Comminution and liberation of minerals

Minerals Engineering, Vol. 7, Nos 2/3, pp. 129-140, 1994

0892-6875/94 $6.00+0.00 Pergamon Press Ltd

Printed in Great Britain

COMMINUTION AND LIBERATION OF MINERALS

R.P. KING Comminution Center, 306 W.C. Browning Building, University of Utah, Salt Lake City, Utah 84112, USA (Received 20 August 1993; accepted 13 September 1993)

ABSTRACT

Two industrial problems, energy consumption and mineral recovery, have dominated research in comminution during the past five years. Both have required new and fundamental insights into the fracture process, when brittle particles are subject to rapid impact. New experimental procedures using the ultrafast load cell, dual pendulum and high-precision image analysers have produced experimental data that is providing the basis for significantly new models of the fracture process. New generation comminution machines are exploiting the deeper understanding and theoretical models of wide applicability are emerging to provide a sound basis for design, scale-up and optimization of comminution operations. Keywords Comminution, mineral liberation, energy consumption, fracture.

INTRODUCTION Comminution has always occupied center stage in the repertoire of mineral processing operations. It will continue to do so for a long time to come because comminution is a problematical unit operation. It is impossible to exploit the mineral resources of the earth's crust without comminuting the material to the particulate state in which it can be usefully processed and used. The tools of comminution operations are blunt by comparison with other process engineering operations, and as a result, comminution operations deliver products that are never optimal for their subsequent use. In addition to the problems caused by the purely technical difficulties associated with the operations, comminution commands increasing attention because it is a major consumer of energy, and we know that the energy utilization efficiency of industrial comminution machines is uncomfortably low. The cost of energy for comminution is often a determining factor in the economic viability of a mineral producing activity. Research into comminution processes has led to considerable advances in our knowledge and understanding. In particular, the development of population-balance techniques for the description of the size distribution in the charge in a comminution machine and its product has provided an important unifying framework which allows the comparison of different processes and procedures. This has greatly improved design and scale-up procedures and has promoted the introduction of larger and newer comminution machines. Improvements in the understanding of the process engineering of comminution systems have encouraged the introduction of automatic control systems so that automatic control is now the rule rather than the exception. Automatic control has brought with it quite impressive improvements in system productivity, either increased mill capacity for the same product fineness or increased comminution at the same mill capacity or sometimes both together. It is these improvements in productivity from the economy of larger scale and controlled operation that have been the most significant rewards of research in comminution systems over the past three decades. Keynote Address, Minerals Engineering '93, Cape Town, South Africa, August 1993 129

130

R.P. KINO

In more recent times, the focus of research attention has turned increasingly to the question of energy consumption by eomminution operations. It has been known for a long time that comminution operations use energy inefficiently in the sense that they consume much more energy than is actually required to break the material. But we really have very little understanding of why this is so.

COMMINUTION AND ENERGY Energy consumption by industrial processes is becoming a major issue worldwide. Uncertainties in the supply of natural gas and the adverse environmental effects of the other major energy sources, crude oil, coal and uranium, are building economic pressures that pose significant problems for the minerals producing industries. In some cases there is already evidence that the energy costs of comminution will determine the viability of significant sections of the industry. Although crude procedures for the quantification of energy consumption by comminution do exist, it is now very clear that further progress in the understanding of how comminution energy is used can only come from a detailed and fundamental understanding of the physics of brittle particulate material fracture. Any new really significant advances in comminution technology in the forthcoming decade will come particularly from the exploitation of a basic fundamental understanding of the fracture processes that underlie all industrial comminution systems. The precise mechanisms of fracture are unknown when brittle solids, such as particulate mineralogical material, are subjected to rapid impact. This problem has attracted the interest of many scientists since the early sixties. The early work, particularly that undertaken in Europe, was concerned mostly with the determination of the size distribution of progeny particles that resulted from the controlled fragmentation of single brittle particles and with the relationship between that distribution and the specific energy input. The chief difficulty in investigating the phenomena associated with the rapid impact fracture of brittle particulates, is devising experiments that are capable of unravelling the details of the microprocesses that occur in the sub-millisecond range. Weichert and Herbst [1] solved this by adapting a device which has been used successfully by mechanical engineers and material scientists for many years to study impact processes -- namely, the Hopkinson pressure bar. The new device, which became known as the Ultrafast Load Cell, allows the accurate and reproducible measurement of the distribution of fracture energies, particle strengths as well as the distribution of progeny sizes that result from impact fracture. The ultrafast load cell can also be used to study the impact fracture of particle beds under conditions that would correspond, at least approximately, to the conditions that occur in a ball mill. At about the same time, researchers at the Julius Kruttschnitt Mineral Research Centre in Brisbane were developing the dual-pendulum device to study single-particle impact fracture [2]. In recent years, data from both the ultrafast load cell and the dual-pendulum have exhibited a remarkable consistency and have revealed a uniformity and universality that was previously unsuspected, but which enables us to speculate now, with considerable confidence, that the size distribution that results from an impact event on a single particle is predictable under a wide variety of conditions provided that the energy available for the impact event is known. These devices are shown schematically in Figure 1. The figure illustrates the greater utility of the UFLC, which can measure both the distribution of fracture energies as well as the progeny size distribution, while the dual-pendulum measures only the size distribution. The first tangible results of this research was the demonstration by Herbst, Lo and Hrfler [3,4] that some aspects of the performance of a laboratory ball mill could be predicted using data obtained from the fracture of particle beds on the ultrafast load cell. While work at the Julius Kruttschnitt Mineral Research Centre has shown that the energy-specific breakage functions measured using the dual pendulum correlates with the behavior of the same material in the ball mill [2]. More recently work has been concentrated on the investigation of single-particle fracture on the UFLC. The possibility now exists of modeling ball mill perforrnanee in a two-step process from single-particle tests to multiple-particle single-impact tests and from those the behavior of the ball mill is synthesized. An essential ingredient in this two-step process is the understanding of the nature of individual media-

