to a maxi-mum and then decreases. That is why in practice, ball fillings usu-ally range between 20% and 40%.

2.2.2. Mill filling by powderThe fraction of mill filled by powder (fc) is expressed as the

function of the mill volume filled by powder bed, using a formalbed porosity of 0.4. The fraction of mill filled by powder (fc) hasto be determined for every mill operation where a different J isused since the mill should not be under-filled or overfilled.Under-filling the mill leads to energy wasted in steel-to-steel con-tacts. This produces no breakage but rather unwanted materialwear. Overfilling the mill, on the other hand, leads to an effectcalled powder cushioning, which decreases the efficiency of thebreakage action. That is why it is important to fill the mill withthe appropriate volume of powder.

In order to relate the powder loading to ball loading, the formalbulk loading of powder is compared to the formal porosity of theball bed (Austin et al., 1984). This way, the notion of powder fillingU can be introduced, that is, the fraction of the spaces between theballs, at rest that is filled with powder:

U ¼ fc

0:4Jð5Þ

Austin et al. (1984) reported that powder fillings U between 0.6and 1 will generally give the most efficient breakage in the mill.

2.2.3. Critical speedThe critical speed of the mill is the theoretical rotational speed

at which balls just centrifuge on the mill case rather than tumble. Itis given by:

Critical speed /c; rpm ¼ 42:2ffiffiffiffiffiffiffiffiffiffiffiffiD� dp ð6Þ

where D is the internal mill diameter and d is the maximum balldiameter loaded into the mill, both expressed in meters.

The rotational speed of the mill is normally specified as a frac-tion of the critical speed /c. For a ball mill with effective lifters,industrial mill rotational speeds in use are found to be between70% and 80% of critical speed (Shoji et al., 1982).

2.2.4. Ball diameterAustin et al. (1984) presented results which showed the effect

of ball size on the breakage function Bi,j based on the experimentsthey did on dry grinding of quartz. They suggested that theproduction of the majority of the fines was caused by the greaterimpact force of collisions involving larger balls, resulting in a low-er value of the breakage function parameter c and a higher valueof U.

The breakage function parameter c is an indication of the rel-ative amount of fines produced from the breakage of the top sizematerial. It therefore relates to the efficiency of grinding. A highervalue of c means grinding takes place at a slow rate while a lowervalues means more effective breakage action with high produc-tion of fines. The parameter U is dependent on the material usedand is the fraction of fines produced in a single fracture. Thus thelower specific rate of breakage due to larger balls is partially com-pensated for by the production of a bigger portion of finefragments.

2.3. Scale-up procedure for batch grinding data

Scale-up is done on a material whose parameters for breakageand selection function parameters have been determined in thelaboratory. In the development of scale-up models material spe-cific parameters must be distinguished from those that are millspecific (King, 2001). The breakage function parameters do notneed any scale-up if the material is considered normalisable, thatis, if the parameter U is constant for all breaking sizes. The empir-ical equations, which predict how the selection function valueschange with ball and mill diameters (d and D), ball filling (J), pow-der filling (U) and rotational speed (/c) are combined as follows(Austin et al., 1984):

SiðdÞ ¼ aTxi

x0

� �a

� 1

1þ xiC1lT

� �K � C2 � C3 � C4 � C5 ð7Þ

The Constant terms C1 to C5, are correction factors applied dur-ing scale-up and have been discussed in detail elsewhere (Austinet al., 1984, 2007)

Eq. (7) enable one to predict the scaled-up selection function ofparticle size class i for a given ball diameter d based batch grindingtests. The equation allows scale-up of batch laboratory data.

Since a and l depend on the conditions and the geometricalscale of the mill, their values have to be scaled up to the conditionsof the mill to be simulated. The scaled-up value of aT in Eq. (7) forthe new mill condition is a� and the conversion from testconditions to the conditions in another mill is (Austin et al.,2007) as given in Eq. (9):

a� ¼aT � D

DT

� �0:5� 1þ6:6J2:3

T

1þ6:6J2:3

� ��exp½1�c � ðU�UTÞ�; D63:8m

aT � 3:8DT

� �0:5� D

DT

� �0:3� 1þ6:6J2:3

T

1þ6:6J2:3

� ��exp½1�c:ðU�UTÞ�; D>3:8m

8><>:

ð8Þ

In a similar fashion and for a different ball diameter d, the valueof l� is converted to the following (Austin et al., 2007):

l� ¼ lT :ddT

� �g

ð9Þ

where dT is the diameter of the balls used in the laboratory mill, d isthe ball diameter of the simulated industrial mill.

