Modelling the performance of industrial ball mills using single particle breakage data

International Journal of Mineral Processing, 20 (1987) 211-228

211

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

Modelling the Performance of Industrial Ball Mills Using Single Particle Breakage Data S.S. NARAYANAN

Julius Kruttschnitt Mineral Research Centre, Isles Road, IndooroopiUy, Queensland 4068 (Australia) (Received March 17, 1986; accepted after revision October 20, 1986 )

ABSTRACT Narayanan, S.S., 1987. Modelling the performance of industrial ball mills using single particle breakage data. Int. J. Miner. Process., 20: 211-228. Ball milling is an energy-intensive unit operation and usually consumes a major proportion of the power drawn by a typical mineral processing plant. Hence, substantial economic benefits can be achieved by optimal design and by operating ball milling circuits under optimum process conditions. This requires an accurate ball mill modelling technique. In the multi-segment ball mill model, the size-dependent material transport within the mill varies systematically with the amount of coarse particles present in each segment. The ore-specific breakage distribution function can be determined from single particle breakage tests conducted using a computer-monitored twin pendulum apparatus. When the ore-specific breakage distribution function is used in the multi-segment, a constant relationship between the breakage rate parameters and mill diameter is observed. Thus, the performance of an industrial ball mill can be adequately described using the ore-specific breakage distribution function together with the systematic variation of the material transport and the breakage rate functions with process conditions and mill diameter, respectively. This ball mill modelling technique is illustrated using a case study on the design of a ball milling circuit for a particular grinding requirement and another case study on modelling the performance of an industrial ball milling circuit.

INTRODUCTION

Comminution is an energy-intensive process and consumes approximately 3% of the electricity generated in the industrialized countries of the world (Schonert, 1979). In a typical mineral processing plant, comminution can account for 55-70% of the total power consumption ( Cohen, 1983; Lynch and Narayanan, 1986). For example, in the Bougainville concentrator which treats approximately 40 million tonnes/year of porphyry copper ore, ball milling consumes 55.6% of the total power, and crushing consumes 20.1% of the power 0301-7516/87/$03.50

© 1987 Elsevier Science Publishers B.V.

212

(Hincktuss, 1976). Hence, substannal economic benefits can be achieved by designing and operating the grinding circuits at optimum conditions+ Since ball mills are widely used in industrial grinding circuits, a vast amount of' research work over the past; several decades has been directed towards modelling ball mill pertormance t see reviews by Rumpf, 1973; Lynch, 1977: Committee on Comminution and Energy Consumption, 1981; Austin et al_ 1984; Lynch et al., 1986). Since comminution is basically a process of creating new suriace area, there was a strong incentive in the early stages of ball mill modelling to use some form of empirical energy-size reduction relationship (Rittinger, 1867: Kick, 1885: Bond, 1961). The Bond relationship was widely used for selecting a number of industrial ball mills ( Rowland. 1982 ). Howeven serious limitations with this empirical approach in accurate prediction, simu+ lation and control of grinding mill performance have been reported (Blaskett+ 1970; Herbst and Fuerstenau, 1980; Committee on Comminution and Energy Consumption, 1981; Austin and Brame, 1983). This has resulted from the increasing knowledge about the various comminution micro-processes and the energy transfer patterns inside ball mills. The modern ball mill models (Herbst and Fuerstenau, 1980; Austin et al., 1984; Kavetsky and Whiten, 1982; Narayanan and Whiten, 1985a) describe ball mill performance in terms of three mechanisms. These mechanisms are: (1) a breakage distribution tunction to describe the product size distribution from a breakage event; (2) a breakage rate function to describe the sizedependent rate at which particles of various sizes would break in a ball mill; and ( 3 ) a material transport thnction to describe the size-dependent material transport through and out of' the mill. The breakage distribution function is usually assumed to be a constant for a particular material being ground. The material transport function depends on the amount of various particle size fractions present in the mill. The breakage rate function parameters are normally related to ball milling variables such as mill diameter, ball charge volume and percent critical speed. Recently a ball mill modelling scheme to describe the performance of industrial ball mills using an ore-specific breakage distribution function in a multisegment ball mill was developed (Narayanan, 1985; Narayanan and Whiten, 1985a). The ore-specific breakage distribution function is determined from single particle breakage tests. This modelling scheme does not require pilot plant data. The accuracy of the predictions from this ball mill modelling technique is illustrated in this paper using an industrial case study. MULTI-SEGMENT BALL MILL MODEL

In the multi-segment model, the ball mill is hypothetically divided into a number of perfectly mixed segments (Whiten, 1974; Kavetsky and Whiten, 1982; Narayanan and Whiten, 1985a ). The contents of each segment are influ-

213 • . Breakage ....

