International Journal o[ Mineral Processing, 20 (1987) 241-251 241 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

Relat ionship B e t w e e n Breakage Parameters and Process Variables in Ball Mill ing - An Industrial Case Study

S.S. NARAYANAN', P.J. LEAN 2 and D.C. BAKER 2

'Julius Kruttschnitt Mineral Research Centre, Isles Road, Indooroopilly, Qld. 4068 (Australia) (presently with BougainviUe Copper Limited) '~Zinc Corporation Limited, Broken Hill, New South Wales (Australia)

( Received November 4, 1985; accepted after revision November 5, 1986)

ABSTRACT

Narayanan, S.S., Lean, P.J. and Baker, D.C., 1987. Relationship between breakage parameters and process variables in ball milling - an industrial case study. Int. J. Miner. Process., 20: 241-251.

The performance of the secondary ball mill at the New Broken Hill Consolidated Ltd. concen- trator is analysed using the perfect mixing model and an ore-specific breakage distribution func- tion. This function was determined from single-particle breakage tests using a computer-monitored twin pendulum apparatus.

The ratio of the breakage rate to the normalized discharge rate, r/d*, determined for the ball mill using the ore-specific breakage distribution function for a range of grinding conditions is related to the mill power consumption. The mill power consumption is related to the percentage of mill volume occupied by the ball charge and to the percentage of solids in the mill feed.

INTRODUCTION

The New Broken Hill Consolidated Limited (NBHC) concentrator treats 7000 tonnes of lead-zinc-silver ore per day in two parallel rod mill/ball mill circuits. A schematic diagram of the NBHC grinding circuit and the sampling points are given in Fig. 1. The performance of the No. 2 secondary ball mill, 3.2 m diameter and 3.05 m long, was studied in this investigation. The rod mill and ball mill discharges are pumped through a variable speed pump to a cluster of 0.5 m diameter cyclones. The cyclone underflow gravitates to the ball mill while the cyclone overflow is the grinding circuit product.

The circuit is operated under manual control with the water addition to the rod-mill feed controlled in proportion to the new feed and the water addition to the cyclone feed controlled from cyclone-feed mass flow rate. The speed of the cyclone feed pump is controlled from the slurry level in the cyclone-feed

0301-7516/87/$03.50 © 1987 Elsevier Science Publishers B.V.

'24 '2

Ore Feed Ons Feed W a t e r A1 A 3 A 4 ~ L A 2 Water

C5 C 6 ~ F . . . .

t Rod M i l l

I I Z44m [ ~'-44m [

® 04 No.2['~ - ~ ] I . . . . . . . . I ~ I I Rake RGC

; } o PrSmaa~ Lead R~,~I~,II. TO Primary Lead i !Wcder Roughers A6

\ ] ~ . ,1 , i ] c , ~ , 1 ; i ~ t : l , t i ; l i : L" I r,tl J

Fig. 1. A schematic diagram of the NBHC grinding circuit. Analogue outputs: C1, C2 =control of returns to grinding circuit (RGC) ; C5, C6 = control of ore feed rate.

sump. The addition of RGC ( returns to the grinding circuit; Fig. 1 ) was stopped during this investigation. This RGC stream carries small yet variable solid flows such as floor clean up in a large quantity of water. Therefore, control by water addition alone will have a negligible effect on solids flow but allows steady state operation.

The data from the NBHC grinding circuit were collected over a wide range of operating conditions. The flow rate and the size distribution data for the various streams collected during this study were mass balanced using a com- puter program available at the Julius Kruttschnitt Mineral Research Centre. The flow rates and the size distributions of the mass balanced mill feed and mill discharge data from this study were analysed using a perfect mixing model ( Whiten, 1976 ) and an ore-specific breakage distribution function.

The modern ball mill models describe a ball mill performance using the basic concepts of brekage rate, breakage distribution and material transport func- tions ( Austin et al., 1984; Herbst et al., 1982; Whiten, 1976; Narayanan, 1987 ). All these models use arbitrary or experimentally determined breakage distri- bution and material transport functions and correlate the breakage rate parameters with ball milling variables. A detailed comparison of the modern ball mill models is given in another publication ( Lynch et al., 1986). Since the

243

breakage distribution function has been found to be invariant with process conditions, it is usually determined from laboratory breakage experiments on the ore sample processed in the mill (Herbst et al., 1982; Austin et al., 1984).