Comminution and liberation of minerals

131

media impacts in the ball mill. These processes have been opened to study by some remarkable simulation work undertaken by Rajamani and Mishra [5] using the discrete element method. For the first time we have the possibility of a design and scale-up procedure for ball mills that is largely free of empiricism and which is based firmly on the principles of solid mechanics and physics.

SINGLE-PARTICLE

BREAKAGE

m L

40 U d q-

i!,; ~ ~ :~p~ .'~"~:7~i;'~-;; ,. ~!.:~;.;.:~ ..

C

O

01

Fr~c±ure

energy

J/kg

U[t~aPost rood

]

N Ln

<

O

8 -p bl

J

/

Duc~l p e n d u l u m P(~r-±icte size Fig. 1 Two devices have been developed during the past decade to investigate single-impact fracture processes. The ultrafast load cell (upper) can measure the distribution of particle fracture energies, the distribution of particle strengths and the size distribution of the progeny. The dual pendulum device provides the size distribution of the progeny. The impact energy and absorbed energy are measured in both devices. The result of this synthesis is the calculation of the selection and breakage functions that have proved to be so effective in correlating the performance of industrial milling equipment through the application of population-balance modeling techniques. The application of population-balance models to comminution processes introduced radical improvements into the design and control technologies for industrial comminution circuits during the sixties, seventies and early eighties. In spite of the tremendous improvements in our understanding of these technologies that resulted from the use of population-balance formulations, these models have always suffered from a serious disadvantage: the key descriptive functions that characterize both the material to be processed and the equipment, namely the selection and breakage functions, cannot be measured directly but must be inferred by calculation from the result of a comminution test. Recent work using single-impact tests has led to a characterization of the breakage process in terms of the specific energy that is absorbed by individual particles. It is now possible to

132

R . P . KING

calculate the breakage and selection functions from a knowledge of the distribution of fracture energies for particles of the material that is being broken end the distribution and frequency of impact energies in the mill [6]. The distribution of particle fracture energies measured in a single-particle impact test on the ultrafast load cell is shown in Figure 2 [7]. The data are for two different sizes of quartz particles. These data reveal that even for particles as small as half a millimeter the average fracture energy is only a fraction of a Joule per gram! Yet industrial mills routinely use in excess of 10 kWhr/ton to break similar material, and this is a sure indication that significant energy savings should be possible by the use of more efficient machine design.

Quartz particles 1.0 •~

0.9 0.8

~

~

0.7 0.6

0

o.s

~ "C

0.4

~

0.3 0.2

~

0.1 0.0 ° 0.0 0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Fracture energy Jig

Fig.2 The distribution of fracture energies measured using quartz particles in the ultrafast load cell. The lines represent the best-fitting log-normal distribution. The single-impact fracture tests have given us a new and fascinating insight into the problem of energy usage. The energy required for comminution has fascinated researchers for much more than a century, and the earliest attempts to quantify the relationship between the amount of breakage and the energy consumed were incorporated into a differential equation, which has been associated with the names of Rittinger, Kick and Bond. This classic equation has been used often to show how some index of the product size (usually the 80 % passing size) varies with the amount of energy consumed. This was the best description of the phenomenon until the early 1970's when Herbst and Fuerstenau [8] demonstrated that the entire size distribution is shifted in a way that tracks the accumulated energy input. Their formulation is shown as Eq. (1), and this equation is now replacing the Rittinger-Kick-Bond equation as the classic description of energy usage in ball mills and is destined to take its place among the fundamental equations that describe the laws of grinding.