I

rl

] ri

,qJ.,[

I Product b P

""'°°

Segment

I

. 2

. . . . . .

n-I

n

THE SAME BREAKAGE RATE FACTOR, ri, APPLIES TO ALL SEGMENTS, FOR PARTICLE SIZE INTERVAL~i. EACH SEGMENTI j ~HAS A DIFFERENT DISCHARGE RATE FACTOR, di(J)l FL~REACH PARTICLE SIZE INTERVAL,i. THE VALUES OF di(j) DEPEND ON THE PROPORTION OF COARSE MATERIAL IN SEGMENT j. MIXING RATE BETWEEN SEGMENTS IS PROPORTIONALTO di(j). THE PRODUCT FROM A SEGMENTj t Pj(j) ~ IS CALCULATED FROM rn p i(j) = d i(j) Si(J) = Fi ( j ) - r i Si(J)+~-==I bjkrkSk(j) WHERE Si(j} iS THE AMOUNT OF THE iTH SIZE FRACTION MATERIAL IN SEGMENT j AND m IS THE NUMBER OF SIZE FRACTtONS.

Fig. 1. A schematicillustrationof the multi-segmentball mill model. enced by the following four mechanisms: (1) the appearance of smaller particles resulting from breakage of larger particles, which is described by the orespecific breakage distribution function, bij; (2) the disappearance of larger particles breaking into smaller particles, which is described by a breakage rateparticle size relationship; (3) the discharge of material from a segment, which is described by a discharge-rate-particle-size relationship. This relationship varies from segment to segment depending on the amount of coarse, say plus 2 mm, material present in each segment; and (4) backward and forward mixing of material between adjacent segments, described by a mixing factor, Hp, which is related to the length of a segment. For a given mill, the number of segments are usually chosen to give a segment length of 0.7 m so that a good degree of mixing is achieved within a segment. Hence, the choice of the mixing constant does not greatly influence the model behaviour and predictions (Kavetsky and Whiten, 1982 ). A schematic illustration of this model is shown in Fig. 1. The breakage distribution function describes the fundamental breakage characteristics of a material and is assumed to be invariant with process conditions. It is usually expressed in a form which is independent of particle size. The ore-specific breakage distribution function for these studies was determined from single particle breakage tests conducted using a computer-monitored twin pendulum apparatus (Narayanan, 1985; Narayanan and Whiten,

214 24 60 [

0 [] ,~

25 05

DIAMETER : 320m DIAMETER: 5.81rn DIAMETER: 472m DIAMETER 5 49m

7~)~ 1147

uJ

5'26

24I

i

ill

L 01~

0 60

1"67

473

19"~0

PARTICLE SIZE, (mm)

Fig. 2. The systematic variation of the breakage-rate-particle-size relationship with ball mill diameter,

1985b). A brief description of these pendulum tests is given in the following section. The breakage rate function, ri (time-1), describes the rate at which particles from a size i are broken. When particle sizes are expressed as narrow discrete size intervals (say, ,,/2 series), the breakage-rate-particle-size relationship can be expressed as a continuous function. For finer particle sizes, experience shows that the function usually increases with particle size. Above a certain particle size, the function may level off or even decrease depending on the process conditions. It has been shown that the breakage-rate-particle-size relationship varies systematically with mill diameter for industrial ball mills operating under similar process conditions such as ball charge volume and percent critical speed (Narayanan, 1985; Narayanan and Whiten, 1985a). These relationships are shown in Fig. 2. In the multi-segment ball mill model, the breakage rate function, ri, is considered to be the same for all the segments. The rate of breakage (tonnes/ hour) of the ith size fraction in the j t h segment, Yi(J), is proportional to the amount of ith size fraction in that segment, Si (j) (tonnes). This relationship is expressed as:

Yi(j) =ri Si(j)

(1)

The material content in a segment, Si (j), is influenced by the amount of the ith size fraction material discharged from that segment, p~ (j) (tonnes/hour).