The exponent value g varies between 1 and 2 depending on thematerial used. Kelsall et al. (1968), for instance, proposed the valueof 2 based on experiments they did using quartz; Yildirim et al.(1999) later found the value of 1 to simulate well a quartz grindingcircuit. Austin et al. (2007), on the other hand, used 1.2 in the anal-ysis of an iron ore grinding circuit. By the same token, Austin et al.(1984) and Napier-Munn et al. (1996) both reported the value of 2to be the best default value as far as g is concerned. Also note thatKatubilwa and Moys (2009) found that g = 2 was a reasonable va-lue for a South African coal tested on a laboratory scale.

3. Experimental method

3.1. Laboratory-scale batch grinding tests

To determined the breakage and selection function parameters,laboratory experiments were conducted in a batch mill with aninternal diameter (DT) of 0.30 m and length (LT) of 0.28 m at the Uni-versity of the Witwatersrand. The mill is fitted with 8 90�, 13 mmhigh and 25 mm wide lifters. Mono-sized ball size and a single feedsize class were used in each of the tests. Media sizes used were10 mm, 20 mm and 30 mm steel balls and the feed size classes usedwere (�850 + 600 lm,�600 + 425 lm and�425 + 300 lm) of PGM

N. Chimwani et al. / Minerals Engineering 63 (2014) 100–109 103

ore. The tests conditions were as follows: ball filling JT = 20%, slurryfilling UT = 0.75 and mill speed /cT = 75% of critical speed. Thebreakage function parameters (b, c and U) and the selection func-tion parameter (a, aT, K and lT) were determined from the batchmill experiments. These are shown in Table 1 (Chimwani et al.,2012).

3.2. The industrial mill

The batch mill test parameters were scaled-up to an industrialmill using a methodology already discussed (Austin et al., 1984).For validation of the scale-up, data provided by Makokha (2011)from an open circuit mill processing similar ore were used. Thisis an overflow mill with an inside diameter (D) of 7.312 m andlength (L) of 9.6 m and is run in an open circuit. It is fitted with44 rubber lifters bars with heights of 100 mm and typicallycharged with 40 mm forged steel balls. The wet overflow mill pro-cesses fine platinum feeds (�1 mm) and ground product goes tothe flotation circuit. However, slimes (i.e. material less than10 lm) are to be avoided in order not to compromise the flotationprocess. For the purpose of scaling-up and modelling, the mill wasassumed to behave as a plug flow reactor; consequently, the grind-ing time was regarded as the mean residence time of the mill (Aus-tin et al., 1984).

3.3. The scale-up procedure

The scale-up methodology proposed by Austin et al. (1984) isonly applicable to materials whose breakage and selection functionparameters have been determined in the laboratory. In the presentstudy, the parameters determined by Chimwani et al. (2012) wereused.

The scale-up algorithm for the selection function parametersfrom the laboratory test conditions to the industrial mill was pro-grammed in Matlab�; the ore was assumed to be normalisable sothe breakage function was unchanged.

The selection function parameters in Table 1 were the initialinput parameters. Other inputs to the Matlab� code were the grind-ing conditions pertaining to the batch mill (DT, JT, UT, /cT, dT) and tothe simulated industrial mill (D, J, U, /c and d). For a given set ofuser-defined industrial mill conditions, specific rates of breakagefor the industrial mill were predicted using Eq. (7). Once thescaled-up selection function values were determined, they wereincorporated into the PBM (Eq. (3)) together with the breakage func-tion values from the laboratory test (see Table 1) under the normal-isation assumption. PSDs were then generated for different grindingtimes (also termed mean residence times in a continuous millingsetup). Finally, the attainable region for the desired size class(�106 + 9 lm) were deduced from the generated PSDs for each setof industrial mill conditions considered. The effect of each millingfactor on the production rate of the desired size class was explored.

