8
P,,, .... n Minerais Engineering, Vol. 14, No, 10, pp. 1321-1328,2001 © 2001 Published by Elsevier Science Ltd Ali rights reserved 0892-6875(01)00146-7 0892-6875/01/$ - see front matter INDUS TRIAL TUMBLING MILL POWER PREDICTION USING THE DIS CRETE ELEMENT METHOD M.K. ABD EL-RAHMAN', B.K. MISHRA§, andR.K. RAJAMANI§ 1 Central Metallurgical R & D Institute (CMRDI), PO Box 87, Helwan, Cairo, Egypt § Department of Metallurgical Engineering, University of Utah, Salt Lake City, UT 84112, U,S,A. E-mail: rajamani@mines,utah.edu (Received 21 December 2000; accepted 5 June 2001) ABSTRACT The discrete element method models the motion of charge in tumbling mills using the physics of individual collision between balls, rock partie/es, and mil! shell. Thus it differs from the approaches of the past wherein the power draft of the mill was based on the torque-arm formula, The discrete element method takes into account the number and dimensions of lifters and the number, density, and size of balls and rock partie/es in the charge, 1t is shown that this method successfully predicts power draft in a variety of semi-autogenous plant operations, Renee the simulation is an effective tool for designing lifters or evaluating the make-up bail size, © 2001 Published by Elsevier Science Ltd. Ali rights reserved. Keywords Modelling; SAG milling INTRODUCTION In the mineraI concentrators around the world, the operation of semi-autogenous grinding (SAG) mills is of prime importance, because the grinding circuit provides the necessary feed material to the rest of the concentration and extraction process, Therefore, the capacity of the grinding circuit is constantly monitored, and various ways and means to increase the capacity are sought. The power draft of the tumbling mill is closely related to the capacity, and hence, while maintaining the power draft at the peak value, the mill operator expects highest production out of the circuit. In particular, the first week after the mill shell has been relined with new shell plates and new lifters, the power draw is near the maximum, and as the sharp edges of the lifters wear out the power draw reaches a peak, at which time the capacity is also at its maximum, As the lifter wears out to less than 30 percent of its original dimension, the capacity as well as mill power draft drops off considerably, Therefore, it is the intention on the part of mill designers to keep the peak power draw period as long as possible during the 6- to 12-month life of a lifter. Hence in the last severa! decades the mineraI processing literature includes several attempts at predicting mill power from design and operating variables. Up until 1980, the mill power draft was modeled from the point of view of a concept called "torque arm formula." As per this concept the mill power is viewed as the torque necessary to keep the entire mass of the mill charge at a slightly higher elevation from the bottom of the mill shell due to the rotation of the mill. Beginning with the Bond power equation, several power relationships have been formulated, and they are quite successful in predicting power even today. With the advances in high-speed computing, coupled with 1321

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Page 1: Industrial tumbling mill power prediction using the discrete element method

~ P,,, .... n Minerais Engineering, Vol. 14, No, 10, pp. 1321-1328,2001

© 2001 Published by Elsevier Science Ltd Ali rights reserved

0892-6875(01)00146-7 0892-6875/01/$ - see front matter

INDUS TRIAL TUMBLING MILL POWER PREDICTION USING THE DIS CRETE ELEMENT METHOD

M.K. ABD EL-RAHMAN', B.K. MISHRA§, andR.K. RAJAMANI§

1 Central Metallurgical R & D Institute (CMRDI), PO Box 87, Helwan, Cairo, Egypt § Department of Metallurgical Engineering, University of Utah,

Salt Lake City, UT 84112, U,S,A. E-mail: rajamani@mines,utah.edu (Received 21 December 2000; accepted 5 June 2001)

ABSTRACT

The discrete element method models the motion of charge in tumbling mills using the physics of individual collision between balls, rock partie/es, and mil! shell. Thus it differs from the approaches of the past wherein the power draft of the mill was based on the torque-arm formula, The discrete element method takes into account the number and dimensions of lifters and the number, density, and size of balls and rock partie/es in the charge, 1t is shown that this method successfully predicts power draft in a variety of semi-autogenous plant operations, Renee the simulation is an effective tool for designing lifters or evaluating the make-up bail size, © 2001 Published by Elsevier Science Ltd. Ali rights reserved.