Comminution and liberation of minerals

133

i-I

dE

j.i

The parameters in these equations all lead to the calculation of 10 and more kWhr/ton of ore milled in typical industrial mills. This is to be expected since they reflect the completely uncontrolled and unselective random impact fracture that occurs in a tumbling mill. It is now clear that a radically new method with controlled application of energy will be required in the next generation of comminution machines. The distribution of progeny sizes that results from single-particle impact fracture is shown as a function of the impact energy in Figure 3. It is interesting to note that these data are completely consistent whether determined in the UFLC, the dual-pendulum [2] or using conventional drop-weight tests [9]. These curves are significant and their accurate prediction is important because they show how the size distribution of the comminution progeny is related to the energy that is dissipated in the mill.

SINGLE-PARTICLE BREAKAGE FUNCTION Variation of breakage function with impact energy 100 90 80 o~

70

60 so 40

m

30 20 10 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Ratio of progeny size to parent size

Fig.3 The generalized model for the breakage function that results from a single impact on a single particle. The progeny particles suffer successive impacts through succeeding generations during the same event until all of the original impact energy has been dissipated or all progeny have been ejected from the impact zone. However we must ask whether we are, in fact, moving towards more energy-efficient comminution machines or whether we have stalled in a morass of antiquated technology weighed down by an ultraconservative industry that accepts innovation only slowly. I believe that it is abundantly clear that major new advances can come only from new research, and very basic research at that.

134

R . P . KING

What evidence do we have that the present research effort is in fact making progress toward more energyefficient comminution technologies? Two examples will suffice. The first is the increasing application of stirred and tower mills for the production of very free particulates. The essence of the stirred mill is that the grinding action is achieved by media-particle impacts that are considerably more gentle but very much more frequent than in a comparable ball mill. This is exactly what is required if we analyze the data coming from the single-impact fracture studies. Comminution is energy efficient only when the intensity of impact is matched optimally to the particle size that is to be broken. High frequencies of impact are needed to maintain the rate of production of progeny and therefore maintain adequate throughput. The stirred mill exploits these ideas effectively. Another example is the high-pressure roll mill. In the early stages of comminution, the high-pressure roll mill can achieve the same degree of comminution as the equivalent ball mill with the expenditure of only 50 % to 75 % of the energy [8]. Of course, as the size reduction ratio becomes greater, the high-pressure roll mill loses its advantage. Consequently, a combination of high-pressure roll mill followed by a ball mill offers potential savings of energy of 25 % for applications that are now handled by ball mill circuits. Major industrial applications of this technology should not be long delayed.

COMMINUTION AND LIBERATION When considering comminution for mineral processing operations, the question of mineral liberation must always be considered. The random fragmentation of multicomponent mineral ores to a size scale that is comparable to the size of the mineral grains gives rise to the phenomenon of liberation. Many particles in the resultant product will consist of only a single mineral but, because the fracture process and the mineral texture are to a considerable extent independent of each other, complete separation of the phases is never achieved. It turns out that the calculation of the liberation spectrum of the resulting product population is a geometrical problem of very considerable complexity. It is however an important problem because rational design of comminution operations can be possible only when the liberation of the product spectrum is properly described. This is particularly true when regrind circuits are being designed or analyzed or when there is a definite relationship between the liberation and the performance of subsequent mineral separation and recovery operations. This problem has been recognized for many years and the early work of Gaudin initiated serious study of this problem. However the liberation phenomenon has defied quantitative analysis, and it has only been relatively recently that a sufficient theoretical understanding of the problem has been achieved, which permits a technically reliable calculation of the liberation spectrum even for ores that have only simple mineralogical textures. The difficulty in this case has been due to the inappropriateness of classical mathematical analysis to provide adequate descriptions and to the difficulties in the way of undertaking valid measurements in the laboratory. Fortunately both of these difficulties have to a large extent been overcome -- the first by using some of the latest mathematical results from the fields of stochastic processes, geometrical probability, random set theory and stereology -- the second by the use of effective and reliable image-analysis systems based on the scanning-electron microscope preferably equipped with X-ray microanalysis such as in QEM*SEM. The tremendous increases in power, particularly for graphical display devices, and the steadily reducing cost of computer equipment have made it easy to build an effective image-analysis system for no more than the price of a very modest scanning- electron microscope and a good computer. The very essence of the liberation problem is the need to recognize that it is largely geometrical in nature and that the geometry that is involved is very irregular. The texture of the ore minerals is very varied and petrologists and mineralogists have devised elaborate descriptions of these textures. However really quantitative descriptions of the textures have been very difficult to achieve. Almost all of the early work was based on modal analysis generated by point counting, which does not do a good job of characterizing the three-dimensional multiphase textures. Modem theoretical models have overcome these deficiencies and the modem approach was initiated in 1979 when King [10] showed that the mineralogical texture

