simulation of wet ball milling of iron ore at carajas, brazil

84 (2007) 157–171www.elsevier.com/locate/ijminpro

Int. J. Miner. Process.

Simulation of wet ball milling of iron ore at Carajas, Brazil

Leonard G. Austin 1, Kesley Julianelli a,Alan Sampaio de Souza a, Claudio L. Schneider b,⁎

a Companhia Vale do Rio Doce, Brazilb CETEM, Brazil

Received 1 July 2006; received in revised form 19 September 2006; accepted 26 September 2006Available online 28 November 2006

Abstract

Laboratory-scale batch grinding tests were performed onffiffiffi2

psieve size fractions of the almost pure hematite mined at the

Carajas mine of CVRD Brasil. Good first-order kinetics of grinding were obtained, both for dry grinding and for grinding withwater, using the Bond test levels of ball load and powder load. However, the primary breakage distribution functions were notdimensionally normalized, as smaller particles produced proportionally more fines on primary breakage, possibly due to internalporosity of the ore. The test results were scaled to the conditions of a pilot-scale mill in normal closed circuit, and a ball millsimulator was used to predict the capacity, circulating load and size distributions, employing the size selectivity values calculatedfrom the balance around the hydrocyclone. The simulation results were in reasonable agreement with the observed values. Thetechnique was then extended to predict the behavior of a full-scale mill at a capacity of 308 metric tons/h and a circulating load ofnearly 600%. Agreement between simulated results and plant test results was reasonable except that the simulator predicted aproduct (overflow) size distribution with a higher sharpness index than observed. The reason for this discrepancy is not known.© 2006 Elsevier B.V. All rights reserved.

Keywords: Iron ore; Ball mill simulation; Grinding kinetics; Sub-sieve size measurement

1. Introduction

Companhia Vale do Rio Doce (CVRD) is a worldleader in production of iron ore, from its deposits inBrazil. It registered a profit of about $5 billion during2005, more than a 60% increase from 2004, from salesof $16 billion. This resulted from increased prices ofiron ore products, 71.5% higher for lump ore and 86%for pellets, plus increases in production and sales. These

⁎ Corresponding author. Fax: +1 801 880 2645.E-mail address: [email protected] (C.L. Schneider).

1 Consultant, USA.

0301-7516/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.minpro.2006.09.010

sales of iron ore amounted to about 70% of the com-pany's income and included exports of $7 billion.

Brazil has one of the largest iron ore reserves in theworld, including an estimated 17 billion tons of high-grade iron ore located in Carajas. The formation origi-nated from jaspelite; according to the currently acceptedtheory, the ore bodies were formed from meteorologicalalteration of the jaspelite, leaching out the silica andenriching the hematite horizons. The supergenic enrich-ing process, with a low degree of metamorphism pro-duced fragile hematites with small crystals and a highdegree of porosity.

Due to the nature of this ore, the mine produces al-most exclusively a crushed material designated as “sinter

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http://dx.doi.org/10.1016/j.minpro.2006.09.010

Table 1Mill characteristics

Batch mill Pilot-scale mill Full-scale mill

Internal dimensions:Diameter m 0.25 1.2 5.9Length m 0.25 2.42 10.3Volume 13080 cm3 2.737 m3 281.6 m3

Ball charge:Weight kg 12.31 3800 485000

Weight fraction of ball charge in each ball sizeBall diameter mm76 0.16 065 0.19 055 0.28 052.5 0 0.65940.1 0 0.22539 0.185 030.7 0 0.07730 0.165 025.4 1.0 0.02 023.5 0 0.02617.9 0 0.00913.7 0 0.003

158 L.G. Austin et al. / Int. J. Miner. Process. 84 (2007) 157–171

feed”. This material (8 mm×0.15 mm) is readily shippedto the coast by rail, as it contains only a small propor-tion of fine dust. It is then shipped by boat to buyers,mostly in Europe and Asia.

However, some fraction of the output of the mine isnow ground, pelletized and sintered by CVRD. Becauseof the operational facilities of the mine complex, andenvironmental concerns, the decision was made to buildthe grinding mill for production of “pellet feed” inCarajas, instead of at the CVRD sintered pellet produc-tion terminal on the east coast of Brasil. This requiredthe production of a de-slimed material with an initialspecific surface area of about 1400 cm2/g, so that thematerial could be more easily dewatered and transportedin the existing railroad system. The specific surface areais then increased to meet sinter feed specifications(Blaine specific surface area of 2000 cm2/g, particle sizedistribution of about 70% less than 45 μm), by re-grinding using a high pressure grinding roll installationat the terminal.

At Carajas, the new grinding plant was operational in2002, with two ball mill lines followed by a de-slimingplant. The closed circuit, wet grinding mills were de-signed to operate at 600% circulating load, in order tominimize the production of fine particles and conse-quently give increased recovery at the de-sliming plant.Even before the plant was commissioned, CVRD starteda development project aimed at optimizing recovery,reducing overall energy consumption, and increasingplant capacity. This project included the detailed char-acterization of alternative ball mill feeds derived fromthe Carajas ore. Pilot-scale grinding tests were carriedout, as well as detailed batch grinding tests. In particular,a screened product (Crusher Product 1) from the crusherline with a top size of about 12 mm was determined tohave a Bond Work Index of 12.6 kW/metric ton. Asecond crusher product (Crusher Product 2) with a muchfiner size distribution had an estimated (non-standard)Work Index of 15.2 kW/metric ton.

A plant sampling campaign was carried out after thestart-up phase of the first mill (using Crusher Product 1as feed) was completed, and all relevant data was care-fully measured, checked and recorded. The data includedthose reported in this paper, with the objective ofconstructing amill simulator that could be used to predictthe effect of changing mill feed and mill conditions. Theend result was a calibrated simulator for the plant, im-plemented on Modsim™ (Ford and King, 1984) andbased on particle “population balance” modeling tech-niques, developed over many years of engineering ex-perience and laboratory and pilot-scale experimentation(Austin et al., 1984a,b,c,d).

The plant simulator became an important require-ment for this project not only because it contains themodel coding, but also because it becomes a practicalinterface between the plant operators, CVRD engineersand the advanced modeling engineer. A custom-madeModsim™ simulator was delivered to CVRD, enablingthe engineers to perform a number of studies such as theeffects of ball charge and ball size distribution, millpower, cyclone configuration, plant capacity and de-sliming circuit configuration.