Keywords Modelling; SAG milling

INTRODUCTION

In the mineraI concentrators around the world, the operation of semi-autogenous grinding (SAG) mills is of prime importance, because the grinding circuit provides the necessary feed material to the rest of the concentration and extraction process, Therefore, the capacity of the grinding circuit is constantly monitored, and various ways and means to increase the capacity are sought. The power draft of the tumbling mill is closely related to the capacity, and hence, while maintaining the power draft at the peak value, the mill operator expects highest production out of the circuit. In particular, the first week after the mill shell has been relined with new shell plates and new lifters, the power draw is near the maximum, and as the sharp edges of the lifters wear out the power draw reaches a peak, at which time the capacity is also at its maximum, As the lifter wears out to less than 30 percent of its original dimension, the capacity as well as mill power draft drops off considerably, Therefore, it is the intention on the part of mill designers to keep the peak power draw period as long as possible during the 6- to 12-month life of a lifter. Hence in the last severa! decades the mineraI processing literature includes several attempts at predicting mill power from design and operating variables.

Up until 1980, the mill power draft was modeled from the point of view of a concept called "torque arm formula." As per this concept the mill power is viewed as the torque necessary to keep the entire mass of the mill charge at a slightly higher elevation from the bottom of the mill shell due to the rotation of the mill. Beginning with the Bond power equation, several power relationships have been formulated, and they are quite successful in predicting power even today. With the advances in high-speed computing, coupled with

1321

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1322 M. K. Abd El-Rahman et al.

the availability of better numerical tools, researchers began ta model the mill charge in greater physical detail to arrive at power draft. The earliest work is that of McIvor (1983), who considered a single ball as a separate element ta deduce the trajectory of the bail as it leaves the lifter at the shoulder of the charge. Essentially, the baIl is held pressed to the mill shell as the centrifugai force component combined with frictional force exceeds the gravitational component. As the gravitation al component exceeds the other two components, the baIl leaves the mill shell and follows a parabolic trajectory until it impacts the lower mill shell. Powell and Nurick (1996 a, b, c) refined this approach by explicitly defining the geometry of the lifter. The baIl starts to move on the lifter at the point defined by McIvor (1993), but it begins to roll or slide on the slanting face of the lifter, and then slides off the lifter, following a parabolic trajectory into the toe of the charge. Thus the static and rolling friction of the baIl against the lifter face cornes into play in determining the baIl trajectory. Next came the volume element scheme of Radziszewski (1989) and Radziszewski and Tarasiewicz (1990). In this scheme the baIl charge is viewed as a number of larger volume elements, thus implying that each volume element is comprised of a larger mass than individual balls. These volume elements are chosen by constant radial division of the mill circle intersecting with constant angular radial. vectors from the mill center ta the periphery. Ta extend the single ball motion described earlier, this scheme assumed that the principle of conservation of momentum applies. Essentially, the mean trajectory of the center of mass of the volume elements does not change as a result the collision.

One can conceive the charge motion in a baIl mill as a combination of the motion of individual layers as opposed ta the previously discussed single baIl or torque-arm approach. Here, the motion of each layer is due to the motion of the next layer towards the mill shell. The motion of the outermost layer of balls is due ta the motion of the mill shell itself. Morrell (1992) developed a model by considering the ball charge motion as successive layers of balls moving up from the toe to the shoulder of the mill. The momentum of the charge was integrated, in the radial direction, from the inner radius, starting at the kidney of the charge and ending at the radius of the mill shell. In the circumferential-angular direction, the momentum was integrated from the toe to the shoulder of the charge. Viewing the charge as successive layers moving up and falling down is a scheme that is readily amen able to mathematical analysis. Recently Hu et al. (2000) analyzed such a mathematical description via nonlinear optimization. Although not directly predicting power, they were able to determine the speed at which power would be maximum.

The discrete element method (Mishra and Rajamani, 1992; Cleary, 1998) does not prescribe a specifie pattern to the baIl charge motion. Instead the balls can cascade, interpenetrate layers while cascading, and also cataract. The balls can bounce off the mill shell and lifters, and moreover, balls of different sizes can coll ide with each other at oblique angles. Therefore the calculations require many hours of computing. Here the collision of a baIl with the mill wall or another baIl is modeled. The energy consumed in each of the thousands of collisions is summed to arrive at power draft. For these reasons this is a powerful technique for computing power draft. Since the internai geometry of the mill shell is explicitly taken into account in the calculation of ball charge motion, the method allows prediction of power draft for variations in lifter designs as weIl as variations in ball size distribution.