Comminution and liberation of minerals

135

could be completely characterized by the linear-intercept distributions measured on polished sections of the ore. This development was influenced significantly by the excellent work that was being done at that time by M.P. Jones in London in measuring the linear-intercept distributions accurately [11,12,13]. Perhaps the most significant difficulty associated with the linear-intercept characterization was the need to convert the calculated apparent linear-liberation spectrum to the true three-dimensional volumetric-grade distribution. This problem was solved, at least in principle, by King in 1982 [14], but the practical implementation only became possible in 1985 with the generation by Baba, Herbst and Miller [15] of a transformation matrix that was calculated by simulating the linear-intercept analysis of simulated twophase particles using digital techniques. The utility of this transformation technique was demonstrated by comparing the calculated liberation spectrum with that measured using laborious serial sectioning techniques. The transformation was shown to be effective for different ore textures and for samples that showed very poor as well as good liberation characteristics. The transformation of measured linearliberation spectra to volumetric spectra has since been shown to be accurate by comparison with spectra measured using carefully fractionated samples obtained by magnetic-fluid separation [16]. The transformation technique involves the solution of the Fredholm integral equation 1

P(gLI D) = f P(gL Ig ,D) p(g [D) dg

(2)

0

for the volumetric liberation spectrum p(glD) in a sample of particles of mesh size D. This requires careful numerical analysis because of the notorious instability of the solution procedure. The interested reader is referred to the excellent text by Wing [17] for a discussion of this problem. It turns out, however, that least squares solutions to this equation can be generated quickly and accurately provided that the apparent linear-liberation spectrum is measured adequately. This can be done with modern imageanalysis equipment [18]. This procedure is generally referred to as the stereological correction of the apparent linear-liberation spectrum because it is usually applied to the linear-liberation spectrum measured in the ore in particulate form to generate the measured liberation spectrum. The liberation prediction problem is stated as follows. Given an ore texture which is characterized by the linear-intercept distributions through each phase, what is the volumetric particle-grade distribution to be expected when this ore is fractured randomly? This problem is solved in two steps. First, the apparent linear-liberation spectrum P(gLID) is calculated exactly, then this calculated distribution is stereologically transformed into the volumetric particle-grade distribution.

THE STEREOLOGICAL PROCEDURE TO CALCULATE MINERAL LIBERATION An exact procedure that avoids all assumptions other than the assumption of random fracture was developed by King [10,14] to predict the liberation spectrum from the measured linear-intercept distributions through each phase in the unbroken ore

Let p(glD) = .~r[p(gt.lg),P(gLID)] be the solution to the integral Eq. (2). Here gz represents the apparent linear grade reported by a single-line probe traverse through a particle and P(gL) the associated cumulative distribution function. Then the liberation spectrum of the three-dimensional particles is given by p(g) = sr[p(gL l g), (u tH~ + ~0H0)]

(3)

where the functions H 1 and H 0 are determined entirely by the linear-intercept distributions of the two phases in the unbroken ore together with the linear-intercept distribution for the particles p(llD)

-- i ~,(gLlOp(llD)dl 0

(4)

136

The functions

R . P . KING

¢i(gt.I/)

can be evaluated independently of the particle shape or size O0(gLI/) = 1 _ 1<'-'t)'~ ! [1-F0(u)-~n.~

~(gL[/) = lpt !

F('(gLl)l~'(u)]du

1 -F,(u)-~_~n.,F~°n'(Z-gLl)f~n'(u) du

(s)

(6)

In these equations, Fi(l) is the cumulative distribution of linear intercepts through phase i in the unbroken ore, Fi(n)(u) is the n-fold convolution of Fi(u) with itself and Fi(O)(u) m 1 and

(7)

~°~(u) = F~°-~(u) - 2F~n~(u) + F~cn'~(u)

Pi is the mean

of the distribution

Fi.