2. Experimental methods

2.1. Laboratory-scale batch grinding tests

The tests were made in a steel cylindrical ball mill of250 mm internal diameter (DT) and 250 mm length (LT)fitted with eight symmetrically located horizontal lifters(see Table 1) and smooth end walls, with one end wallremovable and locked in place with a quick releaselocking clamp. The media were stainless steel ballbearings of 25.4 mm (1 inch) diameter, with a density(ρb) of (7.84)(10

3) kg/m3. The mill was filled with aformal ball filling (JT) of 0.2 volume fraction (20%) androtated on a roller table at 70% of critical speed. Theformal definition of J assumes that the porosity of theball bed at rest in the mill is 0.4 so

Mass of balls ¼ 0:6JVqb ð1Þ

159L.G. Austin et al. / Int. J. Miner. Process. 84 (2007) 157–171

where V is the mill volume. A formal mill filling U isdefined as the fractional filling of the interstices of theball bed at rest by powder. Thus

Mass of material ðholdupÞ ¼ 0:4JVUqm ð2ÞThe formal bulk density of the powder, ρm, was

determined by measuring the weight of 16×20 mesh(1.18×0.85 mm) of the test iron ore in a known volumeof a measuring cylinder. The use of these formal defini-tions (Austin et al., 1984a) means that the mass of mediain a mill is always proportional to J and V and the massof material is always proportional to J, V and U, irre-spective of the mill and ball diameters and any variationof the actual powder density for different size distribu-tions of powders.

The conditions chosen for test (J=0.2, U=0.5, dry)are those of the standard Bond test (Bond, 1960). Thesewere chosen because (i) dry testing for the breakageparameters is much quicker and more reproducible thanwet ball milling tests (ii) the results generally show goodfirst-order kinetics of grinding and (iii) the low value ofUreduces thework required for preparation of feed samples.

To determine the parameters of grinding kinetics it isnecessary to test samples of the feed material screenedinto single

ffiffiffi2

psieve fractions, e.g. 4 × 6 mesh,

16×20 mesh, etc. For the size of mill and the test con-ditions used here the mass of each test sample was 1.13 kgof oven-dried iron ore, plus a sample of 200 g retained fora blank screen analysis. The 1.13 kg of material wasplaced in the mill with the media, ground for a suitableshort period of time, the mill emptied through a screen toretain the balls, and the powder cleaned from the mill andballs with a brush. The collected powder was weighed todetermine weight loss on handling, and then mixed andcone-and-quartered on a flexible plastic or hard rubbersheet to give suitable samples for size analysis, e.g. 200 gfor the 4×6 mesh feed, 100 g for smaller sized feeds.

A screening of the blank (without grinding) sampleusing the lower screen size of the sample always gives asmall percentage amount passing through this screen, sothat the feed material appears to be, for example, 97% of16×20 mesh and 3% less than 20 mesh. After the initialgrinding period t1, screening of a sample of product willgive w1(t1) as the weight fraction retained in the feed sizeinterval (the interval indexed by 1). The first-order grindingrelation for the loss of weight from this size interval is

w1ðt1Þ ¼ w1ð0Þe−S1t ð3ÞThe value of w1(0) is, in the example, 0.97 and a plot

of w1(t) on a log scale versus t on a linear scale will be astraight line. Drawing the line through the two available

points enables an estimate to be made of the nextgrinding time required to get a desired degree of break-age of the feed in the next grinding step.

The dried product of the sample screened after t1 isthen added back to the mill contents and the materialreturned to the mill with the ball charge. There is alwayssome small loss in the handling process but this loss is(with care) negligible compared to the initial charge(1.13 kg) of material. The mill is then operated for theadditional time period required to get the desired w1(t2)and the above steps repeated. This procedure is per-formed for a total of four or five times that give values ofw1(t) from, approximately, 1 to start, to 0.75, to less than0.05, so that a good straight line according to Eq. (3) canbe drawn and used to determine the S value, that is, thespecific rate of breakage of this size of feed materialunder the test milling conditions.

2.2. Sieve size analysis

As discussed above, the act of screening will gen-erally produce some size reduction, and this effect, thesieving error, must be taken into account to get validresults. As we had no prior experience with this iron ore,a sieving kinetic test was performed on a sample mainlyof 16×20 mesh material. The result showed that aftersix 5 min steps of sieving the rate of loss for each stepreached about 0.4% per step, with most of the materialpassing the 20 mesh screen appearing in the 20×30mesh size interval. This type of result is normal andshowed no excessive abrasion of near-screen sizes.Making a balance between a sufficient time of sieving topass all less-than-size material but allowing for aconstant abrasion rate to less-than-size material, gave adry screening time of about 20 min.

The complete size analyses were made using a stack offfiffiffi2

pspaced sieves, startingwith the bottom size of the feed

size interval and ending with the pan, and screening for20 min. The pan containing fines were then removed andthe stack washed well to remove adhered fines, and thewash water collected and vacuum filtered throughweighed filter paper. The sieves were oven-dried and thedried weight of material retained on each sieve deter-mined, and the weight collected on the filter was added tothat of the pan. The total weight was checked against thestarting sample to ensure that weight losses were less thanabout half a percent. This loss was also added to the pan.

2.3. Wet grinding tests

It is known that wet grinding under the test millconditions used above, and a low volume concentration

Fig. 1. A. First-order plots for batch dry grinding of feeds of 16×20,35×45, 45×60 mesh iron ore. B. First-order plot for dry grinding of4×6 mesh (oversized particles) iron ore: effective S value based on95% breakage at a grinding time of 8.1 min.

Fig. 2. Specific rates of breakage of ore inffiffiffi2

psieve intervals plotted

versus the upper size of the interval: 0.5×0.355 mm material isCrusher Product 2, the other sizes are from Crusher Product 1.


of slurry, gives first-order kinetics of grinding and thesame breakage parameters as dry grinding, except thatthe specific rates of breakage are higher by a constantfactor. The formal fractional filling of the ball bed by thematerial being ground (U) is still calculated from Eq. (2)irrespective of the presence of water. Attempts tocalculate U based on the volume of slurry destroy theusefulness of the formal definition and void the validityof the scaling equations that have been derived usingthis formal definition.

