a discrete element method investigation of the charge motion.pdf

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.Int. J. Miner. Process. 61 2001 7792

www.elsevier.nlrlocaterijminpro

A discrete element method investigation of thecharge motion and power draw of an experimental

two-dimensional mill

M.A. van Nieropa,)

, G. Glover b

, A.L. Hindeb

, M.H. Moysa

aSchool of Process and Materials Engineering, Uniersity of the Witwatersrand, P.O. WITS,

Johannesburg 2050, South Africab

Mintek, Priate Bag X3015, Randburg, 2125, South Africa

Received 11 February 2000; accepted 26 June 2000

Abstract

.The Discrete Element Method DEM has the potential to be a powerful tool for the design andoptimisation of mills. However, for DEM to gain acceptance within the minerals processing

industry, it is necessary to show that the results obtained from a DEM simulation are valid, and

that this validity extends over a wide range of mill operating conditions. Real grinding mills are

complex multi-phase devices with a range of particle dynamics and material processes that depend

on the exact operating point of the mill. Mill conditions will generally vary statistically over time.

It is therefore difficult in this type of environment to systematically verify DEM, where some

degree of precision in the mill operation is required. With these considerations in mind a

programme of both experimental and DEM simulation work was developed. A Atwo-dimensionalB

laboratory mill was built in such a way that precise power measurement and monitoring of charge

motion was possible. DEM simulation runs were matched to the experimental conditions. In thisaccount of the work, particular attention is given to the effect of mill speed on power and charge

motion, and also of particle behaviour at mill speeds above the critical. DEM predicts the power

draft and charge motion of the mill well at speeds below the critical speed. At super-critical

speeds, the centrifuging of material in the load was predicted, but power predictions were not as

accurate. q2001 Elsevier Science B.V. All rights reserved.

Keywords:Discrete Element Method; Mill power; Charge motion

)

Corresponding author.

0301-7516r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. .P I I : S 0 3 0 1 - 7 5 1 6 0 0 0 0 0 2 8 - 4


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( )M.A. an Nierop et al.rInt. J. Miner. Process. 61 2001 77 9278

1. Introduction

Grinding mills are an important unit operation in the field of minerals processing. .The Discrete Element Method DEM , which can model the motions and interactions of

.a set of individual particles and moving walls Cundall and Strack, 1979 , is a potentialtechnique for improving knowledge of factors that strongly influence charge motion andmill power. The charge motion inside a mill affects the energy and forces experienced

by particles, and hence the efficiency of fragmentation or abrasion to achieve a required

particle size, while mill power determines plant operation costs. However, for DEM to

gain acceptance within the minerals processing industry as a viable design tool, it is

necessary to show that the results obtained from DEM simulations are valid, and that

this validity extends over a wide range of operating conditions. An additional require-

ment from the DEM modellers perspective, is to attempt to isolate the material

properties, such as coefficient of friction, coefficient of restitution, stiffness and shape ofthe particles, which potentially could have a significant influence on the results of a

DEM simulation. .Mishra and Rajamani 1992, 1996 pioneered the application of DEM to grinding

mills and demonstrated that the technique is able to predict the power draw of

production mills with reasonable accuracy over a wide range of mill diameters. This is

despite the fact that most DEM simulations, for reasons of computational efficiency, are . two-dimensional 2D that is, a slice perpendicular to the mill axis, one particle

.diameter in thickness is modelled , use perfectly spherical particles and do not account

for the possible fluid dynamic effects of slurry. Furthermore, mill power is just one of arange of parameters that can be obtained from a DEM simulation. As noted above,

impact energies, contact forces and abrasion are all important for measuring the

Fig. 1. A diagrammatic representation of the springsliderdashpot contact interface, which was used formodelling the interactions between elements in a DEM simulation. See text for an explanation of the

.notation.


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( )M.A. an Nierop et al.rInt. J. Miner. Process. 61 2001 77 92 79

performance of a mill and need to be investigated to ensure that DEM has proper

predictive capabilities.

Real grinding mills are complex multi-phase devices with a range of particle

dynamics and material processes that depend on the exact operating point of the mill.