4. Results and discussions

In order to gain more confidence in the scale-up procedure,PSDs of mill product from the scaled up data were compared with

Table 1Laboratory tests parameters.

Breakage function parameters b 6.2c 0.8U 0.6

Selection function parameters a 1.40K 4.74a 0.32l 4.2

the PSDs obtained from the industrial mill. Since milling kineticsare essentially dependent on the following factors: mill diameter,mill speed, ball filling, powder filling and ball size, all the afore-mentioned factors were evaluated on scaled-up data except thediameter of the mill.

Having verified the scale-up method and shown that it wasapplicable to the platinum ore and the full-scale mill consideredin this thesis, the mill operating conditions used to describe theindustrial mill operation were varied. This made it possible touse AR approach to optimise the operating conditions by consider-ing the full range of options.

4.1. Validation of the scale-up procedure

The values of a and l were scaled up to the conditions of thesimulated industrial mill since they depend on the conditionsand the geometrical scale of the mill. The scaled-up value of aT inEq. (7) for the new mill condition is a� and that of lT in Eq. (7) isl�. The set of values of the scaled-up parameters used to generatethe simulated PSDs are presented in Table 2.

The validation results of prediction of industrial mill productare given in Fig. 1. It can be seen that there is a good match be-tween the predicted PSDs and the measured plant data. Havingsuccessfully validated the scale-up parameters against real plantdata, the next step was to use the model to explore various factorsthat affect the milling kinetics that are discussed in the sectionsthat follow.

4.2. Using modelling to explore effects of varying operationalconditions

In the following sections, the model was used to generate largemill outputs under various operating conditions and the ARscheme was used to recommend optimal operating conditions.

4.2.1. Effects of mill speed on milling kineticsFig. 2 shows the predicted PSD for four mill speeds (40%, 60%,

75%, and 90% of critical speed) and for three mean residence times(2, 4, and 15 min). It is clear that finer grinding is achieved as millspeed increases from 40% to 75% and then becomes coarser as thespeed is increased to 90% of critical. Common practices recom-mend speeds around 75% of critical to guarantee high power drawand therefore ensure grinding at a faster rate (Austin et al., 1984;Wills and Napier-Munn, 2005). This is supported by the PSDs inFig. 2 in which a coarser material is produced at speeds, both lowerand higher than 75% of the critical speed.

The application of the AR method to milling gave a whole newapproach to optimize milling (Khumalo et al., 2006, 2007, 2008).Fig. 3 shows the mass fraction of material in the intermediate (de-sired) size class termed m2 (�106 + 9 lm) for the following speeds:40%, 60%, 75% and 90% of critical. Different maximum amounts ofthe intermediate size class m2 were achieved at different speeds;in addition, the maximum achievable mass fraction of the desiredsize class decreased with the increase in mill speed. The lowestfractional speed, that is /c of 40% of critical or 6.24 rpm, produced

Table 2Breakage parameters as scaled-up to industrial mill.

Breakage function parameters b 6.2c 0.8U 0.6

Selection function parameters a 1.40K 4.74a� 3.74l� 9.65

Fig. 1. Comparison between experimentally measured data and the predicted PSD using the scale up method for the large scale mill: (a) J = 24.6% and 74.5% solids, (b)J = 32.9% and 67.7% solids, (c) J = 32.8% and 65.1% solids, and (d) J = 32.9% and 72.1% solids.

0

20

40

60

80

100

0.001 0.01 0.1 1

Per

cent

pas

sing

Particle size xi in microns

F Speed 0.4 2 min F Speed 0.6 2 min F Speed 0.75 2 min F Speed 0.9 2 min

4 min 4 min 4 min 4 min

15 min 15 min 15 min 15 min

Fig. 2. Predicted PSDs for four mill speeds: J = 40%, U = 0.75, d = 40 mm and varying residence times.

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80 90 100

Mas

s fr

acti

on p

rodu

ced

in s

ize

clas

s m

2

Number of Revolutions

Fraction of speed 0.4

Fraction of speed 0.6

Fraction of speed 0.75

Fraction of speed 0.9

Fig. 3. Plot of mass fraction of m2 versus number of revolutions for different speeds /c; J = 40%, U = 0.75, and d = 40 mm.