One test was made with these conditions (J=0.2,U=0.5) using a feed of 16×20 mesh ore and a slurryconcentration of 35 vol.% of solid in water, in order toget the factor of change from dry to wet grinding. Itshould be noted that the shear forces in a small mill aremuch less than those in a pilot-scale or full-scale mill

because of the small weight of the charge. Consequent-ly, wet grinding tests in a laboratory-scale mill usingslurry densities corresponding to those in full-scalepractice usually give non first-order breakage and othercomplications, due to the production of high viscosityslurry as the size distribution becomes finer. Anothertest was made to illustrate this, using J=0.3, U=1 and aconcentration of the slurry of 45 vol.% of solid in water.

3. Results and analysis of the batch grinding tests

3.1. Dry tests

The data is first plotted as w1(t) on a log scale versust on a linear scale and a straight line drawn through thepoints. Extrapolation back to t=0 gives the value of w1

(0). This value is always less than 1 because of thesieving error and, in addition, there is generally a smallfraction of the feed material that is so weak that itdisappears in the first few seconds of grinding. In thedata reported here the measured sieving error was closeto 0.05 fraction and the weak material also about 0.05fraction, so that the effective w1(0) was 0.09 for all thefeed sizes.

The first-order plots of the data according to Eq. (3)are shown in Fig. 1, which also shows the best-fit straightlines through the experimental points. It is seen that goodfirst-order kinetics were obtained (Fig. 1A), with smallersizes having lower rates of breakage (smaller slopes)than larger, except for the largest tested size (4×6 mesh,4.75×3.35 mm) which gave slower breakage than that ofthe 16×20 mesh ore and an obvious non first-order(Fig. 1B). This result is typical for a feed that has someparticles too large to be nipped by the ball size in the mill:

Fig. 3. Cumulative primary breakage distribution calculated from shorttime 16×20 mesh feed data, dry and wet grinding.


the rate of breakage is less than expected and decreasesas grinding proceeds, probably due to the accumulationof the harder (stronger) material in the remaining feedmaterial. An effective mean S value can be defined usingthe grinding time to 95% breakage since the degree ofnon first-order is not large.

Fig. 2 shows the S values plotted according to thefitting equation (Austin et al., 1984b)

Si ¼A xi

x0

� �a

1þ xil

� �kð4Þ

In this equation, x0 is a standard size (taken here as1 mm), xi is the upper size of the screen interval indexedby i, and A, α and μ are constants that depend on thematerial and the milling conditions. Experience hasshown that λ can be satisfactorily taken as 3 for mostmaterials, especially when the top size of the feed to themill to be simulated is not much larger than μ. The fittedvalues were A=0.72 min−1, α=0.9, and μ=2.5 mm.

As was noted in the Introduction, there was alreadyan indication that the breakage properties of the second,finer product from the crushing system were differentfrom those of the first product, since the Bond WorkIndex was different. The point for the 35×45 mesh(500×355 μm) material lies above the fitting curve,meaning that the Product 2 material is easier to breakthan the same size of Product 1, in accordance with theBond test results. Thus the Product 2 material not onlygives a much finer size distribution as feed to the millbut it also breaks more easily, with different breakageparameters. Therefore it was necessary to repeat the setof batch tests on this material in order to be able topredict the behavior of the mill with this feed, althoughthese results will not be presented here.

The primary breakage distribution values were firstestimated using the BII method (Austin and Luckie,1972):

Bi;1 ¼ log½ð1−FiÞ=ð1−PiÞ�log½ð1−F2Þ=ð1−P2Þ� ; i N1 ð5Þ

where Bi,1 is “the cumulative mass fraction of fragmentsformed on primary fracture of feed material in the sizeinterval indexed by 1 that is less than or equal to sievesize xi″. Obviously B1,1=1 and B2,1=1. As usual, Fi isthe mass fraction of the feed material ≤xi, Pi that of theproduct ≤xi. It must be remembered that the definitionof primary fracture is that the fragments formed fromfracture mix back into the charge and are not re-selectedfor a second breakage while the first breakage is in

process. Thus Eq. (5) only applies as an approximationthat is closer to correct as the fraction broken is smaller,for then the primary fragments have a smaller chance ofre-breakage. Therefore Eq. (5) is applied to resultswhere less than about 30% of the feed has been broken.

Fig. 2 shows that the specific rate of breakage of the4×6 mesh material is to the right of the maximum in thecurve and Fig. 1B shows that the breakage is non first-order in this region, so Eq. (5) cannot be applied to thissize. On the other hand, if a small feed size is used thenumber of points available to the finest sieve size (400or 325 mesh) can be too small to show the full shape ofthe primary breakage distribution. In the case consi-dered here, as in many such cases, the preferred feedmaterial to use for the determination of the B vector is16×20 mesh.

The results were fitted to the equation

Bi;1 ¼ Uxix2

� �g

þð1−UÞ xix2

� �b

; i N1 ð6Þ

where Φ, β and γ are characteristic of the material.The result is shown in Fig. 3. The values found wereΦ=0.30, β=3.5 and γ=0.62.

It has been shown by many investigators that Bvalues are often dimensionally normalized, that is, Eq.(6) can be generalized as

Bi; j ¼ Uxixj−1

� �g

þð1−UÞ xixj−1

� �b

; i Nj ð6aÞ

where j indexes the breaking size interval. When this isapplicable the cumulative product size distributionsfrom batch grinding of a single

ffiffiffi2

pfeed size for various

times have, when plotted on log–log scales, a constantslope for the fine sizes. Fig. 4 shows this type of plot for

Fig. 4. Comparison of simulated and experimental size distributionsfrom dry batch grinding of 16×20 mesh iron ore (AT=0.72 min−1,α=0.90, Λ=3.0, μT=2.5 mm, Φ=0.3, β=3.5, γ0=0.60, δ1=0.25,δ2=−0.30).

Fig. 5. First-order plots for dry and wet batch grinding of 16×20 meshfeed: dry, wet.


the 16×20 mesh data. Detailed examination of the dataindicated that the slopes show a consistent change tosmaller values as the grinding time increases. Thismeans that the B values are not dimensionally nor-malized. Calculating B values for the breakage ofsmaller feeds showed the same trend. Thus Eq. (6a) wasmodified by making γ a function of the breaking size j.