Mill conditions will generally vary statistically over time. It is therefore difficult in this

type of environment to systematically verify DEM, where some degree of precision inthe mill operation is required. To overcome this problem, and at the same time produce

results that are meaningful, consideration can be given to the use of small laboratory

mills. If mill conditions can be precisely controlled, and if the DEM simulation

parameters can be better matched to the mill geometry and particle load, then a clear

advantage in predictive capabilities is obtained.

With these considerations in mind, a programme of both experimental and DEM

simulation work was developed. A Atwo-dimensionalB laboratory mill was built that

could measure power precisely and which enabled the charge motion to be monitored.

DEM simulation runs were matched to the experimental conditions. In this account ofthe work, particular attention is given to the effect of mill speed on power and charge

motion, and also to particle behaviour at mill speeds above the critical.

2. The DEM

The DEM simulation code used in the present investigation uses a springslider

.dashpot configuration Fig. 1 to model the force displacement interactions betweenparticles that are in contact with each other, and between the particles in contact with the

.mill wall liner and lifter elements. The spring accounts for the elastic interactions,which depend on the material stiffness K and K in the normal and shear directions,n t

. respectively , the slider accounts for any surface motion that takes place determined by.the coefficient of friction m and the dashpot allows for energy loss during collisions

which is a measure of the coefficient of restitution, e, the ratio of the velocity of a.particle before and after a collision event .

A linear relationship between particle overlap and force was used in the simulation

. .code. In this case, the normal F and tangential F force acting between two particlesn tin contact is given by:

FsK dqC n n n n n

FsmF F)mF .t n t n

FsK d tqC FFmF . .Ht t t t t t n

d is the particle overlap,C and C are the dashpot damping coefficients, and n and nn n t n t

the relative velocities of the particles in the normal and tangential directions. n hastcomponents of both translational and rotational motion. d t is the timestep of the

simulation, which is a measure of the time interval between any two successive

calculations.


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The normal damping coefficient is related to the coefficient of restitution by:

)K m( nCsy2ln e ,n 2 2( ln e qp .

) .where m is the reduced mass of the two particles mass m and m :1 2

m m1 2)m s .

m qm1 2

.In this work, the tangential damping coefficient C was taken as equal to C .t nFor a contact between a particle and a wall element, the same basic force equations

are used, except that the reduced mass of the system is put equal to the mass of the

)

.particle m sm , and appropriate modifications are made for the wall velocity1components to obtain and .n t

Since a particle may be in contact with a number of neighbouring particles, or with

wall elements, the net force acting on the particle is taken as the sum of the normal andtangential contact forces. This force is then resolved into the x- and y-directions and

. .z-direction for the three-dimensional 3D simulations and integrated using a finite-dif-

. .Fig. 2. DEM torque calculation. a A particle is shown in contact with a wall element. The normal F andn . . .tangential force F are resolved in the x- and y-directions F and F , respectivley . b The resolved forcest x y

are used in a torque arm calculation.


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ference technique to obtain the new translational and rotational velocities, and displace-

ment of the particle.

The power drawn by the mill is determined from the ball contact forces acting on the

mill lifters and liners. At any wall element the contact normal and tangential force is

resolved into the x and y components, and the distance from the mill centre determined

.Fig. 2 . A torque arm calculation is performed, and the torque integrated over allcontacts and all timesteps. The power is calculated from:

Powers2pNTorque,

where N is the rotational speed of the mill in revolutions per second. .As noted by Zhang and Whiten 1996 , for the form of the DEM equations generally

used, the contact normal force can become negative just before the complete separation

of the particles. In effect the particles experience an attractive, rather than a repulsive

force. To overcome this problem, the normal force was set equal to zero when it became

negative. Although the particles would still be in contact, this situation would be quite

realistic of actual contact processes: plastic deformation at the contact point during an

impact results in blunting or a reduction in local particle radius. While the present DEM

simulations cannot account for this blunting as the simulation proceeds, the setting of the

normal force to zero does give the particle dynamics greater realism.