Relatively fine feeds, with less than 2% plus 2 mm material, are encountered in secondary milling circuits. Therefore, a simple perfect mixing model (Whi- ten, 1976) was chosen for this analysis. The ore-specific breakage distribution function was determined from single particle breakage tests conducted on the cyclone under flow ( mill feed) sample. A computer monitored twin pendulum device was used for conducting these single particle breakage tests (Naray- anan, 1985; Narayanan and Whiten, 1985).

One of the objectives of this analysis is to determine the effect of process variables on the breakage rates of the various size fractions and on the ball mill power consumption. From this analysis, a relationship to quantify the effect of ball charge volume, Cv, on the performance of the ball mill is devel- oped. The ball charge volume, Cv, is defined as the percentage of mill volume occupied by the ball charge and this includes volume of the slurry trapped in the voids between balls.

RANGE OF OPERATING VARIABLES STUDIED

The normal operating feed rate of this grinding circuit was 130 tonnes/hour and the normal ball charge volume was 42.0%. The grinding mill performance at three ball charge volumes, 37.43, 42.03 and 45.16%, was investigated. Three feed rates, 105, 130 and 155 tonnes/hour, were studied at each ball charge vol- ume. These feed rates are the new feed rates to the grinding circuit, i.e. the rod mill feed rate. A summary of the process variables tested in this study is given in Table I.

In each survey, the rod mill feed rate, water additions, cyclone overflow den- sity and cyclone pressure were logged using a Data General computer system. They were monitored for two hours before the commencement of sampling. When the circuit reached the steady state condition, indicated by the above variables remaining constant over the two hour period, the rod mill discharge, cyclone underflow, ball mill discharge and cyclone overflow streams were sam- pled over a period of one hour. The ball charge volume, Cv, was determined by stopping the mill and measuring the distance between the level of the ball charge and the overflow trunnion. From this measurement and the inside diameter of the mill, the angle subtended by the ball charge and the percent of the mill volume occupied by the ball charge were determined using the method proposed by Arbiter and Harris (1982).

An important observation was that with the increase in the fresh feed rate to the grinding circuit, the percent solids in the cyclone underflow also increased. This may be due to the increase in the flow rates of solids which results in lower residence times for solids. Hence, some particles were not broken at higher

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TABLE 1

A summary of the grinding circuit surveys conducted at the concentzator of New Broken Hill Consolidated Limited

Test Circuit feed Cyclone Ball mill feed Ball mill Charge volume no. rate underflow rate power (', ~

( tonnes /h ) ( % solids ) ( t onnes /h ) t kWh ) ( ~; !

1 130.0 81.79 418.5 427.50 37.43 2 155.0 82.33 392.5 349.56 37.43 3 105.0 79.40 288.7 432.11 37.43 4 130.0 81.32 303.2 442.95 42.03 5 155.0 83.38 342.6 365.66 42.0:~ 6 105.0 79.53 265.0 472.39 42.28 *~ 7 130.0 81.17 349.3 475.23 45.16 8 155.0 82.29 307.7 430.26 45.16 9 105.0 78.43 271.3 495.91 45.16

* 1Percentage of the mill volume occupied by the ball charge including voids. *~Additional balls inadvertent ly added to the ball mill.

feed rates and were recirculating in the circuit which influenced the cyclone underflow density. Another observation was that the power consumption decreased with increase in the fresh feed rate as shown in Fig.2.

P E R F E C T M I X I N G MODEL

The simplest case of modelling ball mills is by considering the mill as one perfectly mixed segment ( Whiten, 1976). The grinding process in such a per-

550.00 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~ 500.00

k--

450.00

4 o 0 0 0

* C l ' l ~ l ~ VOLUI~= 4~..05% - ~ o CI..I~RGE VOLUME= 45.16%

3 0 0 0 0 I00.00 120"00 140.0 160-00

CIRCUIT NEW FEED RATE (TONNES/HOUR)

Fig. 2. The relat ionship between the ball mill power consumpt ion and the grinding circuit new feed rate at three ball charge volumes for the NBHC ball mill.

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fectly mixed mill can be described in terms of the mass balance of each size fraction. If n size fractions, usually in a geometric series, are considered, with 1 representing the largest size fraction and n representing the submesh frac- tions, the equation for the rate of change of the ith size fraction is:

d S i / d t = f i - r i S i -~- ~ bijrjS j - d i S i -~0 (1)

where [i is the mass flow rate of that size fraction in the feed ( tonnes/hour) ; ri is the breakage rate of that size fraction (1/time); si is the mass of that size fraction in the mill contents (tonnes); b~j is the mass fraction of the particles appearing from the breakage of the j th size fraction and distributed to ith size fraction; and di is the discharge rate of that size fraction (1/time).