A suitable function was found to be

gj ¼ g0x

x0

� �d1

; d1N0 ð7Þ

In this equation x0 is the standard size=1 mm and γ0is the value of gamma for the standard size of 1 mm. Avalue of δ1=0.25 was found to give the appropriatedecrease of γ as the breaking size became smaller, withγ0=0.60.

The mass fraction of primary fracture from breakageof the material in the size interval indexed by j thatappears in the size interval indexed by i is

bi; j ¼ Bi; j−Biþ1; j ; nNiNjbn; j ¼ Bn; j ; j ¼ n

ð8Þ

In this equation, n indexes the size interval below thesmallest sieve size used, that is, the material sized from 0to 400 or 325 mesh. This size interval is called the “sink”interval because it can receive material from breakage oflarger sizes but it cannot lose material, as it is implicitthat re-growth to larger sizes does not occur. For thisreason Sn=0 and bn,j=Bn,j.

Solution of the batch grinding equation (Austin andCho, 2002) using these S and B values gives results thatcan be compared to the experimental values. It is alwaysnecessary to confirm that the S and B values deduced

from analysis of the breakage do indeed reproduce thebatch grinding size distributions. In fact, for the caseconsidered here of non-normalized primary breakagedistributions, the only way to obtain the B values is byback calculating the B parameters from the complete setof size distributions. This was done using the BII para-meters only as starting estimates in the search.

Fig. 4 shows the result of the simulation. It is seenthat the simulated size distribution at 0.4 min of grindingtime is coarser than the experimental result, but the sizedistributions at longer times are in good agreement. Thisillustrates the difficulty of obtaining satisfactory B val-ues from a single experimental result at a short grindingtime: it is difficult to get a representative sample whenthe amount of broken material is small compared to theunbroken feed, yet it is impractical to screen the whole1.13 kg of mill contents.

The simulated size distributions compared to the ex-perimental values for batch dry grinding of 45×60 meshfeed, using the same S and B values as in Fig. 4, gave thesame kind of agreement as in Fig. 4. Once again, the shorttime result was not closely matched by the simulation, butthe agreement at longer grinding times was satisfactory.

3.2. Wet tests

Fig. 5 shows the first-order plot (Eq. (3)) of the wetgrinding test on 16×20 mesh material, as compared tothe equivalent result for dry grinding. Under thesegrinding conditions the wet grinding gives clearly first-order grinding kinetics, with a specific rate of breakageof 1.2 times that for dry grinding. In accordance withprevious experience, it can be assumed that all para-meters are the same for these otherwise identicalgrinding conditions except that the A value of Eq. (4)


is a factor of 1.2 higher for wet grinding than for drygrinding, that is, A=0.86(4) for wet grinding.

Fig. 6 shows the comparison of simulated to exper-imental size distributions for this batch wet grinding.The agreement is not generally as good as that for drygrinding but the short time result is better and the fit isgood at the longest time. This suggests that the poor fitat the short grinding time seen in Fig. 4 is due tosampling error. The reasonable overall prediction is avalidation of the assumptions that have been made andshows that the very time-consuming and poorly repro-ducible wet grinding tests can be replaced by easier andquicker dry grinding tests plus only one wet test.

4. Scale-up equations and the mill model

4.1. Mill diameter, ball filling, material filling

The equation to convert the specific rates of breakagefrom the test conditions to the conditions in another millis (Austin et al., 1984c)

A ¼AT

DDT

� �0:51þ 6:6J 2:3T

1þ 6:6J 2:3e1−1:32ðU−UT Þ ;DV3:8m

AT3:8DT

� �0:5 DDT

� �0:31þ 6:6J 2:3T

1þ 6:6J 2:3e1−1:32ðU−UT Þ ;DN3:8m

8>>><>>>:

ð9Þ

where A is the scaled up value of A in Eq. (4) for thenew mill conditions D, J and U, and AT is the value forthe test conditions DT, JT and UT. This equation is basedon experimental observations (Shoji et al., 1979, 1982)and uses the formal definitions of J and U.

Fig. 6. Comparison of simulated and experimental size distributionsfrom wet batch grinding of 16×20 mesh iron ore (AT=0.864 min−1,α=0.90, Λ=3.0, μT=2.5 mm, Φ=0.3, β=3.5, γ0=0.60, δ1=0.25,δ2=−0.30).

4.2. Ball diameter, ball mix for S values

The value of μ in Eq. (4) is converted to a differentball diameter by

l ¼ lTddT

� �1:2

ð10Þ

It should be noted that an exponent of 1.2 (Magne,1987) has replaced the value of 2 used previously(Austin et al., 1984c).

For a known mix of ball sizes, the weighted sum of Sivalues, SMi say, is given by

SMi ¼Xk

mkSi;k ; nziz1 ð11Þ

where k indexes the ball diameter and mk is the massfraction of the ball charge that is of the diameter indexedby k. For an equilibrium ball charge where the wear rateis linear (Austin and Klimpel, 1985)

SMi ¼ 4ATdTd41

1

1−dmin

d1

� �4

0BBB@

1CCCA

xix0

� �a Zd1

dmin

d2

1þ xigT

d

dT

� �1:2 dðdÞ ; nNiz1

0 ; i ¼ n

8>>>>><>>>>>:

ð12Þ

This equation applies for a single size d1 of make-upball diameter, and dmin is the minimum size of ball thatexists in the mill, usually about 12 to 18 mm (1 /2 to 3 /4 inch) for a full-scale mill. The exact value of dmin isnot crucial because it occurs as the ratio to d1 raised tothe 4th power.

4.3. Ball diameter, ball mix for B values

The allowance for non-normalized B values isdone with the following logic. It is incorrect to extra-polate Eq. (7) to large particle sizes because it isknown that particles with sizes to the right of the sizewhere the maximum S occurs not only have lowerspecific rates of breakage and abnormal breakagekinetics but they also have lower gamma values. Thelarger the size compared to the ball diameter the loweris the gamma value. This occurs because breakage ofparticles too large to be nipped between the balls willhave a larger component of chipping and abrasionas compared to disintegrative fracture, leading to aproportionally greater production of fine particles.


