3D simulation of charge motion in tumbling mills by the discrete element method

R. Venugopal, R.K. Rajamani
Department of Metallurgical Engineering
University of Utah, Salt Lake City,
UT 84112-0114,
USA
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Abstract

The dynamics of charge motion in tumbling mills has been a challenging problem both experimentally and theoretically. The harsh environment within the mill precluded sophisticated sensors. On the other hand, first principle modeling of charge motion was only marginally successful since the motion involves hundreds of bodies colliding with each other. The numerical scheme known as discrete element method DEM. is the fitting solution to this modeling problem. The three-dimensional 3D. discrete element modeling of tumbling mills is described first. The simulation results are verified against both still images of ball charge motion and power draft in experimental mills. The excellent agreement implies that the simple spring–dashpot collision model is adequate for this problem. The simulation scheme can be extended for studying the life cycle of the mill itself. 2001 Elsevier Science B.V. All rights reserved.

Introduction

A realistic simulation of charge motion in autogenous AG., semi-autogenous SAG. and ball mills is desirable since such a study would shed light on shell wear, ball wear and mill capacity besides the much sought after power draft. Measurement of charge motion with sensor has yielded very little information and sophisticated sensors are precluded due to the harsh environment within the mill. The simulation problem seemed intractable until two decades ago when the discrete element method DEM. began to appear in the engineering literature. Since then, the solution to this problem has been leaping forward than ever before. The earliest analyses of ball motion within a tumbling mill by White w1x and Davies w2x dates back to early 1900. These authors considered that spherical balls are carried along the mill shell to a point where a component of the gravitational force overcomes the centrifugal force, letting the charge fall in a parabolic path till it hits the mill shell. The process of balls rolling down the surface is called cascading while that of projected out stream is called cataracting. Rose and Sullivan w3x, in a review of these studies, criticized that the frictional characteristic of the charge was ignored in predicting the ball trajectories. Barth w4x predicted the presence of an Aequilibrium surfaceB within the charge on which particles are in dynamic equilibrium under the action of centripetal, frictional and gravitational forces. This surface is a plane that divides the charge into two regions: an inner static region where there is no motion relative to the mill shell and a shear region where balls either flow down the free surface of the charge or are projected out of the surface and follow parabolic paths. It is quite obvious that these depictions of charge profile relied on oversimplifying assumptions. In the next decade, researchers pursued the mill power draft empirically since charge motion was beyond mathematical tools available at that time. Austin et al. w5x developed a semi-empirical formula to estimate the power draft, which finds use even today. Later researchers developed models to predict mill power based on the torque–arm principle, which presumes that the charge profile is completely cascading between the toe and shoulder of the charge. Harris et al. w6x proposed an equation based on the torque–arm principle, which used industrial mill data and some empirical relations. However, the formula did not have any speed dependent term. Hogg and Furstenau w7x assumed the charge profile of the load to be a chord from the toe to the shoulder and derived an expression for mill power draw. This model gave favorable power predictions at low mill speeds, however it failed to do so at high speeds because it did not account for contributions due to the cataracting charge. Liddel and Moys w8x modified Hogg and Furstenau’s model equation to include a speed correction function. This model faired well against data collected in a 55-cm diameter mill. Kapur et al. w9x, recognizing the cataracting charge motion, divided the mill charge into cascading and cataracting regions and considered the energy consumption in both the regions to arrive at an expression for the power draft. Although better than the Atorque–armB formulae, a perfect partitioning of the mill charge into cascading and cataracting regions is highly idealized. In summary, these power draft approaches had to evolve gradually since the cataracting part of the charge could not be modeled in any simplified manner and there was much uncertainty in fixing shoulder position of the charge. Next came a spurt of activity in experimental schemes to measure impact force between balls. Vermulen et al. w10x studied the impact forces within a ball mill using piezoelectric sensors. Rolf and Vongluekiet w11x used a sophisticated experimental setup using hollow balls with embedded sensors to measure impact energies. A considerably improved version of the same idea was implemented by Rothkegel w12x. While such experiments were the first of their kind, some difficulties were inevitable in the implementation. A number of low energy impacts went unrecorded and the high-energy impacts were underestimated because of the fixed orientation of the sensor within the instrumented ball. Also the lifetime of the instrumented ball was limited as it is exposed to a harsh environment. These studies came to an abrupt end since the sensors could not be improved in any meaningful way and the transmission of data from inside a ball by wireless means was too complex. Next came advances in technology and computing that led researchers to take a fresh look at this problem. Powell and Nurick w13,14x studied single ball trajectories using a radioactive ball and filming its motion through the mill charge using a gamma camera. These individual ball trajectories led to the determination of the charge center of mass, the angle of repose and its variation with mill operating conditions. Also the power draft was computed directly from charge center of mass. This study led to a clear understanding of the motion of balls, charge location, charge interaction, charge segregation and the influence of lifters. Morrell w15x measured the tangential velocity of ball layers moving in the upward direction. Then, integrating the lift force across the plane of lift, an expression for power draft was obtained. Furthermore, the velocities of ball layers in larger mills were obtained by empirically scaling the measured velocity in the lab mill. Excellent agreement between predicted and measured power draft in a variety of mills was shown. The only drawback of this approach is the empirical scaling. The upward velocities depend on friction between layers as well as the geometry of the lifters. Only a detailed physical model can predict tangential velocities accurately. Recently, a numerical simulation scheme called the DEM developed to model the mechanics of discrete objects in space w22x, has been adapted by Mishra and Rajamani w16x to suit the tumbling mill problem. Perhaps this effort opened up uncharted grounds in the simulation problem. Studies on power draft, mill speed and the effect of lifter bars on the charge profile w17–19x have been carried out for mills ranging in size from lab mills to large ball, AG and SAG mills. A slice of the mill of width equal to the maximum ball diameter is simulated in the two-dimensional 2D. numerical scheme. The power draw is estimated by a sum of the normal and shear energy lost in all impacts occurring over all such slices. Although this approach has met with considerable success in predicting charge profiles and power draws for a wide range of mills, it is still a 2D approximation to the collision problem. The motion of balls in the axial direction is completely omitted. In this respect, Cleary w20x presents a detailed investigation of ball mill power by the DEM. In particular non-spherical particles are modeled. Acharya w21x and Kano et al. w22x have formulated this numerical scheme in 3D. These two works did not explore the tumbling mill problem in any great detail. Rather, they solved the 3D DEM problem for a limited number of bodies in a tumbling mill. Acharya simulated the motion of balls in a 25=28 cm mill and Kano et al. w22x too dealt with a laboratory mill of 15=15 cm size.



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