104 N. Chimwani et al. / Minerals Engineering 63 (2014) 100–109

N. Chimwani et al. / Minerals Engineering 63 (2014) 100–109 105

the highest mass fraction of m2 (74.8% mass fraction) followed by/c = 60% (9.38 rpm) which achieved a maximum m2 of 66.5%. Thespeed that had the lowest maximum amount of m2 was /c = 75%(11.7 rpm) which produced only 62.1% mass fraction. This is con-trary to the widely held belief in comminution practices which rec-ommends industrial grinding mills to be run at /c = 75%, though itis not uncommon to find mills being run at speeds as low as 69% ofcritical speed to prevent cataracting balls from damaging theliners.

As mill speed was increased to /c = 90% (14 rpm) the maximummass fraction of the desired product m2 rose to 65.6%. This observa-tion confirms the ability of the AR technique to ascertain the opti-mal operating condition of a preselected industrial mill. Thegrinding profiles in Fig. 3 also confirms the findings by Metzgeret al. (2009) that low mill speeds produce more intermediate classat a higher rate than high speed mills. Based on the experimentsthey conducted on silica sand, Metzger et al. (2009) found thatoperation at the slowest rotation is the policy which utilizes energyto the fullest extent and therefore enables optimal production ofthe desired size class with the most efficient use of energy. Howeverit has been pointed out that though it is desirable to operate mills atlow speeds, lower than optimum speed may lead to the excessiveamount of wear on the shell liners and production of undesirablelarge amounts of slimes (Wills and Napier-Munn, 2005).

4.2.2. Effect of ball filling on millingThe PSDs for the ball fillings (J) of 5% and 40% for the grinding

times (2, 4, and 15 min) are shown in Fig. 4. All the other factorswere held constant with mill speed at 75% of critical, 40 mm balldiameter and U of 0.75. It is seen from the PSD that coarser grind-ing is achieved at a higher ball filling (J = 40%) for all grindingtimes. A possible explanation for that behaviour is that at a givenmill rotational speeds, lower media load shows quite a bit morecataracting action than the higher media load (Fortsch, 2006).

Interesting to note is that J = 40% produced the maximumamount of the desired size class m2 as shown in Fig. 5. This isnot very different from J = 35% proposed by Fortsch (2006) as opti-mal. Previous work by Metzger et al. (2009) suggests that the low-est ball filling produce the highest amount of desired product,which differs from our suggestion which shows that J = 40% pro-duces the highest amount of m2, which agrees with Austin et al.(1984) who suggested the same percentage as optimal media fill-ing level. This could be ascribed to less cataracting and more of cas-cading action of grinding media at such a media filling level andthat situation provides high grinding efficiency due to more con-tact surfaces (Fortsch, 2006).

0

20

40

60

80

100

0.001 0.01

Per

cent

pas

sing

Particle size

J = 40%

4 min

15 min

Fig. 4. Effect of ball filling J on PSD

4.2.3. Effect of ball diameter on millingThe influence of the size of balls on the breakage rate is shown in

Fig. 6. The general expected behaviour is that effective breakage ispromoted if bigger balls are used to break coarser particles. Chimw-ani et al. (2012) observed that when grinding a mono-sized feed ofsize (�850 + 600 lm), the breakage rate increased with ball sizewhereas a decrease was recorded for a feed of (�600 + 425 lm).As expected, the PSDs in Fig. 6 shows a coarser product for40 mm balls and the increase in fineness of the product as balldiameter decreases. This is owing to the fact that, as the diameterof the ball decreases, the number of balls in the mill increases andthe relative surface area available for grinding increases, which inturn leads to the increase of the specific rates of breakage for smallparticle sizes (Austin et al., 1984).

The effect of the ball diameter on the grinding profiles of the de-sired size class at a single speed, media and powder load is shownin Fig. 7. The three ball sizes considered (i.e. 20 mm, 30 mm and40 mm) all achieved different maximum amounts of the desiredsize class. It can be seen that as the ball size decreases, the maxi-mum achievable mass fraction of size class two decreases (seeFig. 7). 40 mm balls produced the highest maximum amount ofthe desired size class (62%), 30 mm balls produced 55.7% while20 mm balls produced 45.3%. The results in Fig. 7 are also desirablesince they promote the use of larger diameter balls. This is espe-cially more preferable considering that the cost of smaller balls ishigher than that of bigger balls. However, it is worth mentioningthat Fig. 6 shows that the highest breakage rate is achieved by20 mm balls followed 30 mm balls and finally 40 mm balls.