215

~I'01

N-~~:-IUUJI

X-DISCHARGE FUNCTION (ram)

_~WIDTH

/ ........ PARTICSILEZE(ram)

X

Fig. 3. The variationofthe normalizeddischargerate factor-particlesizerelationshipwith process conditions. The material discharged from a segment depends on the size-dependent normalized discharge rate, di (time -1), of material from that segment. This relationship is described by:

Pi(J) = k ( j ) di(j) Si(j) (2) where k(j) is a dimensionless constant which is adjusted to give the correct volume of pulp in a segment. Usually, 19% of the mill volume (inside liners at normal ball charge conditions) is assumed to be occupied by the pulp. Data collected from two industrial ball mills operating over a wide range of process conditions indicate that the normalized discharge rate factor varies consistently with the material contents of each segment (Kavetsky and Whiten, 1982). The normalized discharge factor is unity for zero particle size including submesh sizes and water. It approaches zero at larger particle sizes. Hence, its variation due to process conditions is simulated by varying the particle size at which the function has a value of 0.5. This particle size is called the width of the discharge function. Typical variation of the normalized discharge-rate-particle-size relationship with the width of the discharge rate function is shown in Fig. 3. Analysis of data collected from two industrial ball mills operating over a wide range of grinding conditions (Kavetsky and Whiten, 1982; Narayanan and Whiten, 1985a) indicates that the width of the discharge rate function varies systematically with the amount of coarse material present in a segment. This coarse material is nominally plus 2 mm in size. As the amount of coarse material in a segment increases, the space between the balls in which new coarse particles can be retained decreases. Hence, these coarse particles are rapidly discharged to the adjacent segment and eventually are pushed out of the mill with little or no breakage. A similar effect results from high feed rates of coarse material. The variation of the discharge function width with the percentage of plus 2 mm material in the contents of a segment is shown in Fig. 4 (Kavetsky and Whiten, 1982; Narayanan and Whiten, 1985a). The relationship shown in Fig. 4 was obtained from the analysis of data from a 5.03 m diameter and 6.71 m long ball mill at Philex Mining Corporation

216 500

o•

.

.

.

.

.

.

.

.

.

.

.

.

]

4OO

f J 3oo

z

~

20.0

N

g g

000 000

36

72

IC"8

~44

180

DISCHARf..~ FUNCTION WIDTH (mm}

Fig. 4. The variation of the discharge function width with the amount of plus 2 mm material present in a segment.

(Kavetsky and Whiten, 1982). The feed sizing to this mill was varied from 8.2% plus 5.6 mm to 29.9% plus 5.6 mm. The mill feed rate ranged from 670 to 1730 tonnes/hour. Data from a 5.36 m diameter and 6.25 m long ball mill at the Bougainville concentrator was also used in this analysis. A detailed description of the development of this discharge rate function has been published (Kavetsky and Whiten, 1982). In summary, the breakage-rate-particle-size relationship varies systematically with ball mill diameter and the material transport can be described from the amount of coarse material present in a segment. These relationships have been developed by using the ore-specific breakage distribution function to describe accurately the variation of ball mill performance due to changes in ore breakage characteristics. Hence, for a given ball mill feed, the mill performance can be simulated using the ore-specific breakage distribution function together with the breakage rate and material transport functions determined from the systematic variations with mill diameter and process conditions respectively in the multi-segment model (Narayanan, 1985; Narayanan and Whiten, 1985a). This simulation procedure is schematically shown in Fig. 5. DETERMINATION OF ORE-SPECIFIC BREAKAGE DISTRIBUTION FUNCTION

The experimental procedure for the determination of the ore-specific breakage distribution function from single particle breakage tests is relatively sim-

217 I GIVEN BALL MILL DIAMETER]

DETERMINE BREAKAGE RATE FUNCTION FROM BREAKAGE RATE- PARTICLE SIZE - MILL DIAMETER RELATIONSHfP (FIGURE 2)

1

I DETERMINEORE- SPECIFIC BREAKAGE DISTRIBUTION FUNCTIONFROM PENDULUM TESTS

I

J MOLT,S-EGME=T I 7 BALL MILL

I

DETERMINE MATERIAL TRANSPORT CHARACTERIST)CS FROM THE AMOUNTOF PLUS P_mm MATERIAL IN EACHSEGMENT I (FIGURE 4]