Although these equations look rather unpromising, they are actually quite easy to evaluate from the measured or theoretical linear-intercept distributions for each of the phases [14,19,18]. However the generation of the solution (3) to the integral Eq. (2) is not a simple task. The numerical solution is notoriously unstable and requires very careful treatment [20]. In addition, an appropriate kernel P(gt.I g) must be used. At the present time, the best kernels available are those generated by simulated linearintercept analysis of computer-generated two-phase particles [15,21], although recent work at the Comminution Center has produced kernels directly from carefully fractionated mineral ores. An alternative, but closely related approach to mineral liberation, has been taken by Barbery [22] and his collaborators. They have relied more heavily on modeling the texture of the ore and have used the interesting Poisson-Boolean models for two-phase systems that have been described and made popular by authors such as Serra [23] and Stoyan, Kendall and Mecke [24]. These models, which can be calibrated against measured linear-intercept distributions in the ore, provide a geometrical description of the texture, which in turn can be used to predict how the random-fracture pattern that is induced by comminution will produce particles having compositions ranging from completely liberated gangue to completely liberated mineral with every composition between. The essence of this predictive method is the spatial-correlation function which provides a second-order statistical description of the texture. The spatial-correlation function for the texture is defined by C 12(L) =

probability that two points separated by a distance L in space are in phases 1 and 2 respectively.

This correlation function is related to the distribution of linear intercepts through the two phases in the ore and as a result can be measured using image-analysis techniques. The relationship between C12(L) and the two linear-intercept distributions can be given explicitly only in the Laplace Transform domain by [19]

C'2(s) = LaP

1

In Eq. (8), f/(/) is the distribution density corresponding to

-- i e-'~(l)dl o

(8)

(1

trans[C~2] - s2(Pl + It2)

Fi(l)

1-f:2

]

and

(9)

Comminution and liberation of minerals

137

THE INTEGRAL GEOMETRY PROCEDURE TO CALCULATE MINERAL LIBERATION The integral geometric method to calculate mineral liberation was developed by Barbery and his colleagues [22]. This method is based on the approximation that the distribution of compositions of unliberated particles in the population is described by an incomplete beta function p(g) = (l-u1~ - Uo~) g"1(l-g)S'~

B(a,~)

(I0)

where g is the volumetric grade of a particle. In Eq. (I0) ,~I and '~0 represent the fractional liberation of mineral (phase I) and gangue (phase 0) respectively. B(~,fl)is the beta function and the parameters t~ and fl are related to the first two moments of the liberation spectrum, n 1, and n 2, by the following equations M

nI

=

nl(1 -~o)

(11)

1-n,~ -0-n,N,

u n2 - n , ~ n2 = l_nl~ _(l_nt)~£° M n2 nl

(13)

M

i -n

"t =

(12)

j'

n r -n~ 4

(14)

_ n~v

(is)

- (1-n~)¥

(16)

Equation (10) is essentially a four-parameter representation of the liberation spectrum. By virtue of the subsidiary Eqs. (ll) - 06) the four parameters are ~'o and '~1 the fractional liberation of each phase, and n I and n 2, the first two moments of the liberation spectrum. The calculation of the liberation spectrum is therefore reduced to the determination of these four parameters. The~ parameters can in fact be estimated from measurements made on sections of the unbroken ore. All that is needed are the measured linear-intercept distributions through each phase Fi(l) and through the particles p(/). The,so measurements are easy to make using any good image analyzer. The reduction of the linear-intercept distributions to the required four parameters is done by noting that the mineralogical structure is completely characterized by the two density functions Fl(l) and F2(l) and the fracture pattern is completely characterized by the particle linear-intercept density function p(l). The first moment of the distribution is just the average volumetric fraction of phase 1 n I = D|

Pl P, +Po

(17)

The second moment of the liberation spectrum is estimated using an equation developed by Davy [26]

n2 = th _ 4xE[V]]: L2P(L)(u,u ° _ C,,(L))dL

E[ V~

where V is the volume of a particle.