None of the other methods are capable of calculating power for variations in these two variables. The only limitation in the discrete element method is that we limit ourselves ta two-dimensional simulation, Le., a thin slice of the mill as wide as the largest baIl is simulated to keep the computing time to a reasonably tolerable level. Finally, since the method allows each individual baIl to take a trajectory depending on its collision with balls and shell elements in the neighborhood, it is a highly reliable method for calculating mill power draft. However, there are four modeling parameters involved in the calculations, which will be outlined in the next section.

The objective of this study is to evaluate the predictive capability in terms of mill power draft, using the discrete element method in its simplest form where the contact model is assumed to be linear. In particular, the predictive capability is tested with industrial semi-autogenous mil! data.

THE DIS CRETE ELEMENT METHOD

The discrete element method (DEM) models the ball-to-ball collision with a linear spring and dashpot. The spring provides the repulsive force, and the dashpot dissipates a portion of the relative kinetic energy.

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Industrial tumbling mill power prediction using the discrete element method 1323

During collision the balls are allowed a virtual overlap L1x, and the normal Vn and tangential Vt relative velocities determine the collision forces. The normal force is given by:

(1)

where Kn is the normal spring constant and Cn is the normal damping coefficient. Another pair of spring and dashpot with spring constant Kt and damping coefficient Ct at right angles to the normal direction serves to model oblique collisions. In this direction the tangential force is given by:

(2)

The balls rebounding after collision follow a free-flight trajectory as per Newton's law until the next collision. The spring stores the energy from the relative tangential motion, while the dashpot dissipates the energy from the same motion. The total tangential force, given by the sum of spring and dashpot forces. Its maximum limit is the frictional force at which the surface contact shears and the balls begin to slide over the contact surface.

The complexity of computations and inordinate amount of computing time are avoided by confining the scheme to simulation in two dimensions. In other words, a thin section of mill as wide as the largest bail diameter is simulated. In two dimensions, a bail is represented as a disc with a mass equal to that of the sphere, and the circular mill shell including the lifters is represented by a series of straight lines (walls). Here, the spring-and-dashpot model equally applies for collision of balls against mill shell and lifters.

The two quantities of interest within the sc ope of the current work are the mill power draft and the impact energy distribution. The power supplied to the mill is expended to maintain the bail motion. While doing this work, energy is also spent in friction and collisions. At each collision, part of the energy is spent, which is modeled by the dashpot. Thus, the addition of the product of normal and shear force on the dashpot and respective overlap (Llxn in the normal direction and Llxt in the tangential direction) gives the energy spent at that contact. The energy spent is given by:

(3)

As shown in Equation (3), the energy-spent term is summed up over aIl the collisions (k) and for aIl the time steps (t). A record of the energy associated with each of these collisions, which at the end constitute the total energy loss, leads to the impact energy distribution and power draft.

POWER DRA W PREDICTIONS

The power draw of the tumbling mill can be predicted by properly accounting for the energy dissipation within the mi Il. According to the mode!, during an impact, the available energy is only dissipated through the viscous dashpot and friction. Therefore, if aIl the impacts during one revolution are monitored, then the total energy dissipated over aIl such impacts should account for the net energy loss during one revolution. This idea is implemented through the data structure of the computer program, where for each collision the corresponding energy for a given time step is stored. After the completion of ail calculations during any time step, aIl the collision energies are summed to calculate the total energy loss over a period of time.

Bail mill simulation

The computer pro gram was originally developed for the simulation of baIl rnills to understand the motion of the charge under various operating and design conditions. It has been extensively used for predicting power draft of bail mills over a wide range of diameters. For the purpose of illustration, in Figure 1 we show the predictive capability of the model (Datta et al., 1999) for a wide range of mills from 0.25 m to 4.8 m in diameter. The best-fit straight lines in log-scale have a si ope of 2.5 and 2.3 for laboratory- and industrial-scale mills respectively, which are close to the Bond exponent of 2.5. Such predictions verify that

Page 4: Industrial tumbling mill power prediction using the discrete element method

1324 M. K. Abd El-Rahman et al.

the total energy loss summed over ail the individual collisions is an accurate indication of power integrated over a specified time.

1.0E+07 r----------------------,

1.0E+06

1.0E+05

~ ~ 1.0E+04

o Il.

1.0E+03

1 JE+02

1.0E+01

0.1

i • Meas~red--­iDDEM Prediction

Mill Diameter lm]

10

Fig.1 Comparisons of power draw for bail mills of different diameters.