7~

Fig. 5. The ball millmodellingschemeusing ore-specificbreakagedistribution function. ple. It has been shown that these results can be used to describe accurately the performance of industrial comminution systems (Rumpf, 1973; Schonert, 1979, 1981; Awachie, 1983; Narayanan, 1985; Narayanan and Whiten, 1985a; Narayanan et al., 1987). The common single particle breakage tests are: single impact, dynamic loading ( drop weight and pendulum) and slow compression. A review of the principles involved in single particle breakage tests and the application of their results to comminution modelling has concluded that the product size distributions from the tests are very similar (Awachie, 1983; Narayanan, 1986 ). A computer-monitored twin pendulum apparatus developed at the Julius Kruttschnitt Mineral Research Centre (Narayanan, 1985) was used in this investigation to conduct single particle breakage tests. The pendulum apparatus consists of an input pendulum and a rebound pendulum suspended from a frame using cotton ropes. A schematic diagram of this apparatus is given in Fig. 6. The rebound pendulum swings between a laser source and a detector which are mounted on an optical bench. The rebound pendulum and the optical bench arrangement are kept inside a box to collect the fragments from the breakage process. Transparent perspex sheets are hung from the frame used for suspending the pendulum to arrest the flying particles resulting from breakage. The principles and the theory of the twin pendulum device used in this study have been described elsewhere (Narayanan, 1985; Narayanan and Whiten, 1985b). The input pendulum can be released from a known height to swing down and break a particle which is loosely attached to the rebound pendulum. After the first impact, the input pendulum is manually stopped and the subsequent motion of the rebound pendulum is monitored using a PDP 11/03 computer system. This is achieved by recording the time taken by a multiple fin arrangement attached to the rebound pendulum to pass through the laser beam. The energy transmitted to the rebound pendulum, ER2 (joules), can be

/

3 4 5 6 7

RELEASEPULLEY COTTONROPE COLLECTION BOX LASERSOURCE FINS

[

J ~~

~

///2////////////=/////

Fig. 6. Schematic arrangement of the computer-monitored twin pendulum apparatus.

determined from the computer-logged timing signals recorded while breaking a closely sized specimen. The principles of conservation of m o m e n t u m (Narayanan and Whiten, 1985b) allow the velocity of the input pendulum after impact and hence its residual energy, ER1 (joules), to be computed. The energy balance during a collision of the input pendulum with a particle attached to the rebound pendulum can be written as: Er = ER1 + ER2 + E,

(3 )

where E1 is the input energy (joules) and Ec (joules) is the total energy loss representing the energy consumed by the specimen for breakage and other losses such as acoustic, thermal and strain energy losses. This energy loss term, Ec, is defined as the comminution energy and is a lumped representation of all losses occurring in single particle breakage. A number of closely sized particles from three size ranges, -9.5+8.0 mm, -5.6 + 4.75 mm and -2.8 + 2.36 mm, are tested using this pendulum apparatus (Narayanan, 1985). A summary of conditions for tests conducted on a lead-zinc-silver ore sample at three levels of input energy per particle size and similar tests conducted on a gold ore sample processed at the Kalgoorlie Mining Associates (KMA) concentrator is given in Table I. The average value of the comminution energy for a size range and its standard deviation are calculated from the comminution energies determined for particles in that size range. The specific comminution energy E¢~, defined as the comminution energy per unit mass ( k W h / t o n n e ) , can be computed from the value of average comminution energy. A detailed description of the comminution energy calculations is published elsewhere (Narayanan, 1985; Narayanan and Whiten, 1985b).

219 TABLE I Summary of the pendulum tests conducted for the present study Ore sample

Particle size (mm)

Input energy levels (J }

Lead-zincsilver ore

-9.5 + 8.00 -5.6 + 4.75 -2.8+2.36 -9.5 + 8.00 -5.6 + 4.75 -2.8+2.36

4.38, 4.10, 0.55, 5.04, 3.47, 1.13,

KMA gold ore

7.05, 5.75, 1.11, 7.48, 5.04, 1.88,

9.40 7.73 1.47 9.64 7.48 3.14

1 joule = 2.72391 • 10-s kWh.