The ith size fraction in the mill product Pi ( tonnes/h) is determined from:

gi ~- di Si ( 2 )

The discharge rate d~ is normalized using the volumetric flow rate of the mill feed V ( solids and water, m3/h ), mill length L ( m ) and mill diameter D ( m ) in an equation of the form:

di =di*(4V/D2L) (3)

The mill contents term S~ can be eliminated by combining the eqns. 1, 2 and 3. Then the ball mill performance can be described by the ratio rJd~* and bii values.

Usually, a normalized breakage distribution vector with components bi is used. This concept is based on the assumption that particles of various sizes break in a similar manner and the b~ values determined for one particle size can be used for all other particle sizes. In this study, an ore-specific breakage distribution function determined from single particle breakage tests (Naray- anan, 1985; Narayanan and Whiten, 1985 ) was used in the analysis. Therefore, the variation in the ball mill performance can be described by the systematic changes in the ri/di* function with process conditions.

DETERMINATION OF BREAKAGE PARAMETERS USING PERFECT MIXING MODEL

Usually, the r/d*-particle size relationship is represented as a spline function with three or more 'spline knots'. Each spline knot represents the r/d* value corresponding to a chosen particle size. The particle sizes encountered in the secondary mill are less than 4 mm and the percentage of material greater than 2 mm is in the order of 1-2%. Hence, 0.106 mm, 0.596 mm and 1.673 mm particles are chosen to represent the feed size range processed in these mills. Therefore, the spline knots are the three r/d* values corresponding to these three particle sizes. Between two consecutive knots, the relationship between

24t~

r/d* and the particle size is described by a smooth cubic spline function with continuous first and second derivatives. The extrapolation above the maxi- mum knot value (1.673 ram) or below the minimum knot value (0.106 ram) is linear with a zero second derivative.

Using the mass balanced data for the mill teed and the mill discharge, the r/d* values at these three knots are determined by a non-linear fitting program (Kavetsky and Whiten, pets. commun., 1982). This computer program cal- culates the particle size distribution in the mill product from the mill teed by adjusting the three r/d* values at the spline knots such that the sum of' squares of the residuals between the observed and the predicted particle size distribu- tions in the mill discharges are minimised.

ANALYSIS USING ORE-SPECIFIC BREAKAGE DISTRIBUTION FUNCTION

Single particle breakage tests were conducted on -5 .6+4.75 mm and - 2.8 + 2.36 mm particles from the cyclone underflow samples collected during this study. A computer monitored twin pendulum apparatus (Narayanan, 1985; Narayanan and Whiten, 1985) comprising an input pendulum and a rebound pendulum was used for these tests. A closely sized particle was broken at a known level of energy by the impact of the input pendulum on the particle tenuously attached to the rebound pendulum. The comminution energy, defined as the energy available for breakage of the particle, was measured by monitor- ing the motion of the rebound pendulum after impact using a computer.

A number of' particles from each size range was tested at four levels of input energy per particle size. From the results of these tests, the values of the ore- specific breakage distribution function, bi, were determined (Narayanan and Whiten, 1985). The values of' the ore-specific breakage distribution function determined for the NBHC ore are given in Table II.

The breakage parameters R1, R2 and R3 were determined from the param- eter estimation program for the nine surveys using the ore-specific bi values. The development of a correlation between the breakage parameters and the process variables is described in the following sections.

DEVELOPMENT OF A CORRELATION BETWEEN THE BREAKAGE PARAMETERS AND THE PROCESS VARIABLES

It was found that the breakage parameters R1, R2 and R3 are related to the mill power consumption, P (kW), by the following regression equations:

R1=ln(r/d*)o.lo6mm=1.9868*10-4(+/-0.0000382) * (P) (4)

R2=ln(r/d*)o.596mm=O.04401(+/-0.00355) * (P) (5)

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TABLE II

Ore-specific breakage distribution function for the NBHC ore determined from single particle breakage tests

Size Breakage distribution function interval (bi) values

1 0.000 2 0.051 3 0.097 4 0.128 5 0.128 6 0.113 7 0.095 8 0.078 9 0.064

10 O.053 11 0.044 12 0.037 13 0.029

Note: size interval is in ~ series.

R3=ln(r/d* ) 1.673 mm =0 .11090(+ / - -0 .00936) * (P ) (6)

The t-test showed that these relationships were significant at the 90% level. The mill power consumption, P, was related to the ball charge volume, Cv,

and percent solids in the mill feed, S%, by a multiple linear regression of the form:

P = 7 . 8 5 7 ( _+2.07)* Cv - 2 4 . 0 5 ( _+4.13)* S% +2043.0(_+348.3) (7)

The level of significance of this relationship, determined from the t-test, was 90%.

