Marhl Compur. Modeliing, Vol. 14, pp. 1072-1074, 1990 0895-7177/90 $3.00 + 0.00

Printed in Great Britain Pergamon Press plc

GRINDING PROCESS SIMULATION IN A BALLMILL

Stenislaw Tereslewicz Peter Radziszewski

Ddpartement de genie mdcanlque, Universltd Lava1 Ste-Fey, Qudbec, CANADA ClK 7P4

Abstract. As a combination of various elements that are both interdependent and interactive, the ballmlll can be described by its physical parameters along with the parameters of the material ground. Together these elements transform energy to produce an evolving material granulometry. Different from black box models, these energy transformations are functions of physl- cal parameters (mill diameter, length, lifter configuration, charge volume, ball distribution), control parameter (feed, rotation) and material parame- ters (granulometry, tensile strength, Youngs Modulus) of the process. After a description of the fundamental model of commlnutlon, this study presents results showing the effects of varying material hardness on product granulo- metry, as well as, how these variations may be compensated for by using changing rotation speed.

Keywords. Ballmill simulation; energetics; material hardness.

INTRODUCTION

In the past, ballmills have been studied using a black box philosophy, where by the lnterac- tlons and lnterelationships of ballmill ele- ments have been neglected and mill output stud- ied as a function of mill input. Such approach- es gave rise to the development of the comminu- tlon laws of Kick, Rittinger and Bond summa- rized by Charles equation CLowrlson, 19741 :

(1)

where setting the exponent n to 1, 14, 2 and integrating defines these laws. This approach was also used to develope the batch grinding equation [Austin et al., 19841 presented as :

&dY)

dt =xb Sm-sm

a,Yaa YY

where material breakage is defined by the para- meters b and S. Using either of both develop- ments it is difficult to answer the question : How can a commlnution process be optimized ?

These work presents a fundamental model as a function of process physical, material and operating parameters, followed by a case exam- ple for the compensation of material hardness effects using mill rotation speed.

BACKGROUND

The commlnutlon event can be described by the following relationship derived from a breakage energy balance ITaraslewicz, Radziszewskl, CEII

Y-l

ti(Y) _ x 1 de(a) ~ - dt e(y) dt 1 q(a) m(a) lost out - e(y)

(3)

We shall call this the breakage energy (bren) function. Negating the energy lost term {(@lost leaves only two constltuative equa- t ions I the specific energy term e(y) and the specific energy rate of change dt de

proc. 7th Int. Con/ on Mathematical and Computer Modelling

BALLMILL SIMULATION Charge dynamics

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Charge dynamics can generally be pictured as in figure 1, where three distinct zones can be identified : crushing, tumbling and grinding.

As most ballmills in the litterature are not described sufficiently [Austin et al., 1984; Lynch, 19771 with a number of parameters being approximated. The input-output feed granulome- tries for copper are ground in a 3.20m x 3.05m ballmill are found in figure 2.

FIG. 1 Charge Profile.

The mathematical description of the charge motion evolves from the application of single particle motion CMcIvor, 19831 in a mill to a system of particles defining the mill charge [Tarasiewicz, Radziszewski, BASII. In essence, charge motion is determined from a force bal- ance on mill lifters which defines the point- of-flight or the point-of-slippage as a func- tion of static and dynamic friction factors. Discretizing the charge and defining each ele- ment in time and space permits the simulation of charge motion. During simulation, it is then possible to calculate the energy rate consumed by the mill and how energy is redis- tributed in crushing, tumbling and grinding.

Axial evolution

Axial evolution of particle granulometry de- scribes the mass transport phenomena of mate- rial in the mill. It can be described as [Tarasiewicz, 19851 :

D a2 Ii(Y) H a* r;l(Y) ax* + pA ayat - V&y) *

+ y (n;(Y)in & - m(Yy - at (7)

where the bran function (3) integro-differen- tial form after simplification becomes :

g.(Y) Z(Y)

ay at I (8) Tying charge motion to breakage mechanics com- pletes the necessary constituative equations for the fundamental model description. It is

now possible to simulate a real case scenario.

3.2t 1.985 E.?lb

FIG. 2 Input-Output Granulometry for Copper ore.

Mill lifters are of the Sniplap type [Dunn, 19761 with charge volume being 40% of mill volume filled with an average ball size of 2% inches, rotation speed is 80% of critical and ore feed is 208 t/hr.

Model Tuning

Model tuning is accomplished using a systematic trial and error shooting method where the tar- get curve is the output granulometry (figure 2). After simulating the charge motion (figure 3), it is possible to determine the axial mate- rial size evolution found in figure 4.

Material Hardness Effect

Material hardness effect on output granulometry is shown in figure 5. The simulation results show that as material hardness increased output granulometry cumulatif oversize increases in the larger size intervals. The opposite trend is true for decreasing hardness.

FIG. 3 Simulated Charge Motion.

1074 Proc. 7th Int. Conf on Mathematical and Computer Modelling

FIG. 4 Material Size Evolution.

FIG. 5 Material Hardness Effect On Output Granulometry.

Rotation Speed Compensation

Rotation speed compensation is shown in figure 6. There, for varying shifts in material hard- ness, the desired output granulometry is main- tained at a constant feed rate using the com- pensating effect of changing ballmill rotation speed.

CONCLUSION

Ballmlll modelllng using a first principles approach provides a better understanding of comminutlon dynamics as a function of both physical end material parameters. After tuning the model to a typical grinding situation, it becomes possible to determine the effect of these parameters on a desired output. Slmuls- tlon results show that as mill feed hardness evolves so does output granulometry forcing a needed decrease in input feed. Ballmlll output feed and granulometry can be maintained regard- less of material hardness shifts with the ap- propriate variations in mill rotation speed. Process optimization becomes a distinct possl- blllty using this approach.

REFERENCES

Austin, L.G.; Kllmpel, R.P.; Luckle, P.T.; (1984). Process Engineering of Size Reduction: Ball Milling; Sot. Min. Eng., New York.

Dunn, D.J. (1976). "Optimizing Ball Mill Liners for Production and Economy"; Mining Eng., Dec., 32-34.

Lowrlnson, G.C. (1974). Crushing and Grinding, Butterworths, London.

Lynch, A.J. (1977). Mineral Crushing and Grinding Circuits; Elsevler, New York.

McIvor, R.E. (1983). "Effect of Speed and Liner Configuration on Ballmill Perfor- mance-; Mining Eng., June, 617-622.

Taraslewlcz, S. (1985). "A Dynamic Model for Grinding in a Ballmlll~, IASTED Lugano, June, 406-409.

Taraslewlcz, S.; Radzlszewski, P.; (BASI); "Ballmill Simulation Part I : A Kinetic Model to Ballmlll Charge Motion", Trans. Sot. Comp. Slm., (to appear).

Taraslewicz, S.; Radziszewski, P.; (CEI); "Com- mlnutlon Energetlcs Part I: Breakage Ener- gy Model, (in preparation).

Ylglt, E. (1976). "Three Mathematical Models Based on Strain Energy"; Int. J. Min. Pro- cess, 3, 365-374.

FIG. 6 Rotation Speed Compensation.