Thus it will be assumed that Eq. (7) applies up toxj= xmax and then is replaced by

gj ¼ gmaxxj

xmax

� �d2; xjNxmax ð13aÞ

where it is convenient to define γmax as the value ofγ at the particle sieve size xmax and δ2 will benegative to cause to decrease as particle size increasesabove xmax. Eq. (7) with this definition becomes

gj ¼ gmaxxj

xmax

� �d1; xjV xmax ð13bÞ

where γmax is related to γ0 by

gmax ¼ g0x0

xmax

� �d1 ð13cÞ

The value of xmax is dependent on ball diameter sinceEq. (10) shows thatμ is a function of d ,and differentiatingEq. (4) and setting dS / dx=0 gives (Austin et al., 1984c):

xmaxT ¼ lTa

k−a

� �1=kð14Þ

where T refers to the test ball diameter, 25.4 mm in thiscase.

Thus Eq. (14) for the values of μ, α and λ for the testball diameter of 25.4 mm gives xmaxT=1.885 and Eq.(13c) with x0=1 mm gives γmaxT=0.70. Then thegeneralization is

lk ¼ lTdkdT

� �1:2

ð10aÞ

xmaxk ¼ lka

k−a

� �1=kð14aÞ

gmaxk ¼ gmaxTxmaxkxmaxT

� �d1

ð13dÞ

gj;k ¼gmaxk

xjxmaxk

� �d1

; xjV xmaxk

gmaxkxj

xmaxk

� �d2

; xjNxmaxk

8>>><>>>:

ð15Þ

where k indexes the ball diameter class. Then the valueof the Bi,j matrix for ball class k is

Bði; j; kÞ ¼ Uxixjþ1

� �gj;k

þð1−UÞ xixjþ1

� �B

; xib xjþ1

1 ; xizxjþ1

8<:

ð16Þ

The weighted value of the amount produced belowparticle sieve size xi from the size interval indexed by jfrom breakage by all ball size classes, BMi,j say, is

BMi; j ¼

Xk

mkSj; kBði; j; kÞ

SMjð17Þ

The values of bi,j are, using BMn,j=0,

bi; j ¼ BMi; j−BMiþ1; j ; nzizjz1 ð18Þ

4.4. Residence Time Distributions (RTD)

Data on residence time distributions have been fittedby a number of different RTD models (Austin et al.,1984d), usually two or three parameter models, wherethe mean residence time is one of the parameters. It hasbeen argued (Cho and Austin, 2001) that it is not nec-essary to have models with more than two parametersand that the measured RTD's can generally be suf-ficiently well fitted by the semi-infinite solution to theaxial dispersion equation (Mori et al., 1964; Austinet al., 1984d; Austin and Concha, 1994). This model hasthe mean residence time τ and the dimensionless axialdispersion coefficient as parameters.

A detailed study of a number of mills of differentdiameters and length/diameter ratio grinding high den-sity slurry (Shoji et al., in press) also found the RTD's tofit the Mori et al. model and the axial dispersion co-efficient was determined for each mill. They give therelation for the dimensionless dispersion coefficient E⁎

as

E⁎ ¼ E⁎0ðD=LÞ ;D=Lb0:83 ð19Þ

In this equation E⁎0 is a constant for the system understudy. For their material the values of E⁎ versus D /Lextrapolated to D /L=1 gave E⁎0=0.81. It should benoted that the Mori et al. model does not reduce the fullymixed case for large values of E⁎, due to the approximateboundary condition at the exit that is used in the deri-vation. Thus Eq. (19) must not be used for large values ofD /L, the constraint must be obeyed and the largest validvalue of E⁎ for their material is 0.67.

However, it has been shown (Cho and Austin, 2001)that a model consisting of one large fully-mixed millsection followed in series by two equal smaller fully-mixed sections will produce almost the same degree ofgrinding as the Mori et al. model providing that therelative size of the large to small sections is calculated


from the axial dispersion coefficient. The relation givenis

h ¼ 0:86

1þ eð−14:13E*þ3:07Þ ; hV0:859 ð20Þ

θ is the fraction of mean residence time assigned to thelarge section and (1−θ) / 2 is the fraction for each smallsection. The constraint on the value of θ means that E⁎must be less than 0.688 but that covers the range of theresults of Shoji et al.

4.5. Exit classification

The use of first-order grinding kinetics in a simu-lation model will always predict some small percentagesof material in the top size intervals of the mill product.However, it is often found in wet ball milling thatinstead of the predicted small values the experimentalresult is actually zero percentage. There are severalpossible physical processes that could give rise to thiseffect (Cho and Austin, 2004), but the net effect appearsas if the specific rates of breakage of the larger sizesincrease when they are present in small amounts. This isequivalent to a classification at the mill exit that returnslarger sizes back into the mill (Austin, 1999). The ana-lysis by Cho and Austin of a series of tests on opencircuit wet ball milling of phosphate ore led to anequivalent classification of

si ¼ 1

1þ x50xi

� �3:5 ð21Þ

The controlling parameter is the particle sieve sizex50 at which 50% is returned to the mill feed and 50%exits the mill. They suggest that an initial estimate of x50is given by

x50 ¼ 2:9ðx80Þ1:07 ð22Þwhere x80 is the 80% passing size of the mill product inmillimeters.

In a normal closed circuit milling this exit classifi-cation will be in series with the external classification,and the external classification will generally dominateover the exit classification effect so that the size dis-tribution of the circuit product (e.g., the cyclone over-flow) is not much affected by exit classification.However, if the exit classification is not included inthe simulation model, the model will predict a wrongmill product size distribution and, hence, a wrongcirculating load.

5. Analysis of pilot-scale results

5.1. Circulating load and classification

Table 1 gives the details of the pilot-scale overflowmill of 1.2 m internal diameter and D /L ratio 0.5,operating in normal closed circuit at a solid make-upfeed rate of 1.31×103 kg/h. The test had been made atthe Vitoria plant of CVRD and included size analysis ofsamples of the mill product, and cyclone underflow andoverflow. The top sieve size of the make-up feed was11.2 mm but the mill product was less than 1 mm insieve size and the cyclone overflow less than 250 μm.