ME 7-2/3---B

"o

(IS)

R.P. KING

138

The fractional liberation of each of the phases is estimated using an approximate relationship suggested by Barbery [22]

= (f~t))4

(19)

where ~(1) is the apparent liberation of phase i that would be reported by a one-dimensional probe. The apparent linear liberation can be predicted from the linear-intercept distribution measured on the unbroken ore by a formula derived by King [10] ~) = 1 - 1 where

f (1-/'(/))(1-F~(l))dl

(20)

Fi(l) is the cumulative distribution of linear intercepts through phase i in the unbroken ore.

The sequence of Eqs. (10) - (19) for the calculation of the liberation spectrum was developed by Barbery [22], and it is appropriate that the procedure should be identified as the Barbery Procedure to recognize his pioneering work.

LIBERATION IN CONTINUOUSLY OPERATING MILLING CIRCUITS It is remarkable that both Barbery's procedure and the stereologieal procedure rely entirely on data that can be readily measured on polished sections of ore using standard image-analysis procedures. Image analyzers that use electron microscopy as the imaging source are particularly suitable for this purpose. The generation of these complete solutions to the liberation problem must rank as one of the notable successes of comminution research during the past decade. Further development of both the integral geometry and stereological methods is providing models that can predict the liberation that can be expected in primary milling circuits and also regrind mills. This information can in turn be used to assess the effect of incomplete liberation on the performance of the mineral processing operations and ultimately will greatly improve the possibility of optimizing cowminution circuits for enhanced mineral recovery and optimum energy utilization. In recent times emphasis in liberation research has consequently shifted to the analysis of liberation in continuous milling circuits [25,16]. The two predictive procedures described previously cannot be used because the feed to the mill is generally a composite including fresh ore feed plus recycled material from various separation operations. These operations do not return particles uniformly to the mill. For example, a hydrocyclone will preferentially return particles that contain heavier minerals while flotation and other separation operations will preferentially return unliberated particles to primary and regrind mills. Quantitative models for this phenomenon have been generated using the population-balance approach with the Andrews-Mika diagram providing the framework for accounting for particles of varying grade. The internal structure of the Andrews-Mika diagram is now being investigated and constructed using two-phase mineral ores that can be fractionated into narrow composition classes which are then fractured and the resulting progeny analyzed by image analysis. These are tedious experiments because a large number of mono-grade, mono-size particles covering the entire region of interest must be broken and the progeny screened and analyzed. A typical set of data is shown in Figure 4, which shows the measured sphalerite grade distribution in each of six size fractions that result from the breakage of particles in the size range 710 I~m - 1000 I~m and in the narrow density range 3.5 - 3.7 g/cc. The gangue mineral is dolomite. Various models for the distribution of mineral within the fan-shaped product region below the parent particle class are being evaluated. Within a comparatively short while, some fairly good models should be available. The application of the Andrews-Mika diagram in the form shown in Figure 4 requires some subtle changes in the algorithms that are used to implement the population-balance model for comminution circuits. These modifications have already been made and successfully implemented in the MODSIM simulator [25].

Comminution and liberation of minerals

139

ACKOWLEDGEMENT The preparation of this paper has been supported by the Department of the Interior's Mineral Institute program administered by the United States Bureau of Mines through the Generic Mineral Technology Center for Comminution under grant number G1105149.

Andrews -Mika, -3.7+3.5 g/cc particles

j

It

Fig.4 The liberation spectra measured in the progeny particles that result from fracture of dolomite/sphalerite ore particles in the narrow density range 2.5-3.7 g/co and size range 710-1000 i~m. This shows an approximation to the internal structure of the Andrews-Mika diagram for one parent particle type.

REFERENCES I,

2. 3.

.

Weichert, R. & Herbst, J.A., An Ultrafast Load Cell Device for Measuring Particle Breakage. Preprint 1st Worm Congress on Particle Techology, Nfirnberg, II, 3 (1986). Narayanan, S.S., Modelling the Performance of Industrial Ball Mills Using Single-Particle Breakage Data. lnt. J. Miner. Process., 20, 211 (1987). Htfler, A. & Herbst, J.A., Ball Mill Modeling through Microscale Fragmentation Studies: Fully Monitored Particle Bed Comminution Versus Single Particle Impact Tests. Prec. 7th European Symposium Comminution. Lyubljana, Yugoslavia (1990). Herbst, J.A. & Lo, Y.C., Microscale Comminution Studies for Ball Mill Modeling. Comminution - Theory and Practice Symposium. S.K. Kawatra Ed., SME, 137 (1992).

140 .

6.

7. 8.

,

10. 11.

12. 13.

14. 15.

16. 17. 18.

19. 20.

21. 22. 23. 24. 25. 26.

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