3. Experimental apparatus

.The experimental mill Fig. 3 had a diameter of 550 mm, with a nominal axial lengthof 23 mm. Glass end plates were used to observe and photograph the tumbling load.

Fig. 3. Schematic diagram of the experimental 2D mill.


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Table 1

Standard material parameters used in the DEM simulations

Parameter Value Units

.Coefficient of friction ballball 0.142 .ballwall 0.188

.Coefficient of restitution ballball 0.66 .ballwall 0.36

Ball diameter 0.02224 my3Ball density 7800 kg my1 .Normal stiffness ball 400,000 N my1 .wall 400,000 N my1 .Tangential stiffness ball 300,000 N my1 .wall 300,000 N m

y3

.Precision steel ball bearings diameter 22.24 mm and density 7800 kg m were usedas the particle load. Twelve equally spaced mild steel lifters of square cross-section

.22=22 mm were attached to the mill shell at equal intervals.The mill and motor were attached to a rigid frame using low-friction bearings. The

motor is free to rotate on the shaft, but is held in position by the load beam, which

therefore makes accurate measurement of torque possible. The mill motor has a variable

speed drive and can be accurately controlled at any speed up to 200% of the critical

speed. Mill speed and torque data were logged automatically. A high-speed digital video .camera framing speed of up to 1000 frames per second was used to collect and store

high quality images of the mill load behaviour.The mill variables that were manipulated were the charge filling and rotational speed.

The charge fillings used were 20%, 35% and 45% of the mill volume. The speed of the

mill was varied in increments ranging from 60% to 200% of the critical speed, and

conditions allowed to stabilise before measurements were taken. The mill charge was

operated dry.

4. DEM simulations

Both a 2D DEM and a 3D DEM program was used for the simulations. The charge

filling and rotational speed of the simulations covered the same range as the experimen-

Table 2

The effect of normal stiffness on mill power

y1 . . .Normal stiffness N m Timestep ms Power W

5

4=

10 10 26.4264=10 10 26.8574=10 3.5 26.0284=10 1 27.47


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. ..Fig. 4. Load motion comparison as a function of charge filling J and percentage of critical speed N from . . .experimental measurements left column , 2D DEM middle column and 3D DEM right column predictions.


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tal programme. With the 3D simulations, it is of course necessary to fix a mill length.The experimental mill had a nominal length of just larger than a ball diameter 22.24

.mm but clearly, there must be sufficient space between the mill end plates to allow theballs to tumble without becoming jammed. A base case was therefore chosen with mill

length equal to 24.24 mm, which gives an effective 1 mm gap between a ball and each

end plate. The 3D DEM does not generate the balls exactly in a plane, allowing balls tohave contact with the end plate.

.Values of the material parameters ball and wall used in the simulations aresummarised in Table 1. The coefficient of friction and coefficient of restitution data

were obtained by experiment. The coefficient of restitution is assumed to be a single .value it is a function of the approach velocity of the particles to reduce computational

complexity. The stiffness coefficient presented a problem in terms of the need to keep

the simulation running time to as short a period as possible. To ensure computational .stability, the timestep t of the simulation needs to be chosen according to thes

following criterion:

tF0.1 mrK(s n

The steel balls used in the experiments have a mass of 0.045 kg. The stiffness of steel

balls under impact conditions has been experimentally measured by Mishra and . y1Rajamani 1992 , and has a value of about 140 MN m . This gives a timestep off1

ms, and would result in a simulation time approaching 50 h or more. This is impractical,

and it would be desirable to reduce the timestep by a factor of 10. A series of DEM

simulations were therefore made with a mill operating at 80% of critical speed and 35%

Fig. 5. Comparison of mill power, determined using 2D DEM, with the experimentally measured power.


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Fig. 6. Comparison of mill power, determined using 3D DEM, with the experimentally measured power.

charge filling to test the effect on mill power of changing the normal stiffness. The

timestep was adjusted to meet the stability criterion. The shear stiffness was 2r3 of the .

normal stiffness. The results Table 2 indicated that there is little variation in power

Fig. 7. The effect of mill length on mill power draw. 3D DEM simulation, 35% charge filling, 80% of critical.speed.


