Liberation modelling using a dispersion equation

Minerals Engineering, Vol. 12, No. 2, pp. 219-227, 1999

Pergamon 0892--6875(98)00133-2

© 1999 Elsevier Science Ltd All rights reserved 0892-6875/99IS--see front matter

LIBERATION MODELLING USING A DISPERSION EQUATION

X. WE1 and S. GAY JKMRC, The University of Queensland, Isles Rd., Indooroopilly, 4068, Brisbane, Australia E-mail: X.Wei@ mailbox.uq.edu.au (Received 30 June 1998; accepted 14 October 1998)

ABSTRACT

This paper presents a new approach to model liberation for comminution. The liberation distribution is characterised by a dispersion rate function, which is related to the texture of the ore. Once this function is determined, the dispersion model predicts the liberation of the mill product when the feed or operating conditions changes. This model is validated using computer-simulated data and the dolomite-sphalerite data made available by Schneider (1995). The results show that the dispersion model is capable of describing and predicting the liberation of particles as size changes when breakage is non-preferential. © 1999 Elsevier Science Ltd. All rights reserved

Keywords Liberation; liberation analysis

INTRODUCTION One of the objectives of liberation modelling is to predict the distribution of valuable mineral in particles that have been subjected to comminution. This distribution is called the liberation distribution. One of the major applications of a liberation study is to determine the optimal grind size for a comminution unit. If the grind size is too large, then a relatively large proportion of the valuable minerals will not be extracted, leading to a low recovery rate and a loss of potential revenue. If the grind size is too small, the plant will incur unnecessarily high energy costs. A liberation study is also useful to improve the performance in other mineral processing operations. The product particles of the comminution stage will be subjected to classification according to one or more of its properties, for example, density, magnetic susceptibility or chemical affinity. In order to study the behaviour of particles in the classification process, it is important to know the liberation distribution of the particles.

219

220

X. Wei and S. Gay

History of liberation modelling The importance of liberation to mineral processing operations has been recognized since the late 1930's [1]; and liberation has become an active field in mineral processing since the mid-1980's because of the maturity of mineral processing simulation, the improvements in measurement technology and the increase in computing power. Most liberation models, including the first quantitative model proposed by Gaudin [1], are based on the analysis of the mineral texture of an ore. In these models, the mineralogical texture of an ore is simplified and characterised in such a way so that the liberation distribution of the particles can be predicted as a function of size. For example, regularly arranged cubes are used to model ore texture and fracture patterns in Gaudin's model. Although this model was simple, the idea of superimposing fracture patterns on an ore texture seeded much of the work that followed. Meloy [2], Barbery [3] and King [4] developed the most significant models along this line. Meloy (1984) devised a texture transformation to transform the original ore texture to a simple geometry, i.e. cubic or sphere. The regular geometric model is then broken into small particles of simple shape, and the liberation distribution calculated using simple geometrical formulas. B arbery (1987) modelled the ore texture by using a Boolean model. Under the assumption of non-preferential breakage, the first two moments of the liberation distribution, and the fraction of liberated particles can be estimated. The liberation distribution is fitted to a Beta distribution. King (1979) used a slightly different approach. Instead of transforming the original texture, a linear probe across the image of a polished section of an ore generates the linear intercept distribution for each phase of the ore, which characterises the ore texture. The linear sample is then broken in a pattern consistent with non-preferential breakage. By modelling the linear fragments by an alternating renewal process King (1979) derived an equation to predict the linear liberation distribution as a function of particle size. The true volumetric liberation distribution is estimated by applying a stereological correction to the predicted linear liberation distribution. The masking method [5] provided a practical method to study liberation. The idea is that if we have an image of ore texture, an estimate of the liberation for a particular size can be achieved by constructing a mask using particle sections of the desired size. By superimposing the mask on the texture, the grade of each section can be calculated, hence, the liberation can be obtained by stereological correction. The liberation models based on ore texture are useful for mill design but they are not very useful for grinding circuits and plant simulation. This is because once concentration takes place, the liberation distribution of comminution products is no longer the natural distribution produced by the breakage of the original texture of the ore. The model that can be used in the plant simulation is the extended grinding model. Andrews and Mika [6] proposed the first extended grinding model that predicts both the size and liberation distribution of the product particles. The number of arbitrary parameters required by the model is huge; and it appeared that to obtain these parameters, intensive experimental work was required. This experimental work involved separating particles into sets of several narrow size and narrow grade samples that upon breakage will generate hundreds of combinations of progeny sub-samples; this intense experimental work rendered the method impractical. Thus the Andrews-Mika model was theoretically valid, but practically limited. This practical limitation led researchers to develop experimental methods that were not as intense. Such a model is the size-reduction/liberation model due to Schneider [7]. The breakage function was determined for dolomite-sphalerite ore for a specific size class (-1000+710 microns). The breakage function over the whole size-grade plane was then estimated by assuming that the breakage function is normalisable with respect to the parent particle size. This data set is also used to validate the dispersion model, and is discussed in more detail in section 6.