It has been shown that the product size distributions resulting from the breakage of single particles at different levels of input energy can be described as a one parameter family of curves (Narayanan and Whiten, 1983; Narayanan, 1985). This parameter, t, is defined as the cumulative percent passing size Y/IO, where Y is geometric mean size of the test particle. The results from pendulum tests on three narrow size ranges of particles from a number of ore samples and a raw feed coal sample have shown that the size distribution parameter, t, is related to specific comminution energy (Narayanan, 1985; Narayanan and Whiten, 1985b; Narayanan et al., 1987). This relationship for the two ore samples tested in this study is shown in Fig. 7. The particle size effects are normalized in this relationship. This shows that, at a given level of specific comminution energy, the product size distribution resulting from breakage of various sizes of ore particles remains a constant. A linear regression analysis is carried out on each set of pendulum results to describe the relationship between the size distribution parameter t and the specific comminution energy, Ecs. This relationship is expressed as:

t=aln(Ecs) + b

(4)

where a and b are constants. The results from the regression analysis for the two ore samples tested in this investigation are given in Table II. The results from pendulum tests conducted on a number of ore samples has shown that (Narayanan, 1985; Narayanan and Whiten, 1985b) t, Ec~ and Operating Work Index, WIo (kWh/ton), are related by the following equation: t = 13.77 In (Ec~) - 2.009 WIo ÷ 49.0

(5)

Further work to extend this relationship by analysing a wider range of ball mill data is in progress. Since the pendulum tests show that particles of various sizes break in a similar manner (Fig. 7) the product size distribution from the breakage of -5.6 + 4.75 mm particles at a chosen level of input energy is used to determine

220

42 5

0

LEAD-ZINC

0

LEAD-zrNc- SILVER ORE

5 6+4 75mm~

LEAD-ZINC - SILVER ORE

4 ~5* 80rnm ]

[]

GOLD ORE ( - Z 8 + 2 36mm)

0

GOLOORE t( 5 6 ' 4 7 5 m m

0

GOLD ORE - 9 5 '~B Ornrr)

SILVER ORE

!8*256mm) j~,[O ~ ' J [ ~ •

J

• /,

~

~,(

[] /

/ "/

350

o C~

/o

275

200

jo

J

12 5

OI

0"2

0'5

I-0

30

50

I0-0

SPECIFIC COMMINUTION ENERGY (kWh/fonne)

Fig. 7. The relationship between the size distribution parameter and specific comminution energy for two ore samples.

the material-specific breakage distribution function (Narayanan, 1985; Narayanan and Whiten, 1985a). The ore-specific breakage distribution functions determined for the two ore samples discussed in this study are given in Table III. TABLE II Results from multiple linear regression analysis of the pendulum test results Ore Sample

Lead-zincsilver ore K M A gold ore

Regression coefficients a

b

10.56

27.31

_+ 1.201

_+ 1.273

12.24

18.09

+

+

1.714

Eq. res. s.e.

Significance level from ttest ( % )

2.352

99.9

2.126

99.0

1.218

Regression type: t = a In (Ecs) + b Eq. res. s.e.: residual standard error from linear regression analysis. Ec,=specific comminution energy (kWh/tonne)

221 TABLE III

Ore-specific breakage distribution function values determined for the two ore samples tested in this study Size interval 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Lead-zinc-silver ore

KMA gold ore

0.000 0.059 0.105 0.132 0.130 0.113 0.093 0.075 0.061 0.050 0.042 0.035 0.027 0.021 0.015 0.011 0.009

0.000 0.114 0.150 0.150 0.125 0.099 0.078 0.063 0.049 0.038 0.030 0.024 0.019 0.014 0.011 0.009 0.007

The size interval is in ~/2 series. PREDICTIONS OF T H E PERFORMANCE OF VARIOUS DIAMETER BALL MILLS: A HYPOTHETICAL CASE STUDY

Acceptable predictions can now be made of the performance of various diameter ball mills using single particle breakage data in the multi-segment ball mill model (Narayanan, 1985; Narayanan and Whiten, 1985a). These predictions can be illustrated by using a typical design problem in which a grinding circuit treating 10,000 tonnes per day of lead-zinc-silver ore is to be selected. The size distributions of the feed to this grinding circuit and the required product are given in Table IV. This feed has an 80% passing size of 9.1 mm, and the required circuit product is 65% passing 75tt at 38% by weight of solids. The usual approach to this problem is to determine the capacities and operating conditions of circuits containing different sizes of single stage ball mills operated in closed circuit with hydrocyclones. The classification characteristics of the ore can be determined from tests using laboratory (150 mm diameter) hydrocyclones. The cyclone model parameters can then be estimated from these cyclone feed, overflow and underflow data. The cyclone model used is a regression model based on dimensionless groups (Nageswararao, 1978). Hence, the estimated cyclone model