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Fig. 8. The effect of the mill end-plate friction coefficient on mill power draw. 3D DEM simulation, 35%.charge filling, 80% of critical speed.

with changing stiffness. Therefore, for the present simulations, a stiffness coefficient of

4.10 5 N my1 and a timestep of 10 ms was used.

.Fig. 9. At mill speeds above the critical speed 20% mill charge filling , the 2D DEM and experimental power

diverges significantly. The DEM power is determined from the second of two revolutions. Note the drop in

experimental power for critical speeds in excess of 140%.


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5. Results

5.1. Load behaiour

The mill charge motion predicted by the simulations can be compared with experi- . . .mental data in several ways: i instantaneous ball positions; ii ball trajectories; iii

ball velocities. In this paper only the qualitative overall load positions instantaneous.ball positions will be used. Fig. 4 compares AsnapshotsB of the mill charge captured by

the digital video camera, with those of the 2D DEM and 3D DEM predictions. The

match between measured and predicted load motion at a mill speed of 60% of critical .for both 20% and 35% mill filling is very good. At higher speeds 80% of critical the

load is lifted somewhat higher in the experimental mill than is predicted by the DEM

results. This is, however, a marginal difference that could probably be eliminated by an

increase in the ballwall coefficient of friction. There is no qualitative differencebetween the 2D and 3D DEM charge motion predictions.

5.2. Mill power

5.2.1. 2D DEM: mill speed range 60100% of critical speed

A comparison of the mill power obtained from the 2D DEM simulations with the .experimental values Fig. 5 indicates that the results for the 20% charge filling are in

excellent agreement. However, for greater mill charge fillings, and mill speeds below

Fig. 10. By increasing the number of mill revolutions to 10, and plotting the power of the last revolution, the .2D DEM simulations give better agreement with the experimental results 20% charge filling .


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approximately 90% of critical, the DEM results are lower than the corresponding

experimental values. The experimental and DEM results also show a difference in the

rotational speed at which the mill power is at a maximum.

5.2.2. 3D DEM: mill speed range 60100% of critical speedResults obtained with the base-case mill length of 24.24 mm and with the end-plate

coefficient of friction equal to that of the mill wall, gave better overall agreement with .the experimental mill power Fig. 6 . The positions of the peak power for the charge

fillings of 45% and 35% now agree. The only significant discrepancy is at a charge

filling of 45% where, at mill speeds in excess of 90%, the DEM simulation results show

a consistent upward trend while experimentally power is falling. .The effect of changing the mill length Fig. 7 and end plate coefficient of friction

.Fig. 8 indicates that these factors do influence the power drawn by the mill. An

increase in mill length or coefficient of friction increases the power.

5.2.3. Mill speed range 100200% of critical speed

At mill speeds above the critical speed, the experimental and 2D DEM power results .diverged considerably Fig. 9 . Experimentally, power drops to very low levels by 150%

of critical speed. It was noted during the running of the experimental mill at these high

speeds, that the balls would coalesce into several layers around the mill shell and

centrifuge. Therefore, as the balls form layers and lock, the mill power will decrease.These layers develop over time, and the standard DEM simulations which were run for

.Fig. 11. Mill torque behaviour 2D DEM simulations over a number of revolutions for different mill speeds.

In situations where the balls form layers and centrifuge, a drop in torque is evident.


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.only two revolutions , do not allow enough time for this effect to fully take place.However, by increasing the number of revolutions of a DEM simulation to 10, and

taking the power as the average over the last revolution, it was now possible to observe .the same effect Fig. 10 . Plotting mill torque as a function of the number of mill

.revolutions Fig. 11 shows the influence of time on the process. The corresponding

.charge motions at 200% of critical speed Fig. 12 indicate the manner in which the ballsbehave as they form layers and centrifuge.

To determine why the DEM results gave the onset of power reduction at mill speeds

higher than those observed experimentally, the charge motion for the simulation run at

160% of critical speed was examined in detail. Examination of the charge motion .pictures Fig. 13 shows that a type of charge surging occurs. The balls are swept around

with the mill shell, but they eventually lose traction and fall back into the mill charge.

