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Reprinted from Proceedings of the Sixth International MINERAL PROCESSING CONGRESS held at Cannes, 26th May - 2nd June, 1963 PERGAMON PRESS OXFORD. LONDON. EDINBURGH. NEW YORK PARIS. FRANKFURT 1965 COMMINUTION KINETICS

.~ D. W. FUERSTENAU. AND P. SOMASUNDARANt

1 ." INTRODUCTION where Yj is the percentage of the fragments b} The object of this paper is to present an analysis weight from the single comminution event that are of the kinetics of comminution of quartz and lime- finer than size x, kj is the size modulus of the frag- stone in ball mills and rod mills. In this investi- ments(the theoretical m~imumsize in the assembly gation, the kinetics of size reduction has been of fragments),and IXis the distribution modulus that studied for thesematerials being ground separately is characteristic of the material and comminution as well as part of a mixture. method. The size modulus kj is determined by the Arbiter and Bhrant1) recently observedthat the energy expended per unit mass of the particle initial rate of production of fine particles in a ball being broken during the comminution event.(3) miU is a zero order rate phenomenon, that is the For a given material and comminution method, amount of fine particles produced is directly the distribution modulus is constant.<2) The size proportional to the time of grinding. As pointed distribution of the total product must then be the out by theseauthors, if fine particles are producedat summation of the products from all of the individual a constant rate, comminution of thesefine particles comminution events: themselvesmust be negligible. Arbiter and Bhrany suggestthat evenin ball milling the coarsestparticles are ground selectively, these particles acting as shields for the finer particles. Intermediate sized wherey is the cumulative percentageof total material particles are produced at a constant rate initially finer than size x, ZI is the number of comminution L. but as the coarserparticles disappear, these particles events of kind i, and 1511'1is the weight of particle themselvesare ground. Measurementof the initial broken by the given event. The more nearly alike rates of formation of particles provides an exceed- the different comminution events are, the less will ingly useful tool for studying the comminution be the deviation of the size distribution from action in a tumbling mill and this will be the basis equation (2a): for the investigation reported in this paper. y = l00(~)« (2a)

BASIC PRINCIPLES where k is the size modulus of the total product. Schuhmann(2) has considered that a complex For comminution eventsthat are nearly alike in a comminution operation, such as ball milling, can mill that consumes power at a constant rate, the be considered to be the summation of many rate of formation of particles finer than any individual comminution events. Schuhmann has stated size must be constant, as observed experi- assumedthat each event producesfragments whose mentally by Arbiter and Bhrany.(IJ Considering size distribution is characterized by the following the case where uniformly sized feed particles are size distribution: being comminuted, the rate at which particles finer than sizex are being formed is given in equation (3): Wx= Kxt (3) . Professor of Metallurgy, University of California, where Wx is the cumulativeweight of particles Berkeley4, California. t ResearchAssistant, University of California, Berkeley4, finer than size .'\",Kx is the cumulative rate constant California. for particles finer than size x, and 1 is the grinding

2S -~=-

26 D. W. FUERSTENAU AND P. SOMASUNDARAN

time. If W0 js the weight of material jn the mill, laboratory ball mills and rod mills have been re- )' is simply 100W,xl Wo. Thus, with comminution ported.(3,6) These investigations showed that the events that are quite similar, K,x must be related to distribution modulus of quartz and limestone is x«. Arbiter and Bhranyball mill. it is quite probable that any particle milling, however, these deviations from linearity or particlescan be impacted by a ball and be broken. would be expected to be less. Thus, not only feed particles but also the fragments To test thesehypotheses, this investigation will be from broken feed will be comminuted by the balls. concernedwith the rate of formation of fine quartz Thus the product from a ball mill will be a mixture and limestonein a laboratory rod mill and ball mill of the products from these different kinds of under conditions where each material is ground comminution events. and may even contain un- separatelyand as part of a binary mixture. Since l-:. touched feed.(2,4)On the other hand, in a rod mill we are concerned primarily with initial rntes of the rods are wedged apart by the largest particles formation, these experiments will have to be for and there is little chance of finer particles being short grinding times. ground until the coarse particles have been broken. Thus, in the rod mill the kinds of comminution MATERIALS AND METHOD events are more nearly alike. and the size distri- bution of the rod mill product then follows For ease of differentiating the comminution equation (2a) over a much greater range. Because characteristics of two minerals in a mixture, of the wide variation in kinds of comminution Brazil quartz and California limestone were selected events in the ball mill and because of possible for this study for several reasons. Quartz has a untouched feed, the ball mill product follows distribution modulus of about 0.9 whereasthat of equation (2a) only in the finer sizes, with large limestoneis about 0.6. Sincethe limestoneanalyzed deviations in the coarser sizes. In terms of the 99.8 per cent CacoJ, 0.1 per cent Sial, and 0.1 per kinetics of particle formation in a ball mill. one cent RlOJ, decompositionwith acid affords an easy might then expect deviations from a straight line method for differentiating betweenthe two minerals. on the log-log plot of K%versus x in the coarser In each test, I kg of 4 x 8 mesh material was sizes. whereasin the case of rod milling. the log- ground at 60 per cent solids by weight(7)for short log plot of K%versus .y may be linear up to coarse time periods (I, 2, 3, and 4 min) in an 8 x 9! in. sizes. mill that contained either 17.6 kg of rods (26 rods Recently extensive investigations of the com- t in. diameter, 10 rods i in. diameter and 5 rods i minution (If mixtures of limestone and quartz in in. diameter) or 17.6 kg of balls ranging from i to 28 D. W. FUERSTENAU AND P. SOMASUNDARAN

