-,~ 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 Bhrany<l)experimentally found that the samewhether it is ground separatelyor as part ( log-log plot of the initial rate of formation of of a mixture, which is in accord with Schuhmann's particles finer than a given size versusthe sizeyields single comminution event hypothesis. The size a straight line of slope !X. The equation of this modulus of each component in the ground product straight line is: dependsupon the fraction of energy consumed by the particular mineral. In the caseof ball milling, the K~ = KO(~). (4) fraction of energy that each component consumes is approx.imatelydetermined by the volume fraction where Ko and xo refer to some referencesize. of that component in the mixture.(6) On the other For a particular grinding time, 1" for all sizesthat hand, in rod milling, the wedge action of coarse follow equation (3), the followil)g can be written by particles betweenthe rods causesthe material with substituting equation (4) into equation (3) the least grindability to consumea greater quantity of energy. Thus, in a rod mill, quartz will consume an increasingly greater amount of energy when ground as a mixture with lirnestone.(3) Hence, in This equation gives the weight-size relationship the caseof rod milling mixtures, the rate of forma- among the various sizes after a grinding time 19. tion of fine quartz might be expectedto increasewith Equation (5) becomesequation (2a). if the quantity grinding time, with the rate of formation of fine Ko', is 100, making '\"0equal to k. limestone decreasingwith time. In the caseof ball In a ball 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.