The breakage function; What is it really?

Minerals Engineering, Vol. 3, No. 5, pp. 405--414, 1990

0892-6875/90 $3.00 + 00 © 1990 Pergamon Press pie

Printed in Great Britain

THE BREAKAGE FUNCTION; WHAT IS IT REALLY?

E.G. KELLY§ and D.J. SPOTTISWOODt § Dept. of Chemical & Materials Engineering, University of Auckland, New Zealand t Dept. of Metallurgical and Materials Engineering, Colorado School of Mines, U.S.A. (Received 16 October 1989; accepted 12 November 1989)

ABSTRACT This paper discusses the breakage function (better called the breakage distribution function) used in the population balance method o/analysing size reduction. It is concluded that the breakage distribution function can be considered to result from two different breakage processes; shatter and cleavage. In size reduction equipment in practice, an individual fracture event usually involves both processes, and may also involve rebreakage o f the progeny products. The overall result is, however, a relatively stable breakage distribution function.

Keywords Breakage distribution function, breakage mechanisms, shatter breakage, cleavage breakage. INTRODUCTION Given the inefficiency of the size reduction process, the vastness of the literature on the subject is not suprising. Because of the complexity of the process, progress has not been easy or rapid. Early attempts to study the "fundamentals" achieved comparatively little at the time, and led to an emphasis on empirical methods, in particular the so-called "laws" of size reduction. Nevertheless, the value of this, and particularly Bond's work, cannot be belittled. It could be argued that much of the confusion in the literature arises because, in studying size reduction, there was often too little distinction made between fracture mechanisms and the size reduction process. Over the last 30 years or so, the population balance method of describing size reduction has been developed - to the extent that today it is capable of accurately describing the performance of the renowned Bougainville mills that appeared to be so inadequately designed by the older Bond method [1]. (Although it could well be argued that the failure of Bond's method in this case was due to the fact that it was being extrapolated beyond its empirical data base.) A notable feature of the population balance method is that, in essence, it considers size reduction to consist of two basic components: the fracture event (represented by the breakage distribution function), and the fracture process (represented by the rate or selection function). It is ironic that these functions (and the others that are required to fully represent practical size reduction equipment) must still be evaluated by empirical methods, the very criticism that is so often used to damn Bond's method. A number of workers have attempted to derive the breakage distribution function from fundamental principles, but without much success [2],[3],[4]. This work attempts to address this problem in a qualitative manner, in the hope of initiating more discussion on the topic. ~E 3/~--A

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THE BREAKAGE DISTRIBUTION FUNCTION The concept of the breakage distribution function was adopted by a number of workers [5],[6]. It can be defined as the average size distribution resulting from the fracture of a single particle. Implicit in this definition is that a number of particles must be broken to obtain the average result. Initially, a number of methods were used to determine it. Gardener and Austin [7] originally used radioacive tracers, while Kelsall et. al. [6] used mineral tracers. Whiten [8] assumed what must be considered an arbitary mathematical function. More recently, back calculation techniques have become the norm [9],[10]. Most of the work, and certainly the earlier work, was carried out with tumbling mills, particularly ball mills. It was generally considered that the breakage distribution function was relatively insensitive to most variables [11 ],[12]. In particular, it was commonly found to be independent of the initial particle size; that is, it could be normalised, and could be fitted by an expression of the form of