Fig. 12. 2D DEM charge motions indicating the mechanism by which balls form layers and centrifuge at mill . . .speeds in excess of the critical speed. a Revolution number 2; b revolution number 3; c revolution number

. .4; d Revolution number 7. 20% charge filling, 200% of critical speed. The mill rotates counter-clockwise.

Compare with Fig. 11 for the corresponding average torque.


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. .Fig. 13. Ball surging effect noted using 2D DEM at high critical speeds. a Torque plot, indicating regions

.of low and high torque where different ball behaviours were observed. b Charge motion giving rise to low . . . torque balls dropping . c Charge motion giving rise to high torque balls ascending . 160% of critical speed,.20% mill charge filling. Mill rotates counter-clockwise.

This suggested that the coefficient of friction of the balls could be an important

parameter, since slip between the balls did not allow them to attain the speed necessary

for centrifuging. .A series of DEM simulations where the ballball coefficient of friction m wasbb

.varied shows Fig. 14 that this is indeed an important parameter. By increasing thevalue of m to 0.5, the power now drops to low values at 160% of critical speed Fig.bb

.15 . Higher values of m did not give any better agreement with the experimentalbbresults. Using the 3D DEM program, the results were only marginally improved.


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Fig. 14. At 160% of critical speed, the extent to which the balls form layers and centrifuge is stronglydependent on the choice of the coefficient of friction of the ball ball interactions. 20% charge filling. Mill

.power is measured over the tenth revolution of the 2D DEM simulation.

The above discussion shows the need to consider transient effects when the validation

of DEM is considered. Experimental data of these transient effects are needed.

.Fig. 15. An increase in the ballball coefficient of friction m results in a reduction in the mill speed atbbwhich the balls form layers and completely centrifuge. 20% charge filling. Mill power is measured over the

.tenth revolution of the 2D DEM simulation.


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6. Discussion and conclusions

A precision laboratory mill will have a high sensitivity to changes in operating

parameters. This makes the use of such a mill ideal for testing the extent to which the

DEM is able to accurately predict mill power and charge motion. The present work has

indicated that for the best results it is necessary to incorporate 3D effects into the DEM .code. Although the experimental mill is nominally 2D one ball diameter in length the

end plates fitted to the mill clearly influence the power draw. The mill length and the

coefficient of friction of the end plates both need to be taken into account.

Overall, the DEM power draw results, at mill speeds below critical, are in good

agreement with those obtained experimentally. The DEM simulations were also able to

predict the centrifuging of all the balls in the mill charge at speeds well above critical. In

this case, although the DEM simulation results were not able to accurately track the

experimentally measured drop in power as the balls locked into layers and centrifuged,

sensitivity to the coefficient of friction of the ballball interactions was noted. Increas-ing the coefficient of friction improved agreement with the experimental results.

Our results suggest that values for the various material properties that are used in the

DEM simulations need to be accurately measured. The coefficient of friction appears to

play a significant role and needs to be experimentally determined under conditions of

particle speed and load typical of the operating mill. Coefficient of restitution and

particle stiffness would seem to be less critical, but interactions between parameters

cannot be discounted, and may be important when considering particle impact energy

and contact force. It is a relatively straightforward procedure to incorporate into the

DEM code more complex, but realistic, relationships between particle motion and thefundamental material properties.

Acknowledgements

The authors acknowledge the financial support from Mintek for this work. This paper

is published with permission from Mintek

References

Cundall, P.A., Strack, O.D.L., 1979. A discrete model for granular materials. Geotechnique 1, 4765.Mishra, B.K., Rajamani, R.K., 1992. The discrete element method for the simulation of ball mills. Appl. Math.

Modell. 16, 598604.

Mishra, B.K., Rajamani, R.K., 1996. Analysis of power draw in tumbling mill. Proc. Optimisation of

Comminution, Johannesburg, 57 August.

Zhang, D., Whiten, W.J., 1996. The calculation of contact forces between particles using spring and damping

models. Powder Technol. 88, 5964.

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