Liberation modelling using a dispersion equation

221

S E P A R A T I O N OF L I B E R A T I O N F R O M C O M M I N U T I O N When minerals within a particle are indistinguishable as far as breakage is concerned, the breakage is described as non-preferential. The complexity of liberation modelling is simplified significantly if the nonpreferential breakage assumption is valid. Consider, as illustrated in Figure 1, that the parent particles are broken with either: high energy generating the high-energy progeny set, or low energy generating the low-energy progeny set.

Q IIgDOO

IImDwO Fig. 1

4Q|eBeJi

Schematic diagram showing the effects of breaking particles with high and low energy when breakage is nonpreferential.

The resulting size distribution is affected by the amount of input energy; the average size of the high-energy progeny set is smaller compared to that of the low-energy progeny set. However, if the breakage is nonpreferential, then for the same size-classes of the low-energy and high-energy progeny sets, the liberation distributions are the same. As illustrated in Figure 1, the smaller particles in the low-energy set have the same size as the larger particles in the high-energy set. For non-preferential breakage, these particles have the same liberation distribution. Thus if breakage is nonpreferential then for the purpose of modelling liberation, breakage can be considered as a continuous process even though it is discontinuous process.

A dispersion model for liberation From the discussion in section 2, we can model breakage as a continuous process even though breakage is in fact a discontinuous process. Thus one can consider, for modelling purposes, that if one were to consider a point within a particle of size s, one can suppose that this point also previously belonged to a particle of size s+~s where ~Ss indicates a very small change in size. As particles change size they also change grade; the possible change of grade is directly proportional to the change in size, and also directly proportional to another factor called the dispersion rate. The dispersion rate depends on the size and grade of the particles. For example, Figure 2 shows two particles with different dispersion rates. The particle shown in Figure 2b has a larger dispersion rate than the particle shown in Figure 2a. Similarly, the dispersion rate changes as a function of grade. The progeny of liberated particles will always be liberated particles. Thus for grade =0 or 100%, the dispersion rate is zero. Figure 3 shows a schematic diagram of the dispersion function as a function of grade and size.

222

x. Wei and S. Gay

To model l a grid system is used with grade represented by i and size represented by j, as shown in Figure 4. A variable k represents the dispersion rate. The equation for the change in the liberation distribution with size is given by:

li- l,/ : li,i÷ kl,i, lli,i, 1+ ki,i- 1li,/- 1- 2k¢/li/ This equation has an analytical equivalent:

Ol(s,g\ s / ) ad

a2k(s,g)l(s,g\ ag 2

s /)

with initial condition:

l(s--s ~ , g - - g ~ \ s ~ ) -- lp(s/,g ~)

Co) Fine particle

(a) Coarse particle Fig.2

Two particles typifying a coarse particle and a fine particle. For the coarse particle, the grain size is much smaller than the particle size. For the fine particle the grain size is of the same order as the particle size. If the coarse particle is split into two, the grades of the progeny will be very similar to the parent; the dispersion rate is slow. If the fine particle is split into two, the grades of the progeny are likely to be much different to the parent; the dispersion rate is fast.

A

Low

Size

o Fig.3

1o0

Schematic diagram showing how the dispersion rate changes with size and grade. For liberated particles, the dispersion rate is zero. The dispersion rate increases as particle size decreases.

Liberation modellingusing a dispersionequation

223

The dispersion rate k is itself a function of grade and size as demonstrated by Figure 3. The usefulness of the dispersion method is that once k is determined the dispersion equation is applied to the parent particles to determine the liberation distribution of their respective progeny particles.