4.2.4. Effects of slurry filling on millingThe ratio of volume of powder loaded to the volume of ball

interstices available within the bed at rest is the slurry filling(U). Past works (Shoji and co-workers, 1980, 1982; Tangs-athitkulchai, 2003) showed that milling rate is affected in thesame manner (size distributions produced from the same feedare the same) regardless of whether it is the slurry or powderthat fills the interstices of the mill bed. Slurry firstly occupiesthe interstices between grinding balls before forming a layer onthe bed of balls at rest. Fig. 8 shows the PSD for slurry fillingsU = 0.6, 0.75 and 1 and for mean residence times t = 2, 4, and15 min. It can be seen that the slurry filling U = 0.6 producedthe finest product. As slurry filling was increased, the product be-came coarser. This could be attributed to the fact that furtheraddition of slurry increases the mass hold-up of slurry in the mill.This then leads to deadening of the collisions by powder cushion-ing, resulting from the expansion of the ball-powder bed to give

0.1 1

xi in microns

2 min J = 5% 2 min

4 min

15 min

: /c = 75%, U = 0.75, d = 40 mm.

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Mas

s fr

acti

on p

rodu

ced

in s

ize

clas

s m

2

Number of Revolutions

J = 5%

J = 20%

J = 40%

Fig. 5. Plot of mass fraction of m2 versus number of revolutions for ball filling J on PSD: /c = 75%, U = 0.75, d = 40 mm.

0

20

40

60

80

100

0.001 0.01 0.1 1

Per

cent

pas

sing

Particle size xi in microns

20 mm balls 2 min 30 mm balls 2 min 40 mm balls 2 min4 min 4 min 4 min15 min 15 min 15 min

Fig. 6. PSD for different media sizes;/c = 75%, U = 0.75, J = 40%.

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

Mas

s fr

acti

on p

rodu

ced

in s

ize

clas

s m

2

Number of Revolutions

Ball size = 20mm

Ball size = 30mm

Ball size = 40mm

Fig. 7. Plot of mass fraction of m2 versus number of revolutions for different ball sizes; J = 40%, /c = 75%, U = 0.75.

106 N. Chimwani et al. / Minerals Engineering 63 (2014) 100–109

poor ball-ball-powder nipping collisions thereby lowering break-age rates. (Austin et al., 1984).

Fig. 9 shows the effect of slurry filling on the grinding profiles.The grinding profile of U = 1 produced the highest throughput,

followed by U = 0.75 and lastly U = 0.6. From this observation, itis clear that operating mills at higher U enables us to maximizethe production of the desired size class, further addition of slurrybeyond U = 1 leads to the formation of slurry pool. Katubilwa

Fig. 8. PSDs for three slurry fillings; /c = 75%, J = 40%, d = 40 mm.

Table 3Individual milling parameters and corresponding optimum throughput.

Milling parameter Optimum value Production of (�106 + 9 lm)

Fraction of speed /c = 40% 74.75Ball filling J = 40% 56.78Ball size d = 40 mm 64.53Slurry filling U = 1 68.5

N. Chimwani et al. / Minerals Engineering 63 (2014) 100–109 107

(2012) concluded that U = 1 was the optimal slurry filling afterrealising that a slurry pool forms around the toe of the mediacharge once the mill is supplied with more slurry than the mediainterstices can take. This was also recommended by Latchireddiand Morrell (2003) who found that the condition required for thebest grinding efficiency corresponds to the maximum slurryhold-up that can be held in grinding media without a pool.

Table 3 shows the compilation of all factors with the grindingprofiles that had the highest throughput.

Fig. 10 shows profiles of the desired size class m2 for all operat-ing factors considered in this work. Also included is the profile forthe operational conditions typically used in industry (Ball sized = 40 mm, Fraction of speed /c = 75%, Ball load J = 30%. Underthese typical industrial conditions, the maximum amount of m2

produced is 63.9%. It can be seen that a higher rate of productionof m2 per revolution as well as a higher maximum amount of m2

(74.75%) can be produced by using a lower mill speed, (i.e./c = 40%).