'2"2 "2 TABLF IX,` Results of single stage bali milling circuit simulat tons to treat a lead-zinc-silver ore

Cumulative percent passing size t mm ):

Stream

F80 { rIln~

New feed Product

Ball mill D (m)

4,0 4.0 4.5 5.0 5.5

L (m)

5.0 6.0 5.63 6.25 6.9

16.0

8.0

4.()

2.o

1.~)

1).5

98.55 100,0

72.56 100.0

41.43 160.(I

25.27 100.0

16.07 100.0

10.14 6.32 <,)9,99 97.84

New feed (tph)

70.0 83.0 110.0 155.0 245.0

0.25

~!.91 8t.09

Stream data cyc. teed

cyc. UF

i

11.125 0.062 2.41 58,61

P ( Ci )

Cyclone pressure (kPa)

64.4 64.7 64.7 64.9 64.8

27 31 41 62 113

cyc. OF

(%S)

(tph)

(%S)

(tph)

(%S)

(tph)

55.8 56.0 57.2 58.3 60.1

'375.0 405.0 480.0 602.0 850.0

62.4 63.6 66.8 71.0 77.5

305.0 322.0 370.0 447.0 605.0

38.1 38.2 38.5 38.5 38.7

70.0 83.0 110.1) 155.0 245.0

9.1 --

Cyclones: Numbers 2; Diameter 0.66 m; Vortex finder 0.192 m; Spigot 0.133 m.D = ball mill diameter ( m ) ; F80 = 80% passing size ( mm ) ; L = ball mill length ( m ) ; P = percent passing 75tt; % S = percent solids by weight; tph = solids flow rate (tonnes/hour); c y c . f e e d - cyclone feed; cyc.UF = cyclone underflow; cyc.OF = cychme overflow.

parameters are ore-dependent and can be used to simulate the performance of different diameter cyclones. If sufficient ore sample (approximately 100-500 kg depending on the nature of investigation) is not available to conduct classification tests, suitable assumptions can be made by using the classification data available for ores with similar mineral composition. However, if clays or other minerals which have a significant effect on the pulp viscosity are present, it is necessary to conduct classification tests. For this case study, the cyclone model parameters from the data available on a similar ore type are used. The performance of various diameter ball mills can now be simulated using the following mechanisms: (1) an ore-specific breakage distribution function determined from pendulum tests (the results are given in Table III); (2) a breakage-rate-particle-size relationship for a given ball mill diameter obtained from the known constant relationship between mill diameter and breakage rate parameters illustrated in Fig. 2; and (3) a material transport mechanism determined from the systematic variation of the size-dependent discharge rate with the amount of coarse material present in each segment, as shown in Fig. 4. For a certain ball mill diameter, standard mill lengths are available in the

223 literature (Rowland, 1982). An integral number of segments for a standard mill length is chosen such that the length of a segment is approximately 0.7 m. The performance of various diameter ball mills operated in closed circuit with hydrocyclones can now be simulated using the scheme illustrated in Fig. 5. Simulations were performed to determine the capacities of various diameter ball mills operated in closed circuit to produce the desired product from the given feed size distribution. The hydrocyclone design parameters such as cyclone diameter, vortex finder and spigot were not changed during these simulations. However, for optimizing the total circuit design, changes in the cyclone design parameters would usually be necessary. The results for the lead-zinc-silver ore simulations are given in Table IV. The following observations can be made from the results given in Table IV. (a) The cyclone underflow percent solids and the cyclone operating pressure fbr the 5.5 m diameter mill are high and further simulations with changes in cyclone parameters such as cyclone diameter, vortex finder and spigot need to be carried out. (b) For the 4.0 m and 4.5 m diameter mills, the cyclone underflow percent solids and the cyclone operating pressure are low; these circuits can be optimized by changing the cyclone design parameters. The number of mills of a given diameter required to achieve a capacity of 10,000 tonnes/day can be selected from these simulation results. Therefore, a mill diameter can be chosen to obtain the optimum capital, maintenance and operating costs. The performance of this chosen milling circuit can be simulated to determine the optimum operating conditions for treating different types of ores that are likely to be processed and therefore minimize projected capital and operating costs. It has been shown ( Narayanan, 1985 ) that for a certain degree of size reduction, the throughput per unit length for the various diameter mills to achieve the required size reduction varies non-linearly with mill diameter. This relationship is illustrated in Fig. 8 and shows a decrease in mill performance of ball mills greater than approximately 3.8 m diameter. The relationship between breakage rate parameters, particle size and ball mill diameter, shown in Fig. 2, was developed for mills operating under similar process conditions such as percent critical speed, normal double wave liners, seasoned ball charge and ball charge volume (Narayanan, 1985). Furthermore, the mill diameters studied were from 3.2 m to 5.5 m. Further work to extend this relationship to smaller diameter mills, and to mills operating under different process conditions is in progress. SIMULATIONOF INDUSTRIALBALL MILLING CIRCUIT:A CASE STUDY The ball milling circuit at the KMA concentrator was simulated using the techniques described in the previous section. This circuit comprises a 3.5 m