Size

~q

i

J

\

Grade

j

Fig.4 Numerical scheme of the dispersion model. To estimate the dispersion function k, the dispersion function is first separated into a grade component G(g), and a size component D(s).

K(g,s) = G(g)*D(s) Keith [8] theoretically investigated the dispersion approach for very small particles. For such particles, the change of dispersion with size can be ignored. He was able to obtain a dispersion equation for these small particles given by:

G(g) = gr (l_g)~ where T is a constant. Initial attempts by the authors to find an appropriate analytical or numerical function for the size component D(s) proved unsuccessful. Instead of using a function, the following scheme was employed in the model fitting and prediction. Liberation was simulated without initially trying to estimate D(s). That is, for the computer simulation D(s) was taken as an arbitrary constant. Once the computer simulations were completed, for each actual-size the variance of the liberation distribution was determined. We then chose the computer simulated-step for which the variance was the same as for the actual size. The matching computer-simulated step was then taken as corresponding to the actual size. In this way a transform function was developed to relate computer-steps to actual sizes. This transform function fulfils the same role as D(s). The transform function is denoted by N s which represents the computer-simulated size or step, for an actual size s.

NUMERICAL VERIFICATION A computer simulation was carried out to schematically break an ore into cubic particles. The liberation was then determined for the respective progeny particles and the dispersion model was calibrated. The model

224

X. Wei and S. Gay

calibration results are shown in Figure 5. The agreement between the actual liberation distribution and the fitted liberation distribution is excellent.

I

Model Fitting Results

¢ Observation ,4"- Calculation

-

8 loo 8O

4O

|

20

o

0 0

10

20

30

40

50

60

70

80

100

90

Compo~Uon

Fig.5 Calibration results of the dispersion model for liberation. After calibrating the model, its predictive ability was tested. This was achieved by generating a set of particles of some large size, and then to take only particles which had a composition within a certain range. In this case the composition range was 40-60%. The breakage of these particles is then simulated, and the liberation distribution of the progeny particles is calculated. To determine the predictive ability of the dispersion model, the liberation distribution of the progeny particles are calculated using that of the largest size particles and the dispersion rate function determined in the model calibration. Figure 6 shows the model prediction results. The agreement between the prediction and observation is excellent.

Observation " Predction

Model Prediction Results "

lOO

jl

]

•

.°t

°T o

o

10

20

30

40

50

60

70

80

90

100

ComposiUon Fig.6 Application results of the dispersion model for liberation.

EXPERIMENTAL VERIFICATION The experimental verification was conducted using the liberation data made available by Schneider [7]. In that liberation study, a sample of dolomite-sphalerite particles in the size range of 1000 to 710 microns was

Liberation modellingusing a dispersionequation

225

separated into grade-classes. These samples were then broken into smaller sized particles. The liberation distributions of both the parent and progeny particles were determined. A resultant liberation distribution was constructed using this liberation data set, which is shown in Figure 7, and used to fit the dispersion rate function for dolomite-sphalerite ore. The fitted liberation distribution is shown in Figure 8. The agreement between the observation and model-fitted distribution is good.

Resultant Distribution 100

.

.

.

.

.

.

.

80 60

|

¢ -710+500 .It. -500+355 -355+250 .--I.---250+180 /~-180+106 0

10

20

30

40

50

60

70

80

90

L--A-

100

Compoeition

Fig.7 Resultant liberation distribution obtained from Schneider's Dolomite-Sphalerite data.

1

[T,-,=ol

M o d e l fitting r e s u l t s

60

I / I 0

10

20

30

40

50

60

70

80

90

100 L' - ' I ~ ' 1 0 6

/ / / J

Coml~ailion Fig.8 Fitted liberation distribution using the dispersion model. Predictions for the liberation distributions of the sub-samples were made using the dispersion rate function determined, and compared with the experimental data. Figure 9 shows the experimental data and the prediction for the particles with a specific gravity of -3.7+3.5. In most cases, the prediction of the dispersion model is reliable. The liberation distribution of the largest size is taken as the parent particles, and those of smaller sizes were calculated using the dispersion model. Therefore the largest discrepancy was expected to occur at the smallest size. However, the largest discrepancy occurred at the second largest size, and this discrepancy appears systematic for the seven grade classes. The magnitude of the discrepancy is much larger than that in the simulation results. The major cause of this inaccuracy is either that the breakage of dolomite-sphalerite is preferential or the data are subject to large measurement errors.