A Matlab� search engine was then used to search for the param-eters which would produce the highest amount of the desired sizeclass over the range of 0 < J < 0.6; 0.3 < U < 3; 0.4 < /c < 0.9;0.01 < d < 0.1. The optimal set of parameters from the search, thatmaximises the amount of m2 formed was found to be J = 0.4,U = 1.86, /c = 40% and d = 40 mm; these produced the maximumthroughput of 79.1%.

0

10

20

30

40

50

60

70

80

0 10 20 30 40

Mas

s fr

acti

on p

rodu

ced

in s

ize

clas

s m

2

Number of

Fig. 9. Plot of mass fraction of m2 versus number of revolutions fo

This clearly shows that there is still room to improve thethroughput hence maximising the production of the desired sizeclass.

When mill speed was varied, the lowest speed produced thehighest maximum amount of the desired size class m2. This didnot agree with the common practice which recommends 75% ofcritical speed as the optimal. Ball filling J = 40% produced the high-est throughput and that corresponds to the results of previous re-search (Austin et al., 1984). A ball size of d = 40 mm as well as U = 1maximised the production of the desired size class m2. Interestingto note though is that all these factors, though producing the high-est amount of desired product, also produced the coarsest size. TheAR analysis also suggests that it is always worthwhile to considerspeed as the first factor to consider adjusting when optimizingmilling to achieve a given desired size class, although in industry,speed is seldom adjusted.

50 60 70 80 90 100

Revolutions

U = 0.6

U = 0.75

U = 1

r for different slurry fillings; J = 40%, /c = 75%, and d = 40 mm.

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80 90 100

Mas

s fr

acti

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ced

in s

ize

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Number of Revolutions

Ball filling (J) ranges 5 - 40 Ball sizes 20 - 40Powder filling (U) ranges 0.6 - 1 Fraction of speed 0.4Fraction of speed 0.6 Fractional speed 0.75Fraction of speed 0.9 Industrial operational conditionsoptimal conditions

Fig. 10. Summary of various simulations, optimised solution and industrial operating conditions.

108 N. Chimwani et al. / Minerals Engineering 63 (2014) 100–109

5. Conclusion

In this paper, empirical models were successfully used to scale-up the breakage rate from laboratory tests to an industrial mill withsufficient validation using both batch experimental data and indus-trial mill sampling data. Upon validation it was thus possible to usethe simulator to explore a number of variables that would affectproduct size distribution. The main objective being the maximisa-tion of the m2 size fraction (�106 lm to + 9 lm). Using the AR tech-nique, a graphical method for exploring output limits of a system, ithas been shown that our objective is achieved when the rotationalspeed /c = 40% of critical, ball filling J = 40%, ball size d = 40 mmand slurry filling U = 1. Though product was coarser, the greatestamount of the desired size class m2 was produced. The AR can easilybe used to determine operational limits once the objective productdesired is set and this can be flexibly adjusted if the objectiveschange. It has thus been demonstrated how the AR can be integratedwith population balance modelling to make optimisation of grind-ing easier. There is no reason why in future this technique cannotbe integrated with other models such as grinding liberation models.

The findings described in this paper regarding AR reinforce thesuggestions made by Metzger et al. (2009) that running the mill atlow speeds saves energy and maximizes the production of the de-sired size class, however the throughput dimension was not con-sidered and will be tackled in future. Our findings also suggestthat the current operational conditions in industrial mills leavemuch room for improvement. If the aim is to achieve maximumthroughput of the desired size class, much can be done to remedyunderperformance and energy inefficiency. The AR technique hasbeen shown to have potential to provide an alternative way of opti-mizing milling.

For future work, the energy requirements of the full-scale millwill be investigated as a function of operating parameters suchas powder filling, media charge filling, mill speed and ball diameterfrom an AR perspective. The objective will be to determine ways ofreducing the energy consumption under given constraints.

Acknowledgements

The authors wish to thank the Amandelbult mine of AngloAmerican for supporting this project by supplying the ore thatwas used in this work.

The University of South Africa (MaPS/UNISA) is also acknowl-edged for encouraging this collaborative work.

Professor M.H. Moys of the University of the Witwatersrand andPippa Lange are acknowledged for their advice and contributions tothis investigation.

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