224 ¢',00

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g ~_ 200¸0

g I

J iooo J

500

.

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.

.

.

• 3 25

.

. _ L

. . . . . .

4.00

~ 4.50

. . . .

L_ 500

_l . . . . . . . . . . 5.50

BALL MILL DIAMETER (m)

Fig. 8. The variation of ball mill throughput per unit length with mill diameter for a specific grinding requirement.

diameter (inside liners) by 5.18 m (inside liners) long ball mill operated in closed circuit with a 0.508 m diameter hydrocyclone. The new feed enters the ball mill along with the cyclone underflow. The mill discharge is pumped to the cyclone and the cyclone overflow is the circuit product. A grinding circuit survey was conducted at a new feed rate of 55 tonnes per hour (B.A. O'Brien, private communication, 1985). The circuit was operated under steady state conditions for approximately 2 h and then the new feed, cyclone feed, cyclone underftow and cyclone overflow streams were sampled over a period of 2 h at 30-min intervals. The percent solids and the size distribution of these streams were determined from these samples. The ore-specific breakage distribution function determined from the pendulum tests conducted on particles from the new feed sample is given in Table III. The Operating Work Index estimated from the pendulum test results using eq. 5 was 16.04 kWh/ton. The Operating Work Index calculated from the standard Bond Work Index test (Rowland, 1982) on the new feed sample was 16.14 kWh/ton. The cyclone model parameters were estimated from the cyclone feed, overflow and underflow data. The KMA ball milling circuit can now be simulated using the ore-specific breakage distribution function and these cyclone model parameters for comparison with the observed data. The breakage-rate-particlesize relationship for this ball mill was computed from the relationship with mill diameter (Fig. 2). The material transport mechanisms of this ball mill

225 TABLE V Comparison of the observedand predicted data for KMA ball milling circuit Stream

CumulativePercent Passing size (mm): 16.0

8.0

New feed Observed cyc. OF Predicted cyc. OF

100.0 100.0

81.94 5 5 . 5 9 3 8 . 8 0 27.14 21.07 100.0 100.0 100.0 100.0 100.0

100.0

100.0

Data type

New feed (%S)

Observed 98.2 value Predicted 98.2 value

4.0

2.0

100.0

100.0

Cyc. feed

1.0

100.0

0.5

100.0

Cyc. UF

0.25

0.125

0.062

17.13 14.37 100.0 90.0

12.21 68.5

99.38 88.67

66.72

Cyc. OF

(tph)

(%S)

(tph)

(%S)

(tph)

(%S)

P80(/~) P75(%)

55.3

61.6

183.75

75.7

130.5

42.3

90.0

74.83

55.3

63.1

191.2

79.1

135.9

42.1

92.7

73.0

% S = percent solids by weight; tph = solids flow rate (tonnes per hour); P80 = 80% passing size (micrometres); P75= % passing 75 micrometres; ball mill=3.5m diameter X5.18m length; cyclone=one 0.51 m diameter hydrocyclonewith 0.171 m vortex finder and 0.076 m spigot; cyc. feed= cyclonefeed; cyc. UF = cycloneunderflow;cyc. OF = cycloneoverflow. were described using the systematic variation in discharge rates with the am ount of coarse material pr e s ent in each segment (Fig. 4). A comparison of the observed and simulated data for all the pulp streams in the K M A grinding circuit is given in Table V. T hese results show t h a t this ball mill modelling scheme using the ore-specific breakage distribution function gives acceptable predictions for an industrial ball milling circuit. A detailed analysis of the accuracy of such predictions for a n u m b e r of industrial ball milling circuits is described elsewhere ( N a r a y a n a n , 1985; N a r a y a n a n and Whiten, 1985a). Since the ball mill modelling scheme uses an ore-specific breakage distribution function, the pe r f or m a nc e of the K M A circuit can be predicted and optimized for different ore types. This is achieved by using the ore-specific breakage distribution function d e t e r m i n e d from pendul um tests on samples of the ore types. I n f o r m at i on on the classification characteristics of different ore types is also necessary and can be d e t e r m i n e d with small cyclones. T h e optim u m operating conditions can t h e n be de t e r m i ned using the above simulation technique. Predictions of the response of the grinding circuit to ore type changes are not possible if an arbitrary constant breakage distribution function (Lynch, 1977) is used for all ores. T hi s requires t h a t a selection function be calculated