This relationship shows that the power consumption decreases with increase in the percent solids in the mill feed. A similar relationship has been reported by Laplante and Redstone (1983). The observed power consumption data are given in Table I. The observed power consumption and the predicted values from eq. 7 are compared in Fig. 3.

ACCURACY OF THE RELATIONSHIP BETWEEN THE BREAKAGE PARAMETERS AND THE PROCESS VARIABLES

The accuracy and the application of the two relationships outlined in the previous section need to be considered separately. The percentage of relative errors in the power draft measurements is usually lower than the errors encountered in the measurement of other ball milling variables. However, the

248

I - O~ ~ 450.00

o

400.00

t- ~_, 3500O

i e ~ ~

30000 L I L

300.00 550-00 400. O0 450.00 SO0.O0 OBSERVED POWER CONSUMPTION (KW)

J ! Q /"

j r J / / ~ . J

7

i

!

f~0.00

Fig. 3. Comparison of the observed ball mill power consumption with the predictions from the relationship between the mill power consumption, ball charge volume and percent solids in the mill feed.

major variable influencing the power consumption of a ball mill is the ball charge volume. Hence two separate relationships have been proposed from this analysis.

The first relationship predicts the variation of the breakage parameter, r/d*, values with the ball mill power consumption through the eqs. 4, 5 and 6. This relationship was found to give acceptable predictions for the performance of the NBHC ball mill. An analysis of the accuracy of the mill discharges pre- dicted from this relationship is presented in a later section.

The second relationship is between the power consumption and the grinding process variables such as ball charge volume and percent solids. This relation- ship is expressed by eq. 7. The standard deviations of the coefficients of the ball charge volume, Cv, and the percent solids, S%, are relatively high at approximately 20%. This can be due to the inherent errors in the measurement of the ball charge volume and the difficulty experienced in assessing the num- ber of various diameter balls present in an industrial ball mill which influences the nature of ball load. Even though eq. 7 can be used to predict the variation of mill power with the grinding process variables, further work to develop a general relationship by analysing similar data from a number of secondary ball mills is in progress.

ACCURACY OF THE PREDICTIONS FROM THE RELATIONSHIP BETWEEN BREAK- AGE PARAMETERS AND MILL POWER CONSUMPTION

The eqs. 4, 5 and 6 predict the spline knot values for the r/d*-particle size relationship from the mill power consumption. Such a relationship can be used

249

for simulation and control purposes. The accuracy of this relationship can be analysed by comparing the mill discharges predicted from this relationship with the observed data.

The In (r/d*) spline knot values determined from the eqs. 4, 5 and 6 and the laboratory bi values were used in the perfect mixing model to predict the mill discharge for the nine surveys. The relative error on the volumetric flow rates of the ith size fraction for j th data set, Eo, was calculated from:

Eii = [ VPij( O ) - VPij( C ) ] / V P o (O) (8)

where V P 0 ( 0 ) and V P o ( C ) are the volumetric flow rates of the ith size frac- tion in the observed and the calculated mill discharges, respectively, for t he j th data set.

For each size fraction, there will be j errors from the j data sets. In this case j is equal to nine. The mean and the standard deviation of the j errors for each size fraction were calculated. If the proposed model is accurate and does not have any bias, the mean of the errors for each size fraction will be zero. Oth- erwise, a bias between the model and the observed data can be suspected and the magnitude of the bias depends on the magnitude of the difference. How- ever, small differences between the model predictions and the experimental data may not necessarily be significant.

Since the observed data will contain small random errors, the mean of the errors for a size fraction should be close to zero. The standard deviation of the errors for a size fraction should be small, but larger than the corresponding mean value. Under these conditions, the model predictions are acceptable. However, if the mean of the errors for a size fraction is approximately zero but the standard deviation is large relative to the magnitude of the data, it may suggest that the observed data contains large random errors in that size fraction.

The mean and standard deviations of the errors E 0- for the 14 size fractions from the nine grinding circuit surveys are given in Table III. Except for the first two coarsest fractions, the mean of the errors for each size fraction is close to zero. The standard deviation of each of the errors is larger than its corre- sponding mean value. In the nine surveys conducted under a wide range of conditions, the amount of material in the two coarsest fractions was less than 2%. Thus, the mass flow rates of these two coarser fractions is relatively insig- nificant. The presence of errors in the two coarsest fractions will not signifi- cantly influence the accuracy of the model predictions. It can be concluded from this analysis that the relationship developed between the breakage parameters and the mill power consumption, using the ore-specific breakage distribution function, gives acceptable predictions and does not have any sig- nificant bias.