226

X. Wei and S. Gay

Actual distribution for the particles with specific gravitY •

100 >.~

.

l-'e~'1000+710 -..,l~-710+500

8o

x

i 60

-soo+~s

~ -355+250

~.~ 40

.-..IIII--~0+180

¢~

-..e---180+106 ---~---106

20 0 0

10

20

30

40 50 60 Compo+ilion

70

80

90

100

Prediction for the particles with specific gravity -3.7+3.5

118~t

~

120

-4k'-710+500

I'--~---1000+710

l-e-.Im+,.1o+

|

I:-.---.10+

0

10

20

30

40

50

60

70

80

90

100

Composition Fig.9 Prediction results for the particles with specific gravity -3.7+3.5

DISCUSSION Although the dispersion model is only verified for the breakage of single sized particles, it applies to multisize class particles provided that the comminution model is established. The following notation is used in the discussion: is the size class of parent particles, is the size class of progeny particles, is the mass proportion of particles in size class i for the feed, lS the mass proportion of particles in size class i for the product, Is the mass proportion in size class i' and grade class j ' for the feed lS the mass proportion in size class i and grade class j the product and is the mass proportion in size class i and grade class j in the product resulting from feed particles of size i'.

i'

i

FI, Pi l0 Lij L6,i'

A comminution model, for example the perfect mixing ball mill model [9] can be used to relate the size distribution of the feed to that of the product. This relationship is here written as: in~x

P,=

~ it..i

Tii,,Fi,

where Tii, is the proportion of the feed in size class i' which breaks into size class i in the product. If the liberation distribution of the feed is known, and the dispersion rate function determined for the ore, then

Liberation modellingusinga dispersionequation

227

the liberation distribution of the product can be predicted. First the liberation distribution of the product of size class i that is produced from size i' is determined by the dispersion model (Lij,i,, ). The liberation distribution of the total product is calculated by summing the liberation distributions for the different sizes as follows:

Tii,Fi,Lij, il i' L..

"-

Tii,Fi,

Furthermore, if the liberation distributions for the feed and product are determined in a survey, it should be possible to use this equation to determine the dispersion rate function using the survey data for a comminution device. The method would be to change the dispersion rate function Kij (which relates ll,,j, to Lij ' i) so as to fit Lij. This is the subject of further research.

CONCLUSION The results of the numerical simulations and experimental validation show that the dispersion model provides a useful framework for liberation modelling.

ACKNOWLEDGEMENT We wish to thank Bill Whiten, Jonathan Keith, and Michal Andrusiewicz for stimulating discussions; and Peter King and Claudio Schneider for the assistance regarding the dolomite-sphalerite data. The financial support by the AMIRA P9L - - "Mineral Processing" project sponsors and ARC is also acknowledged.

REFERENCES .

2. 3. 4. 5. 6.

.

8.

9.

Gaudin, A.M., Principles of Mineral Dressing. New York: McGraw-Hill, 1939 Meloy, T.P. and Gotoh, K., Liberation in a homogeneous two-phase ore. International Journal of Mineral Processing, 1984, 13, p. 313-324. Barbery, G., Random sets and integral geometry in comminution and liberation of minerals. Minerals and Metallurgical Processing, 1987, 4, 96--102. King, R.P., A model for the quantitative estimation of mineral liberation by grinding, Int. J. Min. Process, 1979, 6, p. 207-220. Gay, S.L., Liberation modelling using particle sections. PhD thesis, University of Queensland, 1994. Andrews, J.R.G and Mika, T.S., Comminution of a heterogeneous material: development of a model for liberation phenomena. Proceedings, I1 ~ International Mineral Processing Congress, Cagliari, 1975, 59-88. Schneider, C.L. Measurement and calculation of liberation in continuous milling circuits. PhD thesis, University of Utah, 1995. Keith, J., 1997 The dispersion function for the invariance of Meloy' s distribution. JKMRC Internal report. Whiten, W.J., A matrix theory of comminution machines, Chem. Eng. Sci. 1974, 29, p. 589-599.

C o r r e s p o n d e n c e on p a p e r s p u b l i s h e d in Minerals Engineering is invited, p r e f e r a b l y b y em a i l to m i n . e n g @ n e t m a t t e r s . c o . u k , or b y F a x to + 4 4 - ( 0 ) 1 3 2 6 - 3 1 8 3 5 2