T: 'z (0) ( w60 I u a: w ~i~~¥,~~ Q. ~50 (!)z . « ~401 -+'0 0. I ..- I :I: ~ 30 30 w 3: w 2.0 > ..- ~ -I ~ :> fa "r 10 2 ~. ::> u 0 0 I 2 3 4 0 I 2 3 4 5 GRINDING TIME, MINUTE S FIG. 2. Plot of cumulativepercentage of limestonefiner than eachsieve size versusgrinding time in (a) the ball mill and (b) the rod mill.

in Fig. 4a, the initial rates of formation (given as kinds of comminution eventsoccurring in the mill, cumulative percentageof material finer than any that is, the grinding of fragments while untouched stated size per minute) of limestone and quartz feed remains. As mentioned earlier, the wedge are plotted semilogarithmically as a function of action of the largest particles between the rods particle size in mesh(Tyler scale)for thesematerials within a rod mill prevents appreciable grinding of ground separately in the ball mill and rod mill. any particles finer than the coarsestmaterial in the The slope of the log-log plot of initial rate of for- mill. These experimentsshow that during grinding mation, Kx. versusparticle size (a linear plot of size a certain size fraction of material is formed at a in mesh being equivalent to a logarithmic plot of constant rate which is characteristicof the material, size in microns) is 0.61and 0.63 for limestoneground the size being considered, and the comminution in the ball mill and rod mill. respectively,and 0.87 method. for quartz ground in the ball mill and rod mill. Observation of Fig. 4a shows that the rate of These slopes compare quite well experimentally formation of fine limestone particles in the rod mill with the values of IXfrom the size distribution of exceedsthe rate in the ball mill whereasthe rate of thesematerials, as would be expectedfrom equations formation of fine quartz particles in the ball mill (4) and (5). The log Kx -log.~ plot for rod milling exceedsthe rate in the rod 'mill. The increased is a straight line to about 20 mesh for both quartz grindability of limestone in the rod mill could and limestone. whereaseach curve is a straight line possibly be due to the greater attrition effect in the only up to about 65 mesh for the ball milling of rod mill. Where the grindability of the material these materials. This deviation in the case of ball (quartz) is about the same in either type of mill, milling can be ascribed to wide variation in the the rate of formation of fine particles in the ball mill COMMINunON KJNEnCS 29 70 .- z 6u ~ Co) ~ \ ~50

~ c5 t z U) U) 40 ~ a.. .- a 30

~ I ~ 20 :2: .- oct 5 10 10 ~ :) Co)

0 I 2 3 4 0.-- I 2 3 4 5- GRINDING TIME, MINUTES FIG. 3. Plot or cumulativepcrccntaac or quartz finer than eachsieve size venus grindina time in (a) the ball mill and (b) the rod mill.