SAG mill simulation

SAG mills as opposed to bail mills pose different problems, particularly with respect to the grinding media that is composed of both spherical steel balls and non spherical rocks. Conceptually, there is no problem to representing rocks as irregularly shaped objects within the framework of DEM. However, difficulties arise since an excessive amount of computational time is required for collision detection between two irregular bodies. To simplify the problem one can consider rocks as elliptical objects (Rajamani and Mishra,1996), but even then the computational time and other associated numerical problems restrict the SAG mill simulation to spherical objects only. Thus the computer code developed for SAG miII simulation only needs the material properties to differentiate between rocks and steel balls.

The design and operating data for a variety of mills were taken from the Proceedings of the International Autogenous and Semi-autogenous Grinding Technology (SAG 96) (Mular, et al., 1996), held at Vancouver, Canada in 1996. This is a rich source of practical information collected from many concentrators around the world. While the data on mill diameter, length, critical speed, filling level, and power draft are readily available from the published articles in SAG 96, invariably the articles fail to mention the size and the number of lifters. The general trend in the mining industry is to associate power with miII speed and neglect the role played by the lifters in transmitting the peripheral motion of the mill into motion of the mill charge. In such case the number of lifters is taken as twice the diameter of the mill measured in feet. Almost all of the mill manufacturers follow this rule-of-thumb. Next, most lifters in use today are of top-hat shape; that is, in geometrical terms, the design is a trapezoid in cross-section. The height of the trapezoid is anywhere from 0.15 m to 0.225 m, depending on the diameter of the mill. While most articles in the SAG 96 mention the mill filling, they do not go into the details of the bail size distribution and rock size distribution in the mill charge. In such a case the baIl size distribution is assumed to be a Gaudin-Schuhmann distribution with a slope of 3.8, which corresponds to the equilibrium baIl charge distribution found in industrial mills. In other words, a mill that is continuously fed with balls to make up for the lost volume of grinding balls attains an equilibrium size distribution. Regarding the rock size distribution, the DEM simulation requires only information regarding larger pieces of rock. It would be impossible to take into account the smaller sizes, because it would exponentially increase the number of particles in the simulation, consequently increasing the computing time. Therefore, a typical rock size distribution is assumed as a Gaudin-

Page 5: Industrial tumbling mill power prediction using the discrete element method

lndustrial tumbling mil! power prediction using the discrete element method 1325

Schuhmann distribution with a slope of unity. The simulations require the internai shell diameter, and hence a shell plate thickness of 0.075 m is subtracted from the stated mill diameter.

The DEM model is parameter-dependent. Table 1 gives the DEM simulation parameters, which were arrived at from experience in predicting carefully measured data in a laboratory environment (Datta et al., 1999). In fact, the same parameter set was used to predict the power draw values of ball mills that are presented in Figure 1.

TABLE 1 Parameters used in discrete element numerical method

Parameter Value

Normal Stiffness 400,000

Shear Stiffness 300,000

Coefficient of Friction 0.7

Coefficient of Restitution 0.5

The principal aim here is to show that power predictions by DEM agree with plant observations in a practical sense, and hence no attempt is made here to tune the DEM parameter shown in Table 1 to fit a particular data set. However, more realistic parameters can be obtained and implemented in a non-linear DEM model as suggested by Mishra and Murty (2000), for better estimates of power. Again, considering the numerical problems and the computational speed, a linear model with fixed parameters was used. It must also be reckoned that the data available in SAG 96 (Mular, et al., 1996) is only an average over a period of several months. Table 2 shows the list of available and unavailable data in each mill operation. Nevertheless, the intent is to show that the discrete element methodology produces a prediction in the neighborhood of the observed power, and as a result with more accurate data sets the model can produce much closer predictions.

Fifteen different mine sites employing mills from as small as 3.26 m diameter to as large as 10.8 m diameter are listed in Table 3. In the cases of Chuquicamata, Kanowna Belle, Mount Isa, Leeurdon, and Fimiston, where sorne measured power draft was reported, the predictions are as close as one can expect, given the lack of measured design and operating data (see Table 2 on corresponding rows). In ail other SAG mill operations the predicted power is between 40 and 80% of installed power, which shows the cataracting and cascading action of the charge. A typical snapshot of the charge motion of the Chuquicamata SAG mill is shown in Figure 2.

Figure 3 shows a comparison of the calculated power draw of the mill for a set of operating conditions with the installed power draw of the mil!. It is known that if the mill motor is correctly sized the net mill power draft is 85% of the installed power; the remainder is spent on overcoming transmission losses. In most concentrators, however, the mill motor is oversized to accommodate unforeseen events. In light of these facts the power of predictions can be claimed as far surpassing practical expectations.