In the standard symbolism used here, P refers to millproduct, Q to cyclone overflow (circuit product) and Tto cyclone underflow (tailings), and Pi is the sizedistribution as cumulative weight fraction (or percenta-ge)≤ size xi, pi is the fraction in each size interval, etc.

Analysis around the cyclone gave a circulation ratioC of 2.67 calculated from

C ¼

Xi

ðqi−piÞXi

ðpi−tiÞð23Þ

Size selectivity values (fraction sent to underflow)were calculated from

si ¼ 1−qi

ð1þ CÞpi

� �ð24Þ

The data was quite erratic but it was possible todetermine a reasonably good classifier function of

k ¼ 0:9553

log1

Sharpness Index

� �

si ¼ aþ ð1−aÞ 1

1þ xid50

� �−k

0BBB@

1CCCA

ð25Þ

The fitting values found were: bypass fractiona=0.125, d50=0.046 mm and the Sharpness Index=0.42.

Table 2 shows the values of C calculated using thebalance in each size interval

Ci ¼ qi−pipi−ti

ð23aÞ

If the sampling and measurements of size distribu-tions are good then the Ci values will scatter randomly

Table 2Variation of Ci values (Eq. (23a)) with sieve size interval

Sieve interval Pilot mill Full-scale mill

Upper size mm Ci Ci

1.0 – 7.50.707 – 7.50.5 1.55 6.630.354 1.55 6.630.25 2.61 6.00.177 2.51 5.150.125 2.42 4.610.088 3.13 4.470.063 5.62 4.370.044 3.60 4.480.031 6.60 5.160.022 2.32 5.900.016 – 6.94C from Eq. (23) = 2.67 4.95C from Eq. (26) = 2.87 4.67


about the value of C. It can be seen that there is a cleartrend for the Ci values to increase as the size of theinterval decreases. This indicates that one or more of thesize distributions has systematic error and not onlyrandom error. When this is true the programs for dis-tributing error among the streams do not work well sincethey are based on the assumption that errors occur ran-domly, so this type of program was not used here.

5.2. Extrapolation of size intervals

It should be noted that the calculation of sn (that is,the fraction of the feed in the sink interval that is sent tounderflow) using Eq. (24) is not valid if the valueobtained is much larger than the bypass value. In thiscase the simulation must be run with a sufficiently largenumber of size intervals to reach the bypass value. Thisnecessitates the extrapolation of the make-up feed sizedistribution to get the values for the extra size intervals.As will be seen later, the measured make-up sizedistribution showed 11.4%b62 μm, no material in thenext intervals and 11.4%b22 μm. This is not onlyunlikely from a physical point of view but also it makesimpossible to perform an extrapolation to smaller inter-val sizes. It was assumed therefore that values below22 μm could be extrapolated with the same slope as thatseen for the sizes above 62 μm.

5.3. Residence time distribution and exit classification

For the D /L value of 0.5, Eq. (19) gives a dimension-less axial dispersion coefficient of 0.405. Eq. (20) thengives the large fraction in the one large- two small RTDmodel as 0.80.

Since the interpolated 80% passing size of the millproduct is 0.140 mm, Eq. (22) gives an x50 value of0.354 mm. This was used as the starting estimate and thevalue adjusted to make the simulated result fit as closelyas possible to the experimentally observed values of thesize distribution at the mill exit.

5.4. Simulation results

The solid holdup in the mill was estimated using Eq.(2) assuming U to be unity and using the experimentallydetermined bulk density of (2.16)(103) kg/m3. Thevalue obtained was 720 kg. At the make-up feed rate of1310 kg/h and an estimated circulation ratio of 2.66 (acirculating load of 3.66) the estimated mean residencetime of solid in the mill was 9.07 min.

It is not possible to run the simulation directly with thisvalue of τ since the simulated value of circulating load

will not generally be the same and the effect of exitclassification is not known a priori. It is necessary topostulate a false τ and run the simulator to give thecorrection to allow for the exit classification effect. Thisgives an estimate of the size distribution leaving the mill,and closing the circuit through the external classificationgives an estimate of the tailings size distribution and,hence, a new estimate of the mill feed. This process isiterated to steady state using the make-up feed size dis-tribution as the initial estimate of mill feed. The wholecomputation is repeated with a search on the false τ untilthe closest possible match is obtained between the millcapacity and the size distribution of the cyclone overflowfor the simulation as compared to the experimental values.

Fig. 7 shows the comparisons between size distribu-tions and Table 3 shows the other results. It was necessaryto change the x50 of exit classification from the initialestimate of 0.354 mm to 0.5 mm in order to get a rea-sonable fit to themill exit size distribution. This representsa somewhat smaller exit classification effect thanexpected. Fig. 8 and Table 3 show the comparable resultswithout any exit classification. In this case the fit of modelto experiment was made to generate the best agreementbetween the overflow size distributions and this fit wasquite good, but agreement was poor for the other sizedistributions, the circulating load and the mill capacity.

6. Full-scale mill simulation

6.1. Preliminary (design) simulation

This simulation is different from that of the pilot-scale mill given above because only the size distribution

Table 3Simulation and experimental comparisons for the pilot-scale mill

With exit classification Without

Simulated mill capacitykg/h

1.34×103 1.20×103

Experimental 1.30×103 1.30×103

Simulated circulating load 3.535 4.07Experimental 3.64 3.64

Parameters: Exit classification Externalclassification

x50=0.5 mm d50=0.046 mm

SI=0.533 SI=0.42

AT=0.864 min−1 a=0.125α=0.9μT=2.5 mmβ=3.5ϕ=0.3


of the make-up feed and the desired capacity are knownin addition to the size and details of the mill and thebreakage characteristics. Before a mill is brought onload it is not possible to pre-specify the classificationexcept with the use of a hydrocyclone model, such asthat in the Modsim™ mineral processing simulator.However, the bulk density of (2.16)(103) kg/m3 foundfor 16×20 mesh particles (see the analysis of batchtests) gives a material density of (3.6)(103) if it isassumed that the bed porosity of the column of particleswas close to 0.4 (i.e. 40%) as found in tests of similarbeds using a mercury porosimeter (Austin et al., 1963).Since the specific gravity of pure hematite is 5.26 thisimplies that the internal porosity of the particles wasabout 0.32. It might be expected that smaller particleswould have some of this porosity broken out, withparticle densities increasing toward (5.26)(103) kg/m3,as size became smaller. It is difficult get reliable valuesfrom a cyclone model if the density of the particlesvaries with particle size, especially at high circulationratios.