might be expectedto be greater than that in a rod formation of fine quartz increaseswith time and mill becauseof intermediate sized particles being that of fine limestone decreaseswith the time for ground themselves. which the mixtures are ground. Although Fuer- The size distributions of quartz and limestone, stenauand Sullivan's data for the extendedgrinding when ground as I : I mixtures. are given in Figs. 5a of quartz and limestoneseparately are not presented and 5b for ball milling and rod milling, respectively. in Figs. 2 and 3, their data lie on these same Tl1e valucs of IXfor quartz ground as a component straight lines. In the caseof grinding mixtures, the of a mixture in the ball mill and rod mill are 0.87 :i: initial rate of formation is linear because com- 0.03 and 0.92 :i: 0.02. respectively. For limestone. minution will be independentof the grindability of the corresponding values are 0.61 ::t 0.01 in both the constituents in the beginning. Becauseof the cases. This is in accord with Schuhmann's single greater grindability of limestone,quartz will remain comminution event hypothesis. larger in size as comminution proceeds and these Figures 6a. 6b. 7a. and 7b presentthe cumulative larger quartz particles, therefore,will consumemore percentageof material finer than each sieve size as of the energythan the protected,finer limestonepar- a function of time of grinding of mixtures in the ticles. This effect becomes more pronounced as ball mill and rod mill. Data for ball milling the grinding progresses,with the rate of fine quartz mixtures for 5 and 10 min are taken from the formation increasing and that of fine limestone de- experimentsof Fuerstenauand Sullivan(6)and data creasingwith time. The wedgeaction in the rod mill for rod milling the mixtures for 5, 7.5.and 10min should augment this phenomenon. It also occurs in are also taken from experimentsof Fuerstenauand a ball mill, becauseof the existenceof a protective Sullivan.(') These figures show that the rate of action, to some extent. - 30 D. W. FUERSTENAUAND P. SOMASUNDARAN

40 ,(0( us too :> to) {b) z ~ ~~~..- ~~ a: 20 20 us z ~ too Z W 10 (,) 10 0.61' , a: v w ". 8 , ,g:", ..-.,6 8 Q. &. ~ ~, ~/ .6/ ,,~ / z b / ~/ /. A / Q // p' / / too ,: ~ 0.q7" c( / . ~ i ,0'...' IS. / 2 41 . p...'.' 0.81 ~ a: I // / ,. , ,0.61 ~I ~/ .' / i ~ 0.87. . .A I ~ ~/ 0 / / .Aa //,,10.81 ,. / fo.87 8ALL .A ~ W / ,. ROD / ' BALL ROD too / N ILL 2 c( MILL ,A /. MIll MILL a: . ~. LlM ESTONE ..0- LIMESTONE .0- -- -J /A' .. c( QUARTZ ~ -A.- /1 QUARTZ -4- -A- , ~ MIXTURE --0-- .,8:-- ~ 400 200 100 48 28 14 400 200 100 48 28 14 PARTICLE SIZ E, ME SH FIG.4. Variation of initial ratesof formation with particlesizein the ball mill and rod mill for (a) grinding limestone ( and quartz separatelyand (b) grindirIg limestoneand quartz as mixtures.

The initial rates of formation of fine quartz and log-log plot of rate of formation of particles versus fine limestone when ground as mixtures in the ball size yields lines of slopes of about (1.. Such plots are mill and rod mill are presented in Fig. 4b as a func- not given in this paper. tion of particle size. These results are somewhat The initial rates of formation of both limestone similar to those obtained when the minerals are and quartz particles are found to be less when ground alone in that the slopes of the linear portions ground as a mixture than when ground alone in a of the log K - log x plots are again about equal to rod mill. This possibly could be due to the protec- IX. This shows that the mode of comminution of one tion of limestone particles by coarser quartz particles mineral is indep~ndent of the presence of other and protection of quartz particles by the limestone minerals. However, because more energy may be going to the harder material, even in the initial dust coating on the quartz particles. The kinetics of stages, the rate of formation of the softer mineral grinding mixtures may be difficult to interpret (limestone) in the rod mill has decreased. Because because of the possible change in power required the initial rate of formation of quartz particles is to move the ball or rod charge, dependent upon the relatively low, changes in the rate may be difficult size and hardness of material in the mill.<8) Accurate to detect. determination of power consumed in comminution Figure 4b gives only the initial rates of formation may help in analyzing comminution during short of limestone and quartz particles in mixtures. If grinding times, since this would tell us how valid is the instantaneous rate of formation of limestone and the assumption that energy expended is perfectly quartz particles is taken at higher grinding times, a proportional to the grinding time. COMMINU110N KINEnCS 31 The rate of particle formation of a mixture as a m3.terial. For this equation to hold, the grinding whole behaves similarly to the caseobtained when of thesefine particles themselvesmust be negligible. minerals are ground separately, that is a plot of In the ball mill, where probability has a great role cumulative percent finer versus grinding time is a governing a ball striking any particle, deviations straight line. The slope of the log K versus log x occur for sizes greater than 65 mesh. However, plot for the mixture as a whole is 0.67 (seeFig. 4b). in the rod mill where coarse particles wedge the \, This agreeswith the distribution modulus IXfor the rods apart, a particle cannot be ground until the mixture if a weighted averageis taken basedon the coarse one has been considerably reduced in size. i relative grindabilities of the two constituents. -} Consequently, the initial rate of formation of particll:s in rod mills satisfies the above equation SUMMARY to the region of 20 mesh. This study of the initial rates of formation of In the caseof grinding of mixtures, the material quartz and limestone particles, when ground with least grindability remains coarser and hence separately in ball mills and rod mills, shows that more of the energy will go for grinding quartz than each mineral follows the relation: limestone after the first stagesof grinding. Thus the K~ = Ko(~)2 rate of formation of fine quartz increasesas grinding of the mixture proceedsand the rate of formation of limestone particles in the mixture decreases. where K~ is the initial rate of formation of particles This effect is greater for the rod mill than for the finer than size x, Ko and .'t"orefer to some reference ball mill. size, and IX is the distribution modulus of the This work has shown that during grinding, a 32 D. W. FUERSTENAUAND P. SOMASUNDARAN