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1326 M. K. Abd El-Rahman et al.

TABLE 2 Mill data presented in SAG 96 Proceedings Cv = data presented, X = data not presented)

Mine site

Grinding Mill

Chuquicamata

Kanowna Bellt!

\'loCo,

.linshandan

Ellimon

Palabora

Mount Isa

Moun! lsa

Cyprus Baghdad

I.t:t.!udoom

Ht!ndc!rson

Forrestania Nick~1

Vaal Reets

FimislOn

Amandelbult

Page No. in SAG 96

49

81

97

108

J33

ID

145

145

164

177

200

217

207

233

289

Lifter Dimension

X

X

X

X

X

X

X

X

V

V

X

V X

X

X

Maximum Bali size

X

V X

X

X

X

X

V V X

V X

X

X

V

Maximum rock Size

x v X

X

V V

V

V V X

X

X

V V X

Mill Filling

X

X

V X

X

X

V V V X

V X

X

V ,;

Mill Speed

Measured Power Draft

v v X

X

X

X

V V V

V X

V X

V X

Installed Power

TABLE 3 A comparison power prediction with industrial semi-autogenous grinding mill data

~1ine site

Cirinding Mill

Chuquicamata

Kano ... na Belle

\lcCo)"

Jlnshandan

Ellimon

Palabora

\Ioun! Isa

\lnunl Isa

Cyprus Baghdad

1.f..'l.:udourn

Henderson

Forrestania Nickel

Vaal Reefs

Fimiston

Fimiston

Amandelbult

Dimension (m)

Diamctcr

9.60

7.35

6.40

5.34

9.76

9.75

9.75

9.76

5.00

8.53

3.62

4.85

10.80

10.80

4.27

Lellgth

4.57

2.85

3.36

1.80

1.78

4.73

4.85

4.85

3.96

Il.00

4.26

5.62

9.15

5.65

5.65

4.27

1- high lifter. 2- low lifter.

Number of Lifter Dimension (m) Mill Speed Percent Filling

Lifter

64

48

42

36

36

64

64

64

64

34

56

24

48

72

72

28

Hcight * Width * angleO

0.1' 0.15 * 7.5

0.15 * 0.15 * 7.5

0.15 * 0.15 ·7.5

0.15 • 0.15 * 7.5

0.15 * 0.15 * 7.5

0.15 * 0.15 * 0.0

0.15 * 0.15 * 7.5

0.15 * 0.15 * 7.5

0.25 * 0.25 * 0.0

0.11 * 0.11 * 0.0

0.15 * 0.15. 7.5 1

0.15' 0.15 *7.5 2

0.13 * 0.13 * 0.0

0.15 * 0.075 * 0.0

0.15 * 0.15 * 7.5

0.15 * 0.15 * 7.5

0.15 • 0.10 * 0.0

rpm

10.2

10.9

13.8

13.4

13.6

10.0

10.7

10.7

10.3

16.7

10.9

15.6

17.2

9.3

9.9

15.6

Bali % : Rock %

12: 13

Il: 19

12: 16

10: 15

12: 13

27: 0.0

27: 0.0

5: 22

7: 20

8: 17

10 15

00: 35

13 : 12

13: 8.6

13: 12.2

12: 13

Power (kW)

Expected Calculated

5890 5406

2134 1652

2150 1714

1000 514

596 516

7000 3398

5200 3810

5700 4798

407J 4166

2660-4000 2848

5222 3547

550

3000

9255

10,374

1250

430

2593

8766

9432

776

The success of the discrete element method lies, however, in setting and tuning design, operation, and model parameters as accurately as possible to achieve very accurate power predictions. Then the model is used to answer questions, such as what would be the power draft when (a) the lifters wear down to 50% of their original height, or (b) along with the lifter wear, mill speed were increased by 2% in critical speed, or (c) 0.125 m balls constituting 40% of the bail charge were used instead of 0.1 m balls. While DEM is much more capable than other models in predicting power for ail these practical situations, certain points are still lacking in DEM. It is computationally intensive, and hence rock sizes below 0.025-m diameter have to be neglected to keep the number of particles in a simulation to a reasonable value. The spring-and-dashpot model embedded in DEM models aIl energy-consuming events in the mill. The spring and dashpot parameters are extrapolated from steel-to-steel collisions_ There are no experimental schemes to determine these parameters for identical milling conditions. Despite these shortcomings DEM stands as the only method that incorporates ail of the physics of the tumbling-mill problem.