It was decided to use as much information from thepilot-scale test as possible applied to the full scale,including the Sharpness Index and bypass fraction ofthe external classifier. Also, the exit classification wasassumed to be the same and the size distribution of thepilot-scale circuit product (cyclone overflow) was usedas a suitable criterion for the required degree of fineness,i.e. a one-point fit of 74.9%≤44 μm ; was chosen.Three changes were made for the full scale. First, thecirculating load was desired to be close to the originaldesign specification of 6. To use the simulator to reachboth a desired product size and a desired circulating load

Fig. 7. Comparison of simulated (including exit classification) andexperimental size distributions from wet grinding of iron ore in thepilot-scale mill: lines are simulated, symbols of experimental values,see Table 3.

requires two degrees of freedom, the mean residencetime and the d50 of the hydrocyclone classification.Second, the different mill diameter-to-length ratiorequires a change in the relative size of the first sectionin the residence time distribution model. Eqs. (19) and(20) give θ=0.835 for D /L=0.573. Thirdly, the sizedistribution of the material to be the make-up was takenas the standard values found at Carajas, not the valuesused in the pilot-scale tests. The standard size distri-bution did not have small size intervals that contained nomaterial and it was easy to do the required extrapolationto finer sizes.

Table 4 and Fig. 9 show the results of the simulationcompared to the actual results found when the mill cameinto operation. The experimentally determined value ofthe percentage of the product size less than 44 μm was73% not 74.9% but the circulating load was close to theCarajas design value of 6, and the simulated capacity of310 metric tons per hour was close to the 308 found inthe test. However, the agreement between the other sizedistributions around the circuit was poor, as discussed inthe next section.

6.2. Analysis of the full-scale mill

When the experimental values became available afterthe mill was brought on load it was possible to modifythe simulation using the same approach as used for thepilot-scale mill simulation. Instead of postulating thecirculating load and overflow size distribution and ad-justing the classifier d50, the actual size distributionswere used to calculate both circulating load and a clas-sifier function, and the mean residence time τ is then the

Fig. 8. Comparison of simulated (without exit classification) andexperimental size distributions from wet grinding of iron ore in thepilot-scale mill.

Table 4Simulation (preliminary design) and experimental comparisons for thefull-scale mill, based on pilot-scale results: τ and d50 varied to getcircuit product of 74.9%≤44 μm and circulating load of 6

With exit classification

Simulated mill capacity kg/h 3.10×105

Experimental 3.08×105

Simulated circulating load 6.001Design 6.0Experimental 5.67Simulated % ≤44 μm 74.9Experimental 73.0

Parameters: Exit classification External classification

x50=0.5 mm d50=0.046 mm

SI=0.533 SI=0.42

AT=0.864 min−1 a=0.125α=0.9μT=2.5 mmβ=3.5ϕ=0.3

Fig. 9. Initial (design) simulated and experimental size distributions forthe full-scale mill, see Table 4.


major adjustable parameter in the simulation. However,it is not easy to get good representative samples around afull-scale cyclone bank and a detailed analysis of thedata is necessary to indicate the reliability of the sizedistributions.

First, Eq. (23) was used to calculate the circulationratio, giving a value of C=4.95. Second, the weightpercent of solid in samples of the hydrocyclone feed,underflow and overflow was determined by weighingand drying. Designating these percentages by wC, wT

and wQ,

C ¼100−wc

wc

� �− 100−wQ

wQ

� �100−wT

wT

� �− 100−wc

wc

� � ð26Þ

The values found were wC=47.1%, wQ=15.31%and wT=84.8%, giving C=4.67 (a circulating load of5.67). It was felt that this was the most reliable value ofC and it was used to calculate classifier selectivityvalues using Eq. (24). The values of Ci calculated fromEq. (23a) are shown in Table 2 and it is again seen thatthere is a consistent change in the values as size de-creases, indicating a consistent error in one (or more) ofthe size distributions. In addition, the values of si wereerratic, and it was difficult to use Eq. (25) to get a precisefitting equation. The estimates found were SI=0.55,a=0.15 and d50=0.050 mm but other combinations ofparameters gave fits that were statistically valid.

The simulation was then performed to give73%≤44 μm and two adjustments made to get a closeragreement with the experimental circulating load: thex50 of exit classification was set at 1 mm (instead of

0.5 mm) and the searched value of the d50 of thehydrocyclone classification was 0.04885 mm. ??Fig. 10and Table 4 show the results. The ratio of the simulatedmill capacity to the experimental value was 1.009, ingood agreement. However, the agreement between theexperimental size distributions and the simulated valueswas not as good as those for the pilot-scale mill. Inparticular, it is clear that the circuit product size distribu-tion Q(x) is significantly steeper than the experimentalsize distribution even though both have 73%≤44 μm.This discrepancy was not seen in the pilot-scale case(Fig. 7).

Fig. 10. Comparison of simulated and experimental size distributionsfor the full-scale mill, see Table 5.


If the product specification was only to fit a singlepoint, as is often the case, the simulation would be con-sidered to be in satisfactory agreement with the exper-imental result. However, in the case considered here it isimportant to be able to predict the specific surface areadeveloped in the product, and this cannot be done if theshape of the product size distribution is not predictedaccurately. An attempt was made to back-calculate thevector of si values that would produce the correct sizedistribution, using a definition of, from re-arrangement ofEq. (24),

si ¼ 1−ðqi=CLpcalciÞ ð27Þ

where pcalc is the simulated value of the mill product sizedistribution. This worked well for larger sizes but theseback-calculated values of s became negative for the

Table 5Simulation and experimental comparisons for the full-scale mill, usingexternal classification from full-scale results, adjusting τ to get circuitproduct of 73%≤44 μm, with exit classification

Simulated mill capacity kg/h 3.104×105

Experimental 3.08×105

Simulated circulating load 5.64Experimental 5.67

Parameters: Exit classification External classification

x50=1 mm d50=0.0489 mm

SI=0.533 SI=0.55

AT=0.864 min−1 a=0.15α=0.9μT=2.5 mmβ=3.5ϕ=0.3

smaller size intervals and negative values have nophysical meaning.