'.~ 60 :(b) 48 I- Z ~ 'U ~ ~ 5 0 Q. ISO

~ If ~40 z ~o ~ I- -/, ~30 ~ ~JO/ ~o ~ ~OO

~ 20 I- 120

! - I I . I . I I I . /0 0' ~ 4 ei 8 0 2 - 4 6 8 10 GRINDING- TIME. MINUTES FIG. 6. Plot of cumulative ~ta&e of material finer than e8d11iew size venus grindina time for (a) quartz fractions and (b) limestone fractions of the I: I mixtures !found in a ball mill.

t- (0) (b) 14 , ~60 120 160 l u ~ w ~c, a.. 20 28 48 ~50 ~5 so ~ 3,5 100 w 0 z 48 u. 40 !50 ~o t- 6.5 :I: 200 (!)

w3Q 270 ~ '00 I:» W ,ISO

t-~20 120

. -, i I I I I I I I I I '0 0.) 2 4 6 8 O. 2 4 6 8 10 GRINDING TIME, MINUTES FIG. 7. Plot of cumulative pc~nL1.aeuf materialfi~ thatl eachsie\Ie - versuslrindinc titne for (a) quiirtz (ractions a:1d (bl Ij'"l'~,i(\"e fr;J .tior.s of tht" i: j mixiurt1 gro'.nd in a roo mill COMMINunON KINETICS 33