226

from known feed and product data tbr each ore type. '['he technique using con stant breakage distribution function cannot lhereibre be predictive for change:, in ore types. FUTURE WORK ON SINGLE PARTICLE BREAKAGE STUDIES

The ball mill modelling technique described in this paper has been developed for mills operating under 'similar' process conditions (Narayanan, 1985). Data from a number of mills treating a wide range of ore types need to be analysed to incorporate the effects of process variables such as ball size, ball charge volume, operating speed and linear configurations into this modelling scheme. Since data from ball mills of' diameters ranging from 3.2 m to 5.5 m were used for developing this technique, performance of' mills smaller than 3.2 m should be studied to extend the modelling scheme to smaller diameter ball mills. The ball mill modelling technique developed using all this information can lead to the development of a robust procedure which is capable of describing ball mill performance under all operating conditions. The single particle breakage tests have shown I Fig. 7 ) that particles of various sizes give a constant product distribution at a certain specific comminution energy. However, it is a well known fact that energy consumption increases as the particle size decreases. Hence, further research is necessary to understand the mechanisms responsible for the inefficient energy utilization as the particle size decreases. This could lead to the development of' energy-efficient fine grinding processes. Further studies are also necessary to relate the energy utilization measured in single particle breakage tests to the energy consumption patterns in ball mills. This could lead to the identification of the mechanisms responsible for efficient breakage in ball mills which could enable the optimization of energy usage with substantial economic benefits. Application of single particle breakage results to modelling autogenous grinding processes is in progress. A model of crusher performance using single particle breakage data has been developed (Awachie, 1983). Further work to incorporate ore-specific breakage characteristics and the effects of operating variables such as closed side setting, throughput and coarseness of' feed on crusher performance are in progress ( Narayanan, 1986; Narayanan et al., 1987 ). CONCLUSIONS

Ball milling operations usually account for a significant proportion of the capital and operating costs in a typical mineral processing operation. Hence it is very important to optimise the ball milling operations in the design and operating stages to achieve economic benefits. An accurate procedure for modelling industrial ball mill performance at various operating conditions is

227 r e q u i r e d to achieve this objective. T h e ball mill modelling s c h e m e using a n orespecific b r e a k a g e d i s t r i b u t i o n f u n c t i o n in a m u l t i - s e g m e n t ball mill model has p r o v i d e d an a d e q u a t e r e p r e s e n t a t i o n of i n d u s t r i a l ball mill p e r f o r m a n c e . Furt h e r work to e x t e n d this t e c h n i q u e to ball mills less t h a n 3.2 m in d i a m e t e r a n d to mills o p e r a t i n g at d i f f e r e n t process c o n d i t i o n s is in progress. ACKNOWLEDGEMENTS T h e a u t h o r would like to t h a n k Mr. B.A. O ' B r i e n a n d t h e s t a f f of the K M A c o n c e n t r a t o r for p r o v i d i n g the d a t a for a case study. T h a n k s also go to Dr. W.J. W h i t e n a n d Dr. A. K a v e t s k y for p r o v i d i n g the m u l t i - s e g m e n t model programs. M a n y helpful discussions with P r o f e s s o r A.J. L y n c h , Dr. D.J. M c K e e , a n d Dr. P.D. B u s h are t h a n k f u l l y acknowledged. T h e work was m a d e possible b y t h e finance f r o m t h e A u s t r a l i a n M i n e r a l I n d u s t r i e s R e s e a r c h Association a n d t h e c o n t i n u e d s u p p o r t f r o m the various sponsors is gratefully acknowledged.

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