Similarly, the makeup feed size distribution of thepilot-scale test was used instead of the size distributiondetermined at Carajas. On iterating to the steady state,the simulated circuit product size distribution was againtoo steep as compared to the experimental values and theresult was only slightly different from that using theprevious size distribution. Thus the difference betweenthe good fit of the pilot-scale test and the poor fit of thefull-scale test was not due to the different feeds used.

7. Sub-sieve size measurement

To investigate further the discrepancy discussedabove, tests were made to check the assumptionsinherent in the use of laser diffractometer sizes con-verted to equivalent sieve sizes (Austin and Shah, 1983).A material density is entered into the MasterSizerprogram but when it is used to go from volumetric sizedistribution to size distribution by mass fraction it willcancel out if the density was constant for differentparticle sizes. Similarly, when the measured mean par-ticle cross section determined by the laser beam isconverted to equivalent spherical volume and then toequivalent mass, the shape factor of the conversionprocess will cancel out if it was constant for differentparticle sizes. It has already been noted that the porosityof the ore might change with particle size, which couldalter both factors.

The constancy of the factors was tested by mea-suring the laser diffractometer size distributions of care-fully sieved fractions of 60×100, 100×140, 140×200,200×270 and 270×400 mesh iron ore. The results wereplotted in the log-logistic form

P ¼ 1

1þ xx50

� �−k

that is

1P−1 ¼ x50

x

� �kð28Þ

Plotting x versus (1 /P)−1 on log–log scales enablesx50 and λ to be determined over the region where astraight line is obtained, see Fig. 11. Table 6 shows thatthere is no consistent change in λ or the ratio of x50 to thegeometric sieve size with particle size (Austin et al.,1988) over this size range (250 to 38 μm). It wasassumed that laser diffractometer size could be convertedto equivalent sieve size by dividing by 1.23, giving thevalues for the sub-sieve sizes given in the tables.

Fig. 12. Laser diffractometer size distribution of the minus 325 mesh(≤45 μm) fraction of the circuit product (hydrocyclone overflow) ofthe full-scale mill test.

Fig. 11. Size distributions offfiffiffi2

psieve intervals of iron ore as measured

by the laser diffractometer, plotted according to Eq. (27).


Fig. 12 shows another interesting finding. The productsize distribution (cyclone overflow) showed a clearchange in slope at about 7 μm. This again would have alarge effect on calculations of specific surface area fromthe size distribution. The results given previously indicatethat there was no change in particle properties (shape anddensity) down to 38 μm but it is possible that there werechanges at smaller sizes. The simpleAustin–Shahmethodof detecting such changes cannot be used at sizes belowthe sieve range, and investigation using microscopicshape analysis was beyond the scope of the project.

8. Discussion of results and conclusions

What was expected to be a straightforward applica-tion of known techniques to construct a mill simulatorapplicable to ball milling of iron ore has shown anumber of unanticipated problems. The large differencebetween the measured bulk density of a

ffiffiffi2

pscreened

sample of the material and that expected from the theo-retical density of the ore suggested that the material hasa considerable porosity and this has been confirmed.Depending on the size distribution of the pores it is

Table 6Values of λ and the ratio of x50 to the geometric sieve size for eachsquare root 2 size intervals measured by the laser diffractometer

Sieve size, μm λ Ratio x50/mean x

250×150 6.77 1.053150×106 5.91 1.19106×75 7.12 1.2275×53 6.69 1.2353×45 6.69 1.2345×38 6.69 1.2338×20 5.56 1.37

possible that the porosity varies as the material is brokenfiner. This could be investigated by helium pycnometryand mercury porosimetry but this is a research project inits own right and further work remains to be done(Schneider et al., 2005).

Measurements by the Warman Cyclosizer and by laserdiffractometry both require knowledge of the effectivedensity of each tested size in order to get accurate sub-sieve size distributions. For a product that has the majorpart of its mass in sizes less than the sieve range it must beexpected that accurate sub-sieve measurement will beimportant in defining the properties of the product.

Unlike many other examples, the primary breakagedistributions were not dimensionally normalized. Thismay be due to changing porosity but this has not beenproved. An empirical method of describing the non-normalization was developed but more detailed studiesare certainly necessary. Apart from this anomaly, thebatch grinding tests showed excellent first-order break-age kinetics and reasonably good simulations of theproduct size distributions were obtained.

Scaling up to a pilot-scale test gave simulation resultsthat were in reasonable agreement with the experimentalresults, including product size distribution, circulatingload and mill capacity. However, extension to the full-scale mill gave reasonable matches between simulatedand experimental circulating load and capacity but aninadequate match for the product size distribution. Al-though the ball size distributions were different betweenthe pilot and full-scale mills, both distributions gave thesame variations of specific rates of breakage withparticle size and almost the same mean primary break-age distributions (B values), so that it does not seemlikely that the steeper product size distribution seen withthe full-scale mill is due to this difference.


It is generally expected in mill modeling studies thatan increase in circulating load will lead to a steeperproduct size distribution by removing fine materialbefore it can be re-ground to too fine sizes. Although thesimulation predicted this, we cannot explain why theexperimental product size distributions are almost iden-tical from the pilot test with circulating load=3.66 to thefull scale with circulating load=5.67 and why, therefore,the product size distribution from the full-scale mill issignificantly shallower than the simulated result.

The finding that the slope of the product size dis-tribution changed at about 7 μm suggests that sub-sievesize measurement should be used in the analysis of batchgrinding tests in order to look for the same phenomenon,and we recommend that this be done for all futurestudies of this kind.

Acknowledgements

The authors thank the Companhia Vale do Rio Doce,Brasil, for permission to publish this work.

Two of the authors, Austin and Schneider, would alsolike to thank Professor R. Peter King for his many yearsof encouragement and friendship. He has been a simula-tion enthusiast for most of his career and his develop-ment of simulation techniques started early, almost assoon as computers became available, leading eventuallyto the Modsim™ simulator.

References

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simulation of wet ball milling of iron ore at carajas, brazil

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