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Industrial tumbling mill power prediction using the discrete element method 1327

Fig.2 A snapshot of the Chuquicamata SAG mill operating at 72% critical speed with 25% mill filling.

-~ -c. ~ Q)

~ 0 C.

"C Q) .... 0 "0

~ D-

12000

10000

8000

6000

4000

2000

O

0

p = O.85P

2000 4000 6000 8000 10000 12000

Installed power, P, (kw)

Fig.3 Comparison between predicted power versus the installed power.

CONCLUSIONS

In mineraI processing plant operation the power draft of tumbling mills is a key variable; hence, the need for predicting power draft frequently arises. Whenever lifter dimensions are altered or ball load is increased, there is a need to know if the existing motor will be capable of delivering the needed power draft. In such situations a mathematical model is quite useful for predicting power draft. The empirical power draft models primarily address the influence of mill speed and total mill filling on power draft. A descriptive model is required to address the influence of lifter dimensions, ball size distribution, or number

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1328 M. K. Abd El-Rahman et al.

of lifters. The discrete element method provides the detailed description posed here. It is shown that this method pro vides more than adequate prediction of power in a wide variety of plant operations where the available data are only approximate. The predictions fall in the vicinity of actual plant data. These predictions can be readily refined by getting accurate design and operating plant data. Furthermore, the simulation parameters could be tuned to correctly predict the existing, operation and subsequently the model becomes a practical tool for investigating other proposed design variations. The simulation tool discussed here is a two-dimensional approximation of the industrial mill. Obviously, there is much to be gained by doing the discrete element simulation in three dimensions to include the effect of end lifters, shape of the rock, axial transport, segregation of the grinding media, etc. Such a scheme is not out of the question since desktop-computing power has been leaping ahead each year.

REFERENCES

Cleary, P.W., Predicting charge motion power draw, segregation and wear in ball mills using discrete element methods. MineraIs Engineering, 1998,11(11), 1061-1080.

Datta, A., Mishra, B. K., and Rajamani, R. K., Analysis of power draw in ball mills by discrete element method. Canadian Metallurgical Quarterly, 1999,38(2), 133-140.

Hu, G., Otaki, H., and Watanuki, K., Motion analysis of a tumbling ball mill based on non-linear optimization. MineraIs Engineering, 2000, 13(8-9),933-947.

McIvor, R.E., Effects of speed and liner configuration on ball mill performance. Mining Engineering, 1983, 35(7),617-622.

Mishra, B.K., and Murty, e. V. R., On the deterrnination of contact parameters for the realistic DEM simulations ofball mills. Powder Technology, 2001,115,290-297.

Misbra, B.K., and Rajamani, R.K., The discrete element method for the simulation of ball mills. Applied Mathematical Modeling, 1992,16,598-604.

Morrell, S., Prediction of grinding-mill power. Transactions of the Institution of Mining and Metallurgy (Section C: Mineral Processing and Extractive Metallurgy), 1992,101, C25-C32.

Mular, A., Barratt, D., and Knight, D., eds., Proceedings of an International conference on Autogenous and Semiautogenous Grinding Technology held in Vancouver, B.e., Canada, Vol. 1, October 6-9, 1996, pp. 1-390.

Powell, M.S., and Nurick, G.N., A study of charge motion in rotary mills-Part 1 extension of the theory. MineraIs Engineering, 1996a, 9(2) 259-268.

Powell, M.S., and Nurick, G.N., A study of charge motion in rotary mills-Part 2 experimental work. MineraIs Engineering, 1996b, 9(3), 259-268.

Powell, M.S., and Nurick, G.N., A study of charge motion in rotary mills-Part 3 analysis of results. MineraIs Engineering, 1996c, 9(2) 259-268.

Rajamani, R.K., and Mishra, B. K., Dynamics of ball and rock charge in SAG mills, ln Proceedings of an International conference on Autogenous and Semiautogenous Grinding Technology, Eds: Mular, A., Barratt, D., and Knight, D., held in Vancouver, B.C., Canada, Vol. 1, October 6-9, 1996, pp. 700-712

Tarasiewicz S. and Radziszewski P., Comminution energetic. Part l, II and III, Material Chemistry and Physics, 1990,25,1-25.

Tarasiewicz, S., and Radziszewski, P., Bali mill simulation: Part 1 - A kinetic model of baIl mill charge motion, Transactions of the Society for Computer Simulation, 1989,6(2),61-73.

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