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Yet even in these early studies on ball mills, there were instances where the breakage distribution function was found to be non-normalisable, that is, dependent on the initial particle size [13]. At the time, no attempt was made to explain this phenomena, nor the form of Eq. 1. The initial qualitative description relevant to this equation can be considered that given by Lynch et. al.[14] in their initial (and one of the few) analysis of crushing. They attributed the two components implicit in Eq. 1 as being a fine size distribution, resulting from localised fracture of the large particle against the crusher face, and a coarse distribution, resulting from the main fracture of the particle. There have been a number of papers which have described fracture in terms of mechanisms such as cleavage, shatter, abrasion, chipping, etc.[15]. Interpretation of this work is complicated by two factors; different authors have used different terms for what may be the same mechanism, but, more significantly, some have been considering single fracture events, while others were considering multi-fracture events. In 1982, the present authors introduced the concept that the fracture mechanism, and the resulting size distribution, were dependent on the energy intensity applied to the particle [16]. Now that more information is available, it seems an appropriate time to review the value of this concept. THE BREAKAGE PROCESSES In considering energy intensity, there are a number of factors of significance. Most significant is that as particles get smaller, they become stronger. This was most graphically described by Hukki [17], and has been substantiated by later work [ 18]. In general, this can be attributed to the flaws that cause fracture in brittle materials. Firstly, as particles are reduced in size, the weaker flaws are consumed, making further fracture more difficult (it must be appreciated that in reality flaws may be generated during the fracture process; the assumption here refers to the average number of flaws existing at a given initial size.). Secondly, as the particle becomes very small (and the work of Yashima et. al. [18] suggests that this is about 500#m), plastic deformation in the vicinity of the crack tip becomes significant; eventually to the extent that a fracture limit is reached at about l#m. The other factor of significance regarding energy intensity is the rate at which the energy is applied (which must of course be dependent on particle size). If the energy is applied slowly to a single (relatively large) particle, then primary facture can occur just after the weakest flaw is overloaded. The resulting fracture will cause unloading of the product particles, and the size distribution will be a few particles of size close to that of the original particle. Such fracture is best described as cleavage. (Chipping can then be thought of as a special ease of cleavage whereby a relatively small piece is cleaved off the particle, leaving a particle of essentially the original size.) Consider such a fracture in terms of the stress-strain curve of a brittle material as shown in Fig. 2 (although in reality, particle shape [ 19], and/or plastic deformation [18] may result in non-linear relationships, but these do not alter the general argument). The initial loading of the particle causes elastic deformation up to the fracture point 'F', where typically a spherical particle fails by tensile fracture from within [20]. Once fracture occurs and the particle falls apart as the loading is released, it contracts elastically; that is, virtually all the energy that was put into elastically deforming the particle (i.e., the area under the curve in Fig. 2) remains in the particle until it is lost, essentially by dissipation as heat. Only a small amount of energy is ultimately retained by that particle and can be considered as useful energy; that converted to surface energy during crack propogation at a stress relatively close to the fracture stress. Schematically, this "useful" energy is represented as the small shaded area 'A' on Fig. 2, and the information available indicates that it is typically about 1% of the applied energy. Thus, unless the elastic deformation can be harnessed, fracture is inherently inefficient. This point will be considered again below.

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Fig.2 Schematic illustration of the stress-strain diagram of a brittle material. 'F' is the point of fracture, and the shaded area 'A' is the effective work done in opening the crack to generate the new surface With higher loading rates, three changes are possible. Firstly, loading may increase significantly beyond the point 'F' (i.e., along the dashed line) before fracture and ,subsequent unloading occurs (i.e., more "unused" energy which further lowers the efficiency). (At first sight this indicates qualitatively why crushers are supposedly more energy efficient than tumbling mills: the loading rate is slower, and there is less chance of causing wasteful elastic deformation beyond the fracture point,) Secondly, with high rates of loading under compressive conditions, there is the possibility that the energy can, initially, be "absorbed" in the particle, in that, before fracture can occur at the weakest flaw and unloading takes place, sufficient energy is applied to the particle for fracture to occur at a number of flaws of increasing strength. The product size distribution will then contain many particles of varying size, and the fracture mechanism can be described as shatter. Such behaviour is more likely in grinding environments, and is substantiated by the work of Krogh which shows that impact energy, and not the rate of impact, determines the breakage distribution function [21 ]. A shatter mechanism will also occur from the third possibility: high loading rates under tensile conditions. In this situation, the crack velocity reaches a maximum, whereby energy can be released only by bifurcation of the crack into two branches which in turn bifurcate again and again [4]. This situation can be envisaged as occurring under two sets of conditions. Under pure impact conditions involving a particle striking (or being struck by) a surface, a compressive shock wave propogates through the specimen. Although the compressive stresses may be insufficent to cause fracture, the tensile wave reflected [22] from the unloaded surface of the particle can generate sufficient stresses to cause tensile failure that could involve bifurcation. Alternatively, when a relatively large particle breaks, because the material is so much weaker in tension than in compression, the fracture occurs predominantly by cleavage, as a consequence of tensile failure from within [20]. However, in reaching sufficient loading to produce tensile failure, large sections of the particle are subjected to appreciably higher compressive stresses (Fig. 3). Consequently, if rapid unloading occurs as a result of the cleavage fracture, some of the progeny fragments could experience tensile shock waves as a result of the rapid unloading; a situation that could produce shatter fracture in these progeny fragments, again by bifurcation.