CONCLUSIONS

The performance of the NBHC secondary ball mill was successfully ana- lysed using the perfect mixing model and the ore-specific breakage distribution

25t~

' F A B L E III

A s u m m a r y of t he accuracy ana lys i s of t he power r/d*modetpredietionsfortheNBHCsecondar} ball mil l data

Size fract ion M e a n of t he errors S t a n d a r d d e v i a t i o n o~

( r a m ) f rom n i ne da ta the errors f rom n i n e

se ts da ta se ts

- 4 . 0 0 +2 .80 0.210 o.368 - 2 . 8 0 + 2 . 0 0 0.151 0.268 - 2 . 0 0 + 1.40 0.014 0.223 - 1 . 4 0 + 1 . 0 0 0.013 0.113

- 1.00 + 0 . 7 0 0.007 0.097 - 0 . 7 0 + 0 . 5 0 0.002 0.123

- 0 . 5 0 + 0 . 3 5 0.014 0.085

- 0 . 3 5 + 0 . 2 5 0.005 0.043 - 0 . 2 5 + 0 . 1 7 5 - 0 . 0 1 4 0.020 - 0 . 1 7 5 + 0 . 1 2 5 - 0 . 0 0 4 0.018 - 0 . 1 2 5 + 0 . 0 8 7 5 0.003 0.035

- 0.0875 + 0.0625 - 0.039 0.053 - 0.0625 + 0.0438 - 0.009 O.068 - 0.0438 + 0.0 - 0.021 0.078

function determined from pendulum tests. Two relationships were proposed from this analysis.

(1) The r/d* parameter was related to the mill power consumption. This relationship described the ball mill performance over a wide range of operating conditions with acceptable accuracy.

( 2 ) The ball mill power consumption was related to the ball charge volume and the percent solids in the mill feed.

The relationship developed between the breakage parameters and process variables for the NBHC secondary ball mill can be extended to a more general relationship by analysing the performances of a number of secondary ball mills.

A C K N O W L E D G E M E N T S

The authors would like to thank the staff of the New Broken Hill Limited for their co-operation. Thanks also go to Professor A.J. Lynch, Dr. W.J. Whi- ten, Dr. D.J. Mckee, Dr. A. Kavetsky and Dr. P.D. Bush for many valuable discussions. The research work was carried out from the Australian Mineral Industries Research Association's grant to the Julius Kruttschnitt Mineral Research Centre and their continued support is gratefully acknowledged.

251

REFERENCES

Arbiter, N. and Harris, C.C., 1982. Scale-up and dynamics of large grinding mills - a case study. In: A.L. Mular and G.V. Jergensen (Editors), Design and Installation of Comminution Cir- cuits. SME/AIME, New York, N.Y., pp. 491-508.

Austin, L.G, Klimpel, R.R. and Luckie, P.T., 1984. Process engineering of size reduction: ball milling. SME/AIME, New York, N.Y., pp. 181-230.

Herbst, J.A., Siddique, M., Rajamani, K. and Sanchez, E., 1982. Population balance approach to ball mill scale-up: bench and pilot scale investigations. Trans. SME/AIME, 272: 1945-1954.

Laplante, A.R. and Redstone, J., 1983. Modelling of grinding kinetics in Sidbec-Normines Port- Cartier pelletizing plant. Mini symposium on grinding performance of large ball mills II, Salt Lake City, Utah, SME/AIME, New York, N.Y., pp. 13-26,

Lynch, A.J., Whiten, W.J. and Narayanan, S.S., 1986. Ball mill models: their evolution and pres- ent status. In: P. Somasundaran (Editor), Advances in Mineral Processing, Arbiter Sympo- sium. SME/AIME, Littleton, Colo., pp. 48-66.

Narayanan, S.S., 1985. Development of a laboratory single particle breakage technique and its application to ball mill modelling and scale-up. Ph.D. thesis, University of Queensland, 209 pp., unpublished.

Narayanan, S.S., 1987. Modelling the performance of industrial ball mills using single particle breakage data. Int. J. Miner. Process., 20: 211-228.

Narayanan, S.S. and Whiten, W.J., 1985. Determination of comminution characteristics of ores from single particle breakage tests. Trans. Inst. Min. Metall., Section C, submitted manuscript.

Whiten, W.J., 1976. Ball mill simulation using small calculators. Proc. Australas. Inst. Min. Metall., 258: 47-53.