certain size fraction of material is formed at a Thus the larae and small scale grindabilities, as defined above. were not the same, because of the differences between the constant rate, which is characteristicof the material, mills, but nevertheless were exactly proportional. the size being considered and the comminution method. DR. G. AGAR (International Minerals altd Chemicals Corporation, U.S.A.) REFERENCES This paper is based on the excellent work of Arbiter and Bbrany which was presented under a title that disauised the I. ARBITtR, N. and BHRANY,U. N. Correlation of product tnJe worth of their contribution. We have herc one of the size, capacity and power in tumbliol mills, TralU. few public applications of their findings to the practical j\,~ A.1.M.E. 117, 24S (I~). problem of comminution. One of the significant points, that 2. ScHUHMANN,R., Jr. Energy input and size distribution has not been stressed in the presentation here, is the unique in comminution, TraIlS. A.1.M.E., 117, 22 (19W). value of the parameter « associated with each mineral. This 3. BERosn<*, B. H., SoLUNBERGER,C. L. and MITCHELL, conflicts radically with some ~nt proposals on fracture W., Jr. Energy aspects of sioaic particle CnlSbing, TraIlS. and particle size distributions. The authors' comments would A.1.M.E., 110, 361 (1961). be appreciated. - 4. AGAR, G. and CHARU:S, R. J. Size distribution shift in Also this analysis can be e;.tended, as Arbiter and Bhrany grinding, TraIlS. A.1.M.E., no, 390 (1961). did, to show that the energy consumed E is equal to Ak-A. S. FUERSTENAU,D. W. and SULUVAN, D. A., Jr. Analysis The analysis seems to indicate therefore, that the size of the or the comminution of mixtura, CtlnQ(/.J. CMm. &g., feed is without effect on the energy consumed. 40, 46 (1962). Can the writers reconcile this with the proposal of Svensson 6. FUERSTENAU,D. W. and SULLIVAN, D. A., Jr. Com- and Murkes(l) who account for the feed size in their energy- minution of mixtures in ball mills, Trmrs. A.1.M.E., 123, particle size relationship? IS2 (1962). 7. FUERSTENAU, D. W. and SuWVAN, D. A., Jr. Size distnoution and cncray consumption in wet and dry AUTHORS'REPLY TO G. AoAR's COMMENTS grinding, Trans. A.1.M.E., 110, 391 (1961). 8. CoGHILL,W. H. and DEV ANEY,F. D. Ballarinding, Tech. Our views on the parameter «, as obtained from size dis. Paper S81, U.S. Butt8u of Mines (1931). tribution or from kinetic data, are that the value is not unique for any material but that it depends on the mineral and com. minution method. For example, we have found in our labo- ratories that « for numerous minerals is about 1.0 for impact DISCUSSION and compression fracture of individual specimens; but when PROFESSORN. ARBITER the same minerals are ground in a rod mill or a ball mill, the values of the parameter « become less than unity. In (Columbia Un;~slty, New York) sorne cases, the values obtained in a ball mill may differ This is an important contribution to further understanding considerably from those in a rod mill. The explanation for of grindina kinetics. In the investigation by Bhrany, data these differences is that they appear to result from differences were obtained only on single minerals, although data by in comminution mechanisms. Although Schuhmann implied others on heterogeneous malerials did appear to exhibit only impact events, we ha\ e now postulated that comminu- conSlancy of initial grinding rates. Now in the present study tion events other than impact events occur in a mill, these it has been shown that a mineral has the same fragment size events being abrasion events (surface attrition) and chipping dispersion characteristic in a mixture as it has alone (Figs. events (nonfruitful impact that fractures only comers and 4, ~). Also important is the authors' finding that the initial edges). When abrasion and chipping are appreciable, the rates for the mixtures are constant althouah obviously dif- value of « decreases because of the production of 8reat~f ferent from those for the components. Thus the validity of quantities of fines. Hence, the size distribution produced by Bhrany's thesis that the rates of formation or the product any device will depend upon the relative abundance of impact, sizes are constant over practical arinding times receives chipping, and abrasion events. This in turn depcnds upon powerful support. the machine and such properties as hardness, cleavage, and The implication of this constancy is large, both theoreti- &rain size of the material being gfound. cally and practically. Thus, the slope in Fig. 4b is the proper The problem of feed size in analysis of comminution sys- exponent ror the energy versus product size relationship and tems is a difficult one. However, we designed our experiments this exponent is not constant for different materials but varies so that the feed size eff~ct (if there were any) would be con. approximately rrom 0.3 to 1.0. depending on the material stant by using narrowly sind (4 x 8 mesh) feed particles. and. to some extent. on the method or grindina. Dr. Agar himself has written a paper in which he suggested The constant in equation 3 is a true grindability, with the that a feed size term does not enter into comminution of dimensions: weight per unit time and per unit energy. fairly uniformly siz~d materials and he suggested that the The original observation of the simplicity or arind kinetics observed effects in his own experiments w~re due to efficiency and its userulness was made about 1~ years ago in a large of energy transfer from crinding balls to feed particles (see Arizona copper ore grinding circuit. The capacity of 10 x 10 Trans. A/ME,120. 390 (1961». For those who are interested ft grate mills to produce a desired constant grind varied rrom in this problem ,,~ would like to suaest that they read the 1SOO to 2(XX)lons per day. Samples of mill feed. showing this following papcrs: Trans. A/ME, 217, 22 (1900) and TrUII.f. variation. were tested in a 1 foot batch mill. The rates or A/ME, 123, 62 (1962). rormalion of sizes from 65 to 200 mesh plotted against cor- responding daily mill capacities showed a linear relationship, III Stockholm Congress (19S7). 34 D. W. FUERSTENAU AND P. SOMASUNDARAN

DR. G. AGAR. d'action. Dans Ie domaine des grenus Ie broyeur a barres donne une deviation faible et Ie broyeur a boulets une devia- The authors' reply to one of the questions indicated that they believe that all single particle fracture results in a size tion importante, correspondant a un reliquat de grenus qui distribution with

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