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Fig.3 Distribution of principle stresses in a spherical particle under localised compressive loading In the case of crushers, there is a situation described above, proposed by Lynch et. al., that could lead to a similar result. The energy intensity (energy/unit volume) is (relatively) low, as is the loading rate. The resulting size distribution is a narrow cleavage size distribution. However, where the particle is in contact with the crushing surfaces, the energy intensity is high, and some material breaks to give a shatter size distribution. This latter component of the product distribution could be termed abrasion fracture. This is, in fact, the term that is usually used in autogenous grinding, for the process that initially wears away the largest particles until they undergo what is probably cleavage fracture. However, because the cumulative size distributions for shatter and abrasion typically have similar slopes, it is better to restrict the number of titles, and to consider the "abrasion" fracture events in autogenous grinding to be situations where the energy has been insufficient to initiate a cleavage fracture, and what is occurring is a localised shatter fracture. Thus, there are a range of situations that can occur in practical size reduction equipment, and consideration of the two expressions on the right hand side of Eq. I as being due to two basic fracture mechanisms, shatter and cleavage, is not unreasonable. As mentioned above, attempts to date to predict breakage distributions from theoretical analyses have had little success. However, if the distribution is considered in terms of two mechanisms, then it can be argued that some analyses have some validity. Typical measured values of ~ lie between 0.6 and 1.3, while values of 13 lie in the range 3 to 5. Viswanathan's models [23] predict values between 2 to 6, depending on the particular model. This would suggest that some of his models are reasonable descriptions of the cleavage process. If one considers simple successive bifurcation whereby a single particle was broken in two equal parts, with one of each progeny particle being rebroken, then this would give a maximum value of 2. If however, the rebreaking particle of any pair was larger than the nonbreaking particle (more likely if it is considered that the smaller particle would be stronger), then it is easy to see that o~ could easily be appreciably less than 2; that is, capable of representing the shatter distribution.

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It also follows from the discussion so far that as the (feed) particle size decreases there will be an increase in the energy intensity (energy/mass of particle), and we would expect to see the proportion of cleavage fracture decrease, and the portion of shatter fracture increase. Breakage distribution function data for crushers are still limited. Figure 4a shows some data from a laboratory crusher [24], where it can be seen that cleavage fracture dominates. A considerable amount of data has been published on the breakage distribution functions occurring in tumbling mills [25]. It shows the breakage distribution function to be reasonably consistent, with cleavage fracture dominating (Fig. 4b). (These two sets of data suggest that the fracture mode, rather than than the conventional use of particle size, may be a better distinction between crushing and grinding.) 100

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Fig.4 (a) Breakage distribution function for 20,560#m calcite particles broken in a (Blake) laboratory jaw crusher [24] (b) Breakage distribution for North Carolina Quartz, broken in a laboratory ball mill [25] At first sight, the consistency of the breakage distribution function in tumbling mills suggests that the above concept of energy intensity is invalid. However, this is not the case. The inconsistency can be attributed to the method used for determining the breakage distribution function. Most tumbling mill breakage distribution functions described in the literature have been determined from mill charges, and represent average conditions. More recent work [26],[27],[28] on the direct breakage of single particles shows that as the energy intensity is raised, the overall product size distribution becomes finer. Further, if this data is broken into two product size distributions (Fig. 2), it can be seen that the increasing energy intensity is essentially changing only the proportions of the two products, not their size distributions. It is worth noting two things about these determinations of the breakage distribution functions; firstly, that they must be a satisfactory representation of the fracture behaviour in real equipment, and secondly, that even though at very high energy intensities the concept of single fracture must be no longer valid (because progeny fragments must be being rebroken), the concept of the two fracture processes remains an acceptable descriptor, as does the concept of the breakage distribution function itself. Some further comments on rebreakage must be made. If the cleavage of a single particle into two pieces is considered the "perfect" fracture (in that it minimises over-breakage), the best that will be achieved in practice is the type of situation illustrated in Fig. 5. In the situation where the loading is sufficiently rapid, two things can happen: either a relatively

The breakage function

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large amount of energy is absorbed in the particle, and breakage of the whole particle occurs by shatter fracture, or the cleavage progeny fragments are rebroken as the breaking surfaces (either crusher faces, or media particles) continue towards each other. From the arguments above, the products of these two alternatives will be indistinguishable, and it is convenient to consider the rebreakage of cleavage progeny as having arisen from single breakage cleavage fracture. Eventually however, a point is reached where the proportion of cleavage fracture exceeds 1.0, that is • > 1.0. At this stage it can be considered that rebreakage of shatter progeny fragments is occurring, and the extent of this can be given by (~ ° l).

Fig.5 Combination of fracture mechanisms, as occurs in practice TUMBLING MILLS From the discussion so far, it follows that the non-normalisable breakage distribution function should be the norm, rather than the exception. Why then is it so remarkably constant for all particle sizes in tumbling mills? To answer this we must consider the fracture environment in a tumbling mill. Many interpretations have been presented. It is likely that most fracture occurs in tumbling mills by the media sliding or rolling past each other, at which time particles are nipped and broken - in effect by mini-crushers. Thus, virtually all fracture occurs in the tumbling charge. The speed at which two pieces of media pass each other will be important, but probably most important is the "head" of tumbling media above the given pieces of media. If there is sufficient head of charge above the two given pieces of media, fracture will occur. Thus, at the top of the charge, the head will be insufficient for fracture to occur. In the upper section there will be only sufficient energy intensity for cleavage fractures to occur, but near the bottom of the charge the excessive head will cause very high energy intensity shatter fracture, with rebreakage, to dominate. Thus, in a real tumbling mill there is a wide range of breakage distribution functions (actually only two functions, cleavage and shatter, but in differing proportions) applying, and the effective result is truly an average. Given that most mills balance the media mass to the particle mass, it is not suprising that there is little variation in breakage distribution functions in tumbling mills. Non-normalisable behaviour then is likely to be associated with non-ideal brittle materials (e.g., sandstones) or mills that are not operating at optimum conditions (e.g., mis-matched media and ore). It also follows that the non first order kinetic behaviour that has been

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found to occur [29] should be associated with an increasing tendency to a higher proportion of cleavage fracture. The above comments also indicate a contribution to the poorer efficiency of very large ball mills. These mills will have a wider range of breakage distribution functions. In particular, grinding at the bottom of the charge will involve very high energy intensities. Given that the work of Pauw and Mare [21] indicates that for -12.7 +9.5mm particles, the most energy efficient breakage occurs when the fracture is about 70:30 cleavage:shatter, it suggests that the particles at the bottom of a deep charge are not only likely to be too fine, their energy efficiency is undesirably low. HIGH PRESSURE ROLLS On the basis of the comments above, it is also possible to explain much of the controversy that has arisen over the efficiency of the new high pressure rolls. At first glance, the operation of these rolls contradicts the recommendations of Steenberg [30]. In his analysis of size reduction using stress analysis of bead matrices, he argued that when a network of particles is loaded, fracture will occur first, as expected, at the weakest flaw. If the loading is continued, subsequent fracture has to occur at increasingly stronger flaws that have less favourably orientated loadings, which would not be so efficient. To maximise energy efficiency, he claimed the bed should be unloaded after the initial fracture, then rearranged and loaded again to produce fracture at a new "weakest" flaw. While this analysis may have some value in the type of breakage used for wood fibres, it is hardly typical of most mineral size reduction. More significantly, it is not valid applied to the high pressure rolls. What Steenberg's analysis neglects is the fact that by not unloading the bed after the initial fracture, the work used in elastically deforming the fractured particles is being retained, and contributes to the loading of as yet unbroken particles. Because the work of elastic deformation is the major contribution to inefficiencies in size reduction, the retaining of this work in the bed more than compensates for the extra work necessary to break the less favourably loaded flaws. Thus, it follows that high pressure rolls cannot be compared with conventional size reduction equipment, and the application of what is relatively unconstrained single particle fracture data is not valid, because the fracture mechanism and process are markedly different. CONCLUSIONS It is considered that cleavage fracture occurs with low energy intensities, and shatter occurs with high energy intensities. Any single fracture event can be considered to be made up of these two distinctly different fracture mechanisms. In any actual single particle fracture, the product size distribution from the two mechanisms remains fairly constant; it is primarily the proportion of the two mechanisms that change. In reality, of course, this may not be so (for example, shatter and abrasion may arise from different processes). However, the consistency of the experimental data to date suggests that it is a more than adequate assumption, implying that the two distributions are characteristic of an ideal brittle material. The slight changes in the slopes of the mechanism size distibutions that do occur in practice can be attributed to minor variations in the properties of a brittle material, and/or the way in which any individual particle is loaded in the different types of equipment. Although this discussion emphasises the probability that the breakage distribution function is not constant, it does not negate the point that size reduction can be adequately analysed by assuming that it is constant. This is firstly because the variations, in a reasonably efficient mill, will cancel out, and secondly, because the breakage distribution function is probably interrelated to the breakage rate. It is, however, possible that the effect may be important when extrapolations are made.

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REFERENCES

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26. Narayanan S.S. & Whiten W.J. Trans (C) IMM, 97, CI15(1988). 27. Flavel M.D., Rimmer H.W., Woody R.N., & Schmalzel M.O. Preprint 88-199, 177th SME-AIME Annual Meeting, Pheonix (1988). 28. Pauw O.G. & Mare M.S. Powder Technol. 54, 3(1988). 29. Austin L.G., Shoji K., & Everett M.D. Powder Technol. 7, 3(1973). 30. Steenberg B. Powder Technol. 37, 289(1984).